Categories of partial algebras for critical points between varieties of algebras

We denote by Con_c A the ${(\vee, 0)}$ -semilattice of all finitely generated congruences of an algebra A. A lifting of a ${(\vee, 0)}$ -semilattice S is an algebra A such that ${S \cong {\rm Con}_{\rm c} A}$ . The assignment Conc can be extended to a functor. The notion of lifting is generalized to diagrams of ${(\vee, 0)}$ -semilattices. A gamp is a partial algebra endowed with a partial subalgebra together with a semilattice-valued distance; gamps form a category that lends itself to a universal algebraic-type study. The raison d’être of gamps is that any algebra can be approximated by its finite subgamps, even in case it is not locally finite. Let ${\mathcal{V}}$ and ${\mathcal{W}}$ be varieties of algebras (on finite, possibly distinct, similarity types). Let P be a finite lattice. We assume the existence of a combinatorial object, called an ${\aleph_0}$ -lifter of P, of infinite cardinality ${\lambda}$ . Let ${\vec{A}}$ be a P-indexed diagram of finite algebras in ${\mathcal{V}}$ . If ${{\rm Con}_{\rm c} \circ \vec{A}}$ has no partial lifting in the category of gamps of ${\mathcal{W}}$ , then there is an algebra ${A \in \mathcal{V}}$ of cardinality ${\lambda}$ such that Con_c A is not isomorphic to Con_c B for any ${B \in \mathcal{W}}$ . This makes it possible to generalize several known results. In particular, we prove the following theorem, without assuming that ${\mathcal{W}}$ is locally finite. Let ${\mathcal{V}}$ be locally finite and let ${\mathcal{W}}$ be congruence-proper (i.e., congruence lattices of infinite members of ${\mathcal{W}}$ are infinite). The following equivalence holds. Every countable ${(\vee, 0)}$ -semilattice with a lifting in ${\mathcal{V}}$ has a lifting in ${\mathcal{W}}$ if and only if every ${\omega}$ -indexed diagram of finite ${(\vee, 0)}$ -semilattices with a lifting in ${\mathcal{V}}$ has a lifting in ${\mathcal{W}}$ . Gamps are also applied to the study of congruence-preserving extensions. Let ${\mathcal{V}}$ be a non-semidistributive variety of lattices and let n ≥ 2 be an integer. There is a bounded lattice ${A \in \mathcal{V}}$ of cardinality ${\aleph_1}$ with no congruence n-permutable, congruence-preserving extension. The lattice A is constructed as a condensate of a square-indexed diagram of lattices.     

Rational preimages in families of dynamical systems

Let ${\phi}$ be a rational function of degree at least two defined over a number field k. Let ${a \in \mathbb{P}^1(k)}$ and let K be a number field containing k. We study the cardinality of the set of rational iterated preimages Preim ${(\phi, a, K) = \{x_{0} \in \mathbb{P}^1(K) | \phi^{N} (x_0) = a {\rm for some} N \geq 1\}}$ . We prove two new results (Theorems 2 and 4) bounding ${|{\rm Preim}(\phi, a, K)|}$ as ${\phi}$ varies in certain families of rational functions. Our proofs are based on unit equations and a method of Runge for effectively determining integral points on certain affine curves. We also formulate and state a uniform boundedness conjecture for Preim ${(\phi, a, K)}$ and prove that a version of this conjecture is implied by other well-known conjectures in arithmetic dynamics.     

