Lim Xn Research Articles

Topological classification of function spaces with the Fell topology IV

For a Tychonoff space X, let ↓CF(X) denote the collection of the hypographs of all continuous maps from X to [0,1] with the Fell topology. We show that, for a Tychonoff non-discrete k-space X, the function space ↓CF(X) is homeomorphic to c0∪(Q∖Σ) if ↓CF(X) is metrizable and the set of isolated points of X is dense in X, where Q=[−1,1]N is the Hilbert cube, Σ={(xn)∈Q:sup⁡|xn|<1} and c0={(xn)∈Σ:lim⁡xn=0} are its subspaces. Combining results in the previous papers of the series, we give the topological classification for all metrizable function spaces ↓CF(X) of k-spaces X.

Topological classification of function spaces with the Fell topology II

For a Tychonoff space X, let ↓CF(X) denote the collection of the hypographs of all continuous maps from X to [0,1] with the Fell topology. We show that for a Tychonoff k-space X, ↓CF(X) is homeomorphic to c0 if and only if ↓CF(X) is metrizable and not Baire if and only if X is a weakly locally compact and hemicompact ℵ0-space without dense set of isolated points, where Q=[−1,1]ω is the Hilbert cube, Σ={(xn)∈Q:sup⁡|xn|<1} and c0={(xn)∈Σ:lim⁡xn=0} are the subspaces of it.

Iterated nets of weakly contractive operators

Let K be a subset of a Banach space X. An operator T:K→K is said to be weakly contractive, if for every point x∈K there is a number C(x) and for any sequence {xn} with lim xn = x there exists a number N such that ∥Tx−Txn∥ ≤ C(x)∥x−xn∥ for all n&gt;N. Every contractive operator is weakly contractive.The investigation of transfinite iteration processes leads to the study of the sequence space Σ(X) with Tikhonov's topology, to the study of fundamental and uniformly fundamental nets in the space Σ(X). In Theorem 2, by using regular operators summing divergent sequences, necessary and sufficient conditions for the existence of a fixed point of a weakly contractive operator are given. In particular, if T is a contractive operator, then by a theorem of Kirk and Massa [1], TΩ(x)=z for all x∈K, and by Theorem 2 T(z)=z.

Asymptotic behaviour of unbounded nonexpansive sequences in Banach spaces

Let X be a real Banach space, (x0 )n>O a nonexpansive sequence in X (i.e., llxil xj+1 11 0), and C the closed convex hull of the sequence (x,+i x0),>o We prove that lim0+OO llxn/nll = infn>l II(xn xo)/nll = infZEC lizil and deduce a simple short proof for the following result. (i) If X is reflexive and strictly convex, then x,,/n converges weakly in X to the element of minimum norm PCO in C with IIPCOII = inf Xn -XO = lim Xn n>l fl n-'+0O fl (ii) If X* has Frechet differentiable norm, then X/nl converges strongly to PCO. This result contains previous results by Pazy, Kohlberg and Neyman, Plant and Reich, and Reich and is also optimal since the assumptions made on X in (i) or (ii) are also necessary for the respective conclusion to hold.

On the spectrum of C, as an operator on bv

In 1985 John Reade determined the spectrum of the Cesâro Operatör which is repre- sented by the matrix: / I 0 0 . . . \ C ı — ı 2 0 ... I regarded as an operatör on the space co of ali null sequences normed by || x j| = sup [ xn [. n> 0 It is the purpose of this paper to determine the spectrum of regarded as an operatör on the 00 space bv of ali sequences x such that lim xn exists and ][ x || = lim | xn 4- 2 ]xn + l-xn | n—>Oo n->00 n=0 < 00 We do so by proving that (C , - XI)-1 GB (bv) for ali XeC such that [ X- | > i •

Phyllotactic Iterations

AbstractThe origin of the golden angle γ = 137.5° found in most leaf arrangements is investigated by an inhibitor diffusion model. A recursive iteration scheme is set up generating inhibitor concentrations on the meristematic ring and angles xn between succeeding leaves. The universality of γ is reflected by existence of a broad parameter range with lim xn ≈ γ. Likewise, the golden angle is shown to occur in a modified Gierer‐Meinhardt model of nonlinear activator and inhibitor reactions.

Almost convergent and almost summable sequences

2. Definitions. Let m be the Banach space of real bounded sequences x = { x, } with the usual norm I|I x = sup Ix,| . There exist continuous linear functionals on m called Banach limits [1]. It is well known that any Banach limit of x lies between lim inf xn and lim sup xn. An element x of m is said to be almost convergent, or F-summable, if all of its Banach limits are equal. We let F denote the set of all almost convergent sequences and we write F-lim x = s if s is the common value of all Banach limits of x. Lorentz [3] showed that F-lim x = s if and only if limp (xn + * * * +x=+, )/p =s uniformly in n. He extended F-limits to bounded complex sequences by means of this condition. Let A = (ank) be a real infinite matrix. We say that x is FA-summable to s if and only if lim >,k kankXk+p = S uniformly in p. If A (x) is the sequence >k ankXk}, x is said to be almost A-summable to s if and only if A (x) is in F and F-lim A (x) = s. The matrix A is said to be regular if A (x) converges to lim xn for all convergent sequences x. If F-lim A (x) = lim xn for all convergent sequences x, then the matrix A is said to be almost regular. According to [2 ], a matrix A is almost regular if and only if (1) >k Iankl <Hforalln, (2) a(k) = { ank } F and F-lim a(k) = O for every k and (3) a= { k kank} E F and F-lim a=1.

