Arithmetic properties of 7-regular partitions

Let \(b_{\ell }(n)\) denote the number of \(\ell \)-regular partitions of n. By employing the modular equation of seventh order, we establish the following congruence for \(b_{7}(n)\) modulo powers of 7: for \(n\ge 0\) and \(j\ge 1\),

$$\begin{aligned} b_{7}\left( 7^{2j-1}n+\frac{3\cdot 7^{2j}-1}{4}\right) \equiv 0 \pmod {7^j}. \end{aligned}$$

We also find some infinite families of congruences modulo 2 and 7 satisfied by \(b_{7}(n)\).

     

Congruences for 5-regular partitions modulo powers of 5

Let \(b_{5}(n)\) denote the number of 5-regular partitions of n. We find the generating functions of \(b_{5}(An+B)\) for some special pairs of integers (A, B). Moreover, we obtain infinite families of congruences for \(b_{5}(n)\) modulo powers of 5. For example, for any integers \(k\ge 1\) and \(n\ge 0\), we prove that

$$\begin{aligned} b_{5}\left( 5^{2k-1}n+\frac{5^{2k}-1}{6}\right) \equiv 0 \quad (\mathrm{mod}\, 5^{k}) \end{aligned}$$

and

$$\begin{aligned} b_{5}\left( 5^{2k}n+\frac{5^{2k}-1}{6}\right) \equiv 0 \quad (\mathrm{mod}\, 5^{k}). \end{aligned}$$

     

On $$\ell $$-regular bipartitions modulo $$\ell $$

Let \(B_\ell (n)\) denote the number of \(\ell \)-regular bipartitions of n. In this paper, we prove several infinite families of congruences satisfied by \(B_\ell (n)\) for \(\ell \in {\{5,7,13\}}\). For example, we show that for all \(\alpha >0\) and \(n\ge 0\),

$$\begin{aligned} B_5\left( 4^\alpha n+\frac{5\times 4^\alpha -2}{6}\right)\equiv & {} 0 \ (\text {mod}\ 5),\\ B_7\left( 5^{8\alpha }n+\displaystyle \frac{5^{8\alpha }-1}{2}\right)\equiv & {} 3^\alpha B_7(n)\ (\text {mod}\ 7) \end{aligned}$$

and

$$\begin{aligned} B_{13}\left( 5^{12\alpha }n+5^{12\alpha }-1\right) \equiv B_{13}(n)\ (\text {mod}\ 13). \end{aligned}$$

     

On $$(\ell,m)$$-regular partitions with distinct parts

Let \(a_{\ell ,m}(n)\) denote the number of \((\ell ,m)\)-regular partitions of a positive integer n into distinct parts, where \(\ell \) and m are relatively primes. In this paper, we establish several infinite families of congruences modulo 2 for \(a_{3,5}(n)\). For example,

$$\begin{aligned} a_{3, 5}\left(2^{6\alpha +4}5^{2\beta }n+\frac{ 2^{6\alpha +3}5^{2\beta +1}-1}{3}\right) \equiv 0 , \end{aligned}$$

where \(\alpha , \beta \ge 0\).

     

Large conformal metrics with prescribed sign-changing Gauss curvature

Let \((M,g)\) be a two dimensional compact Riemannian manifold of genus \(g(M)>1\). Let \(f\) be a smooth function on \(M\) such that

$$\begin{aligned} f \ge 0, \quad f\not \equiv 0, \quad \min _M f = 0. \end{aligned}$$

Let \(p_1,\ldots ,p_n\) be any set of points at which \(f(p_i)=0\) and \(D^2f(p_i)\) is non-singular. We prove that for all sufficiently small \(\lambda >0\) there exists a family of “bubbling” conformal metrics \(g_\lambda =e^{u_\lambda }g\) such that their Gauss curvature is given by the sign-changing function \(K_{g_\lambda }=-f+\lambda ^2\). Moreover, the family \(u_\lambda \) satisfies

$$\begin{aligned} u_\lambda (p_j) = -4\log \lambda -2\log \left( \frac{1}{\sqrt{2}} \log \frac{1}{\lambda }\right) +O(1) \end{aligned}$$

and

$$\begin{aligned} \lambda ^2e^{u_\lambda }\rightharpoonup 8\pi \sum _{i=1}^{n}\delta _{p_i},\quad \text{ as } \lambda \rightarrow 0, \end{aligned}$$

where \(\delta _{p}\) designates Dirac mass at the point \(p\).

