A Probabilistic Characterization of the Dominance Order on Partitions

We present a probabilistic characterization of the dominance order on partitions. Let ν be a partition and Y _ν its Ferrers diagram, i.e. a stack of rows of cells with row i containing ν _i cells. Let the cells of Y _ν be filled with independent and identically distributed draws from the random variable X = B i n(r, p) with r ≥ 1 and p ∈ (0, 1). Given j, t ≥ 0, let P(ν, j, t) be the probability that the sum of all the entries in Y _ν is j while the sum of the entries in each row of Y _ν is no more than t. It is shown that if ν and μ are two partitions of n, ν dominates μ if and only if P(ν, j, t) ≤ P(μ, j, t) for all j, t ≥ 0. It is shown that the same result holds if X is any log-concave integer valued random variable with {i : P(X = i) > 0} = {0, 1,…,r} for some r ≥ 1.     

Optimal boundary control by an elastic force at one end and a displacement at the other end for an arbitrary sufficiently large time interval in the string vibration problem

We consider a boundary value problem for the wave equation with given initial conditions and with boundary conditions of the second kind at one end of the string and boundary conditions of the first kind at the other end of the string. We assume the boundary conditions to ensure that the solution of the problem (in the class of generalized functions) satisfying the initial conditions at the initial time t = 0 satisfies given terminal conditions at the terminal time t = T. We clarify the relationship between the functions µ(t) and ν(t) in the boundary conditions and the given functions specifying the initial and terminal states. We obtain closed-form analytic expressions for the functions µ(t) and ν(t) minimizing the boundary energy functional.     

Ergodic properties of skew products in infinite measure

Let (Ω, µ) be a shift of finite type with a Markov probability, and (Y, ν) a non-atomic standard measure space. For each symbol i of the symbolic space, let Φ_i be a non-singular automorphism of (Y, ν). We study skew products of the form (ω, y) ? (σω, Φ_ω0 (y)), where σ is the shift map on (Ω, µ). We prove that, when the skew product is recurrent, it is ergodic if and only if the Φ_i’s have no common non-trivial invariant set.     

Polyboxes,Cube Tilings and Rigidity

A non-empty subset A of X=X ₁×???×X _d is a (proper) box if A=A ₁×???×A _d and A _i ?X _i for each i. Suppose that for each pair of boxes A, B and each i, one can only know which of the three states takes place: A _i =B _i , A _i =X _i ?B _i , A _i ?{B _i ,X _i ?B _i }. Let F and G be two systems of disjoint boxes. Can one decide whether ∪F=∪G? In general, the answer is ‘no’, but as is shown in the paper, it is ‘yes’ if both systems consist of pairwise dichotomous boxes. (Boxes A, B are dichotomous if there is i such that A _i =X _i ?B _i .) Several criteria that enable to compare such systems are collected. The paper includes also rigidity results, which say what assumptions have to be imposed on F to ensure that ∪F=∪G implies F=G. As an application, the rigidity conjecture for 2-extremal cube tilings of Lagarias and Shor is verified.     

Prime and composite integers close to powers of a number

We prove that, for any real numbers ξ ≠ 0 and ν, the sequence of integer parts [ξ2 ⁿ  + ν], n = 0, 1, 2, . . . , contains infinitely many composite numbers. Moreover, if the number ξ is irrational, then the above sequence contains infinitely many elements divisible by 2 or 3. The same holds for the sequence [ξ( ? 2) ⁿ  + ν _n ], n = 0, 1, 2, . . . , where ν ₀, ν ₁, ν ₂, . . . all lie in a half open real interval of length 1/3. For this, we show that if a sequence of integers x ₁, x ₂, x ₃, . . . satisfies the recurrence relation x _n+d  = cx _n  + F(x _n+1, . . . , x _n+d-1) for each n  ≥  1, where c ≠ 0 is an integer, \({F(z_1,\dots,z_{d-1}) \in \mathbb {Z}[z_1,\dots,z_{d-1}],}\) and lim _{n→ ∞}|x _n | = ∞, then the number |x _n | is composite for infinitely many positive integers n. The proofs involve techniques from number theory, linear algebra, combinatorics on words and some kind of symbolic computation modulo 3.     

