Constructive factorization of some almost periodic triangular matrix functions with a quadrinomial off diagonal entry

Using an abbreviation eμ to denote the function eiμx on the real line R, let , where f is a linear combination of the functions eα, eβ, eα−λ, eβ−λ with some . The criterion for G to admit a canonical factorization was established recently by Avdonin, Bulanova and Moran (2007) [1]. We give an alternative approach to the matter, proving the existence (when it does take place) via deriving explicit factorization formulas. The non-existence of the canonical factorization in the remaining cases then follows from the continuity property of the geometric mean.     

Blow-up and stability of semilinear PDEs with gamma generators

We investigate finite-time blow-up and stability of semilinear partial differential equations of the form , w₀(x)=φ(x)?0, x∈R₊, where Γ is the generator of the standard gamma process and ν>0, σ∈R, β>0 are constants. We show that any initial value satisfying c₁x−a₁?φ(x), x>x₀, for some positive constants x₀, c₁, a₁, yields a non-global solution if a₁β<1+σ. If , where x₀,c₂,a₂>0, and a₂β>1+σ, then the solution wt is global and satisfies , for some constant C>0. This complements the results previously obtained in [M. Birkner et al., Proc. Amer. Math. Soc. 130 (2002) 2431; M. Guedda, M. Kirane, Bull. Belg. Math. Soc. Simon Stevin 6 (1999) 491; S. Sugitani, Osaka J. Math. 12 (1975) 45] for symmetric α-stable generators. Systems of semilinear PDEs with gamma generators are also considered.     

The oscillation of differential transforms - entire Jacobi expansions

Let L=(1−x²)D²−((β−α)−(α+β+2)x)D with , and . Let f∈C^∞[−1,1], , with normalized Jacobi polynomials and the Cn decrease sufficiently fast. Set Lk=L(Lk−1), k?2. Let ρ>1. If the number of sign changes of (Lkf)(x) in (−1,1) is O(k1/(ρ+1)), then f extends to be an entire function of logarithmic order . For Legendre expansions, the result holds with replaced with .     

A general iterative algorithm for nonexpansive mappings in Hilbert spaces

Let H be a real Hilbert space. Suppose that T is a nonexpansive mapping on H with a fixed point, f is a contraction on H with coefficient 0<α<1, and F:H→H is a k-Lipschitzian and η-strongly monotone operator with k>0,η>0. Let . We proved that the sequence {x_n} generated by the iterative method x_n+1=α_nγf(x_n)+(I−μα_nF)Tx_n converges strongly to a fixed point , which solves the variational inequality , for x∈F_ix(T).     

A complete determination of Rabinowitsch polynomials

Let m be a positive integer and fm(x) be a polynomial of the form fm(x)=x²+x−m. We call a polynomial fm(x) a Rabinowitsch polynomial if for and consecutive integers x=x₀,x₀+1,…,x₀+s−1, |fm(x)| is either 1 or prime. In this paper, we show that there are exactly 14 Rabinowitsch polynomials fm(x).     

Rectifiable oscillations in second-order linear differential equations

We study the linear differential equation , on I=(0,1), where the coefficient f(x) is strictly positive and continuous on I, and satisfies the Hartman-Wintner condition at x=0. The four main results of the paper are: (i) a criterion for rectifiable oscillations of (P), characterized by the integrability of on I; (ii) a stability result for rectifiable and unrectifiable oscillations of (P), in terms of a perturbation on f(x); (iii) the s-dimensional fractal oscillations (for which we assume also f(x)∼cx−α when x→0, α>2, and s=max{1,3/2−2/α}); and (iv) the co-existence of rectifiable and unrectifiable oscillations in the absence of the Hartman-Wintner condition on f(x). Explicit examples related to the above results are given.     

On the number of directions determined by a pair of functions over a prime field

A three-dimensional analogue of the classical direction problem is proposed and an asymptotically sharp bound for the number of directions determined by a non-planar set in AG(3,p), p prime, is proved. Using the terminology of permutation polynomials the main result states that if there are more than pairs with the property that f(x)+ag(x)+bx is a permutation polynomial, then there exist elements c,d,e∈Fp with the property that f(x)=cg(x)+dx+e.     

Maximum principles for bounded solutions of the telegraph equation in space dimensions two and three and applications

A maximum principle is proved for the weak solutions of the telegraph equation in space dimension three u_tt−Δ_xu+cu_t+λu=f(t,x), when c>0, λ∈(0,c²/4] and (Theorem 1). The result is extended to a solution and a forcing belonging to a suitable space of bounded measures (Theorem 2). Those results provide a method of upper and lower solutions for the semilinear equation u_tt−Δ_xu+cu_t=F(t,x,u). Also, they can be employed in the study of almost periodic solutions of the forced sine-Gordon equation. A counterexample for the maximum principle in dimension four is given.     

Class numbers of quadratic fields, Hasse invariants of elliptic curves, and the supersingular polynomial

Using the theory of elliptic curves, we show that the class number h(−p) of the field appears in the count of certain factors of the Legendre polynomials , where p is a prime >3 and m has the form (p−e)/k, with k=2,3 or 4 and . As part of the proof we explicitly compute the Hasse invariant of the Hessian curve y²+αxy+y=x³ and find an elementary expression for the supersingular polynomial ss_p(x) whose roots are the supersingular j-invariants of elliptic curves in characteristic p. As a corollary we show that the class number h(−p) also shows up in the factorization of certain Jacobi polynomials.     

