Odd Positive Integer Research Articles

Acoustic radiation force of high-order Bessel beam standing wave tweezers on a rigid sphere

Background and objective Particle manipulation using the acoustic radiation force of Bessel beams is an active field of research. In a previous investigation, [F.G. Mitri, Acoustic radiation force on a sphere in standing and quasi-standing zero-order Bessel beam tweezers, Annals of Physics 323 (2008) 1604–1620] an expression for the radiation force of a zero-order Bessel beam standing wave experienced by a sphere was derived. The present work extends the analysis of the radiation force to the case of a high-order Bessel beam (HOBB) of positive order m having an angular dependence on the phase ϕ. Method The derivation for the general expression of the force is based on the formulation for the total acoustic scattering field of a HOBB by a sphere [F.G. Mitri, Acoustic scattering of a high-order Bessel beam by an elastic sphere, Annals of Physics 323 (2008) 2840–2850; F.G. Mitri, Equivalence of expressions for the acoustic scattering of a progressive high order Bessel beam by an elastic sphere, IEEE Transactions on Ultrasonics Ferroelectrics and Frequency Control 56 (2009) 1100–1103] to derive the general expression for the radiation force function Y J m , st ( ka , β , m ) , which is the radiation force per unit characteristic energy density and unit cross-sectional surface. The radiation force function is expressed as a generalized partial wave series involving the half-cone angle β of the wave-number components and the order m of the HOBB. Results Numerical results for the radiation force function of a first and a second-order Bessel beam standing wave incident upon a rigid sphere immersed in non-viscous water are computed. The rigid sphere calculations for Y J m , st ( ka , β , m ) show that the force is generally directed to a pressure node when m is a positive even integer number (i.e. Y J m , st ( ka , β , m ) > 0 ), whereas the force is generally directed toward a pressure antinode when m is a positive odd integer number (i.e. Y J m , st ( ka , β , m ) < 0 ). Conclusion An expression is derived for the radiation force on a rigid sphere placed along the axis of an ideal non-diffracting HOBB of acoustic standing (or stationary) waves propagating in an ideal fluid. The formulation includes results of a previous work done for a zero-order Bessel beam standing wave ( m = 0). The proposed theory is of particular interest essentially due to its inherent value as a canonical problem in particle manipulation using the acoustic radiation force of a HOBB standing wave on a sphere. It may also serve as the benchmark for comparison to other solutions obtained by strictly numerical or asymptotic approaches.

Nonoscillation for second order sublinear dynamic equations on time scales

Consider the Emden–Fowler sublinear dynamic equation (0.1) x Δ Δ ( t ) + p ( t ) f ( x ( σ ( t ) ) ) = 0 , where p ∈ C ( T , R ) , T is a time scale, f ( x ) = ∑ i = 1 m a i x β i , where a i > 0 , 0 < β i < 1 , with β i the quotient of odd positive integers, 1 ≤ i ≤ m . When m = 1 , and T = [ a , ∞ ) ⊂ R , (0.1) is the usual sublinear Emden–Fowler equation which has attracted the attention of many researchers. In this paper, we allow the coefficient function p ( t ) to be negative for arbitrarily large values of t . We extend a nonoscillation result of Wong for the second order sublinear Emden–Fowler equation in the continuous case to the dynamic equation (0.1). As applications, we show that the sublinear difference equation Δ 2 x ( n ) + b ( − 1 ) n n − c x α ( n + 1 ) = 0 , 0 < α < 1 , has a nonoscillatory solution, for b > 0 , c > α , and the sublinear q-difference equation x Δ Δ ( t ) + b ( − 1 ) n t − c x α ( q t ) = 0 , 0 < α < 1 , has a nonoscillatory solution, for t = q n ∈ T = q 0 N , q > 1 , b > 0 , c > 1 + α .

