Odd Positive Integer Research Articles

OSCILLATION THEOREMS FOR CERTAIN SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS

In this paper, we consider the oscillation of the following certain second order nonlinear differential equations >, where and are ratios of positive odd integers. New oscillation theorems are established, which are based on a class of new functions defined in the sequel. Also, we establish some interval oscillation criteria for this equation.

Generalized fast mixed-radix algorithm for the computation of forward and inverse MDCTs

This paper presents a generalized mixed-radix decimation-in-time (DIT) fast algorithm for computing the modified discrete cosine transform (MDCT) of the composite lengths N=2× q m , m≥2, where q is an odd positive integer. The proposed algorithm not only has the merits of parallelism and numerical stability, but also needs less multiplications than that of type-IV discrete cosine transform (DCT-IV) and type-II discrete cosine transform (DCT-II) based MDCT algorithms due to the optimized efficient length-( N/ q) modules. The computation of MDCT for composite lengths N= q m ×2 n , m≥2, n≥2, can then be realized by combining the proposed algorithm with fast radix-2 MDCT algorithm developed for N=2 n . The combined algorithm can be used for the computation of length-12/36 MDCT used in MPEG-1/-2 layer III audio coding as well as the recently established wideband speech and audio coding standards such as G.729.1, where length-640 MDCT is used. The realization of the inverse MDCT (IMDCT) can be obtained by transposing the signal flow graph of the MDCT.

The convergence classes of Collatz function

The Collatz conjecture, also known as the 3 x + 1 conjecture, can be stated in terms of the reduced Collatz function R ( x ) = ( 3 x + 1 ) / 2 h (where 2 h is the larger power of 2 that divides 3 x + 1 ). The conjecture is: Starting from any odd positive integer and repeating R ( x ) we eventually get to 1. G k , the k -th convergence class, is the set of odd positive integers x such that R k ( x ) = 1 . In this paper an infinite sequence of binary strings s h of length 2 ⋅ 3 h − 1 (the seeds) are defined and it is shown that the binary representation of all x ∈ G k is the concatenation of k periodic strings whose periods are s k , … , s 1 . More precisely x = s k , d k , 1 [ n 1 ) … s 1 , d k , k [ n k ) where s k , d k , i [ n i ) is the substring of length n i that starts in position d k , i in a sufficiently long repetition of the seed s i . Finally, starting positions d k , i and lengths n i for which s k , d k , 1 [ n 1 ) … s 1 , d k , k [ n k ) ∈ G k are defined, thus giving a complete characterization of classes G k .

Closed polylines inscribed in a closed space curve

Let n be an odd positive integer. It is proved that if n + 2 is a power of a prime number and C is a regular closed non-self-intersecting curve in $ {\mathbb{R}^n} $ ,then C contains vertices of an equilateral (n + 2)-link polyline with n + 1 vertices lying in a hyperplane. It is also proved that if C is a rectifiable closed curve in $ {\mathbb{R}^n} $ ,then C contains n + 1 points that lie in a hyperplane and divide C into parts one of which is twice as long as each of the others. Bibliography: 6 titles.

A particular Hamiltonian cycle on middle levels in the De Bruijn digraph

In this paper, it is proved that for every positive odd integer n , there exists a Hamiltonian cycle in the De Bruijn digraph B ( 2 , n ) on middle levels.

Transcendental values of certain Eichler integrals

We study the transcendence of certain Eichler integrals associated to Eisenstein series and more generally to modular forms using functional identities due to Ramanujan, Grosswald, Weil et al. The special values of such integrals at algebraic points in the upper half-plane are linked to Riemann zeta values at odd positive integers.

On The Bounded Oscillation of Certain Second-order Nonlinear Neutral Delay Dynamic Equations with Oscillating Coefficients

We investigate the bounded oscillation of the second-order nonlinear neutral delay dynamic equation with oscillating coefficients \[\Big(r(t)\Big|\big[x(t)+p(t)x(\tau(t))\big]^\Delta\Big|^{\alpha-1}\big[x(t)+p(t)x(\tau(t))\big]^\Delta\Big)^\Delta+q(t)|x(t)|^{\beta-1}x(t)=0 \] on an arbitrary time scale $\mathbb{T}$, where $p$ is an oscillating function defined on $\mathbb{T}$ and $\alpha,\beta>0$ are constants, and obtain several sufficient conditions for the oscillation of all bounded solutions of the equation when $\beta>\alpha, \beta=\alpha$ and $\beta<\alpha$, respectively. Our results extend and complement some known results where $p(t)\equiv 0$ and $\alpha,\beta$ are quotients of odd positive integers.

