Classical Theorem Research Articles

An infinite-dimensional representation of the Ray-Knight theorems

The classical Ray-Knight theorems for Brownian motion determine the law of its local time process either at the first hitting time of a given value a by the local time at the origin, or at the first hitting time of a given position b by Brownian motion. We extend these results by describing the local time process jointly for all a and all b, by means of stochastic integral with respect to an appropriate white noise. Our result applies to µ-processes, and has an immediate application: a µ-process is the height process of a Feller continuous-state branching process (CSBP) with immigration (Lambert [10]), whereas a Feller CSBP with immigration satisfies a stochastic differential equation driven by a white noise (Dawson and Li [7]); our result gives an explicit relation between these two descriptions and shows that the stochastic differential equation in question is a reformulation of Tanaka's formula.

Analyzing in-plane mechanics of a novel honeycomb structure with zero Poisson's ratio

The superior performance of honeycomb structure has facilitated its extensive application in various fields. In this paper, a novel honeycomb structure is proposed, which exhibits zero Poisson's ratio behavior. Firstly, the in-plane mechanical properties are analyzed based on the classical Castigliano's theorem to obtain expressions connecting the three elastic constants of the structure and geometric parameters. Then, the theoretical expressions are verified by homogenized finite element methods and 3D-printed experimental models, which help to determine the relationship between the in-plane mechanical properties and the geometric parameters. The results show that the mechanical properties of the honeycomb structure can be tailored by changing the geometric parameters, while ensuring that the zero Poisson's ratio property is maintained within a relatively small fluctuation range. This study reveals that the mechanical properties of the novel honeycomb structure are tunable, which has outperformed some honeycomb structures on adjusting range.

Dynamic shakedown limits of the pavement on saturated subgrade subjected to traffic rolling-sliding loading

This paper establishes a semi-analytical lower-bound dynamic shakedown method to investigate the shakedown of the pavement on saturated subgrade based on dynamic Biot's consolidation theory, considering a rolling-sliding contact between vehicle tire and pavement. This analysis has demonstrated that properties of the saturated subgrade generally have negative effects on the shakedown limit of the pavement system under high-speed traffic loadings. In general, the classical single-phased shakedown theorem overestimates the shakedown limit of the pavement on the soft ground of low permeability, particularly under high-speed traffic loadings. The shakedown limit of the pavement on a saturated subgrade gradually reduces with decreasing permeability of the subgrade. The sliding-induced friction significantly affects the shakedown limit and the location of critical shakedown failure. The permeability of the saturated subgrade has considerable effects on the shakedown limit for the case of low tire-pavement friction. The shakedown limit of the pavement on a saturated subgrade and a single-phased subgrade shows a considerable difference at the optimum modulus ratio, at which the critical shakedown failure tends to shift from the saturated subgrade to the pavement layer. The proposed method will be more desirable to evaluate the performance of the pavement in the saturated soft ground.

An Upper Bound Energy Formulation of Free-Chip Machining with Flat Chips and an Alternative Method of Determination of Cutting Forces without Using the Merchant’s Circle Diagram

An upper bound analysis of free-chip machining has been carried out, where the tool cutting and friction forces were determined from the deformation energy dissipated during the chip separation process. The method employed was based on the classical upper bound theorem, as formulated by Prager and Hodge, and Drucker, Prager, and Greenberg, and its modification by Collins, to deal with the metal forming processes involving coulomb friction. A straight shear plane and coulomb friction at the chip/tool interface were assumed and the energy required for cutting was calculated from a strain rate/velocity field that was constructed using the method proposed by Collins. Cutting forces, thrust forces, tool/chip contact lengths, and chip thickness ratios were determined for different tool rake angles and friction conditions. The theoretical results were also compared with some experimental results that are available in the published literature. The comparison between the two was not found to be satisfactory. This may be due to the non-unique nature of the machining process, as stated by Hill and demonstrated by other authors. The results calculated from the present method of ”energy balance” were also found to be in agreement with those obtained by Merchant using the principle of ”force balance”.

Total torsion of three-dimensional lines of curvature

A curve gamma in a Riemannian manifold M is three-dimensional if its torsion (signed second curvature function) is well-defined and all higher-order curvatures vanish identically. In particular, when gamma lies on an oriented hypersurface S of M, we say that gamma is well positioned if the curve’s principal normal, its torsion vector, and the surface normal are everywhere coplanar. Suppose that gamma is three-dimensional and closed. We show that if gamma is a well-positioned line of curvature of S, then its total torsion is an integer multiple of 2pi ; and that, conversely, if the total torsion of gamma is an integer multiple of 2pi , then there exists an oriented hypersurface of M in which gamma is a well-positioned line of curvature. Moreover, under the same assumptions, we prove that the total torsion of gamma vanishes when S is convex. This extends the classical total torsion theorem for spherical curves.

