Some new families of partial difference sets in finite fields

In this paper we present some new cyclotomic families of partial difference sets. The argument rests on a general procedure for constructing cyclotomic difference sets or partial difference sets in Galois domains due to Ott (Des Codes Cryptogr, doi:10.1007/s10623-015-0082-6, 2015). Definitions and various properties of partial difference sets can be found for instance in Ma (Des Codes Cryptogr 4:221–261, 1994).     

Weighted Solyanik Estimates for the Hardy–Littlewood Maximal Operator and Embedding of into

Let Open image in new window denote a weight in Open image in new window which belongs to the Muckenhoupt class Open image in new window and let Open image in new window denote the uncentered Hardy–Littlewood maximal operator defined with respect to the measure Open image in new window . The sharp Tauberian constant of Open image in new window with respect to Open image in new window , denoted by Open image in new window , is defined by

Open image in new window

In this paper, we show that the Solyanik estimate

$$\begin{aligned} \lim _{\alpha \rightarrow 1^-}\mathsf{C}_{w}(\alpha ) = 1 \end{aligned}$$

holds. Following the classical theme of weighted norm inequalities we also consider the sharp Tauberian constants defined with respect to the usual uncentered Hardy–Littlewood maximal operator Open image in new window and a weight Open image in new window :

Open image in new window

We show that we have Open image in new window if and only if Open image in new window . As a corollary of our methods we obtain a quantitative embedding of Open image in new window into Open image in new window .

     

Serre’s modularity conjecture (I)

This paper is the first part of a work which proves Serre’s modularity conjecture. We first prove the cases \(p\not=2\) and odd conductor, and p=2 and weight 2, see Theorem 1.2, modulo Theorems 4.1 and 5.1. Theorems 4.1 and 5.1 are proven in the second part, see Khare and Wintenberger (Invent. Math., doi: 10.1007/s00222-009-0206-6, 2009). We then reduce the general case to a modularity statement for 2-adic lifts of modular mod 2 representations. This statement is now a theorem of Kisin (Invent. Math., doi: 10.1007/s00222-009-0207-5, 2009).     

A note “On <Emphasis Type="Italic">ss</Emphasis>-quasinormal or weakly <Emphasis Type="Italic">s</Emphasis>-permutably embedded subgroups of finite groups”

We point out some gaps in the paper (Monatsh Math doi: 10.1007/s00605-016-0877-1, 2016) by Kong and Guo, and give new brief proofs of the main results.     

$$C_0$$-semigroups associated with uniquely ergodic Kantorovich modifications of operators

We extend to the context of \(L^p\) spaces and \(C_0\)-semigroups of operators our previous results from Heilmann and Ra?a (Positivity 21:897–910, 2017.  https://doi.org/10.1007/s11117-016-0441-1), concerning the eigenstructure and iterates of uniquely ergodic Kantorovich modifications of linking operators.     

Filtration de monodromie et cycles évanescents formels

Berkovich (Invent. Math. 125(2):367–390, 1996), Fujiwara (Duke Math. J. 80(1):15–57, 1995) and Huber (J. Algebraic Geom. 7(2):359–403, 1998) have proved that the fiber of the vanishing cycles at a point of the special fiber depends only on the formal completion at this point. We refine this result and prove the invariance under formal completion of the perverse monodromy filtration on the fiber of vanishing cycles. This result is used in an essential way by Boyer (Invent. Math. doi: 10.1007/s00222-009-0183-9, 2009).     

Rankin–Cohen brackets on Jacobi forms and the adjoint of some linear maps

Given a fixed Jacobi cusp form, we consider a family of linear maps between the spaces of Jacobi cusp forms using the Rankin–Cohen brackets, and then we compute the adjoint maps of these linear maps with respect to the Petersson scalar product. The Fourier coefficients of the Jacobi cusp forms constructed using this method involve special values of certain Dirichlet series associated to Jacobi cusp forms. This is a generalization of the work due to Kohnen (Math Z, 207:657–660, 1991) and Herrero (Ramanujan J,  10.1007/s11139-013-9536-5, 2014) in case of elliptic modular forms to the case of Jacobi cusp forms which is also considered earlier by Sakata (Proc Japan Acad Ser A, Math Sci 74, 1998) for a special case.     

