Additionally, the MUMPS language design requires that all subscripts of variables are automatically kept in sorted order. Numeric subscripts (including floating-point numbers) are stored from lowest to highest. All non-numeric subscripts are stored in alphabetical order following the numbers. In MUMPS terminology, this is canonical order.
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Subscripts are not limited to numerals—any ASCII character or group of characters can be a subscript identifier. While this is not uncommon for modern languages such as Perl or JavaScript, it was a highly unusual feature in the late 1970s. This capability was not universally implemented in MUMPS systems before the 1984 ANSI standard, as only canonically numeric subscripts were required by the standard to be allowed. Thus, the variable named 'Car' can have subscripts "Door", "Steering Wheel" and "Engine", each of which can contain a value and have subscripts of their own.
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The second congruence is proved similarly, by exchanging the subscripts 1 and 2.
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There is only one data structure - multi-dimensional sparse arrays (key-value nodes, sub-trees, and associative memory are all equally valid descriptions) with up to 32 subscripts. A scalar can be thought of as an array element with zero subscripts. Nodes with varying numbers of subscripts (including one node with no subscripts) can freely co- exist in the same array. For example, if one wanted to represent the national capitals of the United States: :Set Capital("United States")="Washington" :Set Capital("United States",1774,1776)="Philadelphia" :Set Capital("United States",1776,1777)="Baltimore" Variables are created on demand when first assigned to.
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Subscripts f and g refer to saturated liquid and saturated gas respectively, and fg refers to vaporisation.
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However, because there is no repetition of wildcard symbols, partial words may be written more simply by omitting the subscripts on the wildcard symbols.
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Iliffe vectors are contrasted with dope vectors in languages such as Fortran, which contain the stride factors and offset values for the subscripts in each dimension.
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These digital versions also include accented Latin characters, mathematical symbols, and Latin ligatures. In the URW/Nebiolo version, there are also extended Latin, subscripts and superscripts, and extended Latin ligatures.
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The discussion below takes their status as ellipses largely for granted. The example sentences below employ the convention whereby the elided material is indicated with subscripts and smaller font size.
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In 2007–09, Berthold released OpenType Pro version of City called City Pro, which supports Central European, Latin Extended A characters. OpenType features include ordinals, proportional lining figures, subscripts and superscripts, fractions.
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In terms of volume charge densities, the total charge density is: :\rho = \rho_f + \rho_b\,. as for surface charge densities: :\sigma = \sigma_f + \sigma_b\,. where subscripts "f" and "b" denote "free" and "bound" respectively.
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Coxeter uses δ4 for a cubic honeycomb, hδ4 for an alternated cubic honeycomb, qδ4 for a quarter cubic honeycomb, with subscripts for other forms based on the ring patterns of the Coxeter diagram.
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The four common locations of subscripts and superscripts. The typeface is Myriad Pro. A single typeface may contain sub- and superscript glyphs at different positions for different uses. The four most common positions are listed here.
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The author(s) will usually make it clear whether a subscript is intended as an index or as a label. For example, in 3-D Euclidean space and using Cartesian coordinates; the coordinate vector shows a direct correspondence between the subscripts 1, 2, 3 and the labels . In the expression , is interpreted as an index ranging over the values 1, 2, 3, while the subscripts are not variable indices, more like "names" for the components. In the context of spacetime, the index value 0 conventionally corresponds to the label .
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Superscripts and Subscripts is a Unicode block containing superscript and subscript numerals, mathematical operators, and letters used in mathematics and phonetics. The use of subscripts and superscripts in Unicode allows any polynomial, chemical and certain other equations to be represented in plain text without using any form of markup like HTML or TeX. Other superscript letters can be found in the Spacing Modifier Letters, Phonetic Extensions and Phonetic Extensions Supplement blocks, while the superscript 1, 2, and 3, inherited from ISO 8859-1, were included in the Latin-1 Supplement block.
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In engineering, double-subscript notation is notation used to indicate some variable between two points (each point being represented by one of the subscripts). In electronics, the notation is usually used to indicate the direction of current or voltage, while in mechanical engineering it is sometimes used to describe the force or stress between two points, and sometimes even a component that spans between two points (like a beam on a bridge or truss). Note that, although there are many cases where multiple subscripts are used, they are not necessarily called double subscript notation specifically.
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The second typeface is Myriad Pro; the superscript is about 60% of the original characters, raised by about 44% above the baseline.) A subscript or superscript is a character (such as a number or letter) that is set slightly below or above the normal line of type, respectively. It is usually smaller than the rest of the text. Subscripts appear at or below the baseline, while superscripts are above. Subscripts and superscripts are perhaps most often used in formulas, mathematical expressions, and specifications of chemical compounds and isotopes, but have many other uses as well.
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A particle may be distinguished by multiple subscripts, such as for the triple bottom omega particle. Similarly, subscripts are also used frequently in mathematics to define different versions of the same variable: for example, in an equation x0 and xf might indicate the initial and final value of x, while vrocket and vobserver would stand for the velocities of a rocket and an observer. Commonly, variables with a zero in the subscript are referred to as the variable name followed by "naught" (e.g. v0 would be read, "v-naught").
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The subscripted designation is now universally used where typography allows for subscripts. L2 Puppis was discovered to be variable by Benjamin Apthorp Gould in 1872, and was listed in Uranometria Argentina as 73 G. Puppis with magnitude 5.10v.
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The principal database feature in this project is the global array which permits direct, efficient manipulation of multi-dimensional arrays of effectively unlimited size. A global array is a persistent, sparse, undeclared, multi-dimensional, string indexed data disk based structure. A global array may appear anywhere an ordinary array reference is permitted and data may be stored at leaf nodes as well as intermediate nodes in the data base array. The number of subscripts in an array reference is limited only by the total length of the array reference with all subscripts expanded to their string values.
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Given their sorting and naming features, it's not uncommon for subscripts and variable names to be used as data stores themselves, independent of any data stored at their locations. This feature is often used for database indexes. E.g., `SET ^INDEX(lastname,firstname,SSNumber)=RecordNum`.
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For this b-y ion pair, the sum of their subscripts is equal to the total number of amino acid residues in the unknown peptide. If the C-terminus is Arg or Lys, y1-ion can be found in the spectrum to prove it.
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Magnetic inequivalence may occur with two symmetry-related HA-C-C-HB fragments (where the different subscripts indicate chemical inequivalence) that may or may not be contiguous. In order to distinguish the resulting coupling relationships, the symmetry-related pair would be labelled HA′-C-C-HB′.
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In the tables of quantities and their units, the ISO 31-8 standard shows symbols for substances as subscripts (e.g., cB, wB, pB). It also notes that it is generally advisable to put symbols for substances and their states in parentheses on the same line, as in c(H2SO4).
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The sum of the subscripts in a monomials is equal to the total number of elements. Thus, the number of monomials that appear in the partial Bell polynomial is equal to the number of ways the integer n can be expressed as a summation of k positive integers.
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The first one is the measured diffusion ellipsoid sitting at an angle determined by the axons, and the second one is perfectly aligned with the three Cartesian axes. The term "diagonalize" refers to the three components of the matrix along a diagonal from upper left to lower right (the components with red subscripts in the matrix at the start of this section). The variables ax2, by2, and cz2 are along the diagonal (red subscripts), but the variables d, e and f are "off diagonal". It then becomes possible to do a vector processing step in which we rewrite our matrix and replace it with a new matrix multiplied by three different vectors of unit length (length=1.0).
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The only common use of these subscripts is for the denominators of diagonal fractions, like ½ or the signs for percent %, permille ‰, and basis point ‱. Certain standard abbreviations are also composed as diagonal fractions, such as ℅ (care of), ℀ (account of), ℁ (addressed to the subject), or in Spanish ℆ (cada uno/una, "each one").
