Infinite Set Research Articles

On the Emergent Dynamics of the Infinite Set of Kuramoto Oscillators

On the Emergent Dynamics of the Infinite Set of Kuramoto Oscillators

Physically motivated moment-tensor decomposition for mining-induced seismicity

SUMMARY Compared to existing schemes, the decomposition of moment tensors for mining-induced seismic events into closing-crack and double-couple components has the advantage that each can be interpreted in terms of a physical source process (excavation convergence and slip/shear, respectively). Obviously, not every moment tensor permits such a decomposition, and we translate existing bounds on those that do to the Hudson source-type plot. For moment tensors falling within these bounds, it has previously been noted that there will be a infinite set of possible decompositions in general. We derive an implicit equation defining this set and suggest physically motivated criteria that can be used to select a single decomposition from it. Furthermore, for moment tensors falling outside the source-type bounds, we present a simple geometric scheme for determining the closest moment tensor within them (which can then be decomposed). To demonstrate the methods developed in this paper, we apply them to two catalogues of mining-induced seismicity.

Exploring congruent diversification histories with flexibility and parsimony

Abstract Using phylogenies of present‐day species to estimate diversification rate trajectories—speciation and extinction rates over time—is a challenging task due to non‐identifiability issues. Given a phylogeny, there exists an infinite set of trajectories that result in the same likelihood; this set has been coined a congruence class. Previous work has developed approaches for sampling trajectories within a given congruence class, with the aim to assess the extent to which congruent scenarios can vary from one another. Based on this sampling approach, it has been suggested that rapid changes in speciation or extinction rates are conserved across the class. Reaching such conclusions requires to sample the broadest possible set of distinct trajectories. We introduce a new method for exploring congruence classes that we implement in the R package CRABS. Whereas existing methods constrain either the speciation rate or the extinction rate trajectory, ours provides more flexibility by sampling congruent speciation and extinction rate trajectories simultaneously. This allows covering a more representative set of distinct diversification rate trajectories. We also implement a filtering step that allows selecting the most parsimonious trajectories within a class. We demonstrate the utility of our new sampling strategy using a simulated scenario. Next, we apply our approach to the study of mammalian diversification history. We show that rapid changes in speciation and extinction rates need not be conserved across a congruence class, but that selecting the most parsimonious trajectories shrinks the class to concordant scenarios. Our approach opens new avenues both to truly explore the myriad of potential diversification histories consistent with a given phylogeny, embracing the uncertainty inherent to phylogenetic diversification models, and to select among these different histories. This should help refining our inference of diversification trajectories from extant data.

Elusive properties of infinite graphs

AbstractA graph property is said to be elusive (or evasive) if every algorithm testing this property by asking questions of the form “is there an edge between vertices and ?” requires, in the worst case, to ask about all pairs of vertices. The unsettled Aanderaa–Karp–Rosenberg conjecture is that every nontrivial monotone graph property is elusive for any finite vertex set. We show that the situation is completely different for infinite vertex sets: the monotone graph properties “every vertex has degree at least ” and “every connected component has size at least ,” where and are natural numbers, are not elusive for infinite vertex sets, but the monotone graph property “the graph contains a cycle” is elusive for arbitrary vertex set. On the other hand, we also prove that every algorithm testing some natural monotone graph properties, for example, “every vertex has degree at least ” or “connected” on the vertex set should check “lots of edges,” more precisely, all the edges of an infinite complete subgraph.

Nonlinear instability and scalar clouds of spherical exotic compact objects in scalar-Gauss-Bonnet theory

In this work, we present a new type of scalar clouds supported by spherically symmetric horizonless compact objects in the scalar-Gauss–Bonnet theory. Unlike the previous spontaneous scalarization that is triggered by the tachyonic instability, our scalarization arises from a nonlinear instability that is non-spontaneous. We explore two types of boundary conditions for the scalar field at the surface of the compact objects and find an infinite countable set of scalar clouds characterized by the number of nodes for both cases. Our study demonstrates that boundary conditions have a significant impact on the formation of scalar clouds. Specifically, for the Dirichlet boundary condition, scalarization is more likely to occur for compact objects with medium radii and becomes harder for ultra-compact and large ones. Conversely, for the Robin boundary condition, scalarization is easier for more compact objects.

