Odd Positive Integer Research Articles

Fekete-like polynomials

In 2001, Borwein, Choi, and Yazdani looked at an extremal property of a class of polynomial with ±1 coefficients. Their key result was: Theorem (See Borwein, Choi, Yazdani, 2001.) Let f ( z ) = ± z ± z 2 ± ⋯ ± z N − 1 , and ζ a primitive Nth root of unity. If N is an odd positive integer then max i | f ( ζ i ) | ⩾ N with equality if and only if N is an odd prime. Moreover, if equality holds, they gave an explicit construction for f ( z ) . In this paper, we look at the case when N is even. In particular, we investigate the following Conjecture Let f ( z ) and ζ be as above. If N > 2 is an even positive integer then max i | f ( ζ i ) | ⩾ N + 1 with equality if and only if N + 1 is a power of an odd prime. This conjecture was made after extensive computations. Partial results towards proving this conjecture are given.

Asymptotic behavior of solutions for third-order half-linear delay dynamic equations on time scales

By means of Riccati transformation technique and integral averaging technique, we establish some new sufficient conditions which guarantee that every solution of the third-order half-linear delay dynamic equations $$(r(t)(x^{\Delta\Delta}(t))^\gamma)^\Delta+p(t)x^\gamma(\tau(t))=0$$ oscillates or converges to zero on a time scale \(\mathbb{T}\), here γ>0 is a quotient of odd positive integers with r and p real-valued positive rd-continuous functions defined on \(\mathbb{T}\). Our results not only extend and improve the results given in literatures but also unify the asymptotic behavior results of the third order nonlinear delay differential equation and the third order nonlinear delay difference equation. Some examples are considered to illustrate the importance of our results.

On the reducibility of certain quadrinomials

In 2007 West Coast Number Theory conference Walsh asked to determine all irreducible polynomials of the form P(x) = x i + x j + x k + 4 with integer exponents i > j > k > 0, such that for some positive integer l the polynomial P(x l ) is reducible in Z(x): In this paper we prove that such polynomials are quadrinomials x 4m + x 3m + x 2m + 4, where m is an odd positive integer. In addition, Walsh asked for the examples of reducible quadrinomials x i + x j + x k + n, n > 4 with no linear or quadratic factors. We compute the examples of reducible polynomials of the form above with non-trivial factors and negative coefficient n. Let P(x) be a polynomial with integer coefficients. The polynomialP(x) is called primitive, if one cannot write P(x) = P1(x l ) for some polynomial P1 ∈ Z(x) and integer l > 1. The reciprocal polynomial x deg P P(1/x) of the polynomial P(x) is denoted by P � (x). Through the paper, reducibility shall

The period of the Bell numbers modulo a prime

We discuss the numbers in the title, and in particular whether the minimum period of the Bell numbers modulo a prime p can be a proper divisor of Np = (p P — 1)/(p - 1). It is known that the period always divides Np. The period is shown to equal Np for most primes p below 180. The investigation leads to interesting new results about the possible prime factors of Np. For example, we show that if p is an odd positive integer and m is a positive integer and q = 4m 2 p + 1 is prime, then q divides p m2p - 1. Then we explain how this theorem influences the probability that q divides Np.

New oscillation criteria for second-order neutral functional dynamic equations via the generalized Riccati substitution

In this paper, we establish some new sufficient conditions for oscillation of the second-order neutral functional dynamic equation ( p ( t ) ( [ y ( t ) + r ( t ) y ( τ ( t ) ) ] Δ ) γ ) Δ + f ( t , y ( θ ( t ) ) = 0 , t ∈ [ t 0 , ∞ ) T , on a time scale T , where | f ( t , u ) | ⩾ q ( t ) | u γ | , r, p and q are real valued rd-continuous positive functions defined on T , γ ⩾ 1 is the quotient of odd positive integers. Our results improve existence results in the literature in the sense that our results do not require p Δ ( t ) ⩾ 0 , and ∫ t 0 ∞ θ γ ( s ) q ( s ) [ 1 - r ( θ ( s ) ) ] γ Δ s = ∞ . Some examples are given to illustrate the main results.

Oscillation of third-order neutral differential equations

The objective of this paper is to study asymptotic properties of the couple of third-order neutral differential equations ( E ± ) [ a ( t ) ( [ x ( t ) ± p ( t ) x ( δ ( t ) ) ] ″ ) γ ] ′ + q ( t ) x γ ( τ ( t ) ) = 0 , t ≥ t 0 , where a ( t ) , q ( t ) , p ( t ) are positive functions, γ > 0 is a quotient of odd positive integers and τ ( t ) ≤ t , δ ( t ) ≤ t . We will establish some sufficient conditions which ensure that all nonoscillatory solutions of ( E ± ) converge to zero. Some examples are considered to illustrate the main results.

Some curious congruences modulo primes

Let n be a positive odd integer and let p > n + 1 be a prime. We mainly derive the following congruence: ∑ 0 < i 1 < ⋯ < i n < p ( i 1 3 ) ( − 1 ) i 1 i 1 ⋯ i n ≡ 0 ( mod p ) .

