The method of infinite ascent applied on A 4 ± nB 3 = C 2

Each of the Diophantine equations A ⁴ ± nB ³ = C ² has an infinite number of integral solutions (A,B,C) for any positive integer n. In this paper, we will show how the method of infinite ascent could be applied to generate these solutions. We will investigate the conditions when A, B and C are pair-wise co-prime. As a side result of this investigation, we will show a method of generating an infinite number of co-prime integral solutions (A,B,C) of the Diophantine equation aA ³ +cB ³ = C ² for any co-prime integer pair (a, c).     

The proof of a conjecture concerning the intersection of k-generalized Fibonacci sequences

For k ≥ 2, the k-generalized Fibonacci sequence (F _n ^(k) ) _n is defined by the initial values 0, 0, …, 0,1 (k terms) and such that each term afterwards is the sum of the k preceding terms. In 2005, Noe and Post conjectured that the only solutions of Diophantine equation F _m ^(k) = F _n ^(?) , with ? > k > 1, n > ? + 1, m > k + 1 are $(m,n,\ell ,k) = (7,6,3,2)and(12,11,7,3)$ . In this paper, we confirm this conjecture.     

Existence of triple solutions of discrete (n,p) boundary value problems

We consider the following boundary value problem: −Δⁿy = F(k,y, Δy,…,Δⁿ⁻¹y), k ϵ Z[n − 1, N], Δⁱy(0) = 0, 0 ≤ i ≤ n − 2, Δ^py(N + n - p) = 0, where n ≥ 2 and p is a fixed integer satisfying 0 ≤ p ≤ n − 1. Using a fixed-point theorem for operators on a cone, we shall yield the existence of at least three positive solutions.     

Pointwise estimates for the fundamental solutions of a class of singular parabolic problems

We deal with the Cauchy problem associated to a class of quasilinear singular parabolic equations with L ^∞ coefficients whose prototypes are the p-Laplacian (2N/(N + 1) < p < 2) and the porous medium equation (((N ? 2)/N)₊ < m < 1). We prove existence of and sharp pointwise estimates from above and from below for the fundamental solutions. Our results can be extended to general non-negative L ¹ initial data.     

On weighted hyperbolic polynomials

The results by Görding, Larsson, Cattabriga, Rodino, Calvo on correctness of Cauchy problem for N-hyperbolic equations are generalized. We prove that in the general case where the vector N = (N ₁, …,N _n ) is different from the vector (1, 0, …, 0), for the correctness of the Cauchy problem more stronger condition is required, which we call weighted hyperbolicity condition. We also discuss the properties of polynomials possessing weighted hyperbolicity property.     

Permanence and Positive Periodic Solutions for Kolmogorov Competing Species Systems

We prove the existence of periodic solutions in a compact attractor of (R⁺)ⁿ for the Kolmogorov system x′_i = x_if_i(t, x₁, …, x_n), i = l, …, n in the competitive case. Extension to differential delay equations are con- sidered too. Applications are given to Lotka-Volterra systems with periodic coefficients.     

On the diophantine equation x 2 + 2 a · 3 b · 11 c = y n

In this note, we find all the solutions of the Diophantine equation x ² + 2 ^a · 3 ^b · 11 ^c = y ⁿ , in nonnegative integers a, b, c, x, y, n ≥ 3 with x and y coprime.     

The interval number of a complete multipartite graph

The interval number of a graph G, denoted i(G), is the least positive integer t for which G is the intersection graph of a family of sets each of which is the union of at most t closed intervals of the real line $\text{R}$. Trotter and Harary showed that the interval number of the complete bipartite graph K(m,n) is $\text{?}\text{(mn + 1)}\text{(m + n)}\text{?}$. Matthews showed that the interval number of the complete multipartite graph K(n₁,n₂,…,n_p) was the same as the interval number of K(n₁,n₂) when n₁ = n₂ = ? = n_p. Trotter and Hopkins showed that i(K(n₁,n₂,…,n_p)) ≤ 1 + i(K(n₁,n₂)) whenever p ≥ 2 and n₁≥n₂≥ ? ≥n_p. West showed that for each n ≥ 3, there exists a constant c_n so that if p ≥ c_n,n₁ = n²?n ?1, and n₂ = n₃ = ? n_p = n, then i(K(n₁,n₂,…,n_p) = 1 + i(K(n₁, n₂)). In view of these results, it is natural to consider the problem of determining those pairs (n₁,n₂) with n₁ ≥ n₂ so that i(K(n₂,…,n_p)) = i(K(n₁,n₂)) whenever p ≥ 2 and n₂ ≥ n₃ ≥ ? ≥ n_p. In this paper, we present constructions utilizing Eulerian circuits in directed graphs to show that the only exceptional pairs are (n² ? n ? 1, n) for n ≥ 3 and (7,5).     