     

Mean curvature bounds and eigenvalues of Robin Laplacians

We consider the Laplacian with attractive Robin boundary conditions,

$$\begin{aligned} Q^\Omega _\alpha u=-\Delta u, \quad \dfrac{\partial u}{\partial n}=\alpha u \text { on } \partial \Omega , \end{aligned}$$

in a class of bounded smooth domains \(\Omega \in \mathbb {R}^\nu \); here \(n\) is the outward unit normal and \(\alpha >0\) is a constant. We show that for each \(j\in \mathbb {N}\) and \(\alpha \rightarrow +\infty \), the \(j\)th eigenvalue \(E_j(Q^\Omega _\alpha )\) has the asymptotics

$$\begin{aligned} E_j(Q^\Omega _\alpha )=-\alpha ^2 -(\nu -1)H_\mathrm {max}(\Omega )\,\alpha +{\mathcal O}(\alpha ^{2/3}), \end{aligned}$$

where \(H_\mathrm {max}(\Omega )\) is the maximum mean curvature at \(\partial \Omega \). The discussion of the reverse Faber-Krahn inequality gives rise to a new geometric problem concerning the minimization of \(H_\mathrm {max}\). In particular, we show that the ball is the strict minimizer of \(H_\mathrm {max}\) among the smooth star-shaped domains of a given volume, which leads to the following result: if \(B\) is a ball and \(\Omega \) is any other star-shaped smooth domain of the same volume, then for any fixed \(j\in \mathbb {N}\) we have \(E_j(Q^B_\alpha )>E_j(Q^\Omega _\alpha )\) for large \(\alpha \). An open question concerning a larger class of domains is formulated.

     

The effect of the domain topology on the number of solutions of fractional Laplace problems

In this paper we are concerned with the multiplicity of solutions for the following fractional Laplace problem

$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}u= \mu |u|^{q-2}u + |u|^{2^*_s-2}u &{}\quad \text{ in } \Omega \\ u=0 &{}\quad \text{ in } {\mathbb {R}}^n{\setminus } \Omega , \end{array}\right. \end{aligned}$$

where \(\Omega \subset {\mathbb {R}}^n\) is an open bounded set with continuous boundary, \(n>2s\) with \(s\in (0,1),(-\Delta )^{s}\) is the fractional Laplacian operator, \(\mu \) is a positive real parameter, \(q\in [2, 2^*_s)\) and \(2^*_s=2n/(n-2s)\) is the fractional critical Sobolev exponent. Using the Lusternik–Schnirelman theory, we relate the number of nontrivial solutions of the problem under consideration with the topology of \(\Omega \). Precisely, we show that the problem has at least \(cat_{\Omega }(\Omega )\) nontrivial solutions, provided that \(q=2\) and \(n\geqslant 4s\) or \(q\in (2, 2^*_s)\) and \(n>2s(q+2)/q\), extending the validity of well-known results for the classical Laplace equation to the fractional nonlocal setting.

     

Spectrality of infinite convolutions with three-element digit sets

Let \(0< \rho <1\) and let \(\{a_n, b_n\}_{n=1}^\infty \) be a sequence of integers with bounded from upper and lower. Associated with them there exists a unique Borel probability measure \(\mu _{\rho , \{0, a_n, b_n\}}\) generated by the following infinite convolution product

$$\begin{aligned} \mu _{\rho , \{0, a_n, b_n\}}=\delta _{\rho \{0, a_1, b_1\}} *\delta _{\rho ^2 \{0, a_2, b_2\}} *\delta _{\rho ^3 \{0, a_3, b_3\}} *\cdots \end{aligned}$$

in the weak convergence, where \(\delta _E=\frac{1}{\# E}\sum _{e \in E} \delta _e\) and \(\hbox {gcd}(a_n, b_n)=1\) for all \(n \in {{\mathbb {N}}}\). In this paper, we show that \(L^2(\mu _{\rho , \{0, a_n, b_n\}})\) admits an exponential orthonormal basis if and only if \(\rho ^{-1} \in 3{{\mathbb {N}}}\) and  \(\{a_n, b_n\} \equiv \{1, 2\} \ (\mathrm {mod} \ 3)\) for all \(n \in {{\mathbb {N}}}\).