On the functional equation f(x+y)=g(xy)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {f}(\\hbox {x}+\\hbox {y})=\\hbox {g}(\\hbox {xy})$$\\end{document}

The functional equation f(x+y)=g(xy)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(x+y)=g(xy)$$\\end{document} is investigated with unknown functions f:A+A→Y\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:A+A\\rightarrow Y$$\\end{document}, g:A·A→Y\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g:A\\cdot A\\rightarrow Y$$\\end{document} in the following cases: A:=α,β⊆F+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A:=\\left]\\alpha ,\\beta \\right[\\subseteq \\mathbb {F}_{+}$$\\end{document} where F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}$$\\end{document} is an Archimedean ordered field; A is the set of all positive integers; A is the set of all positive dyadic rational numbers. The set Y is an arbitrarily fixed (infinite) set. The main result of the paper shows that there exists a set A⊆R+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A\\subseteq \\mathbb {R}_{+}$$\\end{document} that is closed under addition and multiplication and there exist functions f, g:A→Y\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g:A\\rightarrow Y$$\\end{document} which satisfy the equation f(x+y)=g(xy)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(x+y)=g(xy)$$\\end{document} for all x,y∈A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x,y\\in A$$\\end{document} such that the range of the function f is infinite. Finally, some application of the above results is also given.

Cantorův diagonální důkaz

Cantor's diagonal proof is sig­nificant both because the central method of proof used in it has been subsequently applied in a number of other proofs, and because it is considered to confirm the existence of infinite sets whose size fun­ damentally and by an order of magnitude exceeds the size of the "classical" infinite set represented by all natural numbers, while their size can theoretically exceed every conceivable limit. Although Can­tor's proof is generally accepted by the scientific community, some experts are somewhat reserved about it. The aim of this paper is to present Cantor's proof in an accessible way, while pointing out its (hidden) assumptions and possible problematic points, and pointing out that some of its underlying assumptions are not indisputable mathematical truths, but rather postulated propositions that may or may not be accepted.

The Complex Impedance of the Hemispherical Ground Electrode: An Open Analytical Problem

The hemispherical ground electrode is a basic electrode whose analysis appears in many textbooks on electromagnetics in chapters dedicated to steady currents. Considering a soil with a given resistivity and an electrode with a given perimeter, the electrode DC resistance is simply calculated from the ratio resistivity/perimeter. Strangely, the generalization of this result to AC regimes is missing. The issue of the frequency-domain impedance of the hemispherical ground electrode has been avoided in the literature despite its trivial geometry. But the problem is indeed not easy; electromagnetic field calculation involves Legendre and Bessel functions; the application of boundary conditions involves an infinite set of points, and some integrals involved need to be calculated recursively. We analyzed the math and physics of the problem but failed to find a closed-form solution. This article with “negative results” can, however, be useful; on one hand it may prevent researchers from wasting their time following the same steps, and, on the other hand, it may attract the interest of new researchers to the subject, ultimately, accelerating its analytical solution (if the solution exists).

Infinite orthogonal exponentials for a class of Moran measures

For a sequence [Formula: see text] of expanding matrices with [Formula: see text] and a sequence [Formula: see text] of finite digit sets with [Formula: see text], the Moran measure [Formula: see text] is defined by the infinite convolution [Formula: see text] and the convergence is in the weak sense. Under some additional assumptions, we show that [Formula: see text] contains an infinite orthogonal set of exponential functions if and only if there exists an infinite subsequence [Formula: see text] of [Formula: see text] such that [Formula: see text] for any [Formula: see text], where [Formula: see text]. This extends the results of [J. L. Li, A necessary and sufficient condition for the finite [Formula: see text]-orthogonality, Sci. China Math. 58 (2015) 2541–2548].

Computation of the Hausdorff Distance between Two Compact Convex Sets.

The Hausdorff distance between two closed sets has important theoretical and practical applications. Yet apart from finite point clouds, there appear to be no generic algorithms for computing this quantity. Because many infinite sets are defined by algebraic equalities and inequalities, this a huge gap. The current paper constructs Frank-Wolfe and projected gradient ascent algorithms for computing the Hausdorff distance between two compact convex sets. Although these algorithms are guaranteed to go uphill, they can become trapped by local maxima. To avoid this defect, we investigate a homotopy method that gradually deforms two balls into the two target sets. The Frank-Wolfe and projected gradient algorithms are tested on two pairs of compact convex sets, where: (1) is the box translated by and is the intersection of the unit ball and the non-negative orthant; and (2) is the probability simplex and is the unit ball translated by . For problem (2), we find the Hausdorff distance analytically. Projected gradient ascent is more reliable than the Frank-Wolfe algorithm and finds the exact solution of problem (2). Homotopy improves the performance of both algorithms when the exact solution is unknown or unattained.

