Convergent Sums Research Articles

The Trace Formula for Quantum Graphs with General Self Adjoint Boundary Conditions

We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing spectral properties of Laplacians, we prove several versions of trace formulae, relating Laplace spectra to sums over periodic orbits on the graph. This includes trace formulae with, respectively, absolutely and conditionally convergent periodic orbit sums; the convergence depending on properties of the test functions used. We also prove a trace formula for the heat kernel and provide small-t asymptotics for the trace of the heat kernel.

A 2-adic formula for Bernoulli numbers of the second kind and for the Nörlund numbers

We give a formula expressing Bernoulli numbers of the second kind as 2-adically convergent sums of traces of algebraic integers. We use this formula to prove and explain the formulas and conjectures of Adelberg concerning the initial 2-adic digits of these numbers. We also give analogous results for the Nörlund numbers.

On the volume of a tilting module

We consider a finite dimensional k-algebraA and associate to each tilting module a cone in the Grothendieck groupK0 of finitely generated A-modules. We prove that the set of cones associated to tilting modules of projective dimension at most one defines a, not necessarily finite, fan Σ(A). IfA is of finite global dimension, the fan Σ(A) is smooth. Moreover, using the cone of a tilting module, we can associate a volume to each tilting module. Using the fan and the volume, we obtain new proofs for several classical results; we obtain certain convergent sums naturally associated to the algebraA and obtain criteria for the completeness of a list of tilting modules. Finally, we consider several examples.

Uniqueness for multiple trigonometric and Walsh series with convergent rearranged square partial sums

If at each point of a set of positive Lebesgue measure every rearrangement of a multiple trigonometric series square converges to a finite value, then that series is the Fourier series of a function to which it converges uniformly. If there is at least one point at which every rearrangement of a multiple Walsh series square converges to a finite value, then that series is the Walsh-Fourier series of a function to which it converges uniformly.

On the Convergence Acceleration of Slowly Convergent Sums Involving Oscillating Terms

In this paper, an acceleration technique based on a Kummer’s transformation method is developed for some slowly convergent series. The original series is decomposed into two parts; one part being rapidly convergent and the other part being slowly convergent. Then the series in the slowly convergent part is expressed as integrals of some auxiliary functions and subsequently they are written in terms of polynomials whose coefficients are given by the zeta functions. The given method is computationally oriented and does not involve much analytic effort. A numerical example is provided to illustrate the usage and the efficiency of the method.

On the Weak Limiting Behavior of Almost Surely Convergent Row Sums from Infinite Arrays of Rowwise Independent Random Elements in Banach Spaces

For an array {Vnk,k≥1,n≥1} of rowwise independent random elements in a real separable Banach space \(X\) with almost surely convergent row sums \(S_n = \sum {_{k = 1}^\infty {\text{ }}V_{nk} ,n \geqslant 1} \), we provide criteria for Sn−An to be stochastically bounded or for the weak law of large numbers \(S_n - A_n \xrightarrow{P}0\) to hold where {An,n≥1} is a (nonrandom) sequence in \(X\).

Fermionic vacuum energy from a Nielsen-Olesen vortex

We calculate the vacuum energy of a spinor field in the background of a Nielsen-Olesen vortex. We use the method of representing the vacuum energy in terms of the Jost function on the imaginary momentum axis. We use zeta-functional regularization. Renormalization is then carried out with the help of the heat kernel expansion. This method gives rise to well convergent sums and integrals which allow for an efficient numerical calculation of the vacuum energy even in the case when the background is not known analytically but only numerically. The vacuum energy is calculated for several choices of the parameters and, as it turns out, is small compared to the classical energy.

How do bootstrap and permutation tests work?

Resampling methods are frequently used in practice to adjust critical values of nonparametric tests. In the present paper a comprehensive and unified approach for the conditional and unconditional analysis of linear resampling statistics is presented. Under fairly mild assumptions we prove tightness and an asymptotic series representation for their weak accumulation points. From this series it becomes clear which part of the resampling statistic is responsible for asymptotic normality. The results leads to a discussion of the asymptotic correctness of resampling methods as well as their applications in testing hypotheses. They are conditionally correct iff a central limit theorem holds for the original test statistic. We prove unconditional correctness iff the central limit theorem holds or when symmetric random variables are resampled by a scheme of asymptotically random signs. Special cases are the m (n) out of k (n) bootstrap, the weighted bootstrap, the wild bootstrap and all kinds of permutation statistics. The program is carried out for convergent partial sums of rowwise independent infinitesimal triangular arrays in detail. These results are used to compare power functions of conditional resampling tests and their unconditional counterparts. The proof uses the method of random scores for permutation type statistics.

Exponentially convergent lattice sums

For any oblique incidence and arbitrarily high order, lattice sums for one-dimensional gratings can be expressed in terms of exponentially convergent series. The scattering Green's function can be efficiently evaluated also in the grating plane. Numerical implementation of the method is 200 times faster than for the previous best result.

