This paper addresses three key objectives. First, weighted Sobolev trace embeddings are studied for a class of function spaces defined in the upper half-space. Second, Liouville-type theorems are proved, providing essential insights into the nonexistence of solutions for a class of quasilinear elliptic problems with indefinite boundary conditions under specific constraints. Third, by combining the derived embedding results with the fibering method, the existence of solutions for such problems is established. These findings contribute to a deeper understanding of the analytical and variational properties of these elliptic equations.

Let $\Omega$ be a lattice in $\mathbb C$ with invariants $g_2,g_3$ and let $\wp(z), \zeta(z)$ be the associated Weierstrass elliptic and zeta functions, respectively. In this paper, we prove that if $\omega$ is any non-zero period of $\wp(z)$ and $u_1,u_2$ complex numbers such that $u_1,u_2, \omega$ are $\mathbb{Q}$-linearly independent with $({\mathbb{Z}} u_1+{\mathbb{Z}} u_2)\cap\Omega=(0),$ then at least two of the numbers

\begin{equation*}g_2,g_3,\omega, \eta,u_1,u_2,\wp(u_i),\zeta(u_i)~~(1\leq i\leq 2)\end{equation*}

$\mathbb{Q},$

$\eta$

$\zeta(z)$

$\omega.$

Two-sort species yield differential equations for functional digraphs of Cayley permutations. From these, we obtain an explicit formula for fixed-point-free Cayley permutations and prove that their proportion tends to $1/e$, as for permutations and endofunctions. Our approach also yields counting formulas when the functional digraph is a tree, forest, or connected.

Cohomological equation is of special interest because it concerns the study of time change for flows, topological stability and topological conjugacy in dynamical systems, which is also an auxiliary equation to study the problem of linearization. In this paper, we consider a general form of cohomological equation for planar contractions. By using the ideas of invariant manifold and estimations in [W. Zhang and W. Zhang, $C^1$ linearization for planar contractions, J. Funct. Anal. 260 (2011), 2043–2063.], we present new criteria on eigenvalues of the linear parts for the existence of $C^1$ solutions in the Poincaré domain. Our results are a generalization of $C^1$ linearization for contractions.

Let $r_s(n)$ denote the number of representations of $n$ as a sum of $s$ squares. Hurwitz established eleven identities expressing the generating function of $r_3(an+b)$ as a simple infinite product. Cooper and Hirschhorn (Discrete Math 274 (1-3):9–24, 2004) proved that for any $k\geq0$, the generating functions $\sum_{n=0}^\infty r_3\big(3^{2k}n\big)q^n$ and $\sum_{n=0}^\infty r_3\big(3^{2k+1}n\big)q^n$ can be written as linear combinations of two specified generalized eta-quotients. In this paper, we substantially extend these results to high dimensions. Specifically, we prove that for any $k\geq0$ and $3\leq s\leq 100$, the generating functions $\sum_{n=0}^\infty r_s\big(3^{2k+1}n\big)q^n$ and $\sum_{n=0}^\infty r_s\big(3^{2k+2}n\big)q^n$ can also be expressed as linear combinations of certain generalized eta-quotients. Motivated by these results, we conjecture that this phenomenon holds for $r_s(n)$ for all $s\geq3$, and further that $r_s(n)$ satisfies an infinite family of internal congruences modulo high powers of $3$.

In this paper, we introduce and study a new kind of generalized Hilbert matrix operators, induced by a positive finite Borel measure on $(0,1)$, acting on weighted sequence spaces. We establish a sufficient and necessary condition for the boundedness of these operators. These results extend some related ones obtained recently in [Bull. Lond. Math. Soc. 55(6) (2023), 2598–2610].

Let M be a compact three-dimensional Riemannian manifold with non-negative Ricci curvature and a non-empty boundary $\partial M$. Fraser and Li [2] established a compactness theorem for the space of compact, properly embedded minimal surfaces of fixed topological type in M with a free boundary on $\partial M$, assuming that $\partial M$ is strictly convex with respect to the inward unit normal. In this paper, we show that the strict convexity condition on $\partial M$ cannot be relaxed.