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Centre for Multilevel Modelling, University of Bristol. MLwiN represents multilevel models using mathematical notation including Greek letters and multiple subscripts, so the user needs to be (or become) familiar with such notation. For a tutorial introduction to multilevel models and their applications in medical statistics illustrated using MLwiN, see Goldstein et al.
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In the second volume, subscript numbers were added, some early symbols were modified (e.g. what was initially D1 became D01, D2 became D02, etc.), and the categories modified (types I, K,S added). In the third volume, the class I was renamed J. Later designations began to use a lower case letter in subscripts as well.
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North American and British/Australian convention reverse the usage of S & Z. Elastic modulus is S in North America, but Z in Britain/Australia, and vice versa for the plastic modulus. Eurocode 3 (EN 1993 - Steel Design) resolves this by using W for both, but distinguishes between them by the use of subscripts - Wel and Wpl.
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Variable names, function names, and statement labels have the same form, a letter followed by zero to five letters or digits. Function names end with a period. All names can be subscripted (the name followed by parentheses, with multiple subscripts separated by commas). For GOM names may be up to 24 characters long and may include the underscore (_) character.
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Lucida Sans Unicode Based on Lucida Sans Regular, this version added characters in Arrows, Block Elements, Box Drawing, Combining Diacritical Marks, Control Pictures, Currency Symbols, Cyrillic, General Punctuation, Geometric Shapes, Greek and Coptic, Hebrew, IPA Extensions, Latin Extended-A, Latin Extended-B, Letterlike Symbols, Mathematical Operators, Miscellaneous Symbols, Miscellaneous Technical, Spacing Modifier Letters, Superscripts and Subscripts regions.
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The point in maps to in , so is identified as the (representation) vector of the point at the origin. A vector in with a nonzero coefficient, but a zero coefficient, must (considering the inverse map) be the image of an infinite vector in . The direction therefore represents the (conformal) point at infinity. This motivates the subscripts and for identifying the null basis vectors.
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Iota subscripts in the word , ("ode", dative) The iota subscript is a diacritic mark in the Greek alphabet shaped like a small vertical stroke or miniature iota placed below the letter. It can occur with the vowel letters eta , omega , and alpha . It represents the former presence of an offglide after the vowel, forming a so‐called "long diphthong". Such diphthongs (i.e.
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Characters are classified with a Numeric type. Characters such as fractions, subscripts, superscripts, Roman numerals, currency numerators, encircled numbers, and script-specific digits are type Numeric. They have a numeric value that can be decimal, including zero and negatives, or a vulgar fraction. If there is not such a value, as with most of the characters, the numeric type is "None".
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Let be a Hilbert space and the bounded operators on . Consider a self-adjoint unital subalgebra of (this means that contains the adjoints of its members, and the identity operator on ). The theorem is equivalent to the combination of the following three statements: :(i) :(ii) :(iii) where the and subscripts stand for closures in the weak and strong operator topologies, respectively.
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The "x" denotes either sea level pressure or potential temperature (θ) and the subscripts 1–3 denote stations running from west to east along the line, while the "d" represents the distance between two stations. Negative Laplacian values are typically associated with pressure maxima at the center station, while positive Laplacian values usually correspond to colder temperatures in the center of the section.
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An early matrix-generator for LP was developed around 1969 at the Mathematisch Centrum (now CWI), Amsterdam.Jac. M. Anthonisse, An input system for linear programming problems, Statistica Neerlandica 24 (1970), 143-153. Its syntax was very close to the usual mathematical notation, using subscripts en sigmas. Input for the generator consisted of separate sections for the model and the data.
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A uses the point R_A to create an isogeny mapping \phi_A: E\rightarrow E_A and curve E_A isogenous to E. 4A. applies \phi_A to P_B and Q_B to form two points on E_A: \phi_A(P_B) and \phi_A(Q_B). 5A. A sends to B E_A, \phi_A(P_B), and \phi_A(Q_B). 1B - 4B: Same as A1 through A4, but with A and B subscripts swapped. 5B.
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Commonly used block sizes are 4KB, 8KB and 16KB - so, with an 8KB block size, an individual global variable can grow to 1,792GB. A global variable node (global variable, subscripts plus value) must fit in one database block and each block has a 16 byte overhead. So, the largest node that will fit in a database with a 4KB block size is 4,080 bytes.
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Palatino Linotype is the version of the Palatino family included with modern versions of Microsoft software. It incorporates extended Latin, Greek, Cyrillic characters, as well as currency signs, subscripts and superscripts, and fractions. The family includes roman and italic in text and bold weights. Palatino Linotype was notable as being the first western OpenType font that Microsoft shipped; Palatino Linotype was bundled with Windows 2000.
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Given any valid primary arithmetic expression, insert into one or more locations any number of Latin letters bearing optional numerical subscripts; the result is a primary algebra formula. Letters so employed in mathematics and logic are called variables. A primary algebra variable indicates a location where one can write the primitive value Image:Laws of Form - cross.gif or its complement Image:Laws of Form - double cross.gif.
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Which story1 has John told Peter that Mary likes t1? \- Movement indicated using a trace Subscripts help indicate the constituent that is assumed to have left a trace in its former position, the position marked by t.For examples of t used in this manner, see for instance Ouhalla (1994:63) and Haegeman and Guéron (1999:172). The other means of indicating movement is in terms of copies.
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Without subscripts, one ends up with the odd- sounding conclusion that a body is weightless when the only force acting on it is its weight. The apocryphal apple that fell on Newton's head can be used to illustrate the issues involved. An apple weighs approximately . This is the weight1 of the apple and is considered to be a constant even while it is falling.
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Let be a compact Hausdorff space and k= \R or \Complex. Then K_k(X) is defined to be the Grothendieck group of the commutative monoid of isomorphism classes of finite-dimensional -vector bundles over under Whitney sum. Tensor product of bundles gives -theory a commutative ring structure. Without subscripts, K(X) usually denotes complex -theory whereas real -theory is sometimes written as KO(X).
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Globals stored as persistent sparse arrays), gives the MUMPS database the characteristics of a document-oriented database. All variable names which are not prefixed with caret character ("^") are temporary and private. Like global variables, they also have a hierarchical storage model, but are only "locally available" to a single job, thus they are called "locals". Both "globals" and "locals" can have child nodes (called subscripts in MUMPS terminology).
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Suppose that the vertices of the quadrilateral Q are given by Q_1,Q_2,Q_3,Q_4 . Let b_1,b_2,b_3,b_4 be the perpendicular bisectors of sides Q_1Q_2,Q_2Q_3,Q_3Q_4,Q_4Q_1 respectively. Then their intersections Q_i^{(2)}=b_{i+2}b_{i+3} , with subscripts considered modulo 4, form the consequent quadrilateral Q^{(2)} . The construction is then iterated on Q^{(2)} to produce Q^{(3)} and so on.
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Discontinuous-constituent Phrase Structure Grammar (DCPSG) (distinct from Discontinuous Phrase Structure Grammar/DPSG) is a formalism for describing discontinuous phrase structures in natural language, such as verb phrases in VSO languages. The formalism was introduced in the slightly more constrained form of Discontinuous-constituent Phrase Structure Grammar with Subscripts and Deletes (DCPSGsd) in Harman (1963).Harman, Gilbert H. 1963. Generative Grammars without Transformation Rules: A Defense of Phrase Structure.
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When data objects are stored in an array, individual objects are selected by an index that is usually a non-negative scalar integer. Indexes are also called subscripts. An index maps the array value to a stored object. There are three ways in which the elements of an array can be indexed: ; 0 (zero-based indexing): The first element of the array is indexed by subscript of 0.
