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1
BARRAL, JULIEN, and STÉPHANE SEURET. "Ubiquity and large intersections properties under digit frequencies constraints." Mathematical Proceedings of the Cambridge Philosophical Society 145, no. 3 (November 2008): 527–48. http://dx.doi.org/10.1017/s030500410800159x.
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AbstractWe are interested in two properties of real numbers: the first one is the property of being well-approximated by some dense family of real numbers {xn}n≥1, such as rational numbers and more generally algebraic numbers, and the second one is the property of having given digit frequencies in some b-adic expansion.We combine these two ways of classifying the real numbers, in order to provide a finer classification. We exhibit sets S of points x which are approximated at a given rate by some of the {xn}n, those xn being selected according to their digit frequencies. We compute the Hausdorff dimension of any countable intersection of such sets S, and prove that these sets enjoy the so-called large intersection property.
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Yurkov, V. "Images of Linear Conditions on a Manhattan Plane." Geometry & Graphics 8, no. 1 (April 20, 2020): 3–14. http://dx.doi.org/10.12737/2308-4898-2020-3-14.
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In this paper are considered planar point sets generated by linear conditions, which are realized in rectangular or Manhattan metric. Linear conditions are those expressed by the finite sum of the products of distances by numerical coefficients. Finite sets of points and lines are considered as figures defining linear conditions. It has been shown that linear conditions can be defined relative to other planar figures: lines, polygons, etc. The design solutions of the following general geometric problem are considered: for a finite set of figures (points, line segments, polygons...) specified on a plane with a rectangular metric, which are in a common position, it is necessary to construct sets that satisfy any linear condition. The problems in which the given sets are point and segment ones have been considered in detail, and linear conditions are represented as a sum or as relations of distances. It is proved that solution result can be isolated points, broken lines, and areas on the plane. Sets of broken lines satisfying the given conditions form families of isolines for the given condition. An algorithm for building isoline families is presented. The algorithm is based on the Hanan lattice construction and the isolines behavior in each node and each sub-region of the lattice. For isoline families defined by conditions for relation of distances, some of their properties allowing accelerate their construction process are proved. As an example for application of the described theory, the problem of plane partition into regions corresponding to a given set of points, lines and other figures is considered. The problem is generalized problem of Voronoi diagram construction, and considered in general formulation. It means the next: 1) the problem is considered in rectangular metric; 2) all given points may be integrated in various figures – separate points, line segments, triangles, quadrangles etc.; 3) the Voronoi diagram’s property of proximity is changed for property of proportionality. Have been represented examples for plane partition into regions, determined by two-point sets.
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BOMFIM, THIAGO, and PAULO VARANDAS. "Multifractal analysis for weak Gibbs measures: from large deviations to irregular sets." Ergodic Theory and Dynamical Systems 37, no. 1 (October 12, 2015): 79–102. http://dx.doi.org/10.1017/etds.2015.46.
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In this article we prove estimates for the topological pressure of the set of points whose Birkhoff time averages are far from the space averages corresponding to the unique equilibrium state that has a weak Gibbs property. In particular, if$f$has an expanding repeller and$\unicode[STIX]{x1D719}$is a Hölder continuous potential, we prove that the topological pressure of the set of points whose accumulation values of Birkhoff averages belong to some interval$I\subset \mathbb{R}$can be expressed in terms of the topological pressure of the whole system and the large deviations rate function. As a byproduct we deduce that most irregular sets for maps with the specification property have topological pressure strictly smaller than the whole system. Some extensions to a non-uniformly hyperbolic setting, level-2 irregular sets and hyperbolic flows are also given.
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Sun, Chia-Liang. "Weak Approximation for Points with Coordinates in Rank-one Subgroups of Global Function Fields." Canadian Mathematical Bulletin 61, no. 4 (November 20, 2018): 878–90. http://dx.doi.org/10.4153/cmb-2018-008-3.
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AbstractFor every affine variety over a global function field, we show that the set of its points with coordinates in an arbitrary rank-one multiplicative subgroup of this function field satisfies the required property of weak approximation for finite sets of places of this function field avoiding arbitrarily given finitely many places.
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Polyrakis, Ioannis A. "Strongly exposed points in bases for the positive cone of ordered Banach spaces and characterizations of l1(Г)." Proceedings of the Edinburgh Mathematical Society 29, no. 2 (June 1986): 271–82. http://dx.doi.org/10.1017/s0013091500017648.
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The study of extreme, strongly exposed points of closed, convex and bounded sets in Banach spaces has been developed especially by the interconnection of the Radon–Nikodým property with the geometry of closed, convex and bounded subsets of Banach spaces [5],[2] . In the theory of ordered Banach spaces as well as in the Choquet theory, [4], we are interested in the study of a special type of convex sets, not necessarily bounded, namely the bases for the positive cone. In [7] the geometry (extreme points, dentability) of closed and convex subsets K of a Banach space X with the Radon-Nikodým property is studied and special emphasis has been given to the case where K is a base for acone P of X. In [6, Theorem 1], it is proved that an infinite-dimensional, separable, locally solid lattice Banach space is order-isomorphic to l1 if, and only if, X has the Krein–Milman property and its positive cone has a bounded base.
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Kent, Steven L., Roy A. Mimna, and Jamal K. Tartir. "A Note on Topological Properties of Non-Hausdorff Manifolds." International Journal of Mathematics and Mathematical Sciences 2009 (2009): 1–4. http://dx.doi.org/10.1155/2009/891785.
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The notion of compatible apparition points is introduced for non-Hausdorff manifolds, and properties of these points are studied. It is well known that the Hausdorff property is independent of the other conditions given in the standard definition of a topological manifold. In much of literature, a topological manifold of dimension is a Hausdorff topological space which has a countable base of open sets and is locally Euclidean of dimension . We begin with the definition of a non-Hausdorff topological manifold.
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Martínez, Teresa, and José L. Torrea. "Boundedness of vector-valued martingale transforms on extreme points and applciations." Journal of the Australian Mathematical Society 76, no. 2 (April 2004): 207–22. http://dx.doi.org/10.1017/s1446788700008909.
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AbstractLet Β1, Β2be a pair of Banach spaces andTbe a vector valued martingale transform (with respect to general filtration) which maps Β1-valued martingales into Β2-valued martingales. Then, the following statements are equivalent:Tis bounded fromintofor somep(or equivalently for everyp) in the range 1 <p< ∞;Tis bounded fromintoBMOB2;Tis bounded fromBMOB1intoBMOB2;Tis bounded frominto. Applications toUMDand martingale cotype properties are given. We also prove that the Hardy spacedefined in the case of a general filtration has nice dense sets and nice atomic decompositions if and only if Β has the Radon-Nikodým property.
