Odd Positive Integer Research Articles

Oscillation results for higher order nonlinear neutral delay difference equations

In this work, we shall consider higher order nonlinear neutral delay difference equation of the type Δ m ( y n + p n y n − l ) + q n y n − k α = 0 , where { p n } , { q n } are sequences of nonnegative real numbers, k and l are positive integers and α ∈ ( 0 , 1 ) is a ratio of odd positive integers. We obtain sufficient conditions for the oscillations of all solutions of this equation.

Imaginary quadratic fields with noncyclic ideal class groups

Let g be an odd positive integer and X be a positive real number. We shall show that for any ∊ > 0, the number of imaginary quadratic fields with discriminant ≥ − X and ideal class group having a subgroup isomorphic to ℤ/g ℤ× ℤ/g ℤ is ≫ X1/g−∊.

The Exceptional Set in the Two Prime Squares and a Prime Problem

In this paper we prove that, with at most \( O{\left( {N^{{\frac{5} {{12}} + \in }} } \right)} \) exceptions, all positive odd integers n ≤ N with n ≡ 0 or 1(mod 3) can be written as a sum of a prime and two squares of primes.

Some results about the global attractivity of bounded solutions of difference equations with applications to periodic solutions

Abstract We obtain some results about the global attractivity of bounded solutions of difference equation x n + 1 = f ( x n , x n - 1 , … , x n - k ) , n = 0 , 1 , … where f is non-increasing or non-decreasing in each argument and every point in I is an equilibrium point of above equation where I is an invariant interval for this equation. By our results we prove that when k is an odd positive integer and p ⩾ 1 is a real number, every positive solution of x n + 1 = p + x n - k 1 + x n , n = 0 , 1 , … converges to a period-two solution of this equation. We also apply our results to the rational difference equation x n + 1 = 1 + x n - 2 k + 1 x n - 2 l , n = 0 , 1 , … where k,l ∈ {0, 1, …}, and we show that every positive solution of this equation converges to a period-two solution of this equation.

Testing numbers of the form N = 2kpm − 1 for primality

We suggest an algorithm to test numbers of the form N = 2kpm − 1 for primality, where 2k < pm, k is an odd positive integer, 2k < pm, p is a prime number, and p = 3 (mod 4). The algorithm makes use of the Lucas functions. First we present an algorithm to test numbers of the form N = 2k3m − 1. Then the same technique is used in the more general case where N = 2kpm − 1. The algorithms suggested here are of complexity O((log N)2 log log N log log log N).

Hille-Kneser-type criteria for second-order dynamic equations on time scales

We consider the pair of second-order dynamic equations, (r(t)(xΔ) γ )Δ + p(t)x γ (t) = 0 and (r(t)(xΔ) γ )Δ + p(t)x γσ (t) = 0, on a time scale , where γ > 0 is a quotient of odd positive integers. We establish some necessary and sufficient conditions for nonoscillation of Hille-Kneser type. Our results in the special case when involve the well-known Hille-Kneser-type criteria of second-order linear differential equations established by Hille. For the case of the second-order half-linear differential equation, our results extend and improve some earlier results of Li and Yeh and are related to some work of Doslý and Řehak and some results of Řehak for half-linear equations on time scales. Several examples are considered to illustrate the main results.

Riesz potential operators and inverses via fractional centredderivatives

Fractional centred differences and derivatives definitions are proposed, generalizing to real orders the existing ones valid for even and odd positive integer orders. For each one, suitable integral formulations are obtained. The computations of the involved integrals lead to new generalizations of the Cauchy integral derivative. To compute this integral, a special two‐straight‐line path was used. With this the referred integrals lead to the well‐known Riesz potential operators and their inverses that emerge as true fractional centred derivatives, but that can be computed through summations similar to the well‐known Grünwald‐Letnikov derivatives.

Oscillation criteria for first-order forced nonlinear difference equations

Some new criteria for the oscillation of first-order forced nonlinear difference equations of the form Δx(n)+q1(n)xμ(n+1) = q2(n)xλ(n+1)+e(n), where λ, μ are the ratios of positive odd integers 0 <μ < 1 and λ > 1, are established.

