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Products of Snowflaked Euclidean Lines Are Not Minimal for Looking Down

Matthieu Joseph, Tapio Rajala

Analysis and Geometry in Metric Spaces > 2017 > 5 > 1 > 78-97

We show that products of snowflaked Euclidean lines are not minimal for looking down. This question was raised in Fractured fractals and broken dreams, Problem 11.17, by David and Semmes. The proof uses arguments developed by Le Donne, Li and Rajala to prove that the Heisenberg group is not minimal for looking down. By a method of shortcuts, we define a new distance d such that the product of snowflaked...

article

Groups and monoids of Pythagorean triples connected to conics

Nadir Murru, Marco Abrate, Stefano Barbero, Umberto Cerruti

Open Mathematics > 2017 > 15 > 1 > 1323-1331

We define operations that give the set of all Pythagorean triples a structure of commutative monoid. In particular, we define these operations by using injections between integer triples and 3 × 3 matrices. Firstly, we completely characterize these injections that yield commutative monoids of integer triples. Secondly, we determine commutative monoids of Pythagorean triples characterizing some Pythagorean...

article

Structured Matrix Methods Computing the Greatest Common Divisor of Polynomials

Dimitrios Christou, Marilena Mitrouli, Dimitrios Triantafyllou

Special Matrices > 2017 > 5 > 1 > 202-224

This paper revisits the Bézout, Sylvester, and power-basis matrix representations of the greatest common divisor (GCD) of sets of several polynomials. Furthermore, the present work introduces the application of the QR decomposition with column pivoting to a Bézout matrix achieving the computation of the degree and the coeffcients of the GCD through the range of the Bézout matrix. A comparison in terms...

article

Sign patterns of J-orthogonal matrices

Frank J. Hall, Zhongshan Li, Caroline T. Parnass, Miroslav Rozložník

Special Matrices > 2017 > 5 > 1 > 225-241

This paper builds upon the results in the article “G-matrices, J-orthogonal matrices, and their sign patterns", Czechoslovak Math. J. 66 (2016), 653-670, by Hall and Rozloznik. A number of further general results on the sign patterns of the J-orthogonal matrices are proved. Properties of block diagonal matrices and their sign patterns are examined. It is shown that all 4 × 4 full sign patterns...

article

Spectral properties of a sequence of matrices connected to each other via Schur complement and arising in a compartmental model

Evan C. Haskell, Vehbi E. Paksoy

Special Matrices > 2017 > 5 > 1 > 242-249

We consider a sequence of real matrices An which is characterized by the rule that An−1 is the Schur complement in An of the (1,1) entry of An, namely −en, where en is a positive real number. This sequence is closely related to linear compartmental ordinary differential equations. We study the spectrum of An. In particular,we show that An has a unique positive eigenvalue λn and {λn} is a decreasing...

article

Euler-Lagrange equation for a delay variational problem

Joël Blot, Mamadou I. Koné

Nonautonomous Dynamical Systems > 2017 > 4 > 1 > 52-61

We establish Euler-Lagrange equations for a problem of Calculus of Variations where the unknown variable contains a term of delay on a segment

article

Inverse Problems for Parabolic Equation with Discontinuous Coefficients

V. Dinakar, N. Barani Balan, K. Balachandran

Nonautonomous Dynamical Systems > 2017 > 4 > 1 > 40-51

We consider the reaction-diffusion equation with discontinuities in the diffusion coefficient and the potential term. We start by deriving the Carleman estimate for the discontinuous reaction-diffusion operator which is deployed in the inverse problems of finding the stability result of the two discontinuous coefficients from the internal observations of the given parabolic equation.

article

Non-exponential and polynomial stability results of a Bresse system with one infinite memory in the vertical displacement

Aissa Guesmia

Nonautonomous Dynamical Systems > 2017 > 4 > 1 > 78-97

The asymptotic stability of one-dimensional linear Bresse systems under infinite memories was obtained by Guesmia and Kafini [10] (three infinite memories), Guesmia and Kirane [11] (two infinite memories), Guesmia [9] (one infinite memory acting on the longitudinal displacement) and De Lima Santos et al. [6] (one infinite memory acting on the shear angle displacement). When the kernel functions have...

article

Existence results for fractional integro-differential inclusions with state-dependent delay

Giovana Siracusa, Hernán R. Henríquez, Claudio Cuevas

Nonautonomous Dynamical Systems > 2017 > 4 > 1 > 62-77

In this paper we are concerned with a class of abstract fractional integro-differential inclusions with infinite state-dependent delay. Our approach is based on the existence of a resolvent operator for the homogeneous equation.We establish the existence of mild solutions using both contractive maps and condensing maps. Finally, an application to the theory of heat conduction in materials with memory...

article

Poisson-Fermi Formulation of Nonlocal Electrostatics in Electrolyte Solutions

Jinn-Liang Liu, Dexuan Xie, Bob Eisenberg

Molecular Based Mathematical Biology > 2017 > 5 > 1 > 116-124

We present a nonlocal electrostatic formulation of nonuniform ions and water molecules with interstitial voids that uses a Fermi-like distribution to account for steric and correlation efects in electrolyte solutions. The formulation is based on the volume exclusion of hard spheres leading to a steric potential and Maxwell’s displacement field with Yukawa-type interactions resulting in a nonlocal...

article

Relative dielectric constants and selectivity ratios in open ionic channels

Bob Eisenberg, Weishi Liu

Molecular Based Mathematical Biology > 2017 > 5 > 1 > 125-137

We investigate the effects of the relative dielectric coefficient on ionic flows in open ion channels, using mathematical analysis of reasonably general Poisson-Nernst-Planck type models that can include the finite sizes of ions. The value of the relative dielectric coefficient is of course a crucial parameter for ionic behavior in general. Using the powerful theory of singularly perturbed problems...