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Particle reinforcement a highly advantageous method of tuning mechanical properties of materials since it is very easy implement while being low cost. The elastic modulus of particle- reinforced composites can be expressed as, E_c = V_m E_m + K_c V_p E_p where E is the elastic modulus, V is the volume fraction. The subscripts c, p and m are indicating composite, particle and matrix, respectively. K_c is a constant can be found empirically.
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Lorentz force on a charged particle (of charge q) in motion (velocity v), used as the definition of the E field and B field. Here subscripts e and m are used to differ between electric and magnetic charges. The definitions for monopoles are of theoretical interest, although real magnetic dipoles can be described using pole strengths. There are two possible units for monopole strength, Wb (Weber) and A m (Ampere metre).
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The modifier "cohomological" indicates that the δn raise the index on the T. A covariant homological δ-functor between A and B is similarly defined (and generally uses subscripts), but with δn a morphism Tn(M ) → Tn-1(M). The notions of contravariant cohomological δ-functor between A and B and contravariant homological δ-functor between A and B can also be defined by "reversing the arrows" accordingly.
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The variable ^Car("Door") could have a nested variable subscript of "Color" for example. Thus, you could say SET ^Car("Door","Color")="BLUE" to modify a nested child node of ^Car. In MUMPS terms, "Color" is the 2nd subscript of the variable ^Car (both the names of the child-nodes and the child-nodes themselves are likewise called subscripts). Hierarchical variables are similar to objects with properties in many object oriented languages.
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By using only non-negative integer subscripts, the MUMPS programmer can emulate the arrays data type from other languages. Although MUMPS does not natively offer a full set of DBMS features such as mandatory schemas, several DBMS systems have been built on top of it that provide application developers with flat- file, relational and network database features. Additionally, there are built- in operators which treat a delimited string (e.g., comma-separated values) as an array.
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An event is something that happens at a certain point in spacetime, or more generally, the point in spacetime itself. In any inertial frame an event is specified by a time coordinate ct and a set of Cartesian coordinates to specify position in space in that frame. Subscripts label individual events. From Einstein's second postulate of relativity (invariance of c) it follows that: in all inertial frames for events connected by light signals.
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Russell and Whitehead (1927). To resolve one of these paradoxes means to pinpoint exactly where our use of language went wrong and to provide restrictions on the use of language which may avoid them. This family of paradoxes can be resolved by incorporating stratifications of meaning in language. Terms with systematic ambiguity may be written with subscripts denoting that one level of meaning is considered a higher priority than another in their interpretation.
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The World Wide Web Consortium and the Unicode Consortium have made recommendations on the choice between using markup and using superscript and subscript characters: > When used in mathematical context (MathML) it is recommended to consistently > use style markup for superscripts and subscripts.... However, when super and > sub-scripts are to reflect semantic distinctions, it is easier to work with > these meanings encoded in text rather than markup, for example, in phonetic > or phonemic transcription.
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The game features a virtual Clue Sheet. When suggestions are made and answered, ticks and crosses are automatically added to show which characters do or do not have particular cards. Players are also able to manually make their own notations, including numeric subscripts, that allow them to use advanced deduction techniques. The game can be played against 3 to 5 AI opponents, each of which can be set to Easy, Medium or Hard difficulty.
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The image can be vectorized manually. A person could look at the image, make some measurements, and then write the output file by hand. That was the case for the vectorization of a technical illustration about neutrinos. The illustration has a few geometric shapes and a lot of text; it was relatively easy to convert the shapes, and the SVG vector format allows the text (even subscripts and superscripts) to be entered easily.
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The number of words can vary depending upon aspects such as whether the Hebrew alphabet in Psalm 119, the superscriptions listed in some of the Psalms, and the subscripts traditionally found at the end of the Pauline epistles, are included. Except where stated, the following apply to the King James Version of the Bible in its modern 66-book Protestant form including the New Testament and the protocanonical Old Testament, not the deuterocanonical books.
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The stress on the composite can be expressed in terms of the volume fraction of the fiber and the matrix. \sigma_c = V_f \sigma_f + V_m \sigma_m where \sigma is the stress, V is the volume fraction. The subscripts c, f and m are indicating composite, fiber and matrix, respectively. Although the stress-strain behavior of fiber composites can only be determined by testing, there is an expected trend, three stages of the stress-strain curve.
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The first stage is the region of the stress-strain curve where both fiber and the matrix are elastically deformed. This linearly elastic region can be expressed in the following form. \sigma_c - E_c \epsilon_c = \epsilon_c (V_f E_f + V_m E_m) where \sigma is the stress, \epsilon is the strain, E is the elastic modulus, and V is the volume fraction. The subscripts c, f, and m are indicating composite, fiber, and matrix, respectively.
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Let S0 = S and let Sk+1 be the derived set of Sk. If there is a finite number n for which Sn is finite, then all the coefficients are zero. Later, Lebesgue proved that if there is a countably infinite ordinal α such that Sα is finite, then the coefficients of the series are all zero. Cantor's work on the uniqueness problem famously led him to invent transfinite ordinal numbers, which appeared as the subscripts α in Sα .
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Moreover, important formulas like Paul Lévy's inversion formula for the characteristic function also rely on the "less than or equal" formulation. If treating several random variables X,Y,\ldots etc. the corresponding letters are used as subscripts while, if treating only one, the subscript is usually omitted. It is conventional to use a capital F for a cumulative distribution function, in contrast to the lower-case f used for probability density functions and probability mass functions.
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Finite spherical symmetry groups are also called point groups in three dimensions. There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral, octahedral, and icosahedral symmetry. This article lists the groups by Schoenflies notation, Coxeter notation,Johnson, 2015 orbifold notation,Conway, 2008 and order. John Conway uses a variation of the Schoenflies notation, based on the groups' quaternion algebraic structure, labeled by one or two upper case letters, and whole number subscripts.
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The amount of the delay depends upon the volume fraction of the strong phase. Thus, the tensile strength of the composite can be expressed in terms of the volume fraction. (T.S.)_c=V_f(T.S.)_f+V_m \sigma_m(\epsilon_m) where T.S. is the tensile strength, \sigma is the stress, \epsilon is the strain, E is the elastic modulus, and V is the volume fraction. The subscripts c, f, and m are indicating composite, fiber, and matrix, respectively.
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While the terms generally apply to arterial blood delivered to the kidneys, both RBF and RPF can be used to quantify the volume of venous blood exiting the kidneys per unit time. In this context, the terms are commonly given subscripts to refer to arterial or venous blood or plasma flow, as in RBFa, RBFv, RPFa, and RPFv. Physiologically, however, the differences in these values are negligible so that arterial flow and venous flow are often assumed equal.
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The most common practice throughout human history has been to start counting at one, and this is the practice in early classic computer programming languages such as Fortran and COBOL. However, in the late 1950s LISP introduced zero-based numbering for arrays while Algol 58 introduced completely flexible basing for array subscripts (allowing any positive, negative, or zero integer as base for array subscripts), and most subsequent programming languages adopted one or other of these positions. For example, the elements of an array are numbered starting from 0 in C, so that for an array of n items the sequence of array indices runs from 0 to . This permits an array element's location to be calculated by adding the index directly to address of the array, whereas 1-based languages precalculate the array's base address to be the position one element before the first. There can be confusion between 0- and 1-based indexing, for example Java's JDBC indexes parameters from 1 although Java itself uses 0-based indexing.
