scholarly journals On the special semigroup “at infinity”

2014 ◽  
Vol 23 (2) ◽  
pp. 131-136
Author(s):  
ABDOL MOHAMMAD AMINPOUR ◽  
◽  
MEHRDAD SEILANI ◽  
Keyword(s):  
Topological Semigroup ◽  
Compact Semigroup ◽  
New Technique ◽  
The One ◽  
One Point Compactification ◽  
Positive Integers ◽  
Quotient Semigroup ◽  
Right Topological Semigroup

This paper presents an important new technique for studying a particular compact semigroup, N∪{∞}, the one-point compactification of positive integers with usual addition, which is an important semigroup. Indeed, the semigroup N ∪ {∞} is constructed as the quotient semigroup of a particular compact right topological semigroup. In the study of such a semigroup, a major role is played by the substructures called standard oids. For instance, some of the already known results on the structure of N ∪ {∞} are obtained as immediate consequences.

Rearrangement of vector series. I

2001 ◽  
Vol 130 (1) ◽  
pp. 89-109 ◽  
Author(s):  
C. St. J. A. NASH-WILLIAMS ◽  
D. J. WHITE
Keyword(s):  
Euclidean Space ◽  
Partial Sums ◽  
Combinatorial Characterization ◽  
Cluster Set ◽  
The One ◽  
One Point Compactification ◽  
Zero Sum ◽  
Positive Integers

Let ℝd* = ℝd ∪ {[midast ]} be the one-point compactification of Euclidean space ℝd and d [ges ] 2. Given a permutation f of the set ℕ of positive integers, let [Cscr ]f(ℝd*) denote the set of all sets C ⊆ ℝd* for which there is a series [sum ]an in ℝd with zero sum such that C is the cluster set in ℝd* of the sequence of partial sums of [sum ]af(n). Every C ∈ [Cscr ]f(ℝd*) is non-empty, connected and closed in ℝd*. We give a combinatorial characterization of the permutations f for which all non-empty closed connected subsets of ℝd* belong to [Cscr ]f(ℝd*). For every permutation f of ℕ, we determine all C ∈ [Cscr ]f(ℝd*) which contain [midast ].

scholarly journals Study of a Forwarding Chain in the Category of Topological Spaces between T0 and T2 with respect to One Point Compactification Operator

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Fatemah Ayatollah Zadeh Shirazi ◽  
Meysam Miralaei ◽  
Fariba Zeinal Zadeh Farhadi
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
The One ◽  
One Point Compactification

In the following text, we want to study the behavior of one point compactification operator in the chain Ξ := {Metrizable, Normal, T2, KC, SC, US, T1, TD, TUD, T0, Top} of subcategories of category of topological spaces, Top (where we denote the subcategory of Top, containing all topological spaces with property P , simply by P). Actually we want to know, for P∈Ξ and X∈P, the one point compactification of topological space X belongs to which elements of Ξ. Finally we find out that the chain {Metrizable, T2, KC, SC, US, T1, TD, TUD, T0, Top} is a forwarding chain with respect to one point compactification operator.

scholarly journals The Fourier transform of thick distributions

2020 ◽  
pp. 1-26
Author(s):  
Ricardo Estrada ◽  
Jasson Vindas ◽  
Yunyun Yang
Keyword(s):  
Fourier Transform ◽  
Dual Space ◽  
Canonical Projection ◽  
Test Functions ◽  
Tempered Distributions ◽  
Delta Functions ◽  
The Fourier Transform ◽  
The One ◽  
One Point Compactification ◽  
Logarithmic Type

We first construct a space [Formula: see text] whose elements are test functions defined in [Formula: see text] the one point compactification of [Formula: see text] that have a thick expansion at infinity of special logarithmic type, and its dual space [Formula: see text] the space of sl-thick distributions. We show that there is a canonical projection of [Formula: see text] onto [Formula: see text] We study several sl-thick distributions and consider operations in [Formula: see text] We define and study the Fourier transform of thick test functions of [Formula: see text] and thick tempered distributions of [Formula: see text] We construct isomorphisms [Formula: see text] [Formula: see text] that extend the Fourier transform of tempered distributions, namely, [Formula: see text] and [Formula: see text] where [Formula: see text] are the canonical projections of [Formula: see text] or [Formula: see text] onto [Formula: see text] We determine the Fourier transform of several finite part regularizations and of general thick delta functions.

scholarly journals A note on primes in certain residue classes

2018 ◽  
Vol 14 (08) ◽  
pp. 2219-2223
Author(s):  
Paolo Leonetti ◽  
Carlo Sanna
Keyword(s):  
Positive Integer ◽  
Asymptotic Density ◽  
Residue Classes ◽  
Euler Totient Function ◽  
The One ◽  
Positive Integers

Given positive integers [Formula: see text], we prove that the set of primes [Formula: see text] such that [Formula: see text] for [Formula: see text] admits asymptotic density relative to the set of all primes which is at least [Formula: see text], where [Formula: see text] is the Euler totient function. This result is similar to the one of Heilbronn and Rohrbach, which says that the set of positive integer [Formula: see text] such that [Formula: see text] for [Formula: see text] admits asymptotic density which is at least [Formula: see text].

scholarly journals Embedding inverse semigroups of homeomorphisms on closed subsets

1977 ◽  
Vol 18 (2) ◽  
pp. 199-207 ◽  
Author(s):  
Bridget Bos Baird
Keyword(s):  
Topological Space ◽  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Topological Spaces ◽  
Locally Compact ◽  
Compact Metric Space ◽  
Countable Space ◽  
The One ◽  
One Point Compactification ◽  
First Countable Space

All topological spaces here are assumed to be T2. The collection F(Y)of all homeomorphisms whose domains and ranges are closed subsets of a topological space Y is an inverse semigroup under the operation of composition. We are interested in the general problem of getting some information about the subsemigroups of F(Y) whenever Y is a compact metric space. Here, we specifically look at the problem of determining those spaces X with the property that F(X) is isomorphic to a subsemigroup of F(Y). The main result states that if X is any first countable space with an uncountable number of points, then the semigroup F(X) can be embedded into the semigroup F(Y) if and only if either X is compact and Y contains a copy of X, or X is noncompact and locally compact and Y contains a copy of the one-point compactification of X.