Monodromie unipotente maximale, congruences “à la Lucas” et indépendance algébrique

Let f ( z ) f(z) be in 1 + z Q [ [ z ] ] 1+z\mathbb {Q}[[z]] and S \mathcal {S} be an infinite set of prime numbers such that, for all p ∈ S p\in \mathcal {S} , we can reduce f ( z ) f(z) modulo p p . We let f ( z ) ∣ p f(z)_{\mid p} denote the reduction of f ( z ) f(z) modulo p p . Generally, when f ( z ) f(z) is D-finite, f ∣ p ( z ) f_{\mid p}(z) is algebraic over F p ( z ) \mathbb {F}_p(z) . It turns out that if f ∣ p ( z ) f_{\mid p}(z) is a solution of a polynomial of the form X − A p ( z ) X p l X-A_p(z)X^{p^l} , we can use this type of equations to obtain results of transcendence and algebraic independence over Q ( z ) \mathbb {Q}(z) . In the present paper, we look for conditions on the differential operators annihilating f ( z ) f(z) to guarantee the existence of these particular equations. Suppose that f ( z ) f(z) is solution of a differential operator H ∈ Q ( z ) [ d / d z ] \mathcal {H}\in \mathbb {Q}(z)[d/dz] having a strong Frobenius structure for all p ∈ S p\in \mathcal {S} and we also suppose that f ( z ) f(z) annihilates a Fuchsian differential operator D ∈ Q ( z ) [ d / d z ] \mathcal {D}\in \mathbb {Q}(z)[d/dz] such that zero is a regular singular point of D \mathcal {D} and the exponents of D \mathcal {D} at zero are equal to zero. Our main result states that, for almost every prime p ∈ S p\in \mathcal {S} , f ∣ p ( z ) f_{\mid p}(z) is solution of a polynomial of the form X − A p ( z ) X p l X-A_p(z)X^{p^l} , where A p ( z ) A_p(z) is a rational function with coefficients in F p \mathbb {F}_p of height less than or equal to C p 2 l Cp^{2l} with C C a positive constant that does not depend on p p . We also study the algebraic independence of these power series over Q ( z ) \mathbb {Q}(z) .

Almost Everywhere Convergence Questions of Series of Translates of Non-Negative Functions

This survey paper is based on a talk given at the 44th Summer Symposium in Real Analysis in Paris. This line of research was initiated by a question of Haight and Weizsäker concerning almost everywhere convergence properties of series of the form $\sum_{n=1}^{{\infty}}f(nx)$. A more general, additive version of this problem is the following: Suppose $\Lambda$ is a discrete infinite set of nonnegative real numbers. We say that $ {\Lambda}$ is of type 1 if the series $s(x)=\sum_{\lambda\in\Lambda}f(x+\lambda)$ satisfies a zero-one law. This means that for any non-negative measurable $f: \mathbb R\to [0,+ {\infty})$ either the convergence set $C(f, {\Lambda})=\{x: s(x)<+ {\infty} \}= \mathbb R$ modulo sets of Lebesgue zero, or its complement the divergence set $D(f, {\Lambda})=\{x: s(x)=+ {\infty} \}= \mathbb R$ modulo sets of measure zero. If $ {\Lambda}$ is not of type 1 we say that $ {\Lambda}$ is of type 2. The exact characterization of type $1$ and type $2$ sets is still not known. The part of the paper discussing results concerning this question is based on several joint papers written at the beginning with J-P. Kahane and D. Mauldin, later with B. Hanson, B. Maga and G. Vértesy. Apart from results from the above project we also cover historic background, other related results and open questions.

Can we recover an integral quadratic form by representing all its subforms?

Let o be the ring of integers of a number field. If f is a quadratic form over o and g is another quadratic form over o which represents all proper subforms of f, does g represent f? We show that if g is indefinite, then g indeed represents f. However, when f is positive definite and indecomposable, then there exists a g which represents all proper subforms of f but not f itself. Along the way we give a new characterization of positive definite decomposable quadratic forms over o and a number-field generalization of the finiteness theorem of representations of quadratic forms by quadratic forms over Z[13, Theorem 3.3] which asserts that given any infinite set S of classes of positive definite integral quadratic forms over o of a fixed rank, there exists a finite subset S0 of S with the property that a positive definite quadratic form over o represents all classes in S if and only if it represents all classes in S0.

Edge colorings avoiding patterns

We say that a pattern is a graph together with an edge coloring, and a pattern P=(H,c) occurs in some edge coloring c′ of G if c′, restricted to some subgraph of G isomorphic to H, is equal to c up to renaming the colors. Inspired by Matoušek’s visibility blocking problem, we study edge colorings that avoid certain patterns.We show that for every pattern P, such that the number of edges in P is at least the number of vertices in P plus the number of colors minus 2, there is a constant C such that every graph with maximum degree Δ admits an edge coloring with CΔ colors avoiding P; the same also holds for infinite sets of such patterns, provided that the number of patterns in the set grows at most exponentially.