Oscillation of second-order neutral delay and mixed-type dynamic equations on time scales

We consider the equation (r(t)(yΔ(t))γ)Δ + f(t, x(δ(t))) = 0, , where y(t) = x(t) + p(t)x(τ(t)) and γ is a quotient of positive odd integers. We present some sufficient conditions for neutral delay and mixed-type dynamic equations to be oscillatory, depending on deviating arguments τ(t) and δ(t), .

An Euler-type formula for [formula omitted

In this short paper, we give several new formulas for ζ ( n ) when n is an odd positive integer. The method is based on a recent proof, due to H. Tsumura, of Euler's classical result for even n. Our results illuminate the similarities between the even and odd cases, and may give some insight into why the odd case is much more difficult.

Vertex-magic total labeling of odd complete graphs

Let n be an odd positive integer. It is shown that the complete graph K n has a vertex magic total labeling with magic constant h if and only if ( n / 4 ) ( n 2 + 3 ) ⩽ h ⩽ ( n / 4 ) ( n + 1 ) 2 . This partially solves the conjecture regarding possible magic constants of complete graphs. In addition, new techniques are introduced to allow for the simultaneous construction of labelings with different magic constants, and to then alter some of these labelings to obtain new ones with slightly lesser magic constants.

Existence theorems for second-order discrete boundary value problems

This article is concerned with the existence of solutions of boundary value problems for nonlinear second-order difference equations of the type Δ [ p n ( Δ x n − 1 ) δ ] + q n x n δ = f ( n , x n ) , where δ > 0 is the ratio of odd positive integers, { p n } and { q n } are real sequences. The authors apply the Linking Theorem and the Mountain Pass Lemma in the critical point theory and give some new results for the existence of solutions.

Subsquare-free Latin squares of odd order

For every odd positive integer m we prove the existence of a Latin square of order 3m having no proper Latin subsquares. Combining this with previously known results it follows that subsquare-free Latin squares exist for all odd orders.

On some properties of the Euler's factor of certain odd perfect numbers

Let n = π α 3 2 β Q 2 β be an odd positive integer, with π prime, π ≡ α ≡ 1 (mod 4), Q squarefree, ( Q , π ) = ( Q , 3 ) = 1 . It is shown that: if n is perfect, then σ ( π α ) ≡ 0 ( mod 3 2 β ) . Some corollaries concerning the Euler's factor of odd perfect numbers of the above mentioned form, if any, are deduced.

EnPiT: filtered back-projection algorithm for helical CT using an n-Pi acquisition

In this paper, we formulate a reconstruction algorithm for an n-Pi acquisition, where n can be any positive odd integer. The algorithm is a generalization of the method presented in (Bontus et al. 2003). It is based on the results obtained by Katsevich (2004). For the algorithm, different sets of filter-lines have to be defined. We describe the variation of these lines along the detector in some detail, before we discuss, how the method gives all Radon-plane contributions the correct weighting. The different sets of filter-lines are all contained within the n-Pi window, such that a practical realization is possible. Reconstruction results, which we present in the final section, show convincing image quality.

A generalized Lucas sequence and permutation binomials

Let p p be an odd prime and q = p m q=p^m . Let l l be an odd positive integer. Let p ≡ − 1 ( mod l ) p\equiv -1~(\textrm {mod}~l) or p ≡ 1 ( mod l ) p\equiv 1~(\textrm {mod}~l) and l ∣ m l\mid m . By employing the integer sequence a n = ∑ t = 1 l − 1 2 ( 2 cos ⁡ π ( 2 t − 1 ) l ) n \displaystyle {a_n=\sum _{t=1}^{\frac {l-1}{2}} {\left (2\cos {\frac {\pi (2t-1)}{l}}\right )}^n} , which can be considered as a generalized Lucas sequence, we construct all the permutation binomials P ( x ) = x r + x u P(x)=x^r+x^u of the finite field F q \mathbb {F}_q .

