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Since the development of higher local class field theory, several explicit reciprocity laws have been constructed. In particular, there are formulas describing the higher-dimensional Hilbert symbol given, among others, by M. Kurihara, A. Zinoviev and S. Vostokov. K. Kato also has explicit formulas for the higher-dimensional Kummer pairing associated to certain (one-dimensional) $p$-divisible groups. In this paper we construct an explicit reciprocity law describing the Kummer pairing associated to any (one-dimensional) formal group. The formulas are a generalization to higher-dimensional local fields of Kolyvagin's reciprocity laws. The formulas obtained describe the values of the pairing in terms of multidimensional $p$-adic differentiation, the logarithm of the formal group, the generalized trace and the norm on Milnor K-groups. In the second part of this paper, we will apply the results obtained here to give explicit formulas for the generalized Hilbert symbol and the Kummer pairing associated to a Lubin-Tate formal group. The results obtained in the second paper constitute a generalization to higher local fields, of the formulas of Artin-Hasse, K. Iwasawa and A. Wiles., The stronger reciprocity laws in this new version cover arbitrary higher local fields, as opposed to only standard higher local fields

The goal of this paper is to provide a categorical framework that leads to the definition of shtukas a la Drinfeld and of excursion operators a la V. Lafforgue. We take as the point of departure the Hecke action of Rep ( G ˇ ) on the category Shv ( Bun G ) of sheaves on Bun G , and also the endofunctor of the latter category, given by the action of the geometric Frobenius. The shtuka construction will be obtained by applying (various versions of) categorical trace.

In this paper, we extend the classical Bohr's inequality to the setting of the non-commutative Hardy space $H^1$ associated with a semifinite von Neumann algebra. As a consequence, we obtain Bohr's inequality for operators in the von Neumann-Schatten class $\cl C_1$ and square matrices of any finite order. Interestingly, we establish that the optimal bound for $r$ in the above mentioned Bohr's inequality concerning von Neumann-Shcatten class is 1/3 whereas it is 1/2 in the case of $2\times 2$ matrices and reduces to $\sqrt{2}-1$ for the case of $3\times 3$ matrices. We also obtain a generalization of our above-mentioned Bohr's inequality for finite matrices where we show that the optimal bound for $r$, unlike above, remains 1/3 for every fixed order $n\times n,\ n\ge 2$.

The aim of this note is to give a formula expressing the trace form associated with the 27 lines of a cubic surface.

In this paper we investigate the nuclear trace of vector‐valued Fourier multipliers on the torus and its applications to the index theory of periodic pseudo‐differential operators. First we characterise the nuclearity of pseudo‐differential operators acting on Bochner integrable functions. In this regards, we consider the periodic and the discrete cases. We go on to address the problem of finding sharp sufficient conditions for the nuclearity of vector‐valued Fourier multipliers on the torus. We end our investigation with two index formulae. First, we express the index of a vector‐valued Fourier multiplier in terms of its operator‐valued symbol and then we use this formula for expressing the index of certain elliptic operators belonging to periodic Hormander classes.

This paper develops nonasymptotic growth and concentration bounds for a product of independent random matrices. These results sharpen and generalize recent work of Henriksen-Ward, and they are similar in spirit to the results of Ahlswede-Winter and of Tropp for a sum of independent random matrices. The argument relies on the uniform smoothness properties of the Schatten trace classes., Comment: 21 pages

Through the Loewner equation, real-valued driving functions generate sets called Loewner hulls. We analyze driving functions that approach 0 at least as fast as \(a (T-t)^r\) as \(t \to T\), where \(r \in (0, 1/2)\), and show that the corresponding Loewner hulls have tangential behavior at time \(T\). We also prove a result about trace existence and apply it to show that the Loewner hulls driven by \(a(T-t)^r\) for \(r \in (0,1/2)\) have a tangential trace curve.

In this paper, we consider $C^*$ -algebras generated by Markov operators induced by a probabilistic dynamical system. It is shown that in some cases these algebras are isomorphic to Toeplitz, Cuntz, and Cuntz–Toeplitz algebras. By the probabilistic dynamic system, the Markov operator is constructed, and it is shown that it can be extended to a quantum channel on a convex set of positive operators with unit trace.