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Palmer, T. N. "Discretization of the Bloch sphere, fractal invariant sets and Bell’s theorem." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, no. 2236 (April 2020): 20190350. http://dx.doi.org/10.1098/rspa.2019.0350.
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An arbitrarily dense discretization of the Bloch sphere of complex Hilbert states is constructed, where points correspond to bit strings of fixed finite length. Number-theoretic properties of trigonometric functions (not part of the quantum-theoretic canon) are used to show that this constructive discretized representation incorporates many of the defining characteristics of quantum systems: completementarity, uncertainty relationships and (with a simple Cartesian product of discretized spheres) entanglement. Unlike Meyer’s earlier discretization of the Bloch Sphere, there are no orthonormal triples, hence the Kocken–Specker theorem is not nullified. A physical interpretation of points on the discretized Bloch sphere is given in terms of ensembles of trajectories on a dynamically invariant fractal set in state space, where states of physical reality correspond to points on the invariant set. This deterministic construction provides a new way to understand the violation of the Bell inequality without violating statistical independence or factorization, where these conditions are defined solely from states on the invariant set. In this finite representation, there is an upper limit to the number of qubits that can be entangled, a property with potential experimental consequences.
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Uehara, Kiyohiko, and Kaoru Hirota. "Multi-Level Interpolation for Inference with Sparse Fuzzy Rules: An Extended Way of Generating Multi-Level Points." Journal of Advanced Computational Intelligence and Intelligent Informatics 17, no. 2 (March 20, 2013): 127–48. http://dx.doi.org/10.20965/jaciii.2013.p0127.
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As an extended way for inference based on multi-level interpolation, the number of multi-level points, generated with the α-cuts of given facts, is increased by making larger the number of levels of α. The conventional way uses the number of the levels adopted to define each given fact by α-cuts. A basic study is performed with triangular membership functions, where it is examined how the accuracy of mapping with fuzzy rules changes in increasing the number of the levels. It is also examined how deduced consequences behave when the number is increased. Moreover, convergent core sets of consequences are theoretically derived in the increase by the effective use of non-adaptive inference operations for core sets. They are used as references in simulation studies. Increasing the number of the levels provides nonlinear mapping with more precise reflection of distribution forms of sparse fuzzy rules to consequences. The basic study here contributes to improving the reflection accuracy. In simulations for the basic study, it is found that the mapping accuracy improves when the number of levels of α is increased. It is also confirmed that deduced core sets converge to those theoretically derived. Support sets are also found to converge in increasing the number of the levels. The core sets and support sets of deduced consequences do not, however, monotonically converge. This property causes difficulty in determining the optimized number of the levels so as to satisfy required mapping accuracy. In order to solve this problem, further discussions may be possible to theoretically derive convergent consequences and to use them in practical fields, in accordance with the theoretically derived core sets mentioned above.
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Lasserre, Jean B. "Level Sets and NonGaussian Integrals of Positively Homogeneous Functions." International Game Theory Review 17, no. 01 (March 2015): 1540001. http://dx.doi.org/10.1142/s0219198915400010.
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We investigate various properties of the sublevel set G = {x : g(x) ≤ 1} and the integration of h on this sublevel set when g and h are positively homogeneous functions (and in particular homogeneous polynomials). For instance, the latter integral reduces to integrating h exp (-g) on the whole space ℝn (a nonGaussian integral) and when g is a polynomial, then the volume of G is a convex function of the coefficients of g. We also provide a numerical approximation scheme to compute the volume of G or integrate h on G (or, equivalently to approximate the associated nonGaussian integral). We also show that finding the sublevel set {x : g(x) ≤ 1} of minimum volume that contains some given subset K is a (hard) convex optimization problem for which we also propose two convergent numerical schemes. Finally, we provide a Gaussian-like property of nonGaussian integrals for homogeneous polynomials that are sums of squares and critical points of a specific function.
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Viaud, Daniel Pierre Loti. "Random perturbations of recursive sequences with an application to an epidemic model." Journal of Applied Probability 32, no. 3 (September 1995): 559–78. http://dx.doi.org/10.2307/3215113.
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We investigate the asymptotic sample path behaviour of a randomly perturbed discrete-time dynamical system. We consider the case where the trajectories of the non-perturbed dynamical system are attracted by a finite number of limit sets and characterize a case where this property remains valid for the perturbed dynamical system when the perturbation converges to zero. For this purpose, no further assumptions on the perturbation are needed and our main condition applies to the limit sets of the non-perturbed dynamical system. When the limit sets reduce to limit points we show that this main condition is more general than the usual assumption of the existence of a Lyapunov function for the non-perturbed dynamical system. An application to an epidemic model is given to illustrate our results.
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Viaud, Daniel Pierre Loti. "Random perturbations of recursive sequences with an application to an epidemic model." Journal of Applied Probability 32, no. 03 (September 1995): 559–78. http://dx.doi.org/10.1017/s0021900200103043.
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We investigate the asymptotic sample path behaviour of a randomly perturbed discrete-time dynamical system. We consider the case where the trajectories of the non-perturbed dynamical system are attracted by a finite number of limit sets and characterize a case where this property remains valid for the perturbed dynamical system when the perturbation converges to zero. For this purpose, no further assumptions on the perturbation are needed and our main condition applies to the limit sets of the non-perturbed dynamical system. When the limit sets reduce to limit points we show that this main condition is more general than the usual assumption of the existence of a Lyapunov function for the non-perturbed dynamical system. An application to an epidemic model is given to illustrate our results.
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Baake, Michael, Holger Koesters, and Robert Moody. "Random point processes and tilings arising from Gaussian analytic functions." Acta Crystallographica Section A Foundations and Advances 70, a1 (August 5, 2014): C523. http://dx.doi.org/10.1107/s2053273314094765.
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Getting a grasp of what aperiodic order really entails is going to require collecting and understanding many diverse examples. Aperiodic crystals are at the top of the largely unknown iceberg beneath. Here we present a recently studied form of random point process in the (complex) plane which arises as the sets of zeros of a specific class of analytic functions given by power series with randomly chosen coefficients: Gaussian analytic functions (GAF). These point sets differ from Poisson processes by having a sort of built in repulsion between points, though the resulting sets almost surely fail both conditions of the Delone property. Remarkably the point sets that arise as the zeros of GAFs determine a random point process which is, in distribution, invariant under rotation and translation. In addition, there is a logarithmic potential function for which the zeros are the attractors, and the resulting basins of attraction produce tilings of the plane by tiles which are, almost surely, all of the same area. We discuss GAFs along with their tilings and diffraction, and as well note briefly their relationship to determinantal point processes, which are also of physical interest.
