Finite Limit Points Research Articles

A note on fixed points of entire function of bicomplex variable

The prime concern of this paper is to derive the bi-complex analogue of the existence theorem of fixed points of order one of entire functions. We discuss some lemmas and prove the existence of an entire function f(z) in C_2 which has the given fixed points a_1,a_2… with given multipliers b_1,b_2… respectively, provided that the sequence {a_n } has no finite limit point in C_2. Finally, we give some examples having different kind of fixed points.

Quasi-periodic solutions for the generalized Benjamin–Bona–Mahony​ equation

A generalized Benjamin–Bona–Mahony (gBBM) equation ut−uxxt+ux+u4ux=0,subject to the periodic boundary condition is studied in this paper. Based on a new infinite dimensional Kolmogorov–Arnold–Moser (KAM) theorem with normal frequencies of finite limit-points, it is shown that the above gBBM equation admits plenty of time-quasi-periodic solutions with two frequencies of high modes.

On eigenfunction expansions of differential equations with degenerating weight

Let A be a symmetric operator. By using the method of boundary triplets we parameterize in terms of a Nevanlinna parameter τ all exit space extensions A˜=A˜⁎ of A with the discrete spectrum σ(A˜) and characterize the Shtraus family of A˜ in terms of abstract boundary conditions. Next we apply these results to the eigenvalue problem for the 2r-th order differential equation l[y]=λΔ(x)y on an interval [a,b),−∞<a<b≤∞, subject to λ-depending separated boundary conditions with entire operator-functions C0(λ) and C1(λ), which form a Nevanlinna pair (C0,C1). The weight Δ(x) is nonnegative and may vanish on some intervals (α,β)⊂I. We show that in the case when the minimal operator of the equation has the discrete spectrum (in particular, in the case of the quasiregular equation) the set of eigenvalues of the eigenvalue problem is an infinite subset of R without finite limit points and each function y∈LΔ2(I) admits the eigenfunction expansion y(x)=∑k=1∞yk(x) converging in LΔ2(I). Moreover, we give an explicit method for calculation of eigenfunctions yk in this expansion and specify boundary conditions on y implying the uniform convergence of the eigenfunction expansion of y. These results develop the known ones obtained for the case of the positive weight Δ and the more restrictive class of λ-depending boundary conditions.

On the absolutely continuous component of a weak limit of measures on ℝ supported on discrete sets

Let μ 1 , μ 2 , … \mu _1,\mu _2,\ldots be a sequence of positive Borel measures on R \mathbb {R} each of which is supported on a set having no finite limit points. Suppose the sequence μ n \mu _n weakly converges to a Borel measure ν \nu . Let ν a c \nu _{\mathrm {ac}} be the absolutely continuous component of ν \nu , and X ⊂ R X\subset \mathbb {R} the essential support of ν a c \nu _{\mathrm {ac}} . We characterize the set X X in terms of the limiting behavior of the Hilbert transforms of the measures μ n \mu _n . Potential applications include those in spectral theory.

Spectral properties of birth–death polynomials

We consider sequences of polynomials that are defined by a three-terms recurrence relation and orthogonal with respect to a positive measure on the nonnegative axis. By a famous result of Karlin and McGregor such sequences are instrumental in the analysis of birth–death processes. Inspired by problems and results in this stochastic setting we present necessary and sufficient conditions in terms of the parameters in the recurrence relation for the smallest or second smallest point in the support of the orthogonalizing measure to be larger than zero, and for the support to be discrete with no finite limit point.

Discrete unbounded sets in a finite dimensional space and beyond

Abstract Discrete infinite sets in a finite dimensional space, i.e., infinite sets without finite limit points appear in various branches of analysis (zero and pole sets of meromorphic functions, various models in the mathematical theory of quasicrystals, and so on). Here we introduce some notions and present some new theorems connected with such sets.

