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At least 55 records · Page 3Linked to original sources
arXiv2007

Cesàro means of orthogonal expansions in several variables

Cesàro $(C,δ)$ means are studied for orthogonal expansions with respect to the weight function $\prod_{i=1}^{d}|x_i|^{2\k_i}$ on the unit sphere, and for the corresponding weight functions on the unit ball and the Jacobi weight on the simplex. A sharp pointwise estimate is established for the $(C,\d)$ kernel with $\d > -1$ and for the kernel of the projection operator, which allows us to derive the exact order for the norm of the Cesàro means and the projection operator on these domains.

math.CA↗
arXiv2007

On the Riemann zeta-function, Part II

An odd meromorphic function f(s) is constructed from the Riemann zeta-function evaluated at one-half plus s. We determine the two-sided Laplace transform representation of f(s) on open vertical strips, V'(4w), disjoint from the (translated) critical strip. V'(4w) consists of all s with real part, Re(s), of absolute value greater than one-half and Re(s) between successive poles 4w, 4(w + 1) of f(s), with w an integer. The corresponding Laplace density is related to confluent hypergeometric functions. That density is shown to be positive for nonzero w other than -1. Those results are obtained without relying on any unproven hypothesis. They are used together with the Riemann hypothesis and hypotheses advanced by the author to obtain conditional results concerning the zeta-function. Those results are presented in Part I. Their proofs are derived in Parts III-V. A metric geometry expression of the positivity of the Laplace densities arising is established in Part VI.

math.GM↗
arXiv2007

On the Riemann zeta-function, Part III

An odd meromorphic function f(s) is constructed from the Riemann zeta-function evaluated at one-half plus s. The partial fraction expansion, p(s), of f(s) is obtained using the conjunction of the Riemann hypothesis and hypotheses advanced by the author. That compound hypothesis and the expansion p(s) are employed in Part IV to derive the two-sided Laplace transform representation of f(s) on the open vertical strip of all s with real part between zero and four.

math.GM↗
arXiv2007

On Stability of Sampling-Reconstruction Models

A useful sampling-reconstruction model should be stable with respect to different kind of small perturbations, regardless whether they result from jitter, measurement errors, or simply from a small change in the model assumptions. In this paper we prove this result for a large class of sampling models. We define different classes of perturbations and quantify the robustness of a model with respect to them. We also use the theory of localized frames to study the frame algorithm for recovering the original signal from its samples.

math.GM↗
arXiv2007

On the Riemann zeta-function, Parts IV-V

In Part I an odd meromorphic function f(s) has been constructed from the Riemann zeta-function evaluated at one-half plus s. The conjunction of the Riemann hypothesis and hypotheses advanced by the author in Part I is assumed. In Part IV we derive the two-sided Laplace transform representation of f(s) on the open vertical strip V of all s with real part between zero and four. An additional hypothesis is used to prove that the Laplace density of f(s) on the strip V is positive. Let z(n) be the nth critical zero of the Riemann zeta-function of positive imaginary part in order of magnitude thereof. In Part V an expression is derived for z(1). A relation is obtained of the pair z(n) and the first derivative thereat of the zeta-function to the preceding such pairs.

math.GM↗
arXiv2007

Proof of generalized Riemann hypothesis for Dedekind zetas and Dirichlet L-functions

A short proof of the generalized Riemann hypothesis (gRH in short) for zeta functions $ζ_{k}$ of algebraic number fields $k$ - based on the Hecke's proof of the functional equation for $ζ_{k}$ and the method of the proof of the Riemann hypothesis derived in [$M_{A}$] (algebraic proof of the Riemann hypothesis) is given. The generalized Riemann hypothesis for Dirichlet L-functions is an immediately consequence of (gRH) for $ζ_{k}$ and suitable product formula which connects the Dedekind zetas with L-functions.

math.GM↗
arXiv2026

On a new approach to the Riemann hypothesis

Suppose that the Riemann hypothesis is false and $ρ_{*} = 1/2 + η_{*} + i γ_{*}$, $η_{*} > 0$, is a nontrivial zero of the Riemann $ζ$-function off the critical line. Under the negation of the Riemann hypothesis for the Riemann $ζ$-function, we establish an asymptotic relation (as $γ_{*} \to \infty$) which relates the residues of the series $\sum_{n \geq 1} Λ(n) e^{- 2πi p n } n^{-s}$ at $s =$ corresponding nontrivial zeros of some Dirichlet $L$-functions to some function, valid for any rational number of the form $p = a/b < 1$ with $b \ll \log γ_{*}$. This related function is continuous in $p$ and we mention its implication to the Riemann hypothesis.