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Muciño-Raymundo, Jesús, and Carlos Valero-Valdés. "Bifurcations of meromorphic vector fields on the Riemann sphere." Ergodic Theory and Dynamical Systems 15, no. 6 (December 1995): 1211–22. http://dx.doi.org/10.1017/s0143385700009883.
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AbstractLet {Xθ} be a family of rotated singular real foliations in the Riemann sphere which is the result of the rotation of a meromorphic vector field X with zeros and poles of multiplicity one. We prove that the set of bifurcation values, in the circle {θ}, is for each family a set with at most a finite number of accumulation points. A condition which implies a finite number of bifurcation values is given. We also show that the property of having an infinite set of bifurcation values defines open but not dense sets in the space of meromorphic vector fields with fixed degree.
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Kamilya, Supreeti, and Jarkko Kari. "Nilpotency and periodic points in non-uniform cellular automata." Acta Informatica 58, no. 4 (July 19, 2021): 319–33. http://dx.doi.org/10.1007/s00236-020-00390-7.
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AbstractNilpotent cellular automata have the simplest possible dynamics: all initial configurations lead in bounded time into the unique fixed point of the system. We investigate nilpotency in the setup of one-dimensional non-uniform cellular automata (NUCA) where different cells may use different local rules. There are infinitely many cells in NUCA but only a finite number of different local rules. Changing the distribution of the local rules in the system may drastically change the dynamics. We prove that if the available local rules are such that every periodic distribution of the rules leads to nilpotent behavior then so do also all eventually periodic distributions. However, in some cases there may be non-periodic distributions that are not nilpotent even if all periodic distributions are nilpotent. We demonstrate such a possibility using aperiodic Wang tile sets. We also investigate temporally periodic points in NUCA. In contrast to classical uniform cellular automata, there are NUCA—even reversible equicontinuous ones—that do not have any temporally periodic points. We prove the undecidability of this property: there is no algorithm to determine if a NUCA with a given finite distribution of local rules has a periodic point.
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Herbort, Gregor. "Localization lemmas for the Bergman metric at plurisubharmonic peak points." Nagoya Mathematical Journal 171 (2003): 107–25. http://dx.doi.org/10.1017/s0027763000025538.
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AbstractLet D be a bounded pseudoconvex domain in ℂn and ζ ∈ D. By KD and BD we denote the Bergman kernel and metric of D, respectively. Given a ball B = B(ζ, R), we study the behavior of the ratio KD/KD∩B(w) when w ∈ D ∩ B tends towards ζ. It is well-known, that it remains bounded from above and below by a positive constant. We show, that the ratio tends to 1, as w tends to ζ, under an additional assumption on the pluricomplex Green function D(·, w) of D with pole at w, namely that the diameter of the sublevel sets Aw :={z ∈ D | D(z, w) < −1} tends to zero, as w → ζ. A similar result is obtained also for the Bergman metric. In this case we also show that the extremal function associated to the Bergman kernel has the concentration of mass property introduced in [DiOh1], where the question was discussed how to recognize a weight function from the associated Bergman space. The hypothesis concerning the set Aw is satisfied for example, if the domain is regular in the sense of Diederich-Fornæss, ([DiFo2]).
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Cholaquidis, Alejandro, and Antonio Cuevas. "Set estimation under biconvexity restrictions." ESAIM: Probability and Statistics 24 (2020): 770–88. http://dx.doi.org/10.1051/ps/2020019.
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A set in the Euclidean plane is said to be biconvex if, for some angle θ ∈ [0, π∕2), all its sections along straight lines with inclination angles θ and θ + π∕2 are convex sets (i.e., empty sets or segments). Biconvexity is a natural notion with some useful applications in optimization theory. It has also be independently used, under the name of “rectilinear convexity”, in computational geometry. We are concerned here with the problem of asymptotically reconstructing (or estimating) a biconvex set S from a random sample of points drawn on S. By analogy with the classical convex case, one would like to define the “biconvex hull” of the sample points as a natural estimator for S. However, as previously pointed out by several authors, the notion of “hull” for a given set A (understood as the “minimal” set including A and having the required property) has no obvious, useful translation to the biconvex case. This is in sharp contrast with the well-known elementary definition of convex hull. Thus, we have selected the most commonly accepted notion of “biconvex hull” (often called “rectilinear convex hull”): we first provide additional motivations for this definition, proving some useful relations with other convexity-related notions. Then, we prove some results concerning the consistent approximation of a biconvex set S and the corresponding biconvex hull. An analogous result is also provided for the boundaries. A method to approximate, from a sample of points on S, the biconvexity angle θ is also given.
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Shcherbakov, Gleb V. "Laffer points, area of fiscal contradictions and taxpayers’ acceptance power." RUDN Journal of Economics 27, no. 1 (December 15, 2019): 49–62. http://dx.doi.org/10.22363/2313-2329-2019-27-1-49-62.
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The Laffer curve is the eternal problem of mathematical economics. Attempts to find the Laffer curve functions lead to new results that do not give the function in coordinates “tax burden - tax revenues” but give results in larger dimensions. Purpose of the article is developing tools to access the excessive tax burden on organizations. The general methods used in the article are analysis, generalization, synthesis. Special methods are mathematical induction, mathematical methods. In the study previously proposed mathematical models of Laffer curves by V.G. Papava (Ananiashvili, Papava, 2010) and E.V. Balatskii (Balatskii, 2000) are generalized and clarified. Taxation limit concept is expanded and necessity of determining the lower taxation limit is shown. The new approach to determining the values of Laffer points based on the use of tax burden and current assets turnover ratio is proposed. The determination of taxpayers’ acceptance power (in meaning “exponent”) is introduced and the property linking it with area of fiscal contradictions is shown. The constancy of the location of the first and second kind Laffer points is proved. Conditions limiting the sets of values of Laffer points are given. As a result the concept of the area of fiscal contradictions is divided with concepts of Laffer curves and Laffer points.
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YOSHINAGA, TETSUYA, YOSHIHIRO IMAKURA, KEN'ICHI FUJIMOTO, and TETSUSHI UETA. "BIFURCATION ANALYSIS OF ITERATIVE IMAGE RECONSTRUCTION METHOD FOR COMPUTED TOMOGRAPHY." International Journal of Bifurcation and Chaos 18, no. 04 (April 2008): 1219–25. http://dx.doi.org/10.1142/s021812740802094x.
