Infinitely many congruences modulo 5 for 4-colored Frobenius partitions

In his 1984 AMS Memoir, Andrews introduced the family of functions \(c\phi _k(n),\) which denotes the number of generalized Frobenius partitions of \(n\) into \(k\) colors. Recently, Baruah and Sarmah, Lin, Sellers, and Xia established several Ramanujan-like congruences for \(c\phi _4(n)\) relative to different moduli. In this paper, employing classical results in \(q\)-series, the well-known theta functions of Ramanujan, and elementary generating function manipulations, we prove a characterization of \(c\phi _4(10n+1)\) modulo 5 which leads to an infinite set of Ramanujan-like congruences modulo 5 satisfied by \(c\phi _4.\) This work greatly extends the recent work of Xia on \(c\phi _4\) modulo 5.     

Some congruences for 6-colored generalized Frobenius partitions

We prove some congruences discovered by Baruah and Sarmah and by Xia for \(c\phi _6(n)\), the number of 6-colored generalized Frobenius partitions of n.     

Generalized Frobenius partitions with 6 colors

We present the generating function for \(c\phi _6(n)\), the number of generalized Frobenius partitions of \(n\) with \(6\) colors, in terms of Ramanujan’s theta functions and exhibit \(2\), and \(3\)-dissections of it that yield the congruences \(c\phi _6(2n+1)\equiv 0~(\text {mod}~4)\), \(c\phi _6(3n+1)\equiv 0~(\text {mod}~3^2)\) and \(c\phi _6(3n+2)\equiv 0~(\text {mod}~3^2)\).     

Improved upper bounds for partial spreads

A partial \((k-1)\)-spread in \({\text {PG}}(n-1,q)\) is a collection of \((k-1)\)-dimensional subspaces with trivial intersection. So far, the maximum size of a partial \((k-1)\)-spread in \({\text {PG}}(n-1,q)\) was known for the cases \(n\equiv 0\pmod k\), \(n\equiv 1\pmod k\), and \(n\equiv 2\pmod k\) with the additional requirements \(q=2\) and \(k=3\). We completely resolve the case \(n\equiv 2\pmod k\) for the binary case \(q=2\).     

An unexpected Ramanujan-type congruence modulo 7 for 4-colored generalized Frobenius partitions

In this paper, we present an unexpected Ramanujan-type congruence modulo 7 for \(c\phi _4(n)\), which denotes the number of generalized Frobenius partitions of n with 4 colors. This work extends the recent work of Lin on \(c\phi _4\) modulo 7.     

The crank moments weighted by the parity of cranks

In this note, we introduce the 2kth crank moment \(\mu _{2k}(-1,n)\) weighted by the parity of cranks and show that \((-1)^n \mu _{2k}(-1,n)>0\) for \(n\ge k \ge 0\). When \(k=0\), the inequality \((-1)^n \mu _{2k}(-1,n)>0\) reduces to Andrews and Lewis’s inequality \((-1)^n(M_e(n)-M_o(n))>0\) for \(n\ge 0\), where \(M_e(n)\) (resp. \(M_o(n)\)) denotes the number of partitions of n with even (resp. odd) crank. Several generating functions of \(\mu _{2k}(-1,n)\) are also studied in order to show the positivity of \((-1)^n\mu _{2k}(-1,n)\).     

On Weak Invariance Principles for Partial Sums

Given a sequence of random functionals \(\bigl \{X_k(u)\bigr \}_{k \in \mathbb {Z}}\), \(u \in \mathbf{I}^d\), \(d \ge 1\), the normalized partial sums \(\check{S}_{nt}(u) = n^{-1/2}\bigl (X_1(u) + \cdots + X_{\lfloor n t \rfloor }(u)\bigr )\), \(t \in [0,1]\) and its polygonal version \({S}_{nt}(u)\) are considered under a weak dependence assumption and \(p > 2\) moments. Weak invariance principles in the space of continuous functions and càdlàg functions are established. A particular emphasis is put on the process \(\check{S}_{nt}(\widehat{\theta })\), where \(\widehat{\theta } \xrightarrow {\mathbb {P}} \theta \), and weaker moment conditions (\(p = 2\) if \(d = 1\)) are assumed.     

