On the Diophantine equation $$X^{2N}+2^{2\alpha }5^{2\beta }{p}^{2\gamma } = Z^5$$

We prove that for each prime p, positive integer \(\alpha \), and non-negative integers \(\beta \) and \(\gamma \), the Diophantine equation \(X^{2N} + 2^{2\alpha }5^{2\beta }{p}^{2\gamma } = Z^5\) has no solution with N, X, \(Z\in \mathbb {Z}^+\), \(N > 1\), and \(\gcd (X,Z) = 1\).     

Generalized Drazin invertibility and local spectral theory

If \(A\in B(\mathcal{X})\) is an upper triangular Banach space operator with diagonal \((A_1,A_2)\), \(A_1\) invertible and \(A_2\) quasinilpotent, then \(A_1^{-1}\oplus A_2\) satisfies either of the single-valued extension property, Dunford’s condition (C), Bishop’s property \((\beta )\), decomposition property \((\delta )\) or is decomposable if and only if \(A_1\) has the property. The operator \(A^{-1}_1\oplus 0\) is subscalar (resp., left polaroid, right polaroid) if and only if \(A_1\) is subscalar (resp., left polaroid, right polaroid). For Drazin invertible operators A, with Drazin inverse B, this implies that B satisfies any one of these properties if and only if A satisfies the property.     

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

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

     

Ratio sets of random sets

We study the typical behaviour of the size of the ratio set A / A for a random subset \(A\subset \{1,\dots , n\}\). For example, we prove that \(|A/A|\sim \frac{2\text {Li}_2(3/4)}{\pi ^2}n^2 \) for almost all subsets \(A\subset \{1,\dots ,n\}\). We also prove that the proportion of visible lattice points in the lattice \(A_1\times \cdots \times A_d\), where \(A_i\) is taken at random in [1, n] with \(\mathbb P(m\in A_i)=\alpha _i\) for any \(m\in [1,n]\), is asymptotic to a constant \(\mu (\alpha _1,\dots ,\alpha _d)\) that involves the polylogarithm of order d.     

Anderson Polymer in a Fractional Brownian Environment: Asymptotic Behavior of the Partition Function

We consider the Anderson polymer partition function

$$\begin{aligned} u(t):=\mathbb {E}^X\left[ e^{\int _0^t \mathrm {d}B^{X(s)}_s}\right] \,, \end{aligned}$$

where \(\{B^{x}_t\,;\, t\ge 0\}_{x\in \mathbb {Z}^d}\) is a family of independent fractional Brownian motions all with Hurst parameter \(H\in (0,1)\), and \(\{X(t)\}_{t\in \mathbb {R}^{\ge 0}}\) is a continuous-time simple symmetric random walk on \(\mathbb {Z}^d\) with jump rate \(\kappa \) and started from the origin. \(\mathbb {E}^X\) is the expectation with respect to this random walk. We prove that when \(H\le 1/2\), the function u(t) almost surely grows asymptotically like \(e^{\lambda t}\), where \(\lambda >0\) is a deterministic number. More precisely, we show that as t approaches \(+\infty \), the expression \(\{\frac{1}{t}\log u(t)\}_{t\in \mathbb {R}^{>0}}\) converges both almost surely and in the \(\hbox {L}^1\) sense to some positive deterministic number \(\lambda \). For \(H>1/2\), we first show that \(\lim _{t\rightarrow \infty } \frac{1}{t}\log u(t)\) exists both almost surely and in the \(\hbox {L}^1\) sense and equals a strictly positive deterministic number (possibly \(+\infty \)); hence, almost surely u(t) grows asymptotically at least like \(e^{\alpha t}\) for some deterministic constant \(\alpha >0\). On the other hand, we also show that almost surely and in the \(\hbox {L}^1\) sense, \(\limsup _{t\rightarrow \infty } \frac{1}{t\sqrt{\log t}}\log u(t)\) is a deterministic finite real number (possibly zero), hence proving that almost surely u(t) grows asymptotically at most like \(e^{\beta t\sqrt{\log t}}\) for some deterministic positive constant \(\beta \). Finally, for \(H>1/2\) when \(\mathbb {Z}^d\) is replaced by a circle endowed with a Hölder continuous covariance function, we show that \(\limsup _{t\rightarrow \infty } \frac{1}{t}\log u(t)\) is a deterministic finite positive real number, hence proving that almost surely u(t) grows asymptotically at most like \(e^{c t}\) for some deterministic positive constant c.