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Of the iterative image reconstruction algorithms for computed tomography (CT), the power multiplicative algebraic reconstruction technique (PMART) is known to have good properties for speeding convergence and maximizing entropy. We analyze here bifurcations of fixed and periodic points that correspond to reconstructed images observed using PMART with an image made of multiple pixels and we investigate an extended PMART, which is a dynamical class for accelerating convergence. The convergence process for the state in the neighborhood of the true reconstructed image can be reduced to the property of a fixed point observed in the dynamical system. To investigate the speed of convergence, we present a computational method of obtaining parameter sets in which the given real or absolute values of the characteristic multiplier are equal. The advantage of the extended PMART is verified by comparing it with the standard multiplicative algebraic reconstruction technique (MART) using numerical experiments.
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Chentsov, Aleksandr G. "Maximal linked systems and ultrafilters: main representations and topological properties." Russian Universities Reports. Mathematics, no. 129 (2020): 68–84. http://dx.doi.org/10.20310/2686-9667-2020-25-129-68-84.
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Questions connected with representation of the ultrafilter (UF) set for widely understood measurable space are investigated; this set is considered as a subspace of bitopological space of maximal linked systems (MLS) under equipment with topologies of Wallman and Stone types (measurable structure is defined as a π -system with “zero” and “unit”). Analogous representations connected with generalized variant of cohesion is considered also; in this variant, for corresponding set family, it is postulated the nonemptyness of intersection for finite subfamilies with power not exceeding given. Conditions of identification of UF and MLS (in the above-mentioned generalized sense) are investigated. Constructions reducing to bitopological spaces with points in the form of MLS and 𝑛-supercompactness property generalizing the “usual” supercompactness are considered. Finally, some characteristic properties of MLS and their corollaries connected with the MLS contraction to a smaller π -system are being studied. The case of algebras of sets is selected separately.
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Reddy, H. Venkateswara, S. Viswanadha Raju, B. Suresh Kumar, and C. Jayachandra. "An Approach for Data Labelling and Concept Drift Detection Based on Entropy Model in Rough Sets for Clustering Categorical Data." Journal of Information & Knowledge Management 13, no. 02 (June 2014): 1450020. http://dx.doi.org/10.1142/s0219649214500208.
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Clustering is an important technique in data mining. Clustering a large data set is difficult and time consuming. An approach called data labelling has been suggested for clustering large databases using sampling technique to improve efficiency of clustering. A sampled data is selected randomly for initial clustering and data points which are not sampled and unclustered are given cluster label or an outlier based on various data labelling techniques. Data labelling is an easy task in numerical domain because it is performed based on distance between a cluster and an unlabelled data point. However, in categorical domain since the distance is not defined properly between data points and data points with cluster, then data labelling is a difficult task for categorical data. This paper proposes a method for data labelling using entropy model in rough sets for categorical data. The concept of entropy, introduced by Shannon with particular reference to information theory is a powerful mechanism for the measurement of uncertainty information. In this method, data labelling is performed by integrating entropy with rough sets. This method is also applied to drift detection to establish if concept drift occurred or not when clustering categorical data. The cluster purity is also discussed using Rough Entropy for data labelling and for outlier detection. The experimental results show that the efficiency and clustering quality of this algorithm are better than the previous algorithms.
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Castillo, Enrique, Diego Peteiro-Barral, Bertha Guijarro Berdiñas, and Oscar Fontenla-Romero. "Distributed One-Class Support Vector Machine." International Journal of Neural Systems 25, no. 07 (August 27, 2015): 1550029. http://dx.doi.org/10.1142/s012906571550029x.
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This paper presents a novel distributed one-class classification approach based on an extension of the ν-SVM method, thus permitting its application to Big Data data sets. In our method we will consider several one-class classifiers, each one determined using a given local data partition on a processor, and the goal is to find a global model. The cornerstone of this method is the novel mathematical formulation that makes the optimization problem separable whilst avoiding some data points considered as outliers in the final solution. This is particularly interesting and important because the decision region generated by the method will be unaffected by the position of the outliers and the form of the data will fit more precisely. Another interesting property is that, although built in parallel, the classifiers exchange data during learning in order to improve their individual specialization. Experimental results using different datasets demonstrate the good performance in accuracy of the decision regions of the proposed method in comparison with other well-known classifiers while saving training time due to its distributed nature.
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AUJOGUE, JEAN-BAPTISTE. "Pure point/continuous decomposition of translation-bounded measures and diffraction." Ergodic Theory and Dynamical Systems 40, no. 2 (July 10, 2018): 309–52. http://dx.doi.org/10.1017/etds.2018.38.
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In this work we consider translation-bounded measures over a locally compact Abelian group$\mathbb{G}$, with a particular interest in their so-called diffraction. Given such a measure$\unicode[STIX]{x1D714}$, its diffraction$\widehat{\unicode[STIX]{x1D6FE}}$is another measure on the Pontryagin dual$\widehat{\mathbb{G}}$, whose decomposition into the sum$\widehat{\unicode[STIX]{x1D6FE}}=\widehat{\unicode[STIX]{x1D6FE}}_{\text{p}}+\widehat{\unicode[STIX]{x1D6FE}}_{\text{c}}$of its atomic and continuous parts is central in diffraction theory. The problem we address here is whether the above decomposition of$\widehat{\unicode[STIX]{x1D6FE}}$lifts to$\unicode[STIX]{x1D714}$itself, that is to say, whether there exists a decomposition$\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{\text{p}}+\unicode[STIX]{x1D714}_{\text{c}}$, where$\unicode[STIX]{x1D714}_{\text{p}}$and$\unicode[STIX]{x1D714}_{\text{c}}$are translation-bounded measures having diffraction$\widehat{\unicode[STIX]{x1D6FE}}_{\text{p}}$and$\widehat{\unicode[STIX]{x1D6FE}}_{\text{c}}$, respectively. Our main result here is the almost sure existence, in a sense to be made precise, of such a decomposition. It will also be proved that a certain uniqueness property holds for the above decomposition. Next, we will be interested in the situation where translation-bounded measures are weighted Meyer sets. In this context, it will be shown that the decomposition, whether it exists, also consists of weighted Meyer sets. We complete this work by discussing a natural generalization of the considered problem.
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Peng, Yong, Wanzeng Kong, Feiwei Qin, and Feiping Nie. "Manifold Adaptive Kernelized Low-Rank Representation for Semisupervised Image Classification." Complexity 2018 (2018): 1–11. http://dx.doi.org/10.1155/2018/2857594.
