Negative entropy,zero temperature and Markov chains on the interval

We consider ergodic optimization for the shift map on the modified Bernoulli space σ: [0, 1]^? → [0, 1]^?, where [0, 1] is the unit closed interval, and the potential A: [0, 1]^? → ? considered depends on the two first coordinates of [0, 1]^?. We are interested in finding stationary Markov probabilities µ_∞ on [0, 1]^? that maximize the value ∫ Adµ, among all stationary (i.e. σ-invariant) probabilities µ on [0, 1]^?. This problem correspond in Statistical Mechanics to the zero temperature case for the interaction described by the potential A. The main purpose of this paper is to show, under the hypothesis of uniqueness of the maximizing probability, a Large Deviation Principle for a family of absolutely continuous Markov probabilities µ _β which weakly converges to µ_∞. The probabilities µ _β are obtained via an information we get from a Perron operator and they satisfy a variational principle similar to the pressure in Thermodynamic Formalism. As the potential A depends only on the first two coordinates, instead of the probability µ on [0, 1]^?, we can consider its projection ν on [0, 1]². We look at the problem in both ways. If µ_∞ is the maximizing probability on [0, 1]^?, we also have that its projection ν _∞ is maximizing for A. The hypothesis about stationarity on the maximization problem can also be seen as a transhipment problem. Under the hypothesis of A being C ² and the twist condition, that is,

$\frac{{\partial ^2 A}}{{\partial x\partial y}}(x,y) \ne 0, for all (x,y) \in [0,1]^2 ,$

we show the graph property of the maximizing probability ν on [0, 1]². Moreover, the graph is monotonous. An important result we get is: the maximizing probability is unique generically in Mañé’s sense. Finally, we exhibit a separating sub-action for A.

     

Holomorphic injective extensions of functions in <Emphasis Type="Italic">P</Emphasis>(<Emphasis Type="Italic">K</Emphasis>) and algebra generators

We present necessary and sufficient conditions on planar compacta K and continuous functions f on K in order that f generate the algebras P(K), R(K), A(K) or C(K). We also unveil quite surprisingly simple examples of non-polynomial convex compacta K ? C and f ∈ P(K) with the property that f ∈ P(K) is a homeomorphism of K onto its image, but for which f ^?1 ? P(f(K)). As a consequence, such functions do not admit injective holomorphic extensions to the interior of the polynomial convex hull \(\widehat K\). On the other hand, it is shown that the restriction f*|_G of the Gelfand-transform f* of an injective function f ∈ P(K) is injective on every regular, bounded complementary component G of K. A necessary and sufficient condition in terms of the behaviour of f on the outer boundary of K is given in order that f admit a holomorphic injective extension to \(\widehat K\). We also include some results on the existence of continuous logarithms on punctured compacta containing the origin in their boundary.     

Realizability of Two-dimensional Linear Groups over Rings of Integers of Algebraic Number Fields

Given the ring of integers O _K of an algebraic number field K, for which natural numbers n there exists a finite group G???GL(n, O _K ) such that O _K G, the O _K -span of G, coincides with M(n, O _K ), the ring of (n?×?n)-matrices over O _K ? The answer is known if n is an odd prime. In this paper we study the case n?=?2; in the cases when the answer is positive for n?=?2, for n?=?2m there is also a finite group G???GL(2m, O _K ) such that O _K G?=?M(2m, O _K ).     

A Minimum Principle for Potentials with Application to Chebyshev Constants

For “Riesz-like” kernels K(x,y) = f(|x?y|) on A×A, where A is a compact d-regular set \(A\subset \mathbb {R}^{p}\), we prove a minimum principle for potentials \(U_{K}^{\mu }=\int K(x,y)\textup {d}\mu (x)\), where μ is a Borel measure supported on A. Setting \(P_{K}(\mu )=\inf _{y\in A}U^{\mu }(y)\), the K-polarization of μ, the principle is used to show that if {ν _N } is a sequence of measures on A that converges in the weak-star sense to the measure ν, then P _K (ν _N )→P _K (ν) as \(N\to \infty \). The continuous Chebyshev (polarization) problem concerns maximizing P _K (μ) over all probability measures μ supported on A, while the N-point discrete Chebyshev problem maximizes P _K (μ) only over normalized counting measures for N-point multisets on A. We prove for such kernels and sets A, that if {ν _N } is a sequence of N-point measures solving the discrete problem, then every weak-star limit measure of ν _N as \(N \to \infty \) is a solution to the continuous problem.     

Error estimates for a projection-difference method for a linear differential-operator equation

We study a projection-difference method for approximately solving the Cauchy problem u′(t) + A(t)u(t) + K(t)u(t) = h(t), u(0) = 0 for a linear differential-operator equation in a Hilbert space, where A(t) is a self-adjoint operator and K(t) is an operator subordinate to A(t). Time discretization is based on a three-level difference scheme, and space discretization is carried out by the Galerkin method. Under certain smoothness conditions on the function h(t), we obtain estimates for the convergence rate of the approximate solutions to the exact solution.     