Some convexity properties of Dirichlet series with positive terms

AbstractSome basic results for Dirichlet series ψ with positive terms via log‐convexity properties are pointed out. Applications for Zeta, Lambda and Eta functions are considered. The concavity of the function 1/ψ is explored and, as a main result, it is proved that the function 1/ζ is concave on (ζ–1(e), ∞). As a consequence of this fundamental result it is noted that Zeta at the odd positive integers is bounded above by the harmonic mean of its immediate even Zeta values which are known explicitly (© 2009 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Enumeration of Unrooted Odd-Valent Regular Planar Maps

We derive closed formulae for the numbers of rooted maps with a fixed number of vertices of the same odd degree except for the root vertex and one other exceptional vertex of degree 1. The same applies to the generating functions for these numbers. Similar results, but without the vertex of degree 1, were obtained by the first author and Rahman. We also show, by manipulating a recursion of Bouttier, Di Francesco and Guitter, that there are closed formulae when the exceptional vertex has arbitrary degree. We combine these formulae with results of the second author to count unrooted regular maps of odd degree. In this way we obtain, for each even n, a closed formula for the function f n whose value at odd positive integers r is the number of unrooted maps (up to orientation-preserving homeomorphisms) with n vertices and degree r. The formula for f n becomes more cumbersome as n increases, but for n > 2 each has a bounded number of terms independent of r.

Counting squarefree discriminants of trinomials under abc

For an odd positive integer $n\ge 5$, assuming the truth of the $abc$ conjecture, we show that for a positive proportion of pairs $(a,b)$ of integers the trinomials of the form $t^n+at+b \ (a,b\in \mathbb Z)$ are irreducible and their discriminants are squarefree.

On the constructions of constant-composition codes from perfect nonlinear functions

A new construction of constant-composition codes based on all known perfect nonlinear functions from F q m to itself is presented, which provides a kind of unified constructions of constant-composition codes based on all known perfect nonlinear functions from F q m to itself. It is proved that the new constant-composition codes are optimal with respect to the Luo-Fu-Vinck-Chen bound, when m is an odd positive integer greater than 1. Finally, we point out that two constructions of constant-composition codes, proposed by Ding Cunsheng et al. in 2005, are equivalent to two special types of the new constant-composition codes.

Gracefulness of union of cycle with parallel chords and complete bipartite graphs or paths

A graph denoted Hn , is called a cycle with parallel chords if Hn is obtained from the cycle Cn : u 1, u 2,…, un (n ≥ 6) by adding the chords u 2 un , u 3 u n−1 ,…, uαuβ , where and if n is odd or if n is even (see Figure 1). In this paper we prove the following results. (1) For all p, q ≥ 1 and for all odd positive integers n ≥ 5, Hn ∪ Kp,q is graceful, where Kp,q is a complete bipartite graph. (2) For n ≥ 6, Hn ∪ Sm is graceful, where m ≥ 1 when n is odd and while when n is even, or . Here Sm denotes a star of size m. (3) For n ≥ 6, Hn ∪ Pm is graceful, where m = n or n − 2 depends on n = 1 or 3(mod 4) or m = n − 1 or n − 3 depends on n = 0 or 2 (mod 4). Here Pm denotes a path of order m.

Selmer groups and Tate–Shafarevich groups for the congruent number problem

We study the distribution of the sizes of the Selmer groups arising from the three 2-isogenies and their dual 2-isogenies for the elliptic curve E_n: y^2 = x^3 − n^2x . We show that three of them are almost always trivial, while the 2-rank of the other three follows a Gaussian distribution. It implies three almost always trivial Tate–Shafarevich groups and three large Tate–Shafarevich groups. When combined with a result obtained by Heath-Brown, we show that the mean value of the 2-rank of the large Tate–Shafarevich groups for square-free positive odd integers n ≤ X is \frac 1 2 \log \log X + O(1) , as X → ∞ .

