scholarly journals Explicit and asymptotic formulae for Vasyunin-cotangent sums

2017 ◽  
Vol 102 (116) ◽  
pp. 155-174 ◽  
Author(s):  
Mouloud Goubi ◽  
Abdelmejid Bayad ◽  
Mohand Hernane
Keyword(s):  
Asymptotic Formula ◽  
Explicit Formula ◽  
Continued Fraction ◽  
Reciprocity Law ◽  
Asymptotic Formulae

For coprime numbers p and q, we consider the Vasyunin-cotangent sum V(q, p)= ?p?1 k=1 {kq/p} cot (?k/p). First, we prove explicit formula for the symmetric sum V(p,q)+ V(q,p) which is a new reciprocity law for the sums above. This formula can be seen as a complement to the Bettin-Conrey result [13, Theorem 1]. Second, we establish an asymptotic formula for V(p,q). Finally, by use of continued fraction theory, we give a formula for V(p,q) in terms of continued fraction of p/q.

The log-normal approximation in financial and other computations

2004 ◽  
Vol 36 (03) ◽  
pp. 747-773 ◽  
Author(s):  
Daniel Dufresne
Keyword(s):  
Brownian Motion ◽  
Explicit Formula ◽  
Normal Approximation ◽  
Random Variables ◽  
Exact Distribution ◽  
Financial Mathematics ◽  
Asian Options ◽  
Basket Options ◽  
Asymptotic Formulae ◽  
Log Normal

Sums of log-normals frequently appear in a variety of situations, including engineering and financial mathematics. In particular, the pricing of Asian or basket options is directly related to finding the distributions of such sums. There is no general explicit formula for the distribution of sums of log-normal random variables. This paper looks at the limit distributions of sums of log-normal variables when the second parameter of the log-normals tends to zero or to infinity; in financial terms, this is equivalent to letting the volatility, or maturity, tend either to zero or to infinity. The limits obtained are either normal or log-normal, depending on the normalization chosen; the same applies to the reciprocal of the sums of log-normals. This justifies the log-normal approximation, much used in practice, and also gives an asymptotically exact distribution for averages of log-normals with a relatively small volatility; it has been noted that all the analytical pricing formulae for Asian options perform poorly for small volatilities. Asymptotic formulae are also found for the moments of the sums of log-normals. Results are given for both discrete and continuous averages. More explicit results are obtained in the case of the integral of geometric Brownian motion.

Asymptotic Formulae for Pairs of Diagonal Cubic Equations

2011 ◽  
Vol 63 (1) ◽  
pp. 38-54 ◽  
Author(s):  
Jörg Brüdern ◽  
Trevor D. Wooley
Keyword(s):  
Asymptotic Formula ◽  
Linear Combination ◽  
Asymptotic Formulae ◽  
Local Densities ◽  
Integral Solutions

Abstract We investigate the number of integral solutions possessed by a pair of diagonal cubic equations in a large box. Provided that the number of variables in the system is at least fourteen, and in addition the number of variables in any non-trivial linear combination of the underlying forms is at least eight, we obtain an asymptotic formula for the number of integral solutions consistent with the product of local densities associated with the system.

High frequency asymptotic analysis of a string with rapidly oscillating density

2000 ◽  
Vol 11 (6) ◽  
pp. 595-622 ◽  
Author(s):  
C. CASTRO ◽  
E. ZUAZUA
Keyword(s):  
Asymptotic Expansion ◽  
Eigenvalue Problem ◽  
Asymptotic Analysis ◽  
Explicit Formula ◽  
High Frequency ◽  
Critical Size ◽  
The Other ◽  
Full Description ◽  
Eigenvalues And Eigenfunctions ◽  
Asymptotic Formulae

We consider the eigenvalue problem associated with the vibrations of a string with rapidly oscillating periodic density. In a previous paper we stated asymptotic formulae for the eigenvalues and eigenfunctions when the size of the microstructure ε is shorter than the wavelength of the eigenfunctions 1/√λε. On the other hand, it has been observed that when the size of the microstructure is of the order of the wavelength of the eigenfunctions (ε ∼ 1/√λε) singular phenomena may occur. In this paper we study the behaviour of the eigenvalues and eigenfunctions when 1/√λε is larger than the critical size ε. We use the WKB approximation which allows us to find an explicit formula for eigenvalues and eigenfunctions with respect to ε. Our analysis provides all order correction formulae for the limit eigenvalues and eigenfunctions above the critical size. Each term of the asymptotic expansion requires one more derivative of the density. Thus, a full description requires the density to be C∞ smooth.

