"The characters encoded for these calendrical symbols in Unicode have compatibility decompositions, and those decompositions depend on the actual name chosen for the era," Whistler wrote.
|
|
The upshot is that engineers have a new, practical tool that they can use to test for nonnegativity and find sum of squares decompositions quickly.
|
|
Dr. Tate laid the groundwork for a wide range of abstract but fundamental concepts that now bear his name, among them the Tate module, the Tate curve, Tate cycles, Hodge-Tate decompositions, Tate cohomology, the Serre-Tate parameter, Lubin-Tate formal groups, the Tate trace, the Shafarevich-Tate group and the Néron-Tate height.
|
|
Cell decompositions in the form of triangulations are the representations used in 3d finite elements for the numerical solution of partial differential equations. Other cell decompositions such as a Whitney regular stratification or Morse decompositions may be used for applications in robot motion planning.
|
|
On classical computers, ear decompositions of 2-edge-connected graphs and open ear decompositions of 2-vertex-connected graphs may be found by greedy algorithms that find each ear one at a time. A simple greedy approach that computes at the same time ear decompositions, open ear decompositions, st-numberings and -orientations in linear time (if exist) is given in . The approach is based on computing a special ear decomposition named chain decomposition by one path-generating rule. shows that non-separating ear decompositions may also be constructed in linear time.
|
|
Decompositions, decomposition type, canonical combining class, composition exclusions, and more.
|
|
Series parallel graphs may also be characterized by their ear decompositions.
|
|
The width of a problem is the minimal width of its decompositions.
|
|
A very simple bridge-finding algorithm. uses chain decompositions. Chain decompositions do not only allow to compute all bridges of a graph, they also allow to read off every cut vertex of G (and the block-cut tree of G), giving a general framework for testing 2-edge- and 2-vertex-connectivity (which extends to linear-time 3-edge- and 3-vertex-connectivity tests). Chain decompositions are special ear decompositions depending on a DFS-tree T of G and can be computed very simply: Let every vertex be marked as unvisited.
|
|
On estimates of the stability measure for decompositions of probability distributions into components. Theory of probability and its applications, 1978, vol. 23, N 3, pp. 507–520. which considered the stability for decompositions of probability distributions into components.
|
|
Applications include embeddings,Gross, Tucker 1987 computing genus distribution,Gross 2011 and Hamiltonian decompositions.
|
|
It follows that the only graphs whose Hamiltonian decompositions are unique are the cycle graphs.
|
|
Analogously, we can define QL, RQ, and LQ decompositions, with L being a lower triangular matrix.
|
|
Several important classes of graphs may be characterized as the graphs having certain types of ear decompositions.
|
|
Several important problems can be phrased in terms of eigenvalue decompositions or singular value decompositions. For instance, the spectral image compression algorithmThe Singular Value Decomposition and Its Applications in Image Compression is based on the singular value decomposition. The corresponding tool in statistics is called principal component analysis.
|
|
Emmanuel Giroux is a blind French geometer known for his research on contact geometry and open book decompositions...
|
|
Later texts use the title Krull–Schmidt (Hungerford's Algebra) and Krull–Schmidt–Azumaya (Curtis–Reiner). The name Krull–Schmidt is now popularly substituted for any theorem concerning uniqueness of direct products of maximum size. Some authors choose to call direct decompositions of maximum-size Remak decompositions to honor his contributions.
|
|
Among such models are mixture models and the recently popular methods referred to as "causal decompositions" or Bayesian networks.
|
|
Generally speaking, paradoxical decompositions arise when the group used for equivalences in the definition of equidecomposability is not amenable.
|
|
There are several decompositions of the Brier score which provide a deeper insight on the behavior of a binary classifier.
|
|
Handle decompositions of manifolds arise naturally via Morse theory. The modification of handle structures is closely linked to Cerf theory.
|
|
The group of area-preserving transformations contains such subgroups, and this opens the possibility of performing paradoxical decompositions using these subgroups. The class of groups von Neumann isolated in his work on Banach–Tarski decompositions was very important in many areas of mathematics, including von Neumann's own later work in measure theory (see below).
|
|
Product decompositions of matrix functions (which occur in coupled multi-modal systems such as elastic waves) are considerably more problematic since the logarithm is not well defined, and any decomposition might be expected to be non-commutative. A small subclass of commutative decompositions were obtained by Khrapkov, and various approximate methods have also been developed.
|
|
Wagon presented an algorithm for calculating such decompositions in 1990, based on work by Serret and Hermite (1848), and Cornacchia (1908)..
|
|
Properties of the VAR model are usually summarized using structural analysis using Granger causality, impulse responses, and forecast error variance decompositions.
|
|
Bulletin of the American Mathematical Society, vol. 84 (1978), no. 5, pp. 832–866.J. W. Cannon, Shrinking cell-like decompositions of manifolds.
|
|
25 44P. Scott and G. A. Swarup. "Regular neighbourhoods and canonical decompositions for groups." Electronic Research Announcements of the American Mathematical Society, vol.
|
|
2, pp. 145–186K. Fujiwara, and P. Papasoglu, "JSJ- decompositions of finitely presented groups and complexes of groups." Geometric and Functional Analysis, vol.
|
|
Every 4-regular undirected graph has an even number of Hamiltonian decompositions. More strongly, for every two edges e and f of a 4-regular graph, the number of Hamiltonian decompositions in which e and f belong to the same cycle is even. If a 2k-regular graph has a Hamiltonian decomposition, it has at least a triple factorial number of decompositions, :(3k-2)\cdot(3k-5)\cdots 7\cdot 4 \cdot 1. For instance, 4-regular graphs that have a Hamiltonian decomposition have at least four of them; 6-regular graphs that have a Hamiltonian decomposition have at least 28, etc.
|
|
The theorem does not assert the existence of a non-trivial decomposition, but merely that any such two decompositions (if they exist) are the same.
|
|
Metastable and kinetically persistent species or systems are not considered truly stable in chemistry. Therefore, the term chemically stable should not be used by chemists as a synonym of unreactive because it confuses thermodynamic and kinetic concepts. On the other hand, highly chemically unstable species tend to undergo exothermic unimolar decompositions at high rates. Thus, high chemical instability may sometimes parallel unimolar decompositions at high rates.
|
|
Lieven De Lathauwer from the KU Leuven, Belgium was named Fellow of the Institute of Electrical and Electronics Engineers (IEEE) in 2015 for contributions to signal processing algorithms using tensor decompositions. He was elected as a fellow of the Society for Industrial and Applied Mathematics in 2017, "for fundamental contributions to theory, computation, and application of tensor decompositions".SIAM Fellows: Class of 2017, retrieved 2017-04-25.
|
|
Ear decompositions may be used to characterize several important graph classes, and as part of efficient graph algorithms. They may also be generalized from graphs to matroids.
|
|
Unrooted binary trees are also used to define branch-decompositions of graphs, by forming an unrooted binary tree whose leaves represent the edges of the given graph. That is, a branch-decomposition may be viewed as a hierarchical clustering of the edges of the graph. Branch-decompositions and an associated numerical quantity, branch-width, are closely related to treewidth and form the basis for efficient dynamic programming algorithms on graphs..
|
|
This scheme follows from the combinatoric (algebraic topological) descriptions of solids detailed above. A solid can be represented by its decomposition into several cells. Spatial occupancy enumeration schemes are a particular case of cell decompositions where all the cells are cubical and lie in a regular grid. Cell decompositions provide convenient ways for computing certain topological properties of solids such as its connectedness (number of pieces) and genus (number of holes).
|
|
Many algorithmic problems that are NP-complete for arbitrary graphs may be solved efficiently for partial k-trees by dynamic programming, using the tree decompositions of these graphs.; ; .
|
|
In the 1980s, Šiljak and his collaborators developed a large number of new and highly original concepts and methods for the decentralized control of uncertain large-scale interconnected systems. He broadened new notions of overlapping sub-systems and decompositions to formulate the inclusion principle. The principle described the process of expansion and contraction of dynamic systems that serve the purpose of rewriting overlapping decompositions as disjoint, which, in turn, allows the standard methods for control design. Structurally fixed modes, multiple controllers for reliable stabilization, decentralized optimization, and hierarchical, epsilon, and overlapping decompositions laid the foundation for a powerful and efficient approach to a broad set of problems in control design of large complex systems.
|
|
The same method can also be used to color the edges of the graph with four colors in linear time.. Quartic graphs have an even number of Hamiltonian decompositions..
|
|
This ensures that each variable appears at the most once on the left of a conditioning bar, which is the necessary and sufficient condition to write mathematically valid decompositions.
|
|
This would make creating two unit squares out of one impossible. But von Neumann realized that the trick of such so-called paradoxical decompositions was the use of a group of transformations that include as a subgroup a free group with two generators. The group of area-preserving transformations (whether the special linear group or the special affine group) contains such subgroups, and this opens the possibility of performing paradoxical decompositions using them.
|
|
NMath contains vector and matrix classes, complex numbers, factorizations, decompositions, linear programming, minimization, root-finding, structured and sparse matrix, least squares, polynomials, simulated annealing, curve fitting, numerical integration and differentiationing.
|
|
The nonnegative rank of a matrix can be determined algorithmically.J. Cohen and U. Rothblum. "Nonnegative ranks, decompositions and factorizations of nonnegative matrices". Linear Algebra and its Applications, 190:149–168, 1993.
|
|
Regular chains also appear in Chou and Gao (1992). Regular chains are special triangular sets which are used in different algorithms for computing unmixed-dimensional decompositions of algebraic varieties. Without using factorization, these decompositions have better properties that the ones produced by Wu's algorithm. Kalkbrener's original definition was based on the following observation: every irreducible variety is uniquely determined by one of its generic points and varieties can be represented by describing the generic points of their irreducible components.
|
|
The examples of the section are designed for illustrating some properties of primary decompositions, which may appear as surprising or counter-intuitive. All examples are ideals in a polynomial ring over a field .
|
|
The holonomy method appears to be relatively efficient and has been implemented computationally by A. Egri-Nagy (Egri-Nagy & Nehaniv 2005). Meyer and Thompson (1969) give a version of Krohn–Rhodes decomposition for finite automata that is equivalent to the decomposition previously developed by Hartmanis and Stearns, but for useful decompositions, the notion of expanding the state-set of the original automaton is essential (for the non-permutation automata case). Many proofs and constructions now exist of Krohn–Rhodes decompositions (e.g.
|
|
The width of a problem is the width of its minimal- width decomposition. While decompositions of fixed width can be used to efficiently solve a problem, a bound on the width of instances does necessarily produce a tractable structural restriction. Indeed, a fixed width problem has a decomposition of fixed width, but finding it may not be polynomial. In order for a problem of fixed width being efficiently solved by decomposition, one of its decompositions of low width has to be found efficiently.
