As is well known, a standard random walk is approximate to the stochastic process corresponding to the heat equation. Lachal constructed the approximate pseudorandom walk which is accompanied by the pseudostochastic process corresponding to an even-order heat-type equation. We have two purposes for this article. The first is to construct the approximate pseudorandom walk which is accompanied by the pseudostochastic process corresponding to an odd-order heat-type equation. The other is to propose a construction method for the approximate pseudorandom walk which is accompanied by the pseudostochastic process corresponding to an even-order heat-type equation. This method is different from that of Lachal.

Given a classical semisimple complex algebraic group $G$ and a symmetric pair $(G,K)$ of non-Hermitian type, we study the closures of the spherical nilpotent $K$-orbits in the isotropy representation of $K$. For all such orbit closures, we study the normality, and we describe the $K$-module structure of the ring of regular functions of the normalizations.

In this article we make a few remarks about the geometry of the holomorphic symplectic manifold $Z$ constructed by Lehn, Lehn, Sorger, and van Straten as a two-step contraction of the variety of twisted cubic curves on a cubic fourfold $Y\subset {\mathbb{P}}^5$. We show that $Z$ is birational to a component of the moduli space of stable sheaves in the Calabi–Yau subcategory of the derived category of $Y$. Using this description we deduce that the twisted cubics contained in a hyperplane section $Y_H=Y\cap H$ of $Y$ give rise to a Lagrangian subvariety $Z_H\subset Z$. For a generic choice of the hyperplane, $Z_H$ is birational to the theta-divisor in the intermediate Jacobian $\mathrm{J}\left(Y_H\right)$.

We discuss generalizations of some results on lattice polygons to certain piecewise linear loops which may have a self-intersection but have vertices in the lattice ${\mathbb{Z}}^2$. We first prove a formula on the rotation number of a unimodular sequence in ${\mathbb{Z}}^2$. This formula implies the generalized twelve-point theorem of Poonen and Rodriguez-Villegas. We then introduce the notion of lattice multipolygons, which is a generalization of lattice polygons, state the generalized Pick’s formula, and discuss the classification of Ehrhart polynomials of lattice multipolygons and also of several natural subfamilies of lattice multipolygons.

There are six types of $2$-dimensional representations in general. For any groups and any monoids, we can construct the moduli of $2$-dimensional representations for each type: the moduli of absolutely irreducible representations, representations with Borel mold, representations with semisimple mold, representations with unipotent mold, representations with unipotent mold over ${\mathbb{F}}_2$, and representations with scalar mold. We can also construct them for any associative algebras.