The log-normal approximation in financial and other computations

2004 ◽  
Vol 36 (3) ◽  
pp. 747-773 ◽  
Author(s):  
Daniel Dufresne
Keyword(s):  
Brownian Motion ◽  
Explicit Formula ◽  
Normal Approximation ◽  
Random Variables ◽  
Exact Distribution ◽  
Financial Mathematics ◽  
Asian Options ◽  
Basket Options ◽  
Asymptotic Formulae ◽  
Log Normal

Sums of log-normals frequently appear in a variety of situations, including engineering and financial mathematics. In particular, the pricing of Asian or basket options is directly related to finding the distributions of such sums. There is no general explicit formula for the distribution of sums of log-normal random variables. This paper looks at the limit distributions of sums of log-normal variables when the second parameter of the log-normals tends to zero or to infinity; in financial terms, this is equivalent to letting the volatility, or maturity, tend either to zero or to infinity. The limits obtained are either normal or log-normal, depending on the normalization chosen; the same applies to the reciprocal of the sums of log-normals. This justifies the log-normal approximation, much used in practice, and also gives an asymptotically exact distribution for averages of log-normals with a relatively small volatility; it has been noted that all the analytical pricing formulae for Asian options perform poorly for small volatilities. Asymptotic formulae are also found for the moments of the sums of log-normals. Results are given for both discrete and continuous averages. More explicit results are obtained in the case of the integral of geometric Brownian motion.

scholarly journals On trees, tanglegrams, and tangled chains

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Sara Billey ◽  
Matjaz Konvalinka ◽  
Frderick Matsen IV
Keyword(s):  
Asymptotic Formula ◽  
Computer Science ◽  
Explicit Formula ◽  
Binary Trees ◽  
Biological Research ◽  
Open Problems ◽  
International Audience ◽  
Enumeration Formula ◽  
New Formula ◽  
Simple Asymptotic Formula

International audience Tanglegrams are a class of graphs arising in computer science and in biological research on cospeciation and coevolution. They are formed by identifying the leaves of two rooted binary trees. The embedding of the trees in the plane is irrelevant for this application. We give an explicit formula to count the number of distinct binary rooted tanglegrams with n matched leaves, along with a simple asymptotic formula and an algorithm for choosing a tanglegram uniformly at random. The enumeration formula is then extended to count the number of tangled chains of binary trees of any length. This work gives a new formula for the number of binary trees with n leaves. Several open problems and conjectures are included along with pointers to several followup articles that have already appeared.

scholarly journals SOME ESTIMATES OF SPECIAL CLASSES OF INTEGRALS

2000 ◽  
Vol 5 (1) ◽  
pp. 127-132
Author(s):  
T. I. Malyutina
Keyword(s):  
Asymptotic Formula ◽  
Asymptotic Expansions ◽  
Entire Functions ◽  
Regular Growth ◽  
Asymptotic Formulas ◽  
Arbitrary Parameter ◽  
Subharmonic Functions ◽  
Completely Regular ◽  
Asymptotic Formulae ◽  
New Variant

We study the integrals fb a f(t) exp(i| ln rt|σ) dt and obtain asymptotic formula for these functions of non‐regular growth. This is a peculiar kind of the theory asymptotic expansions. In particular, we get asymptotic formulae for different entire functions of non‐regular growth. Asymptotic formulas for Levin‐Pfluger entire functions of completely regular growth are well‐known [1]. Our formulas allow to find limiting Azarin's [2] sets for some subharmonic functions. The kernel exp(i| ln rt|σ) contains arbitrary parameter σ > 0. The integrals for σ ∈(0, 1), σ = 1, σ > 1 essentially differ. Our arguments can apply to more general kernels. We give a new variant of the classic lemma of Riemann and Lebesgue from the theory of the transformation of Fourier.