math.GM↗
arXiv2023

Factorization of quadratic polynomials in the ring of formal power series over Z

We establish necessary and sufficient conditions for a quadratic polynomial to be irreducible in the ring $Z[[x]]$ of formal power series with integer coefficients. For $n,m\ge 1$ and $p$ prime, we show that $p^n+p^mβx+αx^2$ is reducible in $Z[[x]]$ if and only if it is reducible in $Z_p[x]$, the ring of polynomials over the $p$-adic integers.

math.AC↗
arXiv2008

On The Universality Of Central Loops

LC-loops, RC-loops and C-loops are collectively called central loops. It is shown that an LC(RC)-loop is a left(right) universal loop. But an LC(RC)-loop is a universal loop if and only if it is a right(left) universal loop. It is observed that not all RC-loops or LC-loops or C-loops are universal loops. But if an RC-loop(LC-loop, C-loop) is universal, then it is a right Bol loop(left Bol loop, Moufang loop) respectively. If a loop and its right or left isotope are commutative then the loop is a C-loop if and only if its right or left isotope is a C-loop. If a C-loop is central square and its right or left isotope is an alternative central square loop, then the latter is a C-loop. Necessary and sufficient condition for an LC-loop(RC-loop) to be a left(right)G-loop is established. Consequently, necessary and sufficient conditions for an LC-loop, and an RC-loop to be a G-loop are established. A necessary and sufficient condition for a C-loop to be a G-loop is established.

math.GM↗
arXiv2009

Some necessary and sufficient conditions for parastrophic invariance of the associative law in quasigroups

Every quasigroup $(S,\cdot)$ belongs to a set of 6 quasigroups, called parastrophes denoted by $(S,π_i)$, $i\in \{1,2,3,4,5,6\}$. It is shown that isotopy-isomorphy is a necessary and sufficient condition for any two distinct quasigroups $(S,π_i)$ and $(S,π_j)$, $i,j\in \{1,2,3,4,5,6\}$ to be parastrophic invariant relative to the associative law. In addition, a necessary and sufficient condition for any two distinct quasigroups $(S,π_i)$ and $(S,π_j)$, $i,j\in \{1,2,3,4,5,6\}$ to be parastrophic invariance under the associative law is either if the $π_i$-parastrophe of $H$ is equivalent to the $π_i$-parastrophe of the holomorph of the $π_i$-parastrophe of $S$ or if the $π_i$-parastrophe of $H$ is equivalent to the $π_k$-parastrophe of the $π_i$-parastrophe of the holomorph of the $π_i$-parastrophe of $S$, for a particular $k\in \{1,2,3,4,5,6\}$.

math.GM↗
arXiv2010

On a Pair of Universal Weak Inverse Property Loops

A new condition called ${\cal T}$ condition is introduced for the first time and used to study a pair of isotopic loops. Under this condition, a loop in the pair is a WIPL if and only if the other loop is a WIPL. Furthermore, such WIPLs are isomorphic. The translation elements $f$ and $g$ of a CIPL with the ${\cal T}$ condition(such that its $f,g$-isotope is an automorphic inverse property loop) are found to be alternative, flexible, centrum and equal elements. A necessary and sufficient condition for a pair WIPLs with a weak ${\cal T}$ condition to be isomorphic is shown. A CIPL and an isomorph are observed to have this weak ${\cal T}$ condition.

math.GM↗
arXiv2025

Study on Hilbert's Eighth Problems

In this paper, we used the principle of sieve function transformation to improve sieve method and the prime number theorem in the arithmetic sequence.For this, we proved General Riemann Hypothesis and Riemann Hypothesis to be true. further, we improved Selberg's line sieve method and proved Goldbach Conjecture and Twin Prime Conjecture to be true. However, we basically solved Hilbert's eighth problems.

math.GM↗
arXiv2007

Solution of the Dirac Equation using the Lanczos Algorithm

Covergent eigensolutions of the Dirac Equation for a relativistic electron in an external Coulomb potential are obtained using the Lanczos Algorithm. A tri-diagonal matrix representation of the Dirac Hamiltonian operator is constructed iteratively and diagonalized after each iteration step to form a sequence of convergent eigenvalue solutions. Any spurious solutions which arise from the presence of continuum states can easily be identified.

math-ph↗