In this work, we investigate the existence and multiplicity as well as regularity of solutions for a huge class of concave–convex problems involving the fractional $\Phi$-Laplacian operator in non-reflexive setting. The main difficulty arises from the lack of homogeneity for the $\Phi$-Laplacian operator. An additional difficulty stems from the interplay between a singular term and a superlinear term. To address these issues, we introduce a suitable class of $N$-functions and employ the Nehari manifold method together with a nonlinear Rayleigh quotient.

In this paper, we investigate the two-dimensional uniform Diophantine approximation in $\beta$-dynamical systems. Let $\beta_i \gt 1(i=1,2)$ be real numbers, and let $T_{\beta_i}$ denote the $\beta_i$-transformation defined on $[0, 1]$. Motivated by Amou and Bugeaud (Exponents of Diophantine approximation and expansions in integer bases, J. London Math. Soc. 81(2) (2010), 297–316), we consider, for each $(x, y) \in[0,1]^2$ in the plane, its asymptotic approximation exponent

\begin{equation*}v_{\beta_1, \beta_2}(x, y)=\sup \left\{0 \leq v \lt \infty: \begin{array}{l}T_{\beta_1}^n x \lt \beta_1^{-n v} \\T_{\beta_2}^n y \lt \beta_2^{-n v}\end{array} \text {for infinitely many } n \in \mathbb{N}\right\} \text {, }\end{equation*}

\begin{align*}\hat{v}_{\beta_1, \beta_2}(x, y)=\sup\Big\{0 \leq \hat{v} \lt \infty&:\text{for every sufficiently large }N, \\&\text{there exists }1 \leq n \leq N ~\text {such that } T_{\beta_1}^n x \lt \beta_1^{-N \hat{v}},T_{\beta_2}^n y \lt \beta_2^{-N \hat{v}}\Big\}.\end{align*}

We then calculate the Hausdorff dimension of the set

\begin{equation*}\left\{(x, y) \in[0,1]^2: \hat{v}_{\beta_1, \beta_2}(x, y)=\hat{v} \text~ {and }\, v_{\beta_1, \beta_2}(x, y)=v\right\},\end{equation*}

We derive combinatorial formulas for invariants of smooth embeddings of $(2\ell-1)$-spheres into $\mathbb{R}^{3\ell}$ for $\ell\geq 2$. In particular, we obtain such a formula for the Haefliger invariant, which classifies smooth knots $S^{4k-1}\hookrightarrow \mathbb{R}^{6k}$ up to isotopy. Our approach is similar in spirit to the work of Goussarov, Polyak, and Viro, who expressed finite-type invariants of classical knots in terms of Gauss diagrams. Analogously, we project higher-dimensional knots and links onto a hyperplane and study the preimages of the loci of double and singular points in the embedded spheres. As an auxiliary result, we show that the space of $n$-dimensional braids with $m$ strands in $\mathbb{R}^{n+q}$ is a homotopy retract of the space of string links $\underset{m}{\sqcup}\mathbb{R}^n\hookrightarrow\mathbb{R}^{n+q}$ for $q\geq 3$, thereby providing a short proof to a conjecture of Komendarczyk, Koytcheff, and Volić.

From the perspective of $\tau$-tilting theory, we study Frobenius–Perron dimensions of finite-dimensional algebras. First, we evaluate the Frobenius–Perron dimensions of $\tau$-tilting finite algebras by a combinatorial method in $\tau$-tilting theory. Second, we give an upper bound for the Frobenius–Perron dimension of $\tau$-tilting finite algebras of tame representation type. Third, we determine the Frobenius–Perron dimensions of Nakayama algebras and generalized preprojective algebras of Dynkin type in the sense of Geiß–Leclerc–Schröer.

