Infinitary Baker–Pixley theorem

An important theorem of Baker and Pixley states that if \(\mathbf {A}\) is a finite algebra with a \((d+1)\)-ary near-unanimity term and f is an n-ary operation on A such that every subalgebra of \(\mathbf {A}^{d}\) is closed under f, then f is representable by a term in \(\mathbf {A}\). It is well known that the Baker–Pixley theorem does not hold when \(\mathbf {A}\) is infinite. We give an infinitary version of the Baker–Pixley theorem which applies to an arbitrary class of structures with a \((d+1)\)-ary near-unanimity term instead of a single finite algebra.     

A Birch–Goldbach theorem

We prove an analogue of a theorem of Birch with prime variables.     

Global Lie–Tresse theorem

We prove a global algebraic version of the Lie–Tresse theorem which states that the algebra of differential invariants of an algebraic pseudogroup action on a differential equation is generated by a finite number of rational-polynomial differential invariants and invariant derivations.     

A chiral Borel–Weil–Bott theorem

We compute the cohomology of modules over the algebra of twisted chiral differential operators over the flag manifold. This is applied to (1) finding the character of G-integrable irreducible highest weight modules over the affine Lie algebra at the critical level, and (2) computing a certain elliptic genus of the flag manifold. The main tool is a result that interprets the Drinfeld–Sokolov reduction as a derived functor.     

A generalized Gleason–Pierce–Ward theorem

The Gleason–Pierce–Ward theorem gives constraints on the divisor and field size of a linear divisible code over a finite field whose dimension is half of the code length. This result is a departure point for the study of self-dual codes. In recent years, additive codes have been studied intensively because of their use in additive quantum codes. In this work, we generalize the Gleason–Pierce–Ward theorem on linear codes over GF(q), q = p ^m , to additive codes over GF(q). The first step of our proof is an application of a generalized upper bound on the dimension of a divisible code determined by its weight spectrum. The bound is proved by Ward for linear codes over GF(q), and is generalized by Liu to any code as long as the MacWilliams identities are satisfied. The trace map and an analogous homomorphism on GF(q) are used to complete our proof.      

Stochastic nonlinear Perron–Frobenius theorem

We establish a stochastic nonlinear analogue of the Perron–Frobenius theorem on eigenvalues and eigenvectors of positive matrices. The result is formulated in terms of an automorphism T of a probability space and a random transformation D of the non-negative cone of an n-dimensional Euclidean space. Under assumptions of monotonicity and homogeneity of D, we prove the existence of scalar and vector measurable functions α > 0 and x > 0 satisfying the equation αTx = D(x) almost surely. We apply the result obtained to the analysis of a class of random dynamical systems arising in mathematical economics and finance (von Neumann–Gale dynamical systems).     

On the Pila–Wilkie theorem

This expository paper gives an account of the Pila–Wilkie counting theorem and some of its extensions and generalizations. We use semialgebraic cell decomposition to simplify part of the original proof. We also include complete treatments of a result due to Pila and Bombieri and of the o-minimal Yomdin–Gromov theorem that are used in this proof. For the latter we follow Binyamini and Novikov.     

On the Jayne–Rogers theorem

J. E. Jayne and C. A. Rogers [3] proved that a mapping \({f \colon {X \rightarrow Y}}\) of an absolute Souslin-\({\mathcal{F}}\) set X to a metric space Y is \({\mathbf{\Delta}^0_2}\)-measurable if and only if it is piecewise continuous. We give a similar result for a perfectly paracompact first-countable space X and a regular space Y.     

The Kolmogorov–Riesz compactness theorem

We show that the Arzelà–Ascoli theorem and Kolmogorov compactness theorem both are consequences of a simple lemma on compactness in metric spaces. Their relation to Helly's theorem is discussed. The paper contains a detailed discussion on the historical background of the Kolmogorov compactness theorem.     

On the Cohen–Lusk theorem

Let G be a finite group and X be a G-space. For a map f: X → ℝ ^m , the partial coincidence set A(f, k), k ≤ |G|, is the set of points x ∈ X such that there exist k elements g ₁,…, g _k of the group G, for which f(g ₁ x) = ⋅⋅⋅ = f(g _k x) holds. We prove that the partial coincidence set is nonempty for G = ℤ _p ⁿ under some additional assumptions. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 8, pp. 61–67, 2007.     

