Let p be a prime, $q=p^n$, and $D \subset \mathbb {F}_q^*$. A celebrated result of McConnel states that if D is a proper subgroup of $\mathbb {F}_q^*$, and $f:\mathbb {F}_q \to \mathbb {F}_q$ is a function such that $(f(x)-f(y))/(x-y) \in D$ whenever $x \neq y$, then $f(x)$ necessarily has the form $ax^{p^j}+b$. In this notes, we give a sufficient condition on D to obtain the same conclusion on f. In particular, we show that McConnel’s theorem extends if D has small doubling.

This paper presents a classification of all simple Harish-Chandra modules for the $N=1$ Heisenberg–Virasoro superalgebra, which turn out to be highest weight modules, lowest weight modules, and evaluation modules of the intermediate series (all weight spaces are one dimensional). Moreover, a characterization of the tensor product of highest weight modules with intermediate series modules is obtained.

Let $(x_n)_{n\geq 0}$ be a linear recurrence sequence of order $k\geq 2$ satisfying

$$ \begin{align*}x_n=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}\end{align*} $$

$n\geq k$

$a_1,\dots ,a_k,x_0,\dots , x_{k-1}\in \mathbb {Z},$

$a_k\neq 0$

$(x_n)_{n\geq 0}$

$\mathbb {Q}_p$

$\pm \alpha $

$\alpha $

$\mathbb {Q}_p$

$(x_n)_{n\geq 0}$

$\mathbb {Q}_p$

$\mathbb {Q}$

In the present paper, we characterize the Fredholmness of Toeplitz pairs on Hardy space over the bidisk with the bounded holomorphic symbols, and hence, we obtain the index formula for such Toeplitz pairs. The key to obtain the Fredholmness of such Toeplitz pairs is the $L^p$ solution of Corona Problem over $\mathbb {D}^2$.

We provide an alternative view of some results in [1, 3, 11]. In particular, we prove that (1) if a continuous self-map of a compact metric space has the shadowing, then the union of the basins of terminal chain components is a dense $G_\delta $-subset of the space; and (2) if a continuous self-map of a locally connected compact metric space has the shadowing, and if the chain recurrent set is totally disconnected, then the map is almost chain continuous.

In this paper, we consider the discrete Orlicz chord Minkowski problem and solve the existence of this problem, which is the nontrivial extension of the discrete $L_{p}$ chord Minkowski problem for ${0<p<1}$.

We prove a result that relates the number of homomorphisms from the fundamental group of a compact nonorientable surface to a finite group G, where conjugacy classes of the boundary components of the surface must map to prescribed conjugacy classes in G, to a sum over values of irreducible characters of G weighted by Frobenius-Schur multipliers. The proof is structured so that the corresponding results for closed and possibly orientable surfaces, as well as some generalizations, are derived using the same methods. We then apply these results to the specific case of the symmetric group.

This paper deals with a 4th-order parabolic equation involving the Frobenius norm of a Hessian matrix, subject to the Neumann boundary conditions. Some threshold results for blow-up or global or extinction solutions are obtained through classifying the initial energy and the Nehari energy. The bounds of blow-up time, decay estimates, and extinction rates are studied, respectively.

The famous Cheng-Shen’s conjecture in Riemann-Finsler geometry claims that every n-dimensional closed W-quadratic Randers manifold is a Berwald manifold. In this paper, first we study the Riemann and Ricci curvatures of homogeneous Finsler manifolds and obtain some rigidity theorems. Then, by using this investigation, we construct a family of W-quadratic Randers metrics which are not R-quadratic nor strongly Ricci-quadratic.

