Bose–Einstein distribution as a problem of analytic number theory: The case of less than two degrees of freedom

The problem of finding the number and the most likely shape of solutions of the system \(\sum\nolimits_{j = 0}^\infty {{\lambda _j}{n_j}} \leqslant M,\;\sum\nolimits_{j = 1}^\infty {{n_j}} = N\), where λ_j,M,N > 0 and N is an integer, as M,N →∞, can naturally be interpreted as a problem of analytic number theory. We solve this problem for the case in which the counting function of λ_j is of the order of λ^d/2, where d, the number of degrees of freedom, is less than two.     

Boundedness and compactness of an integral operator in a mixed norm space on the polydisk

We study the following integral type operator

$T_g (f)(z) = \int\limits_0^{z_{} } { \cdots \int\limits_0^{z_n } {f(\zeta _1 , \ldots ,\zeta _n )} g(\zeta _1 , \ldots ,\zeta _n )d\zeta _1 , \ldots ,\zeta _n } $

in the space of analytic functions on the unit polydisk U ⁿ in the complex vector space ?ⁿ. We show that the operator is bounded in the mixed norm space

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, with p, q ∈ [1, ∞) and α = (α₁, …, α_n), such that α_j > ?1, for every j = 1, …, n, if and only if \(\sup _{z \in U^n } \prod\nolimits_{j = 1}^n {\left( {1 - \left| {z_j } \right|} \right)} \left| {g(z)} \right| < \infty \). Also, we prove that the operator is compact if and only if \(\lim _{z \to \partial U^n } \prod\nolimits_{j = 1}^n {\left( {1 - \left| {z_j } \right|} \right)} \left| {g(z)} \right| = 0\).

     

On the exceptional sets in Sylvester expansions*

For any x ?? (0, 1], let the series \( {\sum}_{n=1}^{\infty }1/{d}_n(x) \) be the Sylvester expansion of x, where {d _j (x),?j?≥?1} is a sequence of positive integers satisfying d₁(x)?≥?2 and d_j?+?1(x)?≥?d _j (x)(d _j (x)???1)?+?1 for j?≥?1. Suppose ? : ? → ?⁺ is a function satisfying ?(n+1) – ? (n) → ∞ as n → ∞. In this paper, we consider the set

$$ E\left(\phi \right)=\left\{x\kern0.5em \in \left(0,1\right]:\kern0.5em \underset{n\to \infty }{\lim}\frac{\log {d}_n(x)-{\sum}_{j=1}^{n-1}\log {d}_j(x)}{\phi (n)}=1\right\} $$

and quantify the size of the set in the sense of Hausdorff dimension. As applications, for any β > 1 and γ > 0, we get the Hausdorff dimension of the set \( \left\{x\in \kern1em \left(0,1\right]:\kern0.5em {\lim}_{n\to \infty}\left(\log {d}_n(x)-{\sum}_{j=1}^{n-1}\log {d}_j(x)\right)/{n}^{\beta }=\upgamma \right\}, \) and for any τ > 1 and η > 0, we get a lower bound of the Hausdorff dimension of the set \( \left\{x\kern0.5em \in \kern0.5em \left(0,1\right]:\kern1em {\lim}_{n\to \infty}\left(\log {d}_n(x)-{\sum}_{j=1}^{n-1}\log {d}_j(x)\right)/{\tau}^n=\eta \right\}. \)     

Matrix polynomial generalizations of the sample variance-covariance matrix when <Emphasis Type="Italic">pn</Emphasis><Superscript>−1</Superscript> → <Emphasis Type="Italic">y</Emphasis> ∈ (0, ∞)

Let {Z _u = ((ε_{u, i, j}))_p×n} be random matrices where {ε_{u, i, j}} are independently distributed. Suppose {A _i }, {B _i } are non-random matrices of order p × p and n × n respectively. Consider all p × p random matrix polynomials \(P = \prod\nolimits_{i = 1}^{k_l } {\left( {n^{ - 1} A_{t_i } Z_{j_i } B_{s_i } Z_{j_i }^* } \right)A_{t_{k_l + 1} } }\). We show that under appropriate conditions on the above matrices, the elements of the non-commutative *-probability space Span {P} with state p^?1ETr converge. As a by-product, we also show that the limiting spectral distribution of any self-adjoint polynomial in Span{P} exists almost surely.     