A note on bireflectional elements of an algebra

A classical theorem of Wonenburger, Djokovič, Hoffmann and Paige [2, 3, 7] states that an element of the general linear group of a finite-dimensional vector space is the product of two involutions if and only if it is similar to its inverse. We give a very elementary proof of this result when the underlying field F is algebraically closed with characteristic other than 2. In that situation, the result is generalized to the group of invertibles of any finite-dimensional algebra over F .

Integral theorems in finite-dimensional commutative algebra

For monogenic (continuous and differentiable in the sense of G\^ateaux) functions given in special real subspaces of an arbitrary finite-dimensional commutative associative algebra over the complex field and taking values in this algebra, we establish basic properties analogous to properties of holomorphic functions of a complex variable. Methods for proving results are based on a representation of monogenic functions via holomorphic functions of complex variables that allows to establish analogues of Cauchy-Riemann conditions and the continuity of G\^ateaux derivatives of all orders for monogenic functions. In such a way, analogues of a number of classical theorems of complex analysis (the Cauchy integral theorem for a curvilinear integral, the Cauchy integral formula, the Morera theorem, the Taylor theorem) are proved and different equivalent definitions for the mentioned monogenic functions are established. An analogue of the Cauchy theorem for an integral over non piecewise smooth surfaces is proved.

Thermal Area Law for Lattice Bosons

A physical system is said to satisfy a thermal area law if the mutual information between two adjacent regions in the Gibbs state is controlled by the area of their boundary. Lattice bosons have recently gained significant interest because they can be precisely tuned in experiments and bosonic codes can be employed in quantum error correction to circumvent classical no-go theorems. However, the proofs of many basic information-theoretic inequalities such as the thermal area law break down for bosons because their interactions are unbounded. Here, we rigorously derive a thermal area law for a class of bosonic Hamiltonians in any dimension which includes the paradigmatic Bose-Hubbard model. The main idea to go beyond bounded interactions is to introduce a quasi-free reference state with artificially decreased chemical potential by means of a double Peierls-Bogoliubov estimate.

Chain-dependent Conditions in Extremal Set Theory

In extremal set theory our usual goal is to find the maximal size of a family of subsets of an n-element set satisfying a condition. A condition is called chain-dependent, if it is satisfied for a family if and only if it is satisfied for its intersections with the n! full chains. We introduce a method to handle problems with such conditions, then show how it can be used to prove three classic theorems. Then, a theorem about families containing no two sets such that A⊂B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A\\subset B$$\\end{document} and λ·|A|≤|B|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda \\cdot |A| \\le |B|$$\\end{document} is proved. Finally, we investigate problems where instead of the size of the family, the number of ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document}-chains is maximized. Our method is to define a weight function on the sets (or ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document}-chains) and use it in a double counting argument involving full chains.

Derivations characterized by monomials x2n in prime rings

Let [Formula: see text] be a prime ring of [Formula: see text] or [Formula: see text] and let [Formula: see text] be an additive map such that [Formula: see text] for all [Formula: see text], where [Formula: see text] is a positive integer and [Formula: see text] is the maximal symmetric ring of quotients of [Formula: see text]. It is shown that there exist a derivation [Formula: see text] and an additive map [Formula: see text] with [Formula: see text] for all [Formula: see text], such that [Formula: see text]. This result is a natural generalization of the classic theorem of Herstein for Jordan derivations on prime rings. Moreover, it gives a purely algebraic version of the theorem recently obtained by Kosi-Ulbl, Rodriguez and Vukman for standard operator algebras on Banach spaces.

Analogue of Kellogg’s Theorem for Piecewise Lyapunov Domains

In weighted Hölder spaces, classes of smooth arcs and piecewise smooth contours are introduced that are invariant under power mappings. The boundary properties of conformal mappings are described in terms of these classes by analogy with Kellogg’s classical theorem.

A Valuation Theorem for Noetherian Rings

Let A⊂B be integral domains. Suppose A is Noetherian and B is a finitely generated A-algebra. Denote by A‾ the integral closure of A in B. We show that A‾ is determined by finitely many unique discrete valuation rings. Our result generalizes Rees’ classical valuation theorem for ideals. We also obtain a variant of Zariski’s main theorem.

Determinantal properties of Boolean graphs using recursive approach

The aim of this paper is to study the determinant and inverse of the adjacency matrices of weighted and directed versions of Boolean graphs. Our approach is recursive. We describe the adjacency matrix of a weighted Boolean graph in terms of the adjacency matrix of a smaller-sized weighted Boolean graph. This allows us to compute the determinant and inverse of the adjacency matrix of a weighted Boolean graph recursively. In particular, we show that the determinant of a directed Boolean graph is 1. Further, using a classical theorem of Cayley which expresses the determinant of any skew-symmetric matrix as a square of its Pfaffian, we show that for any directed Boolean graph, the characteristic polynomial has all its even degree coefficients strictly positive with the odd ones being zero.