     

Congruences for (3,11)-regular bipartitions modulo 11

In this note we investigate the function \(B_{k,\ell }(n)\), which counts the number of \((k,\ell )\)-regular bipartitions of n. We shall prove an infinite family of congruences modulo 11: for \(\alpha \ge 2\) and \(n\ge 0\),

$$\begin{aligned} B_{3,11}\left( 3^{\alpha }n+\frac{5\cdot 3^{\alpha -1}-1}{2}\right) \equiv 0\ (\mathrm{mod\ }11). \end{aligned}$$

     

Semidefinite bounds for nonbinary codes based on quadruples

For nonnegative integers q, n, d, let \(A_q(n,d)\) denote the maximum cardinality of a code of length n over an alphabet [q] with q letters and with minimum distance at least d. We consider the following upper bound on \(A_q(n,d)\). For any k, let \(\mathcal{C}_k\) be the collection of codes of cardinality at most k. Then \(A_q(n,d)\) is at most the maximum value of \(\sum _{v\in [q]^n}x(\{v\})\), where x is a function \(\mathcal{C}_4\rightarrow {\mathbb {R}}_+\) such that \(x(\emptyset )=1\) and \(x(C)=\!0\) if C has minimum distance less than d, and such that the \(\mathcal{C}_2\times \mathcal{C}_2\) matrix \((x(C\cup C'))_{C,C'\in \mathcal{C}_2}\) is positive semidefinite. By the symmetry of the problem, we can apply representation theory to reduce the problem to a semidefinite programming problem with order bounded by a polynomial in n. It yields the new upper bounds \(A_4(6,3)\le 176\), \(A_4(7,3)\le 596\), \(A_4(7,4)\le 155\), \(A_5(7,4)\le 489\), and \(A_5(7,5)\le 87\).     

Wiener–Ingham type inequality for Vilenkin groups and its application to harmonic analysis

An important trigonometric inequality essentially due to Wiener but later on made precise by Ingham concerning the lacunary trigonometric sums \(f(x)=\sum A_ke^{in_kx}\), where \(A_k\)’s are complex numbers, \(n_{-k}=-n_k\) and \(\{n_k\}\) satisfies the small gap condition \((n_{k+1}-n_k)\ge q\ge 1\) for \(k=0,1,2,\ldots \), says that if I is any subinterval of \([-\pi ,\pi ]\) of length \(|I|=2\pi (1+\delta )/q>2\pi /q\) then \(\sum |A_k|^2\le A_{\delta }|I|^{-1}\int _I|f|^2\), \(|A_k|\le A_{\delta }|I|^{-1}\int _I|f|\), wherein \(A_{\delta }\) depends only on \(\delta \). Such an inequality is proved here in the setting of the Vilenkin groups G. The inequality is then applied to generalize the Bernstěin, Szász and Ste?hkin type results concerning the absolute convergence of Fourier series on G.     

Extension of the Set of Complex Hadamard Matrices of Size 8

A complex Hadamard matrix is defined as a matrix H which fulfills two conditions, \(|H_{j,k}|=1\) for all j and k and \(HH^{*}=N \mathbb {I}_N\) where \(\mathbb {I}_N\) is an identity matrix of size N. We explore the set of complex Hadamard matrices \(\mathcal {H}_N\) of size \(N=8\) and present two previously unknown structures: a one-parametric, non-affine family \(T_8^{(1)}\) of complex Hadamard matrices and a single symmetric and isolated matrix \(A_8^{(0)}\).     