On the distribution of integer random variables related by a certain linear inequality: III

We consider tuples {N _jk }, j = 1, 2, ..., k = 1, ..., q _j , of nonnegative integers such that $$ \sum\limits_{j = 1}^\infty {\sum\limits_{k = 1}^{q_j } {jN_{jk} } } \leqslant M. $$ Assuming that q _j ～ j ^d?1, 1 < d < 2, we study how the probabilities of deviations of the sums $ \sum\nolimits_{j = j_1 }^{j_2 } {\sum\nolimits_{k = 1}^{q_j } {N_{jk} } } $ N _jk from the corresponding integrals of the Bose-Einstein distribution depend on the choice of the interval [j ₁,j ₂].     

On <Emphasis Type="Italic">β</Emphasis>-expansions of unity for rational and transcendental numbers <Emphasis Type="Italic">β</Emphasis>

We investigate the sequence of integers x ₁, x ₂, x ₃, … lying in {0, 1, …, [β]} in a so-called Rényi β-expansion of unity 1 = \(\sum\limits_{j = 1}^\infty {x_j \beta ^{ - j} } \) for rational and transcendental numbers β > 1. In particular, we obtain an upper bound for two strings of consecutive zeros in the β-expansion of unity for rational β. For transcendental numbers β which are badly approximable by algebraic numbers of every large degree and bounded height, we obtain an upper bound for the Diophantine exponent of the sequence X = (xj) _j=1 ^∞ in terms of β.     

Estimates for sums of moduli of blocks in trigonometric Fourier series

We consider the following two problems. Problem 1: what conditions on a sequence of finite subsets A _k ? ? and a sequence of functions λ _k : A _k → ? provide the existence of a number C such that any function f ∈ L ₁ satisfies the inequality ‖U _A,Λ(f)‖ _p ≤ C‖f‖₁ and what is the exact constant in this inequality? Here, \(U_{\mathcal{A},\Lambda } \left( f \right)\left( x \right) = \sum\nolimits_{k = 1}^\infty {\left| {\sum\nolimits_{m \in A_k } {\lambda _k \left( m \right)c_m \left( f \right)e^{imx} } } \right|}\) and c _m (f) are Fourier coefficients of the function f ∈ L ₁. Problem 2: what conditions on a sequence of finite subsets A _k ? ? guarantee that the function \(\sum\nolimits_{k = 1}^\infty {\left| {\sum\nolimits_{m \in A_k } {c_m \left( h \right)e^{imx} } } \right|}\) belongs to L _p for every function h of bounded variation?     

Meromorphic Jost functions and asymptotic expansions for Jacobi parameters

We show that the parameters a _n , b _n of a Jacobi matrix have a complete asymptotic expansion

$a_n^2 - 1 = \sum\limits_{k = 1}^{K(R)} {p_k (n)\mu _k^{ - 2n} + O(R^{ - 2n} ),} b_n = \sum\limits_{k = 1}^{K(R)} {p_k (n)\mu _k^{ - 2n + 1} + O(R^{ - 2n} )} $

, where 1 < |µ_j| < R for j ? K(R) and all R, if and only if the Jost function, u, written in terms of z (where E = z + z ^?1) is an entire meromorphic function. We relate the poles of u to the µ_j’s.