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Diffusion from a point source in the anisotropic medium of white matter behaves in a similar fashion. The first pulse of the Stejskal Tanner diffusion gradient effectively labels some water molecules and the second pulse effectively shows their displacement due to diffusion. Each gradient direction applied measures the movement along the direction of that gradient. Six or more gradients are summed to get all the measurements needed to fill in the matrix, assuming it is symmetric above and below the diagonal (red subscripts).
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The array syntax of Fortran is extended with additional trailing subscripts in square brackets to provide a concise representation of references to data that is spread across images. The CAF extension was implemented in some Fortran compilers such as those from Cray (since release 3.1). Since the inclusion of coarrays in the Fortran 2008 standard, the number of implementations is growing. The first open-source compiler which implemented coarrays as specified in the Fortran 2008 standard for Linux architectures is G95.
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The thermal efficiency of a spray pond may be calculated based on its approach to the saturation (wet bulb) temperature of the air: (TH \- TC) / (TH \- TW), where the subscripts H and C refer to the temperatures of the hot and cold water streams, while the subscript W refers to the wet bulb temperature of the air. Typically, spray ponds achieve thermal efficiencies of between 50% and 70%. Further details of performance estimation may be found in the engineering literature.
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In classical continuum mechanics, the space of interest is usually 3-dimensional Euclidean space, as is the tangent space at each point. If we restrict the local coordinates to be Cartesian coordinates with the same scale centered at the point of interest, the metric tensor is the Kronecker delta. This means that there is no need to distinguish covariant and contravariant components, and furthermore there is no need to distinguish tensors and tensor densities. All Cartesian-tensor indices are written as subscripts.
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An explanation of the superscripts and subscripts seen in atomic number notation. Atomic number is the number of protons, and therefore also the total positive charge, in the atomic nucleus. The Rutherford–Bohr model of the hydrogen atom () or a hydrogen-like ion (). In this model it is an essential feature that the photon energy (or frequency) of the electromagnetic radiation emitted (shown) when an electron jumps from one orbital to another be proportional to the mathematical square of atomic charge ().
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Oligosaccharides may be sequenced using tandem mass spectrometry in a similar manner to peptide sequencing. Fragmentation generally occurs on either side of the glycosidic bond (b, c, y and z ions) but also under more energetic conditions through the sugar ring structure in a cross-ring cleavage (x ions). Again trailing subscripts are used to indicate position of the cleavage along the chain. For cross ring cleavage ions the nature of the cross ring cleavage is indicated by preceding superscripts.
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To understand the issue involved, it is necessary to understand how VP-ellipsis works. Consider the following examples, where the expected, but elided, VP is represented with a smaller font and subscripts and the antecedent to the ellipsis is in bold: ::John washed the dishes, and Mary did wash the dishes, too. ::John washed the dishes on Tuesday, and Mary did wash the dishes on Tuesday, too. In both of these sentences, the VP has been elided in the second half of the sentence.
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A skeletal diagram of butane Chemical structures may be written in more compact forms, particularly when showing organic molecules. In condensed structural formulas, many or even all of the covalent bonds may be left out, with subscripts indicating the number of identical groups attached to a particular atom. Another shorthand structural diagram is the skeletal formula (also known as a bond-line formula or carbon skeleton diagram). In a skeletal formula, carbon atoms are not signified by the symbol C but by the vertices of the lines.
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Adobe Caslon is a very popular revival designed by Carol Twombly. It is based on Caslon's own specimen pages printed between 1734 and 1770 and is a member of the Adobe Originals programme. It added many features now standard in high-quality digital fonts, such as small caps, old style figures, swash letters, ligatures, alternate letters, fractions, subscripts and superscripts, and matching ornaments. Adobe Caslon is used for body text in The New Yorker and is one of the two official typefaces of the University of Virginia.
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Variable names may start with any letter except F (F is reserved for functions) and may contain any sequence of letters and numbers. However, only the first two characters are significant. For instance, the following code sample from FOCAL: A New Conversational Language refers to the same variable as DESTINATION and then DES. Internally, both references refer to a variable designated DE: 01.80 ASK DESTINATION 02.30 IF (DES-14) 2.4,3.1,2.4 Any variable may be treated as an array, allowing subscripts from -2048 through 2047.
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This theory, while useful in some contexts, came to be seen as insufficient. Berzelius's work with atomic weights and his theory of electrochemical dualism led to his development of a modern system of chemical formula notation that showed the composition of any compound both qualitatively and quantitatively. His system abbreviated the Latin names of the elements with one or two letters and applied superscripts to designate the number of atoms of each element present in the compound. Later, chemists changed to use of subscripts rather than superscripts.
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Perhaps the most familiar example of subscripts is in chemical formulas. For example, the molecular formula for glucose is C6H12O6 (meaning that it is a molecule with 6 carbon atoms, 12 hydrogen atoms and 6 oxygen atoms). Or the most famous molecule in the world, water, known almost universally by its chemical formula, H2O (meaning it has 2 hydrogen atoms and 1 oxygen atom.) A subscript is also used to distinguish between different versions of a subatomic particle. Thus electron, muon, and tau neutrinos are denoted and .
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Subscripts are often used to refer to members of a mathematical sequence or set or elements of a vector. For example, in the sequence O = (45, −2, 800), O3 refers to the third member of sequence O, which is 800. Also in mathematics and computing, a subscript can be used to represent the radix, or base, of a written number, especially where multiple bases are used alongside each other. For example, comparing values in hexadecimal, denary, and octal one might write Chex = 12dec = 14oct.
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Unicode defines subscript and superscript characters in several areas; in particular, it has a full set of superscript and subscript digits. Owing to the popularity of using these characters to make fractions, most modern fonts render most or all of these as cap height superscripts and baseline subscripts. The same font may align letters and numbers in different ways. Other than numbers, the set of super- and subscript letters and other symbols is incomplete and somewhat random, and many fonts do not contain them.
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The subscripts l, v and i denote liquid phase, vapor phase and vapor-liquid interface, respectively. If the laser intensity is high and pulse duration is short, the so-called Knudsen layer is assumed to exist at the melt-vapor front where the state variables undergo discontinuous changes across the layer. By considering the discontinuity across the Knudsen layer, Yao, et al. (2001) simulated the surface recess velocity Vv distribution, along the radial direction at different times, which indicates the material ablation rate is changing significantly across the Knudsen layer.
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R.J. Chillingworth, Review of Geometric Differentiation, says it is "aimed at advanced undergraduates or beginning graduate students in mathematics..." Chillingworth notes "a peculiar feature of the book is its use of compact notation for differentiation using numerical subscripts that allow tidy presentation of calculations." For instance, Porteous gives Faa di Bruno's formula. Furthermore, the reviewer notes that this mathematics has "connections to optics, kinematics and architecture as well as (more recently) geology, tomography, computer vision and face-recognition." These applications follow from the theories of contact, umbilical points, ridges, germs, and cusps.
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The `PRINT` statement used the comma when printing multiple variables, advancing to the next of five "zones". The comma was not needed in the case where one was printing a prompt and single value, so was valid. A somewhat hidden feature was that all variables were capable of representing arrays (vectors) of up to ten elements (subscripts 1 to 10, changed to 0 to 10 in the Second Edition) without being declared that way using `DIM`. Variable names were limited to a single letter or a letter followed by a digit (286 possible variable names).
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OpenType features include automatic ligature sets, numerals (tabular, proportional, oldstyle and lining), numerator, denominator, scientific inferior subscripts, and small caps. It is also distributed with Microsoft Excel Viewer, Microsoft PowerPoint Viewer,Excel ViewerPowerpoint Viewer the Microsoft Office Compatibility PackMicrosoft Office Compatibility Pack for Word, Excel, and PowerPoint File Formats for Microsoft Windows and the Open XML File Format Converter for Mac.Open XML File Format Converter for Mac 1.2.1 It is not available as a freeware for use in other operating systems such as GNU/Linux, cross-platform use, and web use.