Arithmetic properties for a partition function related to the Ramanujan/Watson mock theta function $$\omega (q)$$

Recently, Andrews, Dixit, and Yee introduced a new partition function \(p_{\omega }(n)\) that denotes the number of partitions of n in which each odd part is less than twice the smallest part. The generating function of \(p_{\omega }(n)\) is associated with the third-order mock theta function \(\omega (q)\). Andrews, Passary, Sellers, and Yee proved three infinite families of congruences modulo 4 and 8 for \(p_{\omega }(n)\) and provided elementary proofs of congruences modulo 5 for \(p_{\omega }(n)\) which were first discovered by Waldherr. In this paper, we prove some new congruences modulo 5 and powers of 2 for \(p_{\omega }(n)\). In particular, we obtain some non-standard congruences for \(p_{\omega }(n)\). For example, we prove that for \(k\ge 0\), \( p_{\omega }\left( \frac{7\times 5^{2k+1}+1 }{3}\right) \equiv (-1)^k \ (\mathrm{mod}\ 5) \) and \( p_\omega \left( \frac{2^{2k+7}+1}{3}\right) \equiv 1251 \times (-1)^k \ (\mathrm{mod}\ 2^{11})\).     

On the approximation of dynamical indicators in systems with nonuniformly hyperbolic behavior

Let f be a \(C^{1+\alpha }\) diffeomorphism of a compact Riemannian manifold and \(\mu \) an ergodic hyperbolic measure with positive entropy. We prove that for every continuous potential \(\phi \) there exists a sequence of basic sets \(\Omega _n\) such that the topological pressure \(P(f|\Omega _n,\phi )\) converges to the free energy \(P_{\mu }(\phi ) = h(\mu ) + \int \phi {d\mu }\). We also prove that for a suitable class of potentials \(\phi \) there exists a sequence of basic sets \(\Omega _n\) such that \(P(f|\Omega _n,\phi ) \rightarrow P(\phi )\).     

Endotrivial modules for the general linear group in a nondefining characteristic

Suppose that \(G\) is a finite group such that \(\mathrm{SL }(n,q)\subseteq G \subseteq \mathrm{GL }(n,q)\), and that \(Z\) is a central subgroup of \(G\). Let \(T(G/Z)\) be the abelian group of equivalence classes of endotrivial \(k(G/Z)\)-modules, where \(k\) is an algebraically closed field of characteristic \(p\) not dividing \(q\). We show that the torsion free rank of \(T(G/Z)\) is at most one, and we determine \(T(G/Z)\) in the case that the Sylow \(p\)-subgroup of \(G\) is abelian and nontrivial. The proofs for the torsion subgroup of \(T(G/Z)\) use the theory of Young modules for \(\mathrm{GL }(n,q)\) and a new method due to Balmer for computing the kernel of restrictions in the group of endotrivial modules.     

On the Extremal Theory of Continued Fractions

Letting \(x=[a_1(x), a_2(x), \ldots ]\) denote the continued fraction expansion of an irrational number \(x\in (0, 1)\), Khinchin proved that \(S_n(x)=\sum \nolimits _{k=1}^n a_k(x) \sim \frac{1}{\log 2}n\log n\) in measure, but not for almost every \(x\). Diamond and Vaaler showed that, removing the largest term from \(S_n(x)\), the previous asymptotics will hold almost everywhere, this shows the crucial influence of the extreme terms of \(S_n (x)\) on the sum. In this paper we determine, for \(d_n\rightarrow \infty \) and \(d_n/n\rightarrow 0\), the precise asymptotics of the sum of the \(d_n\) largest terms of \(S_n(x)\) and show that the sum of the remaining terms has an asymptotically Gaussian distribution.     

