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Higher Transcendental Functions
01 Jan 1981-
About: The article was published on 1981-01-01 and is currently open access. It has received 7882 citations till now. The article focuses on the topics: Transcendental function.
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TL;DR: In this paper, a simple theory is developed that accounts for many of the observed physical properties of micelles, both globular and rod-like, and of bilayer vesicles composed of ionic or zwitterionic amphiphiles.
Abstract: A simple theory is developed that accounts for many of the observed physical properties of micelles, both globular and rod-like, and of bilayer vesicles composed of ionic or zwitterionic amphiphiles. The main point of departure from previous theories lies in the recognition and elucidation of the role of geometric constraints in self-assembly. The linking together of thermodynamics, interaction free energies and geometry results in a general framework which permits extension to more complicated self-assembly problems.
4,563 citations
TL;DR: In this article, a fractional-order PI/sup/spl lambda/D/sup /spl mu/controller with fractionalorder integrator and fractional order differentiator is proposed.
Abstract: Dynamic systems of an arbitrary real order (fractional-order systems) are considered. The concept of a fractional-order PI/sup /spl lambda//D/sup /spl mu//-controller, involving fractional-order integrator and fractional-order differentiator, is proposed. The Laplace transform formula for a new function of the Mittag-Leffler-type made it possible to obtain explicit analytical expressions for the unit-step and unit-impulse response of a linear fractional-order system with fractional-order controller for both open- and closed-loops. An example demonstrating the use of the obtained formulas and the advantages of the proposed PI/sup /spl lambda//D/sup /spl mu//-controllers is given.
2,479 citations
TL;DR: In this paper, it was proved that the counterterm for an arbitrary 4-loop Feynman diagram in an arbitrary model is calculable within the minimal subtraction scheme in terms of rational numbers and the Riemann ζ-function in a finite number of steps via a systematic "algebraic" procedure involving neither integration of elementary, special, or any other functions, nor expansions in and summation of infinite series of any kind.
1,928 citations
TL;DR: In this paper, the authors derived analogues for the Airy kernel of the following properties of the sine kernel: the completely integrable system of P.D.E., the expression of the Fredholm determinant in terms of a Painleve transcendent, the existence of a commuting differential operator, and the fact that this operator can be used in the derivation of asymptotics, for generaln, of the probability that an interval contains preciselyn eigenvalues.
Abstract: Scaling level-spacing distribution functions in the “bulk of the spectrum” in random matrix models ofN×N hermitian matrices and then going to the limitN→∞ leads to the Fredholm determinant of thesine kernel sinπ(x−y)/π(x−y). Similarly a scaling limit at the “edge of the spectrum” leads to theAiry kernel [Ai(x)Ai(y)−Ai′(x)Ai(y)]/(x−y). In this paper we derive analogues for this Airy kernel of the following properties of the sine kernel: the completely integrable system of P.D.E.'s found by Jimbo, Miwa, Mori, and Sato; the expression, in the case of a single interval, of the Fredholm determinant in terms of a Painleve transcendent; the existence of a commuting differential operator; and the fact that this operator can be used in the derivation of asymptotics, for generaln, of the probability that an interval contains preciselyn eigenvalues.
1,923 citations
TL;DR: In this paper, the quantum-mechanical problems of N 1-dimensional equal particles of mass m interacting pairwise via quadratic (harmonical) and/or inverse (centrifugal) potentials is solved.
Abstract: The quantum‐mechanical problems of N 1‐dimensional equal particles of mass m interacting pairwise via quadratic (``harmonical'') and/or inversely quadratic (``centrifugal'') potentials is solved. In the first case, characterized by the pair potential ¼mω2(xi − xj)2 + g(xi − xj)−2, g > −ℏ2/(4m), the complete energy spectrum (in the center‐of‐mass frame) is given by the formula E=ℏω(12N)12[12(N−1)+12N(N−1)(a+12)+ ∑ l=2Nlnl], with a = ½(1 + 4mgℏ−2)½. The N − 1 quantum numbers nl are nonnegative integers; each set {nl; l = 2, 3, ⋯, N} characterizes uniquely one eigenstate. This energy spectrum can also be written in the form Es = ℏω(½N)½ [½(N − 1) + ½N(N − 1)(a + ½) + s], s = 0, 2, 3, 4, ⋯, the multiplicity of the sth level being then given by the number of different sets of N − 1 nonnegative integers nl that are consistent with the condition s=∑l=2Nlnl. These equations are valid independently of the statistics that the particles satisfy, if g ≠ 0; for g = 0, the equations remain valid with a = ½ for Fermi st...
1,454 citations