133 research outputs found

### Iterated Binomial Sums and their Associated Iterated Integrals

We consider finite iterated generalized harmonic sums weighted by the
binomial $\binom{2k}{k}$ in numerators and denominators. A large class of these
functions emerges in the calculation of massive Feynman diagrams with local
operator insertions starting at 3-loop order in the coupling constant and
extends the classes of the nested harmonic, generalized harmonic and cyclotomic
sums. The binomially weighted sums are associated by the Mellin transform to
iterated integrals over square-root valued alphabets. The values of the sums
for $N \rightarrow \infty$ and the iterated integrals at $x=1$ lead to new
constants, extending the set of special numbers given by the multiple zeta
values, the cyclotomic zeta values and special constants which emerge in the
limit $N \rightarrow \infty$ of generalized harmonic sums. We develop
algorithms to obtain the Mellin representations of these sums in a systematic
way. They are of importance for the derivation of the asymptotic expansion of
these sums and their analytic continuation to $N \in \mathbb{C}$. The
associated convolution relations are derived for real parameters and can
therefore be used in a wider context, as e.g. for multi-scale processes. We
also derive algorithms to transform iterated integrals over root-valued
alphabets into binomial sums. Using generating functions we study a few aspects
of infinite (inverse) binomial sums.Comment: 62 pages Latex, 1 style fil

### Exact solutions for a class of integrable Henon-Heiles-type systems

We study the exact solutions of a class of integrable Henon-Heiles-type
systems (according to the analysis of Bountis et al. (1982)). These solutions
are expressed in terms of two-dimensional Kleinian functions. Special periodic
solutions are expressed in terms of the well-known Weierstrass function. We
extend some of our results to a generalized Henon-Heiles-type system with n+1
degrees of freedom.Comment: RevTeX4-1, 13 pages, Submitted to J. Math. Phy

### Multipole structure and coordinate systems

Multipole expansions depend on the coordinate system, so that coefficients of
multipole moments can be set equal to zero by an appropriate choice of
coordinates. Therefore, it is meaningless to say that a physical system has a
nonvanishing quadrupole moment, say, without specifying which coordinate system
is used. (Except if this moment is the lowest non-vanishing one.) This result
is demonstrated for the case of two equal like electric charges. Specifically,
an adapted coordinate system in which the potential is given by a monopole term
only is explicitly found, the coefficients of all higher multipoles vanish
identically. It is suggested that this result can be generalized to other
potential problems, by making equal coordinate surfaces coincide with the
potential problem's equipotential surfaces.Comment: 2 figure

### Quantum effects on Lagrangian points and displaced periodic orbits in the Earth-Moon system

Recent work in the literature has shown that the one-loop long distance quantum corrections to the Newtonian potential imply tiny but observable effects in the restricted three-body problem of celestial mechanics, i.e., at the Lagrangian libration points of stable equilibrium the planetoid is not exactly at equal distance from the two bodies of large mass, but the Newtonian values of its coordinates are changed by a few millimeters in the Earth-Moon system. First, we assess such a theoretical calculation by exploiting the full theory of the quintic equation, i.e., its reduction to Bring-Jerrard form and the resulting expression of roots in terms of generalized hypergeometric functions. By performing the numerical analysis of the exact formulas for the roots, we confirm and slightly improve the theoretical evaluation of quantum corrected coordinates of Lagrangian libration points of stable equilibrium. Second, we prove in detail that also for collinear Lagrangian points the quantum corrections are of the same order of magnitude in the Earth-Moon system. Third, we discuss the prospects to measure, with the help of laser ranging, the above departure from the equilateral triangle picture, which is a challenging task. On the other hand, a modern version of the planetoid is the solar sail, and much progress has been made, in recent years, on the displaced periodic orbits of solar sails at all libration points, both stable and unstable. The present paper investigates therefore, eventually, a restricted three-body problem involving Earth, Moon and a solar sail. By taking into account the one-loop quantum corrections to the Newtonian potential, displaced periodic orbits of the solar sail at libration points are again found to exist

