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1.
Andrew V. Sills 《The Ramanujan Journal》2006,11(3):403-429
A generalized Bailey pair, which contains several special cases considered by Bailey (Proc. London Math. Soc. (2), 50, 421–435 (1949)), is derived and used to find a number of new Rogers-Ramanujan type identities. Consideration of associated
q-difference equations points to a connection with a mild extension of Gordon’s combinatorial generalization of the Rogers-Ramanujan
identities (Amer. J. Math., 83, 393–399 (1961)). This, in turn, allows the formulation of natural combinatorial interpretations of many of the identities
in Slater’s list (Proc. London Math. Soc. (2) 54, 147–167 (1952)), as well as the new identities presented here. A list of 26 new double sum–product Rogers-Ramanujan type
identities are included as an Appendix.
2000 Mathematics Subject Classification Primary—11B65; Secondary—11P81, 05A19, 39A13 相似文献
2.
In this paper, we find by inverse technique two solutions of a system of linear equations which together serve as a sufficient
and necessary condition for well-poised Bailey chains. Using these two solutions, we establish a new well-poised Bailey chain,
two usual Bailey chains, and a well-poised extension of Bailey’s lemma. Their applications to q-series are also investigated.
X. Ma was supported by Natural Science Foundation of China (No. 10771156). 相似文献
3.
We add two sections to [8] and answer some questions asked there. In the first section we give another derivation of Theorem
1.1 of [8], which reveals the relation between the entropy formula, (1.4) of [8], and the well-known Li-Yau ’s gradient estimate.
As a by-product we obtain the sharp estimates on ‘Nash’s entropy’ for manifolds with nonnegative Ricci curvature. We also
show that the equality holds in Li-Yau’s gradient estimate, for some positive solution to the heat equation, at some positive
time, implies that the complete Riemannian manifold with nonnegative Ricci curvature is isometric to ℝ
n
.In the second section we derive a dual entropy formula which, to some degree, connects Hamilton’s entropy with Perelman ’s
entropy in the case of Riemann surfaces. 相似文献
4.
Vivek V. Rane 《Proceedings Mathematical Sciences》2003,113(3):213-221
Euler-Maclaurin and Poisson analogues of the summations ε
a <n ≤b
χ(n)f(n), have been obtained in a unified manner, where (χ(n)) is a periodic complex sequence;d(n) is the divisor function andf(x) is a sufficiently smooth function on [a, b]. We also state a generalised Abel’s summation formula, generalised Euler’s summation formula and Euler’s summation formula
in several variables. 相似文献
5.
Victor Kowalenko 《Acta Appl Math》2010,109(2):413-437
Via a graphical method, which codes tree diagrams composed of partitions, a novel power series expansion is derived for the
reciprocal of the logarithmic function ln (1+z), whose coefficients represent an infinite set of fractions. These numbers, which are called reciprocal logarithm numbers
and are denoted by A
k
, converge to zero as k→∞. Several properties of the numbers are then obtained including recursion relations and their relationship with the Stirling
numbers of the first kind. Also appearing here are several applications including a new representation for Euler’s constant
known as Hurst’s formula and another for the logarithmic integral. From the properties of the A
k
it is found that a term of ζ(2) cannot be eliminated by the remaining terms in Hurst’s formula, thereby indicating that Euler’s
constant is irrational. Finally, another power series expansion for the reciprocal of arctangent is developed by adapting
the preceding material. 相似文献
6.
The two-fold aim of the paper is to unify and generalize on the one hand the double integrals of Beukers for ζ(2) and ζ(3), and of the second author for Euler’s constant γ and its alternating analog ln (4/π), and on the other hand the infinite products of the first author for e, of the second author for π, and of Ser for e
γ
. We obtain new double integral and infinite product representations of many classical constants, as well as a generalization
to Lerch’s transcendent of Hadjicostas’s double integral formula for the Riemann zeta function, and logarithmic series for
the digamma and Euler beta functions. The main tools are analytic continuations of Lerch’s function, including Hasse’s series.
