Affine inequivalence of cubic Maiorana–McFarland type bent functions

We consider cubic Maiorana–McFarland type bent functions having no affine derivatives. By using an invariant proposed by Dillon in 1975 we identify subclasses of inequivalent bent functions within this class. These can also be identified by [4, Theorem B]. However, our technique involves only elementary derivations. We also include some computational results.     

Generalized Maiorana–McFarland class and normality of p-ary bent functions

A class of bent functions which contains bent functions with various properties like regular, weakly regular and not weakly regular bent functions in even and in odd dimension, is analyzed. It is shown that this class includes the Maiorana–McFarland class as a special case. Known classes and examples of bent functions in odd characteristic are examined for their relation to this class. In the second part, normality for bent functions in odd characteristic is analyzed. It turns out that differently to Boolean bent functions, many – also quadratic – bent functions in odd characteristic and even dimension are not normal. It is shown that regular Coulter–Matthews bent functions are normal.     

Wide minimal binary linear codes from the general Maiorana–McFarland class

Designs, Codes and Cryptography - Minimal linear codes form a special class of linear codes that have important applications in secret sharing and secure two-party computation. These codes are...     

An Interpolating Distance Between Optimal Transport and Fisher–Rao Metrics

This paper defines a new transport metric over the space of nonnegative measures. This metric interpolates between the quadratic Wasserstein and the Fisher–Rao metrics and generalizes optimal transport to measures with different masses. It is defined as a generalization of the dynamical formulation of optimal transport of Benamou and Brenier, by introducing a source term in the continuity equation. The influence of this source term is measured using the Fisher–Rao metric and is averaged with the transportation term. This gives rise to a convex variational problem defining the new metric. Our first contribution is a proof of the existence of geodesics (i.e., solutions to this variational problem). We then show that (generalized) optimal transport and Hellinger metrics are obtained as limiting cases of our metric. Our last theoretical contribution is a proof that geodesics between mixtures of sufficiently close Dirac measures are made of translating mixtures of Dirac masses. Lastly, we propose a numerical scheme making use of first-order proximal splitting methods and we show an application of this new distance to image interpolation.     

Relationships Between Convexificators and Greenberg–Pierskalla Subdifferentials for Quasiconvex Functions

In this paper, the relationship between convexificators and Greenberg–Pierskalla-based (GP-based) subdifferentials for quasiconvex functions is proved. The established results lead to a mean value theorem, a chain rule, and the closedness property for GP-based subdifferentials. Furthermore, the connection between Clarke generalized gradient and Mordukhovich subdifferential with GP-based subdifferentials is highlighted.     

Muckenhoupt Ap-properties of Distance Functions and Applications to Hardy–Sobolev -type Inequalities

Potential Analysis - Let X be a metric space equipped with a doubling measure. We consider weights w(x) = dist(x,E)?α, where E is a closed set in X and $alpha in mathbb {R}$ . We...     

The Taylor–Browder Spectrum on Prime C*-Algebras

We provide a formula for the Taylor–Browder spectrum of a pair (L _a , R _b ) of left and right multiplication operators acting on a prime C*-algebra with non-zero socle. We also compute ascent and descent for multiplication operators on a prime ring, characterise Browder elements in a prime C*-algebra and discuss upper semicontinuity for the Browder spectrum.     

The Banach–Mazur Distance to the Cube in Low Dimensions

We show that the Banach–Mazur distance from any centrally symmetric convex body in ? ⁿ to the n-dimensional cube is at most

$\sqrt{n^2-2n+2+\frac{2}{\sqrt{n+2}-1}},$

which improves previously known estimates for “small” n≥3. (For large n, asymptotically better bounds are known; in the asymmetric case, exact bounds are known.) The proof of our estimate uses an idea of Lassak and the existence of two nearly orthogonal contact points in John’s decomposition of the identity. Our estimate on such contact points is closely connected to a well-known estimate of Gerzon on equiangular systems of lines.

     

The Inverse Problem for Ellis–Gohberg Orthogonal Matrix Functions

This paper deals with the inverse problem for the class of orthogonal functions that for the scalar case was introduced by Ellis and Gohberg (J Funct Anal 109:155–198, 1992). The problem is reduced to a linear equation with a special right hand side. This reduction allows one to solve the inverse problem for square matrix functions under conditions that are natural generalizations of those appearing in the scalar case. These conditions lead to a unique solution. Special attention is paid to the polynomial case. A number of partial results are obtained for the non-square case. Various examples are given to illustrate the main results and some open problems are presented.     

The Relation Between the Petits Integral and the McShane Integral of Banach-valued Functions Defined on R~n

R.A.Gordon[1]generalizedthedefinitionoftheMcShaneintegralforrealvaluedfunctionstotheabstractfunctionsfromR1toBanachspaces.Inthispaper,wedefinetheMcShaneintegralforabstractfunctionsfromRntoBanachspacesandprovethatifeveryfunctionffromIoinRntoBanachspacesisMcShaneintegrablethenfisPettisintegrable,andiftheunitballB(X)inX*isrelativelyweaklycompact,fisPetitsintegrablethenfisMcShaneintegrableonIo.ThisgivesapositiveanswertoanopenproblemleftbyR.A.Gordonin[1].ThroughoutthispaperXwilldenoteare…     

The Cauchy–Kowalewski Theorem in the Space of Pseudo Q-holomorphic Functions

In this work, we prove the Cauchy–Kowalewski theorem for the initial-value problem

$$\begin{aligned} \frac{\partial w}{\partial t}= & {} Lw \\ w(0,z)= & {} w_{0}(z) \end{aligned}$$

where

$$\begin{aligned} Lw:= & {} E_{0}(t,z)\frac{\partial }{\partial \overline{\phi }}\left( \frac{ d_{E}w}{dz}\right) +F_{0}(t,z)\overline{\left( \frac{\partial }{\partial \overline{\phi }}\left( \frac{d_{E}w}{dz}\right) \right) }+C_{0}(t,z)\frac{ d_{E}w}{dz} \\&+G_{0}(t,z)\overline{\left( \frac{d_{E}w}{dz}\right) } +A_{0}(t,z)w+B_{0}(t,z)\overline{w}+D_{0}(t,z) \end{aligned}$$

in the space \(P_{D}\left( E\right) \) of Pseudo Q-holomorphic functions.