On the constants in abstract inverse theorems of approximation theory

In the classical inverse theorems of constructive function theory, structural characteristics of an approximated function are estimated in terms of its best approximations. Most of the known proofs of the inverse theorems utilize Bernstein’s idea to expand the function in polynomials of its best approximation. In the present paper, Bernstein’s proof is modified by using integrals instead of sums. With this modification, it turns out that desired inequalities are based on identities similar to Frullani integrals. The considerations here are quite general, which allows one to obtain analogs of the inverse theorems for functionals in abstract Banach or even seminormed spaces. Then these abstract results are specified and inverse theorems in concrete spaces of functions are deduced, including weighted spaces, with explicit constants.

New Banach spaces-based fully-mixed finite element methods for pseudostress-assisted diffusion problems

In this paper we propose and analyze Banach spaces-based fully-mixed approaches yielding new finite element methods for numerically solving the coupled partial differential equations describing the pseudostress-assisted diffusion of a solute into an elastic material. Two mixed formulations employing the diffusive flux as an additional variable are introduced for the diffusion equation, and the concentration gradient is considered as an auxiliary unknown of the second one of them. The resulting coupled systems are rewritten as equivalent fixed point operator equations, so that the respective unique solvabilities are proved by applying the classical Banach theorem along with the Babuška-Brezzi theory. The nonlinear dependency on the elastic variables of the diffusion coefficient and its source term, as well as the nonlinear dependency on the concentration of the elastic source term, suggest, for appropriate continuous and discrete analyses, that the unknowns be sought in suitable Lebesgue spaces. The associated Galerkin schemes are addressed similarly, and the Brouwer theorem yields the existence of discrete solutions. A priori error estimates are derived for both approaches, and rates of convergence for specific finite element subspaces satisfying the required discrete inf-sup conditions, are established in 2D. Finally, several numerical examples illustrating the performance of the two methods and confirming the theoretical findings, are reported.

Insurance–finance arbitrage

AbstractMost insurance contracts are inherently linked to financial markets, be it via interest rates, or—as hybrid products like equity‐linked life insurance and variable annuities—directly to stocks or indices. However, insurance contracts are not for trade except sometimes as surrender to the selling office. This excludes the situation of arbitrage by buying and selling insurance contracts at different prices. Furthermore, the insurer uses private information on top of the publicly available one about financial markets. This paper provides a study of the consistency of insurance contracts in connection with trades in the financial market with explicit mention of the information involved.By defining strategies on an insurance portfolio and combining them with financial trading strategies, we arrive at the notion of insurance–finance arbitrage (IFA). In analogy to the classical fundamental theorem of asset pricing, we give a fundamental theorem on the absence of IFA, leading to the existence of an insurance–finance‐consistent probability. In addition, we study when this probability gives the expected discounted cash‐flows required by the EIOPA best estimate.The generality of our approach allows to incorporate many important aspects, like mortality risk or general levels of dependence between mortality and stock markets. Utilizing the theory of enlargements of filtrations, we construct a tractable framework for insurance–finance consistent valuation.

The fundamental theorem of affine geometry

We deal with a natural generalization of the classical Fundamental Theorem of Affine Geometry to the case of non bijective maps. This extension geometrically characterizes semiaffine morphisms. It was obtained by W. Zick in 1981, although it is almost unknown. Our aim is to present and discuss a simplified proof of this result.

Rigid inner forms over local function fields

We generalize the concept of rigid inner forms, defined by Kaletha in [12], to the setting of a local function field F in order state the local Langlands conjectures for arbitrary connected reductive groups over F. To do this, we define for a connected reductive group G over F a new cohomology set H1(E,Z→G)⊂Hfpqc1(E,G) for a gerbe E attached to a class in Hfppf2(F,u) for a certain canonically-defined profinite commutative group scheme u, building up to an analogue of the classical Tate-Nakayama duality theorem. We define a relative transfer factor for an endoscopic datum serving a connected reductive group G over F, and use rigid inner forms to extend this to an absolute transfer factor, enabling the statement of endoscopic conjectures relating stable virtual characters and s˙-stable virtual characters for a semisimple s˙ associated to a tempered Langlands parameter.

The lively siblings of the Pentagon theorem

The five circles in the classical Pentagon theorem of Miquel are given as circumcircles of five certain triangles in the pentagon. If one chooses instead the circumcircles of five other triangles, one gets a different configuration of circles. This resulting configuration of circles carries three families of five concyclic quadruples of points. Together with the five circumcircles this gives a total of 20 circles. The radical axes of each two of these twenty circles are all concurrent.

A product version of the Hilton-Milner theorem

Two families F,G of k-subsets of {1,2,…,n} are called non-trivial cross-intersecting if F∩G≠∅ for all F∈F,G∈G and ∩{F:F∈F}=∅=∩{G:G∈G}. In the present paper, we determine the maximum product of the sizes of two non-trivial cross-intersecting families of k-subsets of {1,2,…,n} for n≥4k, k≥8, which is a product version of the classical Hilton-Milner Theorem.