     

On Lipschitz continuity of solutions of hyperbolic Poisson’s equation

In this paper, we investigate solutions of the hyperbolic Poisson equation \(\Delta _{h}u(x)=\psi (x)\), where \(\psi \in L^{\infty }(\mathbb {B}^{n}, {\mathbb R}^n)\) and

$$\begin{aligned} \Delta _{h}u(x)= (1-|x|^2)^2\Delta u(x)+2(n-2)\left( 1-|x|^2\right) \sum _{i=1}^{n} x_{i} \frac{\partial u}{\partial x_{i}}(x) \end{aligned}$$

is the hyperbolic Laplace operator in the n-dimensional space \(\mathbb {R}^n\) for \(n\ge 2\). We show that if \(n\ge 3\) and \(u\in C^{2}(\mathbb {B}^{n},{\mathbb R}^n) \cap C(\overline{\mathbb {B}^{n}},{\mathbb R}^n )\) is a solution to the hyperbolic Poisson equation, then it has the representation \(u=P_{h}[\phi ]-G_{ h}[\psi ]\) provided that \(u\mid _{\mathbb {S}^{n-1}}=\phi \) and \(\int _{\mathbb {B}^{n}}(1-|x|^{2})^{n-1} |\psi (x)|\,d\tau (x)<\infty \). Here \(P_{h}\) and \(G_{h}\) denote Poisson and Green integrals with respect to \(\Delta _{h}\), respectively. Furthermore, we prove that functions of the form \(u=P_{h}[\phi ]-G_{h}[\psi ]\) are Lipschitz continuous.

     

Congruences involving $$g_n(x)=\sum \limits _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) x^k$$

Define \(g_n(x)=\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) x^k\) for \(n=0,1,2,\ldots \). Those numbers \(g_n=g_n(1)\) are closely related to Apéry numbers and Franel numbers. In this paper we establish some fundamental congruences involving \(g_n(x)\). For example, for any prime \(p>5\) we have

$$\begin{aligned} \sum _{k=1}^{p-1}\frac{g_k(-1)}{k}\equiv 0\pmod {p^2}\quad \text {and}\quad \sum _{k=1}^{p-1}\frac{g_k(-1)}{k^2}\equiv 0\pmod p. \end{aligned}$$

This is similar to Wolstenholme’s classical congruences

$$\begin{aligned} \sum _{k=1}^{p-1}\frac{1}{k}\equiv 0\pmod {p^2}\quad \text {and}\quad \sum _{k=1}^{p-1}\frac{1}{k^2}\equiv 0\pmod p \end{aligned}$$

for any prime \(p>3\).

     

Uniform Bounds on the Relative Error in the Approximation of Upper Quantiles for Sums of Arbitrary Independent Random Variables

Fix any \(n\ge 1\). Let \(\tilde{X}_1,\ldots ,\tilde{X}_n\) be independent random variables. For each \(1\le j \le n\), \(\tilde{X}_j\) is transformed in a canonical manner into a random variable \(X_j\). The \(X_j\) inherit independence from the \(\tilde{X}_j\). Let \(s_y\) and \(s_y^*\) denote the upper \(\frac{1}{y}{\underline{\text{ th }}}\) quantile of \(S_n=\sum _{j=1}^nX_j\) and \(S^*_n=\sup _{1\le k\le n}S_k\), respectively. We construct a computable quantity \(\underline{Q}_y\) based on the marginal distributions of \(X_1,\ldots ,X_n\) to produce upper and lower bounds for \(s_y\) and \(s_y^*\). We prove that for \(y\ge 8\)