Oscillation of sublinear Emden-Fowler dynamic equations on time scales

Consider the Emden-Fowler sublinear dynamic equation where , where 𝕋 is a time scale, , α is the quotient of odd positive integers. We obtain a Kamenev-type oscillation theorem for (0.1). As applications, we get that the sublinear difference equation where , is oscillatory.

Comparison and oscillatory behavior for certain second order nonlinear dynamic equations

We present some new necessary and sufficient conditions for the oscillation of second order nonlinear dynamic equation $$\bigl(a\bigl(x^{\Delta }\bigr)^{\alpha }\bigr)^{\Delta }(t)+q(t)x^{\beta }(t)=0$$ on an arbitrary time scale \(\mathbb{T}\) , where α and β are ratios of positive odd integers, a and q are positive rd-continuous functions on \(\mathbb{T}\) . Comparison results with the inequality $$\bigl(a\bigl(x^{\Delta }\bigr)^{\alpha }\bigr)^{\Delta }(t)+q(t)x^{\beta }(t)\leqslant 0\quad (\geqslant 0)$$ are established and application to neutral equations of the form $$\bigl(a(t)\bigl(\bigl[x(t)+p(t)x[\tau (t)]\bigr]^{\Delta }\bigr)^{\alpha }\bigr)^{\Delta }+q(t)x^{\beta }\bigl[g(t)\bigr]=0$$ are investigated.

Forced Oscillation of Second‐Order Half‐Linear Dynamic Equations on Time Scales

We will establish a new interval oscillation criterion for second‐order half‐linear dynamic equation (r(t)[xΔ(t)] α) Δ + p(t)xα(σ(t)) = f(t) on a time scale 𝕋 which is unbounded, which is a extension of the oscillation result for second order linear dynamic equation established by Erbe et al. (2008). As an application, we obtain a sufficient condition of oscillation of the second‐order half‐linear differential equation ([x′(t)] α) ′ + csintxα(t) = cos t, where α = p/q, p, q are odd positive integers.

Oscillation of Second-Order Sublinear Dynamic Equations with Damping on Isolated Time Scales

This paper concerns the oscillation of solutions to the second sublinear dynamic equation with damping , on an isolated time scale which is unbounded above. In , α is the quotient of odd positive integers. As an application, we get the difference equation , where , , and is any real number, is oscillatory.

Oscillation for a Class of Second-Order Emden-Fowler Delay Dynamic Equations on Time Scales

By means of Riccati transformation technique, we establish some new oscillation criteria for the second-order Emden-Fowler delay dynamic equations on a time scale ; here is a quotient of odd positive integers with and as real-valued positive rd-continuous functions defined on . Our results in this paper not only extend the results given in Agarwal et al. (2005), Akin-Bohner et al. (2007) and Han et al. (2007) but also unify the results about oscillation of the second-order Emden-Fowler delay differential equation and the second-order Emden-Fowler delay difference equation.

Oscillation Behavior of Third-Order Neutral Emden-Fowler Delay Dynamic Equations on Time Scales

We will establish some oscillation criteria for the third-order Emden-Fowler neutral delay dynamic equations on a time scale , where is a quotient of odd positive integers with , and real-valued positive rd-continuous functions defined on . To the best of our knowledge nothing is known regarding the qualitative behavior of these equations on time scales, so this paper initiates the study. Some examples are considered to illustrate the main results.

Oscillation criteria for third order nonlinear delay dynamic equations on time scales

By means of Riccati transformation technique, we establish some new oscillation criteria for the third-order nonlinear delay dynamic equations x 3 (t) + p(t)x(τ(t)) = 0 on a time scale T unbounded above, here γ > 0 is a quotient of odd positive integers with p real-valued positive rd-continuous function defined on T. Three examples are given to illustrate the main results.

Comments on &amp;#x201C;Fast Radix-9 Algorithm for the DCT-IV Computation&amp;#x201D;

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Recently, a fast radix-<formula formulatype="inline"> <tex Notation="TeX">$q$</tex></formula> algorithm for an efficient computation of the type-IV discrete cosine transform (DCT-IV) has been proposed in <citerefgrp> <citeref refid="ref1"/></citerefgrp>, where <formula formulatype="inline"> <tex Notation="TeX">$q$</tex></formula> is an odd positive integer. In particular, based on the proposed fast algorithm, optimized efficient 3-, 5-, and 9-point scaled DCT-IV (SDCT-IV) modules have been derived in <citerefgrp><citeref refid="ref1"/></citerefgrp>. As a response, an improved efficient optimized 9-point scaled DCT-IV (SDCT-IV) module in terms of the arithmetic complexity is presented. The improved optimized efficient 9-point SDCT-IV module requires 17 multiplications, 53 additions, and three shifts. Consequently, the arithmetic complexity of extended fast mixed-radix DCT-IV algorithm for composite lengths is also significantly improved. </para>

Oscillation criteria for nonlinear functional neutral dynamic equations on time scales

This paper is concerned with the oscillation of the second-order nonlinear functional dynamic equations on a time scale where γ is the quotient of odd positive integers, r(t), p(t), and are positive rd-continuous functions on , , and . We establish some new sufficient conditions for oscillation for the above equation. Several examples illustrating our results will be given.