Averaged Pointwise Bounds for Deformations of Schrödinger Eigenfunctions

Let (M,g) be an n-dimensional, compact Riemannian manifold and ${P_0(\hbar) = -\hbar{^2} \Delta_g + V(x)}$ be a semiclassical Schrödinger operator with ${\hbar \in (0,\hbar_0]}$ . Let ${E(\hbar) \in [E-o(1),E+o(1)]}$ and ${(\phi_{\hbar})_{\hbar \in (0,\hbar_0]}}$ be a family of L ²-normalized eigenfunctions of ${P_0(\hbar)}$ with ${P_0(\hbar) \phi_{\hbar} = E(\hbar) \phi_{\hbar}}$ . We consider magnetic deformations of ${P_0(\hbar)}$ of the form ${P_u(\hbar) = - \Delta_{\omega_u}(\hbar) + V(x)}$ , where ${\Delta_{\omega_u}(\hbar) = (\hbar d + i \omega_u(x))^*({\hbar}d + i \omega_u(x))}$ . Here, u is a k-dimensional parameter running over ${B^k(\epsilon)}$ (the ball of radius ${\epsilon}$ ), and the family of the magnetic potentials ${(w_u)_{u\in B^k(\epsilon)}}$ satisfies the admissibility condition given in Definition 1.1. This condition implies that k ≥ n and is generic under this assumption. Consider the corresponding family of deformations of ${(\phi_{\hbar})_{\hbar \in (0, \hbar_0]}}$ , given by ${(\phi^u_{\hbar})_{\hbar \in(0, \hbar_0]}}$ , where $$\phi_{\hbar}^{(u)}:= {\rm e}^{-it_0 P_u(\hbar)/\hbar}\phi_{\hbar}$$ for ${|t_0|\in (0,\epsilon)}$ ; the latter functions are themselves eigenfunctions of the ${\hbar}$ -elliptic operators ${Q_u(\hbar): ={\rm e}^{-it_0P_u(\hbar)/\hbar} P_0(\hbar) {\rm e}^{it_0 P_u(\hbar)/\hbar}}$ with eigenvalue ${E(\hbar)}$ and ${Q_0(\hbar) = P_{0}(\hbar)}$ . Our main result, Theorem1.2, states that for ${\epsilon >0 }$ small, there are constants ${C_j=C_j(M,V,\omega,\epsilon) > 0}$ with j = 1,2 such that $$C_{1}\leq \int\limits_{\mathcal{B}^k(\epsilon)} |\phi_{\hbar}^{(u)}(x)|^2 \, {\rm d}u \leq C_{2}$$ , uniformly for ${x \in M}$ and ${\hbar \in (0,h_0]}$ . We also give an application to eigenfunction restriction bounds in Theorem 1.3.     

Addition theorems for the Appell polynomials and the associated classes of polynomial expansions

Various interesting and potentially useful properties and relationships involving the Bernoulli, Euler and Genocchi polynomials have been investigated in the literature rather extensively. Recently, the present authors (Srivastava and Pinter in Appl Math Lett 17:375–380, 2004) obtained addition theorems and other relationships involving the generalized Bernoulli polynomials ${B_n^{(\alpha)}(x)}$ and the generalized Euler polynomials ${E_n^{(\alpha)}(x)}$ of order α and degree n in x. The main purpose of this sequel to some of the aforecited investigations is to give several addition formulas for a general class of Appell sequences. The addition formulas, which are derived in this paper, involve not only the generalized Bernoulli polynomials ${B_n^{(\alpha)}(x)}$ and the generalized Euler polynomials ${E_n^{(\alpha)}(x)}$ , but also the generalized Genocchi polynomials ${G_n^{(\alpha)}(x)}$ , the Srivastava polynomials ${\mathcal{S}_{n}^{N}\left( x\right)}$ , several general families of hypergeometric polynomials and such orthogonal polynomials as the Jacobi, Laguerre and Hermite polynomials. Some umbral-calculus generalizations of the addition formulas are also investigated.     

On a conjectured inequality for the largest zero of Jacobi polynomials

P. Leopardi and the author recently investigated, among other things, the validity of the inequality $n\theta_n^{(\alpha,\beta)}\!<\! (n\!+\!1)\theta_{n+1}^{(\alpha,\beta)}$ between the largest zero $x_n\!=\!\cos\theta_n^{(\alpha,\beta)}$ and $x_{n+1}= \cos\theta_{n+1}^{(\alpha,\beta)}$ of the Jacobi polynomial $P_n^{(\alpha,\beta)}(x)$ resp. $P_{n+1}^{( \alpha,\beta)}(x)$ , α?>???1, β?>???1. The domain in the parameter space (α, β) in which the inequality holds for all n?≥?1, conjectured by us, is shown here to require a small adjustment—the deletion of a very narrow lens-shaped region in the square {???1?<?α?<???1/2, ???1/2?<?β?<?0}.     

