Rome survived Caligula and Nero, and the harm now being done to American institutions and even to civic decency and democratic norms may ultimately be rectifiable.
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In mathematics, a rectifiable set is a set that is smooth in a certain measure-theoretic sense. It is an extension of the idea of a rectifiable curve to higher dimensions; loosely speaking, a rectifiable set is a rigorous formulation of a piece-wise smooth set. As such, it has many of the desirable properties of smooth manifolds, including tangent spaces that are defined almost everywhere. Rectifiable sets are the underlying object of study in geometric measure theory.
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The category of smooth manifolds with smooth maps lacks certain desirable properties, and people have tried to generalize smooth manifolds in order to rectify this. Diffeological spaces use a different notion of chart known as a "plot". Frölicher spaces and orbifolds are other attempts. A rectifiable set generalizes the idea of a piece-wise smooth or rectifiable curve to higher dimensions; however, rectifiable sets are not in general manifolds.
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It is continuous everywhere, but differentiable nowhere. It is not rectifiable. It has a Lebesgue measure of 0. The type 1 curve has a dimension of ≈ 1.46.
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If a polyomino is rectifiable, that is, able to tile a rectangle, then it will also be a rep-tile, because the rectangle will then tile a square. This can be seen clearly in the octominoes, which are created from eight squares. Two copies of some octominoes will tile a square; therefore these octominoes are also rep-16 rep-tiles. Rep-tiles based on rectifiable octominoes Four copies of some nonominoes and nonakings will tile a square, therefore these polyforms are also rep-36 rep-tiles.
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The Koch curve. The graph of xsin(1/x). As mentioned above, some curves are non-rectifiable. That is, there is no upper bound on the lengths of polygonal approximations; the length can be made arbitrarily large.
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The relative convex hull of a finite set of points in a simple polygon In discrete geometry and computational geometry, the relative convex hull or geodesic convex hull is an analogue of the convex hull for the points inside a simple polygon or a rectifiable simple closed curve.
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67, available here Due to his intransigence always loved by lampooners, he was first mocked in 1867.Gil Blas 24.11.67, available here Ramón blamed the Carlists for leaving Isabel II no option but to have allied with the liberals; this error, however, was still rectifiable by creating a strong, conservative alliance.Urigüen 1986, pp.
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Produzioni matematiche, 1750 Fagnano is best known for investigations on the length and division of arcs of certain curves, especially the lemniscate; this seems also to have been in his own estimation his most important work, since he had the figure of the lemniscate with the inscription: "Multifariam divisa atque dimensa Deo veritatis gloria", engraved on the title-page of his Produzioni Matematiche, which he published in two volumes (Pesaro, 1750), and dedicated to Pope Benedict XIV. The same figure and words "Deo veritatis gloria" also appear on his tomb. Failing to rectify the ellipse or hyperbola, Fagnano attempted to determine arcs whose difference is rectifiable. The word "rectifiable" meant at that time that the length can be found explicitly, which is different from its modern meaning.
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There is an analogous problem in geometric measure theory which asks the following: under what conditions may a subset E of Euclidean space be contained in a rectifiable curve (that is, when is there a curve with finite length that visits every point in E)? This problem is known as the analyst's travelling salesman problem.
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For example, the perimeter of a regular polygon inscribed in a circle approaches the circumference with increasing numbers of sides (and decrease in the length of one side). In geometric measure theory such a smooth curve as the circle that can be approximated by small straight segments with a definite limit is termed a rectifiable curve.
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Every convex curve that is the boundary of a closed convex set has a well-defined finite length. That is, these curves are a subset of the rectifiable curves. According to the four-vertex theorem, every smooth convex curve that is the boundary of a closed convex set has at least four vertices, points that are local minima or local maxima of curvature..
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Illustration from Acta Eruditorum, 1690 The Tschirnhaus transformation, by which he removed certain intermediate terms from a given algebraic equation, is well known. It was published in the scientific journal Acta Eruditorum in 1683. In 1682, Von Tschirnhaus worked out the theory of catacaustics and showed that they were rectifiable. This was the second case in which the envelope of a moving line was determined.
