A Convolution Formula for the Dickman Function: 10792

2001 ◽  
Vol 108 (3) ◽  
pp. 277 ◽  
Author(s):  
Harold G. Diamond ◽  
Ferrell S. Wheeler
Keyword(s):  
Dickman Function ◽  
Convolution Formula

CONVOLUTION FORMULA AS A STIELTJES RESULTANT

2018 ◽  
Vol 72 (2) ◽  
pp. 429-439
Author(s):  
Soumyarup BANERJEE ◽  
Shigeru KANEMITSU
Keyword(s):  
Convolution Formula

scholarly journals CONVOLUTION FORMULA AND FINITE W BOSON WIDTH EFFECTS IN TOP QUARK WIDTH

2008 ◽  
Vol 23 (22) ◽  
pp. 3525-3533 ◽  
Author(s):  
G. CALDERÓN ◽  
G. LÓPEZ CASTRO
Keyword(s):  
Top Quark ◽  
W Boson ◽  
Matrix Theory ◽  
Finite Width ◽  
Current Limit ◽  
Fermion Masses ◽  
Convolution Formula ◽  
Boson Resonance ◽  
Three Body ◽  
Massless Fermions

In the Standard Model, the top quark decay width Γt is computed from the exclusive t → bW decay. We argue in favor of using the three body decays [Formula: see text] to compute Γt as a sum over these exclusive modes. As dictated by the S-matrix theory, these three body decays of the top quark involve only asymptotic states and incorporate the width of the W boson resonance in a natural way. The convolution formula commonly used to include the finite width effects is found to be valid, in the general case, when the intermediate resonance couples to a conserved current (limit of massless fermions in the case of W bosons). The relation Γt = Γ(t → bW) is recovered by taking the limit of massless fermions followed by the W boson narrow width approximation. Although both calculations of Γt are different at the formal level, their results would differ only by tiny effects induced by light fermion masses and higher-order radiative corrections.

scholarly journals Convolution formula for the sums of generalized Dirichlet $L$-functions

2019 ◽  
Vol 35 (7) ◽  
pp. 1973-1995
Author(s):  
Olga Balkanova ◽  
Dmitry Frolenkov
Keyword(s):  
Convolution Formula ◽  
Generalized Dirichlet

scholarly journals Integers free of small prime factors in arithmetic progressions

2000 ◽  
Vol 157 ◽  
pp. 103-127 ◽  
Author(s):  
Ti Zuo Xuan
Keyword(s):  
Prime Number ◽  
Short Range ◽  
Error Term ◽  
Real Zero ◽  
Prime Number Theorem ◽  
Arithmetic Progressions ◽  
Real Character ◽  
Dickman Function ◽  
Prime Factors ◽  
Positive Integers

For real x ≥ y ≥ 2 and positive integers a, q, let Φ(x, y; a, q) denote the number of positive integers ≤ x, free of prime factors ≤ y and satisfying n ≡ a (mod q). By the fundamental lemma of sieve, it follows that for (a,q) = 1, Φ(x,y;a,q) = φ(q)-1, Φ(x, y){1 + O(exp(-u(log u- log2 3u- 2))) + (u = log x log y) holds uniformly in a wider ranges of x, y and q.Let χ be any character to the modulus q, and L(s, χ) be the corresponding L-function. Let be a (‘exceptional’) real character to the modulus q for which L(s, ) have a (‘exceptional’) real zero satisfying > 1 - c0/log q. In the paper, we prove that in a slightly short range of q the above first error term can be replaced by where ρ(u) is Dickman function, and ρ′(u) = dρ(u)/du.The result is an analogue of the prime number theorem for arithmetic progressions. From the result can deduce that the above first error term can be omitted, if suppose that 1 < q < (log q)A.

scholarly journals Aumann Fuzzy Improper Integral and Its Application to Solve Fuzzy Integro-Differential Equations by Laplace Transform Method

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Elhassan Eljaoui ◽  
Said Melliani ◽  
L. Saadia Chadli
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Convolution Type ◽  
Convolution Product ◽  
Fuzzy Mapping ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Improper Integral ◽  
Convolution Formula

We introduce the Aumann fuzzy improper integral to define the convolution product of a fuzzy mapping and a crisp function in this paper. The Laplace convolution formula is proved in this case and used to solve fuzzy integro-differential equations with kernel of convolution type. Then, we report and correct an error in the article by Salahshour et al. dealing with the same topic.

A convolution involving Bell polynomials

1973 ◽  
Vol 74 (1) ◽  
pp. 97-106 ◽  
Author(s):  
G. P. M. Heselden
Keyword(s):  
Bell Polynomials ◽  
Special Polynomial ◽  
Convolution Formula

AbstractA convolution formula is established for Bell polynomials. This is expressed in seven equivalent ways and used to derive further properties of these polynomials. The application of these results to some twenty-seven special polynomial sets is shown and illustrated in the case of binomial, Hermite, Gegenbauer and generalized Bernoulli sets.

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