We classify all Polish semigroup topologies on the symmetric inverse monoid $I_{\mathbb N}$ on the natural numbers $\mathbb N$. This result answers a question of Elliott et al. There are countably infinitely many such topologies. Under containment, these Polish semigroup topologies form a join-semilattice with infinite descending chains, no infinite ascending chains, and arbitrarily large finite anti-chains. Also, we show that the monoid $I_{\mathbb N}$ endowed with any second countable $T_1$ semigroup topology is homeomorphic to the Baire space $\mathbb N^{\mathbb N}$.

We study the representation of non-weakly compact operators between $AL$-spaces. In this setting, we show that every operator admits a best approximant in the ideal of weakly compact operators. Using duality arguments, we extend this result to operators between $C(L)$-spaces where $L$ is extremally disconnected. We also characterize the weak essential norm for operators between $AL$-spaces in terms of factorizations of the identity on $\ell_1$. As a consequence, we deduce that the weak Calkin algebra $\mathscr{B}(E)/\mathscr{W}(E)$ admits a unique algebra norm for every $AL$-space $E$. By duality, similar results are obtained for $C(K)$-spaces. In particular, we prove that for operators $T: L_{\infty}[0,1] \to L_{\infty}[0,1]$ the weak essential norm, the residuum norm, and the De Blasi measure of weak compactness coincide, answering a question of González, Saksman and Tylli.

We provide four equivalent combinatorial conditions for a simple assembly graph (rigid vertex graph where all vertices are of degree 1 or 4) to have the largest number of Hamiltonian sets of polygonal paths relative to its size. These conditions serve to prove the conjecture that such a maximum, which is equal to $F_{2n+1}-1$, where $F_k$ denotes the $k$th Fibonacci number, is achieved only for special assembly graphs, called tangled cords.

In this paper, we are mainly devoted to the limiting behaviour of smooth inertial manifolds for a class of random retarded differential equations with a singular parameter $\delta$ and their Galerkin approximations, which have not been considered before. Under appropriate conditions, we show not only that the inertial manifolds for this class of random retarded equations converge pointwise to those of the corresponding stochastic equations driven by white noise as $\delta\rightarrow 0$, but also that the inertial manifolds of their Galerkin approximations converge pointwise to those of stochastic equations driven by white noise described above under the simultaneous limits $\delta\rightarrow 0$ and $M\rightarrow +\infty$, where $M$ denotes the dimension index of the orthogonal projection operator $P_M$ in the Galerkin scheme.

We study complete noncompact spacelike mean curvature flow solitons (SMCFS) in a standard static spacetime obeying a suitable constraint on the sectional curvature. In this context, we prove a version of the Omori–Yau generalized maximum principle and apply it to deduce that such an SMCFS must be maximal in the sense that its mean curvature vanishes identically. Next, we use other maximum principles which deal with the notions of convergence to zero at infinity and polynomial volume growth to prove rigidity results for SMCFS. Furthermore, we apply our previous results to establish nonexistence results concerning entire Killing graphs constructed over the Riemannian base of a standard static spacetime. Finally, we also exhibit an example showing the relevance of key hypotheses in our results.

Let $p$ be a prime, let $1 \le t \lt d \lt p$ be integers, and let $S$ be a non-empty subset of $\mathbb{F}_p$. We establish that if a polynomial $P:\mathbb{F}_p^n \to \mathbb{F}_p$ with degree $d$ is such that the image $P(S^n)$ does not contain the full image $A(\mathbb{F}_p)$ of any non-constant polynomial $A: \mathbb{F}_p \to \mathbb{F}_p$ with degree at most $t$, then $P$ coincides on $S^n$ with a polynomial that in particular has bounded degree-$\lfloor d/(t+1) \rfloor$-rank in the sense of Green and Tao. Similarly, we prove that if the assumption holds even for $t=d$, then $P$ coincides on $S^n$ with a polynomial determined by a bounded number of coordinates.

A set $S \subseteq \mathbb{R}$ is almost Erdős if, for every $\varepsilon \gt 0$, there exists a set $E \subseteq \mathbb{R}$ of positive Lebesgue measure such that $\{x \in S : ax+b \notin E\}$ is nonempty for all $|a| \gt \varepsilon$ and $b \in \mathbb{R}$. In this note, we show that any decreasing null sequence $(x_n)$ with decay rate greater than $1/2$ is an almost Erdős set.

