In earlier papers of this series we constructed a sequence of intermediate moduli spaces $\left\{{\mathrm{M̂}}^m\right(c){\}}_{m=0,1,2,\dots }$ connecting a moduli space $M\left(c\right)$ of stable torsion-free sheaves on a nonsingular complex projective surface $X$ and $\mathrm{M̂}\left(c\right)$ on its one-point blow-up $\mathrm{X̂}$. They are moduli spaces of perverse coherent sheaves on $\mathrm{X̂}$. In this paper we study how Donaldson-type invariants (integrals of cohomology classes given by universal sheaves) change from ${\mathrm{M̂}}^m\left(c\right)$ to ${\mathrm{M̂}}^{m+1}\left(c\right)$ and then from $M\left(c\right)$ to $\mathrm{M̂}\left(c\right)$. As an application we prove that Nekrasov-type partition functions satisfy certain equations that determine invariants recursively in second Chern classes. They are generalizations of the blow-up equation for the original Nekrasov deformed partition function for the pure $N=2$ supersymmetric gauge theory, found and used to derive the Seiberg-Witten curves.

We begin a study of the representation theory of quantum continuous ${\mathfrak{gl}}_{\infty }$, which we denote by $\mathcal{E}$. This algebra depends on two parameters and is a deformed version of the enveloping algebra of the Lie algebra of difference operators acting on the space of Laurent polynomials in one variable. Fundamental representations of $\mathcal{E}$ are labeled by a continuous parameter $u\in \mathbb{C}$. The representation theory of $\mathcal{E}$ has many properties familiar from the representation theory of ${\mathfrak{gl}}_{\infty }$: vector representations, Fock modules, and semiinfinite constructions of modules. Using tensor products of vector representations, we construct surjective homomorphisms from $\mathcal{E}$ to spherical double affine Hecke algebras $S{\mathrm{Ḧ}}_N$ for all $N$. A key step in this construction is an identification of a natural basis of the tensor products of vector representations with Macdonald polynomials. We also show that one of the Fock representations is isomorphic to the module constructed earlier by means of the $K$-theory of Hilbert schemes.

We construct a family of irreducible representations of the quantum continuous ${\mathfrak{gl}}_{\infty }$ whose characters coincide with the characters of representations in the minimal models of the ${\mathcal{W}}_n$-algebras of ${\mathfrak{gl}}_n$ type. In particular, we obtain a simple combinatorial model for all representations of the ${\mathcal{W}}_n$-algebras appearing in the minimal models in terms of $n$ interrelating partitions.

Let $(X,C)$ be a germ of a threefold $X$ with terminal singularities along an irreducible reduced complete curve $C$ with a contraction $f:(X,C)\to (Z,o)$ such that $C=f^{-1}(o{)}_{red}$ and $-K_X$ is ample. Assume that a general member $F\in |-K_X|$ meets $C$ only at one point $P$, and furthermore assume that $(F,P)$ is Du Val of type A if index $(X,P)=4$. We classify all such germs in terms of a general member $H\in \left|{\mathcal{O}}_X\right|$ containing $C$.

For some discrete random acoustic operators, we prove Wegner estimates. These estimates are applied to show some regularity of the integrated density of states. Moreover, we prove the generalized eigenvalue-counting estimates by using Combes, Germinet, and Klein’s method. As an application, the multiplicity of the eigenvalues in some interval where the Anderson localization occurs is proven to be finite. For certain models, Poisson statistics for eigenvalues and Lifshitz tails are also studied.

In the present paper, we study a large-time behavior of solutions to a quasi-linear second-order hyperbolic system which describes a motion of viscoelastic materials. The system has dissipative properties consisting of a memory term and a damping term. It is proved that the solution exists globally in time in the Sobolev space, provided that the initial data are sufficiently small. Moreover, we show that the solution converges to zero as time tends to infinity. The crucial point of the proof is to derive uniform a priori estimates of solutions by using an energy method.

Let $A=\mathcal{A}(E,\sigma ),A\text{'}=\mathcal{A}(E\text{'},\sigma \text{'})$ be Noetherian Artin-Schelter regular geometric algebras with ${dim}_k{A}_1={dim}_k{A}_1\text{'}=n$, and let $\nu ,\nu \text{'}$ be generalized Nakayama automorphisms of $A,A\text{'}$. In this paper, we study relationships between the conditions

(A) $A$ is graded Morita equivalent to $A\text{'}$, and

(B) $\mathcal{A}(E,{\nu }^{\ast }{\sigma }^n)$ is isomorphic to $\mathcal{A}(E\text{'},(\nu \text{'}{)}^{\ast }\left(\sigma \text{'}{)}^n\right)$ as graded algebras.

It is proved that if $A,A\text{'}$ are “generic” 3-dimensional quadratic Artin-Schelter regular algebras, then (A) is equivalent to (B), and if $A,A\text{'}$ are $n$-dimensional skew polynomial algebras, then (A) implies (B).