We also use Ramanujan’s polylogarithm formula for the sum of a particular series involving harmonic numbers, and his relations
between certain dilogarithm values.
相似文献
7.
Soichi Okada 《Journal of Algebraic Combinatorics》2010,32(3):399-416
We generalize multivariate hook product formulae for P-partitions. We use Macdonald symmetric functions to prove a (q,t)-deformation of Gansner’s hook product formula for the generating functions of reverse (shifted) plane partitions. (The unshifted
case has also been proved by Adachi.) For a d-complete poset, we present a conjectural (q,t)-deformation of Peterson–Proctor’s hook product formula. 相似文献
8.
Linas Vepštas 《Numerical Algorithms》2008,47(3):211-252
This paper sketches a technique for improving the rate of convergence of a general oscillatory sequence, and then applies
this series acceleration algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may be taken as an extension
of the techniques given by Borwein’s “An efficient algorithm for computing the Riemann zeta function” by Borwein for computing
the Riemann zeta function, to more general series. The algorithm provides a rapid means of evaluating Li
s
(z) for general values of complex s and a kidney-shaped region of complex z values given by ∣z
2/(z–1)∣<4. By using the duplication formula and the inversion formula, the range of convergence for the polylogarithm may be
extended to the entire complex z-plane, and so the algorithms described here allow for the evaluation of the polylogarithm for all complex s and z values. Alternatively, the Hurwitz zeta can be very rapidly evaluated by means of an Euler–Maclaurin series. The polylogarithm
and the Hurwitz zeta are related, in that two evaluations of the one can be used to obtain a value of the other; thus, either
algorithm can be used to evaluate either function. The Euler–Maclaurin series is a clear performance winner for the Hurwitz
zeta, while the Borwein algorithm is superior for evaluating the polylogarithm in the kidney-shaped region. Both algorithms
are superior to the simple Taylor’s series or direct summation. The primary, concrete result of this paper is an algorithm
allows the exploration of the Hurwitz zeta in the critical strip, where fast algorithms are otherwise unavailable. A discussion
of the monodromy group of the polylogarithm is included.
相似文献
9.
Yoshihiro Takeyama 《The Ramanujan Journal》2012,27(1):15-28
We obtain a class of quadratic relations for a q-analogue of multiple zeta values (qMZV’s). In the limit q→1, it turns into Kawashima’s relation for multiple zeta values. As a corollary we find that qMZV’s satisfy the linear relation contained in Kawashima’s relation. In the proof we make use of a q-analogue of Newton series and Bradley’s duality formula for finite multiple harmonic q-series. 相似文献
10.
A. V. Niukkanen 《Mathematical Notes》2000,67(4):487-494
It is shown that the formulas of operator factorization of hypergeometric functions obtained in the author’s previous works
can be extended to hypergeometric series of the most general form. This generalization does not make the technical apparatus
of the factorization method more complicated. As an example illustrating the practical effectiveness of the formulas obtained
in the paper, we analyze transformation properties of the Horn seriesG
3, whose structure is typical for general hypergeometric functions. It is shown that Erdélyi’s transformation formula relating
the seriesG
3 to the Appell functionF
2, contains erroneous expressions in the arguments ofG
3. The correct analog of Erdélyi’s formula is found, and some new transformations of the seriesG
3 are presented.
Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 573–581, April, 2000. 相似文献
11.
Z. Ditzian 《Israel Journal of Mathematics》1971,9(4):541-553
Rate of convergence in terms of the modulus of continuity of eitherT(t)f or ofT(t)Af, whereT(t) is a strongly continuous semi-group of operators, is obtained for Phillips’ and for Widder’s exponential formula. 相似文献
12.
A. V. Shanin 《Journal of Mathematical Sciences》2008,148(5):769-783
The problems of diffraction by a slit or a strip having ideal boundary conditions, and some other problems, can be reduced
to the problem of wave propagation on a multisheet surface by applying the method of reflections. Further simplifications
of the problem can be achieved by applying an embedding formula. As a result, the solution of the problem with a plane wave
incidence becomes expressed in terms of the edge Green’s functions, i.e., in terms of the fields generated by dipole sources
localized at branchpoints of the surface.