On discrete universality of the Riemann zeta-function with respect to uniformly distributed shifts

The Voronin universality theorem asserts that a wide class of analytic functions can be approximated by shifts \(\zeta (s+i\tau )\), \(\tau \in \mathbb {R}\), of the Riemann zeta-function. In the paper, we obtain a universality theorem on the approximation of analytic functions by discrete shifts \(\zeta (s+ix_kh)\), \(k\in \mathbb {N}\), \(h>0\), where \(\{x_k\}\subset \mathbb {R}\) is such that the sequence \(\{ax_k\}\) with every real \(a\ne 0\) is uniformly distributed modulo 1, \(1\le x_k\le k\) for all \(k\in \mathbb {N}\) and, for \(1\le k\), \(m\le N\), \(k\ne m\), the inequality \(|x_k-x_m| \ge y^{-1}_N\) holds with \(y_N> 0\) satisfying \(y_Nx_N\ll N\).     

Monochromatic Solutions for Multi-Term Unknowns

Given integers \(k\ge 2\), \(n \ge 2\), \(m \ge 2\) and \( a_1,a_2,\ldots ,a_m \in {\mathbb {Z}}{\backslash }{\{0\}}\), and let \(f(z)= \sum _{j=0}^{n}c_jz^j\) be a polynomial of integer coefficients with \(c_n>0\) and \((\sum _{i=1}^ma_i)|f(z)\) for some integer z. For a k-coloring of \([N]=\{1,2,\ldots ,N\}\), we say that there is a monochromatic solution of the equation \(a_1x_1+a_2x_2+\cdots +a_mx_m=f(z)\) if there exist pairwise distinct \(x_1,x_2,\ldots ,x_m\in [N]\) all of the same color such that the equation holds for some \(z\in \mathbb {Z}\). Problems of this type are often referred to as Ramsey-type problems. In this paper, it is shown that if \(a_i>0\) for \(1\le i\le m\), then there exists an integer \(N_0=N(k,m,n)\) such that for \(N\ge N_0\), each k-coloring of [N] contains a monochromatic solution \(x_1,x_2,\ldots ,x_m\) of the equation \(a_1x_1+a_2x_2+ \cdots +a_mx_m= f(z)\). Moreover, if n is odd and there are \(a_i\) and \(a_j\) such that \(a_ia_j<0\) for some \(1 \le i\ne j\le m\), then the assertion holds similarly.     

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.     

CCZ equivalence of power functions

Let \(F\simeq {{\mathrm{GF}}}(p^n)\) be a finite field of characteristic p and \(p_k\) and \(p_\ell \) be power functions on F defined by \(p_k(x)=x^k\) and \(p_\ell (x)=x^\ell \) respectively. We show, that \(p_k\) and \(p_\ell \) are CCZ equivalent, if and only if there exists a positive integer \(0\le a< n\), such that \(\ell \equiv p^a k \pmod {p^n-1}\) or \(k\ell \equiv p^a \pmod {p^n-1}\).     

Solvability of Minimal Graph Equation Under Pointwise Pinching Condition for Sectional Curvatures

We study the asymptotic Dirichlet problem for the minimal graph equation on a Cartan–Hadamard manifold M whose radial sectional curvatures outside a compact set satisfy an upper bound

$$\begin{aligned} K(P)\le - \frac{\phi (\phi -1)}{r(x)^2} \end{aligned}$$

and a pointwise pinching condition

$$\begin{aligned} |K(P) |\le C_K|K(P') | \end{aligned}$$

for some constants \(\phi >1\) and \(C_K\ge 1\), where P and \(P'\) are any 2-dimensional subspaces of \(T_xM\) containing the (radial) vector \(\nabla r(x)\) and \(r(x)=d(o,x)\) is the distance to a fixed point \(o\in M\). We solve the asymptotic Dirichlet problem with any continuous boundary data for dimensions \(n=\dim M>4/\phi +1\).

     

Polynomially linked additive functions—II

We continue the study of additive functions \(f_k:R\rightarrow F \;(1\le k\le n)\) linked by an equation of the form \(\sum _{k=1}^n p_k(x)f_k(q_k(x))=0\), where the \(p_k\) and \(q_k\) are polynomials, R is an integral domain of characteristic 0, and F is the fraction field of R. A method is presented for solving all such equations. We also consider the special case \(\sum _{k=1}^n x^{m_k}f_k(x^{j_k})=0\) in which the \(p_k\) and \(q_k\) are monomials. In this case we show that if there is no duplication, i.e. if \((m_k,j_k)\ne (m_p,j_p)\) for \(k\ne p\), then each \(f_k\) is the sum of a linear function and a derivation of order at most \(n-1\). Furthermore, if this functional equation is not homogeneous then the maximal orders of the derivations are reduced in a specified way.     