$$\begin{aligned} 6^{-1} \gamma _{3y/16}\underline{Q}_{3y/16}\le s^*_{y}\le \underline{Q}_y \end{aligned}$$

where

$$\begin{aligned} \gamma _y=\frac{1}{2w_y+1} \end{aligned}$$

and \(w_y\) is the unique solution of

$$\begin{aligned} \Big (\frac{w_y}{e\ln (\frac{y}{y-2})}\Big )^{w_y}=2y-4 \end{aligned}$$

for \(w_y>\ln (\frac{y}{y-2})\), and for \(y\ge 37\)

$$\begin{aligned} \frac{1}{9}\gamma _{u(y)}\underline{Q}_{u(y)}<s_y \le \underline{Q}_y \end{aligned}$$

where

$$\begin{aligned} u(y)=\frac{3y}{32} \left( 1+\sqrt{1-\frac{64}{3y}}\right) . \end{aligned}$$

The distribution of \(S_n\) is approximately centered around zero in that \(P(S_n\ge 0) \ge \frac{1}{18}\) and \(P(S_n\le 0)\ge \frac{1}{65}\). The results extend to \(n=\infty \) if and only if for some (hence all) \(a>0\)

$$\begin{aligned} \sum _{j=1}^{\infty }E\{(\tilde{X}_j-m_j)^2\wedge a^2\}<\infty . \end{aligned}$$

(1)

     

On a Weighted Singular Integral Operator with Shifts and Slowly Oscillating Data

Let \(\alpha ,\beta \) be orientation-preserving diffeomorphism (shifts) of \(\mathbb {R}_+=(0,\infty )\) onto itself with the only fixed points \(0\) and \(\infty \) and \(U_\alpha ,U_\beta \) be the isometric shift operators on \(L^p(\mathbb {R}_+)\) given by \(U_\alpha f=(\alpha ')^{1/p}(f\circ \alpha )\), \(U_\beta f=(\beta ')^{1/p}(f\circ \beta )\), and \(P_2^\pm =(I\pm S_2)/2\) where

$$\begin{aligned} (S_2 f)(t):=\frac{1}{\pi i}\int \limits _0^\infty \left( \frac{t}{\tau }\right) ^{1/2-1/p}\frac{f(\tau )}{\tau -t}\,d\tau , \quad t\in \mathbb {R}_+, \end{aligned}$$

is the weighted Cauchy singular integral operator. We prove that if \(\alpha ',\beta '\) and \(c,d\) are continuous on \(\mathbb {R}_+\) and slowly oscillating at \(0\) and \(\infty \), and

$$\begin{aligned} \limsup _{t\rightarrow s}|c(t)|<1, \quad \limsup _{t\rightarrow s}|d(t)|<1, \quad s\in \{0,\infty \}, \end{aligned}$$

then the operator \((I-cU_\alpha )P_2^++(I-dU_\beta )P_2^-\) is Fredholm on \(L^p(\mathbb {R}_+)\) and its index is equal to zero. Moreover, its regularizers are described.

     

Continued fractions for some transcendental numbers

We consider series of the form

$$\begin{aligned} \frac{p}{q} +\sum _{j=2}^\infty \frac{1}{x_j}, \end{aligned}$$

where \(x_1=q\) and the integer sequence \((x_n)\) satisfies a certain non-autonomous recurrence of second order, which entails that \(x_n|x_{n+1}\) for \(n\ge 1\). It is shown that the terms of the sequence, and multiples of the ratios of successive terms, appear interlaced in the continued fraction expansion of the sum of the series, which is a transcendental number.