The present paper is devoted to finding the edge Green’s functions. For this problem, two sets of differential equations,
namely, the coordinate and spectral equations, are used. The properties of solutions of these equations are studied. Bibliography:
9 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 342, 2007, pp. 233–256. 相似文献
13.
The question whether or not the sum of two maximal monotone operators is maximal monotone under Rockafellar’s constraint qualification—that
is, whether or not “the sum theorem” is true—is the most famous open problem in Monotone Operator Theory. In his 2008 monograph
“From Hahn-Banach to Monotonicity”, Stephen Simons asked whether or not the sum theorem holds for the special case of a maximal monotone linear operator and
a normal cone operator of a closed convex set provided that the interior of the set makes a nonempty intersection with the
domain of the linear operator. In this note, we provide an affirmative answer to Simons’ question. In fact, we show that the
sum theorem is true for a maximal monotone linear relation and a normal cone operator. The proof relies on Rockafellar’s formula for the Fenchel conjugate of the sum as well as some
results featuring the Fitzpatrick function.
相似文献
14.
Marko Uršič 《Acta Analytica》2004,19(33):221-239
In this essay, the author outlines his re-construction of Spinoza’s ontological monism by re-presenting the system of Ethica,
ordine geometrico demonstrata, in an “intuitive” model of the Perfect Diamond, called NATURADEUS. So, for example, ordo et connexio idearum et rerum, is presented to the inner eye in the forms of two parallel structures, of rays and of facets within the NATURADEUS, respectively.
The conceptual background of the proposed model is mostly analytic, the author essays to develop some ideas of Jonathan Bennett’s
Spinoza’s metaphysics (especially “transattributive mode identity”), with strong emphasis on ethical issues of ontological monism or pantheism.
This essay is written as a dialogue between master Bruno and his student John (physically absent at the moment). More philosophical
dialogues of this kind can be found on the author’s web page and in his book Four Seasons (in Slovenian, 2002). 相似文献
15.
The aim of the paper is to relate computational and arithmetic questions about Euler’s constant γ with properties of the values
of the q-logarithm function, with natural choice of~q. By these means, we generalize a classical formula for γ due to Ramanujan, together with Vacca’s and Gosper’s series for
γ, as well as deduce irrationality criteria and tests and new asymptotic formulas for computing Euler’s constant. The main
tools are Euler-type integrals and hypergeometric series.
2000 Mathematics Subject Classification Primary—11Y60; Secondary—11J72, 33C20, 33D15
The work of the second author is supported by an Alexander von Humboldt research fellowship
Dedication: To Leonhard Euler on his 300th birthday. 相似文献
16.
Peter Nyman 《Advances in Applied Clifford Algebras》2011,21(4):799-811
Recently quantum-like representation algorithm (QLRA) was introduced by A. Khrennikov [20]–[28] to solve the so-called “inverse Born’s rule problem”: to construct a representation of probabilistic data by a complex or
hyperbolic probability amplitude or more general complex together with hyperbolic which matches Born’s rule or its generalizations.
The outcome from QLRA is coupled to the formula of total probability with an additional term corresponding to trigonometric,
hyperbolic or hyper-trigonometric interference. The consistency of QLRA for probabilistic data corresponding to trigonometric
interference was recently proved [29]. We complete the proof of the consistency of QLRA to cover hyperbolic interference as well. We will also discuss hyper trigonometric
interference. The problem of consistency of QLRA arises, because formally the output of QLRA depends on the order of conditioning.
For two observables (e.g., physical or biological) a and b, b|a- and a|b-conditional probabilities produce two representations, say in Hilbert spaces H
b|a
and H
a|b
(in this paper over the hyperbolic algebra). We prove that under “natural assumptions” these two representations are unitary
equivalent (in the sense of hyperbolic Hilbert space). 相似文献
17.