Nielsen coincidence,fixed point and root theories of n-valued maps

Let ${(\phi, \psi)}$ be an (m, n)-valued pair of maps ${\phi, \psi : X \multimap Y}$ , where ${\phi}$ is an m-valued map and ${\psi}$ is n-valued, on connected finite polyhedra. A point ${x \in X}$ is a coincidence point of ${\phi}$ and ${\psi}$ if ${\phi(x) \cap \psi(x) \neq \emptyset}$ . We define a Nielsen coincidence number ${N(\phi : \psi)}$ which is a lower bound for the number of coincidence points of all (m, n)-valued pairs of maps homotopic to ${(\phi, \psi)}$ . We calculate ${N(\phi : \psi)}$ for all (m, n)-valued pairs of maps of the circle and show that ${N(\phi : \psi)}$ is a sharp lower bound in that setting. Specifically, if ${\phi}$ is of degree a and ${\psi}$ of degree b, then ${N(\phi : \psi) = \frac{|an - bm|}{\langle m, n \rangle}}$ , where ${\langle m, n \rangle}$ is the greatest common divisor of m and n. In order to carry out the calculation, we obtain results, of independent interest, for n-valued maps of compact connected Lie groups that relate the Nielsen fixed point number of Helga Schirmer to the Nielsen root number of Michael Brown.     

Strong tree properties for two successive cardinals

An inaccessible cardinal κ is supercompact when (κ, λ)-ITP holds for all λ?≥ κ. We prove that if there is a model of ZFC with two supercompact cardinals, then there is a model of ZFC where simultaneously ${(\aleph_2, \mu)}$ -ITP and ${(\aleph_3, \mu')}$ -ITP hold, for all ${\mu\geq \aleph_2}$ and ${\mu'\geq \aleph_3}$ .     

Deformation of extremal metrics, complex manifolds and the relative Futaki invariant

Let ${(\mathcal {X},\Omega)}$ be a closed polarized complex manifold, g be an extremal metric on ${\mathcal {X}}$ that represents the Kähler class Ω, and G be a compact connected subgroup of the isometry group Isom ${(\mathcal {X}, g)}$ . Assume that the Futaki invariant relative to G is nondegenerate at g. Consider a smooth family ${(\mathcal {M}\to B)}$ of polarized complex deformations of ${(\mathcal {X},\Omega)\simeq (\mathcal {M}_0,\Theta_0)}$ provided with a holomorphic action of G which is trivial on B. Then for every ${t\in B}$ sufficiently small, there exists an ${h^{1,1}(\mathcal {X})}$ -dimensional family of extremal Kähler metrics on ${\mathcal {M}_t}$ whose Kähler classes are arbitrarily close to Θ _t . We apply this deformation theory to show that certain complex deformations of the Mukai–Umemura 3-fold admit Kähler–Einstein metrics.     

On the unique representation of very strong algebraic geometry codes

This paper addresses the question of retrieving the triple ${(\mathcal X,\mathcal P, E)}$ from the algebraic geometry code ${\mathcal C = \mathcal C_L(\mathcal X, \mathcal P, E)}$ , where ${\mathcal X}$ is an algebraic curve over the finite field ${\mathbb F_q, \,\mathcal P}$ is an n-tuple of ${\mathbb F_q}$ -rational points on ${\mathcal X}$ and E is a divisor on ${\mathcal X}$ . If ${\deg(E)\geq 2g+1}$ where g is the genus of ${\mathcal X}$ , then there is an embedding of ${\mathcal X}$ onto ${\mathcal Y}$ in the projective space of the linear series of the divisor E. Moreover, if ${\deg(E)\geq 2g+2}$ , then ${I(\mathcal Y)}$ , the vanishing ideal of ${\mathcal Y}$ , is generated by ${I_2(\mathcal Y)}$ , the homogeneous elements of degree two in ${I(\mathcal Y)}$ . If ${n >2 \deg(E)}$ , then ${I_2(\mathcal Y)=I_2(\mathcal Q)}$ , where ${\mathcal Q}$ is the image of ${\mathcal P}$ under the map from ${\mathcal X}$ to ${\mathcal Y}$ . These three results imply that, if ${2g+2\leq m < \frac{1}{2}n}$ , an AG representation ${(\mathcal Y, \mathcal Q, F)}$ of the code ${\mathcal C}$ can be obtained just using a generator matrix of ${\mathcal C}$ where ${\mathcal Y}$ is a normal curve in ${\mathbb{P}^{m-g}}$ which is the intersection of quadrics. This fact gives us some clues for breaking McEliece cryptosystem based on AG codes provided that we have an efficient procedure for computing and decoding the representation obtained.     

