Odd Positive Integer Research Articles

Oscillatory behavior of second order unstable type neutral difference equations

This paper deals with the oscillatory behavior of all bounded/ unbounded solutions of second order neutral type difference equation of the form$$ \Delta (a_n(\Delta_c y_n+py_{n-k}))^\alpha)-g_nf(y_{\sigma(n)})=0, $$where $ p $ is real, $ \alpha $ is a ratio of odd positive integers, $ k $ is a positive integer and $ \{\sigma(n)\} $ is a sequence of integers. Examples are provided to illustrate the results.

On Some Sets with Even Valued Partition Function

Let IN be the set of positive integers, $${\cal B}$$ = {b1 < ⋅s < bh}⊂IN, N∊IN and N≥ bh. $${\cal A}$$ = $${\cal A}$$ 0( $${\cal B}$$ ,N) is the set (introduced by J.-L. Nicolas, I.Z. Ruzsa and A. Sarkozy) such that $${\cal A}$$ {1,..., N} = $${\cal B}$$ and p( $${\cal A}$$ ,n)≡ 0 (mod 2) for n∊IN and n > N, where p( $${\cal A}$$ ,n) denotes the number of partitions of n with parts in $${\cal A}$$ . Let us denote by σ ( $${\cal A}$$ ,n) the sum of the divisors of n belonging to $${\cal A}$$ . In a paper jointly written with J.-L. Nicolas, we have recently proved that, for all k≥ 0, the sequence (σ( $${\cal A}$$ ,2k n))n≥ 1mod 2k+1 is periodic with an odd period qk. In this paper, we will characterize for any fixed odd positive integer q, the sets $${\cal B}$$ and the integers N such that q0 = q, and those for which qk = q for all k≥ 0. Moreover, a set $${\cal A}$$ = $${\cal A}$$ 0( $${\cal B}$$ ,N) is constructed with the property that its period, i.e. the period of (σ( $${\cal A}$$ ,n))n≥ 1mod 2, is 217, and for which the counting function is asymptotically equal to that of $${\cal A}$$ 0({1,2,3,4,5},5) which is a set of period 31.

Computation of abstract groups S(( d + 1)/2, d) and torus knots K( d, 2)

We have developed a computer program to calculate the abstract groups S(( d + 1)/2, d) and the torus knots K( d, 2) which is associated with the same polynomial. When a positive odd integer c ⩾ 1 is given, the program calculates the abstract groups S(( d + 1)/2, d) and the torus knots K( d, 2) having the same polynomial. It is general and easy to use.

Oscillation criteria of second-order half-linear dynamic equations on time scales

In this paper, by using the Riccati transformation technique, chain rule and inequality A λ - λ AB λ - 1 + ( λ - 1 ) B λ ⩾ 0 , λ > 1 , where A and B are positive constants, we will establish some oscillation criteria for the second-order half-linear dynamic equation ( p ( t ) ( x Δ ( t ) ) γ ) Δ + q ( t ) x γ ( t ) = 0 , t ∈ [ a , b ] on time scales, where γ > 1 is an odd positive integer. Our results not only unify the oscillation of half-linear differential and half-linear difference equations but can be applied on different types of time scales and improve some well-known results in the difference equation case. Some examples are considered here to illustrate our main results.

Oscillation criteria for second-order nonlinear neutral delay dynamic equations

In this paper we will establish some oscillation criteria for 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 ; here γ > 0 is a quotient of odd positive integers with r ( t ) and p ( t ) real-valued positive functions defined on T . 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.

Representations of degree three for semisimple Hopf algebras

Let H be a cosemisimple Hopf algebra over an algebraically closed field. It is shown that if H has a simple subcoalgebra of dimension 9 and has no simple subcoalgebras of even dimension, then H contains either a grouplike element of order 2 or 3, or a family of simple subcoalgebras whose dimensions are the squares of each positive odd integer. In particular, if H is odd dimensional, then its dimension is divisible by 3.

Universally bad integers and the 2-adics

In his 1964 paper, de Bruijn (Math. Comp. 18 (1964) 537) called a pair ( a, b) of positive odd integers good, if Z=a S⊖2b S , where S is the set of nonnegative integers whose 4-adic expansion has only 0's and 1's, otherwise he called the pair ( a, b) bad. Using the 2-adic integers we obtain a characterization of all bad pairs. A positive odd integer u is universally bad if ( ua, b) is bad for all pairs of positive odd integers a and b. De Bruijn showed that all positive integers of the form u=2 k +1 are universally bad. We apply our characterization of bad pairs to give another proof of this result of de Bruijn, and to show that all integers of the form u= φ p k (4) are universally bad, where p is prime and φ n ( x) is the nth cyclotomic polynomial. We consider a new class of integers we call de Bruijn universally bad integers and obtain a characterization of such positive integers. We apply this characterization to show that the universally bad integers u= φ p k (4) are in fact de Bruijn universally bad for all primes p>2. Furthermore, we show that the universally bad integers φ 2 k (4), and more generally, those of the form 4 k +1, are not de Bruijn universally bad.