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Constructing a powerful graph that can effectively depict the intrinsic connection of data points is the critical step to make the graph-based semisupervised learning algorithms achieve promising performance. Among popular graph construction algorithms, low-rank representation (LRR) is a very competitive one that can simultaneously explore the global structure of data and recover the data from noisy environments. Therefore, the learned low-rank coefficient matrix in LRR can be used to construct the data affinity matrix. Consider the existing problems such as the following: (1) the essentially linear property of LRR makes it not appropriate to process the possible nonlinear structure of data and (2) learning performance can be greatly enhanced by exploring the structure information of data; we propose a new manifold kernelized low-rank representation (MKLRR) model that can perform LRR in the data manifold adaptive kernel space. Specifically, the manifold structure can be incorporated into the kernel space by using graph Laplacian and thus the underlying geometry of data is reflected by the wrapped kernel space. Experimental results of semisupervised image classification tasks show the effectiveness of MKLRR. For example, MKLRR can, respectively, obtain 96.13%, 98.09%, and 96.08% accuracies on ORL, Extended Yale B, and PIE data sets when given 5, 20, and 20 labeled face images per subject.
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Jeng, Yih-Nen, P. G. Huang, and You-Chi Cheng. "Decomposition of one-dimensional waveform using iterative Gaussian diffusive filtering methods." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 464, no. 2095 (March 13, 2008): 1673–95. http://dx.doi.org/10.1098/rspa.2007.0031.
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The Gaussian smoothing method is shown to have a wide transition zone around the cut-off frequency selected to filter a given dataset. We proposed two iterative Gaussian smoothing methods to tighten the transition zone: one being approximately diffusive and the other being strictly diffusive. The first version smoothes repeatedly the remaining high-frequency parts and the second version requires an additional step to further smooth the resulting smoothed response in each of the smoothing operation. Based on the choice of the criterion for accuracy, the smoothing factor and the number of iterations are derived for an infinite data length in both methods. By contrast, for a finite-length data string, results of the interior points (sufficiently away from the two endpoints) obtained by both methods can be shown to exhibit an approximate diffusive property. The upper bound of the distance affected by the error propagation inward due to the lack of data beyond the two ends is numerically estimated. Numerical experiments also show that results of employing the iterative Gaussian smoothing method are almost the same as those obtained by the strict diffusive version, except that the error propagation distance induced by the latter is slightly deeper than that of the former. The proposed method has been successfully applied to decompose the wave formation of a number of test cases including two sets of real experimental data.
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TAYLOR, M. J., and N. PEAKE. "The long-time impulse response of compressible swept-wing boundary layers." Journal of Fluid Mechanics 379 (January 25, 1999): 333–50. http://dx.doi.org/10.1017/s0022112098003516.
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Following the investigation of the long-time limit of the impulse response of an incompressible swept boundary layer (Taylor & Peake 1998), we now consider the corresponding behaviour of two representative sets of compressible swept-wing profiles, one set in subsonic flow and the other in supersonic flow. The key feature of the incompressible analysis was the occurrence of modal pinch points in the cross-flow wavenumber plane, and in this paper the existence of such pinches over a wide portion of space in high-speed flow is confirmed. We also show that close to the attachment line, no unstable pinches in the chordwise wavenumber plane can be found for these realistic wing profiles, contrary to predictions made previously for incompressible flow with simple Falker–Skan–Cooke profiles (Lingwood 1997). A method for searching for absolute instabilities is described and applied to the compressible boundary layers, and we are able to confirm that these profiles are not absolutely unstable. The pinch point property of the compressible boundary layers is used here to predict the maximum local growth rate achieved by waves in a wavepacket in any given direction. By determining the direction of maximum amplification, we are able to derive upper bounds on the amplification rate of the wavepacket over the wing, and initial comparison with experimental data shows that the resulting N-factors are more consistent than might be expected from existing conventional methods.
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Todorcevic, Stevo. "Some compactifications of the integers." Mathematical Proceedings of the Cambridge Philosophical Society 112, no. 2 (September 1992): 247–54. http://dx.doi.org/10.1017/s0305004100070936.
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Suppose that K = {K0, K1} is a partition of some finite combinatorial power [S]r of a set S. Let X(K) be the compact subspace of the Tychonoff cube [0, 1]S consisting of those functions ƒ whose supports Sƒ = {ƒ > 0} are 0-homogeneous i.e., [Sƒ]r ⊆ K0. It can be said that almost every example of a space witnessing the distinction between various chain conditions is a variation of X(K) (see [2]). This comes from the fact that the subject is closely related to the subject of Partition Calculus. (See [11] for an explanation of this point.) To have examples which are even more topologically interesting one usually tries to make them small, i.e., as closely related to the unit interval as possible. The operation X(K) has a considerable drawback in that respect. For example, if we want X(K) to be ccc this becomes equivalent to the fact that the poset of all finite 0-homogeneous sets is ccc which amounts to the fact that K1 must be very small. Hence K0 is big in the sense that there exist large 0-homogeneous sets. This usually results in X(K) having large size and having points of large character. One attempt to solve this problem was given by van Douwen in [5] by going to the subspace Xm(K) of X(K) consisting of those ƒ for which Sƒ are maximal 0-homogeneous subsets of S. Unfortunately, while Xm(K) usually does have small character it is almost never compact. This might have been the reason for his question ([2], p. 207) whether the Continuum Hypothesis implies that the class of all first countable compacta distinguishes the standard chain conditions that lie between ‘ccc’ and ‘separable’. In this paper we solve this problem completely. Moreover, we shall not go beyond the usual axioms of set theory in constructing the examples. The sequence of examples will start with a compact space of small character whose chain condition is not productive and it will end with a compact space of size c and small character which is ccc in a strong sense but which fails to have calibre θ for some regular uncountable cardinal θ, i.e., it fails to have the property of Shanin. Note that one cannot go further and show that, for example, compact spaces of small character distinguish between ‘the property of Shanin’ and ‘separable’. This follows from an old result of Efimov [3] that, under CH, first-countable spaces of calibre ℵ1 are separable. The combinatorics behind our examples have been developed in a series of papers that deal with the subject of forcing axioms in general and Martin's axiom in particular ([10, 11, 12, 13, 14]). Martin's axiom was originally invented in connection with the Souslin Problem, i.e., to show that certain compact ccc spaces must be separable (see [8]). A result of the aforementioned study of MA showed that this axiom is nothing more than the statement that all compact ccc spaces of π-weight < c must be separable (see [12]). An analysis of the fact that MA implies SH, due to Hajnal and Juhasz[6] (see also [4], §43 for a definite result in that direction due to Shapirovskii), revealed that MA implies that every compact ccc space X with the property χ(X)+ < c must be separable. This result explains why the compact ccc non-separable spaces X that we construct in this paper have the property that χ(x, X) < c for all x m X rather than the stronger property χ(X) < c or even χ(X) = ℵ0. Note that our examples show, answering a question from [6], that the Hajnal–Juhasz result is sharp in the sense that the assumption χ(X)+ < c cannot be weakened to χ(X) < c. The first example to show this was constructed by Bell [1] using a consequence of MA rather than just ZFC for its construction. Another feature of our examples is that they all are remainders of certain compactifications of the integers. This is of independent interest in certain constructions of weak P-points in compact F-spaces. An explanation of this can be found in [1] and [9] where the first examples of ccc non-separable remainders were constructed and used.