scholarly journals Farey Sequences for Thin Groups

2021 ◽  
Author(s):  
Christopher Lutsko
Keyword(s):  
Explicit Formula ◽  
Diophantine Approximation ◽  
Continued Fraction ◽  
Rational Numbers ◽  
Gauss Map ◽  
Gauss Measure ◽  
Farey Sequence ◽  
Farey Sequences ◽  
Partial Quotients

Abstract The Farey sequence is the set of rational numbers with bounded denominator. We introduce the concept of a generalized Farey sequence. While these sequences arise naturally in the study of discrete and thin subgroups, they can be used to study interesting number theoretic sequences—for example rationals whose continued fraction partial quotients are subject to congruence conditions. We show that these sequences equidistribute and the gap distribution converges and answer an associated problem in Diophantine approximation. Moreover, for one example, we derive an explicit formula for the gap distribution. For this example, we construct the analogue of the Gauss measure, which is ergodic for the Gauss map. This allows us to prove a theorem about the associated Gauss–Kuzmin statistics.

On Waring's problem for cubes and biquadrates. II

1988 ◽  
Vol 104 (2) ◽  
pp. 199-206 ◽  
Author(s):  
Jörg Brüdern
Keyword(s):  
Lower Bound ◽  
Asymptotic Formula ◽  
Riemann Hypothesis ◽  
Waring’S Problem ◽  
Waring's Problem ◽  
Asymptotic Formulae ◽  
Sixth Moment ◽  
Almost All

In discussing the consequences of a conditional estimate for the sixth moment of cubic Weyl sums, Hooley [4] established asymptotic formulae for the number ν(n) of representations of n as the sum of a square and five cubes, and for ν(n), defined similarly with six cubes and two biquadrates. The condition here is the truth of the Riemann Hypothesis for a certain Hasse–Weil L-function. Recently Vaughan [8] has shown unconditionally , a lower bound of the size suggested by the conditional asymptotic formula. In the corresponding problem for ν(n) the author [1] was able to deduce ν(n) > 0, as a by-product of the result that almost all numbers can be expressed as the sum of three cubes and one biquadrate. As promised in the first paper of this series we return to the problem of bounding ν(n) from below.

scholarly journals On two sums related to the Lehmer problem over short intervals

2021 ◽  
Vol 6 (11) ◽  
pp. 11723-11732
Author(s):  
Yanbo Song ◽  
Keyword(s):  
Asymptotic Formula ◽  
Error Term ◽  
Short Intervals ◽  
Asymptotic Formulae ◽  
Lehmer Problem ◽  
Lehmer Numbers

<abstract><p>In this article, we study sums related to the Lehmer problem over short intervals, and give two asymptotic formulae for them. The original Lehmer problem is to count the numbers coprime to a prime such that the number and the its number theoretical inverse are in different parities in some intervals. The numbers which satisfy these conditions are called Lehmer numbers. It prompts a series of investigations, such as the investigation of the error term in the asymptotic formula. Many scholars investigate the generalized Lehmer problems and get a lot of results. We follow the trend of these investigations and generalize the Lehmer problem.</p></abstract>

Asymptotic formulae in the theory of partitions

1954 ◽  
Vol 50 (2) ◽  
pp. 225-241 ◽  
Author(s):  
C. B. Haselgrove ◽  
H. N. V. Temperley
Keyword(s):  
Positive Integer ◽  
Asymptotic Formula ◽  
Classical Method ◽  
Contour Integration ◽  
Large Positive Integer ◽  
Asymptotic Formulae ◽  
Positive Integers

It is the object of this paper to obtain an asymptotic formula for the number of partitions pm(n) of a large positive integer n into m parts λr, where the number m becomes large with n and the numbers λ1, λ2,… form a sequence of positive integers. The formula is proved by using the classical method of contour integration due to Hardy, Ramanujan and Littlewood. It will be necessary to assume certain conditions on the sequence λr, but these conditions are satisfied in most of the cases of interest. In particular, we shall be able to prove the asymptotic formula in the cases of partitions into positive integers, primes and kth powers for any positive integer k.

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