     

On the nonexistence of solutions of some anisotropic elliptic and backward parabolic inequalities

We study the nonexistence of weak solutions of higher-order elliptic and parabolic inequalities of the following types: \(\sum {_{i = 1}^N\sum\nolimits_{{e_i} \leqslant {\alpha _i} \leqslant {m_i}} {D_{{x_i}}^{{\alpha _i}}\left( {{A_{{\alpha _i}}}\left( {x,u} \right)} \right)} \geqslant f\left( {x,u} \right),} x \in {\mathbb{R}^N}\), and \({u_t} + \sum {_{i = 1}^N\sum\nolimits_{{k_i} \leqslant {\beta _i} \leqslant {n_i}} {D_{{x_i}}^{{\beta _i}}\left( {{B_{{\beta _i}}}\left( {x,t,u} \right)} \right)} > g\left( {x,t,u} \right),\left( {x,t} \right)} \in {\mathbb{R}^N} \times {\mathbb{R}_ + }\), where l _i , m _i , k _i , n _i ∈ N satisfy the condition l _i , k _i > 1 for all i = 1,..., N, and A _αi (x, u), B _βi (x, t, u), f(x, u), and g(x, t, u) are some given Carathéodory functions. Under appropriate conditions on the functions A _αi , B _βi , f, and g, we prove theorems on the nonexistence of solutions of these inequalities.     

On <Emphasis Type="Italic">s</Emphasis>-Elementary Super Frame Wavelets and Their Path-Connectedness

A super wavelet of length n is an n-tuple (ψ ₁,ψ ₂,…,ψ _n ) in the product space \(\prod_{j=1}^{n} L^{2}(\mathbb{R})\), such that the coordinated dilates of all its coordinated translates form an orthonormal basis for \(\prod_{j=1}^{n} L^{2} (\mathbb{R})\). This concept is generalized to the so-called super frame wavelets, super tight frame wavelets and super normalized tight frame wavelets (or super Parseval frame wavelets), namely an n-tuple (η ₁,η ₂,…,η _n ) in \(\prod_{j=1}^{n}L^{2} (\mathbb{R})\) such that the coordinated dilates of all its coordinated translates form a frame, a tight frame, or a normalized tight frame for \(\prod_{j=1}^{n} L^{2}(\mathbb{R})\). In this paper, we study the super frame wavelets and the super tight frame wavelets whose Fourier transforms are defined by set theoretical functions (called s-elementary frame wavelets). An m-tuple of sets (E ₁,E ₂,…,E _m ) is said to be τ-disjoint if the E _j ’s are pair-wise disjoint under the 2π-translations. We prove that a τ-disjoint m-tuple (E ₁,E ₂,…,E _m ) of frame sets (i.e., η _j defined by \(\widehat{\eta_{j}}=\frac{1}{\sqrt{2\pi}}\chi_{E_{j}}\) is a frame wavelet for L ²(?) for each j) lead to a super frame wavelet (η ₁,η ₂,…,η _m ) for \(\prod_{j=1}^{m} L^{2} (\mathbb{R})\) where \(\widehat{\eta_{j}}=\frac{1}{\sqrt{2\pi}}\chi_{E_{j}}\). In the case of super tight frame wavelets, we prove that (η ₁,η ₂,…,η _m ), defined by \(\widehat{\eta_{j}}=\frac{1}{\sqrt{2\pi}}\chi_{E_{j}}\), is a super tight frame wavelet for ∏_1≤j≤m L ²(?) with frame bound k ₀ if and only if each η _j is a tight frame wavelet for L ²(?) with frame bound k ₀ and that (E ₁,E ₂,…,E _m ) is τ-disjoint. Denote the set of all τ-disjoint s-elementary super frame wavelets for ∏_1≤j≤m L ²(?) by \(\mathfrak{S}(m)\) and the set of all s-elementary super tight frame wavelets (with the same frame bound k ₀) for ∏_1≤j≤m L ²(?) by \(\mathfrak{S}^{k_{0}}(m)\). We further prove that \(\mathfrak{S}(m)\) and \(\mathfrak{S}^{k_{0}}(m)\) are both path-connected under the ∏_1≤j≤m L ²(?) norm, for any given positive integers m and k ₀.     

Information of orderα and typeβ

Information I_α ^β (Q/P) of orderα and typeβ is introduced and it is shown that for every fixedβ, this information is a monotonic increasing function ofα. It is also shown that information of orderα and type 1 is non-negative when\(\sum\limits_{k = 1}^N { q_k } \geqslant \sum\limits_{k = 1}^N { p_k } \), where (q ₁,q ₂ …,q _N) and (p ₁,p ₂, …,p _N) are generalised probability distributions for Q and P respectively.     