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For instance some typographic ligatures like U+FB03 (ffi), Roman numerals like U+2168 (Ⅸ) and even subscripts and superscripts, e.g. U+2075 (⁵) have their own Unicode code points. Canonical normalization (NF) does not affect any of these, but compatibility normalization (NFK) will decompose the ffi ligature into the constituent letters, so a search for U+0066 (f) as substring would succeed in an NFKC normalization of U+FB03 but not in NFC normalization of U+FB03. Likewise when searching for the Latin letter I (U+0049) in the precomposed Roman numeral Ⅸ (U+2168).
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Thus, there are three monomials in B6,2. Indeed, the subscripts of the variables in a monomial are the same as those given by the integer partition, indicating the sizes of the different blocks. The total number of monomials appearing in a complete Bell polynomial Bn is thus equal to the total number of integer partitions of n. Also the degree of each monomial, which is the sum of the exponents of each variable in the monomial, is equal to the number of blocks the set is divided into.
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Most consonants, including a few of the subscripts, form ligatures with the vowel (ា) and with all other dependent vowels that contain the same cane-like symbol. Most of these ligatures are easily recognizable, but a few may not be, particularly those involving the letter . This combines with the a vowel in the form , created to differentiate it from the consonant symbol and also from the ligature for with (). Some more examples of ligatured symbols follow: : បៅ bau Another example with , forming a similar ligature to that described above.
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Vensim provides a graphical modeling interface with stock and flow and causal loop diagrams, on top of a text-based system of equations in a declarative programming language. It includes a patented method for interactive tracing of behavior through causal links in model structure, as well as a language extension for automating quality control experiments on models called Reality Check. The modeling language supports arrays (subscripts) and permits mapping among dimensions and aggregation. Built-in allocation functions satisfy constraints that are sometimes not met by conventional approaches like logit.
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The results of either test can be converted into the more traditional "dependence vector" form, but since this abstraction provides less precision, much of the information about the dependence will be lost. Both techniques produce exact information for programs with affine control and subscript expressions, and must approximate for many programs outside this domain (i.e., in the presence of non-affine subscripts such as index arrays). The original work of Feautrier focused on describing true dependences, which would be referred to as exact value-based flow dependences by the Omega Project.
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For example, in a graph showing how a pressure varies with time, the graph coordinates may be denoted p and t. Each axis is usually named after the coordinate which is measured along it; so one says the x-axis, the y-axis, the t-axis, etc. Another common convention for coordinate naming is to use subscripts, as (x1, x2, ..., xn) for the n coordinates in an n-dimensional space, especially when n is greater than 3 or unspecified. Some authors prefer the numbering (x0, x1, ..., xn−1).
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2007, 27(9) 1533-1539 a large group of researchers who are active in this field agreed to a consensus nomenclature for these terms, with the intent of making the literature in this field more transparent to non-specialists. The convention involves use of the subscripts p for quantities referred to plasma and ND for quantities referred to the free plus nonspecifically bound concentration in brain (NonDisplaceable). Under the consensus nomenclature, the parameters referred to above as f1 and BP1 are now called fp and BPp, while f2 and BP2 are called fND and BPND.
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Superscripts "syn" are associated with synchronized flows. Subscripts "up" and "down" are related respectively to the upstream and downstream fronts of synchronized flow and wide moving jams. Secondly, based on the abovementioned common features of wide moving jams and synchronized flow, the FOTO model tracks the downstream and upstream fronts of synchronized flow denoted by x_{down}^{(syn)}(t), x_{up}^{(syn)}(t), where t is time (Fig. 3). The ASDA model tracks the downstream and upstream fronts of wide moving jams denoted by x_{down}^{(jam)}(t), x_{up}^{(jam)}(t) (Fig. 3).
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Differential forms in R2 and R3 were well known in the mathematical physics of the nineteenth century. In the plane, 0-forms are just functions, and 2-forms are functions times the basic area element , so that it is the 1-forms : \alpha = f(x,y) \, dx + g(x,y) \, dy that are of real interest. The formula for the exterior derivative d here is : d \alpha = (g_x-f_y) \, dx\wedge dy where the subscripts denote partial derivatives. Therefore the condition for \alpha to be closed is : f_y=g_x.
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It is used as an alternate character set of the SUPDUP protocol for terminals with `%TOSAI` and `%TOFCI` bits set. It is also recommended for TeX implementations on systems with large character sets. The default plain TeX macro package sets values (`↑`) and (`↓`) as alternative character codes for superscripts and subscripts, respectively (the default being `^` and `_`). The Knight keyboard is an example of a keyboard capable of inputting all of the defined characters excluding `⋅γδ±⊕◊∫`, as they are mapped to ASCII commands `NUL`, `HT`, `LF`, `FF`, `CR`, `ESC` and `DEL`, respectively.
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The chemical formula for a molecule uses one line of chemical element symbols, numbers, and sometimes also other symbols, such as parentheses, dashes, brackets, and plus (+) and minus (−) signs. These are limited to one typographic line of symbols, which may include subscripts and superscripts. A compound's empirical formula is a very simple type of chemical formula. It is the simplest integer ratio of the chemical elements that constitute it. For example, water is always composed of a 2:1 ratio of hydrogen to oxygen atoms, and ethanol (ethyl alcohol) is always composed of carbon, hydrogen, and oxygen in a 2:6:1 ratio.
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In mathematics, a symmetric polynomial is a polynomial in variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, is a symmetric polynomial if for any permutation of the subscripts one has . Symmetric polynomials arise naturally in the study of the relation between the roots of a polynomial in one variable and its coefficients, since the coefficients can be given by polynomial expressions in the roots, and all roots play a similar role in this setting. From this point of view the elementary symmetric polynomials are the most fundamental symmetric polynomials.
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Monospace is a monospaced Unicode font, developed by George Williams. It is based on the typeface Courier. This font contains 2860 glyphs. It includes characters in the following unicode ranges: Basic Latin, Latin-1 Supplement, Latin Extended-A, Latin Extended-B, IPA Extensions, Spacing Modifier Letters, Combining Diacritical Marks, Greek, Cyrillic, Hebrew, Latin Extended Additional, Greek Extended, General Punctuation, Superscripts and Subscripts, Currency Symbols, Combining Diacritical Marks for Symbols, Letterlike Symbols, Number Forms, Arrows, Mathematical Operators, Miscellaneous Technical, Control Pictures, Enclosed Alphanumerics, Box Drawing, Block Elements, Geometric Shapes, Miscellaneous Symbols, Alphabetic Presentation Forms, Halfwidth and Fullwidth Forms.
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Whereas, metallic fibers have more space to plastically deform, so their composites exhibit a third stage where both fiber and the matrix are plastically deforming. Metallic fibers have many applications to work at cryogenic temperatures that is one of the advantages of composites with metal fibers over nonmetallic. The stress in this region of the stress-strain curve can be expressed as, \sigma_c (\epsilon_c) = V_f \sigma_f \epsilon_c + V_m \sigma_m (\epsilon_c) where \sigma is the stress, \epsilon is the strain, E is the elastic modulus, and V is the volume fraction. The subscripts c, f, and m are indicating composite, fiber, and matrix, respectively.
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In this context using "zeroth" as an ordinal is not strictly correct, but a widespread habit in this profession. Other programming languages, such as Fortran or COBOL, have array subscripts starting with one, because they were meant as high-level programming languages, and as such they had to have a correspondence to the usual ordinal numbers which predate the invention of the zero by a long time. Pascal allows the range of an array to be of any ordinal type (including enumerated types). APL allows setting the index origin to 0 or 1 during runtime programatically.