This paper is concerned with a duality between $r$-regular permutations and $r$-cycle permutations, and a monotone property due to Bóna-McLennan-White on the probability $p_r(n)$ for a random permutation of $\{1,2,\ldots, n\}$ to have an $r$th root, where $r$ is a prime. For $r=2$, the duality relates permutations with odd cycles to permutations with even cycles. For the general case where $r\geq 2$, we define an $r$-enriched permutation as a permutation with $r$-singular cycles coloured by one of the colours $1, 2, \ldots, r-1 $. In this setup, we discover a bijection between $r$-regular permutations and enriched $r$-cycle permutations, which in turn yields a stronger version of an inequality of Bóna-McLennan-White. This leads to a fully combinatorial understanding of the monotone property, thereby answering their question. When $r$ is a prime power $q^l$, we further show that $p_r(n)$ is monotone. In the case that $n+1 \not\equiv 0 \pmod q$, the equality $p_r(n)=p_r(n+1)$ has been established by Chernoff.

In his seminal work on partitions and divisor functions, MacMahon introduced the following two q-series:

\begin{align*}A_k(q):= \sum_{0 \lt s_1 \lt s_2 \lt \cdots \lt s_k} \frac{q^{s_1+s_2+\cdots+s_k}}{(1-q^{s_1})^2(1-q^{s_2})^2\cdots(1-q^{s_k})^2},\qquad\quad\quad\quad\\[6pt]C_k(q):= \sum_{0 \lt s_1 \lt s_2 \lt \cdots \lt s_k}\frac{q^{2(s_1+s_2+\cdots+s_k)-k}}{(1-q^{2s_1-1})^2(1-q^{2s_2-1})^2\cdots(1-q^{2s_k-1})^2}.\end{align*}

In 2013, Andrews and Rose proved that $A_k(q)$ and $C_k(q)$ are quasimodular forms of weight $\leq 2k$. Recently, Ono and Singh proved two interesting identities involving $A_k(q)$ and $C_k(q)$ and showed that the generating functions for the three-coloured partition function $p_3(n)$ and the overpartition function $\overline{p}(n)$ have infinitely many closed formulas in terms of MacMahon’s quasimodular forms $A_k(q)$ and $C_k(q)$. In this paper, we introduce the finite forms $A_{k,n}(q)$ and $C_{k,n}(q)$ of MacMahon’s q-series $A_k(q)$ and $C_k(q)$ and prove two identities which generalize Ono–Singh’s identities. We also prove some new identities involving $A_{k,n}(q)$, $C_{k,n}(q)$ and certain infinite products based on two Bailey pairs. Those identities are analogous to Ono–Singh’s identities.

We prove that for bounded, divergence-free vector fields $\boldsymbol{b}$ in $L^1_{loc}((0,1];BV(\mathbb{T}^d;\mathbb{R}^d))$, there exists a unique incompressible measure on integral curves of $\boldsymbol{b}$. We recall the vector field constructed by Depauw in [8], which lies in the above class, and prove that for this vector field, the unique incompressible measure on integral curves exhibits stochasticity.

We consider a normal operator $T$ on a Hilbert space $H$. Under various assumptions on the spectrum of $T$, we give bounds for the spectrum of $T+A$ where $A$ is $T$-bounded with relative bound less than 1 but we do not assume that $A$ is symmetric or normal. If the imaginary part of the spectrum of $T$ is bounded, then the spectrum of $T+A$ is contained in the region between two hyperbolas whose asymptotic slope depends on the $T$-bound of $A$. If the spectrum of $T$ is contained in a bisector, then the spectrum of $T+A$ is contained in the area between certain rotated hyperbolas. The case of infinitely many gaps in the spectrum of $T$ is studied. Moreover, we prove a stability result for the essential spectrum of $T+A$. If $A$ is even $p$-subordinate to $T$, then we obtain stronger results for the localisation of the spectrum of $T+A$.

We introduce a natural weighted enumeration of lattice points in a polytope, and give a Brion-type formula for the corresponding generating function. The weighting has combinatorial significance, and its generating function may be viewed as a generalization of the Rogers–Szegő polynomials. It also arises from the geometry of the toric arc scheme associated to the normal fan of the polytope. We show that the asymptotic behaviour of thecoefficients at $q=1$ is Gaussian.