Multipartite Ramsey numbers for odd cycles

AbstractIn this paper we study multipartite Ramsey numbers for odd cycles. We formulate the following conjecture: Let n≥5 be an arbitrary positive odd integer; then, in any two‐coloring of the edges of the complete 5‐partite graph K((n−1)/2, (n−1)/2, (n−1)/2, (n−1)/2, 1) there is a monochromatic Cn, a cycle of length n. This roughly says that the Ramsey number for Cn (i.e. 2n−1 ) will not change (somewhat surprisingly) if four large “holes” are allowed. Note that this would be best possible as the statement is not true if we delete from K2n−1 the edges within a set of size (n+ 1)/2. We prove an approximate version of the above conjecture. © 2009 Wiley Periodicals, Inc. J Graph Theory 61:12‐21, 2009

Some oscillation results for second order nonlinear dynamic equations of neutral type

In this paper, some oscillation results for the second order nonlinear neutral delay dynamic equation ( r ( t ) ( ( y ( t ) + p ( t ) y ( t − τ ) ) Δ ) γ ) Δ + q ( t ) | y ( t − δ ) | γ sgn y ( t − δ ) = 0 are obtained on a time scale T , under the assumption ∫ t 0 ∞ ( 1 r ( t ) ) 1 γ Δ t = ∞ , where γ > 0 is a quotient of odd positive integers.

Trace formula and trace identity of twisted Hecke operators on the spaces of cusp forms of weight $k+1/2$ and level $32M$

Let M be an odd positive integer, an even quadratic character defined modulo 32M, and ψ a quadratic primitive character of conductor divisible by 8. Then, we can define twisted Hecke operators R ψ T(n 2 ) on the space of cusp forms of weight k + 1/2, level 32M, and character Χ, under certain conditions on the conductors of Χ and ψ. This is a specific feature of the case of half-integral weight. We give explicit trace formulas of the twisted Hecke operators and their trace identities.

Covers of the integers with odd moduli and their applications to the forms $x^{m}-2^{n}$ and $x^{2}-F_{3n}/2$

In this paper we construct a cover $\{a_{s}(\operatorname {mod} \ n_{s})\}_{s=1}^{k}$ of $\mathbb {Z}$ with odd moduli such that there are distinct primes $p_{1},\ldots ,p_{k}$ dividing $2^{n_{1}}-1,\ldots ,2^{n_{k}}-1$ respectively. Using this cover we show that for any positive integer $m$ divisible by none of $3, 5, 7, 11, 13$ there exists an infinite arithmetic progression of positive odd integers the $m$th powers of whose terms are never of the form $2^{n}\pm p^{a}$ with $a,n\in \{0,1,2,\ldots \}$ and $p$ a prime. We also construct another cover of $\mathbb {Z}$ with odd moduli and use it to prove that $x^{2}-F_{3n}/2$ has at least two distinct prime factors whenever $n\in \{0,1,2,\ldots \}$ and $x\equiv a (\operatorname {mod} M)$, where $\{F_{i}\}_{i\geqslant 0}$ is the Fibonacci sequence, and $a$ and $M$ are suitable positive integers having 80 decimal digits.

On the oscillation of certain second order nonlinear dynamic equations

We establish some new oscillation criteria for solutions to the second order nonlinear dynamic equation ( a ( x Δ ) α ) Δ ( t ) + q ( t ) x β ( t ) = 0 on an arbitrary time scale T , where α and β are ratios of positive odd integers, a and q are positive rd-continuous functions on T . The results are obtained when ∫ ∞ a − 1 / α ( s ) Δ s < ∞ or ∫ ∞ a − 1 / α ( s ) Δ s = ∞ . These criteria unify and extend known results for the corresponding half-linear and nonlinear differential and difference equations. Some of our results are new even in the continuous and discrete cases.

Oscillation for Second-Order Nonlinear Delay Dynamic Equations on Time Scales

By means of Riccati transformation technique, we establish some new oscillation criteria for the second-order nonlinear delay dynamic equations on a time scale ; here is a quotient of odd positive integers with and real-valued positive rd-continuous functions defined on . Our results not only extend some results established by Hassan in 2008 but also unify the oscillation of the second-order nonlinear delay differential equation and the second-order nonlinear delay difference equation.