Rickart $$C^*$$ -algebras are unital and satisfy polar decomposition. We proved that if a unital $$C^*$$ -algebra $${\mathcal {A}}$$ satisfies polar decomposition and admits “good” faithful tracial states then $${\mathcal {A}}$$ is a Rickart $$C^*$$ -algebra. Via polar decomposition we characterized tracial states among all states on a Rickart $$C^*$$ -algebra. We presented the triangle inequality for Hermitian elements and traces on Rickart $$C^*$$ -algebra. For a block projection operator and a trace on a Rickart $$C^*$$ -algebra we proved a new inequality. As a corollary, we obtain a sharp estimate for a trace of the commutator of any Hermitian element and a projection. Also we give a characterization of traces in a wide class of weights on a von Neumann algebra.

Isolating cuspidal automorphic representations from the whole automorphic spectrum is a basic problem in the trace formula approach. For example, matrix coefficients of supercupidal representations can be used as test functions for this, which kills the continuous spectrum, but also a large class of cuspidal automorphic representations. For the case of number fields, multipliers of the Schwartz algebra is used in the recent work [3] to isolate all cuspidal spectrum which provide enough test functions and suitable for the comparison of orbital integrals. These multipliers are then applied to the proof of the Gan-Gross-Prasad conjecture for unitary groups [3,2]. In this article, we prove similar result on isolating the cuspidal spectrum in [3] for the function field case., The paper has been accepted for publication in SCIENCE CHINA Mathematics. A gap in the previous version is fixed

R.G. and Perez proved that under certain conditions the test ideal of a module closure agrees with the trace ideal of the module closure. We use this fact to compute the test ideals of various rings with respect to the closures coming from their indecomposable maximal Cohen–Macaulay modules. We also give an easier way to compute the test ideal of a hypersurface ring in three variables coming from a module with a particular type of matrix factorization.

This paper studies and supplements the proofs of the properties of the Kronecker Product of two matrices of different orders. We observe the relation between the singular value decomposition of the matrices and their Kronecker product and the relationship between the determinant, the trace, the rank and the polynomial matrix of the Kronecker products. We also establish the best least square solutions of the Kronecker product system of equations by using modified QR-algorithm.

We calculate characters of Weil representations of unitary groups over finite fields geometrically. As a byproduct, we deduce another proof of Gerardin’s trace formula for the Weil representation. ...

Let $(M,g)$ be a compact smoothly stratified pseudomanifold with an iterated cone-edge metric satisfying a spectral Witt condition. Under these assumptions the Hodge-Laplacian $\Delta$ is essentially self-adjoint. We establish the asymptotic expansion for the resolvent trace of $\Delta$. Our method proceeds by induction on the depth and applies in principle to a larger class of second-order differential operators of regular-singular type, e.g., Dirac Laplacians. Our arguments are functional analytic, do not rely on microlocal techniques and are very explicit. The results of this paper provide a basis for studying index theory and spectral invariants in the setting of smoothly stratified spaces and in particular allow for the definition of zeta-determinants and analytic torsion in this general setup.

In this paper we study polynomial vector fields on $${\mathbb {C}}^{2}$$ of degree larger than 2 with $$n^{2}$$ isolated singularities. More precisely, we show that if two polynomial vector fields share $$n^{2}-1$$ singularities with the same spectra (trace and determinant) and from these singularities $$n^{2}-2$$ have the same positions, then both vector fields have identical position and spectra at all the singularities. Moreover we also show that if two polynomial vector fields share $$n^{2}-1$$ singularities with the same positions and from these singularities $$n^{2}-2$$ have the same spectra, then both vector fields have identical position and spectra at all the singularities. Moreover we also prove that generic vector fields of degree $$n>2$$ have no twins and that for any $$n>2$$ there exist two uniparametric families of twin vector fields, i.e. two different families of vector fields having exactly the same singular points and for each singular point both vector fields have the same spectrum.

