scholarly journals A combinatorial identity for a problem in asymptotic statistics

2009 ◽  
Vol 3 (1) ◽  
pp. 64-68 ◽  
Author(s):  
Hansjörg Albrecher ◽  
Jozef Teugels ◽  
Klaus Scheicher
Keyword(s):  
Continued Fraction ◽  
Discrete Mathematics ◽  
Variation Theory ◽  
Combinatorial Identity ◽  
Distribution Tail ◽  
Enumeration Problem ◽  
New Formula ◽  
Regularly Varying Distribution ◽  
Orientable Surfaces ◽  
Independent Identically Distributed

Let (Xi)i?1 be a sequence of positive independent identically distributed random variables with regularly varying distribution tail of index 0 < ? < 1 and define Tn = X1?+X2?+???+Xn?/(X1+X2+???+ Xn)?.In this note we simplify an expression for lim n?? E(T kn ), which was obtained by Albrecher and Teugels: Asymptotic analysis of a measure of variation. Theory Prob. Math. Stat., 74 (2006), 1-9, in terms of coefficients of a continued fraction expansion. The new formula establishes an unexpected link to an enumeration problem for rooted maps on orientable surfaces that was studied in Arqu?s and B?raud: Rooted maps of orientable surfaces, Riccati's equation and continued fractions. Discrete Mathematics, 215 (2000), 1-12.

scholarly journals A BURGE TREE OF VIRASORO-TYPE POLYNOMIAL IDENTITIES

1998 ◽  
Vol 13 (29) ◽  
pp. 4967-5012 ◽  
Author(s):  
OMAR FODA ◽  
KEITH S. M. LEE ◽  
TREVOR A. WELSH
Keyword(s):  
Continued Fraction ◽  
Finite Size ◽  
Summation Formula ◽  
Combinatorial Identity ◽  
Size Parameter ◽  
Polynomial Identities ◽  
Infinite Tree ◽  
Coprime Integers ◽  
Virasoro Type ◽  
Virasoro Characters

Using a summation formula due to Burge, and a combinatorial identity between partition pairs, we obtain an infinite tree of q-polynomial identities for the Virasoro characters [Formula: see text], dependent on two finite size parameters M and N, in the cases where: (1) p and p′ are coprime integers that satisfy 0 < p < p′. (2) If the pair (p′:p) has a continued fraction (c1, c2, …, ct-1, ct+2), where t≥1, then the pair (s:r) has a continued fraction (c1, c2, …, cu-1, d), where 1 ≤ u ≤ t, and 1 ≤ d ≤ cu. The limit M → ∞, for fixed N, and the limit N → ∞, for fixed M, lead to two independent boson-fermion-type q-polynomial identities: in one case, the bosonic side has a conventional dependence on the parameters that characterize the corresponding character. In the other, that dependence is not conventional. In each case, the fermionic side can also be cast in either of two different forms. Taking the remaining finite size parameter to infinity in either of the above identities, so that M → ∞ and N → ∞, leads to the same q-series identity for the corresponding character.

A new formula for using the audiogram to predict hearing loss for speech. (Rep. No. 273).

1956 ◽  
Author(s):  
J. Donald Harris ◽  
Henry L. Haines ◽  
Cecil K. Myers
Keyword(s):  
Hearing Loss ◽  
New Formula

Macrosomia – a new formula for optimized fetal weight estimation

2008 ◽  
Vol 68 (S 01) ◽  
Author(s):  
NC Hart ◽  
J Siemer ◽  
B Meurer ◽  
TW Goecke ◽  
RL Schild
Keyword(s):  
Fetal Weight ◽  
Weight Estimation ◽  
Fetal Weight Estimation ◽  
New Formula

Rosen-Chambers Variation Theory of Linearly-Damped Classic and Quantum Oscillator

2014 ◽  
Vol 4 (1) ◽  
pp. 404-426
Author(s):  
Vincze Gy. Szasz A.
Keyword(s):  
Harmonic Oscillator ◽  
Classical Mechanics ◽  
Universal Time ◽  
Variation Theory ◽  
Quantum Oscillator ◽  
Variation Principle ◽  
Poisson Brackets ◽  
Mathematical Meaning ◽  
Damped Harmonic Oscillator ◽  
Damped Oscillator