|
|
L. Chernac Ladislaus Chernac (1742–1816) was a Hungarian scientist who moved to Deventer in the Netherlands. He is the author of the first published table giving decompositions in prime factors up to one million.
|
|
Doubling a fraction with an odd denominator however results in a fraction of the form 2/n. The RMP 2/n table and RMP 36 rules allowed scribes to find decompositions of 2/n into unit fractions for specific needs, most often to solve otherwise un-scalable rational numbers (i.e. 28/97 in RMP 31,and 30/53 n RMP 36 by substituting 26/97 + 2/97 and 28/53 + 2/53) and generally n/p by (n - 2) /p + 2/p. Decompositions were unique.
|
|
Using the QR algorithm, the real Schur decompositions in step 1 require approximately 10(m^3 + n^3) flops, so that the overall computational cost is 10(m^3 + n^3) + 2.5(mn^2 + nm^2).
|
|
He is known for Courcelle's theorem, which combines second-order logic, the theory of formal languages, and tree decompositions of graphs to show that a wide class of algorithmic problems in graph theory have efficient solutions.
|
|
Two different tree- decompositions of the same graph The width of a tree decomposition is the size of its largest set Xi minus one. The treewidth tw(G) of a graph G is the minimum width among all possible tree decompositions of G. In this definition, the size of the largest set is diminished by one in order to make the treewidth of a tree equal to one. Treewidth may also be defined from other structures than tree decompositions, including chordal graphs, brambles, and havens. It is NP-complete to determine whether a given graph G has treewidth at most a given variable k.. However, when k is any fixed constant, the graphs with treewidth k can be recognized, and a width k tree decomposition constructed for them, in linear time.. The time dependence of this algorithm on k is exponential.
|
|
Since in the modern implicit version of the procedure no QR decompositions are explicitly performed, some authors, for instance Watkins, suggested changing its name to Francis algorithm. Golub and Van Loan use the term Francis QR step.
|
|
One of the ways in which amalgamations can be used is to find Hamiltonian Decompositions of complete graphs with 2n + 1 vertices.Bahmanian, Amin; Rodger, Chris 2012 The idea is to take a graph and produce an amalgamation of it which is edge colored in n colors and satisfies certain properties (called an outline Hamiltonian decomposition). We can then 'reverse' the amalgamation and we are left with K_{2n+1} colored in a Hamiltonian Decomposition. In Hilton outlines a method for doing this, as well as a method for finding all Hamiltonian Decompositions without repetition.
|
|
Various other terms have been used including the german Nebengruppen (Weber) and conjugate group (Burnside). Galois was concerned with deciding when a given polynomial equation was solvable by radicals. A tool that he developed was in noting that a subgroup of a group of permutations induced two decompositions of (what we now call left and right cosets). If these decompositions coincided, that is, if the left cosets are the same as the right cosets, then there was a way to reduce the problem to one of working over instead of .
|
|
Generalized hypertree decompositions are defined like hypertree decompositions, but the last requirement is dropped; this is the condition "the variables of the constraints of a node that are not variables of the node do not occur in the subtree rooted at the node". A problem can be clearly solved in polynomial time if a fixed-width decomposition of it is given. However, the restriction to a fixed width is not known to being tractable, as the complexity of finding a decomposition of fixed width even if one is known to exist is not known, .
|
|
Thus every module direct summand of R is generated by an idempotent. If a is a central idempotent, then the corner ring is a ring with multiplicative identity a. Just as idempotents determine the direct decompositions of R as a module, the central idempotents of R determine the decompositions of R as a direct sum of rings. If R is the direct sum of the rings R1,...,Rn, then the identity elements of the rings Ri are central idempotents in R, pairwise orthogonal, and their sum is 1.
|
|
In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems.
|
|
123–123 Therefore, decompositions in Euler chained rotations and Tait–Bryan chained rotations are particular cases of this. The Tait–Bryan case appears when axes 1 and 3 are perpendicular, and the Euler case appears when they are overlapping.
|
|
OpenType has the ccmp "feature tag" to define glyphs that are compositions or decompositions involving combining characters, the mark tag to define the positioning of combining characters onto base glyph, and mkmk for the positionings of combining characters onto each other.
|
|
A partition of a complete graph on 8 vertices into 7 colors (perfect matchings), the case r = 2 of Baranyai's theorem In combinatorial mathematics, Baranyai's theorem (proved by and named after Zsolt Baranyai) deals with the decompositions of complete hypergraphs.
|
|
Theory of Economic Policy, Political Economy, Monetary Integration, Sustainable Fiscal Policies, Demographic Change, Fiscal Rules, Monetary Policy, Inter-institutional and Inter-country Policy Coordination; Dynamic Games and Bargaining Models, Regionalism and Federalism, Time Varying Cyclical Decompositions, Numerical Methods in Econometrics.
|
|
Coxeter decompositions are named after Harold Scott MacDonald Coxeter, an accomplished 20th century geometer. He introduced the Coxeter group, an abstract group generated by reflections. These groups have many uses, including producing the rotations of Platonic solids and tessellating the plane.
|
|
The pathwidth of a graph has a very similar definition to treewidth via tree decompositions, but is restricted to tree decompositions in which the underlying tree of the decomposition is a path graph. Alternatively, the pathwidth may be defined from interval graphs analogously to the definition of treewidth from chordal graphs. As a consequence, the pathwidth of a graph is always at least as large as its treewidth, but it can only be larger by a logarithmic factor. Another parameter, the graph bandwidth, has an analogous definition from proper interval graphs, and is at least as large as the pathwidth.
|
|
The GSVD, formulated as a comparative spectral decomposition, has been successfully applied to signal processing and data science, e.g., in genomic signal processing. These applications inspired several additional comparative spectral decompositions, i.e., the higher-order GSVD (HO GSVD) and the tensor GSVD.
|
|
Moreover, any open ear decomposition of a 2-vertex-connected series-parallel graph must be nested. The result may be extended to series- parallel graphs that are not 2-vertex-connected by using open ear decompositions that start with a path between the two terminals.
|
|
Two important applications are to hyperbolic geometry, where decompositions of closed surfaces into pairs of pants are used to construct the Fenchel-Nielsen coordinates on Teichmüller space, and in topological quantum field theory where they are the simplest non-trivial cobordisms between 1-dimensional manifolds.
|
|
Householder transformations are widely used in numerical linear algebra, to perform QR decompositions and is the first step of the QR algorithm. They are also widely used for transforming to a Hessenberg form. For symmetric or Hermitian matrices, the symmetry can be preserved, resulting in tridiagonalization.
|
|
Householder reflections can be used to calculate QR decompositions by reflecting first one column of a matrix onto a multiple of a standard basis vector, calculating the transformation matrix, multiplying it with the original matrix and then recursing down the (i, i) minors of that product.
|
|
He was included in the 2019 class of fellows of the American Mathematical Society "for contributions to numerical analysis of domain decompositions within computational mathematics and for incubation through his writing and mentorship of a broad international, creative community of practice applied to highly resolved systems simulations".
|
|
Fixing a maximal allowed width is a way for identifying a subclass of constraint satisfaction problems. Solving problems in this class is polynomial for most decompositions; if this holds for a decomposition, the class of fixed-width problems form a tractable subclass of constraint satisfaction problems.
|
|
In 1776 and 1777, Felkel published a table of giving complete decompositions of all integers not divisible by 2, 3, and 5, from 1 to 408,000. Felkel had planned to extend his table to 10 million. A reconstruction of his table is found on the LOCOMAT site.
|
|
The concept of the polyphase matrix allows matrix decomposition. For instance the decomposition into addition matrices leads to the lifting scheme. However, classical matrix decompositions like LU and QR decomposition cannot be applied immediately, because the filters form a ring with respect to convolution, not a field.
|
|
A simple alternative to the above algorithm uses chain decompositions, which are special ear decompositions depending on DFS- trees.. Chain decompositions can be computed in linear time by this traversing rule. Let C be a chain decomposition of G. Then G is 2-vertex-connected if and only if G has minimum degree 2 and C1 is the only cycle in C. This gives immediately a linear-time 2-connectivity test and can be extended to list all cut vertices of G in linear time using the following statement: A vertex v in a connected graph G (with minimum degree 2) is a cut vertex if and only if v is incident to a bridge or v is the first vertex of a cycle in C - C1. The list of cut vertices can be used to create the block-cut tree of G in linear time. In the online version of the problem, vertices and edges are added (but not removed) dynamically, and a data structure must maintain the biconnected components.
|
|
Kolda received a Presidential Early Career Award for Scientists and Engineers in 2003, best paper prizes at the 2008 IEEE International Conference on Data Mining and the 2013 SIAM International Conference on Data Mining, and has been a distinguished member of the Association for Computing Machinery since 2011. She was elected a Fellow of the Society for Industrial and Applied Mathematics in 2015. She was elected a Fellow of the Association for Computing Machinery in 2019 for "innovations in algorithms for tensor decompositions, contributions to data science, and community leadership." She was elected to the National Academy of Engineering in 2020, for "contributions to the design of scientific software, including tensor decompositions and multilinear algebra".
|
|
A polynomial may have distinct decompositions into indecomposable polynomials where f = g_1 \circ g_2 \circ \cdots \circ g_m = h_1 \circ h_2 \circ \cdots\circ h_n where g_i eq h_i for some i. The restriction in the definition to polynomials of degree greater than one excludes the infinitely many decompositions possible with linear polynomials. Joseph Ritt proved that m = n, and the degrees of the components are the same, but possibly in different order; this is Ritt's polynomial decomposition theorem.Capi Corrales-Rodrigáñez, "A note on Ritt's theorem on decomposition of polynomials", Journal of Pure and Applied Algebra 68:3:293–296 (6 December 1990) For example, x^2 \circ x^3 = x^3 \circ x^2.
|
|
The Bolyai–Gerwien theorem is a related but much simpler result: it states that one can accomplish such a decomposition of a simple polygon with finitely many polygonal pieces if both translations and rotations are allowed for the reassembly. It follows from a result of that it is possible to choose the pieces in such a way that they can be moved continuously while remaining disjoint to yield the square. Moreover, this stronger statement can be proved as well to be accomplished by means of translations only. These results should be compared with the much more paradoxical decompositions in three dimensions provided by the Banach–Tarski paradox; those decompositions can even change the volume of a set.
|
|
In mathematics, Lie group decompositions are used to analyse the structure of Lie groups and associated objects, by showing how they are built up out of subgroups. They are essential technical tools in the representation theory of Lie groups and Lie algebras; they can also be used to study the algebraic topology of such groups and associated homogeneous spaces. Since the use of Lie group methods became one of the standard techniques in twentieth century mathematics, many phenomena can now be referred back to decompositions. The same ideas are often applied to Lie groups, Lie algebras, algebraic groups and p-adic number analogues, making it harder to summarise the facts into a unified theory.
