Volume 2016, Issue 715

We consider a nonlinear elliptic problem in divergence form, with nonstandard growth conditions, on a bounded domain. We obtain the global Calderón–Zygmund type gradient estimates for the weak solution of such a problem in the setting of Lebesgue and Sobolev spaces with variable p ( x ) exponents, in the case that the nonlinearity of the coefficients is allowed to be discontinuous and the domain goes beyond the Lipschitz category. We assume that the nonlinearity has small BMO semi-norms and the boundary of the domain satisfies the so-called δ-Reifenberg flatness condition. These conditions on the nonlinearity and the boundary are weaker than those reported in other studies in the literature.

Suppose 𝕆 is an alternative division algebra that is quadratic over some subfield 𝕂 of its center Z(𝕆)$Z(\mathbb {O})$. Then with (𝕆,𝕂)$(\mathbb {O},\mathbb {K})$, there is associated a dual polar space. We provide an explicit representation of this dual polar space into a (6n+7)$(6n+7)$-dimensional projective space over 𝕂, where n=dim𝕂(𝕆)$n = \dim _\mathbb {K}(\mathbb {O})$. We prove that this embedding is the universal one, provided |𝕂|>2$\vert \mathbb {K}\vert >2$. When 𝕆 is not an inseparable field extension of 𝕂, we show that this universal embedding is the unique polarized one. When 𝕆 is an inseparable field extension of 𝕂, then we determine the minimal full polarized embedding, and show that all homogeneous embeddings are either universal or minimal. We also provide explicit generators of the corresponding projective representations of the little projective group associated with the (dual) polar space.

We provide complete characterizations, on Banach spaces with cotype 2, of those linear operators which happen to be weakly mixing or strongly mixing transformations with respect to some nondegenerate Gaussian measure. These characterizations involve two families of small subsets of the circle: the countable sets and the so-called sets of uniqueness for Fourier–Stieltjes series. The most interesting part, i.e. the sufficient conditions for weak and strong mixing, is valid on an arbitrary (complex, separable) Fréchet space.

Let f:X→Z${f\colon X\rightarrow Z}$ be a Mori fibre space. McKernan conjectured that the singularities of Z are bounded in terms of the singularities of X . Shokurov independently proposed a more general conjecture in the setting of pairs: let ( X , B ) be a pair with klt singularities and f:X→Z${f\colon X\rightarrow Z}$ be a contraction such that KX+B∼ℝ0/Z$K_X+B\sim _\mathbb {R}0/Z$ and that the general fibres of f are Fano type varieties; adjunction for fibre spaces produces a discriminant divisor B Z and a moduli divisor M Z on Z . It is then conjectured that the singularities of (Z,BZ+MZ)${(Z,B_Z+M_Z)}$ are bounded in terms of the singularities of ( X , B ). We prove Shokurov's conjecture when (F,SuppBF)${(F,\operatorname{Supp}B_F)}$ belongs to a bounded family where F is a general fibre of f and KF+BF=(KX+B)|F${K_F+B_F=(K_X+B)|_F}$.