Phenomena of damped harmonic oscillator is important in the description of the elementary dissipative processes of linear responses in our physical world. Its classical description is clear and understood, however it is not so in the quantum physics, where it also has a basic role. Starting from the Rosen-Chambers restricted variation principle a Hamilton like variation approach to the damped harmonic oscillator will be given. The usual formalisms of classical mechanics, as Lagrangian, Hamiltonian, Poisson brackets, will be covered too. We shall introduce two Poisson brackets. The first one has only mathematical meaning and for the second, the so-called constitutive Poisson brackets, a physical interpretation will be presented. We shall show that only the fundamental constitutive Poisson brackets are not invariant throughout the motion of the damped oscillator, but these show a kind of universal time dependence in the universal time scale of the damped oscillator. The quantum mechanical Poisson brackets and commutation relations belonging to these fundamental time dependent classical brackets will be described. Our objective in this work is giving clearer view to the challenge of the dissipative quantum oscillator.

Topology knots and hierarchy problem

2019 ◽  
Author(s):  
Vitaly Kuyukov
Keyword(s):  
Quantum Theory ◽  
String Theory ◽  
Quantum Gravity ◽  
Dimensional Space ◽  
Space Time ◽  
Elementary Particles ◽  
Hierarchy Problem ◽  
Topological Quantum ◽  
Dimensional Space Time ◽  
New Formula

Many approaches to quantum gravity consider the revision of the space-time geometry and the structure of elementary particles. One of the main candidates is string theory. It is possible that this theory will be able to describe the problem of hierarchy, provided that there is an appropriate Calabi-Yau geometry. In this paper we will proceed from the traditional view on the structure of elementary particles in the usual four-dimensional space-time. The only condition is that quarks and leptons should have a common emerging structure. When a new formula for the mass of the hierarchy is obtained, this structure arises from topological quantum theory and a suitable choice of dimensional units.

The Solvability of Polynomial Pell’s Equation

2020 ◽  
Vol 25 (2) ◽  
pp. 125-132
Author(s):  
Bal Bahadur Tamang ◽  
Ajay Singh
Keyword(s):  
Trivial Solution ◽  
Continued Fraction ◽  
Laurent Series ◽  
Quadratic Polynomial ◽  
Continued Fraction Expansion ◽  
Pell’S Equation ◽  
Integer Polynomials

This article attempts to describe the continued fraction expansion of ÖD viewed as a Laurent series x-1. As the behavior of the continued fraction expansion of ÖD is related to the solvability of the polynomial Pell’s equation p2-Dq2=1  where D=f2+2g  is monic quadratic polynomial with deg g<deg f  and the solutions p, q  must be integer polynomials. It gives a non-trivial solution if and only if the continued fraction expansion of ÖD  is periodic.

New Approach to the Theory of Circular Dichroism

1998 ◽  
Vol 63 (8) ◽  
pp. 1187-1201 ◽  
Author(s):  
Jaroslav Zamastil ◽  
Lubomír Skála ◽  
Petr Pančoška ◽  
Oldřich Bílek
Keyword(s):  
Electromagnetic Wave ◽  
Circular Dichroism ◽  
Electromagnetic Waves ◽  
Maxwell Equations ◽  
Optically Active ◽  
Semiclassical Approach ◽  
New Approach ◽  
Isotropic Media ◽  
New Formula ◽  
Circular Dichroïsm

Using the semiclassical approach for the description of the propagation of the electromagnetic waves in optically active isotropic media we derive a new formula for the circular dichroism parameter. The theory is based on the idea of the time damped electromagnetic wave interacting with the molecules of the sample. In this theory, the Lambert-Beer law need not be taken as an empirical law, however, it follows naturally from the requirement that the electromagnetic wave obeys the Maxwell equations.

Learning Discrete Mathematics with ISETL

1989 ◽  
Author(s):  
Nancy Baxter ◽  
Ed Dubinsky ◽  
Gary Levin
Keyword(s):  
Discrete Mathematics
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