On the Fermat-Weber Point of a Polygonal Chain and its Generalizations

In this paper, we study the properties of the Fermat-Weber point for a set of fixed points, whose arrangement coincides with the vertices of a regular polygonal chain. A k-chain of a regular n-gon is the segment of the boundary of the regular n-gon formed by a set of k (≤ n) consecutive vertices of the regular n-gon. We show that for every odd positive integer k, there exists an integer N(k), such that the Fermat-Weber point of a set of k fixed points lying on the vertices a k-chain of a n-gon coincides with a vertex of the chain whenever n ≥ N(k). We also show that $\lceil$πm(m + 1) - π 2/4$\rceil$ ≤ N(k) ≤ $\lfloor$πm(m + 1) + 1$\rfloor$, where k (= 2m + 1) is any odd positive integer. We then extend this result to a more general family of point set, and give an O(hk log k) time algorithm for determining whether a given set of k points, having h points on the convex hull, belongs to such a family.

CYCLIC PRESENTATIONS AND TORUS KNOTS K(d, 2)

In this paper, we have shown that the polynomials associated with the cyclically presented groups obtained from the word w generated with Dunwoody parameters 1 1 1 3 2 2 2 2 (1, ,0, 2),(1, k k,0, k),( k ,1,0, k ),( k ,1,0, k ) , where k is an odd positive integer and d  k  2 , coincide (up to sign) with the Alexander polynomial of the torus knot K(d,2) .

Oscillation of Second‐Order Nonlinear Delay Dynamic Equations on Time Scales

In this work, we use the generalized Riccati transformation and the inequality technique to establish some new oscillation criteria for the second‐order nonlinear delay dynamic equation , on a time scale 𝕋, where γ is the quotient of odd positive integers and p(t) and q(t) are positive right‐dense continuous (rd‐continuous) functions on 𝕋. Our results improve and extend some results established by Sun et al. 2009. Also our results unify the oscillation of the second‐order nonlinear delay differential equation and the second‐order nonlinear delay difference equation. Finally, we give some examples to illustrate our main results.

Oscillation Criteria for Certain Second‐Order Nonlinear Neutral Differential Equations of Mixed Type

Some oscillation criteria are established for the second‐order nonlinear neutral differential equations of mixed type , t ≥ t0, where γ ≥ 1 is a quotient of odd positive integers. Our results generalize the results given in the literature.

Packing cycles with modularity constraints

We prove that for all positive integers k, there exists an integer N =N(k) such that the following holds. Let G be a graph and let Γ an abelian group with no element of order two. Let γ: E(G)→Γ be a function from the edges of G to the elements of Γ. A non-zero cycle is a cycle C such that Σ e∈E(C) γ(e) ≠ 0 where 0 is the identity element of Γ. Then G either contains k vertex disjoint non-zero cycles or there exists a set X ⊆ V (G) with |X| ≤N(k) such that G−X contains no non-zero cycle. An immediate consequence is that for all positive odd integers m, a graph G either contains k vertex disjoint cycles of length not congruent to 0 mod m, or there exists a set X of vertices with |X| ≤ N(k) such that every cycle of G-X has length congruent to 0 mod m. No such value N(k) exists when m is allowed to be even, as examples due to Reed and Thomassen show.

Kamenev-type oscillation criteria for second order nonlinear dynamic equations on time scales

The purpose of this paper to establish oscillation criteria for second order nonlinear dynamic equation(r(t)(xΔ(t))γ)Δ+f(t,x(g(t)))=0,on an arbitrary time scale T, where γ is a quotient of odd positive integers and r is a positive rd-continuous function on T. The function g:T→T satisfies g(t)⩾t and limt→∞g(t)=∞ and f∈C(T×R,R). We establish some new sufficient conditions such that the above equation is oscillatory by using generalized Riccati transformation. Our results in the special cases when T=R and T=N involve and improve some oscillation results for second-order differential and difference equations; and when T=hN,T=qN0 and T=N2 our oscillation results are essentially new. Some examples illustrating the importance of our results are included.

Bulk matter fields on a GRS-inspired braneworld

In this paper we investigate the localization and mass spectra of bulk matter fields on a Gergory-Rubakov-Sibiryakov-inspired braneworld. In this braneworld model, there are one thick brane located at the origin of the extra dimension and two thin branes at two sides. For spin 1/2 fermions coupled with the background scalar ϕ via with p a positive odd integer, the zero mode of left-hand fermions can be localized on the thick brane for finite distance of the two thin branes, and there exist some massive bound modes and resonance modes. The resonances correspond to the quasi-localized massive fermions. For free massless spin 0 scalars, the zero mode can not be localized on the thick brane when the two thin branes are located finitely. While for a massive scalar Φ coupled with itself and the background scalar field ϕ, in order to get a localized zero mode on the thick brane, a fine-tuning relation should be introduced. Some massive bound modes and resonances also will appear. For spin 1 vectors, there is no bound KK mode because the effective potential felt by vectors vanishes outside the two thin branes. We also investigate the physics when the distance of the two thin branes tends to infinity.