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In the work of Archimedes it already appears that the length of a circle can be approximated by the length of regular polyhedra inscribed or circumscribed in the circle. In general, for smooth or rectifiable curves their length can be defined as the supremum of the lengths of polygonal curves inscribed in them. The Schwarz lantern shows that surface area cannot be defined as the supremum of inscribed polyhedral surfaces.
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The severity of oral candidiasis is subject to great variability from one person to another and in the same person from one occasion to the next. The prognosis of such infection is usually excellent after the application of topical or systemic treatments. However, oral candidiasis can be recurrent. Individuals continue to be at risk of the condition if underlying factors such as reduced salivary flow rate or immunosuppression are not rectifiable.
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Multidimensional Plateau problem in the class of spectral surfaces (parametrized by the spectra of the manifolds with a fixed boundary) was solved in 1969 by Anatoly Fomenko. The axiomatic approach of Jenny Harrison and Harrison Pugh treats a wide variety of special cases. In particular, they solve the anisotropic Plateau problem in arbitrary dimension and codimension for any collection of rectifiable sets satisfying a combination of general homological, cohomological or homotopical spanning conditions.
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In addition to extending much of Karen Uhlenbeck's analysis to higher dimensions, he studied the interaction of Yang-Mills theory with calibrated geometry. Uhlenbeck had shown in the 1980s that, when given a sequence of Yang-Mills connections of uniformly bounded energy, they will converge smoothly on the complement of a subset of codimension at least four, known as the complement of the "singular set". Tian showed that the singular set is a rectifiable set.
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Consider a set of all oriented lines on a plane. Each line is defined by the parameters \rho and \varphi, where\rho is a distance from the origin to the line, and \varphi is an angle between the line and the x-axis. Then the set of all oriented lines is homeomorphic to a circular cylinder of radius 1 with the area element dS=d \rho \, d\varphi . Let \gamma be a rectifiable curve on a plane.
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Construction of the Gosper curve A fractal curve is, loosely, a mathematical curve whose shape retains the same general pattern of irregularity, regardless of how high it is magnified, that is, its graph takes the form of a fractal. In general, fractal curves are nowhere rectifiable curves — that is, they do not have finite length — and every subarc longer than a single point has infinite length. An extremely famous example is the boundary of the Mandelbrot set.
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The circle is the closed curve of least perimeter that encloses the maximum area. This is known as the isoperimetric inequality, which states that if a rectifiable Jordan curve in the Euclidean plane has perimeter C and encloses an area A (by the Jordan curve theorem) then :4\pi A\le C^2. Moreover, equality holds in this inequality if and only if the curve is a circle, in which case A=\pi r^2 and C=2\pi r.
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Suppose that is a rectifiable curve in the plane and is Borel measurable. Then we may define the length of with respect to the Euclidean metric weighted by ρ to be :\int_a^b \rho(\gamma(t))\,d\ell(t), where \ell(t) is the length of the restriction of to . This is sometimes called the -length of . This notion is quite useful for various applications: for example, in muddy terrain the speed in which a person can move may depend on how deep the mud is.
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Figure 4. A symmetrical Apollonian gasket, also called the Leibniz packing, after its inventor Gottfried Leibniz. By solving Apollonius' problem repeatedly to find the inscribed circle, the interstices between mutually tangential circles can be filled arbitrarily finely, forming an Apollonian gasket, also known as a Leibniz packing or an Apollonian packing. This gasket is a fractal, being self-similar and having a dimension d that is not known exactly but is roughly 1.3, which is higher than that of a regular (or rectifiable) curve (d = 1) but less than that of a plane (d = 2).
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Various specialized forms of the problem were solved, but it was only in 1930 that general solutions were found in the context of mappings (immersions) independently by Jesse Douglas and Tibor Radó. Their methods were quite different; Radó's work built on the previous work of René Garnier and held only for rectifiable simple closed curves, whereas Douglas used completely new ideas with his result holding for an arbitrary simple closed curve. Both relied on setting up minimization problems; Douglas minimized the now-named Douglas integral while Radó minimized the "energy". Douglas went on to be awarded the Fields Medal in 1936 for his efforts.
