In the present paper we discuss the boundedness of the maximal operator in the Lorentz space of variable exponent defined by the symmetric decreasing rearrangement in the sense of Almut [ 1]. As an application of the boundedness of the maximal operator, we establish the Sobolev inequality by using Hedberg's trick in his paper [ 10].

In this article, we study Griess algebras generated by two pairs of Ising vectors $(a_0, a_1)$ and $(b_0,b_1)$ such that each pair generates a $3A$-algebra $U_{3A}$ and their intersection contains the $W_3$-algebra $\mathcal{W}(4/5)\cong L(4/5,0)\oplus L(4/5,3)$. We show that there are only 3 possibilities, up to isomorphisms and they are isomorphic to the Griess algebras of the VOAs $V_{F(1A)}$, $V_{F(2A)}$ and $V_{F(3A)}$ constructed by Höhn–Lam–Yamauchi.

We consider an orientation preserving homeomorphism $h$ of $S^2$ which admits a repellor denoted $\infty$ and an attractor $-\infty$ such that $h$ is not a North-South map and that the basins of $\infty$ and $-\infty$ intersect. We study various aspects of the rotation number of $h:S^2\setminus\{\pm\infty\}\to S^2\setminus\{\pm\infty\}$, especially its relationship with the existence of periodic orbits.

In this paper we study automorphisms of order four on K3 surfaces. We give a classification of the non-symplectic ones when either the square of the automorphism is symplectic, or its fixed locus contains a curve of positive genus, or its fixed locus contains at least a curve and all the curves fixed by its square are rational. We provide partial results in the other cases.

Let $m \ge 2$ and define the multilinear Littlewood–Paley $g$-function by

$$g(\vec{f})(x)=\bigg(\int_{0}^{\infty} \bigg| \frac{1}{t^{mn}}\int_{(\mathbb{R}^n)^m} \psi\bigg(\frac{y_1}{t},\dots,\frac{y_m}{t}\bigg) \prod_{j=1}^mf_j(x-y_j)dy_{j}\bigg|^2 \frac{dt}{t} \bigg)^{1/2}.$$

In this paper, we establish the strong $L^{p_1}(w_1)\times \dots \times L^{p_m}(w_m)$ to $L^p(\nu_{\vec{\omega}}$) boundedness and weak type $L^{p_1}(w_1)\times \dots \times L^{p_m}(w_m)$ to $L^{p,\infty}(\nu_{\vec{\omega}}$) estimate for the multilinear $g$-function. The weighted strong and end-point estimates for the iterated commutators of $g$-function are also given. Here $\nu_{\vec{\omega}} = \prod_{i = 1}^m\omega_i^{{p}/{p_i}}$ and each $w_i$ is a nonnegative function on $\mathbb{R}^n$.

We consider some measure-theoretic properties of functions belonging to a Sobolev-type class on metric measure spaces that admit a Poincaré inequality and are equipped with a doubling measure. The properties we have selected to study are those that are related to area formulas.

In this paper maximal commutators and commutators of maximal functions with functions of bounded mean oscillation are investigated. New pointwise estimates for these operators are proved.

A generalized Euler sequence over a complete normal variety $X$ is the unique extension of the trivial bundle $V \otimes {\mathcal O}_X$ by the sheaf of differentials $\Omega_X$, given by the inclusion of a linear space $V\subset {\rm Ext}^1_X({\mathcal O}_X,\Omega_X)$. For $\Lambda$, a lattice of Cartier divisors, let ${\mathcal R}_\Lambda$ denote the corresponding sheaf associated to $V$ spanned by the first Chern classes of divisors in $\Lambda$. We prove that any projective, smooth variety on which the bundle ${\mathcal R}_\Lambda$ splits into a direct sum of line bundles is toric. We describe the bundle ${\mathcal R}_\Lambda$ in terms of the sheaf of differentials on the characteristic space of the Cox ring, provided it is finitely generated. Moreover, we relate the finiteness of the module of sections of ${\mathcal R}_\Lambda$ and of the Cox ring of $\Lambda.$

The twisted de Rham complex associated with hypergeometric integral of a power product of polynomials is quasi-isomorphic to the corresponding logarithmic complex. We show in this article that the latter has a double filtration with respect to degrees of polynomials and exterior algebras. By a combinatorial method we prove the quasi-isomorphism between the twisted de Rham cohomology and a specially filtered subcomplex in case of polynomials of the same degree. This fact gives a more detailed structure of a basis for the twisted de Rham cohomology.

Given a compact semisimple Lie group $G$ of rank $r$, and a parameter $q$ > 0, we can define new associativity morphisms in ${\rm Rep}(G_q)$ using a $3$-cocycle $\Phi$ on the dual of the center of $G$, thus getting a new tensor category ${\rm Rep}(G_q)^\Phi$. For a class of cocycles $\Phi$ we construct compact quantum groups $G^\tau_q$ with representation categories ${\rm Rep}(G_q)^\Phi$. The construction depends on the choice of an $r$-tuple $\tau$ of elements in the center of $G$. In the simplest case of $G=SU(2)$ and $\tau=-1$, our construction produces Woronowicz's quantum group $SU_{-q}(2)$ out of $SU_q(2)$. More generally, for $G=SU(n)$, we get quantum group realizations of the Kazhdan–Wenzl categories.

