2,088 research outputs found
New families of subordinators with explicit transition probability semigroup
There exist only a few known examples of subordinators for which the
transition probability density can be computed explicitly along side an
expression for its L\'evy measure and Laplace exponent. Such examples are
useful in several areas of applied probability, for example, they are used in
mathematical finance for modeling stochastic time change, they appear in
combinatorial probability to construct sampling formulae, which in turn is
related to a variety of issues in the theory of coalescence models, moreover,
they have also been extensively used in the potential analysis of subordinated
Brownian motion in dimension greater than or equal to 2. In this paper, we show
that Kendall's classic identity for spectrally negative L\'evy processes can be
used to construct new families of subordinators with explicit transition
probability semigroups. We describe the properties of these new subordinators
and emphasise some interesting connections with explicit and previously unknown
Laplace transform identities and with complete monotonicity properties of
certain special functions
The Lagrange inversion formula on non-Archimedean fields. Non-Analytical Form of Differential and Finite Difference Equations
The classical Lagrange inversion formula is extended to analytic and
non--analytic inversion problems on non--Archimedean fields. We give some
applications to the field of formal Laurent series in variables, where the
non--analytic inversion formula gives explicit formal solutions of general
semilinear differential and --difference equations.
We will be interested in linearization problems for germs of diffeomorphisms
(Siegel center problem) and vector fields. In addition to analytic results, we
give sufficient condition for the linearization to belong to some Classes of
ultradifferentiable germs, closed under composition and derivation, including
Gevrey Classes. We prove that Bruno's condition is sufficient for the
linearization to belong to the same Class of the germ, whereas new conditions
weaker than Bruno's one are introduced if one allows the linearization to be
less regular than the germ. This generalizes to dimension some results
of [CarlettiMarmi]. Our formulation of the Lagrange inversion formula by mean
of trees, allows us to point out the strong similarities existing between the
two linearization problems, formulated (essentially) with the same functional
equation. For analytic vector fields of \C^2 we prove a quantitative estimate
of a previous qualitative result of [MatteiMoussu] and we compare it with a
result of [YoccozPerezMarco].Comment: This is the final version in press on DCDS Series A. Some minor
changes have been made, in particular the relation w.r.t. the results of
Perez Marco and Yocco
Analytical properties of the Lambert W function
This research studies analytical properties of one of the special functions, the Lambert W function. W function was re-discovered and included into the library of the computer-algebra system Maple in 1980’s. Interest to the function nowadays is due to the fact that it has many applications in a wide variety of fields of science and engineering.
The project can be broken into four parts. In the first part we scrutinize a convergence of some previously known asymptotic series for the Lambert W function using an experimental approach followed by analytic investigation. Particularly, we have established the domain of convergence in real and complex cases, given a comparative analysis of the series properties and found asymptotic estimates for the expansion coefficients. The main analytical tools used herein are Implicit Function Theorem, Lagrange Inversion Theorem and Darboux’s Theorem.
In the second part we consider an opportunity to improve convergence prop erties of the series under study in terms of the domain of,convergence and rate of convergence. For this purpose we have studied a new invariant transformation defined by parameter p, which retains the basic series structure. An effect of parameter p on a size of the domain of convergence and rate of convergence of the series has been studied theoretically and numerically using M a p l e . We have found that an increase in parameter p results in an extension of the domain of convergence while the rate of convergence can be either raised or lowered.
We also considered an expansion of W(x) in powers of Inx. For this series we found three new forms for a representation of the expansion coefficients in terms of different special numbers and accordingly have obtained different ways ito\u27compute the expansion coefficients. As an extra consequence we have obtained some combinatorial relations including the Carlitz-Riordan identities.
In the third part we study the properties of the polynomials appearing in the expressions for the higher derivatives of the Lambert W function. It is shown that the polynomial coefficients form a positive sequence that is log-concave and unimodal, which implies that the positive real branch of the Lambert W function is Bernstein and its derivative is a Stieltjes function.
In the fourth part we show that many functions containing Ware Stieltjes functions. In terms of the result obtained in the third part, we, in fact, obtain one more way to establish that the derivative of W function is a Stieltjes function. We have extended the properties of the set of Stieltjes functions and also proved a generalization of a conjecture of Jackson, Procacci &; Sokal. In addition, we have considered a relation of W to the class of completely monotonic functions and shown that W is a complete Bernstein function. .
We give explicit Stieltjes representations of functions of W, We also present integral representations of W which are associated with the properties of its being a Bernstein and Pick function. Representations based on Poisson and Burniston- Siewert integrals are given as well. The results are obtained relying on the fact that the all of the above mentioned classes are characterized by their own integral forms and using Cauchy Integral Formula, Stieltjes-Perron Inversion Formula and properties of W itself
Large-Spin Expansions of GKP Strings
We demonstrate that the large-spin expansion of the energy of
Gubser-Klebanov-Polyakov (GKP) strings that rotate in RxS2 and AdS3 can be
expressed in terms of Lambert's W-function. We compute the leading, subleading
and next-to-subleading series of exponential corrections to the infinite-volume
dispersion relation of GKP strings that rotate in RxS2. These strings are dual
to certain long operators of N=4 SYM theory and provide their scaling
dimensions at strong coupling. We also show that the strings obey a short-long
(strings) duality. For the folded GKP strings that spin inside AdS3 and are
dual to twist-2 operators, we confirm the known formulas for the leading and
next-to-leading coefficients of their anomalous dimensions and derive the
corresponding expressions for the next-to-next-to-leading coefficients.Comment: 46 pages, 8 figures; Matches published version; Contains equation
(7.3) that gives the finite-size corrections to the dispersion relation of
giant magnons at strong couplin
Large J expansion in ABJM theory revisited
Recently there has been progress in the computation of the anomalous
dimensions of gauge theory operators at strong coupling by making use of
AdS/CFT correspondence. On string theory side they are given by dispersion
relations in the semiclassical regime. We revisit the problem of large charges
expansion of the dispersion relations for simple semiclassical strings in
background. We present the calculation of the
corresponding anomalous dimensions of the gauge theory operators to an
arbitrary order using three different methods. Although the results of the
three methods look different, power series expansions show their consistency.Comment: 24 pages, 2 figure
A new estimate on Evans' Weak KAM approach
We consider a recent formulation of weak KAM theory proposed by Evans. As
well as for classical integrability, for one dimensional mechanical Hamiltonian
systems all the computations can be explicitly done. This allows us on the one
hand to illustrate the geometric content of the theory, on the other hand to
prove new lower bounds which extend also to the generic n degrees of freedom
case
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