Self-dual representations of some dyadic groups

Let F be a non-Archimedean local field of residual characteristic two and let d be an odd positive integer. Let D be a central F-division algebra of dimension d2. Let π be one of: an irreducible smooth representation of D × , an irreducible cuspidal representation of GLd(F), an irreducible smooth representation of the Weil group of F of dimension d. We show that, in all these cases, if π is self-contragredient then it is defined over \({\mathbb Q}\) and is orthogonal. We also show that such representations exist.

Oscillation Criteria for a Class of Second‐Order Neutral Delay Dynamic Equations of Emden‐Fowler Type

We establish some new oscillation criteria for the second‐order neutral delay dynamic equations of Emden‐Fowler type, on a time scale unbounded above. Here γ &gt; 0 is a quotient of odd positive integers with a and p being real‐valued positive functions defined on 𝕋. Our results in this paper not only extend and improve the results in the literature but also correct an error in one of the references.

Counting Ordered Pairs

Providing an algebraic proof that φ is indeed a bijection is an instructive exercise. By exploiting the multiplicative structure of the codomain, we can construct a map ψ : Z+ × Z+ → Z+ which is immediately recognized as a bijection. (No need to resort to algebraic calculation or a pictorial argument with diagonals.) For each pair of positive integers m and n, let ψ(m, n) = 2m−1(2n − 1). Bijectivity of ψ is equivalent to the fact that every positive integer has a unique representation as the product of an odd positive integer and a non-negative integer power of 2. As one referee noted, this fact is also key to Glaisher’s bijection between partitions of a positive integer into odd parts and partitions with distinct parts [1, Ex. 2.2.6; 4, p. 12].

The large-time development of the solution to an initial-value problem for the generalized Korteweg–de Vries equation

In this work we address an initial-value problem for the generalized Korteweg–de Vries equation. The normalized generalized Korteweg–de Vries (gKdV) equation considered is given by u τ + u k u x + u x x x = 0 , − ∞ < x < ∞ , τ > 0 , where x and τ represent dimensionless distance and time respectively and k ( > 1 ) is an odd positive integer. We consider the case with the initial data having a discontinuous expansive step, where u ( x , 0 ) = u 0 for x ≥ 0 and u ( x , 0 ) = 0 for x < 0 . In particular, we present the large- τ asymptotic structure of the solution to this problem, which exhibits the formation of an expansion wave in x ≥ 0 , while the solution is oscillatory in x < 0 , with the oscillatory envelope being of O ( τ − 1 2 ) as τ → ∞ . This work extends the asymptotic theory developed by Leach and Needham [J.A. Leach, D.J. Needham, The large-time development of the solution to an initial-value problem for the Korteweg–de Vries equation. I. Initial data has a discontinuous expansive step, Nonlinearity 21 (2008) 2391–2408] for this problem when k = 1 .

On almost universal mixed sums of squares and triangular numbers

In 1997 K. Ono and K. Soundararajan [Invent. Math. 130(1997)] proved that under the generalized Riemann hypothesis any positive odd integer greater than 2719 can be represented by the famous Ramanujan form x 2 + y 2 + 1Oz 2 ; equivalently the form 2x 2 + 5y 2 +4T z represents all integers greater than 1359, where T z denotes the triangular number z(z + 1)/2. Given positive integers a, b, c we employ modular forms and the theory of quadratic forms to determine completely when the general form ax 2 + by 2 + cT z represents sufficiently large integers and to establish similar results for the forms ax 2 + bT y + cT z and aT x + bT y + cT z . Here are some consequences of our main theorems: (i) All sufficiently large odd numbers have the form 2αx 2 + y 2 + z 2 if and only if all prime divisors of a are congruent to 1 modulo 4. (ii) The form αx 2 + y 2 + T z is almost universal (i.e., it represents sufficiently large integers) if and only if each odd prime divisor of a is congruent to 1 or 3 modulo 8. (iii) αx 2 +T y + T z is almost universal if and only if all odd prime divisors of a are congruent to 1 modulo 4. (iv) When υ 2 (α) ≠ 3, the form αT x + T y + Tz is almost universal if and only if all odd prime divisors of α are congruent to 1 modulo 4 and υ 2 (α) ≠ 5, 7, ..., where υ 2 (a) is the 2-adic order of α.

The Sums of Alternating Series of Odd Powers of the Reciprocals of Odd Positive Integers

The Sums of Alternating Series of Odd Powers of the Reciprocals of Odd Positive Integers