In linear algebra, the trace of a square matrix is defined as the sum of elements on the main diagonal. The trace of a matrix is the sum of its eigenvalues (counted with multiplicities), and it is invariant under the change of basis. This characterization can be used to define the trace of a tensor in general. Trace inequalities are mathematical relations between different multivariate trace functionals involving linear operators. These relations are straightforward equalities if the involved linear operators commute, however, they can be difficult to prove when the non-commuting linear operators are involved. Given two Hermitian tensors H1 and H2 that do not commute. Does there exist a method to transform one of the two tensors such that they commute without completely destroying the structure of the original tensor? The spectral pinching method is a tool to resolve this problem. In this work, we will apply such spectral pinching method to prove several trace inequalities that extend the Araki–Lieb–Thirring (ALT) inequality, Golden–Thompson(GT) inequality and logarithmic trace inequality to arbitrary many tensors. Our approaches rely on complex interpolation theory as well as asymptotic spectral pinching, providing a transparent mechanism to treat generic tensor multivariate trace inequalities. As an example application of our tensor extension of the Golden–Thompson inequality, we give the tail bound for the independent sum of tensors. Such bound will play a fundamental role in high-dimensional probability and statistical data analysis.

In this paper, we define the localization operator associated with the Riemann–Liouville operator, and show that it is not only bounded, but it is also in the Schatten–von Neumann class. We also give a trace formula when the symbol function is nonnegative.

We study two kinds of categorical traces of (monoidal) dg categories, with particular interest in categories of Soergel bimodules. First, we explicitly compute the usual Hochschild homology, or derived vertical trace, of the category of Soergel bimodules in arbitrary types. Secondly, we introduce the notion of derived horizontal trace of a monoidal dg category and compute the derived horizontal trace of Soergel bimodules in type A. As an application we obtain a derived annular Khovanov-Rozansky link invariant with an action of full twist insertion, and thus a categorification of the HOMFLY-PT skein module of the solid torus., Comment: 74 pages, comments welcome

We give the equations of the n-th symmetric product $$X^n/S_n$$ X n / S n of a flat affine scheme $$X=\mathrm {Spec}\,A$$ X = Spec A over a commutative ring F. As a consequence, we find a closed immersion into the coarse moduli space parameterizing n-dimensional linear representations of A. This is done by exhibiting an isomorphism between the ring of symmetric tensors over A and the ring generated by the coefficients of the characteristic polynomial of polynomials in commuting generic matrices giving representations of A. Using this we derive an isomorphism of the associated reduced schemes over an infinite field. When the characteristic is zero we show that this isomorphism is an isomorphism of schemes and we express it in term of traces.

In this paper, we introduce a new class of additive codes over finite fields, viz. multi-twisted (MT) additive codes, which are generalizations of constacyclic additive codes. We study their algebraic structures by writing a canonical form decomposition and provide an enumeration formula for these codes. By placing ordinary, Hermitian and $$*$$ trace bilinear forms, we further study their dual codes and derive necessary and sufficient conditions under which a MT additive code is self-dual and self-orthogonal. We also derive a necessary and sufficient condition for the existence of a self-dual MT additive code over a finite field, and provide enumeration formulae for all self-dual and self-orthogonal MT additive codes over finite fields with respect to the aforementioned trace bilinear forms. We also obtain several good codes within the family of MT additive codes over finite fields.

A perturbation of the de Rham complex was introduced by Witten for an exact 1-form Θ and later extended by Novikov for a closed 1-form on a Riemannian manifold M. We use invariance theory to show that the perturbed index density is independent of Θ; this result was established previously by J. A. Alvarez Lopez, Y. A. Kordyukov and E. Leichtnam (2020) using other methods. We also show the higher order heat trace asymptotics of the perturbed de Rham complex exhibit nontrivial dependence on Θ. We establish similar results for manifolds with boundary imposing suitable boundary conditions and give an equivariant version for the local Lefschetz trace density. In the setting of the Dolbeault complex, one requires Θ to be a $$\overline \partial $$ closed 1-form to define a local index density; we show in contrast to the de Rham complex, that this exhibits a nontrivial dependence on Θ even in the setting of Riemann surfaces.

We investigate a few special decompositions in arbitrary rings and matrix rings over indecomposable rings into nilpotent and idempotent elements. Moreover, we also define and study the nilpotent sum trace number of nilpotent matrices over an arbitrary ring. Some related notions are explored as well.

