Borel Theorem Research Articles

From pairwise comparisons to consistency with respect to a group operation and Koczkodaj's metric

Important in pairwise comparisons methods (PC) concepts of reciprocal matrices, consistency conditions and priority vectors of consistent PC matrices are investigated in a more general framework where mappings take values in a not necessarily abelian group. A problem about Pauli's matrices is posed. Notions of left and right priority mappings with respect to a group operation of consistent mappings are introduced. Hou's fixed point theorems on consistent matrices are generalized to theorems on consistent mappings with values in an arbitrarily fixed lattice-ordered group. Relevant to topology basic properties of Koczkodaj's metric which induces Koczkodaj's inconsistency index in PC are studied. It is shown in ZF that Koczkodaj's metric is Cantor complete and not totally bounded, it is equivalent with the Euclidean metric on the set of positive real numbers. A version of Heine–Borel Theorem for Koczkodaj's metric is obtained and briefly applied to the compact bornology of R+n.

Nets and reverse mathematics

Nets are generalisations of sequences involving possibly uncountable index sets; this notion was introduced about a century ago by Moore and Smith. They also established the generalisation to nets of various basic theorems of analysis due to Bolzano–Weierstrass, Dini, Arzelà, and others. More recently, nets are central to the development of domain theory, providing intuitive definitions of the associated Scott and Lawson topologies, among others. This paper deals with the Reverse Mathematics study of basic theorems about nets. We restrict ourselves to nets indexed by subsets of Baire space, and therefore third-order arithmetic, as such nets suffice to obtain our main results. Over Kohlenbach’s base theory of higher-order Reverse Mathematics, the Bolzano–Weierstrass theorem for nets implies the Heine–Borel theorem for uncountable covers. We establish similar results for other basic theorems about nets and even some equivalences, e.g. for Dini’s theorem for nets. Finally, we show that replacing nets by sequences is hard, but that replacing sequences by nets can obviate the need for the Axiom of Choice, a foundational concern in domain theory. In an appendix, we study the power of more general index sets, establishing that the ‘size’ of a net is directly proportional to the power of the associated convergence theorem.

Function theory on annuli

ABSTRACTIn this paper, we obtain a local version of Picard's theorem on a sequence of disjointed annuli, which extends the classical Picard's theorem of entire functions. Furthermore, we study the local version of Picard's theorem, Borel's theorem, Nevanlina's five value theorem and uniqueness theorems in class .

Identification of Parameters of Nonlinear Dynamical Systems Simulated by Volterra Polynomials

We study the identification methods for the nonlinear dynamical systems described by Volterra series. One of the main problems in the dynamical system simulation is the problem of the choice of the parameters allowing the realization of a desired behavior of the system. If the structure of the model is identified in advance, then the solution to this problem closely resembles the identification problem of the system parameters. We also investigate the parameter identification of continuous and discrete nonlinear dynamical systems. The identification methods in the continuous case are based on application of the generalized Borel Theorem in combination with integral transformations. To investigate discrete systems, we use a discrete analog of the generalized Borel Theorem in conjunction with discrete transformations. Using model examples, we illustrate the application of the developed methods for simulation of systems with specified characteristics.

Analog of the classical borel theorem for entire harmonic functions in ℝn and generalized orders

The article describes research on the growth of functions that are harmonic in the whole space ℝ n , n≥3, and thus they are called entire harmonic. A relation has been established between the maximum terms of entire functions of finite order in the plane, which are given by power series whose coefficients are somewhat connected. Also, the maximum modulus of a harmonic function in the space ℝ n is evaluated through the maximum modulus of some entire function in the plane, the coefficients which are expressed in terms of the coefficients of the expansion of the harmonic function in a series by Laplace spherical functions. These results made it possible to obtain an analog of the classical Borel theorem for entire harmonic functions of finite order in ℝ n . Besides, the study has revealed the most general characteristics of the growth of entire harmonic functions in ℝ n in terms of the uniform norm of Laplace spherical functions in the expansion of harmonic functions in series. Slow growth of the harmonic functions in the space has also been studied. The obtained results are analogous to the classical results that are known for entire functions of one complex variable. The research findings are important because harmonic functions occupy a special place not only in many mathematical studies but also in the application of mathematical analysis to physics and mechanics, where these functions often describe various stationary processes.