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Baccelli, François, and Bartłomiej Błaszczyszyn. "On a coverage process ranging from the Boolean model to the Poisson–Voronoi tessellation with applications to wireless communications." Advances in Applied Probability 33, no. 2 (June 2001): 293–323. http://dx.doi.org/10.1017/s0001867800010806.
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We define and analyse a random coverage process of the d-dimensional Euclidean space which allows us to describe a continuous spectrum that ranges from the Boolean model to the Poisson–Voronoi tessellation to the Johnson–Mehl model. As for the Boolean model, the minimal stochastic setting consists of a Poisson point process on this Euclidean space and a sequence of real valued random variables considered as marks of this point process. In this coverage process, the cell attached to a point is defined as the region of the space where the effect of the mark of this point exceeds an affine function of the cumulative effect of all marks. This cumulative effect is defined as the shot-noise process associated with the marked point process. In addition to analysing and visualizing this spectrum, we study various basic properties of the coverage process such as the probability that a point or a pair of points be covered by a typical cell. We also determine the distribution of the number of cells which cover a given point, and show how to provide deterministic bounds on this number. Finally, we also analyse convergence properties of the coverage process using the framework of closed sets, and its differentiability properties using perturbation analysis. Our results require a pathwise continuity property for the shot-noise process for which we provide sufficient conditions. The model in question stems from wireless communications where several antennas share the same (or different but interfering) channel(s). In this case, the area where the signal of a given antenna can be received is the area where the signal to interference ratio is large enough. We describe this class of problems in detail in the paper. The results obtained allow us to compute quantities of practical interest within this setting: for instance the outage probability is obtained as the complement of the volume fraction; the law of the number of cells covering a point allows us to characterize handover strategies, and so on.
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Uehara, Kiyohiko, and Kaoru Hirota. "Noise Reduction with Fuzzy Inference Based on Generalized Mean and Singleton Input–Output Rules: Toward Fuzzy Rule Learning in a Unified Inference Platform." Journal of Advanced Computational Intelligence and Intelligent Informatics 23, no. 6 (November 20, 2019): 1027–43. http://dx.doi.org/10.20965/jaciii.2019.p1027.
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A method is proposed for reducing noise in learning data based on fuzzy inference methods called α-GEMII (α-level-set and generalized-mean-based inference with the proof of two-sided symmetry of consequences) and α-GEMINAS (α-level-set and generalized-mean-based inference with fuzzy rule interpolation at an infinite number of activating points). It is particularly effective for reducing noise in randomly sampled data given by singleton input–output pairs for fuzzy rule optimization. In the proposed method, α-GEMII and α-GEMINAS are performed with singleton input–output rules and facts defined by fuzzy sets (non-singletons). The rules are initially set by directly using the input–output pairs of the learning data. They are arranged with the facts and consequences deduced by α-GEMII and α-GEMINAS. This process reduces noise to some extent and transforms the randomly sampled data into regularly sampled data for iteratively reducing noise at a later stage. The width of the regular sampling interval can be determined with tolerance so as to satisfy application-specific requirements. Then, the singleton input–output rules are updated with consequences obtained in iteratively performing α-GEMINAS for noise reduction. The noise reduction in each iteration is a deterministic process, and thus the proposed method is expected to improve the noise robustness in fuzzy rule optimization, relying less on trial-and-error-based progress. Simulation results demonstrate that noise is properly reduced in each iteration and the deviation in the learning data is suppressed considerably.
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Zheng, Guoxiong, Martin Mergili, Adam Emmer, Simon Allen, Anming Bao, Hao Guo, and Markus Stoffel. "The 2020 glacial lake outburst flood at Jinwuco, Tibet: causes, impacts, and implications for hazard and risk assessment." Cryosphere 15, no. 7 (July 9, 2021): 3159–80. http://dx.doi.org/10.5194/tc-15-3159-2021.
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Abstract. We analyze and reconstruct a recent glacial lake outburst flood (GLOF) process chain on 26 June 2020, involving the moraine-dammed proglacial lake – Jinwuco (30.356∘ N, 93.631∘ E) in eastern Nyainqentanglha, Tibet, China. Satellite images reveal that from 1965 to 2020, the surface area of Jinwuco has expanded by 0.2 km2 (+56 %) to 0.56 km2 and subsequently decreased to 0.26 km2 (−54 %) after the GLOF. Estimates based on topographic reconstruction and sets of published empirical relationships indicate that the GLOF had a volume of 10 million cubic meters, an average breach time of 0.62 h, and an average peak discharge of 5602 m3/s at the dam. Based on pre- and post-event high-resolution satellite scenes, we identified a large debris landslide originating from western lateral moraine that was most likely triggered by extremely heavy, south-Asian-monsoon-associated rainfall in June 2020. We back-calculate part of the GLOF process chain, using the GIS-based open-source numerical simulation tool r.avaflow. Two scenarios are considered, assuming a debris-landslide-induced impact wave with overtopping and resulting retrogressive erosion of the moraine dam (Scenario A), as well as retrogressive erosion without a major impact wave (Scenario B). Both scenarios are in line with empirically derived ranges of peak discharge and breach time. The breaching process is characterized by a slower onset and a resulting delay in Scenario B compared to Scenario A. Comparison of the simulation results with field evidence points towards Scenario B, with a peak discharge of 4600 m3/s. There were no casualties from this GLOF, but it caused severe destruction of infrastructure (e.g., roads and bridges) and property losses in downstream areas. Given the clear role of continued glacial retreat in destabilizing the adjacent lateral moraine slopes and directly enabling the landslide to deposit into the expanding lake body, the GLOF process chain can be plausibly linked to anthropogenic climate change, while downstream consequences have been enhanced by the development of infrastructure on exposed flood plains. Such process chains could become more frequent under a warmer and wetter future climate, calling for comprehensive and forward-looking risk reduction planning.
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Perrett, David I., and Mike W. Oram. "Visual Recognition Based on Temporal Cortex Cells: Viewer-Centred Processing of Pattern Configuration." Zeitschrift für Naturforschung C 53, no. 7-8 (August 1, 1998): 518–41. http://dx.doi.org/10.1515/znc-1998-7-807.