We consider a number of nonstandard boundary value problems for the system of Poisson equations on the plane. The statement of these problems is based on the decomposition of the Sobolev space into the sum of kernels of trace functionals and one-dimensional subspaces spanned by a basis vector on which the corresponding trace functional is nontrivial. These problems are nonstandard in the sense that the boundary conditions are nonlocal and may contain the main first-order differential operators of field theory, i.e., the gradient, divergence, and curl. We prove existence and uniqueness theorems for the solutions in the framework of the duality between the Sobolev space and its conjugate space.

Given a binary sequence, its trace representation allows us to reconstruct itself efficiently and to analyze its properties, such as the linear complexity. In this paper, we study a family of the binary sequences derived from Euler quotients modulo pq, where p and q are two distinct odd primes and p divides q − 1. Our main contribution is to give a trace representation of this family within these assumptions by determining the defining pairs of the corresponding subsequences. As a byproduct, we rediscover some known results of linear complexities by using trace representations of the proposed sequences.

Near-bent functions that occur in odd dimensions are the important class of Boolean functions, which are useful functions for cryptography. In this paper, we construct near-bent function with trace term using well known Welch function exponent in polynomial form and various other forms of near-bent functions. We have also investigated some important cryptographical properties of near-bent functions.

We prove that all trace $1$, $2\times 2$ invertible matrices over $\mathbb{Z}$ are nil-clean and, up to similarity, that there are only two trace $1$, $2\times 2$ invertible matrices over $\mathbb{Z}$.

We continue the work of Camano et al. [SIAM J. Math. Anal. 49 (2017) 4376–4401] on electromagnetic Steklov eigenvalues. The authors recognized that in general the eigenvalues do not correspond to the spectrum of a compact operator and hence proposed a modified eigenvalue problem with the desired properties. The present article considers the original and the modified electromagnetic Steklov eigenvalue problem. We cast the problems as eigenvalue problem for a holomorphic operator function A(⋅). We construct a “test function operator function” T(⋅) so that A(λ) is weakly T(λ)-coercive for all suitable λ, i.e. T(λ)*A(λ) is a compact perturbation of a coercive operator. The construction of T(⋅) relies on a suitable decomposition of the function space into subspaces and an apt sign change on each subspace. For the approximation analysis, we apply the framework of T-compatible Galerkin approximations. For the modified problem, we prove that convenient commuting projection operators imply T-compatibility and hence convergence. For the original problem, we require the projection operators to satisfy an additional commutator property which concerns the tangential trace. The existence and construction of such projection operators remain open questions.

[EN] We prove Hardy type inequalities for the fractional relativistic operator by using two different techniques. The first approach goes through trace Hardy inequalities. In order to get the latter, we study the solutions of the associated extension problem. The second develops a non-local version of the ground state representation in the spirit of Frank, Lieb, and Seiringer. This research was supported by the Basque Government through BERC 2018-2021 program, by Spanish Ministry of Economy and Competitiviness through BCAM Severo Ochoa accreditation SEV-2017-2018 and the project MTM2017-82160-C2-1-P funded by AEI/FEDER, UE. She also acknowledges the project RYC2018-025477-I and IKERBASQUE.

An algebra with trace function has exponentially bounded codimensions if and only if it satisfies an ordinary polynomial identity and what we call an inter-trace permuting identity, namely one of t...

We provide sufficient conditions for Toeplitz operators with distributional symbols acting on the Bergman space on the unit disk to be trace class. The Berezin transform of distributions, introduced in the paper, yields a formula for the trace. Several instructive examples are also given.

We prove an almost everywhere convergence result for Bochner-Riesz means of $L^p$ functions on Heisenberg-type groups, yielding the existence of a $p>2$ for which convergence holds for means of arbitrarily small order. The proof hinges on a reduction of weighted $L^2$ estimates for the maximal Bochner-Riesz operator to corresponding estimates for the non-maximal operator, and a `dual Sobolev trace lemma', whose proof is based on refined estimates for Jacobi polynomials., 49 pages, 2 figures

In this paper, we consider a version of the de Rham theorem that clarifies the additional information on the trace of the potential. By using this result, we can improve the existence of a weak solution to a Maxwell–Stokes type system in a multi-connected domain with holes.