On the cohomology of tori over local fields with perfect residue field

If T is an algebraic torus defined over a discretely valued field K with perfect residue field k, we relate the K-cohomology of T to the k-cohomology of certain objects associated to T. When k has cohomological dimension ≤ 1, our results have a particularly simple form and yield, more generally, isomorphisms between Borovoi’s abelian K-cohomology of a reductive group G over K and the k-cohomology of a certain quotient of the algebraic fundamental group of G.     

On Singular Points of Solutions of the Minimal Surface Equation on Sets of Positive Measure

It is shown that, for any compact set K ? ? ⁿ (n ? 2) of positive Lebesgue measure and any bounded domain G ? K, there exists a function in the Hölder class C^1,1(G) that is a solution of the minimal surface equation in G \ K and cannot be extended from G \ K to G as a solution of this equation.     

On the minimum size of binary codes with length 2<Emphasis Type="Italic">R</Emphasis> + 4 and covering radius <Emphasis Type="Italic">R</Emphasis>

The minimum size of a binary code with length n and covering radius R is denoted by K(n, R). For arbitrary R, the value of K(n, R) is known when n ≤  2R +  3, and the corresponding optimal codes have been classified up to equivalence. By combining combinatorial and computational methods, several results for the first open case, K(2R +  4, R), are here obtained, including a proof that K(10, 3) =  12 with 11481 inequivalent optimal codes and a proof that if K(2R +  4, R) <  12 for some R then this inequality cannot be established by the existence of a corresponding self-complementary code.     

On the norm property of the Hilbert symbol for polynomial formal modules in a multidimensional local field

In a two-dimensional local field K containing the pth root of unity, a polynomial formal group F _c (X, Y) = X + Y + cXY acting on the maximal ideal M of the ring of integers б _K and a constructive Hilbert pairing {·, ·} _c : K ₂(K) × F _c (M) → <ξ> _c , where <ξ> _c is the module of roots of [p] _c (pth degree isogeny of F _c ) with respect to formal summation are considered. For the extension of two-dimensional local fields L/K, a norm map of Milnor groups Norm: K ₂(L) → K ₂(K) is considered. Its images are called norms in K ₂(L). The main finding of this study is that the norm property of pairing {·, ·}c: {x,β} _c : = 0 ? x is a norm in K ₂(K([p] _c ^-1 (β))), where [p] _c ^-1 (β) are the roots of the equation [p] _c = β, is checked constructively.     

A BKR Operation for Events Occurring for Disjoint Reasons with High Probability

Given events A and B on a product space \(S={\prod }_{i = 1}^{n} S_{i}\), the set \(A \Box B\) consists of all vectors x = (x₁,…,x_n) ∈ S for which there exist disjoint coordinate subsets K and L of {1,…,n} such that given the coordinates x_i,i ∈ K one has that x ∈ A regardless of the values of x on the remaining coordinates, and likewise that x ∈ B given the coordinates x_j,j ∈ L. For a finite product of discrete spaces endowed with a product measure, the BKR inequality

$$ P(A \Box B) \le P(A)P(B) $$

(1)

was conjectured by van den Berg and Kesten (J Appl Probab 22:556–569, 1985) and proved by Reimer (Combin Probab Comput 9:27–32, 2000). In Goldstein and Rinott (J Theor Probab 20:275–293, 2007) inequality Eq. 1 was extended to general product probability spaces, replacing \(A \Box B\) by the set Open image in new window consisting of those outcomes x for which one can only assure with probability one that x ∈ A and x ∈ B based only on the revealed coordinates in K and L as above. A strengthening of the original BKR inequality Eq. 1 results, due to the fact that Open image in new window . In particular, it may be the case that \(A \Box B\) is empty, while Open image in new window is not. We propose the further extension Open image in new window depending on probability thresholds s and t, where Open image in new window is the special case where both s and t take the value one. The outcomes Open image in new window are those for which disjoint sets of coordinates K and L exist such that given the values of x on the revealed set of coordinates K, the probability that A occurs is at least s, and given the coordinates of x in L, the probability of B is at least t. We provide simple examples that illustrate the utility of these extensions.

     

Minimal non-group-like twisted subsets with involutions

A subset K of some group G is called twisted if 1 ∈ K and xy ^?1 x ∈ K for all x, y ∈ K. We study the finite twisted subsets with an involution which are not subgroups but whose every proper twisted subset is a subgroup. We also consider the groups generated by twisted subsets.     

On the extension of Hölder maps with values in spaces of continuous functions

We study the isometric extension problem for Hölder maps from subsets of any Banach space intoc ₀ or into a space of continuous functions. For a Banach spaceX, we prove that anyα-Hölder map, with 0<α ≤1, from a subset ofX intoc ₀ can be isometrically extended toX if and only ifX is finite dimensional. For a finite dimensional normed spaceX and for a compact metric spaceK, we prove that the set ofα’s for which allα-Hölder maps from a subset ofX intoC(K) can be extended isometrically is either (0, 1] or (0, 1) and we give examples of both occurrences. We also prove that for any metric spaceX, the above described set ofα’s does not depend onK, but only on finiteness ofK.     