Multiple recurrence and convergence for hardy sequences of polynomial growth

We study the limiting behavior of multiple ergodic averages involving sequences of integers that satisfy some regularity conditions and have polynomial growth. We show that for “typical” choices of Hardy field functions a(t) with polynomial growth, the averages

${1 \over N}\sum\nolimits_{n = 1}^N {{f_1}({T^{[a(n)]}}x) \cdots {f_\ell }({T^{\ell [a(n)]}}x)} $

converge in mean and we determine their limit. For example, this is the case if a(t) = t ^3/2, t log t, or t ² + (log t)². Furthermore, if {a ₁(t), …, a _? (t)} is a “typical” family of logarithmico-exponential functions of polynomial growth, then for every ergodic system, the averages

${1 \over N}\sum\nolimits_{n = 1}^N {{f_1}({T^{[{a_1}(n)]}}x) \cdots {f_\ell }({T^{[{a_\ell }(n)]}}x)} $

converge in mean to the product of the integrals of the corresponding functions. For example, this is the case if the functions a _i (t) are given by different positive fractional powers of t. We deduce several results in combinatorics. We show that if a(t) is a non-polynomial Hardy field function with polynomial growth, then every set of integers with positive upper density contains arithmetic progressions of the form {m,m + [a(n)], …, m + ?[a(n)]}. Under suitable assumptions, we get a related result concerning patterns of the form {m,m + [a ₁(n)], …,m + [a _? (n)]}.

     

A class of simple Lie algebras attached to unit forms

Let n ≥ 3. The complex Lie algebra, which is attached to a unit form q(x ₁, x ₂,..., x _n) = \({\sum\nolimits_{i = 1}^n {x_i^2 + \sum\nolimits_{1 \leqslant i \leqslant j \leqslant n} {\left( { - 1} \right)} } ^{j - i}}{x_i}{x_j}\) and defined by generators and generalized Serre relations, is proved to be a finite-dimensional simple Lie algebra of type A _n , and realized by the Ringel-Hall Lie algebra of a Nakayama algebra of radical square zero. As its application of the realization, we give the roots and a Chevalley basis of the simple Lie algebra.     

Divided differences for functions of two variables for irregularly spaced arguments

Divided differences forf (x, y) for completely irregular spacing of points (x _i ,y _i ) are developed here by a natural generalization of Newton's scheme. Existing bivariate schemes either iterate the one-dimensional scheme, thus constraining (x _i ,y _i ) to be at corners of rectangles, or give polynomials Σa _jk x ^j y ^k having more coefficients than interpolation conditions. Here the generalizedn ^th divided difference is defined by (1)\(\left[ {01... n} \right] = \sum\limits_{i = 0}^n {A_i f\left( {x_i , y_i } \right)} \) where (2)\(\sum\limits_{i = 0}^n {A_i x_i^j , y_i^k = 0} \), and 1 for the last or (n+1)^th equation, for every (j, k) wherej+k=0, 1, 2,... in the usual ascending order. The gen. div. diff. [01...n] is symmetric in (x _i ,y _i ), unchanged under translation, 0 forf (x, y) an, ascending binary polynomial as far asn terms, degree-lowering with respect to (X, Y) whenf(x, y) is any polynomialP(X+x, Y+y), and satisfies the 3-term recurrence relation (3) [01...n]=λ{[1...n]?[0...n?1]}, where (4) λ= |1...n|·|01...n?1|/|01...n|·|1...n?1|, the |...i...| denoting determinants inx _i ^j y _i ^k . The generalization of Newton's div. diff. formula is (5)

$$\begin{gathered} f\left( {x, y} \right) = f\left( {x_0 , y_0 } \right) - \frac{{\left| {\alpha 0} \right|}}{{\left| 0 \right|}}\left[ {01} \right] + \frac{{\left| {\alpha 01} \right|}}{{\left| {01} \right|}}\left[ {012} \right] - \frac{{\left| {\alpha 012} \right|}}{{\left| {012} \right|}}\left[ {0123} \right] + \cdots + \hfill \\ + \left( { - 1} \right)^n \frac{{\left| {\alpha 01 \ldots n - 1} \right|}}{{\left| {01 \ldots n - 1} \right|}}\left[ {01 \ldots n} \right] + \left( { - 1} \right)^{n + 1} \frac{{\left| {\alpha 01 \ldots n} \right|}}{{\left| {01 \ldots n} \right|}}\left[ {01 \ldots n} \right], \hfill \\ \end{gathered} $$     