This paper is a sequel to [Adv. Math. 245 (2013), 625–689], where we study the derived affine scheme DRep n(A)${{\mathrm {DRep}}_n(A)}$ parametrizing the n -dimensional representations of an associative k -algebra A . In [Adv. Math. 245 (2013), 625–689], we have constructed canonical trace maps Trn(A)•: HC •(A)→H•[ DRep n(A)] GL n${\operatorname{Tr}_n(A)_{\bullet }: \mathrm {HC}_{\bullet }(A) \rightarrow \mathrm {H}_{\bullet }[{\mathrm {DRep}}_n(A)]^{{\mathrm {GL}}_n} }$ extending the usual characters of representations to higher cyclic homology. This raises the natural question whether a well-known theorem of Procesi [Adv. Math. 19 (1976), 306–381.] holds in the derived setting: namely, is the algebra homomorphism Λ Tr n(A)•:Λk[ HC •(A)]→H•[ DRep n(A)] GL n${\Lambda \mathrm {Tr} _n(A)_{\bullet } : \Lambda _k[\mathrm {HC}_{\bullet }(A)] \rightarrow \mathrm {H}_{\bullet }[{\mathrm {DRep}}_n(A)]^{{\mathrm {GL}}_n}}$ defined by Trn(A)•${ \operatorname{Tr}_n(A)_{\bullet } }$ surjective? In the present paper, we answer this question for augmented algebras. Given such an algebra, we construct a canonical dense subalgebra DRep ∞(A)Tr${{\mathrm {DRep}}_{\infty }(A)^{\operatorname{Tr}}}$ of the topological DG algebra lim← DRep n(A) GL n${\varprojlim {\mathrm {DRep}}_n(A)^{{\mathrm {GL}}_n} }$. Our main result is that on passing to the inverse limit, the family of maps Λ Tr n(A)•${\Lambda \mathrm {Tr} _n(A)_{\bullet }}$ `stabilizes' to an isomorphism Λk( HC ¯•(A))≅H•[ DRep ∞(A)Tr]${\Lambda _k(\overline{\mathrm {HC}}_{\bullet }(A)) \cong \mathrm {H}_{\bullet }[{\mathrm {DRep}}_{\infty }(A)^{\operatorname{Tr}}]}$. The derived version of Procesi's theorem does therefore hold in the limit as n → ∞. However, for a fixed (finite) n , there exist homological obstructions to the surjectivity of Λ Tr n(A)•${ \Lambda \mathrm {Tr} _n(A)_{\bullet }}$, and we show on simple examples that these obstructions do not vanish in general. We compare our result with the classical theorem of Loday, Quillen and Tsygan on stable homology of matrix Lie algebras. We show that the Chevalley–Eilenberg complex 𝒞•(𝔤𝔩∞(A),𝔤𝔩∞(k);k)${{\mathcal {C}}_\bullet ({\mathfrak {gl}}_\infty (A), {\mathfrak {gl}}_\infty (k); k)}$ equipped with a natural coalgebra structure is Koszul dual to the DG algebra DRep ∞(A)Tr${{\mathrm {DRep}}_{\infty }(A)^{\operatorname{Tr}}}$. We also extend our main results to bigraded DG algebras, in which case we show the equality DRep ∞(A)Tr= DRep ∞(A) GL ∞${ {\mathrm {DRep}}_{\infty }(A)^{\operatorname{Tr}} = {\mathrm {DRep}}_{\infty }(A)^{{\mathrm {GL}}_{\infty }}}$. As an application, we compute the Euler characteristics of DRep ∞(A) GL ∞${{\mathrm {DRep}}_{\infty }(A)^{{\mathrm {GL}}_{\infty }}}$ and HC ¯•(A)${\overline{\mathrm {HC}}_{\bullet }(A)}$ and derive some interesting combinatorial identities.

We solve an inverse spectral problem for a star graph of Krein strings, where the known spectral data comprises the spectrum associated with the whole graph, the spectra associated with the individual edges as well as so-called coupling matrices. In particular, we show that these spectral quantities uniquely determine the weight within the class of Borel measures on the graph, which give rise to trace class resolvents. Furthermore, we obtain a concise characterization of all possible spectral data for this class of weights.

In this paper, we show the uniqueness of smooth embedded self-shrinkers asymptotic to infinite order to a generalized cylinder. Also, we construct non-rotationally symmetric self-shrinking ends asymptotic to a generalized cylinder with rate as fast as any given polynomial.

Based on the analysis on the Ocneanu/Groh–Raynaud ultraproducts and the Effros–Maréchal topology on the space vN (H)${\mathrm {vN}(H)}$ of von Neumann algebras acting on a separable Hilbert space H , we show that for a von Neumann algebra M∈ vN (H)${M\in \mathrm {vN}(H)}$, the following conditions are equivalent: (1) M has the Kirchberg's quotient weak expectation property (QWEP). (2) M is in the closure of the set ℱ inj ${\mathcal {F}_{\rm inj}}$ of injective factors on H with respect to the Effros–Maréchal topology. (3) M admits an embedding i into the Ocneanu ultrapower R∞ω${R_{\infty }^{\omega }}$ of the injective III 1${\mathrm {III}_1}$ factor R ∞ with a normal faithful conditional expectation ε:R∞ω→i(M)${\varepsilon \colon R_{\infty }^{\omega }\rightarrow i(M)}$. (4) For every ε>0${\varepsilon &gt;0}$, n ∈ ℕ and ξ1,...,ξn∈𝒫M♮${\xi _1,\ldots ,\xi _n\in \mathcal {P}_M^{\natural }}$, there are k ∈ ℕ and a1,...,an∈Mk(ℂ)+${a_1,\ldots ,a_n\in M_k(\mathbb {C})_+}$ such that |〈ξi,ξj〉-trk(aiaj)|<ε${|\langle \xi _i,\xi _j\rangle -\operatorname{tr}_k(a_ia_j)|&lt;\varepsilon }$ (1≤i,j≤n${1\le i,j\le n}$) holds, where trk${\operatorname{tr}_k}$ is the tracial state on Mk(ℂ)${M_k(\mathbb {C})}$, and 𝒫M♮${\mathcal {P}_M^{\natural }}$ is the natural cone in the standard form of M .