Two-parameter families of uniquely extendable Diophantine triples

Let A and K be positive integers and ε∈ {-2,-1,1,2}. The main contribution of the paper is a proof that each of the D(ε~2)-triples {K, A~2 K+2εA,(A +1)~2 K + 2ε(A+1)} has uniqui extension to a D(ε~2)-quadruple. This is used to slightly strengthen the conditions required for the existencc of a D(1)-quintuple whose smallest three elements form a regular triple.     

Kostka functions associated to complex reflection groups and a conjecture of Finkelberg-Ionov

Kostka functions K_(λ,μ)~±(t), indexed by r-partitions λ and μ of n, are a generalization of Kostka polynomials K_(λ,μ)(t) indexed by partitions λ,μ of n. It is known that Kostka polynomials have an interpretation in terms of Lusztig's partition function. Finkelberg and Ionov(2016) defined alternate functions K_(λ,μ)(t) by using an analogue of Lusztig's partition function, and showed that K_(λ,μ)(t) ∈Z≥0[t] for generic μ by making use of a coherent realization. They conjectured that K_(λ,μ)(t) coincide with K_(λ,μ)~-(t). In this paper, we show that their conjecture holds. We also discuss the multi-variable version, namely, r-variable Kostka functions K_(λ,μ)~±(t_1,…,t_r).     

Influence of <Emphasis Type="Italic">M</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-supplemented subgroups on the structure of <Emphasis Type="Italic">p</Emphasis>-modular subgroups

A subgroup K of G is M _p -supplemented in G if there exists a subgroup B of G such that G = KB and TB < G for every maximal subgroup T of K with |K: T| = p ^α. We study the structure of the chief factor of G by using M _p -supplemented subgroups and generalize the results of Monakhov and Shnyparkov by involving the relevant results about the p-modular subgroup O ^p (G) of G.     

Strict <Emphasis Type="Italic">K</Emphasis>-monotonicity and <Emphasis Type="Italic">K</Emphasis>-order continuity in symmetric spaces

This paper is devoted to strict K-monotonicity and K-order continuity in symmetric spaces. Using a local approach to the geometric structure in a symmetric space E we investigate a connection between strict K-monotonicity and global convergence in measure of a sequence of the maximal functions. Next, we solve an essential problem whether an existence of a point of K-order continuity in a symmetric space E on \([0,\infty )\) implies that the embedding \(E\hookrightarrow {L^1}[0,\infty )\) does not hold. We present a complete characterization of an equivalent condition to K-order continuity in a symmetric space E using a notion of order continuity and the fundamental function of E. We also investigate a relationship between strict K-monotonicity and K-order continuity in symmetric spaces and show some examples of Lorentz spaces and Marcinkiewicz spaces having these properties or not. Finally, we discuss a local version of a crucial correspondence between order continuity and the Kadec–Klee property for global convergence in measure in a symmetric space E.     

Extending Lipschitz maps into <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">K</Emphasis>)-spaces

We show that if K is a compact metric space then C(K) is a 2-absolute Lipschitz retract. We then study the best Lipschitz extension constants for maps into C(K) from a given metric space M, extending recent results of Lancien and Randrianantoanina. They showed that a finite-dimensional normed space which is polyhedral has the isometric extension property for C(K)-spaces; here we show that the same result holds for spaces with Gateaux smooth norm or of dimension two; a three-dimensional counterexample is also given. We also show that X is polyhedral if and only if every subset E of X has the universal isometric extension property for C(K)-spaces. We also answer a question of Naor on the extension of Hölder continuous maps.     

Orthogonal Polynomials Associated with Equilibrium Measures on $\mathbb {R}$

Let K be a non-polar compact subset of \(\mathbb {R}\) and μ _K denote the equilibrium measure of K. Furthermore, let P _n (?;μ _K ) be the n-th monic orthogonal polynomial for μ _K . It is shown that \(\|P_{n}\left (\cdot ; \mu _{K}\right )\|_{L^{2}(\mu _{K})}\), the Hilbert norm of P _n (?;μ _K ) in L ²(μ _K ), is bounded below by Cap(K) ⁿ for each \(n\in \mathbb {N}\). A sufficient condition is given for\(\left (\|P_{n}\left (\cdot ;\mu _{K}\right )\|_{L^{2}(\mu _{K})}/\text {Cap}(K)^{n}\right )_{n=1}^{\infty }\) to be unbounded. More detailed results are presented for sets which are union of finitely many intervals.     

Polar functions of multiparameter bifractional Brownian sheets

Let B^H,K ： （B^H,K（t）, t ∈R＋^N} be an （N,d）-bifractional Brownian sheet with Hurst indices H = （H1,..., HN） ∈ （0, 1）^N and K = （K1,..., KN）∈ （0, 1]^N. The characteristics of the polar functions for B^H,K are investigated. The relationship between the class of continuous functions satisfying the Lipschitz condition and the class of polar-functions of B^H,K is presented. The Hausdorff dimension of the fixed points and an inequality concerning the Kolmogorov＇s entropy index for B^H,K are obtained. A question proposed by LeGall about the existence of no-polar, continuous functions statisfying the Holder condition is also solved.