|
|
Kuznetsov is known for his research in algebraic geometry, mostly concerning derived categories of coherent sheaves and their semiorthogonal decompositions. Kuznetsov received an August Möbius fellowship in 1997. He was awarded a European Mathematical Society prize in 2008. He was an invited speaker at the International Mathematical Congress in Seoul (2014).
|
|
Sometime before the middle of August 1941 he and his sister Stefanja were shot to death in Naujoji Vilnia (Nowa Wilejka), 7 km east of Vilnius, by the occupying German forces or Lithuanian collaborators.Purdy, Robert; Zygmunt, Jan (2018-06-29). "Adolf Lindenbaum, Metric Spaces and Decompositions". The Lvov–Warsaw School.
|
|
At the core of SnapPea are two main algorithms. The first attempts to find a minimal ideal triangulation of a given link complement. The second computes the canonical decomposition of a cusped hyperbolic 3-manifold. Almost all the other functions of SnapPea rely in some way on one of these decompositions.
|
|
We have seen the existence of several decompositions that apply in any dimension, namely independent planes, sequential angles, and nested dimensions. In all these cases we can either decompose a matrix or construct one. We have also given special attention to rotation matrices, and these warrant further attention, in both directions .
|
|
In 2010 Anandkumar joined University of California, Irvine, as an Assistant Professor. At the time, the technology industry was at the beginning of the big data revolution. Here she started working on tensor decompositions of latent variable models. She joined Microsoft Research in New England as a visiting scientist in 2012.
|
|
Oja was born in Tallinn and studied at the Tartu State University (now the University of Tartu), completing her undergraduate studies in 1972 and earning a doctorate (Cand.Sc.) in 1975. Her dissertation, Безусловные шаудеровы разложения в локально выпуклых пространствах (Unconditional Schauder decompositions in locally convex spaces) was supervised by Gunnar Kangro.
|
|
Different ways of representing and storing the same data. Table decompositions may vary, column names (data labels) may be different (but have the same semantics), data encoding schemes may vary (i.e., should a measurement scale be explicitly included in a field or should it be implied elsewhere). Also referred as schematic heterogeneity.
|
|
In mathematical analysis, many generalizations of Fourier series have proved to be useful. They are all special cases of decompositions over an orthonormal basis of an inner product space. Here we consider that of square-integrable functions defined on an interval of the real line, which is important, among others, for interpolation theory.
|
|
His publications covers a wide range of topics in graph theory and combinatorics: convex polyhedra, quasigroups, special decompositions into Hamiltonian paths, Latin squares, decompositions of complete graphs, perfect systems of difference sets, additive sequences of permutations, tournaments and combinatorial games theory. The triakis icosahedron, a polyhedron in which every edge has endpoints with total degree at least 13 One of his results, known as Kotzig's theorem, is the statement that every polyhedral graph has an edge whose two endpoints have total degree at most 13. An extreme case is the triakis icosahedron, where no edge has smaller total degree. Kotzig published the result in Slovakia in 1955, and it was named and popularized in the west by Branko Grünbaum in the mid-1970s.
|
|
One of the pillars of the representation theory of quantum groups (and applications to combinatorics) is Kashiwara's theory of crystal bases. These are highly invariant bases which are well suited for decompositions of tensor products. In a paper with S.-J. Kang and M. Kashiwara, Benkart extended the theory of crystal bases to quantum superalgebras.
|
|
Milgram received his Ph.D. from the University of Pennsylvania in 1937. He worked under the supervision of John Kline (a student of Robert Lee Moore). His dissertation was titled "Decompositions and Dimension of Closed Sets in ". Milgram advised 2 students at Syracuse University in the 1940s and 1950s (Robert M. Exner and Adnah Kostenbauder ).
|
|
Linear algebra took its modern form in the first half of the twentieth century, when many ideas and methods of previous centuries were generalized as abstract algebra. The development of computers led to increased research in efficient algorithms for Gaussian elimination and matrix decompositions, and linear algebra became an essential tool for modelling and simulations. See also and .
|
|
Giroux is known for finding a correspondence between contact structures on three-dimensional manifolds and open book decompositions of those manifolds. This result allows contact geometry to be studied using the tools of low- dimensional topology. It has been called a breakthrough by other mathematicians.. In 2002 he was an invited speaker at the International Congress of Mathematicians..
|
|
QR decompositions can also be computed with a series of Givens rotations. Each rotation zeroes an element in the subdiagonal of the matrix, forming the R matrix. The concatenation of all the Givens rotations forms the orthogonal Q matrix. In practice, Givens rotations are not actually performed by building a whole matrix and doing a matrix multiplication.
|
|
Even when a laser is not operating in the fundamental Gaussian mode, its power will generally be found among the lowest-order modes using these decompositions, as the spatial extent of higher order modes will tend to exceed the bounds of a laser's resonator (cavity). "Gaussian beam" normally implies radiation confined to the fundamental (TEM00) Gaussian mode.
|
|
If we manage to find two sequences x_2,\ldots,x_p and y_2,\ldots,y_q of codewords such that x_2\cdots x_p = wy_2\cdots y_q, then we are finished: For then the string x = x_1x_2\cdots x_p can alternatively be decomposed as y_1y_2\cdots y_q, and we have found the desired string having at least two different decompositions into codewords. In the second round, we try out two different approaches: the first trial is to look for a codeword that has w as prefix. Then we obtain a new dangling suffix w', with which we can continue our search. If we eventually encounter a dangling suffix that is itself a codeword (or the empty word), then the search will terminate, as we know there exists a string with two decompositions.
|
|
An older ancient Egyptian papyrus contained a similar table of Egyptian fractions; the Lahun Mathematical Papyri, written around 1850 BCE, is about the age of one unknown source for the Rhind papyrus. The Kahun 2/n fractions were identical to the fraction decompositions given in the Rhind Papyrus' 2/n table. The Egyptian Mathematical Leather Roll (EMLR), circa 1900 BCE, lists decompositions of fractions of the form 1/n into other unit fractions. The table consisted of 26 unit fraction series of the form 1/n written as sums of other rational numbers.. See in particular pages 21–22. The Akhmim wooden tablet wrote fractions in the form 1/n in terms of sums of hekat rational numbers, 1/3, 1/7, 1/10, 1/11 and 1/13.
|
|
The present-day Krull–Schmidt theorem was first proved by Joseph Wedderburn (Ann. of Math (1909)), for finite groups, though he mentions some credit is due to an earlier study of G.A. Miller where direct products of abelian groups were considered. Wedderburn's theorem is stated as an exchange property between direct decompositions of maximum length. However, Wedderburn's proof makes no use of automorphisms.
|
|
He also found a form of the paradox in the plane which uses area- preserving affine transformations in place of the usual congruences. Tarski proved that amenable groups are precisely those for which no paradoxical decompositions exist. Since only free subgroups are needed in the Banach–Tarski paradox, this led to the long-standing von Neumann conjecture, which was disproved in 1980.
|
|
They are also distinguished by the type of sensitivity measure, be it based on (for example) variance decompositions, partial derivatives or elementary effects. In general, however, most procedures adhere to the following outline: # Quantify the uncertainty in each input (e.g. ranges, probability distributions). Note that this can be difficult and many methods exist to elicit uncertainty distributions from subjective data.
|
|
1937-1963 (2007) i.e., in each grid, a space decomposition based on which the smoothing is applied, has to be constructed so that the null space of the singular part of the nearly singular operator has to be included in the sum of the local null spaces, the intersection of the null space and the local spaces resulting from the space decompositions.
|
|
This included a six-week east coast tour that took the band into Canada for the first time, supported by Arkata and Raise Them And Eat Them. The band's sophomore album, Decompositions: Volume Number One, was released after an 8-year silence on December 21, 2012 as a digital download; physical editions of the album were released in April 2013.
|
|
Theodorus Jozef Dekker (Dirk Dekker, born 1 March 1927) is a Dutch mathematician. Dekker completed his Ph.D degree from the University of Amsterdam in 1958. His thesis was titled "Paradoxical Decompositions of Sets and Spaces". Dekker invented an algorithm that allows two processes to share a single-use resource without conflict, using only shared memory for communication, named Dekker's algorithm.
|
|
By its mild nature, the banskali are distinguished from other decompositions; the same difference is observed in speech. In Bansko and Belitsa there are rumors that there are people who are specifically involved in making money-making money; this craft was old in the place. Note that there are many skillful hardware stores. Carpentry is much advanced: coffins, chairs and more.
|
|
Given a point P_1=(x_1,y_1) on T_{a}, its negation with respect to the neutral element (0:1:0) is -P_1=(x_1,-y_1). There are also other formulas given in Christophe Doche, Thomas Icart, and David R. Kohel, Efficient Scalar Multiplication by Isogeny Decompositions, pag 198-199 for Tripling-oriented Doche–Icart–Kohel curves for fast tripling operation and mixed-addition.
|
|
The book is divided into two parts, the first on the existence of paradoxical decompositions and the second on conditions that prevent their existence. After two chapters of background material, the first part proves the Banach–Tarski paradox itself, considers higher-dimensional spaces and non-Euclidean geometry, studies the number of pieces necessary for a paradoxical decomposition, and finds analogous results to the Banach–Tarski paradox for one- and two-dimensional sets. The second part includes a related theorem of Tarski that congruence-invariant finitely-additive measures prevent the existence of paradoxical decompositions, a theorem that Lebesgue measure is the only such measure on the Lebesgue measurable sets, material on amenable groups, connections to the axiom of choice and the Hahn–Banach theorem. Three appendices describe Euclidean groups, Jordan measure, and a collection of open problems.
|
|
In graph theory, a path decomposition of a graph G is, informally, a representation of G as a "thickened" path graph,. and the pathwidth of G is a number that measures how much the path was thickened to form G. More formally, a path-decomposition is a sequence of subsets of vertices of G such that the endpoints of each edge appear in one of the subsets and such that each vertex appears in a contiguous subsequence of the subsets,. and the pathwidth is one less than the size of the largest set in such a decomposition. Pathwidth is also known as interval thickness (one less than the maximum clique size in an interval supergraph of G), vertex separation number, or node searching number.. Pathwidth and path-decompositions are closely analogous to treewidth and tree decompositions.
|
|
For these reasons they are widely used when it is feasible to calculate them. Typically this calculation involves the use of Monte Carlo methods, but since this can involve many thousands of model runs, other methods (such as emulators) can be used to reduce computational expense when necessary. Note that full variance decompositions are only meaningful when the input factors are independent from one another.
|
|
Dense and sparse matrices are supported. Various matrix decompositions are provided through optional integration with Linear Algebra PACKage (LAPACK), Automatically Tuned Linear Algebra Software (ATLAS), and ARPACK. High-performance BLAS/LAPACK replacement libraries such as OpenBLAS and Intel MKL can also be used. The library employs a delayed-evaluation approach (during compile time) to combine several operations into one and reduce (or eliminate) the need for temporaries.