Domination problem for AM-compact abstract Uryson operators

The Boolean algebra of fragments of a positive abstract Uryson operator recently was described in M. Pliev (Positivity, doi:10.1007/s11117-016-0401-9, 2016). Using this result, we prove a theorem of domination for AM-compact positive abstract Uryson operators from a Dedekind complete vector lattice E to a Banach lattice F with an order continuous norm.     

Optimal Control of Nonlinear Transport-Flow Mixed Variational Problems

Optimal control of nonlinear transport-flow mixed variational problems are studied qualitatively, and the solvability analysis of the transport and flow mixed state systems is performed on the basis of primal and dual evolution duality principles. Corresponding primal and dual mixed optimality conditions are established by the application of some fundamental perturbation conjugate duality results recently proposed [8 G. Alduncin (2013). Optimal control of evolution mixed variational inclusions. Applied Mathematics and Optimization 68:445–473.[Crossref], [Web of Science ®] , [Google Scholar]]. Further, for computational purposes, two- and three-field proximation penalty-duality algorithms in the resolution of the mixed optimality conditions are finally presented and discussed.     

A remark on the result of Radu Miculescu and Alexandru Mihail

In this note, we show that the assumption of continuity considered in the recent result of Miculescu and Mihail (J Fixed Point Theory Appl, doi: 10.1007/s11784-017-0411-7, 2017) can be relaxed further. We also observe that the power convex contraction introduced by Miculescu and Mihail (see condition (1.1)) provides one more solution to the open question of Rhoades (Contemp Math 72:233–245, 1988] regarding existence of a contractive definition which is strong enough to generate a fixed point but does not force the mapping to be continuous at the fixed point.     

Relaxation of Optimal Control Problems Involving Time Dependent Subdifferential Operators

This article provides in the setting of infinite dimensional Hilbert space, a result concerning the existence and uniqueness of solutions for Lipschitz single-valued perturbations of evolution problems associated with time-dependent subdifferential operators. The result is used to extend to optimal control problems associated with such equations the relaxation theorems with Young measures established in Casting et al. [7 C. Castaing , P. Raynaud de Fitte , and M. Valadier , M. ( 2004 ). Young Measures on Topological Spaces with Applications in Control Theory and Probability Theory . Kluwer Academic Publishers , Dordrecht . [Google Scholar], 11 J. F. Edmond and L. Thibault ( 2005 ). Relaxation of an optimal control problem involving a perturbed sweeping process . Math. Program Ser. B 104 : 347 – 373 .[Crossref], [Web of Science ®] , [Google Scholar]] and Edmond and Thibault [11 J. F. Edmond and L. Thibault ( 2005 ). Relaxation of an optimal control problem involving a perturbed sweeping process . Math. Program Ser. B 104 : 347 – 373 .[Crossref], [Web of Science ®] , [Google Scholar]].     

Asymptotic Normality of Some Graph Sequences

For a simple finite graph G denote by Open image in new window the number of ways of partitioning the vertex set of G into k non-empty independent sets (that is, into classes that span no edges of G). If \(E_n\) is the graph on n vertices with no edges then Open image in new window coincides with Open image in new window , the ordinary Stirling number of the second kind, and so we refer to Open image in new window as a graph Stirling number. Harper showed that the sequence of Stirling numbers of the second kind, and thus the graph Stirling sequence of \(E_n\), is asymptotically normal—essentially, as n grows, the histogram of Open image in new window , suitably normalized, approaches the density function of the standard normal distribution. In light of Harper’s result, it is natural to ask for which sequences \((G_n)_{n \ge 0}\) of graphs is there asymptotic normality of Open image in new window . Thanh and Galvin conjectured that if for each n, \(G_n\) is acyclic and has n vertices, then asymptotic normality occurs, and they gave a proof under the added condition that \(G_n\) has no more than \(o(\sqrt{n/\log n})\) components. Here we settle Thanh and Galvin’s conjecture in the affirmative, and significantly extend it, replacing “acyclic” in their conjecture with “co-chromatic with a quasi-threshold graph, and with negligible chromatic number”. Our proof combines old work of Navon and recent work of Engbers, Galvin and Hilyard on the normal order problem in the Weyl algebra, and work of Kahn on the matching polynomial of a graph.     

