Inspired by a construction by Bump, Friedberg, and Ginzburg of a two-variable integral representation on $\text{GS}{{\text{p}}_{4}}$ for the product of the standard and spin $L$-functions, we give two similar multivariate integral representations. The first is a three-variable Rankin-Selberg integral for cusp forms on $\text{PG}{{\text{L}}_{4}}$ representing the product of the $L$-functions attached to the three fundamental representations of the Langlands $L$-group $\text{S}{{\text{L}}_{\text{4}}}\left( \text{C} \right)$. The second integral, which is closely related, is a two-variable Rankin-Selberg integral for cusp forms on $\text{PGU}\left( 2,\,2 \right)$ representing the product of the degree $8$ standard $L$-function and the degree $6$ exterior square $L$-function.

Let $G$ be the $n$-fold covering group of the special linear group of degree two over a non-Archimedean local field. We determine the decomposition into irreducibles of the restriction of the principal series representations of $G$ to a maximal compact subgroup. Moreover, we analyse those features that distinguish this decomposition from the linear case.

This paper gives an equivalent form of Picard’s theorem via entire solutions of the functional equation ${{f}^{2}}\,+\,{{g}^{2}}\,=\,1$ and then its improvements and applications to certain nonlinear (ordinary and partial) differential equations.

We investigate the moduli space of sheaves supported on space curves of degree and having Euler characteristic 1. We give an elementary proof of the fact that this moduli space consists of three irreducible components.

We provide normal forms and the global phase portraits on the Poincaré disk for all Abel quadratic polynomial diòerential equations of the second kind with ${{\mathbb{Z}}_{2}}$-symmetries.

We consider homogeneous multiaffine polynomials whose coefficients are the Plücker coordinates of a point $V$ of the Grassmannian. We show that such a polynomial is stable (with respect to the upper half plane) if and only if $V$ is in the totally nonnegative part of the Grassmannian. To prove this, we consider an action of matrices on multiaffine polynomials. We show that a matrix $A$ preserves stability of polynomials if and only if $A$ is totally nonnegative. The proofs are applications of classical theory of totally nonnegative matrices, and the generalized Pólya-Schur theory of Borcea and Brändén.

We apply our theory of partial flag spaces developed with W. Goldring to study a group-theoretical generalization of the canonical filtration of a truncated Barsotti–Tate group of level 1. As an application, we determine explicitly the normalization of the Zariski closures of Ekedahl–Oort strata of Shimura varieties of Hodge-type as certain closed coarse strata in the associated partial flag spaces.

For a commutative ring $R$, a polynomial$f\,\in \,R[x]$ is called separable if $R[x]/f$ is a separable $R$-algebra. We derive formulae for the number of separable polynomials when $R\,=\,\mathbb{Z}/n$, extending a result of L. Carlitz. For instance, we show that the number of polynomials in $\mathbb{Z}/n[x]$ that are separable is $\phi (n){{n}^{d}}{{\prod }_{i}}(1\,-\,p_{i}^{-d})$, where $n\,=\,\prod p_{i}^{{{k}_{i}}}$ is the prime factorisation of $n$ and $\phi $ is Euler’s totient function.

In this note we prove the following surprising characterization: if $X\,\subset \,{{\mathbb{A}}^{n}}$ is an (embedded, non-empty, proper) algebraic variety deûned over a field $k$ of characteristic zero, then $X$ is a hypersurface if and only if the module ${{T}_{{{O}_{{{\mathbb{A}}^{n\,/k}}}}}}(X)$ of logarithmic vector fields of $X$ is a reflexive ${{O}_{{{\mathbb{A}}^{n}}}}$-module. As a consequence of this result, we derive that if ${{T}_{{{O}_{{{\mathbb{A}}^{n\,/k}}}}}}(X)$ is a free ${{O}_{{{\mathbb{A}}^{n}}}}$-module, which is shown to be equivalent to the freeness of the $t$-th exterior power of ${{T}_{{{O}_{{{\mathbb{A}}^{n\,/k}}}}}}(X)$ for some (in fact, any) $t\,\le \,n$, then necessarily $X$ is a Saito free divisor.