|
|
Gorkin earned bachelor's and master's degrees in statistics from Michigan State University in 1976. She then shifted to pure mathematics for her doctoral studies, completing her Ph.D. at Michigan State in 1982, the same year she joined the Bucknell Faculty. Her dissertation, Decompositions of the Maximal Ideal Space of L, was supervised by Sheldon Axler. At Bucknell, she was Presidential Professor from 2001 to 2004.
|
|
He is the author of four monographs and more than 90 scientific articles. Feldman specializes in the field of abstract harmonic analysis and algebraic probability theory. He constructed a theory of decompositions of random variables and proved analogs of the classical characterization theorems of mathematical statistics in the case when random variables take values in various classes of locally compact Abelian groups (discrete, compact, and others).
|
|
A commutative ring possessing the unique factorization property is called a unique factorization domain. There are number systems, such as certain rings of algebraic integers, which are not unique factorization domains. However, rings of algebraic integers satisfy the weaker property of Dedekind domains: ideals factor uniquely into prime ideals. Factorization may also refer to more general decompositions of a mathematical object into the product of smaller or simpler objects.
|
|
These reactions are often catalyzed by protein enzymes. Enzymes increase the rates of biochemical reactions, so that metabolic syntheses and decompositions impossible under ordinary conditions can occur at the temperatures and concentrations present within a cell. The general concept of a chemical reaction has been extended to reactions between entities smaller than atoms, including nuclear reactions, radioactive decays, and reactions between elementary particles, as described by quantum field theory.
|
|
As cellular decompositions of the projective plane, they have Euler characteristic 1, while spherical polyhedra have Euler characteristic 2. The qualifier "globally" is to contrast with locally projective polyhedra, which are defined in the theory of abstract polyhedra. Non-overlapping projective polyhedra (density 1) correspond to spherical polyhedra (equivalently, convex polyhedra) with central symmetry. This is elaborated and extended below in relation with spherical polyhedra and relation with traditional polyhedra.
|
|
Walecki's Hamiltonian decomposition of the complete graph K_9 In graph theory, a branch of mathematics, a Hamiltonian decomposition of a given graph is a partition of the edges of the graph into Hamiltonian cycles. Hamiltonian decompositions have been studied both for undirected graphs and for directed graphs; in the undirected case, a Hamiltonian decomposition can also be described as a 2-factorization of the graph such that each factor is connected.
|
|
A formula for a certain double transform of the distribution of this area integral is given by Louchard (1984). Groeneboom (1983) and Pitman (1983) give decompositions of Brownian motion W in terms of i.i.d Brownian excursions and the least concave majorant (or greatest convex minorant) of W. For an introduction to Itô's general theory of Brownian excursions and the Itô Poisson process of excursions, see Revuz and Yor (1994), chapter XII.
|
|
He co- founded the International Computer Music Association in 1980 and edited the Computer Music Journal from 1978–2000. He has created software including PulsarGenerator and the Creatovox, both with Alberto de Campo. Since 2004, he has been researching a new method of sound analysis called atomic decompositions, sponsored by the National Science Foundation (NSF). The first movement of his composition Clang-Tint, "Purity", uses intervals from the Bohlen–Pierce scale.
|
|
The treewidth of a problem is the minimal width of its tree decompositions. Bucket elimination can be reformulated as an algorithm working on a particular tree decomposition. In particular, given an ordering of the variables, every variable is associated a bucket containing all constraints such that the variable is the greatest in their scope. Bucket elimination corresponds to the tree decomposition that has a node for each bucket.
|
|
Inductively then, one can also conclude that for any positive integer n. For example, an idempotent element of a matrix ring is precisely an idempotent matrix. For general rings, elements idempotent under multiplication are involved in decompositions of modules, and connected to homological properties of the ring. In Boolean algebra, the main objects of study are rings in which all elements are idempotent under both addition and multiplication.
|
|
The splits of a graph can be collected into a tree-like structure called the split decomposition or join decomposition, which can be constructed in linear time. This decomposition has been used for fast recognition of circle graphs and distance-hereditary graphs, as well as for other problems in graph algorithms. Splits and split decompositions were first introduced by , who also studied variants of the same notions for directed graphs..
|
|
A k-path is a k-tree with at most two leaves, and a k-caterpillar is a k-tree that can be partitioned into a k-path and a set of k-leaves each adjacent to a separator k-clique of the k-path. In particular the maximal graphs of pathwidth one are exactly the caterpillar trees.. Since path-decompositions are a special case of tree-decompositions, the pathwidth of any graph is greater than or equal to its treewidth. The pathwidth is also less than or equal to the cutwidth, the minimum number of edges that cross any cut between lower-numbered and higher-numbered vertices in an optimal linear arrangement of the vertices of a graph; this follows because the vertex separation number, the number of lower-numbered vertices with higher-numbered neighbors, can at most equal the number of cut edges., Lemma 3 p.99; , Theorem 47, p. 24.
|
|
Using the appropriate "angle", and a radial vector, any one of these planes can be given a polar decomposition. Any one of these decompositions, or Lie algebra renderings, may be necessary for rendering the Lie subalgebra of a 2 × 2 real matrix. There is a classical 3-parameter Lie group and algebra pair: the quaternions of unit length which can be identified with the 3-sphere. Its Lie algebra is the subspace of quaternion vectors.
|
|
Surgery theory is a collection of techniques used to produce one manifold from another in a 'controlled' way, introduced by . Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along the cut or boundary. This is closely related to, but not identical with, handlebody decompositions. It is a major tool in the study and classification of manifolds of dimension greater than 3.
|
|
Non-commutative local rings arise naturally as endomorphism rings in the study of direct sum decompositions of modules over some other rings. Specifically, if the endomorphism ring of the module M is local, then M is indecomposable; conversely, if the module M has finite length and is indecomposable, then its endomorphism ring is local. If k is a field of characteristic and G is a finite p-group, then the group algebra kG is local.
|
|
Her dissertation, Regularization of Ill-Posed Problems, was jointly supervised by Dianne P. O'Leary and . After postdoctoral research at Northeastern University, she joined the Tufts faculty in 1999. She was given the William Walker Professorship in 2016, and chaired the Tufts Mathematics Department from 2013 to 2019. In 2019 Kilmer was named a SIAM Fellow "for her fundamental contributions to numerical linear algebra and scientific computing, including ill-posed problems, tensor decompositions, and iterative methods".
|
|
He got his Ph.D. in 1975 and obtained the Candidate degree of the Hungarian Academy of Sciences in 1978, posthumously. He did research in combinatorics; Baranyai's theorem on the decompositions of complete hypergraphs solved a long-standing open problem.. Chapter 38, "Baranyai's theorem", pp. 536–541. As well as being a mathematician, Baranyai was a professional musician, who played the recorder. He died in a car accident after a concert, while touring Hungary with the Bakfark Consort..
|
|
Two types of tensor decompositions exist, which generalise the SVD to multi-way arrays. One of them decomposes a tensor into a sum of rank-1 tensors, which is called a tensor rank decomposition. The second type of decomposition computes the orthonormal subspaces associated with the different factors appearing in the tensor product of vector spaces in which the tensor lives. This decomposition is referred to in the literature as the higher-order SVD (HOSVD) or Tucker3/TuckerM.
|
|
Mahalanobis distance is preserved under full-rank linear transformations of the space spanned by the data. This means that if the data has a nontrivial nullspace, Mahalanobis distance can be computed after projecting the data (non- degenerately) down onto any space of the appropriate dimension for the data. We can find useful decompositions of the squared Mahalanobis distance that help to explain some reasons for the outlyingness of multivariate observations and also provide a graphical tool for identifying outliers.
|
|
In contrast, the opposite is observed in decaying modes: height, vorticity, etc. contours tilt eastward with height, except temperature which tilts westward with height. An equatorward heat flux is induced, decreasing potential vorticity and pressure anomalies and yielding cyclolysis. Making Fourier decompositions on the linearized Eady model equations and solving for the dispersion relation for the Eady Model system allows one to solve for the growth rate of the modes (the imaginary component of the frequency).
|
|
Gyrocommutative gyrogroups are equivalent to K-loopsHubert Kiechle (2002), "Theory of K-loops",Published by Springer,, although defined differently. The terms Bruck loopLarissa Sbitneva (2001), Nonassociative Geometry of Special Relativity, International Journal of Theoretical Physics, Springer, Vol.40, No.1 / Jan 2001 and dyadic symsetJ lawson Y Lim (2004), Means on dyadic symmetrie sets and polar decompositions, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, Springer, Vol.74, No.1 / Dec 2004 are also in use.
|
|
Robert Remak was born in Berlin. He studied at Humboldt University of Berlin under Ferdinand Georg Frobenius and received his doctorate in 1911. His dissertation, Über die Zerlegung der endlichen Gruppen in indirekte unzerlegbare Faktoren ("On the decomposition of a finite group into indirect indecomposable factors") established that any two decompositions of a finite group into a direct product are related by a central automorphism. A weaker form of this statement, uniqueness, was first proved by Joseph Wedderburn in 1909.
|
|
Justus Liebig and his students sought to determine the structure of proteins, but until the methods of Emil Fischer and Franz Hofmeister became available, the amino acid decompositions were unknown.Hubert Bradford Vickery (1942) "Liebig and the Proteins", Journal of Chemical Education Augustus Voelcker was Mulder's assistant for a year from 1846.John Christopher Augustus Voelcker, (1899) Oxford Dictionary of National Biography In 1850, Mulder was elected a foreign member of the Royal Swedish Academy of Sciences. He died in Bennekom.
|
|
In 1993 he proposed an elimination method for triangular decomposition of polynomial systems, which has been referred to as Wang's method and compared with other three methods. Later on he introduced the concepts of regular systems and simple systems and devised algorithms for regular and simple triangular decompositions. He also developed a package, called Epsilon, which implements his methods. Wang popularized the use of methods and tools of computer algebra for symbolic analysis of stability and bifurcation of differential and biological systems.
|
|
In 2002, Emmanuel Giroux published the following result: Theorem. Let M be a compact oriented 3-manifold. Then there is a bijection between the set of oriented contact structures on M up to isotopy and the set of open book decompositions of M up to positive stabilization. Positive stabilization consists of modifying the page by adding a 2-dimensional 1-handle and modifying the monodromy by adding a positive Dehn twist along a curve that runs over that handle exactly once.
|
|
A Heegaard splitting is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies. Every closed, orientable three-manifold may be so obtained; this follows from deep results on the triangulability of three-manifolds due to Moise. This contrasts strongly with higher-dimensional manifolds which need not admit smooth or piecewise linear structures. Assuming smoothness the existence of a Heegaard splitting also follows from the work of Smale about handle decompositions from Morse theory.