Universally bad integers and the 2-adics

In his 1964 paper, de Bruijn (Math. Comp. 18 (1964) 537) called a pair ( a , b ) of positive odd integers good, if Z =a S ⊖2b S , where S is the set of nonnegative integers whose 4-adic expansion has only 0's and 1's, otherwise he called the pair ( a , b ) bad. Using the 2-adic integers we obtain a characterization of all bad pairs. A positive odd integer u is universally bad if ( ua , b ) is bad for all pairs of positive odd integers a and b . De Bruijn showed that all positive integers of the form u =2 k +1 are universally bad. We apply our characterization of bad pairs to give another proof of this result of de Bruijn, and to show that all integers of the form u = φ p k (4) are universally bad, where p is prime and φ n ( x ) is the n th cyclotomic polynomial. We consider a new class of integers we call de Bruijn universally bad integers and obtain a characterization of such positive integers. We apply this characterization to show that the universally bad integers u = φ p k (4) are in fact de Bruijn universally bad for all primes p >2. Furthermore, we show that the universally bad integers φ 2 k (4), and more generally, those of the form 4 k +1, are not de Bruijn universally bad.

Every positive integer is the Frobenius number of a numerical semigroup with three generators

Let n1, . . . , np be positive integers with greatest common divisor (gcd for short) one. Then it is not hard to show that there are finitely many nonnegative integers that cannot be expressed as a nonnegative integer linear combination of n1, . . . , np. The largest nonnegative integer fulfilling this condition is usually known as the Frobenius number of n1, . . . , np and it will be denoted throughout this paper by F(n1, . . . , np). The problem of determining F(n1, . . . , np) appears in the literature as the Frobenius coin-exchange problem (for a complete survey on this problem see [5], [6]). For p = 2, Sylvester proved in [7] that F(n1, n2) = n1n2 −n1 −n2. No general formula has been found so far for the case p ≥ 3. Moreover, as Curtis shows in [1], there is no closed formula of a certain type for p = 3. If we focus our attention on this case, then we can think of F as a correspondence that maps three relatively prime integers n1, n2, n3 to a nonnegative integer F(n1, n2, n3). In this paper we prove that this map is surjective, that is, for every positive integer g there exist positive integers n1, n2 and n3 such that F(n1, n2, n3) = g (Theorem 1.11). One easily realizes that every odd positive integer g can be expressed as F(2, g + 2) (see 1.1 (i)). However no even positive integers can be expressed as the Frobenius number associated to a pair of relatively prime numbers. This follows easily from Sylvester’s formula given above, since this formula can be rewritten as F(n1, n2) = (n1 − 1)(n2 − 1) − 1 and n1, n2 are coprime (in fact, it is well known that every numerical semigroup generated by two elements is symmetric, see for instance [2], [4], and thus its Frobenius number must be odd). Thus in order to express any nonnegative integer as the Frobenius number of a sequence of positive integers, the sequences used should be at least of length

Some new results for bounded and monotone properties of solutions of second-order quasi-linear forced difference equations

Some new results are obtained for the bounded and monotone properties of solutions of second-order quasi-linear forced difference equations δ(P n−1(δY n−1) α)=q nY n β+r n, n≥n 0, where n 0 ϵ N = {0,1,2,3,…}, { p n } n= n 0−1 ∞ is a positive sequence, { q n } n= n 0 ∞ is a nonnegative real sequence with q n ≢ 0, { r n } n= n 0 ∞ is a real sequence, and α and β are quotients of odd positive integers. Some errors in [1] are pointed out and addressed. Examples are given to illustrate the advantages of the new results.

A generalization of Cohen- Eisenstein series and Shimura liftings and some congruences between Cusp forms and Eisenstein series

In this paper we introduce some modular forms of half-integral weight on congruence group Гo(4N) withN an odd positive integer which can be viewed as a natural generalization of Cohen-Eisenstein series. Using these series, we can prove that the restriction of Shimura lifting on Eisenstein spaceEk+1/2+(4N,χl) gives an isomorphism fromEk+1/2+(4N,χl) toE2k(N). We consider some congruence relationships between modular forms in use of Shimura lifting.