In the study of plane curves, one of the problems is to classify the embedded topology of plane curves in the complex projective plane that have a given fixed combinatorial type, where the combinatorial type of a plane curve is data equivalent to the embedded topology in its tubular neighborhood. A pair of plane curves with the same combinatorial type but distinct embedded topology is called a Zariski pair. In this paper, we consider Zariski pairs consisting of conic-line arrangements that arise from Poncelet’s closure theorem. We study unramified double covers of the union of two conics that are induced by a $2m$-sided Poncelet transverse. As an application, we show the existence of families of Zariski pairs of degree $2m+6$ for $m\geq 2$ that consist of reducible curves having two conics and $2m+2$ lines as irreducible components.

We consider uniqueness problems for meromorphic inner functions on the upper half-plane. In these problems, we consider spectral data depending partially or fully on the spectrum, derivative values at the spectrum, Clark measure, or the spectrum of the negative of a meromorphic inner function. Moreover, we consider applications of these uniqueness results to inverse spectral theory of canonical Hamiltonian systems and obtain generalizations of the Borg-Levinson two-spectra theorem for canonical Hamiltonian systems and unique determination of a Hamiltonian from its spectral measure under some conditions.

We present a new proof of the compactness of bilinear paraproducts with CMO symbols. By drawing an analogy to compact linear operators, we first explore further properties of compact bilinear operators on Banach spaces and present examples. We then prove compactness of bilinear paraproducts with CMO symbols by combining one of the properties of compact bilinear operators thus obtained with vanishing Carleson measure estimates and interpolation of bilinear compactness.

Let G be a split connected reductive group defined over $\mathbb {Z}$. Let F and $F'$ be two non-Archimedean m-close local fields, where m is a positive integer. D. Kazhdan gave an isomorphism between the Hecke algebras $\mathrm {Kaz}_m^F :\mathcal {H}\big (G(F),K_F\big ) \rightarrow \mathcal {H}\big (G(F'),K_{F'}\big )$, where $K_F$ and $K_{F'}$ are the mth usual congruence subgroups of $G(F)$ and $G(F')$, respectively. On the other hand, if $\sigma $ is an automorphism of G of prime order l, then we have Brauer homomorphism $\mathrm {Br}:\mathcal {H}(G(F),U(F))\rightarrow \mathcal {H}(G^\sigma (F),U^\sigma (F))$, where $U(F)$ and $U^\sigma (F)$ are compact open subgroups of $G(F)$ and $G^\sigma (F),$ respectively. In this article, we study the compatibility between these two maps in the local base change setting. Further, an application of this compatibility is given in the context of linkage – which is the representation theoretic version of Brauer homomorphism.

We study the Abel-Jacobi image of the Ceresa cycle $W_{k, e}-W_{k, e}^-$, where $W_{k, e}$ is the image of the k-th symmetric product of a curve X with a base point e on its Jacobian variety. For certain Fermat quotient curves of genus g, we prove that for any choice of the base point and $k \leq g-2$, the Abel-Jacobi image of the Ceresa cycle is non-torsion. In particular, these cycles are non-torsion modulo rational equivalence.

For a partially specified stochastic matrix, we consider the problem of completing it so as to minimize Kemeny’s constant. We prove that for any partially specified stochastic matrix for which the problem is well defined, there is a minimizing completion that is as sparse as possible. We also find the minimum value of Kemeny’s constant in two special cases: when the diagonal has been specified and when all specified entries lie in a common row.

Let $g\geqslant 2$. A real number is said to be $g$-normal if its base $g$ expansion contains every finite sequence of digits with the expected limiting frequency. Let $\unicode[STIX]{x1D711}$ denote Euler’s totient function, let $\unicode[STIX]{x1D70E}$ be the sum-of-divisors function, and let $\unicode[STIX]{x1D706}$ be Carmichael’s lambda-function. We show that if $f$ is any function formed by composing $\unicode[STIX]{x1D711}$, $\unicode[STIX]{x1D70E}$, or $\unicode[STIX]{x1D706}$, then the number

$$\begin{eqnarray}0.f(1)f(2)f(3)\cdots\end{eqnarray}$$

$g$

$f$

$g$

$1,2,3,\ldots$

$2,3,5,\ldots$

$0.235711131719\cdots \,$