Positive solutions of a nonlinear initial-value problem on half-line

The nonlinear initial-value problemu″(t)+f(t,u(t))=0,u(t ₀)+bu′(t ₀)=c,t ₀≥0,b≤0,c≥0, is considered for positive solutions on [t ₀, ∞). Existence of positive solutions is proved without the hypothesis thatf(t, ω)≥0 (or ≤0), using the lattice fixed point theorem. A monotonicity condition inf(t, ω) is used to prove the uniqueness of the solution of the initial-value problem. Whenf(t, ω)≥0 (or ≤0), uniqueness is also obtained under a sublinearity condition onf(t, ω).     

The existence of ground states for fourth-order wave equations

Focusing on the fourth-order wave equation u_tt+Δ²u+f(u)=0, we prove the existence of ground state solutions u=u(x+ct) for an optimal range of speeds c∈Rⁿ and a variety of nonlinearities f.     

Weighted Berezin transform in the polydisc

For c>−1, let νc denote a weighted radial measure on C normalized so that νc(D)=1. If f is harmonic and integrable with respect to νc over the open unit disc D, then for every ψ∈Aut(D). Equivalently f is invariant under the weighted Berezin transform; Bcf=f. Conversely, does the invariance under the weighted Berezin transform imply the harmonicity of a function? In this paper, we prove that for any 1?p<∞ and c₁,c₂>−1, a function f∈Lp(D²,νc₁×νc₂) which is invariant under the weighted Berezin transform; Bc₁,c₂f=f needs not be 2-harmonic.     

On local convexity of nonlinear mappings between Banach spaces

We find conditions for a smooth nonlinear map f: U → V between open subsets of Hilbert or Banach spaces to be locally convex in the sense that for some c and each positive ? < c the image f(B _?(x)) of each ?-ball B _?(x) ? U is convex. We give a lower bound on c via the second order Lipschitz constant Lip₂(f), the Lipschitz-open constant Lip_o(f) of f, and the 2-convexity number conv₂(X) of the Banach space X.     

Value distribution and shared sets of differences of meromorphic functions

We investigate value distribution and uniqueness problems of difference polynomials of meromorphic functions. In particular, we show that for a finite order transcendental meromorphic function f with λ(1/f)<ρ(f) and a non-zero complex constant c, if n?2, then fn(z)f(z+c) assumes every non-zero value a∈C infinitely often. This research also shows that there exist two sets S₁ with 9 (resp. 5) elements and S₂ with 1 element, such that for a finite order nonconstant meromorphic (resp. entire) function f and a non-zero complex constant c, Ef(z)(Sj)=Ef(z+c)(Sj)(j=1,2) imply f(z)≡f(z+c). This gives an answer to a question of Gross concerning a finite order meromorphic function f and its shift.     

Meromorphic functions sharing a set with applications to difference equations

This paper is devoted to proving some uniqueness type results for an entire function f(z) that shares a common set with its shift f(z+c) or its difference operator Δcf. We also give some applications to solutions of non-linear difference equations related to a conjecture proposed by C.C. Yang.     

Zeta measures and Thermodynamic Formalism for temperature zero

We address the analysis of the following problem: given a real Hölder potential f defined on the Bernoulli space and μ _f its equilibrium state, it is known that this shift-invariant probability can be weakly approximated by probabilities in periodic orbits associated to certain zeta functions. Given a Hölder function f > 0 and a value s such that 0 < s < 1, we can associate a shift-invariant probability ν _s such that for each continuous function k we have $ \int {kd} v_s = \frac{{\sum\nolimits_{n = 1}^\infty {\sum\nolimits_{x \in Fix_n } {e^{sf^n (x) - nP(f)\frac{{k^n (x)}} {n}} } } }} {{\sum\nolimits_{n = 1}^\infty {\sum\nolimits_{x \in Fix_n } {e^{sf^n (x) - nP(f)} } } }}, $ , where P(f) is the pressure of f, Fix _n is the set of solutions of σ ⁿ (x) = x, for any n ∈ ?, and f ⁿ (x) = f(x) + f(σ (x)) + … + f(σ ^n?1(x)). We call νs a zeta probability for f and s, because it can be obtained in a natural way from the dynamical zeta-functions. From the work of W. Parry and M. Pollicott it is known that ν _s → µ _f , when s → 1. We consider for each value c the potential c f and the corresponding equilibrium state μ _cf . What happens with ν _s when c goes to infinity and s goes to one? This question is related to the problem of how to approximate the maximizing probability for f by probabilities on periodic orbits. We study this question and also present here the deviation function I and Large Deviation Principle for this limit c → ∞, s → 1. We will make an assumption: for some fixed L we have lim _{c→∞, s→1} c(1 ? s) = L > 0. We do not assume here the maximizing probability for f is unique in order to get the L.D.P.     