We construct explicit local systems on the affine line in characteristic $p>2$, whose geometric monodromy groups are the finite symplectic groups $Sp_{2n}(q)$ for all $n \ge 2$, and others whose geometric monodromy groups are the special unitary groups $SU_n(q)$ for all odd $n \ge 3$, and $q$ any power of $p$, in their total Weil representations. One principal merit of these local systems is that their associated trace functions are one-parameter families of exponential sums of a very simple, i.e., easy to remember, form. We also exhibit hypergeometric sheaves on $G_m$, whose geometric monodromy groups are the finite symplectic groups $Sp_{2n}(q)$ for any $n \ge 2$, and others whose geometric monodromy groups are the finite general unitary groups $GU_n(q)$ for any odd $n \geq 3$., 56 pages

Given a prime number p we consider C p the topological completion of the algebraic closure of the field of p-adic numbers with respect to the usual p-adic absolute value. In this paper, we point ou...

In the proof of the BFK-gluing formula for zeta-determinants of Laplacians there appears a real polynomial whose constant term is an important ingredient in the gluing formula. This polynomial is determined by geometric data on an arbitrarily small collar neighborhood of a cutting hypersurface. In this paper we express the coefficients of this polynomial in terms of the scalar and principal curvatures of the cutting hypersurface embedded in the manifold when this hypersurface is 2-dimensional. Similarly, we express some coefficients of the heat trace asymptotics of the Dirichlet-to-Neumann operator in terms of the scalar and principal curvatures of the cutting hypersurface., 28 pages

We generalize von Neumann's well-known trace inequality, as well as related eigenvalue inequalities for hermitian matrices, to Schatten-class operators between complex Hilbert spaces of infinite dimension. To this end, we exploit some recent results on the $C$-numerical range of Schatten-class operators. For the readers' convenience, we sketched the proof of these results in the Appendix., 16 pages

As is known, in the reflection equation (RE) algebra associated with an involutive or Hecke $$R$$ -matrix, the elements $$ \operatorname{Tr} _RL^k$$ (called quantum power sums) are central. Here, $$L$$ is the generating matrix of this algebra, and $$ \operatorname{Tr} _R$$ is the operation of taking the $$R$$ -trace associated with a given $$R$$ -matrix. We consider the problem of whether this is true in certain RE-like algebras depending on a spectral parameter. We mainly study algebras similar to those introduced by Reshetikhin and Semenov-Tian-Shansky (we call them algebras of RS type). These algebras are defined using some current $$R$$ -matrices (i.e., depending on parameters) arising from involutive and Hecke $$R$$ -matrices by so-called Baxterization. In algebras of RS type. we define quantum power sums and show that the lowest quantum power sum is central iff the value of the “charge” $$c$$ in its definition takes a critical value. This critical value depends on the bi-rank $$(m|n)$$ of the initial $$R$$ -matrix. Moreover, if the bi-rank is equal to $$(m|m)$$ and the charge $$c$$ has a critical value, then all quantum power sums are central.

By permutation behavior of certain linearized polynomials, the bentness of quadratic vectorial bent functions of the form F (x)=+Trmkt(ckx1+2kt) is investigated, where n=2kt and m|kt with k, t being positive integers. The numerical results show that there exist new quadratic vectorial bent functions obtained up to extended affine equivalence

Let $${\mathcal {M}}$$ be a diffuse von Neumann algebra with a faithful normal semi-finite trace $$\tau $$ , and let E be a symmetric quasi-Banach space. Then for any Orlicz function $$\varphi $$ , we can define the noncommutative Calderon–Lozanovskiĭ spaces $$E_\varphi ({\mathcal {M}})$$ . These spaces share many properties with their classical counterparts. In particular, new multiplication operator spaces and complex interpolation spaces of such spaces are given under a wide range of conditions. Moreover, letting $${\mathcal {A}}$$ be a maximal subdiagonal algebra of $${\mathcal {M}}$$ , we introduce the noncommutative Calderon–Lozanovskiĭ–Hardy spaces $$H^\varphi ({\mathcal {A}})$$ and transfer the recent results of the noncommutative Hardy spaces to the noncommutative Calderon–Lozanovskiĭ–Hardy spaces.