The splitting relation for Fréchet spaces over non-archimedean fields

Abstract. A complete valued field ( K , | · | ) $(K, |\cdot |)$ is non-archimedean if its valuation satisfies the strong triangle inequality | a + b | ≤ max { | a | , | b | } $|a+b|\le \max \lbrace |a|, |b|\rbrace $ for all a , b ∈ K $a,b \in K$ . We say that a pair (E,F) of Fréchet spaces over a non-archimedean field 𝕂 is splitting if for every Fréchet space G over 𝕂 and for every closed subspace D of G such that D is isomorphic to F and G/D is isomorphic to E, we can infer that the subspace D is complemented in G. In this paper we study when a pair (E,F) of Fréchet spaces of countable type over 𝕂 is splitting. In particular, we show that a pair ( A s ( a ) , A r ( b ) ) $(A_s(a), A_r(b))$ of power series spaces over 𝕂 is splitting if and only if s = ∞ $s=\infty $ or s = 1 $s=1$ and the set M a , b $M_{a,b}$ of all finite limit points of the double sequence ( a p / b q ) p , q ∈ ℕ $(a_p/b_q)_{p,q\in \mathbb {N}}$ is bounded.

On tame operators between non-archimedean power series spaces

Let p ∈ {1,∞}. We show that any continuous linear operator T from A 1(a) to A p (b) is tame, i.e., there exists a positive integer c such that sup x ‖T x ‖ k /|x| ck < ∞ for every k ∈ ℤ. Next we prove that a similar result holds for operators from A ∞(a) to A p (b) if and only if the set M b,a of all finite limit points of the double sequence (b i /a j )i,j∈ℤ is bounded. Finally we show that the range of every tame operator from A ∞(a) to A ∞(b) has a Schauder basis.

Cauchy problem for an extended model of combustion

SynopsisThis paper considers the Cauchy problem for an extended model of combustion (u + qz)t + f(u)x = 0, zt + kg(u)z = 0 with Lp bounded initial data, where g(u) is a piecewise Lipschitz continuous function and its discontinuous points have no finite limit point. The integral representation gives a definition of a weak solution in Lp space. Some existence results are obtained based on a simplified method of compensated compactness in which the weak continuity theorem of 2 * 2 determinants plays a more important role, but the idea of Young measures has been avoided.

Impulse systems with fixed shock times of general disposition: Existence, uniqueness of solution, and the well-posedness of the Cauchy problem

One investigates systems of differential equations with impulse actions at fixed times, when the sequence of the “shock” times may have finite limit points. For such systems one obtains theorems for the existence and uniqueness of the solution, as well as for the well-posedness of the Cauchy problem.

Gaps Between Prime Numbers

Let ${d_n} = {p_{n + 1}} - {p_n}$ denote the $n$th gap in the sequence of primes. We show that for every fixed integer $k$ and sufficiently large $T$ the set of limit points of the sequence $\{ ({d_n}/\log n, \ldots ,{d_{n + k - 1}}/\log n)\}$ in the cube ${[0,T]^k}$ has Lebesgue measure $\geq c(k){T^k}$, where $c(k)$ is a positive constant depending only on $k$. This generalizes a result of Ricci and answers a question of Erdös, who had asked to prove that the sequence $\{ {d_n}/\log n\}$ has a finite limit point greater than 1.

Gaps between prime numbers

Letdn=pn+1−pn{d_n} = {p_{n + 1}} - {p_n}denote thennth gap in the sequence of primes. We show that for every fixed integerkkand sufficiently largeTTthe set of limit points of the sequence{(dn/log⁡n,…,dn+k−1/log⁡n)}\{ ({d_n}/\log n, \ldots ,{d_{n + k - 1}}/\log n)\}in the cube[0,T]k{[0,T]^k}has Lebesgue measure≥c(k)Tk\geq c(k){T^k}, wherec(k)c(k)is a positive constant depending only onkk. This generalizes a result of Ricci and answers a question of Erdös, who had asked to prove that the sequence{dn/log⁡n}\{ {d_n}/\log n\}has a finite limit point greater than 1.