     

A Characterization of Two-Weight Norm Inequality for Littlewood–Paley $$g_{\lambda }^{*}$$-Function

Let \(n\ge 2\) and \(g_{\lambda }^{*}\) be the well-known high-dimensional Littlewood–Paley function which was defined and studied by E. M. Stein,

$$\begin{aligned} g_{\lambda }^{*}(f)(x) =\bigg (\iint _{\mathbb {R}^{n+1}_{+}} \Big (\frac{t}{t+|x-y|}\Big )^{n\lambda } |\nabla P_tf(y,t)|^2 \frac{\mathrm{d}y \mathrm{d}t}{t^{n-1}}\bigg )^{1/2}, \ \quad \lambda > 1, \end{aligned}$$

where \(P_tf(y,t)=p_t*f(y)\), \(p_t(y)=t^{-n}p(y/t)\), and \(p(x) = (1+|x|^2)^{-(n+1)/2}\), \(\nabla =(\frac{\partial }{\partial y_1},\ldots ,\frac{\partial }{\partial y_n},\frac{\partial }{\partial t})\). In this paper, we give a characterization of two-weight norm inequality for \(g_{\lambda }^{*}\)-function. We show that \(\big \Vert g_{\lambda }^{*}(f \sigma ) \big \Vert _{L^2(w)} \lesssim \big \Vert f \big \Vert _{L^2(\sigma )}\) if and only if the two-weight Muckenhoupt \(A_2\) condition holds, and a testing condition holds:

$$\begin{aligned} \sup _{Q : \text {cubes}~\mathrm{in} \ {\mathbb {R}^n}} \frac{1}{\sigma (Q)} \int _{{\mathbb {R}^n}} \iint _{\widehat{Q}} \Big (\frac{t}{t+|x-y|}\Big )^{n\lambda }|\nabla P_t(\mathbf {1}_Q \sigma )(y,t)|^2 \frac{w \mathrm{d}x \mathrm{d}t}{t^{n-1}} \mathrm{d}y < \infty , \end{aligned}$$

where \(\widehat{Q}\) is the Carleson box over Q and \((w, \sigma )\) is a pair of weights. We actually prove this characterization for \(g_{\lambda }^{*}\)-function associated with more general fractional Poisson kernel \(p^\alpha (x) = (1+|x|^2)^{-{(n+\alpha )}/{2}}\). Moreover, the corresponding results for intrinsic \(g_{\lambda }^*\)-function are also presented.

     

Least energy sign-changing solutions for a class of fourth order Kirchhoff-type equations in $$\mathbb {R}^{N}$$

In this article we study the problem

$$\begin{aligned} \Delta ^{2}u-\left( a+b\int _{\mathbb {R}^{N}}\left| \nabla u\right| ^{2}dx\right) \Delta u+V(x)u=\left| u\right| ^{p-2}u\ \text { in }\mathbb {R}^{N}, \end{aligned}$$

where \(\Delta ^{2}:=\Delta (\Delta )\) is the biharmonic operator, \(a,b>0\) are constants, \(N\le 7,\) \(p\in (4,2_{*})\) for \(2_{*}\) defined below, and \(V(x)\in C(\mathbb {R}^{N},\mathbb {R})\). Under appropriate assumptions on V(x), the existence of least energy sign-changing solution is obtained by combining the variational methods and the Nehari method.

     

Generalization of the non-local derangement identity and applications to multiple zeta-type series

The goal of this paper is the study of a transformation concerning the general K-fold finite sums of the form

$$\begin{aligned} \sum _{N\ge n_1\ge \cdots \ge n_K\ge 1}\frac{1}{b_{n_K}}\cdot \prod _{j=1}^{K-1}\frac{1}{a_{n_j}}, \end{aligned}$$

where \((K,N)\in \mathbb {N}^2\) and \(\{a_n\}_{n=1}^{\infty }\), \(\{b_n\}_{n=1}^{\infty }\) are appropriate real sequences. In the application part of our paper we apply the developed transformation to two special parametric multiple zeta-type series that generalize the well-know formula \(\zeta ^\star (\{2\}_K,1)=2\zeta (2K+1)\), \(K\in \mathbb {N}\). As a corollary of our parametric results, we also prove several sum formulas involving multiple zeta-star values.

