The purpose of this paper is to investigate order of contact on real hypersurfaces in $\mathbb{C}^n$ by using Newton polyhedra which are important notion in the study of singularity theory. To be more precise, an equivalence condition for the equality of regular type and singular type is given by using the Newton polyhedron of a defining function for the respective hypersurface. Furthermore, a sufficient condition for this condition, which is more useful, is also given. This sufficient condition is satisfied by many earlier known cases (convex domains, pseudoconvex Reinhardt domains and pseudoconvex domains whose regular types are 4, etc.). Under the above conditions, the values of the types can be directly seen in a simple geometrical information from the Newton polyhedron.

We adapt the commutator theory of universal algebra to the particular setting of racks and quandles, exploiting a Galois connection between congruences and certain normal subgroups of the displacement group. Congruence properties, such as abelianness and centrality, are reflected by the corresponding relative displacement groups, and the global properties, solvability and nilpotence, are reflected by the properties of the whole displacement group. To show the new tool in action, we present three applications: non-existence theorems for quandles (no connected involutory quandles of order $2^k$, no latin quandles of order $\equiv2\pmod4$), a non-colorability theorem (knots with trivial Alexander polynomial are not colorable by solvable quandles; in particular, by finite latin quandles), and a strengthening of Glauberman's results on Bruck loops of odd order.

We derive an explicit formula for the well-known Chern–Moser–Weyl tensor for nondegenerate real hypersurfaces in complex space in terms of their defining functions. The formula is considerably simplified when applying to “pluriharmonic perturbations” of the sphere or to a Fefferman approximate solution to the complex Monge–Ampère equation. As an application, we show that the CR invariant one-form $X_{\alpha}$ constructed recently by Case and Gover is nontrivial on each real ellipsoid of revolution in $\mathbb{C}^3$, unless it is equivalent to the sphere. This resolves affirmatively a question posed by these two authors in 2017 regarding the (non-) local CR invariance of the $\mathcal{I}'$-pseudohermitian invariant in dimension five and provides a counterexample to a recent conjecture by Hirachi.

We study the continuity and the measurability of the solution to Schrödinger's functional equation, with respect to space, kernel and marginals, provided the space of all Borel probability measures is endowed with the weak topology. This is a continuation of our previous result where the space of all Borel probability measures was endowed with the strong topology. As an application, we construct a convex function of which the moment measure is a given probability measure, by the zero noise limit of a class of stochastic optimal transportation problems.

This paper is a continuation of the first paper. The aim of this second paper is to discuss the non-vanishing of the theta lifts to the indefinite symplectic group $GSp(1,1)$, which have been shown to be involved in the Jacquet–Langlands–Shimizu correspondence with some theta lifts to the $\mathbb{Q}$-split symplectic group $GSp(2)$ of degree two. We study an explicit formula for the square norms of the Bessel periods of the theta lifts to $GSp(1,1)$ in terms of central $L$-values. This study involves two aspects in proving the non-vanishing of the theta lifts. One aspect is to apply the results by Hsieh and Chida–Hsieh on “non-vanishing modulo $p$” of central $L$-values for some Rankin $L$-functions. The other is to relate such non-vanishing with studies on some special values of hypergeometric functions. We also take up the theta lifts to the compact inner form $GSp^*(2)$. We provide examples of the non-vanishing theta lifts to $GSp^*(2)$, which are essentially due to Ibukiyama and Ihara.

The $p(x)$-Laplacian Kirchhoff type equation involving the nonlocal term $b \int_{\mathbb{R}^N} (1/p(x)) \lvert\nabla u\rvert^{p(x)}dx$ is investigated. Based on the variational methods, deformation lemma and other technique of analysis, it is proved that the problem possesses one least energy sign-changing solution $u_b$ which has precisely two nodal domains. Moreover, the convergence property of $u_b$ as the parameter $b \searrow 0$ is also obtained.

We introduce a simple, self-dual, rational, and $C_2$-cofinite vertex operator algebra of CFT-type associated with a $\mathbb{Z}_k$-code for $k \ge 2$. Our argument is based on the $\mathbb{Z}_k$-symmetry among the simple current modules for the parafermion vertex operator algebra $K(\mathfrak{sl}_2, k)$. We show that it is naturally realized as the commutant of a certain subalgebra in a lattice vertex operator algebra. Furthermore, we construct all the irreducible modules inside a module for the lattice vertex operator algebra.

Denote by $H(d_1,d_2,d_3)$ the set of all homogeneous polynomial mappings $F=(f_1,f_2,f_3): \mathbb{C}^3 \to \mathbb{C}^3$, such that $\deg f_i=d_i$. We show that if $\gcd(d_i,d_j) \leq 2$ for $1 \leq i < j\leq 3$ and $\gcd(d_1,d_2,d_3)=1$, then there is a non-empty Zariski open subset $U \subset H(d_1,d_2,d_3)$ such that for every mapping $F\in U$ the map germ $(F,0)$ is $\mathcal{A}$-finitely determined. Moreover, in this case we compute the number of discrete singularities (0-stable singularities) of a generic mapping $(f_1,f_2,f_3):\mathbb{C}^3 \to \mathbb{C}^3$, where $\deg f_i=d_i$.

In the late 1980s, Friedlander and Parshall studied the representations of a family of algebras which were obtained as deformations of the distribution algebra of the first Frobenius kernel of an algebraic group. The representation theory of these algebras tells us much about the representation theory of Lie algebras in positive characteristic. We develop an analogue of this family of algebras for the distribution algebras of the higher Frobenius kernels, answering a 30 year old question posed by Friedlander and Parshall. We also examine their representation theory in the case of the special linear group.

We extend a theorem of Haagerup and Kraus in the $\mathrm{C}^{\ast}$-algebra context: for a locally compact group with the approximation property (AP), the reduced $\mathrm{C}^{\ast}$-crossed product construction preserves the strong operator approximation property (SOAP). In particular their reduced group $\mathrm{C}^{\ast}$-algebras have the SOAP. Our method also solves another open problem: the AP implies exactness for general locally compact groups.

Let $X \subset \mathbf{\mathbb{R}}^n$ be a compact semialgebraic set and let $f : X \to \mathbf{\mathbb{R}}$ be a nonzero Nash function. We give a Solernó and D'Acunto–Kurdyka type estimation of the exponent $\varrho\in[0,1)$ in the Łojasiewicz gradient inequality $|\nabla f(x)| \ge C|f(x)|^\varrho$ for $x \in X$, $|f(x)| < \varepsilon$ for some constants $C,\varepsilon > 0$, in terms of the degree of a polynomial $P$ such that $P(x,f(x)) = 0$, $x \in X$. As a corollary we obtain an estimation of the degree of sufficiency of non-isolated Nash function singularities.

We classify Enriques surfaces covered by the supersingular $K3$ surface with the Artin invariant 1 in characteristic 2. There are exactly three types of such Enriques surfaces.