Abel Sum Research Articles

Duelling Idiots and Abel Sums

We investigate a puzzle involving the winning probabilities in a duel of two players. The problem of calculating limiting probabilities leads to the summation of a divergent infinite series. The solution admits a generalization that applies to a wide class of duels.

A note on the finitization of Abelian and Tauberian theorems

AbstractWe present finitary formulations of two well known results concerning infinite series, namely Abel's theorem, which establishes that if a series converges to some limit then its Abel sum converges to the same limit, and Tauber's theorem, which presents a simple condition under which the converse holds. Our approach is inspired by proof theory, and in particular Gödel's functional interpretation, which we use to establish quantitative versions of both of these results.

A high indices theorem without a nontrivial solution

In this paper we will extend the high indices Theorem by Hardy-Littlewood, which says that an Abel summable series, where the exponents fulfil a Hadamard gap criterion, is convergent and it converges to its Abel sum. The focus here is concerning a class of summability methods and their critical rate of convergence, i.e. for a given summability method what is its rate of convergence implying that the series must be identically constant.

Hilbert Transforms on the Sphere with the Clifford Algebra Setting

Through a double-layer potential argument the inner and outer Poisson kernels, the Cauchy-type conjugate inner and outer Poisson kernels, and the kernels of the Cauchy-type inner and outer Hilbert transformations on the sphere are deduced. We also obtain Abel sum expansions of the kernels and prove the Lp-boundedness of the inner and outer Hilbert transformations for 1<p<∞.

CLT for Operator Abel Sums of Random Elements

Abstract Let (ξ 𝑘)𝑘 ≥ 1 be a sequence of independent, identically distributed second order mean zero random elements in a separable Hilbert space 𝐻 and 𝐴 be an element of a certain class of linear continuous operators 𝐻 → 𝐻 such that ‖𝐴‖ &lt; 1. Denote . We prove that if ‖𝐼 – 𝐴‖ tends to zero, where 𝐼 is the identity operator, then the normalized sum (𝐼 – 𝐴2)1/2 η 𝐴 converges in distribution to a Gaussian random element.

On One Sequence of Polynomials and the Radius of Convergence of Its Poisson–Abel Sum

For one sequence of polynomials arising in the construction of the numerical-analytic method for finding periodic solutions of nonlinear differential equations, we determine the explicit form of the Poisson–Abel sum and the exact solution of the equation for finding the radius of convergence of this sum.

Non-uniform rates of convergence for double arrays of independent random variables with applications

Abstract)Let be a double array of consteds and be a sequence of independent random variables. DenoteIt is well known thatunder some conditions, where is the standard normal distributionfunction.This paper studies the rate of convergence of Fn(x) toΦ(x) depending on n and x wherex is in a neighborhood of the origin under minimal conditions. Explicitly, we establish thefollowing theorems.Theorem 1. Suppose thatReceived January 20, 1994. Revised February 22, 1995.where 0 1. Then, there exist n) b 0 (not depending on n) such that for all n alld allx E R with lxl 5 "L.'/',Theorem 2. Suppose that (4) holds, 0 L. - 0 and t.n E N is a sequence ofconstants with t. ￣ co such that t. ￣ o(L.'/'). ThenRemark 1. Noting that L. S sup la.kl/A. ￣ L;, therefore, Theorems 1 and 2 alsokhold if L. whenever it appears, is replaced by L;.For the particular caseSa reduces to the partial sum Z:=, uk, which has been studied extensively in the literature(see [1,2]). In this cajse, A: = n, L. = .--i. It is well known that Theorem 2 can't beremoved under (4).In time series analysis, an important class of covariance stationary models is defined bywhere {Ek, --co k co) is a sequence of independent random variables and Cd areconstants. The sequence {Z j, j 2 1} is called lillear process generated by {Ek, --co k co}. Since Z:=, Zj = ZZ=--co ankEk where ark = Z:=, Cj--k, making use of Theorems 1and 2 we obtain.Theorem 3. Let (4) hold, Cd be constants such thatThen, there exist n, b 0 such that for all n all x E R with lxl 5 un'/6,Theorem 4. Assume that the conditions of Theorem 3 hold and t. is a sequence ofconstants with t. ￣ co such that t. ￣ o(n'/'). ThenRemark 2. for the following Ck,where the function l(x) 2 0 is slowly varying at infinity, a 2 1 and in or = t, ZC=, k--'l(k) co, Halll61 obtained the rate of convergence of F.(x) to .(x). Applying the same methodas that of dealing with aam ￣ Z:=, sj--k in [6], we can obtain some results similar to (10),(11) for Cd satisfying (12).TO conclude this paper, we give some classical examples (the so-called summabilitymethods) including the most importallt ones from the paint of view of probability (see [3--5]for details. In [51, the authors have dealt with the rate of convergence for SummabilityMethods "lider an assumption of finite third moments).1. Cesaro Sum:2. Abel Sum:3. Borel Sum:4. Euler Sum: (0 p 1)5. Delayed Sum (or RI Sum): (e 0)6. Valiron Sum:7. Taylor Sum: (0 or 1)8. Meyer-K5nig Sum: (0 a 1)

