ASYMPTOTICS OF THE WEIGHTED DELANNOY NUMBERS

The weighted Delannoy numbers give a weighted count of lattice paths starting at the origin and using only minimal east, north and northeast steps. Full asymptotic expansions exist for various diagonals of the weighted Delannoy numbers. In the particular case of the central weighted Delannoy numbers, certain weights give rise to asymptotic coefficients that lie in a number field. In this paper we apply a generalization of a method of Stoll and Haible to obtain divisibility properties for the asymptotic coefficients in this case. We also provide a similar result for a special case of the diagonal with slope 2.

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A Gap Principle for Subvarieties with Finitely Many Periodic Points

Canadian Mathematical Bulletin ◽

10.4153/s0008439519000481 ◽

2020 ◽

Vol 63 (2) ◽

pp. 382-392

Author(s):

Keping Huang

Keyword(s):

Number Field ◽

Algebraic Variety ◽

Initial Point ◽

Periodic Points ◽

Gap Principle ◽

Special Case

AbstractLet $f:X\rightarrow X$ be a quasi-finite endomorphism of an algebraic variety $X$ defined over a number field $K$ and fix an initial point $a\in X$. We consider a special case of the Dynamical Mordell–Lang Conjecture, where the subvariety $V$ contains only finitely many periodic points and does not contain any positive-dimensional periodic subvariety. We show that the set $\{n\in \mathbb{Z}_{{\geqslant}0}\mid f^{n}(a)\in V\}$ satisfies a strong gap principle.

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Asymptotic solutions of the Orr—Sommerfeld equation

Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences ◽

10.1098/rsta.1975.0102 ◽

1975 ◽

Vol 280 (1295) ◽

pp. 271-316

Keyword(s):

Differential Equation ◽

Asymptotic Expansions ◽

Asymptotic Properties ◽

Matched Asymptotic Expansions ◽

Asymptotic Solutions ◽

Asymptotic Approximations ◽

General Kind ◽

Work Done ◽

Sommerfeld Equation ◽

Special Case

A rigorous justification is given of work done by Eagles (1969), in which he applied the method of matched asymptotic expansions to the Orr-Sommerfeld equation to obtain formal uniform asymptotic approximations to a certain pair of solutions. (Somewhat more polished formal expansions of the same general kind were subsequently obtained by Reid (1972).) First, a study is made of the asymptotic properties of solutions of a certain differential equation which admits the Orr—Sommerfeld equation as a special case. Previous work on this differential equation by Lin & Rabenstein ( i960, 1969) is extended to develop a theory suited to our main purpose: to prove the validity of Eagles’s approximations. It is then shown how this theory can be used to prove the existence of actual solutions of the Orr—Sommerfeld equation approximated by these formal expansions. In addition, it is verified that these solutions have the properties assumed by Eagles (1969).

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Higher-Order Approximations to the Null Distributions of Test Statistics for Nonlinear Restrictions on Regression Coefficients

Econometric Theory ◽

10.1017/s0266466600012068 ◽

1988 ◽

Vol 4 (2) ◽

pp. 275-299

Author(s):

Kimio Morimune

Keyword(s):

Asymptotic Expansions ◽

Simultaneous Equations ◽

Regression Coefficients ◽

Test Statistics ◽

Lagrange Multiplier Test ◽

Null Distributions ◽

Asymptotic Size ◽

Simultaneous Equations Models ◽

Real Size ◽

Special Case

Asymptotic expansions of the distributions of likelihood ratio and Lagrange multiplier test statistics for nonlinear restrictions on regression coefficients are derived under the null hypothesis. Nonlinear restrictions include, as a special case, the identifiability restrictions in the simultaneous equations models. Our analyses of simultaneous equations deal not only with single equations but also subsystems and complete systems. The asymptotic expansions we derive are informative about deviations of the real size of test from the nominal asymptotic size.

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Derivation of Green-type, transitional and uniform asymptotic expansions from differential equations III. The confluent hypergeometric function U ( a , c , z ) for large | c |

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1965.0078 ◽

1965 ◽

Vol 284 (1399) ◽

pp. 531-539 ◽

Cited By ~ 4

Keyword(s):

Asymptotic Expansions ◽

Confluent Hypergeometric Function ◽

Parabolic Cylinder ◽

Exponential Integral ◽

Parabolic Cylinder Function ◽

Uniform Expansions ◽

Kummer's Equation ◽

Uniform Expansion ◽

Special Case ◽

Uniform Asymptotic Expansions

The methods introduced by Jorna (1964 a , b ) are applied to Kummer’s equation, and Green-type, transitional and uniform expansions derived for solutions of the type denoted in Slater (1960) by U ( a , c , z ) which are valid for large | c |. The subsidiary function in the uniform expansion is essentially a parabolic cylinder function of order ½ a . The general exponential integral is studied as a special case. Here the uniform expansion involves as subsidiary function the extensively tabulated error integral.