Some Choi–Davis–Jensen’s type trace inequalities for convex functions are proved. Also, we generalize these inequalities for any arbitrary operator mean via operator monotone decreasing functions. In particular, we present some new order among $$\mathrm{tr}(\Phi (C)A)$$ and $$\mathrm{tr}(\Phi (C)A^{-1})$$ . New refinements of some power type trace inequalities via reverse and refinement of Young’s inequality are established. Among our results, we obtain new versions of the Holder type trace inequality for any arbitrary operator mean.

We study the kernel function of the operator u $\rightarrow$ L $\mu$ u = --$\Delta$u + $\mu$ |x| 2 u in a bounded smooth domain $\Omega$ $\subset$ R N + such that 0 $\in$ $\partial$$\Omega$, where $\mu$ $\ge$ -- N 2 4 is a constant. We show the existence of a Poisson kernel vanishing at 0 and a singular kernel with a singularity at 0. We prove the existence and uniqueness of weak solutions of L $\mu$ u = 0 in $\Omega$ with boundary data $\nu$ + k$\delta$ 0 , where $\nu$ is a Radon measure on $\partial$$\Omega$ \ {0}, k $\in$ R and show that this boundary data corresponds in a unique way to the boundary trace of positive solution of L $\mu$ u = 0 in $\Omega$.

Strong cell decomposition property has been proved in non-valuational weakly o-minimal expansions of ordered groups. In this note, we show that all o-minimal traces have strong cell decomposition property. Also after introducing the notion of irrational nonvaluational cut in arbitrary o-minimal structures, we show that every expansion of o-minimal structures by irrational nonvaluational cuts is an o-minimal trace.

In this note, we continue to investigate common properties of the products in the setting of rings, bounded linear operators, or Banach algebras. We prove: (i) If a,b,c are elements in a unital associative ring R satisfying aba=aca, then von Neumann regularity (resp. generalized Fredholmness relative to an ideal I of R) of 1−ac is converted into that of 1−ba. (ii) If A,B,C are bounded linear operators satisfying ABA=ACA, then I−AC and I−BA share common complementability of kernels and ranges. (iii) If a,b,c are elements in a unital semisimple Banach algebra A satisfying aba=aca and I is a trace ideal of A such that soc(A)⊆I⊆ kh(soc(A)), then 1−ac and 1−ba share common Fredholmness relative to I and have the same abstract index. A similar result holds for B‐Fredholmness in primitive Banach algebra.

We study the differential-geometric properties of the loci of fixed points of the elliptic isometries of the manifold of positive definite real matrices with the trace metric. We also give an explicit description of such loci and in particular we find their De Rham decomposition.

Made available in DSpace on 2020-12-12T02:12:06Z (GMT). No. of bitstreams: 0 Previous issue date: 2020-05-07 Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) In this work, we present a explicit trace forms for maximal real subfields of cyclotomic fields as tools for constructing algebraic lattices in Euclidean space with optimal center density. We also obtain a closed formula for the Gram matrix of algebraic lattices obtained from these subfields. The obtained lattices are rotated versions of the lattices Λ9, Λ10 and Λ11 and they are images of Z-submodules of rings of integers under the twisted homomorphism, and these constructions, as algebraic lattices, are new in the literature. We also obtain algebraic lattices in odd dimensions up to 7 over real subfields, calculate their minimum product distance and compare with those known in literatura, since lattices constructed over real subfields have full diversity. Department of Mathematics School of Sciences São Paulo State University (Unesp) Department of Mathematics Institute of Biosciences Humanities and Exact Sciences (Ibilce) São Paulo State University (Unesp) São Paulo Federal Institute at Cubatão Department of Mathematics & Statistics San Diego University Department of Mathematics School of Sciences São Paulo State University (Unesp) Department of Mathematics Institute of Biosciences Humanities and Exact Sciences (Ibilce) São Paulo State University (Unesp) CNPq: 429346/2018-2

We prove an optimal extension and trace theorem for Sobolev spaces of nonlocal operators. The extension is given by a suitable Poisson integral and solves the corresponding nonlocal Dirichlet problem. We give a Douglas-type formula for the quadratic form of the Poisson extension., We are now able to add the trace theorem and complete the picture