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The extension of the problem to higher dimensions (that is, for k-dimensional surfaces in n-dimensional space) turns out to be much more difficult to study. Moreover, while the solutions to the original problem are always regular, it turns out that the solutions to the extended problem may have singularities if k \leq n - 2. In the hypersurface case where k = n - 1, singularities occur only for n \geq 8. To solve the extended problem in certain special cases, the theory of perimeters (De Giorgi) for codimension 1 and the theory of rectifiable currents (Federer and Fleming) for higher codimension have been developed.
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During the 15 years or so years prior to that paper, Federer worked at the technical interface of geometry and measure theory. He focused particularly on surface area, rectifiability of sets, and the extent to which one could substitute rectifiability for smoothness in the analysis of surfaces. His 1947 paper on the rectifiable subsets of n-space characterized purely unrectifiable sets by their "invisibility" under almost all projections. A. S. Besicovitch had proven this for 1-dimensional sets in the plane, but Federer's generalization, valid for subsets of arbitrary dimension in any Euclidean space, was a major technical accomplishment, and later played a key role in Normal and Integral Currents.
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By solving Apollonius' problem repeatedly to find the inscribed circle, the interstices between mutually tangential circles can be filled arbitrarily finely, forming an Apollonian gasket, also known as a Leibniz packing or an Apollonian packing. This gasket is a fractal, being self-similar and having a dimension d that is not known exactly but is roughly 1.3, which is higher than that of a regular (or rectifiable) curve (d = 1) but less than that of a plane (d = 2). The Apollonian gasket was first described by Gottfried Leibniz in the 17th century, and is a curved precursor of the 20th- century Sierpiński triangle. The Apollonian gasket also has deep connections to other fields of mathematics; for example, it is the limit set of Kleinian groups.
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More than a decade after Richardson completed his work, Benoit Mandelbrot developed a new branch of mathematics, fractal geometry, to describe just such non-rectifiable complexes in nature as the infinite coastline. His own definition of the new figure serving as the basis for his study is: A key property of the fractal is self- similarity; that is, at any scale the same general configuration appears. A coastline is perceived as bays alternating with promontories. In the hypothetical situation that a given coastline has this property of self- similarity, then no matter how greatly any one small section of coastline is magnified, a similar pattern of smaller bays and promontories superimposed on larger bays and promontories appears, right down to the grains of sand.
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Although the curve is not a circle, and hence does not have any obvious connection to the constant , a standard proof of this result uses Morera's theorem, which implies that the integral is invariant under homotopy of the curve, so that it can be deformed to a circle and then integrated explicitly in polar coordinates. More generally, it is true that if a rectifiable closed curve does not contain , then the above integral is times the winding number of the curve. The general form of Cauchy's integral formula establishes the relationship between the values of a complex analytic function on the Jordan curve and the value of at any interior point of :Joglekar, S.D., Mathematical Physics, Universities Press, 2005, p. 166, .
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Some operating systems, notably Windows Vista, Windows 7, Windows 8, Windows 8.1 and Windows 10, do not configure themselves to load the AHCI driver upon boot if the SATA controller was not in AHCI mode at the time the operating system was installed. Although this is an easily rectifiable condition, it remains an ongoing issue with the AHCI standard. The most prevalent symptom for an operating system (or systems) that are installed in IDE mode (in some BIOS firmware implementations otherwise called 'Combined IDE mode'), is that the system drive typically fails to boot, with an ensuing error message, if the SATA controller (in BIOS) is inadvertently switched to AHCI mode after OS installation. In Microsoft Windows the symptom is a boot loop which begins with a Blue Screen error, if not rectified - and through no fault of Microsoft Windows.
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Let P be a simple polygon or a rectifiable simple closed curve, and let X be any set enclosed by P. A geodesic between two points in P is a shortest path connecting those two points that stays entirely within P. A subset K of the points inside P is said to be relatively convex, geodesically convex, or P-convex if, for every two points of K, the geodesic between them in P stays within K. Then the relative convex hull of X can be defined as the intersection of all relatively convex sets containing X. Equivalently, the relative convex hull is the minimum-perimeter weakly simple polygon in P that encloses X. This was the original formulation of relative convex hulls, by . However this definition is complicated by the need to use weakly simple polygons (intuitively, polygons in which the polygon boundary can touch or overlap itself but not cross itself) instead of simple polygons when X is disconnected and its components are not all visible to each other.
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