Ewald summation technique for one-dimensional charge distributions

We address the numerical evaluation of problematic lattice sums that occur when dealing with long-range electrostatic interactions in systems with one-dimensional periodicity. An Ewald-type solution is developed yielding rapidly convergent sums. Explicit expressions are given for the Madelung matrices that enter the potential and total energy in self-consistent electronic structure calculations for systems such as atomic wires and steps.

Massive 3-loop Feynman diagrams reducible to SC $^*$ primitives of algebras of the sixth root of unity

In each of the 10 cases with propagators of unit or zero mass, the finite part of the scalar 3-loop tetrahedral vacuum diagram is reduced to 4-letter words in the 7-letter alphabet of the 1-forms $\Omega:=dz/z$ and $\omega_p:=dz/ (\lambda^{-p}-z)$, where $\lambda$ is the sixth root of unity. Three diagrams yield only $\zeta(\Omega^3\omega_0)=1/90\pi^4$. In two cases $\pi^4$ combines with the Euler-Zagier sum $\zeta(\Omega^2\omega_3\omega_0)=\sum_{m> n>0}(-1)^{m+n}/m^3n$; in three cases it combines with the square of Clausen's $Cl_2(\pi/3)=\Im \zeta(\Omega\omega_1)=\sum_{n>0}\sin(\pi n/3)/n^2$. The case with 6 masses involves no further constant; with 5 masses a Deligne-Euler-Zagier sum appears: $\Re \zeta(\Omega^2\omega_3\omega_1)= \sum_{m>n>0}(-1)^m\cos(2\pi n/3)/m^3n$. The previously unidentified term in the 3-loop rho-parameter of the standard model is merely $D_3=6\zeta(3)-6 Cl_2^2(\pi/3)-{1/24}\pi^4$. The remarkable simplicity of these results stems from two shuffle algebras: one for nested sums; the other for iterated integrals. Each diagram evaluates to 10 000 digits in seconds, because the primitive words are transformable to exponentially convergent single sums, as recently shown for $\zeta(3)$ and $\zeta(5)$, familiar in QCD. Those are SC$^*(2)$ constants, whose base of super-fast computation is 2. Mass involves the novel base-3 set SC$^*(3)$. All 10 diagrams reduce to SC$^*(3)\cup$SC$^* (2)$ constants and their products. Only the 6-mass case entails both bases.

Source-neutral Green'functions for periodic problems in electrostatics, and their equivalents in electromagnetism

We consider source–neutral problems for periodic problems in electrostatics. These may be used to derive an identity due to Lord Rayleigh entirely within the context of the unit cell, and without use of reasoning based on contributions from charges at an outer boundary of the periodic system, or employing conditionally convergent lattice sums. We also consider the equivalent of source–free Green9functions for the Helmholtz equation, which are Green9functions with their specular parts removed. We show that, as the wavenumber tends to zero, the non–specular dynamic Green9functions tend with no divergences to the source–neutral electrostatic Green9function.

Some new limit theorems for vector space- and group-valued random variables

It is shown that weakly convergent sums (products) of normalized i.i.d. random variables with values in a finite-dimensional vector space or in a group are mixing in the sense of A. Rényi, and limit theorems for random sums with nonindependent indices are obtained. A new version of H. Robbins' limit theorem for random sums is presented.

Summation of leading logarithms at small x

We show how perturbation theory may be reorganized to give splitting functions which include order by order convergent sums of all leading logarithms of x. This gives a leading twist evolution equation for parton distributions which sums all leading logarithms of x and Q 2, allowing stable perturbative evolution down to arbitrarily small values of x. Perturbative evolution then generates the double scaling rise of F 2 observed at HERA, while in the formal limit x → 0 at fixed Q 2 the Lipatov x f- λ behaviour is eventually reproduced. We are thus able to explain why leading order perturbation theory works so well in the HERA region.

Electron nuclear dynamics for a zig–zag chain of nitrogen atoms

We study the nitrogen zig–zag chain with two atoms per unit cell within the electron nuclear dynamics (END) formalism. This amounts to an approximate solution of the time-dependent Schrödinger equation for all the particles in the system. In the present approximation the nuclei are treated classically. The time dependence of the electronic motion is brought in through time-dependent linear combinations of fixed Bloch sums. This implies that the immediate mutual interaction between electronic and nuclear motion is taken into account. We investigate in particular the long-range terms of the interaction so as to arrive at convergent lattice sums. Before going to the general case when electronic and nuclear motion is coupled, we investigate the special cases of END traditional lattice dynamics and the random phase approximation (RPA) for the electrons.

