Expansion Formula Research Articles

Controllability of Some Nonlinear Systems with Drift via Generalized Curvature Properties

We discuss the problem of local attainability for finite-dimensional nonlinear control systems with quite general assumptions on the target set. Special emphasis is given to control-affine systems with a possibly nontrivial drift term. To this end, we provide some sufficient conditions ensuring local attainability, which involve geometric properties both of the target itself (such as a notion of generalized curvature), and of the Lie algebra associated with the control system. The main technique used is a convenient representation formula for the power expansion of the distance function along the trajectories, made at points sufficiently near to the target set.

Atomic Bases and $T$-path Formula for Cluster Algebras of Type $D$

We extend a $T$-path expansion formula for arcs on an unpunctured surface to the case of arcs on a once-punctured polygon and use this formula to give a combinatorial proof that cluster monomials form the atomic basis of a cluster algebra of type $D$. Nous généralisons une formule de développement en $T$-chemins pour les arcs sur une surface non-perforée aux arcs sur un polygone à une perforation. Nous utilisons cette formule pour donner une preuve combinatoire du fait que les monômes amassées constituent la base atomique d’une algèbre amassée de type $D$.

On a Pair of Operator Series Expansions Implying a Variety of Summation Formulas

With the aid of Mullin-Rota＇s substitution rule, we show that the Sheffertypedifferential operators together with the delta operators D and D could be used toconstruct a pair of expansion formulas that imply a wide variety of summation formulasin the discrete analysis and combinatorics. A convergence theorem is establishedfor a fruitful source formula that implies more than 20 noted classical fomulas and identitiesas consequences. Numerous new formulas are also presented as illustrativeexamples. Finally, it is shown that a kind of lifting process can be used to producecertain chains of （∞m） degree formulas for m≥3 with m≡1 （mod 2） and m≡1 （mod3）, respectively.

Two-channel sampling in wavelet subspaces

Abstract We develop two-channel sampling theory in the wavelet subspace V 1 from the multi resolution analysis {Vj } j ∈𝕫. Extending earlier results by G. G. Walter [11], W. Chen and S. Itoh [2] and Y. M. Hong et al [5] on the sampling theory in the wavelet or shift invariant spaces, we find a necessary and sufficient condition for two-channel sampling expansion formula to hold in V 1.

A Multinomial Theorem for Hermite Polynomials and Financial Applications

Different aspects of mathematical finance benefit from the use Hermite polynomials, and this is particularly the case where risk drivers have a Gaussian distribution. They support quick analytical methods which are computationally less cumbersome than a full-fledged Monte Carlo framework, both for pricing and risk management purposes. In this paper, we review key properties of Hermite polynomials before moving on to a multinomial expansion formula for Hermite polynomials, which is proved using basic methods and corrects a formulation that appeared before in the financial literature. We then use it to give a trivial proof of the Mehler formula. Finally, we apply it to no arbitrage pricing in a multi-factor model and determine the empirical futures price law of any linear combination of the underlying factors.

Heat trace asymptotics and boundedness in the second order Sobolev space of isospectral potentials for the Dirichlet Laplacian

Let $\Omega$ be a $C^\infty$-smooth bounded domain of $\mathbb{R}^n$, $n \geq 1$, and let the matrix ${\bf a} \in C^\infty (\overline{\Omega};\R^{n^2})$ be symmetric and uniformly elliptic. We consider the $L^2(\Omega)$-realization $A$ of the operator $-\mydiv ( {\bf a} \nabla \cdot)$ with Dirichlet boundary conditions. We perturb $A$ by some real valued potential $V \in C_0^\infty (\Omega)$ and note $A_V=A+V$. We compute the asymptotic expansion of $\mbox{tr}\left( e^{-t A_V}-e^{-t A}\right)$ as $t \downarrow 0$ for any matrix ${\bf a}$ whose coefficients are homogeneous of degree $0$. In the particular case where $A$ is the Dirichlet Laplacian in $\Omega$, that is when ${\bf a}$ is the identity of $\R^{n^2}$, we make the four main terms appearing in the asymptotic expansion formula explicit and prove that $L^\infty$-bounded sets of isospectral potentials of $A$ are $H^s$-compact for $s <2$.