|
|
His work with Marc Culler related properties of representation varieties of hyperbolic 3-manifold groups to decompositions of 3-manifolds. Based on this work, Culler, Cameron Gordon, John Luecke, and Shalen proved the cyclic surgery theorem. An important corollary of the theorem is that at most one nontrivial Dehn surgery (+1 or −1) on a knot can result in a simply-connected 3-manifold. This was an important piece of the Gordon–Luecke theorem that knots are determined by their complements.
|
|
It is NP-complete to determine whether the pathwidth of a given graph is at most k, when k is a variable given as part of the input.; ; ; . The best known worst-case time bounds for computing the pathwidth of arbitrary n-vertex graphs are of the form O(2n nc) for some constant c.. Nevertheless, several algorithms are known to compute path-decompositions more efficiently when the pathwidth is small, when the class of input graphs is limited, or approximately.
|
|
The proof of the strong perfect graph theorem by Chudnovsky et al. follows an outline conjectured in 2001 by Conforti, Cornuéjols, Robertson, Seymour, and Thomas, according to which every Berge graph either forms one of five types of basic building block (special classes of perfect graphs) or it has one of four different types of structural decomposition into simpler graphs. A minimally imperfect Berge graph cannot have any of these decompositions, from which it follows that no counterexample to the theorem can exist., Conjecture 5.1.
|
|
To understand Einstein's equations as partial differential equations, it is helpful to formulate them in a way that describes the evolution of the universe over time. This is done in "3+1" formulations, where spacetime is split into three space dimensions and one time dimension. The best-known example is the ADM formalism.; for a pedagogical introduction, see These decompositions show that the spacetime evolution equations of general relativity are well-behaved: solutions always exist, and are uniquely defined, once suitable initial conditions have been specified.
|
|
For every positive integer , a primary decomposition in k[x,y] of the ideal I=\langle x^2, xy \rangle is :I = \langle x^2,xy \rangle = \langle x \rangle \cap \langle x^2, xy, y^n \rangle. The associated primes are :\langle x \rangle \subset \langle x,y \rangle. Example: Let N = R = k[x, y] for some field k, and let M be the ideal (xy, y2). Then M has two different minimal primary decompositions M = (y) ∩ (x, y2) = (y) ∩ (x + y, y2).
|
|
The s-cobordism theorem states for a closed connected oriented manifold M of dimension n > 4 that an h-cobordism W between M and another manifold N is trivial over M if and only if the Whitehead torsion of the inclusion M\hookrightarrow W vanishes. Moreover, for any element in the Whitehead group there exists an h-cobordism W over M whose Whitehead torsion is the considered element. The proofs use handle decompositions. There exists a homotopy theoretic analogue of the s-cobordism theorem.
|
|
Geometric topology is a branch of topology that primarily focuses on low- dimensional manifolds (that is, spaces of dimensions 2, 3, and 4) and their interaction with geometry, but it also includes some higher-dimensional topology.R.B. Sher and R.J. Daverman (2002), Handbook of Geometric Topology, North-Holland. Some examples of topics in geometric topology are orientability, handle decompositions, local flatness, crumpling and the planar and higher-dimensional Schönflies theorem. In high-dimensional topology, characteristic classes are a basic invariant, and surgery theory is a key theory.
|
|
He arrived at Stanford in 1962 and became a professor there in 1970. He advised more than thirty doctoral students, many of whom have themselves achieved distinction. Gene Golub was an important figure in numerical analysis and pivotal to creating the NA-Net and the NA- Digest, as well as the International Congress on Industrial and Applied Mathematics.. One of his best-known books is Matrix Computations, co-authored with Charles F. Van Loan. He was a major contributor to algorithms for matrix decompositions.
|
|
In a paper published in 1924, Stefan Banach and Alfred Tarski gave a construction of such a paradoxical decomposition, based on earlier work by Giuseppe Vitali concerning the unit interval and on the paradoxical decompositions of the sphere by Felix Hausdorff, and discussed a number of related questions concerning decompositions of subsets of Euclidean spaces in various dimensions. They proved the following more general statement, the strong form of the Banach–Tarski paradox: : Given any two bounded subsets and of a Euclidean space in at least three dimensions, both of which have a nonempty interior, there are partitions of and into a finite number of disjoint subsets, A=A_1 \cup \cdots\cup A_k, B=B_1 \cup \cdots\cup B_k (for some integer k), such that for each (integer) between and , the sets and are congruent. Now let be the original ball and be the union of two translated copies of the original ball. Then the proposition means that you can divide the original ball into a certain number of pieces and then rotate and translate these pieces in such a way that the result is the whole set , which contains two copies of .
|
|
The thesis of Robert Remak (1911) derived the same uniqueness result as Wedderburn but also proved (in modern terminology) that the group of central automorphisms acts transitively on the set of direct decompositions of maximum length of a finite group. From that stronger theorem Remak also proved various corollaries including that groups with a trivial center and perfect groups have a unique Remak decomposition. Otto Schmidt (Sur les produits directs, S. M. F. Bull. 41 (1913), 161–164), simplified the main theorems of Remak to the 3 page predecessor to today's textbook proofs.
|
|
Broomhead in 2010 Broomhead's main interest was the development of methods for time series analysis and nonlinear signal processing using techniques from the theory of nonlinear dynamical systems. He also championed applying these ideas in interdisciplinary research. In Japan, Broomhead began to work seriously on applied nonlinear dynamics and chaos. With Greg King he developed techniques to determine whether an experimental time series had been generated by a deterministic chaotic system by combining the pure mathematical results on topological embedding due to Takens with the engineering method of singular value decompositions.
|
|
The demonstration of the t and chi-squared distributions for one-sample problems above is the simplest example where degrees-of-freedom arise. However, similar geometry and vector decompositions underlie much of the theory of linear models, including linear regression and analysis of variance. An explicit example based on comparison of three means is presented here; the geometry of linear models is discussed in more complete detail by Christensen (2002). Suppose independent observations are made for three populations, X_1,\ldots,X_n, Y_1,\ldots,Y_n and Z_1,\ldots,Z_n.
|
|
Unlike more traditional techniques of quantum field theory, conformal bootstrap does not use the Lagrangian of the theory. Instead, it operates with the general axiomatic parameters, such as the scaling dimensions of the local operators and their operator product expansion coefficients. A key axiom is that the product of local operators must be expressible as a sum over local operators (thus turning the product into an algebra); the sum must have a non-zero radius of convergence. This leads to decompositions of correlation functions into structure constants and conformal blocks.
|
|
Azumaya's theorem states that if a module has an decomposition into modules with local endomorphism rings, then all decompositions into indecomposable modules are equivalent to each other; a special case of this, especially in group theory, is known as the Krull–Schmidt theorem. A special case of a decomposition of a module is a decomposition of a ring: for example, a ring is semisimple if and only if it is a direct sum (in fact a product) of matrix rings over division rings (this observation is known as the Artin–Wedderburn theorem).
|
|
Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. Flat sides mean that the sides of a (k+1)-polytope consist of k-polytopes that may have (k−1)-polytopes in common. For example, a two- dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. Some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations, decompositions or tilings of curved manifolds including spherical polyhedra, and set-theoretic abstract polytopes.
|
|
The width of a decomposition method is a measure of the size of problem it produced. Originally, the width was defined as the maximal cardinality of the sets of original variables; one method, the hypertree decomposition, uses a different measure. Either way, the width of a decomposition is defined so that decompositions of size bounded by a constant do not produce excessively large problems. Instances having a decomposition of fixed width can be translated by decomposition into instances of size bounded by a polynomial in the size of the original instance.
|
|
The branchwidth of G is the minimum width of any branch-decomposition of G. Branchwidth is closely related to tree-width: for all graphs, both of these numbers are within a constant factor of each other, and both quantities may be characterized by forbidden minors. And as with treewidth, many graph optimization problems may be solved efficiently for graphs of small branchwidth. However, unlike treewidth, the branchwidth of planar graphs may be computed exactly, in polynomial time. Branch-decompositions and branchwidth may also be generalized from graphs to matroids.
|
|
Analogous characterizations of other families of graphs in terms of the summands of their clique-sum decompositions have since become standard in graph minor theory. Wagner conjectured in the 1930s (although this conjecture was not published until later). that in any infinite set of graphs, one graph is isomorphic to a minor of another. The truth of this conjecture implies that any family of graphs closed under the operation of taking minors (as planar graphs are) can automatically be characterized by finitely many forbidden minors analogously to Wagner's theorem characterizing the planar graphs.
|
|
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiable function on a manifold will reflect the topology quite directly. Morse theory allows one to find CW structures and handle decompositions on manifolds and to obtain substantial information about their homology. Before Morse, Arthur Cayley and James Clerk Maxwell had developed some of the ideas of Morse theory in the context of topography.
|
|
On the way, Hui stopped at IPSA in Toronto and obtained a copy of "Operators and Functions" [IBM Research Report No. 7091, 1978]. He has been studying that paper and its successors ever since. In September 1979, Hui entered the Department of Computer Science at the University of Toronto, and received his MSc in May 1981 with a thesis on "The complexity of some decompositions in matrix algebra." After completing his master's degree, Hui worked from 1981 to 1985 as an APL systems analyst and programmer for the Alberta Energy Company in Edmonton.
|
|
Chapter 5 discusses normal surfaces, surfaces that intersect the tetrahedra of a triangulation of a manifold in a controlled way. By parameterizing these surfaces by how many pieces of each possible type they can have within each tetrahedron of a triangulation, one can reduce many questions about manifolds such as the recognition of trivial knots and trivial manifolds to questions in number theory, on the existence of solutions to certain Diophantine equations. The book uses this tool to prove the existence and uniqueness of prime decompositions of manifolds. Chapter 6 concerns Heegaard splittings, surfaces which split a given manifold into two handlebodies.
|
|
In 1999 he returned to Swansea University, where he currently holds a Research Professorship. Williams's research interests encompass Brownian motion, diffusions, Markov processes, martingales and Wiener-Hopf theory. Recognition for his work includes being elected Fellow of the Royal Society in 1984, where he was cited for his achievements on the construction problem for Markov chains and on path decompositions for Brownian motion, and being awarded the London Mathematical Society's Pólya Prize in 1994. He is the author of Probability With Martingales and Weighing the Odds, and co-author (with L. C. G. Rogers) of both volumes of Diffusions, Markov Processes and Martingales.
|
|
In mathematics, p-adic Hodge theory is a theory that provides a way to classify and study p-adic Galois representations of characteristic 0 local fieldsIn this article, a local field is complete discrete valuation field whose residue field is perfect. with residual characteristic p (such as Qp). The theory has its beginnings in Jean-Pierre Serre and John Tate's study of Tate modules of abelian varieties and the notion of Hodge–Tate representation. Hodge–Tate representations are related to certain decompositions of p-adic cohomology theories analogous to the Hodge decomposition, hence the name p-adic Hodge theory.