Existence and asymptotics of traveling waves for nonlocal diffusion systems

In this paper, we deal with the existence and asymptotic behavior of traveling waves for nonlocal diffusion systems with delayed monostable reaction terms. We obtain the existence of traveling wave front by using upper-lower solutions method and Schauder’s fixed point theorem for c > c₁(τ) and using a limiting argument for c = c₁(τ). Moreover, we find a priori asymptotic behavior of traveling waves with the help of Ikehara’s Theorem by constructing a Laplace transform representation of a solution. Especially, the delay can slow the minimal wave speed for ?₂f(0, 0) > 0 and the delay is independent of the minimal wave speed for ?₂f(0, 0) = 0.     

Probabilities of dominant candidates based on first-place votes

Suppose each of an odd number n of voters has a strict preference order on the three ‘candidates’ in {1,2,3} and votes for his most preferred candidate on a plurality ballot. Assume that a voter who votes for i is equally likely to have ijk and ikj as his preference order when {i,j,k} = {1,2,3}.Fix an integer m between $\text{1}\text{2}$(n + 1) and n inclusive. Then, given that n_i of the n voters vote for i, let f_m(n₁,n₂,n₃) be the probability that one of the three candidates is preferred by m or more voters to each of the other two.This paper examines the behavior of f_m over the lattice points in L_n, the set of triples of non-negative integers that sum to n. It identifies the regions in L_n where f_m is 1 and where f_m is 0, then shows that f_m(a,b + 1, c)>f_m(a + 1,b,c) whenever a + b + c + 1 = n, a≤c≤b, a<c<m and c≤n ? m. These results are used to partially identify the points in L_n where f_m is minimized subject to f_m>0. It is shown that at least two of the n_i are equal at minimizing points.     

Ruzsa’s constant on additive functions

A function f : N → R is called additive if f(mn)= f(m)+f(n)for all m, n with(m, n)= 1. Let μ(x)= max n≤x(f(n)f(n + 1))and ν(x)= max n≤x(f(n + 1)f(n)). In 1979, Ruzsa proved that there exists a constant c such that for any additive function f , μ(x)≤ cν(x 2 )+ c f , where c f is a constant depending only on f . Denote by R af the least such constant c. We call R af Ruzsa's constant on additive functions. In this paper, we prove that R af ≤ 20.     

On the successive coefficients of close-to-convex functions

Let f∈S, f be a close-to-convex function, f_k(z)=[f(z^k)]^1/k. The relative growth of successive coefficients of f_k(z) is investigated. The sharp estimate of ||c_n+1|−|c_n|| is obtained by using the method of the subordination function.     

On the distribution of values of certain divisor functions

Let {?_d} be a sequence of nonnegative numbers and f(n) = Σ?_d, the sum being over divisors d of n. We say that f has the distribution function F if for all c ≥ 0, the number of integers n ≤ x for which f(n) > c is asymptotic to xF(c), and we investigate when F exists and when it is continuous.     

Local analytic solutions of the generalized Dhombres functional equation II

We study local analytic solutions f of the generalized Dhombres functional equation f(zf(z))=φ(f(z)), where φ is holomorphic at w₀≠0, f is holomorphic in some open neighborhood of 0, depending on f, and f(0)=w₀. After deriving necessary conditions on φ for the existence of nonconstant solutions f with f(0)=w₀ we describe, assuming these conditions, the structure of the set of all formal solutions, provided that w₀ is not a root of 1. If |w₀|≠1 or if w₀ is a Siegel number we show that all formal solutions yield local analytic ones. For w₀ with 0<|w₀|<1 we give representations of these solutions involving infinite products.     

