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Using Cuntz semigroup techniques, we characterize when limit traces are dense in the space of all traces on a free ultrapower of a C*-algebra. More generally, we consider density of limit quasitraces on ultraproducts of C*-algebras. Quite unexpectedly, we obtain as an application that every simple C*-algebra that is (m , n) -pure in the sense of Winter is already pure. As another application, we provide a partial verification of the first Blackadar–Handelman conjecture on dimension functions. Crucial ingredients in our proof are new Hahn–Banach type separation theorems for noncancellative cones, which in particular apply to the cone of extended-valued traces on a C*-algebra. [ABSTRACT FROM AUTHOR]

In this paper we consider the Laplace operator with Dirichlet boundary conditions on a smooth domain. We prove that it has a bounded H ∞ -calculus on weighted L p -spaces for power weights which fall outside the classical class of A p -weights. Furthermore, we characterize the domain of the operator and derive several consequences on elliptic and parabolic regularity. In particular, we obtain a new maximal regularity result for the heat equation with rough inhomogeneous boundary data. [ABSTRACT FROM AUTHOR]

We give an asymptotic formula for traces of weak Maass forms at CM points with an effective bound on the error term. Upon specializing to the modular j -function, we deduce such a result for traces of singular moduli. Due to work of Zagier, and Bringmann and Ono, these traces of weak Maass forms at CM points appear as Fourier coefficients of half-integral weight weakly holomorphic modular forms. Hence, our results give effective upper bounds for these Fourier coefficients. [ABSTRACT FROM AUTHOR]

Abstract The principal aim in this paper is to develop an effective and unified approach to the computation of traces of resolvents (and resolvent differences), Fredholm determinants, ζ - functions, and ζ -function regularized determinants associated with linear operators in a Hilbert space. In particular, we detail the connection between Fredholm and ζ -function regularized determinants. Concrete applications of our formalism to general (i.e., three-coefficient) regular Sturm–Liouville operators on bounded intervals with various (separated and coupled) boundary conditions, and Schrödinger operators on a half-line, are provided and further illustrated with an array of examples. [ABSTRACT FROM AUTHOR]

Amenability for groups can be extended to metric spaces, algebras over commutative fields and C ⁎ -algebras by adapting the notion of Følner nets. In the present article we investigate the close ties among these extensions and show that these three pictures unify in the context of the uniform Roe algebra C u ⁎ ( X ) over a metric space ( X , d ) with bounded geometry. In particular, we show that the following conditions are equivalent: (1) ( X , d ) is amenable; (2) the translation algebra generating C u ⁎ ( X ) is algebraically amenable (3) C u ⁎ ( X ) has a tracial state; (4) C u ⁎ ( X ) is not properly infinite; (5) [ 1 ] 0 ≠ [ 0 ] 0 in the K 0 -group K 0 ( C u ⁎ ( X ) ) ; (6) C u ⁎ ( X ) does not contain the Leavitt algebra as a unital ⁎-subalgebra; (7) C u ⁎ ( X ) is a Følner C ⁎ -algebra in the sense that it admits a net of unital completely positive maps into matrices which is asymptotically multiplicative in the normalized trace norm. We also show that every possible tracial state of the uniform Roe algebra C u ⁎ ( X ) is amenable. [ABSTRACT FROM AUTHOR]

The paper is concerned with Besov spaces of variable smoothness B p , q φ 0 ( R n , { t k } ) , in which the norms are defined in terms of convolutions with smooth functions. A relation is found between the spaces B p , q φ 0 ( R n , { t k } ) and the spaces B ˜ p , q , r l ( R n , { t k } ) , which were introduced earlier by the author. As an application an equivalent description of the trace space of the weighted Sobolev space with Muckenhoupt weight is given. [ABSTRACT FROM AUTHOR]

We propose a categorification of the Chern character that refines earlier work of Toën and Vezzosi and of Ganter and Kapranov. If X is an algebraic stack, our categorified Chern character is a symmetric monoidal functor from a category of mixed noncommutative motives over X , which we introduce, to S 1 -equivariant perfect complexes on the derived free loop stack L X . As an application of the theory, we show that Toën and Vezzosi's secondary Chern character factors through secondary K -theory. Our techniques depend on a careful investigation of the functoriality of traces in symmetric monoidal ( ∞ , n ) -categories, which is of independent interest. [ABSTRACT FROM AUTHOR]

