Space CP Research Articles

Velichko's notions close to sequential separability and their hereditary variants in Cp-theory

A space X is sequentially separable if there is a countable S⊂X such that every point of X is the limit of a sequence of points from S. In 2004, N.V. Velichko defined and investigated concepts close to sequential separability: σ-separability and F-separability. The aim of this paper is to study σ-separability and F-separability (and their hereditary variants) of the space Cp(X) of all real-valued continuous functions, defined on a Tychonoff space X, endowed with the pointwise convergence topology. In particular, we proved that σ-separability coincides with sequential separability. Hereditary variants (hereditarily σ-separability and hereditarily F-separability) coincide with Fréchet–Urysohn property in the class of cosmic spaces.

Characterizations of the mixed central Campanato spaces via the commutator operators of Hardy type

The purpose of this paper is to establish some characterizations of mixed central Campanato space Cp→,λ(Rn), via the boundedness of the commutator operators of Hardy type. Unlike the case λ⩾0, there are some technical difficulties caused by λ<0 to be overcome. In addition, an extra assumption called as the mixed version of the reverse Hölder class be required in the proof of the converse characterization. Moreover, some further interesting conclusions for the Hardy type operators on mixed λ-central Morrey space Bp→,λ(Rn) are also derived.

On the product of almost discrete Grothendieck spaces

A topological space X is called almost discrete, if it has precisely one nonisolated point. In this paper, we get that for a countable product X=∏Xi of almost discrete spaces Xi the space Cp(X) of all continuous real-valued functions with the topology of pointwise convergence is a μ-space if, and only if, X is a weak q-space if, and only if, t(X)=ω if, and only if, X is functionally generated by the family of all its countable subspaces.This result makes it possible to solve Archangel'skii's problem on the product of Grothendieck spaces. It is proved that in the model of ZFC, obtained by adding one Cohen real, there are Grothendieck spaces X and Y such that X×Y is not weakly Grothendieck space. In (PFA): the product of any countable family almost discrete Grothendieck spaces is a Grothendieck space.

О свойствах пространств Cp(X), близких к свойству Фреше–Урысона

By analogy with the Fréchet-Urysohn property, the properties of n-Fréchet-Urysohn and ω-Fréchet-Urysohn spaces of the spaces Cp(X) are introduced into consideration. The connection between these properties and the properties

New non-supersymmetric flux vacua in string theory

In this note we construct large ensembles of supersymmetry breaking solutions arising in the context of flux compactifications of type IIB string theory. This class of solutions was previously proposed in [1] for which we provide the first explicit examples in Calabi-Yau orientifold compactifications with discrete fluxes below their respective tadpole constraint. As a proof of concept, we study the degree 18 hypersurface in weighted projective space CP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathbbm{CP} $$\\end{document}1,1,1,6,9. Furthermore, we look at 10 additional orientifolds with h1,2 = 2, 3. We find several flux vacua with hierarchical suppression of the vacuum energy with respect to the gravitino mass. These solutions provide a crucial stepping stone for the construction of explicit de Sitter vacua in string theory. Lastly, we also report the difference in the distribution of W0 between supersymmetric and non-supersymmetric minima.

Phase space analysis of the 3-level Lipkin-Meshkov-Glick model using parity adapted U(D)-spin coherent states

We study the phase-space properties of the 3-level Lipkin-Meshkov-Glick as paradigmatic case of critical, parity symmetric, N-quDit systems undergoing a quantum phase transition in the thermodynamic limit N → ∞. We generalize U(2) spin coherent states to U(D) (quDits), and define the coherent state representation Qψ (Husimi function) of a symmetric N- quDit state |ψ〉 in the phase space ℂP D −1 = U(D)/[U(1) × U(D − 1)]. This allows us to define parity adapted U(D) coherent states (𝕔-DCATs), which reproduce accurately the lowest energy Hamiltonian eigenstates obtained by numerical diagonalization. We visualize precursors of the QPTs by plotting localization measures (Husimi function and its moments) of the parity adapted U(D) coherent states and the numerical Hamiltonian eigenstates for a finite number of particles.