     

Scaling Limits for Mixed Kernels

Let \(\mu \) and \(\nu \) be measures supported on \(\left( -1,1\right) \) with corresponding orthonormal polynomials \(\left\{ p_{n}^{\mu }\right\} \) and \( \left\{ p_{n}^{\nu }\right\} \), respectively. Define the mixed kernel

$$\begin{aligned} K_{n}^{{\mu },\nu }\left( x,y\right) =\sum _{j=0}^{n-1}p_{j}^{\mu }\left( x\right) p_{j}^{\nu }\left( y\right) . \end{aligned}$$

We establish scaling limits such as

$$\begin{aligned}&\lim _{n\rightarrow \infty }\frac{\pi \sqrt{1-\xi ^{2}}\sqrt{\mu ^{\prime }\left( \xi \right) \nu ^{\prime }\left( \xi \right) }}{n}K_{n}^{\mu ,\nu }\left( \xi +\frac{a\pi \sqrt{1-\xi ^{2}}}{n},\xi +\frac{b\pi \sqrt{1-\xi ^{2}}}{n}\right) \\&\quad =S\left( \frac{\pi \left( a-b\right) }{2}\right) \cos \left( \frac{\pi \left( a-b\right) }{2}+B\left( \xi \right) \right) , \end{aligned}$$

where \(S\left( t\right) =\frac{\sin t}{t}\) is the sinc kernel, and \(B\left( \xi \right) \) depends on \({\mu },\nu \) and \(\xi \). This reduces to the classical universality limit in the bulk when \(\mu =\nu \). We deduce applications to the zero distribution of \(K_{n}^{{\mu },\nu }\), and asymptotics for its derivatives.

     

Congruences modulo $$10^3$$ for Euler numbers

For a non-negative integer \(n\), let \(E_n\) be the \(n\) th Euler number. In this note, for any positive integer \(n\), we prove the following congruences:

$$\begin{aligned} {\left\{ \begin{array}{ll} E_{4n} \equiv 380n-375 \pmod {10^3}, \\ E_{4n+2} \equiv -460n+399 \pmod {10^3}. \end{array}\right. } \end{aligned}$$

Our proof is based on induction on \(n\) and elementary direct calculations.

     

A remark on exceptional sets in Erdös-Rényi limit theorem

Let \((x_i)_{i=1}^{+\infty }\) be the digits sequence in the unique terminating dyadic expansion of \(x\in [0,1)\). The run-length function \(l_n(x)\) is defined by

$$\begin{aligned} l_n(x):=\max \left\{ j:x_{i+1}=x_{i+2}=\cdots =x_{i+j}=1\ \text {for some}\ 0\le i\le n-j\right\} . \end{aligned}$$

Erdös and Rényi proved that

$$\begin{aligned} \lim _{n\rightarrow +\infty }\frac{l_n(x)}{\log _2{n}}=1, \text {a.e.}\ x\in [0,1). \end{aligned}$$

In this note, we show that for each pair of numbers \(\alpha ,\beta \in [0,+\infty ]\) with \(\alpha \le \beta \), the following exceptional set

$$\begin{aligned} E_{\alpha ,\beta }=\left\{ x\in [0,1):\liminf _{n\rightarrow +\infty }\frac{l_n(x)}{\log _2{n}}=\alpha ,\ \limsup _{n\rightarrow +\infty }\frac{l_n(x)}{\log _2{n}}=\beta \right\} \end{aligned}$$

has Hausdorff dimension one.

     

Complete determination of the zeta function of the Hilbert scheme of <Emphasis Type="Italic">n</Emphasis> points on a two-dimensional torus

We compute the coefficients of the polynomials \(C_n(q)\) defined by the equation

$$\begin{aligned} 1 + \sum _{n\ge 1} \, \frac{C_n(q)}{q^n} \, t^n = \prod _{i\ge 1}\, \frac{(1-t^i)^2}{1-(q+q^{-1})t^i + t^{2i}} \, . \end{aligned}$$

As an application we obtain an explicit formula for the zeta function of the Hilbert scheme of n points on a two-dimensional torus and show that this zeta function satisfies a remarkable functional equation. The polynomials \(C_n(q)\) are divisible by \((q-1)^2\). We also compute the coefficients of the polynomials \(P_n(q) = C_n(q)/(q-1)^2\): each coefficient counts the divisors of n in a certain interval; it is thus a non-negative integer. Finally we give arithmetical interpretations for the values of \(C_n(q)\) and of \(P_n(q)\) at \(q = -1\) and at roots of unity of order 3, 4, 6.