The solution gap of the Brezis–Nirenberg problem on the hyperbolic space

We consider the positive solutions of the nonlinear eigenvalue problem \(-\Delta _{\mathbb {H}^n} u = \lambda u + u^p, \) with \(p=\frac{n+2}{n-2}\) and \(u \in H_0^1(\Omega ),\) where \(\Omega \) is a geodesic ball of radius \(\theta _1\) on \(\mathbb {H}^n.\) For radial solutions, this equation can be written as an ordinary differential equation having n as a parameter. In this setting, the problem can be extended to consider real values of n. We show that if \(2<n<4\) this problem has a unique positive solution if and only if \(\lambda \in \left( n(n-2)/4 +L^*\,,\, \lambda _1\right) .\) Here \(L^*\) is the first positive value of \(L = -\ell (\ell +1)\) for which a suitably defined associated Legendre function \(P_{\ell }^{-\alpha }(\cosh \theta ) >0\) if \(0 < \theta <\theta _1\) and \(P_{\ell }^{-\alpha }(\cosh \theta _1)=0,\) with \(\alpha = (2-n)/2\).     

Line complexity asymptotics of polynomial cellular automata

Cellular automata are discrete dynamical systems that consist of patterns of symbols on a grid, which change according to a locally determined transition rule. In this paper, we will consider cellular automata that arise from polynomial transition rules, where the symbols are integers modulo some prime p. We consider the asymptotic behavior of the line complexity sequence \(a_T(k)\), which counts, for each k, the number of coefficient strings of length k that occur in the automaton. We begin with the modulo 2 case. For a polynomial \(T(x)=c_0+c_1x+\dots +c_nx^n\) with \(c_0,c_n\ne ~0\), we construct odd and even parts of the polynomial from the strings \(0c_1c_3c_5\cdots c_{1+2\lfloor (n-1)/2\rfloor }\) and \(c_0c_2c_4\cdots c_{2\lfloor n/2\rfloor }\), respectively. We prove that \(a_T(k)\) satisfies recursions of a specific form if the odd and even parts of T are relatively prime. We also define the order of such a recursion and show that the property of “having a recursion of some order” is preserved when the transition rule is raised to a positive integer power. Extending to a more general setting, we consider an abstract generating function \(\phi (z)=\sum _{k=1}^\infty \alpha (k)z^k\) which satisfies a functional equation relating \(\phi (z)\) and \(\phi (z^p)\). We show that there is a continuous, piecewise quadratic function f on [1 / p, 1] for which \(\lim _{k\rightarrow \infty }(\alpha (k)/k^2-~f(p^{-\langle \log _p k\rangle })) = 0\) (here \(\langle y\rangle =y-\lfloor y\rfloor \)). We use this result to show that for certain positive integer sequences \(s_k(x)\rightarrow \infty \) with a parameter \(x\in [1/p,1]\), the ratio \(\alpha (s_k(x))/s_k(x)^2\) tends to f(x), and that the limit superior and inferior of \(\alpha (k)/k^2\) are given by the extremal values of f.     

Répartitio n simulta née de S( n) et $$ S( n+1)$$ S( n+1) da ns les progressio ns arithmétiques

If \(q\ge 2\) is an integer, we denote by \(S_q(n)\) the sum of the digits in base q of the positive integer n and by \(v_q(n)\) its q-adic valuation. The goal of this work is to study exponential sums of the form \(\displaystyle \sum \nolimits _{n\le x}\exp \big (2i\pi \big (\frac{l}{m} S_q(n)+\frac{k}{m'}S_q(n+1)+\theta n\big )\big )\) in order to prove some statistical properties of integers n for which \(S_q(n)\) and \(S_q(n+1)\) belong to given arithmetic progressions. This extends the results obtained by Gelfond in 1968 and those obtained by Mauduit–Sárközy in 1996.