Sufficient Conditions of Isolated Minimizers for Constrained Programming Problems

In this article, sufficient optimality conditions for nonsmooth programming problems with inequality constraints are studied. When the objective and constraint functions are ?-stable at some point, a first-order sufficient optimality condition for an isolated local minimizer of order 1 is established. A second-order sufficient optimality condition for an isolated local minimizer of order 2 is obtained in terms of the lower Dini second-order directional derivatives of the Lagrangian function. The obtained results extend and improve the ones found by Ward [21 D.E. Ward ( 1994 ). Characterizations of strict local minima and nacessary conditions for weak sharp minima . Journal of Optimization Theory and Applications 80 : 551 – 571 .[Crossref], [Web of Science ®] , [Google Scholar]].     

Constructions of cyclic constant dimension codes

Subspace codes and particularly constant dimension codes have attracted much attention in recent years due to their applications in random network coding. As a particular subclass of subspace codes, cyclic subspace codes have additional properties that can be applied efficiently in encoding and decoding algorithms. It is desirable to find cyclic constant dimension codes such that both the code sizes and the minimum distances are as large as possible. In this paper, we explore the ideas of constructing cyclic constant dimension codes proposed in Ben-Sasson et al. (IEEE Trans Inf Theory 62(3):1157–1165, 2016) and Otal and Özbudak (Des Codes Cryptogr, doi: 10.1007/s10623-016-0297-1, 2016) to obtain further results. Consequently, new code constructions are provided and several previously known results in [2] and [17] are extended.     

AFFINE EXTENSIONS OF PRINCIPAL ADDITIVE BUNDLES OVER A PUNCTURED SURFACE

The aim of this article is to make a first step towards the classification of complex normal affine

Open image in new window

_α -threefolds X. We consider the case where the restriction of the quotient morphism π: X → S to π^?1 (S _* ), where S _* denotes the complement of some regular closed point in S, is a principal

Open image in new window

_α -bundle. The variety SL₂ will be of special interest and a source of many examples. It has a natural right

Open image in new window

_α -action such that the quotient morphism SL₂ →

Open image in new window

² restricts to a principal

Open image in new window

_α -bundle over the punctured plane

Open image in new window

.

     

Almost Exponential Decay Near Maxwellian

ABSTRACT

By direct interpolation of a family of smooth energy estimates for solutions near Maxwellian equilibrium and in a periodic box to several Boltzmann type equations in Guo (2002 Guo , Y. ( 2002 ). The Landau equation in a periodic box . Comm. Math. Phys. 231 ( 3 ): 391 – 434 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar] 2003a Guo , Y. ( 2003a ). Classical solutions to the Boltzmann equation for molecules with an angular cutoff . Arch. Ration. Mech. Anal. 169 ( 4 ): 305 – 353 . [CSA] [CROSSREF]  [Google Scholar] b Guo , Y. ( 2003b ). The Vlasov-Maxwell-Boltzmann system near maxwellians . Invent. Math. 153 ( 3 ): 593 – 630 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) and Strain and Guo (2004 Strain , R. M. , Guo , Y. ( 2004 ). Stability of the relativistic Maxwellian in a collisional plasma . Comm. Math. Phys. 251 ( 2 ): 263 – 320 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]), we show convergence to Maxwellian with any polynomial rate in time. Our results not only resolve the important open problem for both the Vlasov-Maxwell-Boltzmann system and the relativistic Landau-Maxwell system for charged particles, but also lead to a simpler alternative proof of recent decay results (Desvillettes and Villani, 2005 Desvillettes , L. , Villani , C. ( 2005 ). On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation . Invent. Math. 159 ( 2 ): 245 – 316 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) for soft potentials as well as the Coulombic interaction, with precise decay rate depending on the initial conditions.     

