Sharp Integral Inequalities Based on a General Four-Point Quadrature Formula via a Generalization of the Montgomery Identity

We consider families of general four-point quadrature formulae using a generalization of the Montgomery identity via Taylor’s formula. The results are applied to obtain some sharp inequalities for functions whose derivatives belong to spaces. Generalizations of Simpson’s 3/8 formula and the Lobatto four-point formula with related inequalities are considered as special cases

NEW DOCTORAL DEGREE Montgomery identity, quadrature formulae and derived inequalities

The aim of this PhD dissertation is to give generalizations of classical quadrature formulae with two, three and four nodes using some generalizations of the weighted Montgomery identity. Thereby families of weighted and non-weighted quadrature formulae are considered, some error estimates are derived, and sharp and the best possible inequalities as well as Ostrowski type inequalities are proved. Classes of weighted and non-weighted two-point quadrature formulae are studied and corresponding error estimates are calculated. Two-point Gauss-Chebyshev formulae of the first and of the second kind as well as genera-lizations of the trapezoidal formula, Newton-Cotes two-point formula, Maclaurin two-point formula and midpoint formula are obtained as special cases of these formulae. The dissertation deals with three-point quadrature formulae, generalizations of Simpson\u27s, dual Simpson\u27s and Maclaurin\u27s formula, three-point Gauss-Chebyshev formulae of the first kind and of the second kind that follow from a general formula, as well as corresponding error estimates. It is also dedicated to closed four-point quadrature formulae from which a we-ight-ed and non-weighted generalization of Bullen type inequalities for $(2n)-$ convex functions is obtained. As a special case, Simpson\u27s $3/8$ formula and Lobatto four-point formula with related inequalities are considered. Weighted Euler type identities, which represent weighted integral one-point formulae, are worked out in the dissertation as well. By means of these identities, generalized weighted quadrature formulae are derived in which the integral is estimated by function values in $n$ nodes and generalizations of Gauss-Chebyshev formulae of the first and of the second kind are given. Error estimates are derived and some sharp and best possible inequalities are proved for all given formulae

General quadrature formulae based on the weighted Montgomery identity and related inequalities

In this paper two families of general two-point and closed four-point weighted quadrature formulae are established. Obtained formulae are used to present several Hadamard type and Ostrowski type inequalities for alpha-L-Holder functions. These results are applied to establish error estimates for the Gauss-Chebyshev quadratures

Universal optimality of the E8E_8 and Leech lattices and interpolation formulas

We prove that the $E_8$ root lattice and the Leech lattice are universally optimal among point configurations in Euclidean spaces of dimensions $8$ and $24$, respectively. In other words, they minimize energy for every potential function that is a completely monotonic function of squared distance (for example, inverse power laws or Gaussians), which is a strong form of robustness not previously known for any configuration in more than one dimension. This theorem implies their recently shown optimality as sphere packings, and broadly generalizes it to allow for long-range interactions. The proof uses sharp linear programming bounds for energy. To construct the optimal auxiliary functions used to attain these bounds, we prove a new interpolation theorem, which is of independent interest. It reconstructs a radial Schwartz function $f$ from the values and radial derivatives of $f$ and its Fourier transform $\widehat{f}$ at the radii $\sqrt{2n}$ for integers $n\ge1$ in $\mathbb{R}^8$ and $n \ge 2$ in $\mathbb{R}^{24}$. To prove this theorem, we construct an interpolation basis using integral transforms of quasimodular forms, generalizing Viazovska's work on sphere packing and placing it in the context of a more conceptual theory.Comment: 95 pages, 6 figure

Symmetry in the Mathematical Inequalities

This Special Issue brings together original research papers, in all areas of mathematics, that are concerned with inequalities or the role of inequalities. The research results presented in this Special Issue are related to improvements in classical inequalities, highlighting their applications and promoting an exchange of ideas between mathematicians from many parts of the world dedicated to the theory of inequalities. This volume will be of interest to mathematicians specializing in inequality theory and beyond. Many of the studies presented here can be very useful in demonstrating new results. It is our great pleasure to publish this book. All contents were peer-reviewed by multiple referees and published as papers in our Special Issue in the journal Symmetry. These studies give new and interesting results in mathematical inequalities enabling readers to obtain the latest developments in the fields of mathematical inequalities. Finally, we would like to thank all the authors who have published their valuable work in this Special Issue. We would also like to thank the editors of the journal Symmetry for their help in making this volume, especially Mrs. Teresa Yu