Expected deadlock time in a multiprocessing system

We consider multiprocessing systems where processes make independent, Poisson distributed resource requests with mean arrival time 1. We assume that resources are not released. It is shown that the expected deadlock time is never less than 1, no matter how many processes and resources are in the system. Also, the expected number of processes blocked by deadlock time is one-half more than half the number of initially active processes. We obtain expressions for system statistics such as expected deadlock time, expected total processing time, and system efficiency, in terms of Abel sums. We derive asymptotic expressions for these statistics in the case of systems with many processes and the case of systems with a fixed number of processes. In the latter, generalizations of the Ramanujan Q -function arise. we use singularity analysis to obtain asymptotics of coefficients of generalized Q -functions.

Geometry of KDV (4): Abel sums, Jacobi variety, and theta function in the scattering case

The study of the general projective curve X, defined by the vanishing of an irreducible polynomial p(x, y), originated in the discovery of addition formulas for integrals ∫ ∞ v r(x, y) dx of rational functions on such a curve

Approximation for Abel sums of independent, identically distributed random variables

This note represents a probability inequality for an approximation of Abel sums of independent, identically distributed random variables. This approximation implies a rate for the Prohorov-Lévy distance between Abel sums and their limit process.

On the Abel summability of partial wave amplitudes for Coulomb-type interactions

We prove that the sum of partial wave amplitudes for Coulomb-type potentials [e.g., V(r) =γ/r+O(r−2)] is convergent in the Abel sense although it diverges in the ordinary sense. The method of Abel summation is a generalization of the ordinary summation and allows one to sum certain divergent series explicitly. It is closely connected with analytic continuation; with the help of optimal conformal mappings the convergence of the Abel sum (for long- and short-range interactions) can be improved substantially. This enables us to obtain values of the scattering amplitude for each scattering angle (except forward direction). In particular, we show that the screened scattering amplitude converges in the Abel sense up to a phase factor to the unscreened one if the screening is removed.

A note on an evaluation of Abel Sums

A note on an evaluation of Abel Sums

A log log law for Abel's sum

A log log law for Abel's sum

A general ratio ergodic theorem for Abel sums

A general ratio ergodic theorem for Abel sums

On ‘translated quasi-Cesàro’ summability. II

In a recent paper (1), I considered the summability method (D, α) defined, for α &gt; 0, by the sequence-to-sequence transformationWe note that, as is easily verified (and as was pointed out in (1)) a necessary and sufficient condition for the convergence of (1), and thus for the applicability of (D, α), is thatshould converge. It was proved in (1) that, provided that (2) converges, a sequence summable (C, r) for any r &gt; − 1 is necessarily summable (D, α). We now show that we can strengthen this result by replacing Cesàro by Abel summability. Moreover, we can omit the hypothesis that (2) converges provided that we interpret (1) as an Abel sum.

A summability theorem in countable toral groups

where a (x) is a character and the sum is over all the characters. This paper begins a s tudy of the Abel summabil i ty of this series and developes the theory for countable toral groups. Let G be a countable toral group. Let X C~ be a decomposition of G into an infinite direct product, where each C~ is the group of reals modulo one. The topology and measure of G are the product topology and measure. The characters of G will be isomorphic to the direct product of countably many infinite cyclic groups. Let a = {an} be a sequence of integral multiples of 2 rc with only a finite number of them different from zero. The collection of all a 's is isomorphic to G with addition defined termwise. A point in G is a sequence x = (xn} and each a(x) = exp [-i Xa,,xn]. By the Abel sum of the Fourier series (1) we shall mean: (2) lira • r~l~l/2~ f/(x) ff(y x ) d x . r--~l a

A generalization of Abel summability

The Abel sum of the series can be written in the formwhereThis suggests that we should define the “Abel limit”, as u → ∞, of any function A(u) as being given by the expression (1) whenever this exists. We shall, however, in this paper, restrict ourselves to functions A(u) which are bounded in any finite interval. Since we are concerned only with the behaviour of A(u) as u → ∞, this does not involve any serious loss of generality, while we avoid difficulties arising from the divergence of integrals at finite points. We note that the expression (1) can be written in the formwhereA1(u) may conveniently be described as the “Abel transform” of A(u).