Riesz transform characterization of Hardy spaces associated with Schrödinger operators with compactly supported potentials

Let L=?Δ+V be a Schrödinger operator on ? ^d , d≥3. We assume that V is a nonnegative, compactly supported potential that belongs to L ^p (? ^d ), for some p>d /2. Let K _t be the semigroup generated by ?L. We say that an L ¹(? ^d )-function f belongs to the Hardy space \(H^{1}_{L}\) associated with L if sup?_t>0|K _t f| belongs to L ¹(? ^d ). We prove that \(f\in H^{1}_{L}\) if and only if R _j f∈L ¹(? ^d ) for j=1,…,d, where R _j =(?/? x _j )L ^?1/2 are the Riesz transforms associated with L.     

Asymptotic distribution of singular values of powers of random matrices

Let x be a complex random variable such that \( {\mathbf{E}}x = 0,\,{\mathbf{E}}{\left| x \right|^2} = 1 \), and \( {\mathbf{E}}{\left| x \right|^4} < \infty \). Let \( {x_{ij}},i,j \in \left\{ {1,2, \ldots } \right\} \), be independent copies of x. Let \( {\mathbf{X}} = \left( {{N^{ - 1/2}}{x_{ij}}} \right) \), 1≤i,j≤N, be a random matrix. Writing X ^? for the adjoint matrix of X, consider the product X ^m X ^?m with some m ∈{1,2,...}. The matrix X ^m X ^?m is Hermitian positive semidefinite. Let λ₁,λ₂,...,λ _N be eigenvalues of X ^m X ^?m (or squared singular values of the matrix X ^m ). In this paper, we find the asymptotic distribution function \( {G^{(m)}}(x) = {\lim_{N \to \infty }}{\mathbf{E}}F_N^{(m)}(x) \) of the empirical distribution function \( F_N^{(m)}(x) = {N^{ - 1}}\sum\nolimits_{k = 1}^N {\mathbb{I}\left\{ {{\lambda_k} \leqslant x} \right\}} \), where \( \mathbb{I}\left\{ A \right\} \) stands for the indicator function of an event A. With m=1, our result turns to a well-known result of Marchenko and Pastur [V. Marchenko and L. Pastur, The eigenvalue distribution in some ensembles of random matrices, Math. USSR Sb., 1:457–483, 1967].     

Finite Gap Jacobi Matrices,III. Beyond the Szegő Class

Let \({\frak {e}}\subset {\mathbb {R}}\) be a finite union of ?+1 disjoint closed intervals, and denote by ω _j the harmonic measure of the j left-most bands. The frequency module for \({\frak {e}}\) is the set of all integral combinations of ω ₁,…,ω _? . Let \(\{\tilde{a}_{n}, \tilde{b}_{n}\}_{n=-\infty}^{\infty}\) be a point in the isospectral torus for \({\frak {e}}\) and \(\tilde{p}_{n}\) its orthogonal polynomials. Let \(\{a_{n},b_{n}\}_{n=1}^{\infty}\) be a half-line Jacobi matrix with \(a_{n} = \tilde{a}_{n} + \delta a_{n}\), \(b_{n} = \tilde{b}_{n} +\delta b_{n}\). Suppose

$\sum_{n=1}^\infty \lvert \delta a_n\rvert ^2 + \lvert \delta b_n\rvert ^2 <\infty $

and \(\sum_{n=1}^{N} e^{2\pi i\omega n} \delta a_{n}\), \(\sum_{n=1}^{N} e^{2\pi i\omega n} \delta b_{n}\) have finite limits as N→∞ for all ω in the frequency module. If, in addition, these partial sums grow at most subexponentially with respect to ω, then for z∈???, \(p_{n}(z)/\tilde{p}_{n}(z)\) has a limit as n→∞. Moreover, we show that there are non-Szeg? class J’s for which this holds.