Recently, it was proved by Tikuisis, White and Winter that any faithful trace on a separable, nuclear C ⁎ -algebra in the UCT class is quasidiagonal. Building on their work, we generalise the result, and show that any faithful, amenable trace on a separable, exact C ⁎ -algebra in the UCT class is quasidiagonal. We also prove that any amenable trace on a separable, exact, quasidiagonal C ⁎ -algebra in the UCT class is quasidiagonal. [ABSTRACT FROM AUTHOR]

For a given support L ∈ F q m n and a polynomial g ∈ F q m [ x ] with no roots in F q m , we prove equality between the q -ary Goppa codes Γ q ( L , N ( g ) ) = Γ q ( L , N ( g ) / g ) where N ( g ) denotes the norm of g , that is g q m − 1 + ⋯ + q + 1 . In particular, for m = 2 , that is, for a quadratic extension, we get Γ q ( L , g q ) = Γ q ( L , g q + 1 ) . If g has roots in F q m , then we do not necessarily have equality and we prove that the difference of the dimensions of the two codes is bounded above by the number of distinct roots of g in F q m . These identities provide numerous code equivalences and improved designed parameters for some families of classical Goppa codes. [ABSTRACT FROM AUTHOR]

Abstract: Among ideals of compact operators on a Hilbert space we identify a subclass of those closed with respect to the logarithmic submajorization. Within this subclass, we answer the questions asked by Pietsch [22] and by Dykema, Figiel, Weiss and Wodzicki [7]. In the first case, we show that Lidskii-type formulae hold for every trace on such ideal. In the second case, we provide the description of the commutator subspace associated with a given ideal. Finally, we prove that a positive trace on an arbitrary ideal is spectral if and only if it is monotone with respect to the logarithmic submajorization. [Copyright &y& Elsevier]

Abstract: In a Banach space X endowed with a nondegenerate Gaussian measure, we consider Sobolev spaces of real functions defined in a sublevel set of a Sobolev nondegenerate function . We define the traces at of the elements of for , as elements of where ρ is the surface measure of Feyel and de La Pradelle. The range of the trace operator is contained in for and even in under further assumptions. If is a suitable halfspace, the range is characterized as a sort of fractional Sobolev space at the boundary. An important consequence of the general theory is an integration by parts formula for Sobolev functions, which involves their traces at . [Copyright &y& Elsevier]

Abstract: We report the current state of the graph isomorphism problem from the practical point of view. After describing the general principles of the refinement-individualization paradigm and pro ving its validity, we explain how it is implemented in several of the key implementations. In particular, we bring the description of the best known program nauty up to date and describe an innovative approach called Traces that outperforms the competitors for many difficult graph classes. Detailed comparisons against saucy, Bliss and conauto are presented. [Copyright &y& Elsevier]

Abstract: We provide non-smooth atomic decompositions for Besov spaces , , , defined via differences. The results are used to compute the trace of Besov spaces on the boundary Γ of bounded Lipschitz domains Ω with smoothness s restricted to and no further restrictions on the parameters . We conclude with some more applications in terms of pointwise multipliers. [Copyright &y& Elsevier]

Abstract: We investigate the properties of a class of weighted vector-valued -spaces and the corresponding (an)isotropic Sobolev–Slobodetskii spaces. These spaces arise naturally in the context of maximal -regularity for parabolic initial-boundary value problems. Our main tools are operators with a bounded -calculus, interpolation theory, and operator sums. [Copyright &y& Elsevier]

Abstract: One of the central problems of extremal hypergraph theory is the description of unavoidable subhypergraphs, in other words, the Turán problem. Let be a sequence of positive integers, . An a-partition of a k-set F is a partition in the form with for . An a-cluster with host is a family of k-sets such that for some a-partition of , for and the sets are pairwise disjoint. The family has 2k vertices and it is unique up to isomorphisms. With an intensive use of the delta-system method we prove that for and sufficiently large n, if is a k-uniform family on n vertices with exceeding the Erdős–Ko–Rado bound , then contains an a-cluster. The only extremal family consists of all the k-subsets containing a given element. [Copyright &y& Elsevier]