In this paper, we first discuss the relation between $\text{VB}$-Courant algebroids and $\text{E}$-Courant algebroids, and we construct some examples of $\text{E}$-Courant algebroids. Then we introduce the notion of a generalized complex structure on an $\text{E}$-Courant algebroid, unifying the usual generalized complex structures on even-dimensional manifolds and generalized contact structures on odd-dimensional manifolds. Moreover, we study generalized complex structures on an omni-Lie algebroid in detail. In particular, we show that generalized complex structures on an omni-Lie algebra $\text{gl}\left( V \right)\oplus V$ correspond to complex Lie algebra structures on $V$.

In this paper, we consider the following critical Kirchhoff type equation:

$$\left\{ _{u\,=\,0,\,\,\,\,\,\,\text{on}\,\partial \Omega \text{,}}^{-(a\,+\,b{{\int }_{\Omega }}|\nabla u{{|}^{2}})\Delta u\,=\,\text{Q(}x)|u{{|}^{4}}u\,+\,\lambda |u{{|}^{q-1}}u,\,\,\,\text{in}\,\Omega \text{,}} \right.$$

By using variational methods that are constrained to the Nehari manifold, we prove that the above equation has a ground state solution for the case when $3\,<\,q\,<\,5$. The relation between the number of maxima of $\text{Q}$ and the number of positive solutions for the problem is also investigated.

The study of graph ${{C}^{*}}$-algebras has a long history in operator algebras. Surprisingly, their quantum symmetries have not yet been computed. We close this gap by proving that the quantum automorphism group of a finite, directed graph without multiple edges acts maximally on the corresponding graph ${{C}^{*}}$-algebra. This shows that the quantum symmetry of a graph coincides with the quantum symmetry of the graph ${{C}^{*}}$-algebra. In our result, we use the definition of quantum automorphism groups of graphs as given by Banica in 2005. Note that Bichon gave a different definition in 2003; our action is inspired from his work. We review and compare these two definitions and we give a complete table of quantum automorphism groups (with respect to either of the two definitions) for undirected graphs on four vertices.

For $m,n\,\in \,\mathbb{N},\,1\,<\,m\le \,n$, we write $n={{n}_{1}}+\cdots +{{n}_{m}}$ where $\{{{n}_{1}},\ldots \,,{{n}_{m}}\}\,\subset \,\mathbb{N}$. Let ${{A}_{1}}\,,\,\ldots \,,\,{{A}_{m}}$ be $n\,\times \,n$ singular real matrices such that

$$\underset{i=1\,}{\overset{m}{\mathop{\oplus }}}\,\underset{1\le j\ne i\le m}{\mathop \bigcap }\,\,{{N}_{j}}\,=\,{{\mathbb{R}}^{n}},$$

where ${{N}_{j}}\,=\,\{x\,:\,{{A}_{j}}x\,=\,0\},\,\text{dim(}{{N}_{j}}\text{)}\,=\,n\,-\,{{n}_{j}}$, and ${{A}_{1}}\,,\,\ldots \,,\,{{A}_{m}}$ is invertible. In this paper we study integral operators of the form

$${{T}_{r}}f(x)\,=\,{{\int }_{{{\mathbb{R}}^{n}}}}|x\,-\,{{A}_{1y}}{{|}^{-{{n}_{1}}+{{\alpha }_{{{1}_{\ldots }}}}}}|x\,-\,{{A}_{m}}y{{|}^{-{{n}_{m}}+{{\alpha }_{m}}}}f(y)dy,$$

${{n}_{1}}\,+\,\cdots \,+\,{{n}_{m}}\,=\,n$ , $\frac{{{\alpha }_{1}}}{{{n}_{1}}}\,=\,\cdots \,=\,\frac{{{\alpha }_{m}}}{{{n}_{m}}}\,=\,r$, $0\,<\,r\,<\,1$, and the matrices ${{A}_{i}}\text{ }\!\!'\!\!\text{ s}$ are as above. We obtain the ${{H}^{p}}({{\mathbb{R}}^{n}})\,-\,{{L}^{q}}({{\mathbb{R}}^{n}})$ boundedness of ${{T}_{r}}$ for $0\,<\,p\,<\,\frac{1}{r}$ and $\frac{1}{q}\,=\,\frac{1}{p}\,-\,r$.