Finite-dimensional Perron effect of change of all values of characteristic exponents of differential systems

We obtain a finite-dimensional Perron effect of change of values λ ₁ ≤ … ≤ λ _n < 0 of all arbitrarily specified negative characteristic exponents of the n-dimensional system of linear approximation with infinitely differentiable bounded coefficients to arbitrarily specified, arranged in ascending order, values β _k ≥ λ _k , k = 1, …, n, of characteristic exponents of all nontrivial solutions of an n-dimensional nonlinear differential system with an infinitely differentiable perturbation of arbitrary order m > 1 of smallness in a neighborhood of the origin and growth outside it. Each value β _k is realized by all nontrivial solutions of the perturbed system issuing from the difference R ^k |R ^k?1 of embedded subspaces R ¹ ? R ² ? … ? R ⁿ .     

A maximum principle in Rn + 1

In this paper we establish maximum principles of the Cauchy problem for hyperbolic equations in $\text{R}$³ and $\text{R}$^{n + 1}(n ? 2). Our maximum principles generalize the results of Weinberger [5], and Sather [3, 4] for a class of equations such that the coefficients can be allowed to depend upon t, as well, in {x₁, x₂, t}-space and {x₁, x₂,…, x_n, t}-space. Throughout this paper, the influence of the work of Douglis [1] is apparent. See [2].     

The Connection between the Characteristic Roots and the Corresponding Solutions of a Single Linear Differential Equation with Comparable Coefficients

Given a linear differential equation of the form x (ⁿ) + a₁ (t) x (n-1) + …+ a_n (t) x = 0 with variable coefficients defined on the positive semi -axis for t ? 1. We denote its fundamental set of solutions (FSS) by {exp [∫ ri (t) dt] } (i = 1, 2,…,n). In this paper we look for the asymptotic connection (as t → ∞) between the logarithmic derivatives ri (t) of an FSS and of the roots of the characteristic equation yⁿ + a₁ (t) y^n-1 +… + a_n (t) = 0. We mainly consider the case when the coefficients of the equation and the characteristic roots are comparable and have the power order of growth for t → ∞. We discuss the conditions when the functions λ_ii(t) are equivalent to the corresponding roots λ_ii(t) of the characteristic equation as t → ∞.     

Sums of roots of unity in cyclotomic fields

Let ζ_n denote a primitive nth root of unity, n ≥ 4. For any integer k, 2 ≤ k ≤ n ? 2 it is shown that the diophantine equation 1 + ζ_n + ? + ζ_n^k?1 = ?α^q has no solutions with ?, α in $\text{Q}$(ζ_n), ? a root of unity, α an algebraic integer, and q an integer ≥ 2, except when n = 10, 12, or 30, where the solutions are completely determined.     

Characterization and Transformation of Invariants

We consider difference equations of order k _n+k ≥ 2 of the form: y_n+k = f(yn,…,yn+k-1), n= 0,1,2,… where f: D ^k → D is a continuous function, and D?R. We develop a necessary and sufficient condition for the existence of a symmetric invariant I(x ₁,…,x_k ) ∈C^∞[D^k,D]. This condition will be used to construct invariants for linear and rational difference equations. Also, we investigate the transformation of invariants under invertible maps. We generalize and extend several results that have been obtained recently.     