article

Dynamics of Current, Charge and Mass

Bob Eisenberg, Xavier Oriols, David Ferry

Molecular Based Mathematical Biology > 2017 > 5 > 1 > 78-115

Electricity plays a special role in our lives and life. The dynamics of electrons allow light to flow through a vacuum. The equations of electron dynamics are nearly exact and apply from nuclear particles to stars. These Maxwell equations include a special term, the displacement current (of a vacuum). The displacement current allows electrical signals to propagate through space. Displacement current...

article

New copulas based on general partitions-of-unity and their applications to risk management (part II)

Dietmar Pfeifer, Andreas Mändle, Olena Ragulina

Dependence Modeling > 2017 > 5 > 1 > 246-255

We present a constructive and self-contained approach to data driven infinite partition-of-unity copulas that were recently introduced in the literature. In particular, we consider negative binomial and Poisson copulas and present a solution to the problem of fitting such copulas to highly asymmetric data in arbitrary dimensions.

article

Weighted integral Hankel operators with continuous spectrum

Emilio Fedele, Alexander Pushnitski

Concrete Operators > 2017 > 4 > 1 > 121-129

Using the Kato-Rosenblum theorem, we describe the absolutely continuous spectrum of a class of weighted integral Hankel operators in L2(ℝ+). These self-adjoint operators generalise the explicitly diagonalisable operator with the integral kernel sαtα(s + t)-1-2α, where α > -1/2. Our analysis can be considered as an extension of J. Howland’s 1992 paper which dealt with the unweighted case, corresponding...

article

Generalized n-circular projections on JB*-triples and Hilbert C₀(Ω)-modules

Dijana Ilišević, Chih-Neng Liu, Ngai-Ching Wong

Concrete Operators > 2017 > 4 > 1 > 109-120

Being expected as a Banach space substitute of the orthogonal projections on Hilbert spaces, generalized n-circular projections also extend the notion of generalized bicontractive projections on JB*-triples. In this paper, we study some geometric properties of JB*-triples related to them. In particular, we provide some structure theorems of generalized n-circular projections on an often mentioned...

article

Multipliers of sequence spaces

Raymond Cheng, Javad Mashreghi, William T. Ross

Concrete Operators > 2017 > 4 > 1 > 76-108

This paper is selective survey on the space lAp and its multipliers. It also includes some connections of multipliers to Birkhoff-James orthogonality

article

Integral of Non Positive Functions

Noboru Endou

Formalized Mathematics > 2017 > 25 > 3 > 227-240

In this article, we formalize in the Mizar system [1, 7] the Lebesgue type integral and convergence theorems for non positive functions [8],[2]. Many theorems are based on our previous results [5], [6].

article

Formal Introduction to Fuzzy Implications

Adam Grabowski

Formalized Mathematics > 2017 > 25 > 3 > 241-248

In the article we present in the Mizar system the catalogue of nine basic fuzzy implications, used especially in the theory of fuzzy sets. This work is a continuation of the development of fuzzy sets in Mizar; it could be used to give a variety of more general operations, and also it could be a good starting point towards the formalization of fuzzy logic (together with t-norms and t-conorms, formalized...

article

On Roots of Polynomials and Algebraically Closed Fields

Christoph Schwarzweller

Formalized Mathematics > 2017 > 25 > 3 > 185-195

In this article we further extend the algebraic theory of polynomial rings in Mizar [1, 2, 3]. We deal with roots and multiple roots of polynomials and show that both the real numbers and finite domains are not algebraically closed [5, 7]. We also prove the identity theorem for polynomials and that the number of multiple roots is bounded by the polynomial’s degree [4, 6].

article

Pell’s Equation

Marcin Acewicz, Karol Pąk

Formalized Mathematics > 2017 > 25 > 3 > 197-204

In this article we formalize several basic theorems that correspond to Pell’s equation. We focus on two aspects: that the Pell’s equation x2 − Dy2 = 1 has infinitely many solutions in positive integers for a given D not being a perfect square, and that based on the least fundamental solution of the equation when we can simply calculate algebraically each remaining solution. “Solutions to Pell’s Equation”...

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Search results

Products of Snowflaked Euclidean Lines Are Not Minimal for Looking Down

Groups and monoids of Pythagorean triples connected to conics

Structured Matrix Methods Computing the Greatest Common Divisor of Polynomials

Sign patterns of J-orthogonal matrices

Spectral properties of a sequence of matrices connected to each other via Schur complement and arising in a compartmental model

Euler-Lagrange equation for a delay variational problem

Inverse Problems for Parabolic Equation with Discontinuous Coefficients

Non-exponential and polynomial stability results of a Bresse system with one infinite memory in the vertical displacement

Existence results for fractional integro-differential inclusions with state-dependent delay

Poisson-Fermi Formulation of Nonlocal Electrostatics in Electrolyte Solutions

Relative dielectric constants and selectivity ratios in open ionic channels

Dynamics of Current, Charge and Mass

New copulas based on general partitions-of-unity and their applications to risk management (part II)

Weighted integral Hankel operators with continuous spectrum

Generalized n-circular projections on JB*-triples and Hilbert C₀(Ω)-modules

Multipliers of sequence spaces

Integral of Non Positive Functions

Formal Introduction to Fuzzy Implications

On Roots of Polynomials and Algebraically Closed Fields

Pell’s Equation

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