On the Dual Form of ‘Low <Emphasis Type="Italic">M</Emphasis><Superscript>*</Superscript> Estimate’ in the Quasi-convex Case

Let\(B_{2}^{n}\) denote the Euclidean ball in\({\mathbb R}^n\), and, given closed star-shaped body\(K \subset {\mathbb R}^{n}, M_{K}\) denote the average of the gauge of K on the Euclidean sphere. Let\(p \in (0,1)\) and let\(K \subset {\mathbb R}^{n}\) be a p-convex body. In [17] we proved that for every\(\lambda \in (0,1)\) there exists an orthogonal projection P of rank\((1 - \lambda)n\) such that

$\frac{f(\lambda)}{M_K} PB^{n}_{2} \subset PK,$

where\(f(\lambda)=c_p\lambda^{1+1/p}\) for some positive constant c _p depending on p only. In this note we prove that\(f(\lambda)\) can be taken equal to\(C_p\lambda^{1/p-1/2}\). In terms of Kolmogorov numbers it means that for every\(k \leq n\)

$d_k (\hbox{Id}:\ell^{n}_{2} \to ({\mathbb R}^{n},\|\cdot\|_{K})) \leq C_p \frac{n^{1/p-1}}{k^{1/p-1/2}} \ell(\hbox{ID}: \ell^{n}_{2} \to ({\mathbb R}^{n}, \|\cdot\|_{K})),$

where\(\ell(\hbox{Id})={\bf E}\|\sum\limits^{n}_{i=1}g_i e_i\|_K\) for the independent standard Gaussian random variables\(\{g_i\}\) and the canonical basis\(\{e_i\}\) of\({\mathbb R}^n\). All results do not require the symmetry of K.

     

Heinz mean curvature estimates in warped product spaces $$M\times _{e^{\psi }}N$$

If a graph submanifold (x, f(x)) of a Riemannian warped product space \((M^m\times _{e^{\psi }}N^n,\tilde{g}=g+ e^{2\psi }h)\) is immersed with parallel mean curvature H, then we obtain a Heinz-type estimation of the mean curvature. Namely, on each compact domain D of M, \(m\Vert H\Vert \le \frac{A_{\psi }(\partial D)}{V_{\psi }(D)}\) holds, where \(A_{\psi }(\partial D)\) and \(V_{\psi }(D)\) are the \({\psi }\)-weighted area and volume, respectively. In particular, \(H=0\) if (M, g) has zero-weighted Cheeger constant, a concept recently introduced by Impera et al. (Height estimates for killing graphs. arXiv:1612.01257, 2016). This generalizes the known cases \(n=1\) or \(\psi =0\). We also conclude minimality using a closed calibration, assuming \((M,g_*)\) is complete where \(g_*=g+e^{2\psi }f^*h\), and for some constants \(\alpha \ge \delta \ge 0\), \(C_1>0\) and \(\beta \in [0,1)\), \(\Vert \nabla ^*\psi \Vert ^2_{g_*}\le \delta \), \(\mathrm {Ricci}_{\psi ,g_*}\ge \alpha \), and \({\mathrm{det}}_g(g_*)\le C_1 r^{2\beta }\) holds when \(r\rightarrow +\infty \), where r(x) is the distance function on \((M,g_*)\) from some fixed point. Both results rely on expressing the squared norm of the mean curvature as a weighted divergence of a suitable vector field.     