Determinantal representations of smooth cubic surfaces

For every smooth (irreducible) cubic surface S we give an explicit construction of a representative for each of the 72 equivalence classes of determinantal representations. Equivalence classes (under GL₃ × GL₃ action by left and right multiplication) of determinantal representations are in one to one correspondence with the sets of six mutually skew lines on S and with the 72 (two-dimensional) linear systems of twisted cubic curves on S. Moreover, if a determinantal representation M corresponds to lines (a ₁,...,a ₆) then its transpose M ^t corresponds to lines (b ₁,...,b ₆) which together form a Schläfli’s double-six \(a_1\ldots a_6 \choose b_1\ldots b_6\) . We also discuss the existence of self-adjoint and definite determinantal representation for smooth real cubic surfaces. The number of these representations depends on the Segre type F _i . We show that a surface of type F _i , i = 1,2,3,4 has exactly 2(i?1) nonequivalent self-adjoint determinantal representations none of which is definite, while a surface of type F ₅ has 24 nonequivalent self-adjoint determinantal representations, 16 of which are definite.     

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.

     

Averaging of systems of differential inclusions with slow and fast variables

We consider the system of differential inclusions

$$\dot x \in \mu F(t, x, y, \mu ), x(0) = x_0 , \dot y \in G(t, x, y, \mu ), y(0) = y_0 $$

, where F,G: D → Kυ (\(R^{m_1 } \)), Kυ (\(R^{m_2 } \)) are mappings into the sets of nonempty convex compact sets in the Euclidean spaces \(R^{m_1 } \) and \(R^{m_2 } \), respectively, D = R ₊ × \(R^{m_1 } \) × \(R^{m_2 } \) × [0, a], a > 0, and µ is a small parameter. The functions F and G and the right-hand side of the averaged problem \(\dot u\) ∈ µF ₀(u), u(0) = x ₀, F ₀(u) ∈ Kυ (\(R^{m_1 } \)), satisfy the one-sided Lipschitz condition with respect to the corresponding phase variables. Under these and some other conditions, we prove that, for each ? > 0, there exists a µ > 0 such that, for an arbitrary µ ∈ (0, µ₀] and any solution x _µ(·), y _µ(·) of the original problem, there exists a solution u _µ(·) of the averaged problem such that ∥x _µ(t) ? y _µ(t) ∥ ≤ ? for t ∈ [0, 1/µ]. Furthermore, for each solution u _µ(·)of the averaged problem, there exists a solution x _µ(·), y _µ(·) of the original problem with the same estimate.

     

Series by Haar system with monotone coefficients

Let χ = {χ _n } _n=0 ^∞ be the Haar system normalized in L ²(0, 1) and M = {M _s } _s=1 ^∞ be an arbitrary, increasing sequence of nonnegative integers. For any subsystem of χ of the form {φ _k } = χS = {χ _n } _n∈S , where S = S(M) = {n _k } _k=1 ^∞ = {n ∈ V[p]: p ∈ M}, V[0] = {1, 2} and V[p] = {2 ^p + 1, 2 ^p + 2, …, 2 ^p+1} for p = 1, 2, … a series of the form Σ _i=1 ^∞ a _i φ _i with a _i ↘ 0 is constructed, that is universal with respect to partial series in all classes L ^r (0, 1), r ∈ (0, 1), in the sense of a.e. convergence and in the metric ofL ^r (0, 1). The constructed series is universal in the class of all measurable, finite functions on [0, 1] in the sense of a.e. convergence. It is proved that there exists a series by Haar system with decreasing coefficients, which has the following property: for any ? > 0 there exists a measurable function µ(x), x ∈ [0, 1], such that 0 ≤ µ(x) ≤ 1 and |{x ∈ [0, 1], µ(x) ≠ = 1}| < ?, and the series is universal in the weighted space L _µ[0, 1] with respect to subseries, in the sense of convergence in the norm of L _µ[0, 1].     

Lebesgue decomposition in action via semidefinite relaxations

Given all (finite) moments of two measures μ and λ on \(\mathbb {R}^{n}\), we provide a numerical scheme to obtain the Lebesgue decomposition μ = ν + ψ with ν?λ and ψ ⊥ λ. When ν has a density in \(L_{\infty }(\lambda )\) then we obtain two sequences of finite moments vectors of increasing size (the number of moments) which converge to the moments of ν and ψ respectively, as the number of moments increases. Importantly, no à priori knowledge on the supports of μ,ν and ψ is required.     