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Abstract A model of recognition is described based on cell properties in the ventral cortical stream of visual processing in the primate brain. At a critical intermediate stage in this system, ‘Elaborate’ feature sensitive cells respond selectively to visual features in a way that depends on size (± 1 octave), orientation (± 4 5 °) but does not depend on position within central vision (± 5 °). These features are simple conjunctions of 2-D elements (e.g. a horizontal dark area above a dark smoothly convex area). They can arise either as elements of an object’s surface pattern or as a 3-D component bounded by an object’s external contour. By requiring a combination of several such features without regard to their position within the central region of the visual image, ‘Pattern’ sensitive cells at higher levels can exhibit selectivity for complex configurations that typify objects seen under particular viewing conditions. Given that input features to such Pattern sensitive cells are specified in approximate size and orientation, initial cellular ‘representations’ of the visual appearance of object type (or object example) are also selective for orientation and size. At this level, sensitivity to object view (± 6 0 °) arises because visual features disappear as objects are rotated in perspective. Processing is thus viewer-centred and the neurones only respond to objects seen from particular viewing conditions or ‘object instances’. Combined sensitivity to multiple features (conjunctions of elements) independent of their position, establishes selectivity for the configurations of object parts (from one view) because rearranged configurations of the same parts yield images lacking some of the 2-D visual features present in the normal configuration. Different neural populations appear to be selectively tuned to particular components of the same biological object (e.g. face, eyes, hands, legs), perhaps because the independent articulation of these components gives rise to correlated activity in different sets of input visual features. Generalisation over viewing conditions for a given object can be established by hierarchically pooling outputs of view-condition specific cells with pooling operations dependent on the continuity in experience across viewing conditions. Different object parts are seen together and different views are seen in succession when the observer walks around the object. The view specific coding that characterises the selectivity of cells in the temporal lobe can be seen as a natural consequence of selective experience of objects from particular vantage points. View specific coding for the face and body also has great utility in understanding complex social signals, a property that may not be feasible with object-centred processing.
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Bronshtein, E. M. "On compact convex sets with given extreme points." Siberian Mathematical Journal 36, no. 1 (January 1995): 17–23. http://dx.doi.org/10.1007/bf02113915.
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OPROCHA, PIOTR. "Shadowing, thick sets and the Ramsey property." Ergodic Theory and Dynamical Systems 36, no. 5 (January 9, 2015): 1582–95. http://dx.doi.org/10.1017/etds.2014.130.
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We provide a full characterization of relations between the shadowing property and the thick shadowing property. We prove that they are equivalent properties for non-wandering systems, the thick shadowing property is always a consequence of the shadowing property, and the thick shadowing property on the chain-recurrent set and the thick shadowing property are the same properties. We also provide a full characterization of the cases when for any family ${\mathcal{F}}$ with the Ramsey property an arbitrary sequence of points can be ${\it\varepsilon}$-traced over a set from ${\mathcal{F}}$.
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Paszkiewicz, Adam, and Elżbieta Wagner-Bojakowska. "Fubini Property for Microscopic Sets." Tatra Mountains Mathematical Publications 65, no. 1 (March 1, 2016): 143–49. http://dx.doi.org/10.1515/tmmp-2016-0012.
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Abstract In 2000, I. Recław and P. Zakrzewski introduced the notion of Fubini Property for the pair (I,J) of two σ-ideals in the following way. Let I and J be two σ-ideals on Polish spaces X and Y, respectively. The pair (I,J) has the Fubini Property (FP) if for every Borel subset B of X×Y such that all its vertical sections Bx = {y ∈ Y : (x, y) ∈ B} are in J, then the set of all y ∈ Y, for which horizontal section By = {x ∈ X : (x, y) ∈ B} does not belong to I, is a set from J, i.e., {y ∈ Y : By ∉ I} ∈ J. The Fubini property for the σ-ideal M of microscopic sets is considered and the proof that the pair (M,M) does not satisfy (FP) is given.
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Kaczor, Wieslawa. "Fixed points of asymptotically regular nonexpansive mappings on nonconvex sets." Abstract and Applied Analysis 2003, no. 2 (2003): 83–91. http://dx.doi.org/10.1155/s1085337503205054.
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It is shown that ifXis a Banach space andCis a union of finitely many nonempty, pairwise disjoint, closed, and connected subsets{Ci:1≤i≤n }ofX, and eachCihas the fixed-point property (FPP) for asymptotically regular nonexpansive mappings, then any asymptotically regular nonexpansive self-mapping ofChas a fixed point. We also generalize the Goebel-Schöneberg theorem to some Banach spaces with Opial's property.
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Et.al, Arul Ravi S. "Existence Of Best Proximity Points On Geometrical Properties Of Proximal Sets." Turkish Journal of Computer and Mathematics Education (TURCOMAT) 12, no. 6 (April 10, 2021): 327–30. http://dx.doi.org/10.17762/turcomat.v12i6.1391.
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Halpern, Richard P., David Hobby, and Donald M. Silberger. "Maximal coplanar sets of intersection points." Bulletin of the Australian Mathematical Society 42, no. 1 (August 1990): 41–56. http://dx.doi.org/10.1017/s0004972700028136.
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Let F be any set of five points in R3 so situated that no four of the points are coplanar, and that the line xy through any two x and y of the points has a unique intersection point xy* with the plane determined by the other three. Let F^ denote the family of all such xy*. Let S(F) denote the set of all X ⊆ F^ which are maximal with respect to the property that X is a subset of a plane in R3. For k > 2 an integer, let S(k; F) denote the family of all k-membered elements in S(F).A family 𝒟 of sets is said to be uniformly deep of depth d if and only if for every x ∈ ∪ 𝒟 there are exactly d distinct 𝒜 ∈ 𝒟 for which x ∈ 𝒜.We establish the following result, and extend our ideas to general Euclidean spaces.
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Elias, Juan, and Rosa M. Miró-Roig. "A strange property of the sets of points in uniform position." Communications in Algebra 21, no. 5 (January 1993): 1577–85. http://dx.doi.org/10.1080/00927879308824638.
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Moody, Robert V. "Uniform Distribution in Model Sets." Canadian Mathematical Bulletin 45, no. 1 (March 1, 2002): 123–30. http://dx.doi.org/10.4153/cmb-2002-015-3.
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AbstractWe give a new measure-theoretical proof of the uniform distribution property of points in model sets (cut and project sets). Each model set comes as a member of a family of related model sets, obtained by joint translation in its ambient (the ‘physical’) space and its internal space. We prove, assuming only that the window defining themodel set ismeasurable with compact closure, that almost surely the distribution of points in any model set from such a family is uniform in the sense of Weyl, and almost surely the model set is pure point diffractive.
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Migliore, Juan C. "The Geometry of the Weak Lefschetz Property and Level Sets of Points." Canadian Journal of Mathematics 60, no. 2 (April 1, 2008): 391–411. http://dx.doi.org/10.4153/cjm-2008-019-2.