In this paper, we study Perelman’s W{{\mathcal W}} -entropy formula for the heat equation associated with the Witten Laplacian on complete Riemannian manifolds via the Bakry–Emery
Ricci curvature. Under the assumption that the m-dimensional Bakry–Emery Ricci curvature is bounded from below, we prove an analogue of Perelman’s and Ni’s entropy formula
for the W{\mathcal{W}} -entropy of the heat kernel of the Witten Laplacian on complete Riemannian manifolds with some natural geometric conditions.
In particular, we prove a monotonicity theorem and a rigidity theorem for the W{{\mathcal W}} -entropy on complete Riemannian manifolds with non-negative m-dimensional Bakry–Emery Ricci curvature. Moreover, we give a probabilistic interpretation of the W{\mathcal{W}} -entropy for the heat equation of the Witten Laplacian on complete Riemannian manifolds, and for the Ricci flow on compact
Riemannian manifolds. 相似文献
18.
The effect of managing aircraft movements on the airport’s ground is an important tool that can alleviate the delays of flights,
specially in peak hours or congested situations. Although some strategic design decisions regarding aeronautical and safety
aspects have a main impact on the airport’s topology, there exists a number of other additional factors that must be evaluated
according to the on ground operations, i.e. previous to the taking-off or after landing. Among these factors one can consider
capacities at waiting points and directions of some corridors. These factors are related to the demand situation of a given
period and influence the aircraft’s routing on the ground or short term Taxi Planning problem (or TP-S). While the TP-S problem studies the aircraft routing and scheduling on the airport’s ground under a dynamic point of view, this paper presents
a Taxi Planning network design model (or TPND), attending to these additional factors of the airport’s topology and the conflicting movements of the aircraft on them with
the same modelling approach used in the TP-S problem. The TPND model is formulated as a binary multicommodity network flow problem with additional side constraints under a multiobjective
approach. The side constraints included are the classical limitations due to capacity and also as a distinctive approach,
constraints that restrict the interference of aircraft in order to decrease the intervention of human controllers during the
operations or increase their safety margins. The multiobjective approach adopted for the TPND model balances conflicting objectives: airport’s throughput, travel times, safety of operations and costs. In the paper computational
results are included on two test airports solving the TPND model by “Branch and Bound” showing the effect of the conflicting objectives in the design decisions.
Research supported under Research Project TRA-2005-09068-C03-01/MODAL from the “Ministerio de Educación y Ciencia”. 相似文献
19.
Assaf Libman 《Mathematische Zeitschrift》2007,255(3):515-548
We give a new proof of the Minami–Webb formula for classifying spaces of finite groups by exploiting Symonds’s resolution
of Webb’s conjecture. The methods are applicable to obtain a stable decomposition of Minami’s type for the classifying spaces
of the three exotic p-local finite groups which were introduced by Ruiz and Viruel at the prime 7. We obtain a similar decomposition for the classifying
spaces of a family of exotic p-local finite groups which were constructed by Broto, Levi and Oliver.
The author was supported by the Nuffield Foundation Grant NAL/00735/G. 相似文献
20.
Let X, Y be finite sets and T a set of functions X → Y which we will call “ tableaux”. We define a simplicial complex whose facets, all of the same dimension, correspond to these
tableaux. Such tableau complexes have many nice properties, and are frequently homeomorphic to balls, which we prove using vertex decompositions [BP79].
In our motivating example, the facets are labeled by semistandard Young tableaux, and the more general interior faces are
labeled by Buch’s set-valued semistandard tableaux. One vertex decomposition of this “Young tableau complex” parallels Lascoux’s
transition formula for vexillary double Grothendieck polynomials [La01, La03]. Consequently, we obtain formulae (both old
and new) for these polynomials. In particular, we present a common generalization of the formulae of Wachs [Wa85] and Buch
[Bu02], each of which implies the classical tableau formula for Schur polynomials. 相似文献