|
|
In fixed-point arithmetic, the finite-precision errors accumulated by FFT algorithms are worse, with rms errors growing as O() for the Cooley–Tukey algorithm (Welch, 1969). Achieving this accuracy requires careful attention to scaling to minimize loss of precision, and fixed-point FFT algorithms involve rescaling at each intermediate stage of decompositions like Cooley–Tukey. To verify the correctness of an FFT implementation, rigorous guarantees can be obtained in O(N log N) time by a simple procedure checking the linearity, impulse-response, and time-shift properties of the transform on random inputs (Ergün, 1995).
|
|
This idea was based on previous conjectured structural decompositions of similar type that would have implied the strong perfect graph conjecture but turned out to be false.; ; ; , section 4.6 "The first conjectures". The five basic classes of perfect graphs that form the base case of this structural decomposition are the bipartite graphs, line graphs of bipartite graphs, complementary graphs of bipartite graphs, complements of line graphs of bipartite graphs, and double split graphs. It is easy to see that bipartite graphs are perfect: in any nontrivial induced subgraph, the clique number and chromatic number are both two and therefore both equal.
|
|
The cusp form idea came out of the cusps on modular curves but also had a meaning visible in spectral theory as "discrete spectrum", contrasted with the "continuous spectrum" from Eisenstein series. It becomes much more technical for bigger Lie groups, because the parabolic subgroups are more numerous. In all these approaches there was no shortage of technical methods, often inductive in nature and based on Levi decompositions amongst other matters, but the field was and is very demanding. And on the side of modular forms, there were examples such as Hilbert modular forms, Siegel modular forms, and theta-series.
|
|
Although many other related graph partitioning problems are NP-complete, even for planar graphs, it is possible to find a minimum-width branch-decomposition of a planar graph in polynomial time.. By applying methods of more directly in the construction of branch- decompositions, show that every planar graph has branchwidth at most 2.12√n, with the same constant as the one in the simple cycle separator theorem of Alon et al. Since the treewidth of any graph is at most 3/2 its branchwidth, this also shows that planar graphs have treewidth at most 3.18√n.
|
|
The proposed Ensemble Empirical Mode Decomposition is developed as follows: # add a white noise series to the targeted data; # decompose the data with added white noise into IMFs; # repeat step 1 and step 2 again and again, but with different white noise series each time;and # obtain the (ensemble) means of corresponding IMFs of the decompositions as the final result. The effects of the decomposition using the EEMD are that the added white noise series cancel each other, and the mean IMFs stays within the natural dyadic filter windows, significantly reducing the chance of mode mixing and preserving the dyadic property.
|
|
Multicategories are often incorrectly considered to belong to higher category theory, as their original application was the observation that the operators and identities satisfied by higher categories are the objects and multiarrows of a multicategory. The study of n-categories was in turn motivated by applications in algebraic topology and attempts to describe the homotopy theory of higher dimensional manifolds. However it has mostly grown out of this motivation and is now also considered to be part of pure mathematics. The correspondence between contractions and decompositions of triangles in a multiorder allows one to construct an associative algebra called its incidence algebra.
|
|
Reactions including the use of sodium hydride in DMF as a solvent are somewhat hazardous; exothermic decompositions have been reported at temperatures as low as 26 °C. On a laboratory scale any thermal runaway is (usually) quickly noticed and brought under control with an ice bath and this remains a popular combination of reagents. On a pilot plant scale, on the other hand, several accidents have been reported.UK Chemical Reaction Hazards Forum and references cited therein On the 20 of June 2018, the Danish Environmental Protective Agency published an article about the DMF's use in squishies.
|
|
The transposition (a~b), executed thereafter, then addresses z by the index of b to swap what initially were a and z. In fact, the symmetric group is a Coxeter group, meaning that it is generated by elements of order 2 (the adjacent transpositions), and all relations are of a certain form. One of the main results on symmetric groups states that either all of the decompositions of a given permutation into transpositions have an even number of transpositions, or they all have an odd number of transpositions. This permits the parity of a permutation to be a well-defined concept.
|
|
In mathematics, a Hironaka decomposition is a representation of an algebra over a field as a finitely generated free module over a polynomial subalgebra or a regular local ring. Such decompositions are named after Heisuke Hironaka, who used this in his unpublished master's thesis at Kyoto University . Hironaka's criterion , sometimes called miracle flatness, states that a local ring R that is a finitely generated module over a regular Noetherian local ring S is Cohen–Macaulay if and only if it is a free module over S. There is a similar result for rings that are graded over a field rather than local.
|
|
This hierarchy consists of decomposed low level units of complex actions that could be performed on objects relevant to the domain of computers as assigned in the interface objects hierarchy. Each level in the hierarchy represent different level of decompositions. A high level plan to create a text file might involve mid- level actions such as creating a file, inserting text and saving that file. The mid-level action of saving a file the file can be decomposed into lower level actions such as storing the file with a backup copy and applying the access control rights.
|
|
In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces (for example the Betti numbers) were regarded as derived from combinatorial decompositions of spaces, such as decomposition into simplicial complexes. After the proof of the simplicial approximation theorem this approach provided rigour. The change of name reflected the move to organise topological classes such as cycles-modulo-boundaries explicitly into abelian groups. This point of view is often attributed to Emmy Noether,For example L'émergence de la notion de groupe d'homologie, Nicolas Basbois (PDF), note 41, explicitly names Noether as inventing homology groups.
|
|
Following his exclusion, Wolman continued to develop his own work, and he re-established links with the original Letterist movement and exhibited with them from 1961 to 1964. He devised Scotch Art in 1963, a process which consists in tearing off bands of printed matter and using adhesive tape to reposition them on fabrics or wood. In 1964, however, he split again from Isou's group, to establish the short-lived Second Letterist International with Jean-Louis Brau and François Dufrêne; thereafter, Wolman worked largely in isolation. He later developed the "separatist movement", and series of "dühring dühring", "decompositions" and finally "depicted painting".
|
|
In the area of artificial intelligence, he is best known for his influential early work on the complexity of nonmonotonic logics and on (generalised) hypertree decompositions, a framework for obtaining tractable structural classes of constraint satisfaction problems, and a generalisation of the notion of tree decomposition from graph theory. This work has also had substantial impact in database theory, since it is known that the problem of evaluating conjunctive queries on relational databases is equivalent to the constraint satisfaction problem. His recent work on XML query languages (notably XPath) has helped create the complexity-theoretical foundations of this area.
|
|
In numerical analysis, different decompositions are used to implement efficient matrix algorithms. For instance, when solving a system of linear equations Ax=b, the matrix A can be decomposed via the LU decomposition. The LU decomposition factorizes a matrix into a lower triangular matrix L and an upper triangular matrix U. The systems L(Ux)=b and Ux=L^{-1}b require fewer additions and multiplications to solve, compared with the original system Ax=b, though one might require significantly more digits in inexact arithmetic such as floating point. Similarly, the QR decomposition expresses A as QR with Q an orthogonal matrix and R an upper triangular matrix.
|
|
An LDU decomposition is a decomposition of the form : A = LDU, where D is a diagonal matrix, and L and U are unitriangular matrices, meaning that all the entries on the diagonals of L and U are one. Above we required that A be a square matrix, but these decompositions can all be generalized to rectangular matrices as well. In that case, L and D are square matrices both of which have the same number of rows as A, and U has exactly the same dimensions as A. Upper triangular should be interpreted as having only zero entries below the main diagonal, which starts at the upper left corner.
|
|
If e is an idempotent (e2=e) in an associative algebra A, then the two-sided Peirce decomposition writes A as the direct sum of eAe, eA(1−e), (1−e)Ae, and (1−e)A(1−e). There are also left and right Peirce decompositions, where the left decomposition writes A as the direct sum of eA and (1−e)A, and the right one writes A as the direct sum of Ae and A(1−e). More generally, if e1,...,en are mutually orthogonal idempotents with sum 1, then A is the direct sum of the spaces eiAej for 1≤i,j≤n.
|
|
This is called a pants decomposition for the surface, and the curves are called the cuffs of the decomposition. This decomposition is not unique, but by quantifying the argument one sees that all pants decompositions of a given surface have the same number of curves, which is exactly the complexity. For connected surfaces a pants decomposition has exactly 2g - 2 + k pants. A collection of simple closed curves on a surface is a pants decomposition if and only if they are disjoint, no two of them are homotopic and none is homotopic to a boundary component, and the collection is maximal for these properties.
|
|
The new polygons have the same area as the old polygon, but the two transformed sets cannot have the same measure as before (since they contain only part of the B points), and therefore there is no measure that "works". The class of groups isolated by von Neumann in the course of study of Banach–Tarski phenomenon turned out to be very important for many areas of mathematics: these are amenable groups, or groups with an invariant mean, and include all finite and all solvable groups. Generally speaking, paradoxical decompositions arise when the group used for equivalences in the definition of equidecomposability is not amenable.
|
|
Originally, Hodge structures were introduced as a tool for keeping track of abstract Hodge decompositions on the cohomology groups of smooth projective algebraic varieties. These structures gave geometers new tools for studying algebraic curves, such as the Torelli theorem, Abelian varieties, and the cohomology of smooth projective varieties. One of the chief results for computing Hodge structures is an explicit decomposition of the cohomology groups of smooth hypersurfaces using the relation between the Jacobian ideal and the Hodge decomposition of a smooth projective hypersurface through Griffith's residue theorem. Porting this language to smooth non- projective varieties and singular varieties requires the concept of mixed Hodge structures.
|
|
The additional conditions of the definition of a hinge decomposition are three, of which the first two ensure equivalence of the original problem with the new one. The two conditions for equivalence are: the scope of each constraint is contained in at least one node of the tree, and the subtree induced by a variable of the original problem is connected. The additional condition is that, if two nodes are joined, then they share exactly one constraint, and the scope of this constraint contains all variables shared by the two nodes. The maximal number of constraints of a node is the same for all hinge decompositions of the same problem.
|
|
This means, during deployment, there is no need to carry around a language model making it very practical for applications with limited memory. By the end of 2016, the attention-based models have seen considerable success including outperforming the CTC models (with or without an external language model). Various extensions have been proposed since the original LAS model. Latent Sequence Decompositions (LSD) was proposed by Carnegie Mellon University, MIT and Google Brain to directly emit sub-word units which are more natural than English characters; University of Oxford and Google DeepMind extended LAS to "Watch, Listen, Attend and Spell" (WLAS) to handle lip reading surpassing human-level performance.
|
|
The off-energy shell amplitudes do not coincide with the Feynman amplitudes, and they depend on the orientation of the light-front plane. In the covariant formulation, this dependence is explicit: the amplitudes are functions of \omega. This allows one to apply to them in full measure the well known techniques developed for the covariant Feynman amplitudes (constructing the invariant variables, similar to the Mandelstam variables, on which the amplitudes depend; the decompositions, in the case of particles with spins, in invariant amplitudes; extracting electromagnetic form factors; etc.). The irreducible off-energy- shell amplitudes serve as the kernels of equations for the light-front wave functions.