We prove an interpolation formula for the values of certain $p$-adic Rankin-Selberg $L$-functions associated with non-ordinary modular forms.

For most classical and similitude groups, we show that each element can be written as a product of two transformations that preserve or almost preserve the underlying form and whose squares are certain scalar maps. This generalizes work of Wonenburger and Vinroot. As an application, we re-prove and slightly extend a well-known result of Mœglin, Vignéras, and Waldspurger on the existence of automorphisms of $p$-adic classical groups that take each irreducible smooth representation to its dual.

Let $A$ be a commutative noetherian ring, let $\mathfrak{a}\subseteq A$ be an ideal, and let $I$ be an injective $A$-module. A basic result in the structure theory of injective modules states that the $A$-module ${{\Gamma }_{\alpha }}\left( I \right)$ consisting of $\mathfrak{a}$-torsion elements is also an injective $A$-module. Recently, de Jong proved a dual result: If $F$ is a flat $A$-module, then the $\mathfrak{a}$-adic completion of $F$ is also a flat $A$-module. In this paper we generalize these facts to commutative noetherian $\text{DG}$-rings: let $A$ be a commutative non-positive $\text{DG}$-ring such that ${{\text{H}}^{0}}\left( A \right)$ is a noetherian ring and for each $i\,<\,0,\,\text{the}\,{{\text{H}}^{0}}\left( A \right)$-module ${{\text{H}}^{i}}\left( A \right)$ is finitely generated. Given an ideal $\overline{\mathfrak{a}}\,\subseteq \,{{\text{H}}^{0}}\left( A \right)$, we show that the local cohomology functor $\text{R}{{\Gamma }_{\overline{\mathfrak{a}}}}$ associated with $\overline{\mathfrak{a}}$ does not increase injective dimension. Dually, the derived $\overline{\mathfrak{a}}$-adic completion functor $\text{L}{{\Lambda }_{\overline{\mathfrak{a}}}}$ does not increase flat dimension.

For every affine variety over a global function field, we show that the set of its points with coordinates in an arbitrary rank-one multiplicative subgroup of this function field satisfies the required property of weak approximation for finite sets of places of this function field avoiding arbitrarily given finitely many places.

The Fefferman–Stein type inequalities for strongmaximal operators and directional maximal operators are verified with an additional composition of the Hardy–Littlewood maximal operator in the plane.

A crucial role in the Nyman-Beurling-Báez-Duarte approach to the Riemann Hypothesis is played by the distance

$$d_{N}^{2}:=\underset{{{A}_{N}}}{\mathop{\inf }}\,\frac{1}{2\pi }\int _{-\infty }^{\infty }{{\left| 1-\zeta {{A}_{N}}\left( \frac{1}{2}+it \right) \right|}^{2}}\frac{dt}{\frac{1}{4}+{{t}^{2}}},$$

where the infimum is over all Dirichlet polynomials

$${{A}_{N}}\left( s \right)\,=\,\sum\limits_{n=1}^{N}{\frac{{{a}_{n}}}{{{n}^{s}}}}$$

of length $N$. In this paper we investigate $d_{N}^{2}$ under the assumption that the Riemann zeta function has four nontrivial zeros off the critical line.

In this paper we establish a close connection between three notions attached to a modular subgroup, namely, the set of weight two meromorphic modular forms, the set of equivariant functions on the upper half-plane commuting with the action of the modular subgroup, and the set of elliptic zeta functions generalizing the Weierstrass zeta functions. In particular, we show that the equivariant functions can be parameterized by modular objects as well as by elliptic objects.