Uniform Harmonic Approximation with Continuous Extension to the Boundary

1988 ◽  
Vol 40 (6) ◽  
pp. 1375-1388 ◽  
Author(s):  
M. Goldstein ◽  
W. H. Ow
Keyword(s):  
Complex Plane ◽  
Nonempty Subset ◽  
Harmonic Approximation ◽  
Continuous Extension ◽  
Partial Derivatives ◽  
Extended Plane ◽  
The One ◽  
Image Position ◽  
One Point Compactification

Let G be a domain in the complex plane and F a nonempty subset of G such that F is the closure in G of its interior F0. We will say f ∊ C1(F) if f is continuous on F and possesses continuous first partial derivatives in F which extend continuously to F as finite-valued functions. Let G* – F be connected and locally connected, f ∊ C1(F) be harmonic in F0, and E be a subset of ∂F ∩ ∂G (here G* denotes the one-point compactification of G and the boundaries ∂F, ∂G are taken in the extended plane). Suppose there is a sequence 〈hn〉 of functions harmonic in G such thatuniformly on F as n → ∞.

Spaces which Cannot be Written as a Countable Disjoint Union of Closed Subsets

1973 ◽  
Vol 16 (3) ◽  
pp. 435-437 ◽  
Author(s):  
C. Eberhart ◽  
J. B. Fugate ◽  
L. Mohler
Keyword(s):  
Hausdorff Space ◽  
Disjoint Union ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
The One ◽  
One Point Compactification ◽  
Compact Subsets

It is well known (see [3](1)) that no continuum (i.e. compact, connected, Hausdorff space) can be written as a countable disjoint union of its (nonvoid) closed subsets. This result can be generalized in two ways into the setting of locally compact, connected, Hausdorff spaces. Using the one point compactification of a locally compact, connected, Hausdorff space X one can easily show that X cannot be written as a countable disjoint union of compact subsets. If one makes the further assumption that X is locally connected, then one can show that X cannot be written as a countable disjoint union of closed subsets.(2)

scholarly journals Equations with powers of singular moduli

2019 ◽  
Vol 15 (03) ◽  
pp. 445-468 ◽  
Author(s):  
Antonin Riffaut
Keyword(s):  
Rational Number ◽  
The Other ◽  
Other Hand ◽  
Singular Moduli ◽  
Cm Points ◽  
The One ◽  
Straight Lines ◽  
Positive Integers

We treat two different equations involving powers of singular moduli. On the one hand, we show that, with two possible (explicitly specified) exceptions, two distinct singular moduli [Formula: see text] such that the numbers [Formula: see text], [Formula: see text] and [Formula: see text] are linearly dependent over [Formula: see text] for some positive integers [Formula: see text], must be of degree at most [Formula: see text]. This partially generalizes a result of Allombert, Bilu and Pizarro-Madariaga, who studied CM-points belonging to straight lines in [Formula: see text] defined over [Formula: see text]. On the other hand, we show that, with obvious exceptions, the product of any two powers of singular moduli cannot be a non-zero rational number. This generalizes a result of Bilu, Luca and Pizarro-Madariaga, who studied CM-points belonging to a hyperbola [Formula: see text], where [Formula: see text].

The ℋ-equivalence in a compact semigroup II

1963 ◽  
Vol 3 (3) ◽  
pp. 288-293
Author(s):  
L. W. Anderson ◽  
R. P. Hunter
Keyword(s):  
Compact Semigroup ◽  
The Core ◽  
The Right ◽  
Quotient Semigroup

In [1] we considered various aspects of the quotient semigroup H. · H2 where H is an ℋ-class of a semigroup S. In particular, the action of the Schützenberger group of H upon SH was studied to obtain various results on the existence of subcontinua. Crucial in [1] was the notion of the (right handed) core of an ℋ-class which may be considered as a generalization of the notion of the core of a homogroup, [2].

scholarly journals Dynamical systems, shape theory and the Conley index

1988 ◽  
Vol 8 (8) ◽  
pp. 375-393 ◽  
Keyword(s):  
Dynamical Systems ◽  
Shape Index ◽  
Conley Index ◽  
Invariant Set ◽  
Discrete Time Systems ◽  
Isolated Invariant Set ◽  
The One ◽  
One Point Compactification ◽  
Time Systems ◽  
Construction Works

AbstractThe Conley index of an isolated invariant set is defined only for flows; we construct an analogue called the ‘shape index’ for discrete dynamical systems. It is the shape of the one-point compactification of the unstable manifold of the isolated invariant set in a certain topology which we call its ‘intrinsic’ topology (to distinguish it from the ‘extrinsic’ topology which it inherits from the ambient space). Like the Conley index, it is invariant under continuation. A key point is the construction of a certain ‘index category’ associated with the isolated invariant set; this construction works equally well for flows or discrete time systems, and its properties imply the basic properties of both the Conley index and the shape index.

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