The shape of a number field is a subtle arithmetic invariant arising from the geometry of numbers. It is defined as the equivalence class of the lattice of integers with respect to linear operations that are composites of rotations, reflections, and positive scalar dilations. For a number field of degree $n$, the shape is a point in the space of shapes $\mathcal{S}_{n-1}$, which is the double quotient $\operatorname{GL}_{n-1}(\mathbb{Z}) \backslash \operatorname{GL}_{n-1}(\mathbb{R}) / \operatorname{GO}_{n-1}(\mathbb{R})$. In this paper, we investigate the distribution of shapes in the family of pure quartic fields $K_m = \mathbb{Q}(\sqrt[4]{m})$. We prove that the shape of $K_m$ lies on one of $5$ explicitly described torus orbits in $\mathcal{S}_3$, determined by the sign and residue class of $m \bmod 32$. It is shown that the shape on a given torus orbit is completely determined by two parameters, one of which varies continuously, while the other takes values in a discrete set. As a result, the distribution of shapes in this family is governed by a product of a continuous and a discrete measure. Our results shed new light on a question posed by Manjul Bhargava and Piper H concerning the distribution of shapes in families of non-generic number fields of fixed degree. Notably, the limiting distribution in our case does not arise as the restriction of the natural measure on $\mathcal{S}_3$ induced by the Haar measure on $\operatorname{GL}_3(\mathbb{R})$.

We consider free skew extensions of rings, also known as free multivariate skew polynomial rings, and explore some of the algebraic aspects of this construction. We provide different characterizations of such rings, relate them to Cohn’s construction of a filtered tensor ring on a bimodule, and present conditions under which a free skew extension of a ring is a domain, a prime ring, or can be embedded in a series ring.

For a positive integer $n$ let $\mathcal{X}_n$ be either the algebra $M_n$ of $n \times n$ complex matrices, the set $N_n$ of all $n \times n$ normal matrices, or any of the matrix Lie groups $\mathrm{GL}(n)$, $\mathrm{SL}(n)$ and $\mathrm{U}(n)$. We first give a short and elementary argument that for two positive integers $m$ and $n$ there exists a continuous spectrum-shrinking map $\phi : \mathcal{X}_n \to M_m$ (i.e., $\mathrm{sp}(\phi(X))\subseteq \mathrm{sp}(X)$ for all $X \in \mathcal{X}_n$) if and only if $n$ divides $m$. Moreover, in that case we have the equality of characteristic polynomials $k_{\phi(X)}(\cdot) = k_{X}(\cdot)^\frac{m}{n}$ for all $X \in \mathcal{X}_n$, which in particular shows that $\phi$ preserves spectra. Using this, we show that whenever $n \geq 3$, any continuous commutativity-preserving and spectrum-shrinking map $\phi : \mathcal{X}_n \to M_n$ is of the form $\phi(\cdot)=T(\cdot)T^{-1}$ or $\phi(\cdot)=T(\cdot)^tT^{-1}$, for some $T\in \mathrm{GL}(n)$. The analogous results fail for the special unitary group $\mathrm{SU}(n)$ but hold for the sets of semisimple matrices in either $\mathrm{GL}(n)$ or $\mathrm{SL}(n)$. As a consequence, we also recover (a strengthened version of) Šemrl’s influential characterization of Jordan automorphisms of $M_n$ via preserving properties.

The immanants of combinatorial matrices have many significant properties, including $m$-positivity and Schur positivity. While the immanants of Jacobi–Trudi matrices are known to be both $m$-positive and Schur positive, those of Giambelli matrices have only been proven to be $m$-positive, with Schur positivity remaining a conjecture. These positivity properties rely heavily on lattice path interpretations. In this paper, we study the Newton polytopes of immanants for these two classes of matrices. Using the lattice path method, we verify the saturation property for the Newton polytopes of Jacobi–Trudi matrices in special cases. For Giambelli matrices, we prove that this property holds for all immanants. To achieve this, we obtain the explicit coefficients of the largest monomial (in the dominance order) in the monomial expansion of the immanants of Giambelli matrices.

We talk about the image of the Hilbert map. In particular, we show the necessary and sufficient condition for the Hilbert map to be surjective.

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.

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 $3$.