Approximation of entire functions of exponential type by exponential sums<Superscript>*</Superscript>

Let K be a compact set in \( {{\mathbb R}^n} \). For \( 1 \leqslant p \leqslant \infty \), the Bernstein space \( B_K^p \) is the Banach space of all functions \( f \in {L^p}\left( {{{\mathbb R}^n}} \right) \)such that their Fourier transform in a distributional sense is supported on K. If \( f \in B_K^p \), then f is continuous on \( {{\mathbb R}^n} \) and has an extension onto the complex space \( {{\mathbb C}^n} \) to an entire function of exponential type K. We study the approximation of functions in \( B_K^p \) by finite τ -periodic exponential sums of the form

$ \sum\limits_m {{c_m}{e^{2\pi {\text{i}}\left( {x,m} \right)/\tau }}} $

in the \( {L^p}\left( {\tau {{\left[ { - 1/2,1/2} \right]}^n}} \right) \)-norm as τ → ∞ when K is a polytope in \( {{\mathbb R}^n} \).

     

Existence of positive solution for nonlocal singular fourth order Kirchhoff equation with Hardy potential

This paper is concerned with the existence of positive solution to a class of singular fourth order elliptic equation of Kirchhoff type

$$\begin{aligned} \triangle ^2 u-\lambda M(\Vert \nabla u\Vert ^2)\triangle u-\frac{\mu }{\vert x\vert ^4}u=\frac{h(x)}{u^\gamma }+k(x)u^\alpha , \end{aligned}$$

under Navier boundary conditions, \(u=\triangle u=0\). Here \(\varOmega \subset {\mathbf {R}}^N\), \(N\ge 1\) is a bounded \(C^4\)-domain, \(0\in \varOmega \), h(x) and k(x) are positive continuous functions, \(\gamma \in (0,1)\), \(\alpha \in (0,1)\) and \(M:{\mathbf {R}}^+\rightarrow {\mathbf {R}}^+\) is a continuous function. By using Galerkin method and sharp angle lemma, we will show that this problem has a positive solution for \(\lambda > \frac{\mu }{\mu ^*m_0}\) and \(0<\mu <\mu ^*\). Here \(\mu ^*=\Big (\frac{N(N-4)}{4}\Big )^2\) is the best constant in the Hardy inequality. Besides, if \(\mu =0\), \(\lambda >0\) and h, k are Lipschitz functions, we show that this problem has a positive smooth solution. If \(h,k\in C^{2,\,\theta _0}(\overline{\varOmega })\) for some \(\theta _0\in (0,1)\), then this problem has a positive classical solution.

     

One-sided invertibility of discrete operators and their applications

For \(p\in [1,\infty ]\), we establish criteria for the one-sided invertibility of binomial discrete difference operators \({{\mathcal {A}}}=aI-bV\) on the space \(l^p=l^p(\mathbb {Z})\), where \(a,b\in l^\infty \), I is the identity operator and the isometric shift operator V is given on functions \(f\in l^p\) by \((Vf)(n)=f(n+1)\) for all \(n\in \mathbb {Z}\). Applying these criteria, we obtain criteria for the one-sided invertibility of binomial functional operators \(A=aI-bU_\alpha \) on the Lebesgue space \(L^p(\mathbb {R}_+)\) for every \(p\in [1,\infty ]\), where \(a,b\in L^\infty (\mathbb {R}_+)\), \(\alpha \) is an orientation-preserving bi-Lipschitz homeomorphism of \([0,+\infty ]\) onto itself with only two fixed points 0 and \(\infty \), and \(U_\alpha \) is the isometric weighted shift operator on \(L^p(\mathbb {R}_+)\) given by \(U_\alpha f= (\alpha ^\prime )^{1/p}(f\circ \alpha )\). Applications of binomial discrete operators to interpolation theory are given.     