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This scale signified that some elements had positive and negative charges associated with them and the position of an element on this scale and the element's charge determined how that element combined with others. Berzelius's work on electrochemical atomic theory was published in 1818 as Essai sur la théorie des proportions chimiques et sur l'influence chimique de l'électricité. He also introduced a new chemical nomenclature into chemistry by representing elements with letters and abbreviations, such as O for oxygen and Fe for iron. Combinations of elements were represented as sequences of these symbols and the number of atoms were represented at first by superscripts and then later subscripts.
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In a MUMPS context, this means that there is no requirement for sequential nodes to exist — `A(1), A(99)` and `A(100)` may be used without defining, allocating space for, or using any space for nodes 2 through 98. Indeed, one can even use floating-point numbers and strings (`A(1.2)`, `A(3.3)`, `A("foo")`, etc.), where the subscript names have some meaning external to the program. The access function `$ORDER ( A(1.2) )` returns the next defined key or subscript value, 3.3 in this example, so the program can readily manage the data. Subscripts are always returned (and usually stored) in sorted order.
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We assume that the type, \Omega, has been fixed. Then there are three basic constructions in universal algebra: homomorphic image, subalgebra, and product. A homomorphism between two algebras A and B is a function h: A → B from the set A to the set B such that, for every operation fA of A and corresponding fB of B (of arity, say, n), h(fA(x1,...,xn)) = fB(h(x1),...,h(xn)). (Sometimes the subscripts on f are taken off when it is clear from context which algebra the function is from.) For example, if e is a constant (nullary operation), then h(eA) = eB.
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In the following equations, all subscripts correspond to the different steps in the Fickett–Jacobs cycle as shown in Figure 2. In addition, a representation of the work done by the system and the external work applied on the system is shown is Figure 1. Initially, the work done to the system to begin a cycling detonation is W_i = -P_i A u_p(t-t_0) Figure 2: An schematic representation of the different steps in the Fickett–Jacobs cycle. a) Applying external force to move the piston at velocity up, and simultaneous detonation of the explosives (UCJ is the speed of the wave front of the detonation) .
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In one form, the navigational system of equations acquires linear and angular measurements from the inertial and body frame, respectively and calculates the final attitude and position in the NED frame of reference. 600px Where: f is specific force, \omega is angular rate, a is acceleration, R is position, \dot R and V are velocity, \Omega is the angular velocity of the earth, g is the acceleration due to gravity, \Phi, \lambda and h are the NED location parameters. Also, super/subscripts of E, I and B are representing variables in the Earth centered, Inertial or Body reference frame, respectively and C is a transformation of reference frames.
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The criteria are extended to three-dimensional problems where the maximum stress criteria are used for transverse normal stress component. The failure modes included in Hashin’s criteria are as follows. # Tensile fibre failure for σ11 ≥ 0 # Compressive fibre failure for σ11 < 0 # Tensile matrix failure for σ22 + σ33 > 0 # Compressive matrix failure for σ22 + σ33 < 0 # Interlaminar tensile failure for σ33 > 0 # Interlaminar compression failure for σ33 < 0 where, σij denote the stress components and the tensile and compressive allowable strengths for lamina are denoted by subscripts T and C, respectively. XT, YT, ZT denotes the allowable tensile strengths in three respective material directions.
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For example, letters W, X, Y, Z could be designated to represent predicate variables, whereas letters A, B, C,..., U, V could represent predicate "constants". If these letters are not enough, then numerical subscripts can be appended after the letter in question (as in X1, X2, X3). However, if the predicate variables are not perceived (or defined) as belonging to the vocabulary of the predicate calculus, then they are predicate metavariables, whereas the rest of the predicate letters are just called "predicate letters". The metavariables are thus understood to be used to code for axiom schemata and theorem schemata (derived from the axiom schemata).
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Polyhedral frameworks are designed to support compilers techniques for analysis and transformation of codes with nested loops, producing exact results for loop nests with affine loop bounds and subscripts ("Static Control Parts" of programs). They can be used to represent and reason about executions (iterations) of statements, rather than treating a statement as a single object representing properties of all executions of that statement. Polyhedral frameworks typically also allow the use of symbolic expressions. Polyhedral frameworks can be used for dependence analysis for arrays, including both traditional alias analysis and more advanced techniques such as the analysis of data flow in arrays or identification of conditional dependencies.
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A face of an embedded graph is an open 2-cell in the surface that is disjoint from the graph, but whose boundary is the union of some of the edges of the embedded graph. Let F be a face of an embedded graph G and let v0, v1, ..., vn−1,vn = v0 be the vertices lying on the boundary of F (in that circular order). A circular interval for F is a set of vertices of the form {va, va+1, ..., va+s} where a and s are integers and where subscripts are reduced modulo n. Let Λ be a finite list of circular intervals for F. We construct a new graph as follows.
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A key (global variable plus subscripts) can be up to 255 bytes. The database engine is daemonless and processes accessing the database operate with normal user and group ids - a process has access to a database file if and only if the ownership and permissions of that database file (plus any layered access control such as SELinux) permits access. Each process has within its address space all the logic needed to manage the database, and processes cooperate with one another to manage database files. When a database file is journaled, updates are written to journal files before being written to database files, and in the event of a system crash, database files can be recovered from journal files.
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A symmetric polynomial is a polynomial P(X1, X2, …, Xn) in n variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, P is a symmetric polynomial if for any permutation σ of the subscripts 1, 2, ..., n, one has P(Xσ(1), Xσ(2), …, Xσ(n)) = P(X1, X2, …, Xn). Symmetric polynomials arise naturally in the study of the relation between the roots of a polynomial in one variable and its coefficients, since the coefficients can be given by polynomial expressions in the roots, and all roots play a similar role in this setting. From this point of view, the elementary symmetric polynomials are the most fundamental symmetric polynomials.
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Ricci calculus, and index notation more generally, distinguishes between lower indices (subscripts) and upper indices (superscripts); the latter are not exponents, even though they may look as such to the reader only familiar with other parts of mathematics. It is in special cases (that the metric tensor is everywhere equal to the identity matrix) possible to drop the distinction between upper and lower indices, and then all indices could be written in the lower position – coordinate formulae in linear algebra such as a_{ij} b_{jk} for the product of matrices can sometimes be understood as examples of this – but in general the notation requires that the distinction between upper and lower indices is observed and maintained.
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Dedicated word processors and WP software for general-purpose computers that rose in popularity in the late 1970s and 1980s would use features such as microspacing (usually by 1/120 of an inch horizontally and, possibly, 1/48 of an inch vertically) to implement subscripts, proportional spacing, underlining, and so on. The more rudimentary software packages would implement bold text by overtyping the character in exactly the same spot (for example, using the backspace control code), but better software would print the letter in 3 slightly different positions. Software did exist to (slowly) produce pie charts on such printers (and on some daisywheels the dot was reinforced with metal to cope with extra wear).
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The exponential Bell polynomial encodes the information related to the ways a set can be partitioned. For example, if we consider a set {A, B, C}, it can be partitioned into two non-empty, non- overlapping subsets, which is also referred to as parts or blocks, in 3 different ways: :{{A}, {B, C}} :{{B}, {A, C}} :{{C}, {B, A}} Thus, we can encode the information regarding these partitions as :B_{3,2}(x_1,x_2) = 3 x_1 x_2. Here, the subscripts of B3,2 tells us that we are considering the partitioning of set with 3 elements into 2 blocks. The subscript of each xi indicates the presence of block with i elements (or block of size i) in a given partition.