|
|
At the beginning of the 1970s, it was observed that a large class of combinatorial optimization problems defined on graphs could be efficiently solved by non serial dynamic programming as long as the graph had a bounded dimension, a parameter shown to be equivalent to treewidth by . Later, several authors independently observed at the end of the 1980s; ; . that many algorithmic problems that are NP-complete for arbitrary graphs may be solved efficiently by dynamic programming for graphs of bounded treewidth, using the tree-decompositions of these graphs. As an example, the problem of coloring a graph of treewidth k may be solved by using a dynamic programming algorithm on a tree decomposition of the graph.
|
|
In numerical linear algebra, the Bartels–Stewart algorithm is used to numerically solve the Sylvester matrix equation AX - XB = C. Developed by R.H. Bartels and G.W. Stewart in 1971, it was the first numerically stable method that could be systematically applied to solve such equations. The algorithm works by using the real Schur decompositions of A and B to transform AX - XB = C into a triangular system that can then be solved using forward or backward substitution. In 1979, G. Golub, C. Van Loan and S. Nash introduced an improved version of the algorithm, known as the Hessenberg–Schur algorithm. It remains a standard approach for solving Sylvester equations when X is of small to moderate size.
|
|
Expanding a vertex of a 2k-regular graph into a clique of 2k vertices, one for each endpoint of an edge at the replaced vertex, cannot change whether the graph has a Hamiltonian decomposition. The reverse of this expansion process, collapsing a clique to a single vertex, will transform any Hamiltonian decomposition in the larger graph into a Hamiltonian decomposition in the original graph. Conversely, Walecki's construction can be applied to the clique to expand any Hamiltonian decomposition of the smaller graph into a Hamiltonian decomposition of the expanded graph. This expansion process can be used to produce arbitrarily large vertex-transitive graphs and Cayley graphs of even degree that do not have Hamiltonian decompositions.
|
|
Special hypergeometric functions occur as zonal spherical functions on Riemannian symmetric spaces and semi-simple Lie groups. Their importance and role can be understood through the following example: the hypergeometric series 2F1 has the Legendre polynomials as a special case, and when considered in the form of spherical harmonics, these polynomials reflect, in a certain sense, the symmetry properties of the two-sphere or, equivalently, the rotations given by the Lie group SO(3). In tensor product decompositions of concrete representations of this group Clebsch–Gordan coefficients are met, which can be written as 3F2 hypergeometric series. Bilateral hypergeometric series are a generalization of hypergeometric functions where one sums over all integers, not just the positive ones.
|
|
Compared with SD models, SR models provide a more detailed level of modeling by looking inside actors to model internal, intentional relationships. Intentional elements (goals, soft goals, tasks, resources) appear in the SR model not only as external dependencies, but also as internal elements linked by means-ends relationships and task-decompositions. The means-end links provide understanding about why an actor would engage in some tasks, pursue a goal, need a resource, or want a soft goal; the task-decomposition links provide a hierarchical description of intentional elements that make up a routine. Such a model is used to describe stakeholder interests and concerns, and how they might be addressed by different configurations of systems and environments.
|
|
Some deep learning architectures display problematic behaviors, such as confidently classifying unrecognizable images as belonging to a familiar category of ordinary images and misclassifying minuscule perturbations of correctly classified images. Goertzel hypothesized that these behaviors are due to limitations in their internal representations and that these limitations would inhibit integration into heterogeneous multi-component artificial general intelligence (AGI) architectures. These issues may possibly be addressed by deep learning architectures that internally form states homologous to image-grammar decompositions of observed entities and events. Learning a grammar (visual or linguistic) from training data would be equivalent to restricting the system to commonsense reasoning that operates on concepts in terms of grammatical production rules and is a basic goal of both human language acquisitionMiller, G. A., and N. Chomsky.
|
|
Building on the work of Felix Hausdorff, in 1924 Stefan Banach and Alfred Tarski proved that given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets that can be reassembled together in a different way to yield two identical copies of the original ball. Banach and Tarski proved that, using isometric transformations, the result of taking apart and reassembling a two-dimensional figure would necessarily have the same area as the original. This would make creating two unit squares out of one impossible. But in a 1929 paper, von Neumann proved that paradoxical decompositions could use a group of transformations that include as a subgroup a free group with two generators.
|
|
It has a straightforward extension to modules stating that every submodule of a finitely generated module over a Noetherian ring is a finite intersection of primary submodules. This contains the case for rings as a special case, considering the ring as a module over itself, so that ideals are submodules. This also generalizes the primary decomposition form of the structure theorem for finitely generated modules over a principal ideal domain, and for the special case of polynomial rings over a field, it generalizes the decomposition of an algebraic set into a finite union of (irreducible) varieties. The first algorithm for computing primary decompositions for polynomial rings over a field of characteristic 0Primary decomposition requires testing irreducibility of polynomials, which is not always algorithmically possible in nonzero characteristic.
|
|
The Oberwolfach problem on decompositions of complete graphs into copies of a given 2-regular graph is related, but neither is a special case of the other. If G is a 2-regular graph, with n vertices, formed from a disjoint union of cycles of certain lengths, then a solution to the Oberwolfach problem for G would also provide a decomposition of the complete graph into (n-1)/2 copies of each of the cycles of G. However, not every decomposition of K_n into this many cycles of each size can be grouped into disjoint cycles that form copies of G, and on the other hand not every instance of Alspach's conjecture involves sets of cycles that have (n-1)/2 copies of each cycle.
|
|
Dunwoody works on geometric group theory and low-dimensional topology. He is a leading expert in splittings and accessibility of discrete groups, groups acting on graphs and trees, JSJ-decompositions, the topology of 3-manifolds and the structure of their fundamental groups. Since 1971 several mathematicians have been working on Wall's conjecture, posed by Wall in a 1971 paper,Wall, C. T. C., Pairs of relative cohomological dimension one. Journal of Pure and Applied Algebra, vol. 1 (1971), no. 2, pp. 141-154 which said that all finitely generated groups are accessible. Roughly, this means that every finitely generated group can be constructed from finite and one-ended groups via a finite number of amalgamated free products and HNN extensions over finite subgroups.
|
|
A path decomposition can be described as a sequence of graphs Gi that are glued together by identifying pairs of vertices from consecutive graphs in the sequence, such that the result of performing all of these gluings is G. The graphs Gi may be taken as the induced subgraphs of the sets Xi in the first definition of path decompositions, with two vertices in successive induced subgraphs being glued together when they are induced by the same vertex in G, and in the other direction one may recover the sets Xi as the vertex sets of the graphs Gi. The width of the path decomposition is then one less than the maximum number of vertices in one of the graphs Gi.
|
|
The circuit rank controls the number of ears in an ear decomposition of a graph, a partition of the edges of the graph into paths and cycles that is useful in many graph algorithms. In particular, a graph is 2-vertex-connected if and only if it has an open ear decomposition. This is a sequence of subgraphs, where the first subgraph is a simple cycle, the remaining subgraphs are all simple paths, each path starts and ends on vertices that belong to previous subgraphs, and each internal vertex of a path appears for the first time in that path. In any biconnected graph with circuit rank r, every open ear decomposition has exactly r ears.. See in particular Theorems 18 (relating ear decomposition to circuit rank) and 19 (on the existence of ear decompositions).
|
|
The techniques mentioned above for SONET and WDM networks can also be applied to mesh network architectures provided there are ring decompositions for the mesh architectures; and use well defined protection- switching schemes to restore service when a failure occurs. The three most notable ring-based protection techniques for mesh networks are ring covers, cycle double covers and p-cycles (pre-configured protection cycles). The main goal of the ring cover technique is to find a set of rings that covers all the network links and then use these rings to protect the network against failures. Some network links in the ring cover might get used in more than one ring which can cause additional redundancy in the network and because of this reason, scaling down redundancy is the primary focus of this technique.
|
|
It is NP-complete to determine whether a graph G has a branch-decomposition of width at most k, when G and k are both considered as inputs to the problem. However, the graphs with branchwidth at most k form a minor-closed family of graphs,, Theorem 4.1, p. 164. from which it follows that computing the branchwidth is fixed- parameter tractable: there is an algorithm for computing optimal branch- decompositions whose running time, on graphs of branchwidth k for any fixed constant k, is linear in the size of the input graph.. describe an algorithm with improved dependence on k, (2)k, at the expense of an increase in the dependence on the number of vertices from linear to quadratic. For planar graphs, the branchwidth can be computed exactly in polynomial time.
|
|
Let V and W be handlebodies of genus g, and let ƒ be an orientation reversing homeomorphism from the boundary of V to the boundary of W. By gluing V to W along ƒ we obtain the compact oriented 3-manifold : M = V \cup_f W. Every closed, orientable three-manifold may be so obtained; this follows from deep results on the triangulability of three- manifolds due to Moise. This contrasts strongly with higher-dimensional manifolds which need not admit smooth or piecewise linear structures. Assuming smoothness the existence of a Heegaard splitting also follows from the work of Smale about handle decompositions from Morse theory. The decomposition of M into two handlebodies is called a Heegaard splitting, and their common boundary H is called the Heegaard surface of the splitting.
|
|
Donald G. Higman (September 20, 1928 in Vancouver – February 13, 2006) was an American mathematician known for his discovery, in collaboration with Charles C. Sims, of the Higman–Sims group.. Higman did his undergraduate studies at the University of British Columbia, and received his Ph.D. in 1952 from the University of Illinois Urbana-Champaign under Reinhold Baer. He served on the faculty of mathematics at the University of Michigan from 1956 to 1998. His work on homological aspects of group representation theory established the concept of a relatively projective module and explained its role in the theory of module decompositions. He developed a characterization of rank-2 permutation groups, and a theory of rank-3 permutation groups; several of the later-discovered sporadic simple groups were of this type, including the Higman–Sims group which he and Sims constructed in 1967.
|
|
The next theorem gives necessary and sufficient conditions for a ring to have primary decompositions for its ideals. The proof is given at Chapter 4 of Atiyah–MacDonald as a series of exercises. There is the following uniqueness theorem for an ideal having a primary decomposition. Now, for any commutative ring R, an ideal I and a minimal prime P over I, the pre- image of I RP under the localization map is the smallest P-primary ideal containing I. Thus, in the setting of preceding theorem, the primary ideal Q corresponding to a minimal prime P is also the smallest P-primary ideal containing I and is called the P-primary component of I. For example, if the power Pn of a prime P has a primary decomposition, then its P-primary component is the n-th symbolic power of P.