The Nöther and Riemann-Roch type theorems for piecewise algebraic curve

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. In this paper, the Nöther type theorems for C ^µ piecewise algebraic curves are obtained. The theory of the linear series of sets of places on the piecewise algebraic curve is also established. In this theory, singular cycles are put into the linear series, and a complete series of the piecewise algebraic curves consists of all effective ordinary cycles in an equivalence class and all effective singular cycles which are equivalent specifically to any effective ordinary cycle in the equivalence class. This theory is a generalization of that of linear series of the algebraic curve. With this theory and the fundamental theory of multivariate splines on smoothing cofactors and global conformality conditions, and the results on the general expression of multivariate splines, we get a formula on the index, the order and the dimension of a complete series of the irreducible C ^µ piecewise algebraic curves and the degree, the genus and the smoothness of the curves, hence the Riemann-Roch type theorem of the C ^µ piecewise algebraic curve is established.     

The dual space of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> of a vector measure

For a vector measure ν having values in a real or complex Banach space and \({p \in}\) [1, ∞), we consider L ^p (ν) and \({L_{w}^{p}(\nu)}\), the corresponding spaces of p-integrable and scalarly p-integrable functions. Given μ, a Rybakov measure for ν, and taking q to be the conjugate exponent of p, we construct a μ-Köthe function space E _q (μ) and show it is σ-order continuous when p > 1. In this case, for the associate spaces we prove that L ^p (ν) ^×  = E _q (μ) and \({E_q(\mu)^\times = L_w^p(\nu)}\). It follows that \({L_p (\nu) ^{**} = L_w^p (\nu)}\). We also show that L ¹ (ν) ^×  may be equal or not to E _∞ (μ).     

Extended phase properties and stability analysis of RKN-type integrators for solving general oscillatory second-order initial value problems

In this paper, we study in detail the phase properties and stability of numerical methods for general oscillatory second-order initial value problems whose right-hand side functions depend on both the position y and velocity y ^'. In order to analyze comprehensively the numerical stability of integrators for oscillatory systems, we introduce a novel linear test model y ^?(t) + ? ² y(t) + µ y ^'(t)=0 with µ<2?. Based on the new model, further discussions and analysis on the phase properties and stability of numerical methods are presented for general oscillatory problems. We give the new definitions of dispersion and dissipation which can be viewed as an essential extension of the traditional ones based on the linear test model y ^?(t) + ? ² y(t)=0. The numerical experiments are carried out, and the numerical results showthatthe analysisofphase properties and stability presentedinthispaper ismoresuitableforthenumericalmethodswhentheyareappliedtothe generaloscillatory second-order initial value problem involving both the position and velocity.     

A positive solution of a nonlinear Schrdinger system with nonconstant potentials

In this paper, we study the existence of positive solutions to the following Schr¨odinger system:{-?u + V_1(x)u = μ_1(x)u~3+ β(x)v~2u, x ∈R~N,-?v + V_2(x)v = μ_2(x)v~3+ β(x)u~2v, x ∈R~N,u, v ∈H~1(R~N),where N = 1, 2, 3; V_1(x) and V_2(x) are positive and continuous, but may not be well-shaped; and μ_1(x), μ_2(x)and β(x) are continuous, but may not be positive or anti-well-shaped. We prove that the system has a positive solution when the coefficients Vi(x), μ_i(x)(i = 1, 2) and β(x) satisfy some additional conditions.     

The optimal solution set of the interval linear programming problems

Several methods exist for solving the interval linear programming (ILP) problem. In most of these methods, we can only obtain the optimal value of the objective function of the ILP problem. In this paper we determine the optimal solution set of the ILP as the intersection of some regions, by the best and the worst case (BWC) methods, when the feasible solution components of the best problem are positive. First, we convert the ILP problem to the convex combination problem by coefficients 0 ≤ λ _j , μ _ij , μ _i  ≤ 1, for i = 1, 2, . . . , m and j = 1, 2, . . . , n. If for each i, j, μ _ij  = μ _i  = λ _j  = 0, then the best problem has been obtained (in case of minimization problem). We move from the best problem towards the worst problem by tiny variations of λ _j , μ _ij and μ _i from 0 to 1. Then we solve each of the obtained problems. All of the optimal solutions form a region that we call the optimal solution set of the ILP. Our aim is to determine this optimal solution set by the best and the worst problem constraints. We show that some theorems to validity of this optimal solution set.