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Einstein notation can be applied in slightly different ways. Typically, each index occurs once in an upper (superscript) and once in a lower (subscript) position in a term; however, the convention can be applied more generally to any repeated indices within a term. When dealing with covariant and contravariant vectors, where the position of an index also indicates the type of vector, the first case usually applies; a covariant vector can only be contracted with a contravariant vector, corresponding to summation of the products of coefficients. On the other hand, when there is a fixed coordinate basis (or when not considering coordinate vectors), one may choose to use only subscripts; see ' below.
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In a step-up transformer, conversely, the load is attached across the full winding while the source is connected to a tap across a portion of the winding. For a step-up transformer, the subscripts in the above equations are reversed where, in this situation, N2 and V2 are greater than N1 and V1, respective. As in a two-winding transformer, the ratio of secondary to primary voltages is equal to the ratio of the number of turns of the winding they connect to. For example, connecting the load between the middle of the winding and the common terminal end of the winding of the autotransformer will result in the output load voltage being 50% of the primary voltage.
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It found users at universities and in industry. The main industrial user was the steel maker Hoogovens (now Tata Steel) where it was used for nearly 25 years. A big step towards the modern modelling languages is found in UIMP , where the structure of the mathematical programming models taken from real life is analysed for the first time, to highlight the natural grouping of variables and constraints arising from such models. This led to data-structure features, which supported structured modelling; in this paradigm, all the input and output tables, together with the decision variables, are defined in terms of these structures, in a way comparable to the use of subscripts and sets.
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Additional OpenType features includes rlig for Arabic scripts. Version 5.00 (supplied with Windows Vista and Windows Server 2008) includes 3053 glyphs (2788 characters, 36 blocks), which extended Unicode ranges to include Arabic Supplement, Combining Diacritical Marks Supplement, Combining Half Marks, Latin Extended-C, Latin Extended-D, Phonetic Extensions, Phonetic Extensions Supplement, Specials, Superscripts and Subscripts. New OpenType scripts include Arabic URD (Urdu), Cyrillic (default), Hebrew (default), Latin (default, Romanian), Thai (default). Additional OpenType features includes ccmp, mark, mkmk for Arabic scripts; locl for Arabic URD (Urdu) script; mark, mkmk for default Cyrillic; dlig, ccmp, mark for default Hebrew; ccmp, mark, mkmk for Latin scripts; locl for Romanian Latin; ccmp, mark, mkmk for Thai.
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Any substance consisting of two or more different types of atoms (chemical elements) in a fixed stoichiometric proportion can be termed a chemical compound; the concept is most readily understood when considering pure chemical substances. It follows from their being composed of fixed proportions of two or more types of atoms that chemical compounds can be converted, via chemical reaction, into compounds or substances each having fewer atoms. The ratio of each element in the compound is expressed in a ratio in its chemical formula. A chemical formula is a way of expressing information about the proportions of atoms that constitute a particular chemical compound, using the standard abbreviations for the chemical elements, and subscripts to indicate the number of atoms involved.
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Lexemes with purely grammatical function such as lexically-governed prepositions are not included at this level of representation; values of inflectional categories that are derived from SemR but implemented by the morphology are represented as subscripts on the relevant lexical nodes that they bear on. DSyntR is mapped onto the next level of representation by rules of the deep-syntactic component. The surface-syntactic representation (SSyntR) represents the language-specific syntactic structure of an utterance and includes nodes for all the lexical items (including those with purely grammatical function) in the sentence. Syntactic relations between lexical items at this level are not restricted and are considered to be completely language-specific, although many are believed to be similar (or at least isomorphic) across languages.
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C7 and C8) makes subscripts unnecessary altogether. Although pitch notation is intended to describe sounds audibly perceptible as pitches, it can also be used to specify the frequency of non-pitch phenomena. Notes below E or higher than E are outside most humans' hearing range, although notes slightly outside the hearing range on the low end may still be indirectly perceptible as pitches due to their overtones falling within the hearing range. For an example of truly inaudible frequencies, when the Chandra X-ray Observatory observed the waves of pressure fronts propagating away from a black hole, their one oscillation every 10 million years was described by NASA as corresponding to the B fifty-seven octaves below middle C (B or 3.235 fHz).
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Contrast of helix end views between α (offset squarish) vs 310 (triangular) Similar structures include the 310 helix (i + 3 → i hydrogen bonding) and the π-helix (i + 5 → i hydrogen bonding). The α-helix can be described as a 3.613 helix, since the i + 4 spacing adds three more atoms to the H-bonded loop compared to the tighter 310 helix, and on average, 3.6 amino acids are involved in one ring of α-helix. The subscripts refer to the number of atoms (including the hydrogen) in the closed loop formed by the hydrogen bond. Ramachandran plot (φ, ψ plot), with data points for α-helical residues forming a dense diagonal cluster below and left of center, around the global energy minimum for backbone conformation.
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In general, the governing equations and boundary conditions are transformed and these transforms are used to define a pair of complex functions (typically denoted with '+' and '−' subscripts) which are respectively analytic in the upper and lower halves of the complex plane, and have growth no faster than polynomials in these regions. These two functions will also coincide on some region of the complex plane, typically, a thin strip containing the real line. Analytic continuation guarantees that these two functions define a single function analytic in the entire complex plane, and Liouville's theorem implies that this function is an unknown polynomial, which is often zero or constant. Analysis of the conditions at the edges and corners of the boundary allows one to determine the degree of this polynomial.
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421 polytope of 8-space In geometry, the Gosset–Elte figures, named by Coxeter after Thorold Gosset and E. L. Elte, are a group of uniform polytopes which are not regular, generated by a Wythoff construction with mirrors all related by order-2 and order-3 dihedral angles. They can be seen as one-end-ringed Coxeter–Dynkin diagrams. The Coxeter symbol for these figures has the form ki,j, where each letter represents a length of order-3 branches on a Coxeter–Dynkin diagram with a single ring on the end node of a k length sequence of branches. The vertex figure of ki,j is (k − 1)i,j, and each of its facets are represented by subtracting one from one of the nonzero subscripts, i.e.
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An image that seems improperly rendered It is possible to specify layout details with DOT, although not all tools that implement the DOT language pay attention to the position attributes. Thus, depending on the tools used, users must rely on automated layout algorithms (potentially resulting in unexpected output) or tediously hand-positioned nodes. For example: digraph g { node [shape=plaintext]; A1 -> B1; A2 -> B2; A3 -> B3; A1 -> A2 [label=f]; A2 -> A3 [label=g]; B2 -> B3 [label="g'"]; B1 -> B3 [label="(g o f)'" tailport=s headport=s]; { rank=same; A1 A2 A3 } { rank=same; B1 B2 B3 } } After moving labels and arrows a bit, and changing font size of subscripts, the image looks correct. There are two problems in the image titled "An image that seems improperly rendered".
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SAS was re-designed in SAS 76 with an open architecture that allowed for compilers and procedures. The INPUT and INFILE statements were improved so they could read most data formats used by IBM mainframes. Generating reports was also added through the PUT and FILE statements. The ability to analyze general linear models was also added as was the FORMAT procedure, which allowed developers to customize the appearance of data. In 1979, SAS 79 added support for the CMS operating system and introduced the DATASETS procedure. Three years later, SAS 82 introduced an early macro language and the APPEND procedure. SAS version 4 had limited features, but made SAS more accessible. Version 5 introduced a complete macro language, array subscripts, and a full-screen interactive user interface called Display Manager.