Inequalities

Inequalities appear in various fields of natural science and engineering. Classical inequalities are still being improved and/or generalized by many researchers. That is, inequalities have been actively studied by mathematicians. In this book, we selected the papers that were published as the Special Issue ‘’Inequalities’’ in the journal Mathematics (MDPI publisher). They were ordered by similar topics for readers’ convenience and to give new and interesting results in mathematical inequalities, such as the improvements in famous inequalities, the results of Frame theory, the coefficient inequalities of functions, and the kind of convex functions used for Hermite–Hadamard inequalities. The editor believes that the contents of this book will be useful to study the latest results for researchers of this field

Universal optimality of the E8E_8 and Leech lattices and interpolation formulas

We prove that the $E_8$ root lattice and the Leech lattice are universallyoptimal among point configurations in Euclidean spaces of dimensions $8$ and$24$, respectively. In other words, they minimize energy for every potentialfunction that is a completely monotonic function of squared distance (forexample, inverse power laws or Gaussians), which is a strong form of robustnessnot previously known for any configuration in more than one dimension. Thistheorem implies their recently shown optimality as sphere packings, and broadlygeneralizes it to allow for long-range interactions. The proof uses sharp linear programming bounds for energy. To construct theoptimal auxiliary functions used to attain these bounds, we prove a newinterpolation theorem, which is of independent interest. It reconstructs aradial Schwartz function $f$ from the values and radial derivatives of $f$ andits Fourier transform $\widehat{f}$ at the radii $\sqrt{2n}$ for integers$n\ge1$ in $\mathbb{R}^8$ and $n \ge 2$ in $\mathbb{R}^{24}$. To prove thistheorem, we construct an interpolation basis using integral transforms ofquasimodular forms, generalizing Viazovska's work on sphere packing and placingit in the context of a more conceptual theory.<br

Basis expansions in applied mathematics

Basis expansions are an extremely useful tool in applied mathematics. By using them, we can express a function representing a physical quantity as a linear combination of simpler ``modules'' with well-known properties. They are particularly useful for the applications described in this thesis. Perhaps the best known expansion of this type is the Fourier series of a periodic function, as decomposition into the infinite sum of simple sinusoidal and cosinusoidal elements, originally proposed by Fourier to study heat transfer. This dissertation employs some mathematical tools on problems taken from various areas of Engineering, exploiting their expansion properties: 1) Non-integer bases, which are applied to mathematical models in Robotics (Chapter 2). In this Chapter we study, in particular, a model for snake-like robots based on the Fibonacci sequence. It includes an investigation of the reachableworkspace, a more general analysis of the control system underlying the model, its reachability and local controllability properties. 2) Orthonormal bases, Riesz bases: exponential and cardinal Riesz basis with perturbations (Chapter 3). In this Chapter we obtain also a stability result for cardinal Riesz basis in the case of complex perturbations of the integers. We also consider a mathematical model for energy of the signal at the output of an ideal DAC, in presence of sampling clock jitter. When sampling clock jitter occurs, the energy of the signal at the output of ideal DAC does not satisfies a Parseval identity. Nevertheless, an estimation of the signal energy is here shown by a direct method involving cardinal series. 3) Orthogonal polynomials (Chapter 4). In this Chapter we introduce a new sequence of polynomials, which follow the same recursive rule of the well-known Lucas-Lehmer integer sequence. We show the most important properties of this sequence, relating them to the Chebyshev polynomials of the first and second kind. We discuss some relations between zeros of Lucas-Lehmer polynomials and Gray code. We study nested square roots of 2 applying a "binary code" that associates bits 0 and 1 to + and - signs in the nested form. This gives the possibility to obtain an ordering for the zeros of Lucas-Lehmer polynomials, which take the form of nested square roots of 2. These zeros are used to obtain two new formulas for Pi