Viewpoint Planning for Range Sensors Using Feature Cluster Constrained Spaces for Robot Vision Systems.

The efficient computation of viewpoints for solving vision tasks comprising multi-features (regions of interest) represents a common challenge that any robot vision system (RVS) using range sensors faces. The characterization of valid and robust viewpoints is even more complex within real applications that require the consideration of various system constraints and model uncertainties. Hence, to address some of the challenges, our previous work outlined the computation of valid viewpoints as a geometrical problem and proposed feature-based constrained spaces (C-spaces) to tackle this problem efficiently for acquiring one feature. The present paper extends the concept of C-spaces to consider multi-feature problems using feature cluster constrained spaces (GC-spaces). A GC-space represents a closed-form, geometrical solution that provides an infinite set of valid viewpoints for acquiring a cluster of features satisfying diverse viewpoint constraints. Furthermore, the current study outlines a generic viewpoint planning strategy based on GC-spaces for solving vision tasks comprising multi-feature scenarios effectively and efficiently. The applicability of the proposed framework is validated on two different industrial vision systems used for dimensional metrology tasks.

Non-Zero Sum Nash Game for Discrete-Time Infinite Markov Jump Stochastic Systems with Applications

This paper is to study finite horizon linear quadratic (LQ) non-zero sum Nash game for discrete-time infinite Markov jump stochastic systems (IMJSSs). Based on the theory of stochastic analysis, a countably infinite set of coupled generalized algebraic Riccati equations are solved and a necessary and sufficient condition for the existence of Nash equilibrium points is obtained. From a new perspective, the finite horizon mixed robust H2/H∞ control is investigated, and summarize the relationship between Nash game and H2/H∞ control problem. Moreover, the feasibility and validity of the proposed method has been proved by applying it to a numerical example.

Galois groups of n 0 + n 1 X + ⋯ + n 6 X6

We show that the Galois group of the polynomial in the title is isomorphic to the full symmetric group on six symbols for all but finitely many [Formula: see text]. This complements earlier work of Filaseta and Moy, who studied Galois groups of [Formula: see text] for more general pairs [Formula: see text], but had to admit a possibly infinite exceptional set specifically for [Formula: see text] of at most logarithmic growth in [Formula: see text]. The proof rests upon invoking Faltings’ theorem on a suitable fibration of Galois resolvents to show that this exceptional set is, in fact, finite.

Application of fuzzy logic algorithms for predicting the pipe-culvert technical condition on railways

This article is part of the author’s dissertation research in the field of developing a methodology for predicting the pipe-culvert technical condition on railways using the mathematical apparatus of fuzzy logic. One of the common methods for assessing engineering structures is the point rating method. During the examination, each identified defect is assigned a certain number of points. Then the number of points is summed up according to the developed methodology and, based on the obtained result, a conclusion is drawn about the condition of the object. However, in conditions of significant subjectivity and limited input data, the issue of predicting the structure technical condition is quite labor-intensive. As an addition to existing methods for predicting the structure technical condition, in particular pipe-culvert, it is proposed to apply a method based on the fuzzy logic principles, which is a generalization of fuzzy logic and set theory. In fuzzy logic, unlike classical logic, instead of the values true and false, the truth value degree is used, which takes any value from an infinite set from 0 to 1 inclusive. To predict changes in condition and determine the deterioration rate of the technical characteristics of a pipe-culvert, it is necessary to create a model in which scoring methods are transformed into estimates presented in terms of fuzzy logic. The article presents a methodology for using fuzzy logic algorithms to predict the pipe-culvert technical condition on railways. As a result of the research, numerical gradations of the term «pipe-сulvert Age» were obtained. With limited and subjective input data, expressed numerically in terms of fuzzy logic — «pipe-culvert Age» and «pipe-culvert Condition», it is possible to calculate the values of the term «pipe-culvert Degradation Rate». The author presents a method for predicting the technical condition of railway pipe-culverts, based on the principles of fuzzy logic, founded on fundamental set theory. The developed mathematical model makes it possible to calculate the structure transition time from one technical state to another. To implement fuzzy logical inference of the presented model, the Fuzzy Logic Toolbox package is used in the MATLAB software environment.

The God of Chance and Purpose: Divine Involvement in a Secular Evolutionary World

The God of Chance and Purpose: Divine Involvement in a Secular Evolutionary World

Two disjoint and infinite sets of solutions for a nonlocal problem involving a Hardy potential and critical growth with concave nonlinearities

Two disjoint and infinite sets of solutions for a nonlocal problem involving a Hardy potential and critical growth with concave nonlinearities