The nonlocal Liouville-type equation in $$\mathbb {}$$ and conformal immersions of the disk with boundary singularities

In this paper we perform a blow-up and quantization analysis of the fractional Liouville equation in dimension 1. More precisely, given a sequence \(u_k :\mathbb {R}\rightarrow \mathbb {R}\) of solutions to

$$\begin{aligned} (-\Delta )^\frac{1}{2} u_k =K_ke^{u_k}\quad \text {in} \quad \mathbb {R}, \end{aligned}$$

(1)

with \(K_k\) bounded in \(L^\infty \) and \(e^{u_k}\) bounded in \(L^1\) uniformly with respect to k, we show that up to extracting a subsequence \(u_k\) can blow-up at (at most) finitely many points \(B=\{a_1,\ldots , a_N\}\) and that either (i) \(u_k\rightarrow u_\infty \) in \(W^{1,p}_{{{\mathrm{loc}}}}(\mathbb {R}{\setminus } B)\) and \(K_ke^{u_k} {\mathop {\rightharpoonup }\limits ^{*}}K_\infty e^{u_\infty }+ \sum _{j=1}^N \pi \delta _{a_j}\), or (ii) \(u_k\rightarrow -\infty \) uniformly locally in \(\mathbb {R}{\setminus } B\) and \(K_k e^{u_k} {\mathop {\rightharpoonup }\limits ^{*}}\sum _{j=1}^N \alpha _j \delta _{a_j}\) with \(\alpha _j\ge \pi \) for every j. This result, resting on the geometric interpretation and analysis of (1) provided in a recent collaboration of the authors with T. Rivière and on a classical work of Blank about immersions of the disk into the plane, is a fractional counterpart of the celebrated works of Brézis–Merle and Li–Shafrir on the 2-dimensional Liouville equation, but providing sharp quantization estimates (\(\alpha _j=\pi \) and \(\alpha _j\ge \pi \)) which are not known in dimension 2 under the weak assumption that \((K_k)\) be bounded in \(L^\infty \) and is allowed to change sign.

     

Contributions to a conjecture of Mueller and Schmidt on Thue inequalities

Let \(F(X,Y)=\sum \nolimits _{i=0}^sa_iX^{r_i}Y^{r-r_i}\in {\mathbb {Z}}[X,Y]\) be a form of degree \(r=r_s\ge 3\), irreducible over \({\mathbb {Q}}\) and having at most \(s+1\) non-zero coefficients. Mueller and Schmidt showed that the number of solutions of the Thue inequality

$$\begin{aligned} |F(X,Y)|\le h \end{aligned}$$

is \(\ll s^2h^{2/r}(1+\log h^{1/r})\). They conjectured that \(s^2\) may be replaced by s. Let

$$\begin{aligned} \Psi = \max _{0\le i\le s} \max \left( \sum _{w=0}^{i-1} \frac{1}{r_i-r_w},\sum _{w= i+1}^{s}\frac{1}{r_w-r_i}\right) . \end{aligned}$$

Then we show that \(s^2\) may be replaced by \(\max (s\log ^3s, se^{\Psi })\). We also show that if \(|a_0|=|a_s|\) and \(|a_i|\le |a_0|\) for \(1\le i\le s-1\), then \(s^2\) may be replaced by \(s\log ^{3/2}s\). In particular, this is true if \(a_i\in \{-1,1\}\).

     

Values of general dirichlet series and simultaneous diophantine approximation

We show a joint denseness theorem for values of a general Dirichlet series \( \sum\nolimits_{n = 1}^\infty {{{\text{a}}_n}{{\text{e}}^{ - {\lambda_n}s}}} \) in a certain class and its derivatives, where the numbers {λ _n } are not necessarily linearly independent over \( \mathbb{Q} \).     

Littlewood-Paley Characterizations of Hajłasz-Sobolev and Triebel-Lizorkin Spaces via Averages on Balls

