This article investigates two types of decay structures for linear symmetric hyperbolic systems with nonsymmetric relaxation. Previously, the same authors introduced a new structural condition which is a generalization of the classical Kawashima–Shizuta condition and also analyzed the weak dissipative structure called the regularity-loss type for general systems with nonsymmetric relaxation, which includes the Timoshenko system and the Euler–Maxwell system as two concrete examples. Inspired by the previous work, we further construct in this article two more complex models which satisfy some new decay structure of regularity-loss type. The proof is based on the elementary Fourier energy method as well as the suitable linear combination of different energy inequalities. The results show that the model of type I has a decay structure similar to that of the Timoshenko system with heat conduction via the Cattaneo law, and the model of type II is a direct extension of two models considered previously to the case of higher phase dimensions.

We consider probability measures on ${\mathbb{R}}^{\infty }$ and study optimal transportation mappings for the case of infinite Kantorovich distance. Our examples include (1) quasiproduct measures and (2) measures with certain symmetric properties, in particular, exchangeable and stationary measures. We show in the latter case that the existence problem for optimal transportation is closely related to the ergodicity of the target measure. In particular, we prove the existence of the symmetric optimal transportation for a certain class of stationary Gibbs measures.

For a field $k$, the ring of invariants of an action of the unipotent $k$-group ${\mathbb{G}}_a$ on an affine $k$-variety is quasiaffine, but not generally affine. Cable algebras are introduced as a framework for studying these invariant rings. It is shown that the ring of invariants for the ${\mathbb{G}}_a$-action on ${\mathbb{A}}_k^5$ constructed by Daigle and Freudenburg is a monogenetic cable algebra. A generating cable is constructed for this ring, and a complete set of relations is given as a prime ideal in the infinite polynomial ring over $k$. In addition, it is shown that the ring of invariants for the well-known ${\mathbb{G}}_a$-action on ${\mathbb{A}}_k^7$ due to Roberts is a cable algebra.

We prove sufficient and necessary conditions for the compactness of Calderón–Zygmund operators on the endpoint from $L^{\infty }\left(\mathbb{R}\right)$ into $\mathrm{CMO}\left(\mathbb{R}\right)$. We use this result to prove the compactness on $L^p\left(\mathbb{R}\right)$ with $1<p<\infty$ of a certain perturbation of the Cauchy integral on curves with normal derivatives satisfying a $\mathrm{CMO}$-condition.

We provide a construction of $81$ symplectic resolutions of a $4$-dimensional quotient singularity obtained by an action of a group of order $32$. The existence of such resolutions is known by a result of Bellamy and Schedler. Our explicit construction is obtained via geometric invariant theory (GIT) quotients of the spectrum of a ring graded in the Picard group generated by the divisors associated to the conjugacy classes of symplectic reflections of the group in question. As a result we infer the geometric structure of these resolutions and their flops. Moreover, we represent the group in question as a group of automorphisms of an abelian $4$-fold so that the resulting quotient has singularities with symplectic resolutions. This yields a new Kummer-type symplectic $4$-fold.

Let $L$ be a closed manifold of dimension $n\ge 2$ which admits a totally real embedding into ${\mathbb{C}}^n$. Let $S{T}^{*}L$ be the space of rays of the cotangent bundle $T^{*}L$ of $L$, and let $D{T}^{*}L$ be the unit disk bundle of $T^{*}L$ defined by any Riemannian metric on $L$. We observe that $S{T}^{*}L$ endowed with its standard contact structure admits weak symplectic fillings $W$ which are diffeomorphic to $D{T}^{*}L$ and for which any closed Lagrangian submanifold $N\subset W$ has the property that the map $H_1(N,\mathbb{R})\to H_1(W,\mathbb{R})$ has a nontrivial kernel. This relies on a variation on a theorem by Laudenbach and Sikorav.

We study the connection between the affine degenerate Grassmannians in type $A$, quiver Grassmannians for one vertex loop quivers, and affine Schubert varieties. We give an explicit description of the degenerate affine Grassmannian of type ${GL}_n$ and identify it with semi-infinite orbit closure of type $A_{2n-1}$. We show that principal quiver Grassmannians for the one vertex loop quiver provide finite-dimensional appro- ximations of the degenerate affine Grassmannian. Finally, we give an explicit description of the degenerate affine Grassmannian of type $A_1^{\left(1\right)}$, propose a conjectural description in the symplectic case, and discuss the generalization to the case of the affine degenerate flag varieties.