On Banach Limit of Fourier Series and Conjugate Series I

Let (xn) be a sequence of real numbers. (xn) corresponds to a number Lim xn called the Banach limit of (xn) satisfying the following conditions:(1)Lim (axn + byn) = a Lim xn + b Lim yn(2)If xn ≥ 0 for every n, then Lim xn ≥ 0(3)Lim xn+1 = Lim xn(4)If xn = 1 for every n, then Lim xn = 1The existence of such limits is proved by Banach [1].

Essential cluster sets

1. In 1928, W.H. Young [6] showed that an arbitrary real function of a real variable has a remarkable symmetry property. Let f: R -> R be arbitrary. A number y is a right limit of f at x E R if there is a sequence xn > x, n = 1,... such that lim xn = x and lim f(x,,) = y. A left limit is defined similarly. The theorem of Young asserts that for every f the set of right limits is the same as the set of left limits for every x E R A, where A is a countable set. Young obtained an analogous result for functions of two variables. Here the role of right and left limits is assumed by limits within sectors with the point in question as vertex. Each sector yields a limit set. The result is that all the sectorial limit sets are the same at every p E R2A, where A is of planar measure zero and first category, but not necessarily countable. In 1938, H. Blumberg [2] obtained a variety of results along the lines of those obtained by Young. He pointed out, for example, that in the planar case, if the angles of the sectors considered exceed 1800, then the countable character of the exceptional set A is restored. Returning to the original theorem of Young, Zahorski asked if the result still holds if sets of density zero are neglected. L. Belowska [1] has shown that this is false by constructing a function where in this case the exceptional set is uncountable. However, M. Kulbacka [4] has shown that the exceptional set is always of the first category and measure zero. A simple proof of these results was given by C. Goffman [3]. It is natural to ask whether there are theorems similar to the ones of Young and Blumberg in higher dimensions, but with sets of density zero neglected. This paper gives an affirmative answer to this question by showing that a theorem analogous to the two dimensional result of Young holds if one disregards sets of density zero. In this connection, various examples are given relating to the structure of the exceptional set.

An abstract formulation of some theorems on cluster sets

There are a number of results in the literature which affirm the symmetric structure of functions in terms of Cluster Sets. For example, W. H. Young [5] showed that an arbitrary function f: R-*R has a remarkable symmetry property which may be stated in terms of Cluster Sets. A real number y is a right limit of f at xER if there exists a sequence xn>x, n= 1, 2, * , such that lim xn,=x and limf(xn) =y. A left limit of f at xER is defined similarly. These sets we designate by C+(f, x) and C(f, x), respectively. The result of Young asserts that C+(f, x) = C(f, x) at every xER-A where A is a countable set. Young obtained an analogous result for functions of two variables. By a sector at the origin, we mean either of the closed regions formed by two half-lines emanating from the origin. If ois a sector at the origin, then we designate by o-, the image of ounder the translation taking the origin into p. Letf: R2 >*R be arbitrary and let p CR2. The Cluster Set, C(f, p), of f at p is the set of all real numbers w for which there exists a sequence { pn } CR2 such that lim Pn = p and lim f(pn) = w. If ois a sector at the origin, then The Sectorial Cluster Set, C(f, p, o-,), is defined in the obvious manner. The result Young obtained in this case is that

On a Paper by M. Iosifescu and S. Marcus

In this paper we will construct an example showing that the problem posed in [1] has a negative answer. Two more theorems on the subject treated in [1] will be included.Let Io = [0, 1], R the reals, and let, for A ⊂ R, Ao be the interior of A. Let {xn} be a sequence in [0, 1&gt; such that 0 = x1 &lt; x2 &lt; … and lim xn = 1. For each n, let In be closed interval having x as its midpoint (except for n = 1 in which case x1 is the left endpoint of I1) such that In ∩ Im= ϕ, and the metric density relative to Io of at 1 is zero. Let Jn be a closed interval in In concentric with In (except for n = 1, where J1 has x1 as its left endpoint) whose length is half that of In

Hausdorff transforms of bounded sequences

The sequence is said to be H(ck) evaluable to L if yn-*L as n-oo. The transformation H(Q) is regular (i.e., lim xn=L implies lim yn=L) if and only if c(0+)=0 and c(1)=1. If x is a complex number sequence, then C(x) denotes the set of limit points of x. Among the regular Hausdorff transformations are the methods C, (Re r>0) of Cesaro, Hr (Re r>0) of H 6lder and Er (0<r<1) of Euler for which ck(t) = 1-(1-t)r,