|
|
In 1841 he published his Principles of Mechanism, and in 1851 A System of Apparatus for the Use of Lecturers and Experimenters in Mechanical Philosophy, as well as many works on medieval architecture and the mechanical construction of English cathedrals, notable for his incisive decompositions of these structures' functional and decorative aspects. He willed his manuscript on the Architectural History of the University of Cambridge to his nephew John Willis Clark who completed it. Willis's theory of vowel production assumed a close correspondence between vowel production and the production of musical notes using an organ: the lung acted as a bellows, the vocal folds acted as the reed, and the mouth cavity acted as the organ pipe. Different vowels corresponded to mouth cavities(/organ pipes) of different lengths, which were independent of the properties or vibrations of the vocal folds(/reed).
|
|
However it is possible that cyclic subspaces do allow a decomposition as direct sum of smaller cyclic subspaces (essentially by the Chinese remainder theorem). Therefore, just having for both matrices some decomposition of the space into cyclic subspaces, and knowing the corresponding minimal polynomials, is not in itself sufficient to decide their similarity. An additional condition is imposed to ensure that for similar matrices one gets decompositions into cyclic subspaces that exactly match: in the list of associated minimal polynomials each one must divide the next (and the constant polynomial 1 is forbidden to exclude trivial cyclic subspaces of dimension 0). The resulting list of polynomials are called the invariant factors of (the K[X]-module defined by) the matrix, and two matrices are similar if and only if they have identical lists of invariant factors.
|
|
The typical approach to proving Courcelle's theorem involves the construction of a finite bottom-up tree automaton that acts on the tree decompositions of the given graph. In more detail, two graphs G1 and G2, each with a specified subset T of vertices called terminals, may be defined to be equivalent with respect to an MSO formula F if, for all other graphs H whose intersection with G1 and G2 consists only of vertices in T, the two graphs G1 ∪ H and G2 ∪ H behave the same with respect to F: either they both model F or they both do not model F. This is an equivalence relation, and it can be shown by induction on the length of F that (when the sizes of T and F are both bounded) it has finitely many equivalence classes., Theorem 13.1.1, p. 266.
|
|
To overcome the problems, some applications may simply attempt to replace the decomposed characters with the equivalent precomposed characters. With an incomplete font, however, precomposed characters may also be problematic – especially if they are more exotic, as in the following example (showing the reconstructed Proto-Indo-European word for "dog"): #ḱṷṓn (U+1E31 U+1E77 U+1E53 U+006E) #ḱṷṓn (U+006B U+0301 U+0075 U+032D U+006F U+0304 U+0301 U+006E) In some situations, the precomposed green k, u and o with diacritics may render as unrecognized characters, or their typographical appearance may be very different from the final letter n with no diacritic. On the second line, the base letters should at least render correctly even if the combining diacritics could not be recognized. OpenType has the ccmp "feature tag" to define glyphs that are compositions or decompositions involving combining characters.
|
|
The first two steps of the Gram–Schmidt process In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process is a method for orthonormalizing a set of vectors in an inner product space, most commonly the Euclidean space Rn equipped with the standard inner product. The Gram–Schmidt process takes a finite, linearly independent set S = {v1, ..., vk} for and generates an orthogonal set that spans the same k-dimensional subspace of Rn as S. The method is named after Jørgen Pedersen Gram and Erhard Schmidt, but Pierre-Simon Laplace had been familiar with it before Gram and Schmidt. In the theory of Lie group decompositions it is generalized by the Iwasawa decomposition. The application of the Gram–Schmidt process to the column vectors of a full column rank matrix yields the QR decomposition (it is decomposed into an orthogonal and a triangular matrix).
|
|
The components in the decomposition, however, are not prime automata (with prime defined in a naïve way); rather, the notion of prime is more sophisticated and algebraic: the semigroups and groups associated to the constituent automata of the decomposition are prime (or irreducible) in a strict and natural algebraic sense with respect to the wreath product (Eilenberg, 1976). Also, unlike earlier decomposition theorems, the Krohn–Rhodes decompositions usually require expansion of the state-set, so that the expanded automaton covers (emulates) the one being decomposed. These facts have made the theorem difficult to understand, and challenging to apply in a practical way—until recently, when computational implementations became available (Egri-Nagy & Nehaniv 2005, 2008). H.P. Zeiger (1967) proved an important variant called the holonomy decomposition (Eilenberg 1976).Eilenberg 1976, as well as Dömösi and Nehaniv, 2005, present proofs that correct an error in Zeiger's paper.
|
|
In his later work, Tarski showed that, conversely, non-existence of paradoxical decompositions of this type implies the existence of a finitely-additive invariant measure. The heart of the proof of the "doubling the ball" form of the paradox presented below is the remarkable fact that by a Euclidean isometry (and renaming of elements), one can divide a certain set (essentially, the surface of a unit sphere) into four parts, then rotate one of them to become itself plus two of the other parts. This follows rather easily from a -paradoxical decomposition of , the free group with two generators. Banach and Tarski's proof relied on an analogous fact discovered by Hausdorff some years earlier: the surface of a unit sphere in space is a disjoint union of three sets and a countable set such that, on the one hand, are pairwise congruent, and on the other hand, is congruent with the union of and .
|
|
In the Euclidean plane, two figures that are equidecomposable with respect to the group of Euclidean motions are necessarily of the same area, and therefore, a paradoxical decomposition of a square or disk of Banach–Tarski type that uses only Euclidean congruences is impossible. A conceptual explanation of the distinction between the planar and higher-dimensional cases was given by John von Neumann: unlike the group SO(3) of rotations in three dimensions, the group E(2) of Euclidean motions of the plane is solvable, which implies the existence of a finitely-additive measure on E(2) and R2 which is invariant under translations and rotations, and rules out paradoxical decompositions of non-negligible sets. Von Neumann then posed the following question: can such a paradoxical decomposition be constructed if one allows a larger group of equivalences? It is clear that if one permits similarities, any two squares in the plane become equivalent even without further subdivision.
|
|
Brian Alspach and Heather Gavlas established necessary and sufficient conditions for the existence of a decomposition of a complete graph of even order minus a 1-factor into even cycles and a complete graph of odd order into odd cycles. Their proof relies on Cayley graphs, in particular, circulant graphs, and many of their decompositions come from the action of a permutation on a fixed subgraph. They proved that for positive even integers m and n with 4\leq m\leq n , the graph K_n-I (where I is a 1-factor) can be decomposed into cycles of length m if and only if the number of edges in K_n-I is a multiple of m. Also, for positive odd integers m and n with 3≤m≤n, the graph K_n can be decomposed into cycles of length m if and only if the number of edges in K_n is a multiple of m.
|
|
Two different pants decompositions for the surface of genus 2 The importance of the pairs of pants in the study of surfaces stems from the following property: define the complexity of a connected compact surface S of genus g with k boundary components to be \xi(S) = 3g - 3 + k, and for a non- connected surface take the sum over all components. Then the only surfaces with negative Euler characteristic and complexity zero are disjoint unions of pairs of pants. Furthermore, for any surface S and any simple closed curve c on S which is not homotopic to a boundary component, the compact surface obtained by cutting S along c has a complexity that is strictly less than S. In this sense, pairs of pants are the only "irreducible" surfaces among all surfaces of negative Euler characteristic. By a recursion argument, this implies that for any surface there is a system of simple closed curves which cut the surface into pairs of pants.
|
|
Factor-critical graphs must always have an odd number of vertices, and must be 2-edge-connected (that is, they cannot have any bridges).. However, they are not necessarily 2-vertex- connected; the friendship graphs provide a counterexample. It is not possible for a factor-critical graph to be bipartite, because in a bipartite graph with a near-perfect matching, the only vertices that can be deleted to produce a perfectly matchable graph are the ones on the larger side of the bipartition. Every 2-vertex-connected factor-critical graph with edges has at least different near-perfect matchings, and more generally every factor-critical graph with edges and blocks (2-vertex-connected components) has at least different near-perfect matchings. The graphs for which these bounds are tight may be characterized by having odd ear decompositions of a specific form.. Any connected graph may be transformed into a factor-critical graph by contracting sufficiently many of its edges.
|
|
It is also possible to define a notion of branch-decomposition for matroids that generalizes branch-decompositions of graphs.. Section 12, "Tangles and Matroids", pp. 188–190. A branch- decomposition of a matroid is a hierarchical clustering of the matroid elements, represented as an unrooted binary tree with the elements of the matroid at its leaves. An e-separation may be defined in the same way as for graphs, and results in a partition of the set M of matroid elements into two subsets A and B. If ρ denotes the rank function of the matroid, then the width of an e-separation is defined as , and the width of the decomposition and the branchwidth of the matroid are defined analogously. The branchwidth of a graph and the branchwidth of the corresponding graphic matroid may differ: for instance, the three-edge path graph and the three-edge star have different branchwidths, 2 and 1 respectively, but they both induce the same graphic matroid with branchwidth 1.
|
|
These decompositions may occur during tissue isolation procedures. Recent studies indicate that the metabolism by ALOXE3 of the R stereoisomer of 12-HpETE made by ALOX12B and therefore possibly the S stereoisomer of 12-HpETE made by ALOX12 or ALOX15 is responsible for forming various hepoxilins in the epidermis of human and mouse skin and tongue and possibly other tissues. Human skin metabolizes 12(S)-HpETE in reactions strictly analogous to those of 12(R)-HpETE; it metabolized 12(S)-HpETE by eLOX3 to 8R-hydroxy-11S,12S-epoxy-5Z,9E,14Z-eicosatetraenoic acid and 12-oxo-ETE, with the former product then being metabolized by sEH to 8R,11S,12S-trihydroxy-5Z,9E,14Z-eicosatetraenoic acid. 12(S)-HpETE also spontaneously decomposes to a mixture of hepoxilins and trihydroxy- eicosatetraenoic acids (trioxillins) that possess R or S hydroxy and R,S or S,R epoxide residues at various sites while 8R-hydroxy-11S,12S-epoxy-hepoxilin A3 spontaneously decomposes to 8R,11S,12S-trihydroxy-5Z,9E,14Z-eicosatetraenoic acid.
|
|
It suggested that a machine-readable mapping file between Unicode and KPS 9566 could be provided by the North Korean body itself, and would be more useful than a printed cross-reference in the standard document. Regarding the proposed additional characters, the response stated that characters which would have compatibility decompositions in Unicode should not be added and that logos, including those of political parties, and special characters for names of particular persons should not be added. In July 2000, the North Korean body wrote to WG2, accusing them of developing both versions of the Unicode encoding for Korean on the basis of South Korean proposals only, without consulting North Korea, accusing them putting the commercial interests of companies and fears of international confusion over respect to North Korea's sovereignty, and stating that North Korea would regard further refusal to change the name and order of the Korean characters in Unicode as an insult to their sovereign dignity and as compromising the ISO's claims to impartiality. They re-iterated their demand for WG2 and Unicode to "correct" the order of the Korean characters, and to "correct" the names "Hangul Jamo" and "Hangul Syllable" to "Korean Alphabet" and "Korean Syllable".
|
|