Properties of the set of positive solutions to Dirichlet boundary value problems with time singularities

The paper investigates the structure and properties of the set S of all positive solutions to the singular Dirichlet boundary value problem u″(t) + au′(t)/t ? au(t)/t ² = f(t, u(t),u′(t)), u(0) = 0, u(T) = 0. Here a ∈ (?∞,?1) and f satisfies the local Carathéodory conditions on [0,T]×D, where D = [0,∞)×?. It is shown that S _c = {u ∈ S: u′(T) = ?c} is nonempty and compact for each c ≥ 0 and S = ∪ _c≥0 S _c . The uniqueness of the problem is discussed. Having a special case of the problem, we introduce an ordering in S showing that the difference of any two solutions in S _c ,c≥ 0, keeps its sign on [0,T]. An application to the equation v″(t) + kv′(t)/t = ψ(t)+g(t, v(t)), k ∈ (1,∞), is given.     

The Müntz-Szász theorem and the closure of translates

We study the closure properties in various spaces, of certain sequences of the form {f(t + c_k)} or {exp(c_kti)f(t)}. As an application, we give another proof of the Müntz-Szász theorem for L₂(0, 1).     

Weighted tight frames of exponentials on a finite interval

Given a finite intervalI?R, a characterization is given for those discrete sets of real numbers Λ and associated sequences {c _λ}_λ∈Λ, withc _λ>0, having the properties that every functionf∈L ²(I) can be expanded inL ²(I) as the unconditionally convergent series $$f = \sum\limits_{\lambda \in \Lambda } {\hat f} (\lambda )c_\lambda e^{2\pi i\lambda x} $$ and that the range of the mappingL ²(I)→L _μ ² :f→f has finite codimension inL _μ ² , iff denotes the Fourier transform off and μ is the measure μ = ∑_λ∈Λ c _λ δ_λ.     

Decomposition of the completer-graph into completer-partiter-graphs

Forn ≥ r ≥ 1, letf _r (n) denote the minimum numberq, such that it is possible to partition all edges of the completer-graph onn vertices intoq completer-partiter-graphs. Graham and Pollak showed thatf ₂(n) =n ? 1. Here we observe thatf ₃(n) =n ? 2 and show that for every fixedr ≥ 2, there are positive constantsc ₁(r) andc ₂(r) such thatc ₁(r) ≤f _r (n)?n ^?[r/2] ≤n ₂(r) for alln ≥ r. This solves a problem of Aharoni and Linial. The proof uses some simple ideas of linear algebra.     

Employing applied mathematics to expand the bandwidth of heterodyne carrier signals with a small phase modulation index

Frequency modulation (fm) theory states that heterodyne carrier signals cannot be frequency modulated with modulation frequencies f_m higher than the carrier frequency f_c subtracted by the maximum frequency deviation f_d,max. Thus, f_m ? f_c − f_d,max is considered as maximum limit to obtain the correct modulation signal from the carrier signal by fm demodulation. This paper proves mathematically that this limit can be breached for small phase modulation indices. The result is advanced to a new algorithm to demodulate heterodyne carrier signals with a modulation frequency of up to twice the carrier frequency if a small phase modulation index M can be assumed. This assumption is valid in heterodyne laser interferometers for ultrasonic testing where the phase modulation of a detector signal is originated by vibration amplitudes much smaller than the wavelength of the laser. Simulations of the demodulation demonstrate the functioning of the algorithm. The employment of the presented results in an interferometric system demonstrates the impact in metrology instruments for vibration measurements at ultra-high frequencies. Therefore, the influence of the presented algorithm to the measurement uncertainty of interferometric systems is also derived in this paper.     

LOCAL ANALYTIC SOLUTIONS OF A MORE GENERALIZED DHOMBRES EQUATION

We study the local analytic solutions f of the functional equation f(ψ(zf(z)))=(f(z)) for z in some neighborhood of the origin.Whether the solution f vanishes at z=0 or not plays a critical role for local analytic solutions of this equation.In this paper,we obtain results of analytic solutions not only in the case f(0)=0 but also for f(0)≠0.When assuming f(0) =0,for technical reasons,we just get the result for f’(0)≠0.Then when assuming f(0)=ω0≠0,ψ’(0)=s≠0,ψ(z) is analytic at z=0 and(z)is analytic at z=ω0,we give the existence of local analytic solutions f in the case of 0<|sω0|<1 and the case of |sω0|=1 with the Brjuno condition.