A generalized Hirzebruch Riemann–Roch theorem

This short Note proves a generalization of the Hirzebruch Riemann–Roch theorem equivalent to the Cardy condition. This is done using an earlier result that explicitly describes what the Mukai pairing on Hochschild homology descends to in Hodge cohomology via the Hochschild–Kostant–Rosenberg map twisted by the root Todd genus. To cite this article: A.C. Ramadoss, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

A new discrete Hopf–Rinow theorem

We prove a version of the Hopf–Rinow theorem with respect to path metrics on discrete spaces. The novel aspect is that we do not a priori assume local finiteness but isolate a local finiteness type condition, called essentially locally finite, that is indeed necessary. As a side product we identify the maximal weight, called the geodesic weight, generating the path metric in the situation when the space is complete with respect to any of the equivalent notions of completeness proven in the Hopf–Rinow theorem. As an application we characterize the graphs for which the resistance metric is a path metric induced by the graph structure.     

The Gauss–Wilson theorem for quarter-intervals

We define a Gauss factorial N _n ! to be the product of all positive integers up to N that are relatively prime to n. It is the purpose of this paper to study the multiplicative orders of the Gauss factorials $\left\lfloor\frac{n-1}{4}\right\rfloor_{n}!$ for odd positive integers n. The case where n has exactly one prime factor of the form p≡1(mod4) is of particular interest, as will be explained in the introduction. A fundamental role is played by p with the property that the order of  $\frac{p-1}{4}!$ modulo p is a power of 2; because of their connection to two different results of Gauss we call them Gauss primes. Our main result is a complete characterization in terms of Gauss primes of those n of the above form that satisfy $\left\lfloor\frac{n-1}{4}\right\rfloor_{n}!\equiv 1\pmod{n}$ . We also report on computations that were required in the process.     

Aspects of the Borsuk–Ulam theorem

This is a largely expository account of various aspects of the Borsuk–Ulam theorem, including extensions of the classical theorem to families of maps parametrized by a base space and to multivalued maps. The main technical tool is the Euler class with compact supports.     

Cone theorem via Deligne–Mumford stacks

We prove the cone theorem for varieties with LCIQ singularities using deformation theory of stable maps into Deligne–Mumford stacks. We also obtain a sharper bound on −(K _X + D)-degree of (K _X + D)-negative extremal rays for projective -factorial log terminal threefold pairs (X, D).     

On the Lévy–Raikov–Marcinkiewicz theorem

Let μ be a finite nonnegative Borel measure. The classical Lévy–Raikov–Marcinkiewicz theorem states that if its Fourier transform $\text{μ}\text{?}$ can be analytically continued to some complex half-neighborhood of the origin containing an interval (0,iR) then $\text{μ}\text{?}$ admits analytic continuation into the strip $\text{{t:}\text{0<}\text{I}\text{t<R}}$. We extend this result to general classes of measures and distributions, assuming non-negativity only on some ray and allowing temperate growth on the whole line. To cite this article: I. Ostrovskii, A. Ulanovskii, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Converse Sturm–Hurwitz–Kellogg theorem and related results

We prove that if Vⁿ is a Chebyshev system on the circle and f is a continuous real-valued function with at least n + 1 sign changes then there exists an orientation preserving diffeomorphism of S¹ that takes f to a function L²-orthogonal to V. We also prove that if f is a function on the real projective line with at least four sign changes then there exists an orientation preserving diffeomorphism of that takes f to the Schwarzian derivative of a function on . We show that the space of piecewise constant functions on an interval with values ± 1 and at most n + 1 intervals of constant sign is homeomorphic to n-dimensional sphere. To V. I. Arnold for his 70th birthday     

A note on the Fuglede–Putnam theorem

We prove the following generalization of the Fuglede–Puntam theorem. Let N be an unbounded normal operator in the Hilbert space, and let A be an unbounded self-adjoint operator such that $D(N)\subseteq D(A)$ . Then, $ AN\subseteq N^*A \Rightarrow AN^*\subseteq NA.$