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GENERALIZING DELANNOY NUMBERS VIA COUNTING WEIGHTED LATTICE PATHS

Integers ◽

10.1515/9783110298161.764 ◽

2014 ◽

Author(s):

M. Dziemianczuk

Keyword(s):

Lattice Paths ◽

Delannoy Numbers ◽

Weighted Lattice Paths

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K-Theory of Locally Compact Modules over Rings of Integers

International Mathematics Research Notices ◽

10.1093/imrn/rny083 ◽

2018 ◽

Vol 2020 (6) ◽

pp. 1748-1793 ◽

Cited By ~ 1

Author(s):

Oliver Braunling

Keyword(s):

Cavitating Flow Near the Apex of a Plane Delta Wing

Journal of Ship Research ◽

10.5957/jsr.1984.28.1.65 ◽

1984 ◽

Vol 28 (01) ◽

pp. 65-69

Author(s):

Teruhiko Kida ◽

Takanori Take

Keyword(s):

Approximate Method ◽

Asymptotic Expansions ◽

Velocity Potential ◽

Delta Wing ◽

Apex Angle ◽

Cavitating Flow ◽

Collocation Methods ◽

Physical Insight ◽

Second Eigenvalue ◽

Special Case

The problem of cavitating flow past the apex of an infinite sector, that is, "the delta-wing problem," is treated. This problem is just an inner problem in the sense of the method of matched asymptotic expansions, and the present study might be very important in leading to large increases in the accuracy of collocation methods for cavitating wings with pointed apexes.' A solution of Laplace's equation for this problem, that is, the velocity potential, involves rv where r is the distance from the apex and v is a parameter depending on the angle of the sector. The numerical values of v for various apex angles in the lowest approximation are obtained by Taylor's approximate method. In the special case where the apex angle is x, the first and second eigenvalues are obtained theoretically as m + 1/4 and m + 3/4, respectively, where m is an integer. From a physical insight, it is seen that the second eigenvalue is very important in the cavitating flow.

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Delannoy Numbers and Preferential Arrangements

Mathematics ◽

10.3390/math7030238 ◽

2019 ◽

Vol 7 (3) ◽

pp. 238

Author(s):

Kwang-Wu Chen

Keyword(s):

Lattice Paths ◽

Symmetric Identity ◽

Delannoy Numbers ◽

Combinatorial Methods

A preferential arrangement on [ [ n ] ] = { 1 , 2 , … , n } is a ranking of the elements of [ [ n ] ] where ties are allowed. The number of preferential arrangements on [ [ n ] ] is denoted by r n . The Delannoy number D ( m , n ) is the number of lattice paths from ( 0 , 0 ) to ( m , n ) in which only east ( 1 , 0 ) , north ( 0 , 1 ) , and northeast ( 1 , 1 ) steps are allowed. We establish a symmetric identity among the numbers r n and D ( p , q ) by means of algebraic and combinatorial methods.

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On integral transforms whose kernels are solutions of singular Sturm–Liouville problems

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500014621 ◽

1988 ◽

Vol 108 (3-4) ◽

pp. 201-228 ◽

Cited By ~ 3

Author(s):

Ahmed I. Zayed

Keyword(s):

Fourier Transform ◽

Asymptotic Expansions ◽

Spectral Measure ◽

Liouville Problem ◽

Integral Transforms ◽

Tempered Distributions ◽

Sturm Liouville ◽

The Fourier Transform ◽

Image Position ◽

Special Case

SynopsisIn this paper we investigate integral transforms of type , where φ(x, s) is the solution of the singular Sturm–Liouville problem: y″ + (s2 – q(x))y = 0, 0≦x <∞ with y(0) cos α + y′(0)sin α = 0, y(x) is bounded at ∞, and dp is the spectral measure. If F(s) = sk for some k = 0, 1, 2, …, then f(x) may not exist since, in general, φ(x, s) is not even in . One aim of this paper is to investigate the Abel summability of these integrals. In the special case where q(x) = 0 and α = π/2, then φ(x, s) = cos sx and dp = ds, while if α = 0, then φ(x, s) = −sin sx/s and dp = s2ds. It is known thatwhere the values of these integrals are interpreted as the Abel limits of these integrals or as the Fourier transform of some tempered distributions. Another aim of this paper is to derive the analogue of these results for the general kernel φ(x, s), and then apply that to the theory of asymptotic expansions.

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On the least prime ideal and Siegel zeros

International Journal of Number Theory ◽

10.1142/s1793042116501335 ◽

2016 ◽

Vol 12 (08) ◽

pp. 2201-2229 ◽

Cited By ~ 1

Author(s):

Asif Zaman

Keyword(s):

Prime Ideal ◽

Number Field ◽

Quadratic Forms ◽

Class Group ◽

Binary Quadratic Forms ◽

Integral Ideal ◽

Siegel Zero ◽

Siegel Zeros ◽

Special Case ◽

Exceptional Character

Let [Formula: see text] be a number field, [Formula: see text] be an integral ideal, and [Formula: see text] be the associated narrow ray class group. Suppose [Formula: see text] possesses a real exceptional character [Formula: see text], possibly principal, with a Siegel zero [Formula: see text]. For [Formula: see text] satisfying [Formula: see text] [Formula: see text], we establish an effective [Formula: see text]-uniform Linnik-type bound with explicit exponents for the least norm of a prime ideal [Formula: see text]. A special case of this result is a bound for the least rational prime represented by certain binary quadratic forms.

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