Pullbacks of Saito-Kurokawa lifts

Pullbacks of Siegel Eisenstein series have been studied by Garrett [17], [18], Bocherer [6], [7], Heim [26], and play a key role in the proof of the algebraicity of critical values of certain automorphic L-functions. More generally, one might consider pullbacks of Siegel cusp forms. For example, Ikeda [30] gave a conjectural formula for pullbacks of Ikeda lifts [29] in terms of critical values of L-functions for Spn ×GL2. Also, the Gross-Prasad conjecture [22], [23], [8], [27, §8] would relate pullbacks of Siegel cusp forms of degree 2 to central critical values of L-functions for GSp2 ×GL2 ×GL2. Indeed, Bocherer, Furusawa, and Schulze-Pillot [8] gave an explicit formula for pullbacks of Yoshida lifts [57]. In this paper, we give an explicit formula for pullbacks of Saito-Kurokawa lifts and prove the algebraicity of central critical values of certain L-functions for Sp1 ×GL2. To be precise, let κ be an odd positive integer. Let f ∈ S2κ(SL2(Z)) be a normalized Hecke eigenform and h ∈ S+ κ+1/2(Γ0(4)) a Hecke eigenform associated to f by the Shimura correspondence. Let F ∈ Sκ+1(Sp2(Z)) be the Saito-Kurokawa lift of h. For each normalized Hecke eigenform g ∈ Sκ+1(SL2(Z)), we consider the period integral 〈F|H×H, g× g〉 given by 〈F|H×H, g × g〉 = ∫

Oscillation of perturbed nonlinear dynamic equations on time scales

AbstractIn this paper we consider the oscillatory and asymptotic behavior of bounded solutions of second‐order nonlinear perturbed dynamic equation (α(t)(xΔ(t))r)Δ+ F(t,xσ (t)) = G(t, xσ (t), xΔ (t)), t ∈𝕋 , (0.1) where r = k/l with k even and l odd positive integer. Some new sufficient conditions are obtained for all bounded solutions of (0.1) to be oscillatory. Several examples that dwell upon the importance of our results are also included. In particular, our criteria extend some earlier results.

Oscillation of second-order nonlinear neutral delay dynamic equations on time scales

In this paper, some sufficient conditions for oscillation of the second-order nonlinear neutral delay dynamic equation ( r ( t ) ( [ y ( t ) + p ( t ) y ( t - τ ) ] Δ ) γ ) Δ + f ( t , y ( t - δ ) ) = 0 , on a time scale T are established; here γ ⩾ 1 is an odd positive integer with r ( t ) and p ( t ) are rd -continuous functions defined on T . Our results as a special case when T = R and T = N , involve and improve some well-known oscillation results for second-order neutral delay differential and difference equations. When T = h N and T = q N = { t : t = q k , k ∈ N , q > 1 } , i.e., for generalized neutral delay difference and q -neutral delay difference equations our results are essentially new and also can be applied on different types of time scales, e.g., T = N 2 = { t 2 : t ∈ N } and T = T n = { t n : n ∈ N 0 } where { t n } is the set of harmonic numbers. Some examples illustrating our main results are given.

Classical and quantum dilogarithmic invariants of flatPSL(2,ℂ)–bundles over 3–manifolds

We introduce a family of matrix dilogarithms, which are automorphisms of C^N tensor C^N, N being any odd positive integer, associated to hyperbolic ideal tetrahedra equipped with an additional decoration. The matrix dilogarithms satisfy fundamental five-term identities that correspond to decorated versions of the 2 --> 3 move on 3-dimensional triangulations. Together with the decoration, they arise from the solution we give of a symmetrization problem for a specific family of basic matrix dilogarithms, the classical (N=1) one being the Rogers dilogarithm, which only satisfy one special instance of five-term identity. We use the matrix dilogarithms to construct invariant state sums for closed oriented 3-manifolds $W$ endowed with a flat principal PSL(2,C)-bundle rho, and a fixed non empty link L if N>1, and for (possibly "marked") cusped hyperbolic 3-manifolds M. When N=1 the state sums recover known simplicial formulas for the volume and the Chern-Simons invariant. When N>$, the invariants for M are new; those for triples (W,L,rho) coincide with the quantum hyperbolic invariants defined in [Topology 43 (2004) 1373-1423], though our present approach clarifies substantially their nature. We analyse the structural coincidences versus discrepancies between the cases N=1 and N>1, and we formulate "Volume Conjectures", having geometric motivations, about the asymptotic behaviour of the invariants when N tends to infinity.