Sums of Independent Triangular Arrays and Extreme Order Statistics

Let $X_{n,i}$ denote an infinitesimal array of independent random variables with convergent partial sums $Z_n = \sum^n_{i=1} X_{n,i} -a_n \rightarrow_\mathscr{D}\xi$. Throughout, we find conditions for the convergence of the portion $k_n$ of lower extremes $L_n(k_n) = \sum^{k_n}_{i=1}X_{i:n} - b_n$ given by order statistics $X_{i:n}$. Similarly, $W_n(r_n)$ denotes the sum of the $r_n$ upper extremes and $M_n = Z_n - L_n - W_n$ stands for the middle part of the sum. It is shown that $(L_n, M_n, W_n) \rightarrow_\mathscr{D} (\xi_1, \xi_2, \xi_3)$ jointly converges for various sequences $k_n, r_n \rightarrow \infty$, where the components of the limit law are independent such that $\xi_1 + \xi_2 + \xi_3 =_\mathscr{D} \xi$. The limit of the middle part $\xi_2$ is asymptotically normal and $\xi_1 (\xi_3)$ gives the negative (positive) spectral Poisson part of $\xi$. In the case of a compound Poisson limit distribution we obtain rates of convergence that can be used for applications to insurance mathematics.

Transient nucleation induction time from the birth–death equations

For the set of finite-difference equations of Becker–Döring an exact formula for the induction time, which is expressed in terms of rapidly convergent sums, is presented. The form of the result is particularly amenable for analytical study, and the latter is carried out to obtain approximations of the exact expression in a rigorous manner and to assess its sensitivity to the choice of the nucleation model. The induction time, tind, is found to be governed by two main nucleation parameters, Φ*/kT, the normalized barrier height, and g*, the number of molecules in the critical cluster. The ratio of these two parameters provides an assessment of the importance of discreteness effects. We study the exact expression in both the continuous (g*→∞) and the asymptotic (Φ*/kT→∞) limits. Asymptotic results for tind are compared with those previously reported from simulation studies as well as with tind obtained numerically from the exact expression in the present study. Also, the accuracy of the Zeldovich equation, which is produced in the continuous limit, is discussed for several nucleation models.

A new asymptotic representation for ζ(½ + i t ) and quantum spectral determinants

By analytic continuation of the Dirichlet series for the Riemann zeta function ζ(s) to the critical line s = ½ + i t ( t real), a family of exact representations, parametrized by a real variable K , is found for the real function Z ( t ) = ζ(½ + i t ) exp {iθ( t )}, where θ is real. The dominant contribution Z 0 ( t,K ) is a convergent sum over the integers n of the Dirichlet series, resembling the finite ‘main sum ’ of the Riemann-Siegel formula (RS) but with the sharp cut-off smoothed by an error function. The corrections Z 3 ( t,K ), Z 4 ( t,K )... are also convergent sums, whose principal terms involve integers close to the RS cut-off. For large K , Z 0 contains not only the main sum of RS but also its first correction. An estimate of high orders m ≫ 1 when K < t 1/6 shows that the corrections Z k have the ‘factorial/power ’ form familiar in divergent asymptotic expansions, the least term being of order exp { ─½ K 2 t }. Graphical and numerical exploration of the new representation shows that Z 0 is always better than the main sum of RS, providing an approximation that in our numerical illustrations is up to seven orders of magnitude more accurate with little more computational effort. The corrections Z 3 and Z 4 give further improvements, roughly comparable to adding RS corrections (but starting from the more accurate Z 0 ). The accuracy increases with K , as do the numbers of terms in the sums for each of the Z m . By regarding Planck’s constant h as a complex variable, the method for Z ( t ) can be applied directly to semiclassical approximations for spectral determinants ∆( E, h ) whose zeros E = E j ( h ) are the energies of stationary states in quantum mechanics. The result is an exact analytic continuation of the exponential of the semiclassical sum over periodic orbits given by the divergent Gutzwiller trace formula. A consequence is that our result yields an exact asymptotic representation of the Selberg zeta function on its critical line.

Displaced vortices on lattices

The energy associated with a general displacement of either a single vortex, a cluster, or a superlattice of vortices on an arbitrary lattice is derived in terms of rapidly convergent sums of simple functions. Expansions to second order in the displacement, equivalent to the vibrational modes, are also derived for the single vortex, the superlattice, and an infinite row of vortices.

On the convergence problem for lattice sums

Three-dimensional lattice sums Sigma +or-r-s, where r is the distance to a +or- charged lattice point, arise in classical lattice point problems (s=0) and the Madelung problem of physics and chemistry (s=1). In the latter case the sum over an infinite lattice is purely formal and the value of the Madelung constant must be defined precisely, e.g., via related convergent sums or analytic continuation in s. Indeed, the partial sum over the sphere r(R does not converge as R becomes large. The authors verify a conjecture of J F Delord (1988) that convergence can be obtained by neutralising each sphere with an appropriate surface charge. Specifically, if LR(s) is the sum over the lattice points in the sphere r(R then, for neutrally charged lattices, they show that as R goes to infinity the difference LR(s)-R-sLR(0) approaches L(s), where L(s) is defined by analytic continuation. When s=1 the term L(1) is the Madelung constant and LR(0)/R is the Coulombic correction term.