ON THE CLOSURE OF THE SET OF FINITE-ZONE POTENTIALS

A special class of almost-periodic potentials of the one-dimensional Schr?dinger operator is studied. These potentials are limits of finite-zone potentials. In contrast to previous work of the author in which a class of almost-periodic potentials was considered with lacunae in the spectrum having a single limit point at infinity, in this paper the case is studied where the lacunae in the spectrum also have finite limit points. There may be infinitely many of the latter.Figures: 2. Bibliography: 10 titles.

On the preimage sets of entire functions

From the classical Weierstrass factorization theorem we know that for any given sequence {an} of complex numbers that has no finite limit point, there exists an entire function / such that the sequence {an} is precisely the zeros of / (counted by multiplicities). A natural question to ask is: given a sequence {an} that has no finite limit point, does there exist an entire function / such that the sequence {an} is precisely the zeros and one-points of / ? A similar but more restrictive question was posed by Rubel and Yang [14, p. 289]. The answer to the question here, in general, is no. Gross [4] made the following general definition.

Summability of Matrix Transforms of Subsequences

AbstractD. F. Dawson has considered several questions of the following nature. Suppose T is a regular matrix summability method. If A is a regular matrix and x is a sequence having a finite limit point, then there exists a subsequence y of x such that each finite limit point of x is a T-limit point of Ay. In the present paper, we show the regularity condition for A may be replaced by the requirement that A be a limit preserving bv to c map. This leads to summability characterizations for several classes of sequences.

Limit theorems under the assumption of restricted convergence on a set with a finite limit point

We consider double sequences {{Xnk}} of independent and asymptotically constant random variables. For certain constants An we put The central problem is the following one: Assume that {Fn} for x ϵ I ⊂ R1 converges weakly to a nonconstant limit function ψ where I is a set with a finite limit point (restricted convergence Then additional conditions ensuring the complete convergence Fn ⟹ F to a certain limit distribution function F are given. These additional conditions are weaker than the corresponding sufficient conditions known from the classical theory. Further, these results yield two new versions of the central limit theorem, see § 2. In the case of identically distributed summands with common distribution function V the assumption a sufficient to prove Fn ⟹ Fα, provided that (Fα stands for a stable distribution with characteristic exponent α).

ON THE SINGULAR SPECTRUM IN A SYSTEM OF THREE PARTICLES

Let be the energy operator of a system of three pairwise interacting particles whose pair potentials admit the estimate and suppose the subsystems of two particles have no virtual levels. It is established that the singular continuous spectrum of is empty and its positive eigenvalues have no finite limit points. The considerations of the paper are based on a study of Faddeev's equations in coordinate representation and an application of imbedding theorems for anisotropic Sobolev classes in the space .Bibliography: 13 titles.

Summability of subsequences and rearrangements of sequences

Sufficient conditions have been given that require a matrix A to have the property that every sequence x with a finite limit point have a subsequence y such that each finite limit point of x is a limit point of Ay. In this paper, we show that these conditions may be weakened and obtain an analog in which “subsequence” is replaced with “rearrangement".

The distribution of the poles of the best approximating rational functions and the analytical properties of the approximated function

The following theorem was proved in our paper in Math. USSR-Sb.9 (1969): Iff(z)∈H 2 and the poles of the rational functions of best approximation tend to infinity sufficiently quickly, thenf(z) is an entire function. In the present article we weaken the restrictions on the distribution of poles by assuming only that these poles have no finite limit point (Theorem 1). Some generalizations of this result are also given.

Strong and Weak Limit Points of a Normalized Random Walk

Let $S_n = \sum^n_1 X_i$ be a random walk. A point $b$ is called a strong limit point of $n^{-\alpha}S_n$ if there exists a nonrandom sequence $n_k\rightarrow\infty$ such that $n_k^{-\alpha}S_{n_k}\rightarrow b$ w.p. 1. The possible structures for the set of strong limit points of $n^{-\alpha}S_n$ are determined. We also give a sufficient condition for $n^{-1}S_n$ to be dense in $\mathbb{R}$. In particular $n^{-1}S_n$ is dense in $\mathbb{R}$ when $E|X_1| = \infty$ and $n^{-1}S_n$ has a finite strong limit point.