Abstract: We establish an asymptotic formula with a power savings in the error term for traces of CM values of a family of Maass–Poincaré series which contains the modular j-function as a special case. By work of Borcherds (1998) , Zagier (2002) , and Bringmann and Ono (2007) , these traces are Fourier coefficients of half-integral weight weakly holomorphic modular forms and Maass forms. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we demonstrate how to prove the correctness of systems implemented using low-level imperative features like pointers, files, and socket I/O with respect to high level I/O protocol descriptions by using the Coq proof assistant. We present a web-based course gradebook application developed with Ynot, a Coq library for verified imperative programming. We add a dialog-based I/O system to Ynot, and we extend Ynot’s underlying Hoare logic with event traces to reason about I/O and protocol behavior. Expressive abstractions allow the modular verification of both high level specifications like privacy guarantees and low level properties like data structure pointer invariants. [ABSTRACT FROM AUTHOR]

Abstract: We present weighted Sobolev spaces along with a trace theorem and an interpolation theorem for the spaces. Then we solve nonzero boundary value problems for elliptic equations in . [Copyright &y& Elsevier]

Abstract: In this paper we study conditions guaranteeing that functions defined on a Lipschitz domain Ω have boundary traces in Hardy and Besov spaces on ∂Ω. In turn these results are used to develop a new approach to the theory of compensated compactness and the theory of non-locally convex Hardy and Bergman type spaces. [Copyright &y& Elsevier]

Abstract: In this work we analyze the existence and regularity of the solution of a nonhomogeneous Neumann problem for the Poisson equation in a plane domain Ω with an external cusp. In order to prove that there exists a unique solution in using the Lax–Milgram theorem we need to apply a trace theorem. Since Ω is not a Lipschitz domain, the standard trace theorem for does not apply, in fact the restriction of functions is not necessarily in . So, we introduce a trace theorem by using weighted Sobolev norms in Ω. Under appropriate assumptions we prove that the solution of our problem is in and we obtain an a priori estimate for the second derivatives of the solution. [Copyright &y& Elsevier]

This paper offers evidence for the Isomorphic Mapping Hypothesis, which holds that individuals with agrammatic aphasia tend to have difficulty comprehending sentences in which the order of NPs is not aligned with the structure of the corresponding event. We begin by identifying a set of constructions in English and Korean for which the IMH makes predictions distinct from those of canonical order and trace-based theories of agrammatic comprehension. Then, drawing on data involving the interpretation of those patterns by English-speaking and Korean-speaking agrammatics, we argue for the conceptual and empirical superiority of the isomorphic mapping account. [ABSTRACT FROM AUTHOR]

Let RBp,qs(Rn) be the radial subspace of the Besov space Bp,qs(Rn). We prove the independence of the asymptotic behavior of the entropy numbersek(id : RBp0,q0s0(Rn)↦RBp1,q1s1(Rn))from the difference s0−s1 as long as the embedding itself RBp0,q0s0(Rn)↪RBp1,q1s1(Rn) is compact. In fact, we shall show thatek(id : RBp0,q0s0(Rn)↦RBp1,q1s1(Rn))∼k−n(1/p0−1/p1−1/p1).This is in a certain contrast to earlier results on entropy numbers in the context of Besov spaces Bp,qs(Ω) on bounded domains Ω. [Copyright &y& Elsevier]

In this article, we will give a complete classification of simple C*-algebras which can be written as inductive limits of algebras of the form An=⊕i=1knM[n,i](C(Xn,i)), where Xn,i are arbitrary variable one-dimensional compact metrizable spaces. The results unify and generalize the previous results for the case Xn,i=S1 and for the case of Xn,i being trees. We obtain our classification results by reducing the case of general one-dimensional spaces to the case of circles. The techniques in this paper play important roles in the study of the case of higher-dimensional spaces. [Copyright &y& Elsevier]