Many-Sheeted Versions of the Pólya–Bernstein and Borel Theorems for Entire Functions of Order ρ ≠ 1 and Their Applications

The Puiseux series generated by the power function z = w1/ρ, where ρ > 0,ρ ≠ 1, is considered. A version of the Polya–Bernstein theorem for an entire function of order ρ ≠ 1 and normal type is proposed and applied to describe the domain of analytic continuation of this series. The domain of summability of a “regular” Puiseux series is found (this is a many-sheeted “Borel polygon”); in the case ρ = 1, the “one-sheeted” result of Borel is substantially extended. These results make it possible to describe domains of analytic continuation of the Puiseux expansions of popular many-sheeted functions (such as inverses of rational functions).

A Short Proof of the Bolzano–Weierstrass Theorem

SummaryWe present a short proof of the Bolzano–Weierstrass theorem on the real line which avoids monotonic subsequences, Cantor's intersection theorem, and the Heine–Borel theorem.

A partial analogue of Borel's Fixed Point Theorem for finite p-groups

Borel's Fixed Point Theorem states that a solvable connected algebraic group G on a non-empty complete variety V must have a fixed point. Thus, if V consists of subgroups of G, and G acts on V by conjugation, then some subgroup in V is normal in G. Although G is infinite or trivial in Borel's Theorem, the method of proof yields applications to finite p-groups. They show that in many cases, a family of subgroups of a finite p-group must contain at least one normal subgroup. Here, we obtain some applications of this type.

Peano's Unnoticed Proof of Borel's Theorem

In 1876, P. du Bois-Reymond gave an example of an infinitely differentiable function whose Taylor series diverges everywhere except one point. By generalizing this example, G. Peano proved a theorem in 1884, often credited to É. Borel, which states that every power series is a Taylor series of some infinitely differentiable function. The aim of the paper is to recall Peano's unnoticed contributions to this result.

A Theoretic Analysis Method Research of Load Appling Based on Response Control in Time Domain

With the discretely analysis of Borel theorem and Duhamel integral, a new theoretic analysis method of load appling with the purpose of time domian response control is proposed. Result of numerical simulation indicated the validity of this new method, the research of this paper provided an effective way of time domain load applying in examing the product performance.

On the p-adic Beilinson Conjecture for Number Fields

We formulate a conjectural p-adic analogue of Borel's theorem relating regulators for higher K-groups of number fields to special values of the corresponding zeta-functions, using syntomic regulators and p-adic L-functions. We also formulate a corresponding conjecture for Artin motives, and state a conjecture about the precise relation between the p-adic and classical situations. Parts of the conjectures are proved when the number field (or Artin motive) is abelian over the rationals, and all conjectures are verified numerically in some other cases.

Smooth bumps, a Borel theorem and partitions of smooth functions on p.c.f. fractals

We provide two methods for constructing smooth bump functions and for smoothly cutting off smooth functions on fractals, one using a probabilistic approach and sub-Gaussian estimates for the heat operator, and the other using the analytic theory for p.c.f. fractals and a fixed point argument. The heat semigroup (probabilistic) method is applicable to a more general class of metric measure spaces with Laplacian, including certain infinitely ramified fractals, however the cut off technique involves some loss in smoothness. From the analytic approach we establish a Borel theorem for p.c.f. fractals, showing that to any prescribed jet at a junction point there is a smooth function with that jet. As a consequence we prove that on p.c.f. fractals smooth functions may be cut off with no loss of smoothness, and thus can be smoothly decomposed subordinate to an open cover. The latter result provides a replacement for classical partition of unity arguments in the p.c.f. fractal setting.