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AbstractIn a recent paper, F. Zanello showed that level Artinian algebras in 3 variables can fail to have the Weak Lefschetz Property (WLP), and can even fail to have unimodal Hilbert function. We show that the same is true for the Artinian reduction of reduced, level sets of points in projective 3-space. Our main goal is to begin an understanding of how the geometry of a set of points can prevent its Artinian reduction from having WLP, which in itself is a very algebraic notion. More precisely, we produce level sets of points whose Artinian reductions have socle types 3 and 4 and arbitrary socle degree ≥ 12 (in the worst case), but fail to have WLP. We also produce a level set of points whose Artinian reduction fails to have unimodal Hilbert function; our example is based on Zanello's example. Finally, we show that a level set of points can have Artinian reduction that has WLP but fails to have the Strong Lefschetz Property. While our constructions are all based on basic double G-linkage, the implementations use very different methods.
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Whitfield, J. H. M., and V. Zizler. "Mazur's intersection property of balls for compact convex sets." Bulletin of the Australian Mathematical Society 35, no. 2 (April 1987): 267–74. http://dx.doi.org/10.1017/s0004972700013228.
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We show that every compact convex set in a Banach space X is an intersection of balls provided the cone generated by the set of all extreme points of the dual unit ball of X* is dense in X* in the topology of uniform convergence on compact sets in X. This allows us to renorm every Banach space with transfinite Schauder basis by a norm which shares the mentioned intersection property.
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MILIĆEVIĆ, LUKA. "Sets in Almost General Position." Combinatorics, Probability and Computing 26, no. 5 (April 18, 2017): 720–45. http://dx.doi.org/10.1017/s0963548317000098.
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Erdős asked the following question: given n points in the plane in almost general position (no four collinear), how large a set can we guarantee to find that is in general position (no three collinear)? Füredi constructed a set of n points in almost general position with no more than o(n) points in general position. Cardinal, Tóth and Wood extended this result to ℝ3, finding sets of n points with no five in a plane whose subsets with no four points in a plane have size o(n), and asked the question for higher dimensions: for given n, is it still true that the largest subset in general position we can guarantee to find has size o(n)? We answer their question for all d and derive improved bounds for certain dimensions.
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SHAHBEYK, SHOKOUH, and MAJID SOLEIMANI-DAMANEH. "Limiting proper minimal points of nonconvex sets in finite-dimensional spaces." Carpathian Journal of Mathematics 35, no. 3 (2019): 379–84. http://dx.doi.org/10.37193/cjm.2019.03.12.
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In this paper, limiting proper minimal points of nonconvex sets in Euclidean finite-dimensional spaces are investigated. The relationships between these minimal points and Borwein, Benson, and Henig proper minimal points, under appropriate assumptions, are established. Furthermore, a density property is derived and a linear characterization of limiting proper minimal points is provided.
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Hejduk, Jacek. "On the regularity of topologies in the family of sets having the Baire property." Filomat 27, no. 7 (2013): 1291–95. http://dx.doi.org/10.2298/fil1307291h.
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The paper concerns the topologies introduced in the family of sets having the Baire property in a topological space (X, ?) and in the family generated by the sets having the Baire property and given a proper ?-ideal containing ? -meager sets. The regularity property of such topologies is investigated.
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Migliore, Juan C. "The Geometry of the Weak Lefschetz Property and Level Sets of Points." Journal canadien de mathématiques 60, no. 2 (2008): 391. http://dx.doi.org/10.4153/cjm-2009-019-1.
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Csörnyei, Marianna, David Preiss, and Jaroslav Tišer. "Lipschitz functions with unexpectedly large sets of nondifferentiability points." Abstract and Applied Analysis 2005, no. 4 (2005): 361–73. http://dx.doi.org/10.1155/aaa.2005.361.
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It is known that everyGδsubsetEof the plane containing a dense set of lines, even if it has measure zero, has the property that every real-valued Lipschitz function onℝ2has a point of differentiability inE. Here we show that the set of points of differentiability of Lipschitz functions inside such sets may be surprisingly tiny: we construct aGδsetE⊂ℝ2containing a dense set of lines for which there is a pair of real-valued Lipschitz functions onℝ2having no common point of differentiability inE, and there is a real-valued Lipschitz function onℝ2whose set of points of differentiability inEis uniformly purely unrectifiable.
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Matyja, Janusz. "Sets of primitive words given by fixed points of mappings." International Journal of Computer Mathematics 76, no. 4 (January 2001): 435–46. http://dx.doi.org/10.1080/00207160108805037.
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BRLEK, S., G. LABELLE, and A. LACASSE. "PROPERTIES OF THE CONTOUR PATH OF DISCRETE SETS." International Journal of Foundations of Computer Science 17, no. 03 (June 2006): 543–56. http://dx.doi.org/10.1142/s012905410600398x.
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We consider paths in the square lattice and use a valuation called the winding number in order to exhibit some combinatorial properties on these paths. As a corollary, we obtain a characteristic property of non-crossing closed paths, generalizing in this way a result of Daurat and Nivat (2003) on the boundary properties of polyominoes concerning salient and reentrant points. Moreover we obtain a similar result for hexagonal lattices and show that there is no other regular lattice having that property.
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Zhang, Zhen-Liang. "On the exceptional sets in Sylvester continued fraction expansion." International Journal of Number Theory 11, no. 08 (November 5, 2015): 2369–80. http://dx.doi.org/10.1142/s1793042115501092.
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In this paper, we study some exceptional sets of points whose partial quotients in their Sylvester continued fraction expansions obey some restrictions. More precisely, for α ≥ 1 we prove that the Hausdorff dimension of the set [Formula: see text] is one. In addition, we find that the points whose partial quotients in their Sylvester continued fraction expansions obey some property of divisibility have the same Engel continued fraction expansion and Sylvester continued fraction expansion. And we establish that the set of points whose Engel continued fraction expansion and Sylvester continued fraction expansion coincide is uncountable.
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Kamińska, Anna, Katarzyna Nowakowska, and Małgorzata Turowska. "On sets of points of approximate continuity and ϱ-upper continuity." Mathematica Slovaca 70, no. 2 (April 28, 2020): 305–18. http://dx.doi.org/10.1515/ms-2017-0353.
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Abstract In the paper some properties of sets of points of approximate continuity and ϱ-upper continuity are presented. We will show that for every Lebesgue measurable set E ⊂ ℝ there exists a function f : ℝ → ℝ which is approximately (ϱ-upper) continuous exactly at points from E. We also study properties of sets of points at which real function has Denjoy property. Some other related topics are discussed.