     

Generalized Kloosterman sum with primes

The work is devoted to generalized Kloosterman sums modulo a prime, i.e., trigonometric sums of the form \(\sum\nolimits_{p \leqslant x} {\exp \left\{ {2\pi i\left( {a\bar p + {F_k}\left( p \right)} \right)/q} \right\}} \) and \(\sum\nolimits_{n \leqslant x} {\mu \left( n \right)\exp \left\{ {2\pi i\left( {a\bar n + {F_k}\left( n \right)} \right)/q} \right\}} \), where q is a prime number, \(\left( {a,q} \right) = 1,m\bar m \equiv 1\left( {\bmod {\kern 1pt} q} \right)\), F _k (u) is a polynomial of degree k ≥ 2 with integer coefficients, and p runs over prime numbers. An upper estimate with a power saving is obtained for the absolute values of such sums for x ≥ q^1/2+ε.     

Sobolev spaces and approximation by affine spanning systems

We develop conditions on a Sobolev function \(\psi \in W^{m,p}({\mathbb{R}}^d)\) such that if \(\widehat{\psi}(0) = 1\) and ψ satisfies the Strang–Fix conditions to order m ? 1, then a scale averaged approximation formula holds for all \(f \in W^{m,p}({\mathbb{R}}^d)\) :

$ f(x) = \lim_{J \to \infty} \frac{1}{J} \sum_{j=1}^{J} \sum_{k \in {{\mathbb{Z}}}^d} c_{j,k}\psi(a_j x - k) \quad {\rm in} W^{m, p}({{\mathbb{R}}}^d).$

The dilations { a _j } are lacunary, for example a _j =  2 ^j , and the coefficients c _j,k are explicit local averages of f, or even pointwise sampled values, when f has some smoothness. For convergence just in \({W^{m - 1,p}({\mathbb{R}}^d)}\) the scale averaging is unnecessary and one has the simpler formula \(f(x) = \lim_{j \to \infty} \sum_{k \in {\mathbb{Z}}^d} c_{j,k}\psi(a_j x - k)\) . The Strang–Fix rates of approximation are recovered. As a corollary of the scale averaged formula, we deduce new density or “spanning” criteria for the small scale affine system \(\{\psi(a_j x - k) : j > 0, k \in {\mathbb{Z}}^d \}\) in \(W^{m,p}({\mathbb{R}}^d)\) . We also span Sobolev space by derivatives and differences of affine systems, and we raise an open problem: does the Gaussian affine system span Sobolev space?

     

On the derivative of the Minkowski question mark function ?(<Emphasis Type="Italic">x</Emphasis>)

Let x?=?[0; a ₁ , a ₂ , …] be the regular continued fraction expansion of an irrational number x ∈ [0, 1]. For the derivative of the Minkowski function ?(x) we prove that ?′(x)?=?+∞, provided that \( \mathop {{\lim \sup }}\limits_{t \to \infty } \frac{{{a_1} + \cdots + {a_t}}}{t} < {\kappa_1} = \frac{{2\log {\lambda_1}}}{{\log 2}} = {1.388^{+} } \), and ?′(x)?=?0, provided that \( \mathop {{\lim \inf }}\limits_{t \to \infty } \frac{{{a_1} + \cdots + {a_t}}}{t} > {\kappa_2} = \frac{{4{L_5} - 5{L_4}}}{{{L_5} - {L_4}}} = {4.401^{+} } \), where \( {L_j} = \log \left( {\frac{{j + \sqrt {{{j^2} + 4}} }}{2}} \right) - j \cdot \frac{{\log 2}}{2} \). Constants κ₁, κ₂ are the best possible. It is also shown that ?′(x)?=?+∞ for all x with partial quotients bounded by 4.