Borel theorems for random matrices from the classical compact symmetric spaces

We study random vectors of the form $(\operatorname {Tr}(A^{(1)}V),...,\operatorname {Tr}(A^{(r)}V))$, where $V$ is a uniformly distributed element of a matrix version of a classical compact symmetric space, and the $A^{(\nu)}$ are deterministic parameter matrices. We show that for increasing matrix sizes these random vectors converge to a joint Gaussian limit, and compute its covariances. This generalizes previous work of Diaconis et al. for Haar distributed matrices from the classical compact groups. The proof uses integration formulas, due to Collins and \'{S}niady, for polynomial functions on the classical compact groups.

A generalization of Borel's theorem and microlocal Gevrey regularity in involutive structures

In this paper, we use Borel's procedure to construct Gevrey approximate solutions of an initial value problem for involutive systems of Gevrey complex vector fields. As an application, we describe the Gevrey wave-front set of the boundary values of approximate solutions in wedges W of Gevrey involutive structures ( M , V ) . We prove that the Gevrey wave-front set of the boundary value is contained in the polar of a certain cone Γ T ( W ) contained in R V ∩ T X where X is a maximally real edge of W . We also prove a partial converse.

Infinite magmatic bialgebras

An infinite magmatic bialgebra is a vector space endowed with n-ary operations, and n-ary cooperations, for each n, verifying some compatibility relations. We prove an analogue of the Hopf–Borel theorem for infinite magmatic bialgebras. We show that any connected infinite magmatic bialgebra is of the form Mag ∞ ( Prim H ) , where Mag ∞ ( V ) is the free infinite magmatic algebra over the vector space V.

Deformation quantization and Borel's theorem in locally convex spaces

Deformation quantization and Borel's theorem in locally convex spaces

COMPACT LIE GROUP ACTIONS ON CLOSED MANIFOLDS OF NON-POSITIVE CURVATURE

Borel proved that, if a finite group F acts effectively and continuously on a closed aspherical manifold M with centerless fundamental group π1(M), then a natural homomorphism ψ from F to the outer automorphism group Out π1(M) of π1(M), called the associated abstract kernel, is a monomorphism. In this paper, we investigate to what extent Borel's theorem holds for a compact Lie group G acting effectively and smoothly on a particular orientable aspherical manifold N admitting a Riemannian metric g0 of non-positive curvature in case that π1(N) has a non-trivial center. It turns out that if G attains the maximal dimension equal to the rank of Center π1(N) and the metric g0 is real analytic, then any element of G defining a diffemorphism homotopic to the identity of N must be contained in the identity component G0 of G. Moreover, if the inner automorphism group of π1(N) is torsion free, then the associated abstract kernel ψ : G/G0 → Out π1(N) is a monomorphism. The same result holds for the non-orientable N's under certain technical assumptions. Our result is an application of a theorem by Schoen–Yau [12] on harmonic mappings.

Compactness and finite dimension in asymmetric normed linear spaces

We describe the compact sets of any asymmetric normed linear space. After that, we focus our attention in finite dimensional asymmetric normed linear spaces. In this case we establish the equivalence between T 1 separation axiom and normable spaces. It is proved an asymmetric version of the Riesz Theorem about the compactness of the unit ball. We also prove that the Heine–Borel Theorem characterizes finite dimensional asymmetric normed linear spaces that satisfies the T 2 separation axiom. Finally we focus our attention on the T 0 separation axiom and results that are related to the dual p-complexity spaces.

Equivalents of the (Weak) Fan Theorem

This article presents a weak system of intuitionistic second-order arithmetic, WKV, a subsystem of the one in S.C. Kleene, R.E. Vesley [The Foundations of Intuitionistic Mathematics: Especially in Relation to Recursive Functions, North-Holland Publishing Company, Amsterdam, 1965]. It is then shown that some statements of real analysis, like a version of the Heine–Borel Theorem, and some statements of logic, e.g. compactness of classical proposition calculus, are equivalent to the (Weak) Fan Theorem in this system.

Miscellaneous results on the generating functions of special functions

We use the Borel theorem to find new families of generating functions, involving Hermite, Laguerre and Tchebycheff polynomials of one or more variables. We also use operational methods and integral representations to discuss the properties of the relativistic Laguerre and Bessel polynomials and to derive generating functions involving sums of the type