On newforms for Kohnen plus spaces

In this article, we investigate the plus space of level N, where 4−1 N is a square-free (not necessarily odd) integer. This is a generalization of Kohnen’s work. We define a Hecke isomorphism \({\wp_k}\) of M k+1/2(4M) onto \({M_{k+1/2}^+(8M)}\) for any odd positive integer M. The methods of the proof of the newform theory are this isomorphism, Waldspurger’s theorem, and the dimension identity.

On the representations of integers by the sextenary quadratic form [formula omitted] and 7-cores

In this paper we derive an explicit formula for the number of representations of an integer by the sextenary form x 2 + y 2 + z 2 + 7 s 2 + 7 t 2 + 7 u 2 . We establish the following intriguing inequalities 2 ω ( n + 2 ) ⩾ a 7 ( n ) ⩾ ω ( n + 2 ) for n ≠ 0 , 2 , 6 , 16 . Here a 7 ( n ) is the number of partitions of n that are 7-cores and ω ( n ) is the number of representations of n by the sextenary form ( x 2 + y 2 + z 2 + 7 s 2 + 7 t 2 + 7 u 2 ) / 8 with x, y, z, s, t and u being odd positive integers.

On Generalized Alternating Galileo Sequences

During the course of studying freely falling bodies, Galileo observed a wonderful property that is unique to the sequence of positive odd integers. In this paper, we present and explore a generalized variation of this classical result. We exhibit some properties of our variation and leave the reader with some open questions for possible further exploration.

Covering arrays of strength 3 and 4 from holey difference matrices

A covering array CA(N; t, k, v) is an N × k array with entries from a set X of v symbols such that every N × t sub-array contains all t-tuples over X at least once, where t is the strength of the array. The minimum size N for which a CA(N; t, k, v) exists is called the covering array number and denoted by CAN(t, k, v). Covering arrays are used in experiments to screen for interactions among t-subsets of k components. One of the main problems on covering arrays is to construct a CA(N; t, k, v) for given parameters (t, k, v) so that N is as small as possible. In this paper, we present some constructions of covering arrays of strengths 3 and 4 via holey difference matrices with prescribed properties. As a consequence, some of known bounds on covering array number are improved. In particular, it is proved that (1) CAN(3, 5, 2v) ? 2v 2(4v + 1) for any odd positive integer v with gcd(v, 9) ? 3; (2) CAN(3, 6, 6p) ? 216p 3 + 42p 2 for any prime p > 5; and (3) CAN(4, 6, 2p) ? 16p 4 + 5p 3 for any prime p ? 1 (mod 4) greater than 5.

New comparison and oscillation theorems for second-order half-linear dynamic equations on time scales

Let T be a time scale (i.e., a closed nonempty subset of R ) with sup T = + ∞ . Consider the second-order half-linear dynamic equation ( r ( t ) ( x Δ ( t ) ) α ) Δ + p ( t ) x α ( σ ( t ) ) = 0 , where r ( t ) > 0 , p ( t ) are continuous, ∫ t 0 ∞ ( r ( t ) ) − 1 α Δ t = ∞ , α is a quotient of odd positive integers. In particular, no explicit sign assumptions are made with respect to the coefficient p ( t ) . We give conditions under which every positive solution of the equations is strictly increasing. For α = 1 , T = R , the result improves the original theorem [see: [Lynn Erbe, Oscillation theorems for second-order linear differential equation, Pacific J. Math. 35 (2) (1970) 337–343]]. As applications, we get two comparison theorems and an oscillation theorem for half-linear dynamic equations which improve and extend earlier results. Some examples are given to illustrate our theorems.

OSCILLATION THEOREM FOR SECOND-ORDER DIFFERENCE EQUATIONS

Sufficient and necessary conditions are established for the second-order difference equations \[ \Delta (r_{n - 1} \Delta x_{n - 1} ) + p_n x^\gamma _n = 0,n = 1,2, \ldots \] where $\gamma $ is the quotient of odd positive integers. Our results extend the well known oscillation theorem which was proved in [1,JMAA,91:9-29,1983], and answer an open problem in [2] when $r_n=1,\gamma=1$.