### Closed geodesics and billiards on quadrics related to elliptic KdV solutions

We consider algebraic geometrical properties of the integrable billiard on a
quadric Q with elastic impacts along another quadric confocal to Q. These
properties are in sharp contrast with those of the ellipsoidal Birkhoff
billiards. Namely, generic complex invariant manifolds are not Abelian
varieties, and the billiard map is no more algebraic. A Poncelet-like theorem
for such system is known. We give explicit sufficient conditions both for
closed geodesics and periodic billiard orbits on Q and discuss their relation
with the elliptic KdV solutions and elliptic Calogero systemComment: 23 pages, Latex, 1 figure Postscrip

### Analytic and Numerical Study of Preheating Dynamics

We analyze the phenomenon of preheating,i.e. explosive particle production
due to parametric amplification of quantum fluctuations in the unbroken case,
or spinodal instabilities in the broken phase, using the Minkowski space $O(N)$
vector model in the large $N$ limit to study the non-perturbative issues
involved. We give analytic results for weak couplings and times short compared
to the time at which the fluctuations become of the same order as the tree
level,as well as numerical results including the full backreaction.In the case
where the symmetry is unbroken, the analytic results agree spectacularly well
with the numerical ones in their common domain of validity. In the broken
symmetry case, slow roll initial conditions from the unstable minimum at the
origin, give rise to a new and unexpected phenomenon: the dynamical relaxation
of the vacuum energy.That is, particles are abundantly produced at the expense
of the quantum vacuum energy while the zero mode comes back to almost its
initial value.In both cases we obtain analytically and numerically the equation
of state which turns to be written in terms of an effective polytropic index
that interpolates between vacuum and radiation-like domination. We find that
simplified analysis based on harmonic behavior of the zero mode, giving rise to
a Mathieu equation forthe non-zero modes miss important physics. Furthermore,
analysis that do not include the full backreaction do not conserve energy,
resulting in unbound particle production. Our results do not support the recent
claim of symmetry restoration by non-equilibrium fluctuations.Finally estimates
of the reheating temperature are given,as well as a discussion of the
inconsistency of a kinetic approach to thermalization when a non-perturbatively
large number of particles is created.Comment: Latex file, 52 pages and 24 figures in .ps files. Minor changes. To
appear in Physical Review D, 15 December 199

### The impact of Stieltjes' work on continued fractions and orthogonal polynomials

Stieltjes' work on continued fractions and the orthogonal polynomials related
to continued fraction expansions is summarized and an attempt is made to
describe the influence of Stieltjes' ideas and work in research done after his
death, with an emphasis on the theory of orthogonal polynomials

### An isoperimetric inequality in the plane with a log-convex density

Given a positive lower semi-continuous density $f$ on $\mathbb{R}^2$ the
weighted volume $V_f:=f\mathscr{L}^2$ is defined on the
$\mathscr{L}^2$-measurable sets in $\mathbb{R}^2$. The $f$-weighted perimeter
of a set of finite perimeter $E$ in $\mathbb{R}^2$ is written $P_f(E)$. We
study minimisers for the weighted isoperimetric problem $I_f(v):=\inf\Big\{
P_f(E):E\text{ is a set of finite perimeter in }\mathbb{R}^2\text{ and
}V_f(E)=v\Big\}$ for $v>0$. Suppose $f$ takes the form
$f:\mathbb{R}^2\rightarrow(0,+\infty);x\mapsto e^{h(|x|)}$ where
$h:[0,+\infty)\rightarrow\mathbb{R}$ is a non-decreasing convex function. Let
$v>0$ and $B$ a centred ball in $\mathbb{R}^2$ with $V_f(B)=v$. We show that
$B$ is a minimiser for the above variational problem and obtain a uniqueness
result

### How to generate all possible rational Wilf-Zeilberger pairs?

A Wilf--Zeilberger pair $(F, G)$ in the discrete case satisfies the equation
$F(n+1, k) - F(n, k) = G(n, k+1) - G(n, k)$. We present a structural
description of all possible rational Wilf--Zeilberger pairs and their
continuous and mixed analogues.Comment: 17 pages, add the notion of pseudo residues in the differential case,
and some related papers in the reference, ACMES special volume in the Fields
Institute Communications series, 201

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