On the Diophantine equation x 2 − kxy + y 2 − 2 n = 0

In this study, we determine when the Diophantine equation x ²?kxy+y ²?2 ⁿ = 0 has an infinite number of positive integer solutions x and y for 0 ? n ? 10. Moreover, we give all positive integer solutions of the same equation for 0 ? n ? 10 in terms of generalized Fibonacci sequence. Lastly, we formulate a conjecture related to the Diophantine equation x ² ? kxy + y ² ? 2 ⁿ = 0.     

Quasi-periodic solutions with prescribed frequency in a nonlinear Schrdinger equation

In this paper, one-dimensional (1D) nonlinear Schrdinger equation iut-uxx + Mσ u + f ( | u | 2 )u = 0, t, x ∈ R , subject to periodic boundary conditions is considered, where the nonlinearity f is a real analytic function near u = 0 with f (0) = 0, f (0) = 0, and the Floquet multiplier Mσ is defined as Mσe inx = σne inx , with σn = σ, when n 0, otherwise, σn = 0. It is proved that for each given 0 σ 1, and each given integer b 1, the above equation admits a Whitney smooth family of small-amplitude quasi-periodic solutions with b-dimensional Diophantine frequencies, corresponding to b-dimensional invariant tori of an associated infinite-dimensional Hamiltonian system. Moreover, these b-dimensional Diophantine frequencies are the small dilation of a prescribed Diophantine vector. The proof is based on a partial Birkhoff normal form reduction and an improved KAM method.     

On a Markoff-like chain

Set {I_r = θ|θ = [a₀, a₁, a₂, …, a_n, …] with a_n ≥ r from some point on}. In the context of I₁ (the set of all irrational numbers) the chain of approximation theorems is the well-known Markoff chain (J. W. S. Cassels, “An Introduction to Diophantine Approximation,” Cambridge Tracts No. 45, Cambridge, 1959). The purpose of this paper is to find the chain of approximation theorems when restricted to I_r with r ≥ 2.     

On covering a product of sets with products of their subsets

Suppose that S₁,…,S_N are collections of subsets of X₁,…,X_N, respectively, such that n_i subsets belonging to S_i, and no fewer, cover X_i for all i. the main result of this paper is that to cover X₁ x…x X_N requires no fewer than σ^N_i=1 (n_i–1) + 1 and no more than Π^N_i=1n_i subsets of the form A₁ x…x A_N, where A_i∈S₁foralli. Moreo ver, these bounds cannot be improved. Identical bounds for the spanning number of a normal product of graphs are also obtained.     

The number of large prime divisors of consecutive integers

Let ?(N) > 0 be a function of positive integers N and such that ?(N) → 0 and N^?(N) → ∞ as N → + ∞. Let Nν_N(n:…) be the number of positive integers n ≤ N for which the property stated in the dotted space holds. Finally, let g(n; N, ?, z) be the number of those prime divisors p of n which satisfy N^Z?(N) ? p ? N^?(N), 0 < z < 1 In the present note we show that for each k = 0, ±1, ±2,…, as N → ∞, limv_N(n : g(n; N, ?, z) ? g(n + 1; N, ?z) = k) exists and we determine its actual value. The case k = 0 induced the present investigation. Our solution for this value shows that the natural density of those integers n for which n and n + 1 have the same number of prime divisors in the range (1) exists and it is positive.     

THE EXTREME POINTS OF SEVERAL CLASSES OF ANALYTIC FUNCTIONS

1Intr0ducti0nLetAden0tethesetofallfunctionsanalyticinA={z:Izl<1}.LetB={W:WEAandIW(z)l51}.Aisalocallyconvexlineaztop0l0gicalspacewithrespecttothetopologyofuniformconvergenceon`c0mpact8ubsetsofA-LetTh(c1,'tc.-1)={p(z):p(z)EA,Rop(z)>0,p(z)=1 clz czzz ' c.-lz"-l 4z" ',wherecl,',cn-1areforedcomplexconstants}.LetTh,.(b,,-..,b,-,)={p(z):P(z)'EAwithReP(z)>Oandp(z)=1 blz ' b.-lz"-l 4z" '-,wherebl,-'-jbu-1areffeedrealconstantsanddkarerealnumbersf0rk=n,n 1,'--}-LetTu(l1,'i'tI.-1)={…