Asymptotic properties of MUSIC-Type Imaging in Two-Dimensional Inverse Scattering from Thin Electromagnetic Inclusions

The main purpose of this paper is to study the structure of the well-known non-iterative MUltiple SIgnal Classification (MUSIC) algorithm for identifying the shape of extended electromagnetic inclusions of small thickness located in a two-dimensional homogeneous space. We construct a relationship between the MUSIC-type imaging functional for thin inclusions and the Bessel function of integer order of the first kind. Our construction is based on the structure of the left singular vectors of the collected multistatic response matrix whose elements are the measured far-field pattern and the asymptotic expansion formula in the presence of thin inclusions. Some numerical examples are shown to support the constructed MUSIC structure.

Formal pseudodifferential operators and Witten’s r-spin numbers

Abstract We derive an effective recursion for Witten’s r-spin intersection numbers, using Witten’s conjecture relating r-spin numbers to the Gel’fand–Dikii hierarchy. Consequences include closed-form descriptions of the intersection numbers (for example, in terms of gamma functions). We use these closed-form descriptions to prove Harer–Zagier’s formula for the Euler characteristic of ℳ g , 1 {\mathcal{M}_{g,1}} . Finally, we extend Witten’s series expansion formula for the Landau–Ginzburg potential to study r-spin numbers in the small phase space in genus zero. Our key tool is the calculus of formal pseudodifferential operators, and is partially motivated by work of Brézin and Hikami.

An analogue of the Robinson–Schensted–Knuth correspondence and non-symmetric Cauchy kernels for truncated staircases

We prove a restriction of an analogue of the Robinson–Schensted–Knuth correspondence for semi-skyline augmented fillings, due to Mason, to multisets of cells of a staircase possibly truncated by a smaller staircase at the upper left end corner, or at the bottom right end corner. The restriction to be imposed on the pairs of semi-skyline augmented fillings is that the pair of shapes, rearrangements of each other, satisfies an inequality in the Bruhat order, w.r.t. the symmetric group, where one shape is bounded by the reverse of the other. For semi-standard Young tableaux the inequality means that the pair of their right keys is such that one key is bounded by the Schützenberger evacuation of the other. This bijection is then used to obtain an expansion formula of the non-symmetric Cauchy kernel, over staircases or truncated staircases, in the basis of Demazure characters of type A, and the basis of Demazure atoms. The expansion implies Lascoux expansion formula, when specialized to staircases or truncated staircases, and make explicit, in the latter, the Young tableaux in the Demazure crystal by interpreting Demazure operators via elementary bubble sorting operators acting on weak compositions.

Bivariate Krawtchouk polynomials: Inversion and connection problems with the NAVIMA algorithm

In this paper we present a recurrent procedure to solve an inversion problem for monic bivariate Krawtchouk polynomials written in vector column form, giving its solution explicitly. As a by-product, a general connection problem between two vector column of monic bivariate Krawtchouk families is also explicitly solved. Moreover, in the non monic case and also for Krawtchouk families, several expansion formulas are given, but for polynomials written in scalar form.

Expansions in the Askey–Wilson polynomials

We give a general expansion formula of functions in the Askey–Wilson polynomials and using Askey–Wilson orthogonality we evaluate several integrals. Moreover we give a general expansion formula of functions in polynomials of Askey–Wilson type, which are not necessarily orthogonal. Limiting cases give expansions in little and big q-Jacobi type polynomials. We also give a new generating function for Askey–Wilson polynomials and a new evaluation for specialized Askey–Wilson polynomials.

FACTORIZATIONS OF ANALYTIC FUNCTIONS VIA GENERALIZED INVERSE POWER PRODUCT EXPANSIONS

Given an arbitrary sequence of complex numbers {an} ∞=1 and an arbitrary nonzero sequence of complex numbers {rn} ∞=1 , we study the ex- pansion of the Taylor series 1 + P ∞=1 anx n into infinite products of the form Q ∞ n=1 (1 − hnx n ) −rn . Algebraic properties, convergence criteria, and combina- torial interpretations of the infinite products are investigated. We also provide an asymptotic formula for the majorizing product expansion associated with 1 − P ∞=1 s n x n , s := sup n≥1 |an| 1 n.