Higher-order semiclassical energy expansions for potentials with non-integer powers

In this paper, we present a semiclassical eigenenergy expansion for the potential |x|α when α is a positive rational number of the form2n/m (n is a positive integer and m is an odd positive integer). Remarkably, this expansion is found to be identical to the WKB expansion obtained for the potentialx N(N-even), if2n/m is replaced byN. Taking the limitm → 2 of the above expansion, we obtain an explicit asymptotic energy expansion of symmetric odd power potentials |x|2j+1 (j- positive integer). We then show how to develop approximate semiclassical expansions for potentials |x|α when α is any positive real number.

A Note on Cyclotomic Euler Systems and the Double Complex Method

AbstractLetbe a finite real abelian extension of ℚ. LetMbe an odd positive integer. For every squarefree positive integerrthe prime factors of which are congruent to 1 moduloMand split completely in, the corresponding Kolyvagin classsatisfies a remarkable and crucial recursion which for each prime number ℓ dividingrdetermines the order of vanishing of κrat each place ofabove ℓ in terms of κr/ℓ. In this note we give the recursion a new and universal interpretation with the help of the double complex method introduced by Anderson and further developed by Das and Ouyang. Namely, we show that the recursion satis_ed by Kolyvagin classes is the specialization of a universal recursion independent ofsatisfied by universal Kolyvagin classes in the group cohomology of the universal ordinary distributionà laKubert tensored with ℤ=Mℤ. Further, we show by a method involving a variant of the diagonal shift operation introduced by Das that certain group cohomology classes belonging (up to sign) to a basis previously constructed by Ouyang also satisfy the universal recursion.

Sums of three integral squares in cyclotomic fields

Let m be an odd positive integer greater than 2 and f the smallest positive integer such that 2f ≡ 1 (mod m). It is proved that every algebraic integer in the cyclotomic field ℚ(ζm) can be expressed as a sum of three integral squares if and only if f is even.

A problem of D.H. Lehmer and its mean square value formula

Let q be an odd positive integer and let a be an integer coprime to q. For each integer b coprime to q with 1⩽ b< q, there is a unique integer c coprime to q with 1⩽ c< q such that bc≡a( mod q) . Let N( a, q) denote the number of solutions of the congruence equation bc≡a( mod q) with 1⩽ b, c< q such that b, c are of opposite parity. The main purpose of this paper is to use the properties of Dedekind sums, the properties of Cochrane sums and the mean value theorem of Dirichlet L-functions to study the asymptotic property of the mean square value ∑′ a=1 q(N(a,q)− 1 2 φ(q)) 2 , and give a sharp asymptotic formula.

New results for the asymptotic behavior of a nonlinear second-order difference equation

Some new results are obtained for the asymptotic behavior of the following second-order nonlinear difference equation: δ(r n-1δy n-1)+q n) β-p ny n α= e n, nϵ N, where { r n } n ∞=0 o is a positive sequence, { P n } n ∞ = 1 and { qn n ∞} = 1 are nonnegative sequences, { e n } n ∞ = 1 any real sequence, and α and β are ratios of odd positive integers. Examples are given to illustrate rights reserved.

Oscillation Properties of an Emden-Fowler Type Equation on Discrete Time Scales

In this paper, we explore the oscillation properties of on a time scale T with only isolated points, where p(t) is defined on T and γ is a quotient of odd positive integers. We define oscillation in this setting, and generate conditions on the integral of p(t) which guarantee oscillation and find conditions which give the existence of a nonoscillatory solution. In addition, we consider the case when solutions of this equation has asymptotically positively bounded differences.

Oscillation of second order nonlinear neutral delay difference equations

In this paper, we obtain sufficient conditions for the oscillation of all solutions of the second order nonlinear delay difference equation Δ[a n( Δ(x n+p nx g(n))) α]+f(n,x σ(n))=0, where Δ x n = x n+1 − x n is the forward difference operator, { p n } is a sequence of nonnegative real numbers and α⩾1 is a quotient of positive odd integers. Our results extend and improve the known results in the literature. An example that dwell up the advantage of our results is included.

Irreducible polynomials over GF(2) with three prescribed coefficients

For an odd positive integer n, we determine formulas for the number of irreducible polynomials of degree n over GF(2) in which the coefficients of x n−1 , x n−2 and x n−3 are specified in advance. Formulas for the number of elements in GF(2 n ) with the first three traces specified are also given.

Oscillation of second-order sublinear neutral delay difference equations

We present new oscillation criteria for the second-order sublinear neutral delay difference equation Δ a n Δ x n+p nx n−τ +q nx n−σ γ=0, where 0< γ<1 is a quotient of odd positive integers and ∑ n=0 ∞ 1 a n =∞.