Let \(p\in (1,\infty )\) and \(q\in [1,\infty )\). In this article, the authors characterize the Triebel-Lizorkin space \({F}^{\alpha }_{p,q}(\mathbb {R}^{n})\) with smoothness order α ∈ (0, 2) via the Lusin-area function and the \(g_{\lambda }^{*}\)-function in terms of difference between f(x) and its ball average \(B_{t}f(x):=\frac 1{|B(x,t)|}{\int }_{B(x,t)}f(y)\,dy\) over the ball B(x, t) centered at \(x\in \mathbb {R}^{n}\) with radius t ∈ (0, 1). As an application, the authors obtain a series of characterizations of \(F^{\alpha }_{p,\infty }(\mathbb {R}^{n})\) via pointwise inequalities, involving ball averages, in spirit close to Haj?asz gradients, here some interesting phenomena naturally appear that, in the end-point case when α = 2, some of these pointwise inequalities characterize the Triebel-Lizorkin spaces \(F^{2}_{p,2}(\mathbb {R}^{n})\), while not \(F^{2}_{p,\infty }(\mathbb {R}^{n})\), and that some of other obtained pointwise characterizations are only known to hold true for \(F^{\alpha }_{p,\infty }(\mathbb {R}^{n})\) with \(p\in (1,\infty )\), α ∈ (0, 2) or α ∈ (n/p, 2). In particular, some new pointwise characterizations of Haj?asz-Sobolev spaces via ball averages are obtained. Since these new characterizations only use ball averages, they can be used as starting points for developing a theory of Triebel-Lizorkin spaces with smoothness orders not less than 1 on spaces of homogeneous type.     

New lower bounds for the Shannon capacity of odd cycles

The Shannon capacity of a graph G is defined as \(c(G)=\sup _{d\ge 1}(\alpha (G^d))^{\frac{1}{d}},\) where \(\alpha (G)\) is the independence number of G. The Shannon capacity of the cycle \(C_5\) on 5 vertices was determined by Lovász in 1979, but the Shannon capacity of a cycle \(C_p\) for general odd p remains one of the most notorious open problems in information theory. By prescribing stabilizers for the independent sets in \(C_p^d\) and using stochastic search methods, we show that \(\alpha (C_7^5)\ge 350\), \(\alpha (C_{11}^4)\ge 748\), \(\alpha (C_{13}^4)\ge 1534\), and \(\alpha (C_{15}^3)\ge 381\). This leads to improved lower bounds on the Shannon capacity of \(C_7\) and \(C_{15}\): \(c(C_7)\ge 350^{\frac{1}{5}}> 3.2271\) and \(c(C_{15})\ge 381^{\frac{1}{3}}> 7.2495\).     

On generalized Ramanujan primes

Let \(\Delta = \sum _{m=0}^\infty q^{(2m+1)^2} \in \mathbf {F}_2[[q]]\) be the reduction mod 2 of the \(\Delta \) series. A modular form of level 1, \(f=\sum _{n\geqslant 0} c(n) \,q^n\), with integer coefficients, is congruent modulo \(2\) to a polynomial in \(\Delta \). Let us set \(W_f(x)=\sum _{n\leqslant x,\ c(n)\text { odd }} 1\), the number of odd Fourier coefficients of \(f\) of index \(\leqslant x\). The order of magnitude of \(W_f(x)\) (for \(x\rightarrow \infty \)) has been determined by Serre in the seventies. Here, we give an asymptotic equivalent for \(W_f(x)\). Let \(p(n)\) be the partition function and \(A_0(x)\) (resp. \(A_1(x)\)) be the number of \(n\leqslant x\) such that \(p(n)\) is even (resp. odd). In the preceding papers, the second-named author has shown that \(A_0(x)\geqslant 0.28 \sqrt{x\;\log \log x}\) for \(x\geqslant 3\) and \(A_1(x)>\frac{4.57 \sqrt{x}}{\log x}\) for \(x\geqslant 7\). Here, it is proved that \(A_0(x)\geqslant 0.069 \sqrt{x}\;\log \log x\) holds for \(x>1\) and that \(A_1(x) \geqslant \frac{0.037 \sqrt{x}}{(\log x)^{7/8}}\) holds for \(x\geqslant 2\). The main tools used to prove these results are the determination of the order of nilpotence of a modular form of level-\(1\) modulo \(2\), and of the structure of the space of those modular forms as a module over the Hecke algebra, which have been given in a recent work of Serre and the second-named author.