Phase space Feynman path integrals of higher order parabolic type with general functional as integrand

We give a general class of functionals for which the phase space Feynman path integrals of higher order parabolic type have a mathematically rigorous meaning. More precisely, for any functional belonging to our class, the time slicing approximation of the phase space path integral converges uniformly on compact subsets with respect to the endpoint of position paths and to the starting point of momentum paths. Our class of functionals is rich because it is closed under addition and multiplication. The interchange of the order with the integration with respect to time, the interchange of the order with a limit and the perturbation expansion formula hold in the path integrals.

Analysis of multi-frequency subspace migration weighted by natural logarithmic function for fast imaging of two-dimensional thin, arc-like electromagnetic inhomogeneities

The present study seeks to investigate mathematical structures of a multi-frequency subspace migration weighted by the natural logarithmic function for imaging of thin electromagnetic inhomogeneities from measured far-field pattern. To this end, we designed the algorithm and calculated the indefinite integration of square of Bessel function of order zero of the first kind multiplied by the natural logarithmic function. This is needed for mathematical analysis of improved algorithm to demonstrate the reason why proposed multi-frequency subspace migration contributes to yielding better imaging performance, compared to previously suggested subspace migration algorithm. This analysis is based on the fact that the singular vectors of the collected Multi-Static Response (MSR) matrix whose elements are the measured far-field pattern can be represented as an asymptotic expansion formula in the presence of such inhomogeneities. To support the main research results, several numerical experiments with noisy data are illustrated.

Further results involving a class of generalized Hurwitz-Lerch zeta functions

In this paper, we present several new expansion formulas for a class of generalized Hurwitz-Lerch zeta functions which were introduced by Raina and Chhajed [R. K. Raina and P. K. Chhajed, “Certain Results Involving a Class of Functions Associated with the Hurwitz Zeta Function,” Acta Math. Univ. Comenian. 73, 89–100 (2004)] and (more recently) by Srivastava et al. [H. M. Srivastava, M.-J. Luo, and R. K. Raina, “New Results Involving a Class of Generalized Hurwitz-Lerch Zeta Functions and Their Applications,” Turkish J. Anal. Number Theory 1, 26–35 (2013)]. These expansion formulas are obtained with the help of some fractional calculus theorems such as the generalized Leibniz rules, the Taylorlike expansions in terms of different functions and the generalized chain rule. Several (known or new) special cases are also considered.

The Monge-Ampère trajectory correction for semi-Lagrangian schemes

Requiring that numerical estimates of the flow trajectories comply with the fundamental Euler expansion formula that governs the evolution of a volume of fluid leads to a second-order nonlinear Mon...

Design for Drive Circuit of Solid Micro-Thruster Ignition System

Aiming at the lack of standard design method and the shortcomings of main functions of the existing drive circuit, a “Element-level Design under Transmission Voltage Switch Theory” is set forth and a smallest circuit structure namely as drive-unit, which contains Superstratum-unit, Substratum-unit and LC-unit, is designed in this paper to achieve the necessary functions. The drive circuit of n×n ignition bridges has been got from the structure expansion formula along with the expected life estimation. The Ebers-Moll Model of transistor and the internal circuit of MOSFET were used to make a theoretical analysis and the conclusion of such analysis was verified by simulation subsequently. Actual measurement has been made and the results show that the circuit in simple structure has provided a strong loading ability and accurate output; some elements of the circuit have been heated when the circuit was working; and the response time of the circuit was of the microsecond-level.

Expansion Formulae for Kampe De Feriet and Radial Wave Functions and Heat Conduction

Author evaluates some integrals involving Kampe de Feriet function and uses one of them in obtaining a solution of a problem of heat conduction and a few interesting cases along with the application of one of them, in the radial wave function for the hydrogen like atoms, have also been discussed.