On the prime power factorization of n!

In this paper, we prove two results. The first theorem uses a paper of Kim (J. Number Theory 74 (1999) 307) to show that for fixed primes p₁,…,p_k, and for fixed integers m₁,…,m_k, with , the numbers (e_p1(n),…,e_pk(n)) are uniformly distributed modulo (m₁,…,m_k), where e_p(n) is the order of the prime p in the factorization of n!. That implies one of Sander's conjectures from Sander (J. Number Theory 90 (2001) 316) for any set of odd primes. Berend (J. Number Theory 64 (1997) 13) asks to find the fastest growing function f(x) so that for large x and any given finite sequence , there exists n<x such that the congruences hold for all i?f(x). Here, p_i is the ith prime number. In our second result, we are able to show that f(x) can be taken to be at least , with some absolute constant c₁, provided that only the first odd prime numbers are involved.     

Polynomial almost periodic solutions for a class of Riemann-Hilbert problems with triangular symbols

Let with α,β∈]0,1[ such that α+β<1, αβ⁻¹∉Q and a,b,c∈C?{0}. In this paper the existence of almost-periodic polynomial (APP) solutions to the equation (with and ) is studied. The natural space in which to seek a solution to the above problem is the space of almost periodic functions with spectrum in the group αZ+βZ+Z. Due to the difficulty in dealing with the problem in that generality, solutions are sought with spectrum in the group αZ+βZ. Several interesting and totally new results are obtained. It is shown that, if 1∉αZ+βZ, no polynomial solutions exist, i.e., almost periodic polynomial solutions exist only if αZ+βZ=αZ+βZ+Z. Keeping to this setting, it is shown that APP solutions exist if and only if the function satisfies the simple spectral condition α+β>1/2. The proof of this result is nontrivial and has a number-theoretic flavour. Explicit formulas for the solution to the above problem are given in the final section of the paper. The derivation of these formulas is to some extent a byproduct of the proof of the result on the existence of APP solutions.     

Solutions of non-periodic super-quadratic Dirac equations

This paper is concerned with solutions to the Dirac equation: −i∑αkk_∂u+aβu+M(x)u=Ru(x,u). Here M(x) is a general potential and R(x,u) is a self-coupling which is super-quadratic in u at infinity. We use variational methods to study this problem. By virtue of some auxiliary system related to the “limit equation” of the Dirac equation, we construct linking levels of the variational functional ΦM such that the minimax value cM based on the linking structure of ΦM satisfies , where is the least energy of the “limit equation”. Thus we can show the c(C)-condition holds true for all and consequently obtain one least energy solution to the Dirac equation.     

Interior estimates for solutions of a fourth order nonlinear partial differential equation

Let be a locally strongly convex hypersurface, given by the graph of a convex function xn+1=f(x₁,…,xn) defined in a convex domain Ω⊂Rn. M is called a α-extremal hypersurface, if f is a solution of

     

Strong convergence theorem for uniformly L-Lipschitzian asymptotically pseudocontractive mapping in real Banach space

Let E be a real Banach space. Let K be a nonempty closed and convex subset of E, a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with sequence {kn}_n?0⊂[1,+∞), limn→∞kn=1 such that F(T)≠∅. Let {αn}_n?0⊂[0,1] be such that _∑n?0αn=∞, and _∑n?0αn(kn−1)<∞. Suppose {xn}_n?0 is iteratively defined by xn+1=(1−αn)xn+αnTnxn, n?0, and suppose there exists a strictly increasing continuous function , ?(0)=0 such that 〈Tnx−x^∗,j(x−x^∗)〉?kn‖x−x^∗‖2−?(‖x−x^∗‖), ∀x∈K. It is proved that {xn}_n?0 converges strongly to x^∗∈F(T). It is also proved that the sequence of iteration {xn} defined by xn+1=anxn+bnTnxn+cnun, n?0 (where {un}_n?0 is a bounded sequence in K and {an}_n?0, {bn}_n?0, {cn}_n?0 are sequences in [0,1] satisfying appropriate conditions) converges strongly to a fixed point of T.     

Decay property of solutions for damped wave equations with space-time dependent damping term

We consider the Cauchy problem for the damped wave equation with space-time dependent potential b(t,x) and absorbing semilinear term |u|^ρ−1u. Here, with b₀>0, α,β?0 and α+β∈[0,1). Using the weighted energy method, we can obtain the L² decay rate of the solution, which is almost optimal in the case ρ>ρc(N,α,β):=1+2/(N−α). Combining this decay rate with the result that we got in the paper [J. Lin, K. Nishihara, J. Zhai, L²-estimates of solutions for damped wave equations with space-time dependent damping term, J. Differential Equations 248 (2010) 403-422], we believe that ρc(N,α,β) is a critical exponent. Note that when α=β=0, ρc(N,α,β) coincides to the Fujita exponent ρF(N):=1+2/N. The new points include the estimate in the supercritical exponent and for not necessarily compactly supported data.