A note on flow geometries and the similarity solutions of the boundary layer equations for a nonlinearly stretching sheet

In this note we extend the results of Akyildiz et al. [Similarity solutions of the boundary layer equations for a nonlinearly stretching sheet. Mathematical Methods in the Applied Sciences (www.interscience.wiley.com). doi: 10.1002/mma.1181] for any n > 0, where n is a nonlinear stretching parameter. Thus, the proof presented for the existence of the similarity solutions for the boundary layer equation for a nonlinearly stretching sheet presented in Akyildiz et al. hold not only for positive odd integer values of n, but also for any real value of n > 0: That is, n can be any positive real. We accomplish this by defining the stretching velocity of the sheet as u = csgn(x)|x| n , −∞ < x < ∞, at y = 0 (instead of u = cx n , 0 < x < ∞, y = 0) and accordingly modifying the similarity variables. This definition for u at the stretching surface eliminates the restrictions on n in all future research results related to flow and heat transfer over nonlinear stretching surfaces.

Localization and mass spectra of fermions on symmetric and asymmetric thick branes

A three-parameter (positive odd integer $s$, thickness factor $\ensuremath{\lambda}$, and asymmetry factor $a$) family of asymmetric thick brane solutions in five dimensions were constructed from a two-parameter ($s$ and $\ensuremath{\lambda}$) family of symmetric ones in by R. Guerrero, R. O. Rodriguez, and R. Torrealba in [Phys. Rev. D 72, 124012 (2005).]. The values $s=1$ and $s\ensuremath{\ge}3$ correspond to single branes and double branes, respectively. These branes have very rich inner structure. In this paper, by presenting the mass-independent potentials of Kaluza-Klein (KK) modes in the corresponding Schr\odinger equations, we investigate the localization and mass spectra of fermions on the symmetric and asymmetric thick branes in an anti-de Sitter background. In order to analyze the effect of gravity-fermion interaction (i.e., the effect of the inner structure of the branes) and scalar-fermion interaction to the spectrum of fermion KK modes, we consider three kinds of typical kink-fermion couplings. The spectra of left chiral fermions for these couplings consist of a bound zero mode and a series of gapless continuous massive KK modes, some discrete bound KK modes including zero mode (exist mass gaps), and a series of continuous massive KK modes, infinite discrete bound KK modes, respectively. The structures of the spectra are investigated in detail.

Fermion localization and resonances on a de Sitter thick brane

In C. A. S. Almeida, R. Casana, M. M. Ferreira, Jr., and A. R. Gomes, Phys. Rev. D 79, 125022 (2009), the simplest Yukawa coupling $\ensuremath{\eta}\overline{\ensuremath{\Psi}}\ensuremath{\phi}\ensuremath{\chi}\ensuremath{\Psi}$ was considered for a two-scalar-generated Bloch brane model. Fermionic resonances for both chiralities were obtained, and their appearance is related to branes with internal structure. Inspired on this result, we investigate the localization and resonance spectrum of fermions on a one-scalar-generated de Sitter thick brane with a class of scalar-fermion couplings $\ensuremath{\eta}\overline{\ensuremath{\Psi}}{\ensuremath{\phi}}^{k}\ensuremath{\Psi}$ with positive odd integer $k$. A set of massive fermionic resonances for both chiralities is obtained when provided large coupling constant $\ensuremath{\eta}$. We find that the masses and lifetimes of left and right chiral resonances are almost the same, which demonstrates that it is possible to compose massive Dirac fermions from the left and right chiral resonances. The resonance with lower mass has longer lifetime. For a same set of parameters, the number of resonances increases with $k$ and the lifetime of the lower level resonance for larger $k$ is much longer than the one for smaller $k$.

POLYNOMIAL VARIATIONS ON A THEME OF SIERPIŃSKI

In 1960, Sierpiński proved that there exist infinitely many odd positive integers k such that k · 2n + 1 is composite for all integers n ≥ 0. Variations of this problem, using polynomials with integer coefficients, and considering reducibility over the rationals, have been investigated by several authors. In particular, if S is the set of all positive integers d for which there exists a polynomial f(x) ∈ ℤ[x], with f(1) ≠ -d, such that f(x)xn + d is reducible over the rationals for all integers n ≥ 0, then what are the elements of S? Interest in this problem stems partially from the fact that if S contains an odd integer, then a question of Erdös and Selfridge concerning the existence of an odd covering of the integers would be resolved. Filaseta has shown that S contains all positive integers d ≡ 0 (mod 4), and until now, nothing else was known about the elements of S. In this paper, we show that S contains infinitely many positive integers d ≡ 6 (mod 12). We also consider the corresponding problem over 𝔽p, and in that situation, we show 1 ∈ S for all primes p.