Bernoulli coding map and almost sure invariance principle for endomorphisms of {\mathbb P^k}

Let f be an holomorphic endomorphism of ${\mathbb{P}^k}$ and μ be its measure of maximal entropy. We prove an almost sure invariance principle for the systems ${(\mathbb{P}^k,f,\mu)}$ . Our class ${\mathcal {U}}$ of observables includes the Hölder functions and unbounded ones which present analytic singularities. The proof is based on a geometric construction of a Bernoulli coding map ${\omega: (\Sigma, s, \nu) \to (\mathbb{P}^k,f,\mu)}$ . We obtain the invariance principle for an observable ψ on ${(\mathbb{P}^k,f,\mu)}$ by applying Philipp–Stout’s theorem for ${\chi = \psi \circ \omega}$ on (Σ, s, ν). The invariance principle implies the central limit theorem as well as several statistical properties for the class ${\mathcal {U}}$ . As an application, we give a direct proof of the absolute continuity of the measure μ when it satisfies Pesin’s formula. This approach relies on the central limit theorem for the unbounded observable log ${{\tt Jac}\, f \in \mathcal{U}}$ .     

Stability of a mixed additive and cubic functional equation in several variables in non-Archimedean spaces

Consider the functional equation ${\Im_1(f ) = \Im_2(f )\,\,(\Im)}$ in a certain general setting. A function g is an approximate solution of ${(\Im)}$ if ${\Im_1(g)}$ and ${\Im_2(g)}$ are close in some sense. The Ulam stability problem asks whether or not there is a true solution of ${(\Im)}$ near g. In this paper, we achieve the general solution and the stability of the following functional equation $$\begin{array}{ll}f\left(\sum\limits^{n}_{i=1}x_{i} \right)+f\left(\sum\limits^{n-1}_{i=1} x_{i}-x_{n} \right)\\\quad=2f\left(\sum\limits^{n-1}_{i=1}x_{i} \right)+\sum\limits^{n-1}_{i=1}(f(x_{i}+x_{n}) +f(x_{i}-x_{n})-2f(x_{i}))\end{array}$$ for all x _i (i =? 1,2, . . . , n), in non-Archimedean spaces.     

On functions whose symmetric part of gradient agree and a generalization of Reshetnyak’s compactness theorem

We consider the following question: Given a connected open domain ${\Omega \subset \mathbb{R}^n}$ , suppose ${u, v : \Omega \rightarrow \mathbb{R}^n}$ with det ${(\nabla u) > 0}$ , det ${(\nabla v) > 0}$ a.e. are such that ${\nabla u^T(x)\nabla u(x) = \nabla v(x)^T \nabla v(x)}$ a.e. , does this imply a global relation of the form ${\nabla v(x) = R\nabla u(x)}$ a.e. in Ω where ${R \in SO(n)}$ ? If u, v are C ¹ it is an exercise to see this true, if ${u, v\in W^{1,1}}$ we show this is false. In Theorem 1 we prove this question has a positive answer if ${v \in W^{1,1}}$ and ${u \in W^{1,n}}$ is a mapping of L ^p integrable dilatation for p > n ? 1. These conditions are sharp in two dimensions and this result represents a generalization of the corollary to Liouville’s theorem that states that the differential inclusion ${\nabla u \in SO(n)}$ can only be satisfied by an affine mapping. Liouville’s corollary for rotations has been generalized by Reshetnyak who proved convergence of gradients to a fixed rotation for any weakly converging sequence ${v_k \in W^{1,1}}$ for which $$\int \limits_{\Omega} {\rm dist}(\nabla v_k, SO(n))dz \rightarrow 0 \, {\rm as} \, k \rightarrow \infty.$$ Let S(·) denote the (multiplicative) symmetric part of a matrix. In Theorem 3 we prove an analogous result to Theorem 1 for any pair of weakly converging sequences ${v_k \in W^{1,p}}$ and ${u_k \in W^{1,\frac{p(n-1)}{p-1}}}$ (where ${p \in [1, n]}$ and the sequence (u _k ) has its dilatation pointwise bounded above by an L ^r integrable function, r > n ? 1) that satisfy ${\int_{\Omega} |S(\nabla u_k) - S(\nabla v_k)|^p dz \rightarrow 0}$ as k → ∞ and for which the sign of the det ${(\nabla v_k)}$ tends to 1 in L ¹. This result contains Reshetnyak’s theorem as the special case (u _k ) ≡ Id, p = 1.     