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However, by enlarging the model to a two-sector model, more remarkable features of Ricardian economics emerge. The two sectors include: the basic goods sector (wage goods called ‘corn’) and a luxury goods sector (called ‘gold’). The whole new model appears as: Equations (2.1) to (2.7) are identical to those of the single sector model, but now they bear added subscripts to differentiate the production of ‘corn’ from that of ‘gold’. Equation (2.8) presents the gold production function which, unlike the ‘corn’ production function, exhibits constant returns to scale. The parameter is the physical amount of ‘gold’ produced by a worker in a year. The next two equations show in monetary (nominal terms) the wage rate (‘corn’) per worker and the rate of profit: where p1 and p2 represent the price of ‘corn’ and the price of ‘gold’ respectively.
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Integer programming is NP-complete, and Maydan showed that the problem of checking array aliasing in nested loops with affine bounds and subscripts is equivalent to integer programming; other operations, such as array dataflow analysis, are even more complex (the algorithms of the Omega Library handle the full language of Presburger Arithmetic, which is O(2^2^2^n)). Thus, it is clearly unrealistic to expect exact fast results for arbitrary problems of array aliasing or array data flow, even over the affine domain. Fortunately, many problems fall into a subset of this domain where general algorithms can produce an exact answer in polynomial time. Outside of this domain, the Omega Library, piplib and isl emphasize the production of an exact result (except in the cases of certain uses of uninterpreted function symbols in Omega), despite the high complexity.
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If an equation can be put into the form f(x) = x, and a solution x is an attractive fixed point of the function f, then one may begin with a point x1 in the basin of attraction of x, and let xn+1 = f(xn) for n ≥ 1, and the sequence {xn}n ≥ 1 will converge to the solution x. Here xn is the nth approximation or iteration of x and xn+1 is the next or n + 1 iteration of x. Alternately, superscripts in parentheses are often used in numerical methods, so as not to interfere with subscripts with other meanings. (For example, x(n+1) = f(x(n)).) If the function f is continuously differentiable, a sufficient condition for convergence is that the spectral radius of the derivative is strictly bounded by one in a neighborhood of the fixed point.
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Although it is not encountered in practice, the equations can also apply to the case of two media with a common permittivity but different refractive indices due to different permeabilities. From equations () and (), if ϵ is fixed instead of μ, then becomes inversely proportional to , with the result that the subscripts 1 and 2 in equations () to () are interchanged (due to the additional step of multiplying the numerator and denominator by ). Hence, in () and (), the expressions for and in terms of refractive indices will be interchanged, so that Brewster's angle () will give instead of and any beam reflected at that angle will be p-polarized instead of s-polarized.More general Brewster angles, for which the angles of incidence and refraction are not necessarily complementary, are discussed in C.L. Giles and W.J. Wild, "Brewster angles for magnetic media", International Journal of Infrared and Millimeter Waves, vol.
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Since we know that the real point _xyz_ will remain in the same place in real space from one frame of the image to the next we can make the point a constant even though we do not know where it is. So: : _xyz_ i = _xyz_ j where the subscripts i and j refer to arbitrary frames in the shot we are analyzing. Since this is always true then we know that: :P'( _camera_ i, _XY_ i) ∩ P'( _camera_ j, _XY_ j) ≠ {} Because the value of _XY_ i has been determined for all frames that the feature is tracked through by the tracking program, we can solve the reverse projection function between any two frames as long as P'( _camera_ i, _XY_ i) ∩ P'( _camera_ j, _XY_ j) is a small set. Set of possible _camera_ vectors that solve the equation at i and j (denoted Cij).
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The statement could allocate one- dimensional and two-dimensional arrays of any of the three data types. The range of subscripts always began with 0 (but statements did not set elements in row 0 or column 0). The "virtual DIM" statement could map "virtual data array(s)" or "virtual array(s)" to a disk file, which allowed arrays larger than the computer's available memory (or even its address space), and allowed use of array elements to read, write, and extend disk files (persistent storage). They called this arrangement "virtual data storage" and "virtual core", but it did not use the modern approach of allocating the arrays and a memory-mapped file in the same virtual memory and then allowing the virtual memory manager to handle all access via paging. Instead, a program would first execute the statement, which opened a file with an associated file number and assigned it one one-record (512-byte) file buffer.
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Neutralisation of the precursor ion beam is commonly performed by letting the beam pass through a gas cell.G. Serianni, et al., New Journal of Physics, Volume 19, April 2017 For a precursor negative ion beam at fusion-relevant energies, the key collisional processesIAEA Aladdin database are: : _D −_ \+ D2 -> _D 0_ \+ _e_ \+ D2 (singe electron detachment, with \sigma−10=1.13×10−20 m2 at 1MeV) : _D −_ \+ D2 -> _D +_ \+ _e_ \+ D2 (double electron detachment, with \sigma−11=7.22×10−22 m2 at 1MeV) : _D 0_ \+ D2 -> _D +_ \+ _e_ \+ D2 (reionization, with \sigma01=3.79×10−21 m2 at 1MeV) : _D +_ \+ D2 -> _D 0_ \+ D2+ (charge exchange, \sigma10 negligible at 1MeV) Subscript indicate the fast particles, while subscripts i,j of the cross-section \sigmaij indicate the charge state of fast particle before and after collision. Cross-sections at 1MeV are such that, once created, a fast positive ion cannot be converted into a fast neutral, and this is the cause of the limited achievable efficiency of gas neutralisers.
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Converting ordinary code into SSA form is primarily a simple matter of replacing the target of each assignment with a new variable, and replacing each use of a variable with the "version" of the variable reaching that point. For example, consider the following control flow graph: An example control flow graph, before conversion to SSA Changing the name on the left hand side of "x \leftarrow x - 3" and changing the following uses of x to that new name would leave the program unaltered. This can be exploited in SSA by creating two new variables: x1 and x2, each of which is assigned only once. Likewise, giving distinguishing subscripts to all the other variables yields: An example control flow graph, partially converted to SSA It is clear which definition each use is referring to, except for one case: both uses of y in the bottom block could be referring to either y1 or y2, depending on which path the control flow took.
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The size- and pressure- dependent glass transition temperatures of free-standing films or supported films having weak interactions with substrates decreases with decreasing of pressure and size. However, the glass transition temperature of supported films having strong interaction with substrates increases of pressure and the decrease of size. Different models like two layer model, three layer model, Tg (D, 0) ∝ 1/D and some more models relating specific heat, density and thermal expansion are used to obtain the experimental results on nanopolymers and even some observations like freezing of films due to memory effects in the visco-elastic eigenmodels of the films, and finite effects of the small molecule glass are observed. To describe Tg (D, 0) function of polymers more generally, a simple and unified model recently is provided based on the size-dependent melting temperature of crystals and Lindemann's criterion :: where σg is the root of mean squared displacement of surface and interior molecules of glasses at Tg (D, 0), α = σs2 (D, 0) / σv2 (D, 0) with subscripts s and v denoting surface and volume, respectively.
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Although still in relatively common use, there is limited relevance of these device-specific power supply designations in circuits that use a mixture of bipolar and FET elements, or in those that employ either both NPN and PNP transistors or both n- and p-channel FETs. This latter case is very common in modern chips, which are often based on CMOS technology, where the C stands for complementary meaning that complementary pairs of n- and p-channel devices are common throughout. These naming conventions were part of a bigger picture where, to continue with bipolar transistor examples although the FET remains entirely analogous, DC or bias currents into or out of each terminal may be written IC, IE, and IB. Apart from DC or bias conditions, many transistor circuits also process a smaller audio-, video-, or radio-frequency signal that is superimposed on the bias at the terminals. Lower case letters and subscripts are used to refer to these signal levels at the terminals, either peak-to-peak or RMS as required.
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