We compute the fundamental groups $\pi_1(\mathbb{P}^2\setminus C)$ for all complex curves $C$ of degree $7$ defined by an equation of the form

$$\prod_{j=1}^\ell (Y-\beta_j Z)^{\nu_j} = c\cdot\prod_{i=1}^m (X-\alpha_i Z)^{\lambda_i},$$

where $\sum_{j=1}^\ell \nu_j=\sum_{i=1}^m \lambda_i$ is the degree of the curve, $c\in\mathbb{R}\setminus \{0\}$, and $\beta_1,\ldots,\beta_\ell$ (respectively $\alpha_1,\ldots,\alpha_m$) mutually distinct real numbers.

In this paper, we introduce new combinatorial invariants of any finite simple graph, which arise in toric topology. We compute the $i$-th (rational) Betti number of the real toric variety associated to a graph associahedron $P_{\B(G)}$. It can be calculated by a purely combinatorial method (in terms of graphs) and is denoted by $a_i(G)$. To our surprise, for specific families of the graph $G$, our invariants are deeply related to well-known combinatorial sequences such as the Catalan numbers and Euler zigzag numbers.

In this paper we study fundamental directional properties of sets under the assumption of condition (SSP) (introduced in [ 3]). We show several transversality theorems in the singular case and an (SSP)-structure preserving theorem. As a geometric illustration, our transversality results are used to prove several facts concerning complex analytic varieties in 3.3. Also, using our results on sets with condition (SSP), we give a classification of spirals in the appendix 5.

The (SSP)-property is most suitable for understanding transversality in the Lipschitz category. This property is shared by a large class of sets, in particular by subanalytic sets or by definable sets in an o-minimal structure.

We provide counterexamples to the stable equivalence problem in every dimension $d\geq2$. That means that we construct hypersurfaces $H_1, H_2\subset\C^{d+1}$ whose cylinders $H_1\times\C$ and $H_2\times\C$ are equivalent hypersurfaces in $\C^{d+2}$, although $H_1$ and $H_2$ themselves are not equivalent by an automorphism of $\C^{d+1}$. We also give, for every $d\geq2$, examples of two non-isomorphic algebraic varieties of dimension $d$ which are biholomorphic.

We introduce Ginibre-type point processes as determinantal point processes associated with the eigenspaces corresponding to the so-called Landau levels. The Ginibre point process, originally defined as the limiting point process of eigenvalues of the Ginibre complex Gaussian random matrix, can be understood as a special case of Ginibre-type point processes. For these point processes, we investigate the asymptotic behavior of the variance of the number of points inside a growing disk. We also investigate the asymptotic behavior of the conditional expectation of the number of points inside an annulus given that there are no points inside another annulus.

Motivated by a recent result of Mazurek and Ziembowski in [ 21] that every left distributive ring is Armendariz, in this paper we present methods of constructing Armendariz modules using a distributive module. We prove that, for a bimodule $_RV_A$, $_RV$ being distributive implies that $V_A$ is Armendariz, and that every right module over a right distributive ring is Armendariz. These results can be used to construct new Armendariz rings. Examples are provided to illustrate and delimit the results obtained.

Let $K={\bm Q}(i\sqrt{D_K})$ be an imaginary quadratic field of discriminant $-D_K$. We introduce a notion of an adelic Maass space ${\mathcal S}_{k, -k/2}^{\rm M}$ for automorphic forms on the quasi-split unitary group $U(2,2)$ associated with $K$ and prove that it is stable under the action of all Hecke operators. When $D_K$ is prime we obtain a Hecke-equivariant descent from ${\mathcal S}_{k,-k/2}^{\rm M}$ to the space of elliptic cusp forms $S_{k-1}(D_K, \chi_K)$, where $\chi_K$ is the quadratic character of $K$. For a given $\phi \in S_{k-1}(D_K, \chi_K)$, a prime $\ell$ > $k$, we then construct $(\mod \ell)$ congruences between the Maass form corresponding to $\phi$ and Hermitian modular forms orthogonal to ${\mathcal S}_{k,-k/2}^{\rm M}$ whenever ${\rm val}_{\ell}(L^{\rm alg}({\rm Symm}^2 \phi, k))$ > $0$. This gives a proof of the holomorphic analogue of the unitary version of Harder's conjecture. Finally, we use these congruences to provide evidence for the Bloch–Kato conjecture for the motives ${\rm Symm}^2 \rho_{\phi}(k-3)$ and ${\rm Symm}^2 \rho_{\phi}(k)$, where $\rho_{\phi}$ denotes the Galois representation attached to $\phi$.

Let $(X,L)$ be a smooth polarized variety of dimension $n$. Let $A\in |L|$ be an effective irreducible divisor, and let $\Sigma$ be the singular locus of $A$. We assume that $\Sigma$ is a smooth subvariety of dimension $k\geq 2$, and codimension $c\geq 3$, consisting of non-degenerate quadratic singularities. We study positivity conditions for adjoint bundles $K_X+tL$ with $t\geq n-3$. Several explicit examples motivate the discussion.