     

Distance–regular graphs having the M-property

We analyse when the Moore–Penrose inverse of the combinatorial Laplacian of a distance–regular graph is an M-matrix; that is, it has non-positive off-diagonal elements or, equivalently when the Moore–Penrose inverse of the combinatorial Laplacian of a distance–regular graph is also the combinatorial Laplacian of another network. When this occurs we say that the distance–regular graph has the M-property. We prove that only distance–regular graphs with diameter up to three can have the M-property and we give a characterization of the graphs that satisfy the M-property in terms of their intersection array. Moreover, we exhaustively analyse strongly regular graphs having the M-property and we give some families of distance–regular graphs with diameter three that satisfy the M-property. Roughly speaking, we prove that all distance–regular graphs with diameter one; about half of the strongly regular graphs; only some imprimitive distance–regular graphs with diameter three, and no distance–regular graphs with diameter greater than three, have the M-property. In addition, we conjecture that no primitive distance–regular graph with diameter three has the M-property.     

The Shape of the Borwein–Affleck–Girgensohn Function Generated by Completely Monotone and Bernstein Functions

Fifteen years ago, J. Borwein, I. Affleck, and R. Girgensohn posed a problem concerning the shape (convexity, log-convexity, reciprocal concavity) of a certain function of several arguments that had manifested in a number of contexts concerned with optimization problems. In this paper we further explore the shape of the Borwein–Affleck–Girgensohn function as well as of its extensions generated by completely monotone and Bernstein functions.     

Test Functions, Schur–Agler Classes and Transfer-Function Realizations: The Matrix-Valued Setting

Given a collection of test functions, one defines the associated Schur–Agler class as the intersection of the contractive multipliers over the collection of all positive kernels for which each test function is a contractive multiplier. We indicate extensions of this framework to the case where the test functions, kernel functions, and Schur–Agler-class functions are allowed to be matrix- or operator-valued. We illustrate the general theory with two examples: (1) the matrix-valued Schur class over a finitely-connected planar domain and (2) the matrix-valued version of the constrained Hardy algebra (bounded analytic functions on the unit disk with derivative at the origin constrained to have zero value). Emphasis is on examples where the matrix-valued version is not obtained as a simple tensoring with ${{\mathbb C}^{N}}$ of the scalar-valued version.     

The Finite Spectrum of Sturm–Liouville Problems with Transmission Conditions and Eigenparameter-Dependent Boundary Conditions

We study the finite spectrum of Sturm–Liouville problems with transmission conditions and eigenparameter-dependent boundary conditions. For any positive integers m and n, we construct a class of regular Sturm–Liouville problems with transmission conditions and eigenparameter-dependent boundary conditions, which have at most m + n + 4 eigenvalues.     

Extremal Functions on Sturm–Liouville Hypergroups

We give an application of the theory of reproducing kernels to the Tikhonov regularization on the Sobolev spaces associated with a singular second-order differential operator. Next, we come up with some results regarding the multiplier operators for the Sturm–Liouville transform.     

The Sharp Weighted Bound for Multilinear Maximal Functions and Calderón–Zygmund Operators

We investigate the weighted bounds for multilinear maximal functions and Calderón–Zygmund operators from \(L^{p_1}(w_1)\times \cdots \times L^{p_m}(w_m)\) to \(L^{p}(v_{\vec {w}})\), where \(1<p_1,\cdots ,p_m<\infty \) with \(1/{p_1}+\cdots +1/{p_m}=1/p\) and \(\vec {w}\) is a multiple \(A_{\vec {P}}\) weight. We prove the sharp bound for the multilinear maximal function for all such \(p_1,\ldots , p_m\) and prove the sharp bound for \(m\)-linear Calderón–Zymund operators when \(p\ge 1\).     

The New Extensions of the Henstock–Kurzweil and the McShane Integrals of Vector-Valued Functions

In this paper, we define the Hake–Henstock–Kurzweil and the Hake–McShane integrals of Banach space valued functions defined on an open and bounded subset G of m-dimensional Euclidean space \(\mathbb {R}^{m}\). These are ”natural” extensions of the McShane and the Henstock–Kurzweil integrals from m-dimensional closed non-degenerate intervals to bounded and open subsets of \(\mathbb {R}^{m}\). Our goal is not a generalization for the sake of generalization. Indeed, we will show theorems which reduce the study of our integrals to the study of McShane and Henstock–Kurzweil integrals. As applications, we will present Hake-type theorems for the Henstock–Kurzweil and the McShane integrals in terms of our integrals.     

Doob–Meyer-Decomposition of Hilbert Space Valued Functions

(1) We give here a new proof of the Doob–Meyer-decomposition which is rather quick and elementary. It is more general in some aspects, and weaker in other aspects if compared to other approaches. The functions are defined on a totally ordered set with image in a Hilbert space. (2) We also give a second variant of the Doob–Meyer-decomposition; it is more specialized. (3) We apply (1) by reproving the Doléans-measure in a special setting and demonstrate that the stochastic integral could be defined on more general totally ordered time scales then .