An extended version of the simple and fast Feynman diagram generator EasyFeynDiag

The main improvement in this version of EasyFeynDiag is to prohibit equivalent Feynman diagrams to two-loop order (while the previous version is to one-loop). The symmetry factor calculation with equivalent diagrams prohibition to two-loop is a nontrivial problem, which is well solved in the new program. The code of the program is also reorganized, hopefully in a way more convenient for people to read. New version program summaryProgram title: EasyFeynDiagCatalogue identifier: AEPD_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEPD_v2_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 51148No. of bytes in distributed program, including test data, etc.: 55,295Distribution format: tar.gzProgramming language: C.Computer: Platforms on which an ANSI C compiler is available.Operating system: Operating Systems on which an ANSI C compiler is available.RAM: 236,000 bytes for a typical two-loop order uū→tt̄ process in the QCD model.Classification: 4.4.Does the new version supersede the previous version?: The new version generates Feynman diagrams better, but is more time consuming, than the previous version. Overall, the new version supersedes the previous version.Nature of problem:Automatic generation of Feynman diagrams from given interactions, scattering process, and number of loops.Solution method:The algorithm of EasyFeynDiag is translated from the perturbative expansion formula of the S-matrix [1]. It follows an ordered iterative traversing procedure to find all the possible combinations of the fields and creation/annihilation operators.Two selection rules are adopted in turn, to suppress equivalent Feynman diagrams originating from vertices relabeling, and from multiple identical particles of an interaction term, respectively. This simple algorithm prohibits equivalent diagrams to one-loop order.To further prohibit equivalent diagrams to higher order, more careful treating of the identical particles is required. By introducing one additional “order rule” for the identical particles and one additional “selection rule” for identical vertices, the equivalent diagrams are prohibited to two-loop order. Those two rules are explained briefly below:1. order rule for identical particlesWhen identical particles from the same vertex are to connect to other legs, those connections must be ordered: the second identical particle must connect to a leg equal or behind the leg connected by the first identical particle, and so on the Nth to the (N−1)th identical particle. At the same time, proper symmetry factors need to be generated to count for this ordering.2. selection rule for identical verticesWhen m identical particles from the same vertex have connected to m new vertices that are of the same interaction type, these m vertices are defined as identical vertices. If later another particle is to connect to one of these identical vertices, it is only permitted to connect to the most front one, the other (m−1) connections are represented by a factor m.Reasons for new version:High energy calculations to two-loop order are becoming more and more common, e.g. [2]. To introduce, into EasyFeynDiag, the ability to prohibit equivalent diagrams to two-loop order would make it more attractive to researchers in High Energy Physics.Summary of revisions:1.the code is organized in a totally new way, more convenient for people to read and modify;2.the algorithm for Feynman diagram generation in the program is extended to a version that prohibits equivalent diagrams to two-loop order (the previous version is to one-loop order).Restrictions:1.There are default values for the maximum external particles and internal vertices for a scattering process, and maximum legs or leg types for an interaction term. Users need to modify the corresponding values in the code to get larger limits.2.Equivalent Feynman diagrams arise at three-loop order or higher, although this fact does not hurt the correctness of the Feynman amplitudes to be obtained from those diagrams.Unusual features:Allow for arbitrary physical model and arbitrary number of loops. Symmetry factors are naturally generated. Small and fast.Running time:About 0.139 s to generate all the Feynman diagrams of a typical two-loop order uū→tt̄ process in the QCD model.

Heat Content and Small Time Asymptotics for Schrödinger Operators on ℝ d ${\mathbb {R}}^{d}$

This paper studies the heat content for Schrodinger operators of the fractional Laplacian \({(-{\Delta })^{\frac {\alpha }{2} }}\), 0<α≤2 in \({\mathbb {R}}^{d}\), d≥1. Employing probabilistic and analytic techniques, a small time asymptotic expansion formula is given and the “heat content invariants” are identified. To the best of our knowledge, these results are new even in the case of the Laplacian, α = 2.