The linear complexity of a class of binary sequences with optimal autocorrelation

Binary sequences with optimal autocorrelation and large linear complexity have important applications in cryptography and communications. Very recently, a class of binary sequences of period 4p with optimal autocorrelation was proposed by interleaving four suitable Ding–Helleseth–Lam sequences (Des. Codes Cryptogr.,  https://doi.org/10.1007/s10623-017-0398-5), where p is an odd prime with \(p \equiv 1(\bmod 4)\). The objective of this paper is to determine the minimal polynomial and the linear complexity of this class of binary optimal sequences via a sequence polynomial approach. It turns out that this class of sequences has quite good linear complexity.     

Deformations of Charged Axially Symmetric Initial Data and the Mass–Angular Momentum–Charge Inequality

We show how to reduce the general formulation of the mass–angular momentum–charge inequality, for axisymmetric initial data of the Einstein–Maxwell equations, to the known maximal case whenever a geometrically motivated system of equations admits a solution. It is also shown that the same reduction argument applies to the basic inequality yielding a lower bound for the area of black holes in terms of mass, angular momentum, and charge. This extends previous work by the authors (Cha and Khuri, Ann Henri Poincaré, doi: 10.1007/s00023-014-0332-6, arXiv:1401.3384, 2014), in which the role of charge was omitted. Lastly, we improve upon the hypotheses required for the mass–angular momentum–charge inequality in the maximal case.     

A note on Bridgeland Hall algebras

In this note, let 𝒜 be a finitary hereditary abelian category with enough projectives. By using the associativity formula of Hall algebras, we give a new proof of the main theorem in [17 Yanagida, S. (2016). A note on Bridgeland’s Hall algebra of two-periodic complexes. Math. Z. 282(3):973–991.[Crossref], [Web of Science ®] , [Google Scholar]], which states that the Bridgeland Hall algebra of 2-cyclic complexes of projective objects in 𝒜 is isomorphic to the Drinfeld double Hall algebra of 𝒜.     

Distributional Transformations Without Orthogonality Relations

Distributional transformations characterized by equations relating expectations of test functions weighted by a given biasing function on the original distribution to expectations of the test function’s higher derivatives with respect to the transformed distribution play a great role in Stein’s method and were, in great generality, first considered by Goldstein and Reinert (J Theoret Probab 18(1):237–260, 2005. doi: 10.1007/s10959-004-2602-6). We prove two abstract existence and uniqueness results for such distributional transformations, generalizing their \(X-P\)-bias transformation. On the one hand, we show how one can abandon previously necessary orthogonality relations by subtracting an explicitly known polynomial depending on the test function from the test function itself. On the other hand, we prove that for a given nonnegative integer m, it is possible to obtain the expectation of the m-th derivative of the test function with respect to the transformed distribution in the defining equation, even though the biasing function may have \(k<m\) sign changes, if these two numbers have the same parity. We explain how these results can be used to guarantee the existence of two different generalizations of the zero-bias transformation by Goldstein and Reinert (Ann Appl Probab 7(4):935–952, 1997. doi: 10.1214/aoap/1043862419). Further applications include the derivation of Stein-type characterizations without needing to solve any Stein equation and the presentation of a general framework for estimating the distance from the distribution of a given real random variable X to that of a random variable Z, whose distribution is characterized by some mth-order linear differential operator. We also explain the fact that, in general, the biased distribution depends on the choice of the sign change points, if these are ambiguous. This new phenomenon does not appear in the framework from Goldstein and Reinert (2005).