Elliptic Reflection Structures, K-Loop Derivations and Triangle-Inequality

This paper is a part of our general aim to study properties of elliptic and ordered elliptic geometries and then using some of these properties to introduce new concepts and develop their theories. If ${(P,\mathfrak{G}, \equiv,\tau)}$ denotes an elliptic geometry ordered via a separation ?? then there are polar points o and ?? and on the line ${ \overline{K} := \overline{\infty,o}}$ there can be established an operation ??+?? such that ${(\overline{K},+)}$ becomes a commutative group and the map ${ a^+ :\overline{K}\to \overline{K} ; x \mapsto a + x}$ is a motion on ${\overline{K}}$ . The separation ?? induces on ${\overline{K}}$ a cyclic order ?? with [o, e, ??] = 1 such that ${(\overline{K},+,\omega)}$ becomes a cyclic ordered group. For ${a,b \in K := \overline{K} {\setminus}\{\infty\}}$ we set ${a < b :\Longleftrightarrow [a,b,\infty] =1}$ and for all ${a\in K\,a < \infty}$ . Then (K,?<) is a totally ordered set. We show there is a surjective distance function: $$ \lambda : P \times P \to \overline{K}_+ := \{x \in \overline{K}\,|\,o \leq x\leq\infty\}, $$ with ?? ${\lambda(a,b) = \lambda(c,d) \ \Longleftrightarrow (a,b) \equiv (c,d)}$ ??. We prove in the first part of our project like (cf. Gr?ger in Mitt Math Ges Hamburg 11:441?C457, 1987) the following triangle-inequality: (cf. Theorem 8.2). If (a, b, c) is a triangle consisting of pairwise not polar points with ??(a, c), ??(b, c) < e then ??(a, b) ?? ??(a, c) + ??(b, c) < ??.     

On centralizers of reflexive algebras

Let ${\mathcal{L}}$ be a subspace lattice on a complex Banach space X and δ be a linear mapping from ${alg\mathcal{L}}$ into B(X) such that for every ${A \in alg\mathcal{L}, 2\delta(A^2)=\delta(A)A + A\delta(A)}$ or ${\delta(A^3) = A\delta(A)A}$ . We show that if one of the following holds (1) ${\vee\{L : L \in \mathcal{J}(\mathcal{L})\}=X}$ , (2) ${\wedge\{L_-: L \in \mathcal{J}(\mathcal{L})\}=(0)}$ and X is reflexive, then δ is a centralizer. We also show that if ${\mathcal{L}}$ is a CSL and δ is a linear mapping from ${alg\mathcal{L}}$ into itself, then δ is a centralizer.     

Theta dichotomy for the genuine unramified principal series of {\widetilde{Sp}_2(F)}

Let F be a p-adic field with odd residual characteristic. This work is the continuation of a previous paper that contains some detailed computations of the doubling integral for irreducible constituents ${(\pi, \mathcal{V}_{\pi})}$ of the genuine unramified principal series of ${\widetilde{Sp}_2(F)}$ using various “good test data”. This paper aims to interpret those results in terms of the non-vanishing of local theta lifts. Assuming a technical condition on order of a particular pole for the family of doubling integrals for ${(\pi, \mathcal{V}_{\pi})}$ , we aim to determine the so-called “dichotomy sign” of ${(\pi, \mathcal{V}_{\pi})}$ .     

The cogyrolines of Möbius gyrovector spaces are metric but not periodic

In this paper, we prove that every metric line of a Möbius gyrovector space ${(\mathbb{R}_{1}^{n}, \oplus, \otimes)}$ is exactly a cogyroline of itself, and also we prove the nonexistence of periodic lines in ${(\mathbb{R}_{1}^{n}, \oplus, \otimes)}$ .     

Central limit theorems for hyperbolic spaces and Jacobi processes on [0,\infty [

We present a unified approach to a couple of central limit theorems for the radial behavior of radial random walks on hyperbolic spaces as well as for time-homogeneous Markov chains on $[0,\infty [$ whose transition probabilities are defined in terms of Jacobi convolutions. The proofs of all central limit theorems are based on corresponding limit results for the associated Jacobi functions $\varphi _{\lambda }^{(\alpha ,\beta )}$ . In particular, we consider the limit $\alpha \rightarrow \infty $ , the limit $\varphi _{i\rho -n\lambda }^{(\alpha ,\beta )}(t/n)$ for $n\rightarrow \infty $ , and the behavior of the Jacobi function $\varphi _{i\rho -\lambda }^{(\alpha ,\beta )}(t)$ for small $\lambda $ . The proofs of all these limit results are based on the known Laplace integral representation for Jacobi functions. Parts of the results are known, other improve known ones, and other are new.