Some properties involving feeble compactness, II: Generalized Σ-products and Cp-spaces

In [14], we begin analyzing some classes of topological spaces contained in the class of feebly compact spaces. In this article, which is a second part of that work, we determine, for a compact metrizable topological group G, when Cp(X,G) is selectively pseudocompact, and when it is weakly compact-bounded for a Tychonoff space X. Moreover, we prove that for some Hausdorff extensions XQ of a space X constructed using free filters on X, the space Cp(XQ,G) is selectively sequentially pseudocompact. We therefore obtain a class of spaces having sufficient conditions to affirmatively solve Problem 8.5 posed by Dorantes-Aldama and Shakmatov in [7].

Localization measures of parity adapted U(D)-spin coherent states applied to the phase space analysis of the D-level Lipkin-Meshkov-Glick model.

We study phase space properties of critical, parity symmetric, N-qudit systems undergoing a quantum phase transition (QPT) in the thermodynamic N→∞ limit. The D=3 level (qutrit) Lipkin-Meshkov-Glick model is eventually examined as a particular example. For this purpose, we consider U(D)-spin coherent states (DSCS), generalizing the standard D=2 atomic coherent states, to define the coherent state representation Q_{ψ} (Husimi function) of a symmetric N-qudit state |ψ〉 in the phase space CP^{D-1} (complex projective manifold). DSCS are good variational approximations to the ground state of an N-qudit system, especially in the N→∞ limit, where the discrete parity symmetry Z_{2}^{D-1} is spontaneously broken. For finite N, parity can be restored by projecting DSCS onto 2^{D-1} different parity invariant subspaces, which define generalized "Schrödinger cat states" reproducing quite faithfully low-lying Hamiltonian eigenstates obtained by numerical diagonalization. Precursors of the QPT are then visualized for finite N by plotting the Husimi function of these parity projected DSCS in phase space, together with their Husimi moments and Wehrl entropy, in the neighborhood of the critical points. These are good localization measures and markers of the QPT.

The hyperspace of a semi-Eberlein compact space is semi-Eberlein

We prove, using r-skeletons, that if X is a semi-Eberlein compact space, then its hyperspace of compact subsets is a semi-Eberlein compact space. We prove that c-skeletons can be induced on the hyperspaces of compact subsets of a space admitting a c-skeleton, and we characterize the presence of full c-skeletons on function spaces Cp(X). Finally, we characterize ω-monotone assignments of pseudobases on subspaces of ω1. These results answer some open questions published in [11], [2] and [16].

The Josefson-Nissenzweig property for locally convex spaces

We define a locally convex space E to have the Josefson-Nissenzweig property (JNP) if the identity map (E?, ?(E?, E)) ? (E?, ?*(E?, E)) is not sequentially continuous. By the classical Josefson-Nissenzweig theorem, every infinite-dimensional Banach space has the JNP. A characterization of locally convex spaces with the JNP is given. We thoroughly study the JNP in various function spaces. Among other results we show that for a Tychonoff space X, the function space Cp(X) has the JNP iff there is a weak* null-sequence (?n)n?? of finitely supported sign-measures on X with unit norm. However, for every Tychonoff space X, neither the space B1(X) of Baire-1 functions on X nor the free locally convex space L(X) over X has the JNP.

Approximation by pointwise bounded sets of continuous functions

Assuming that X is a Tychonoff space, we obtain necessary and sufficient conditions for a function f∈RX to belong to the closure in RX of a pointwise bounded family F in C(X), the ring of real-valued continuous functions on X. In other words, if M(X) stands for the bidual of the space Cp(X), equipped with the pointwise topology, we provide necessary and sufficient conditions for a function f∈RX to belong to M(X). As a noteworthy fact, if M, S and R are, respectively, the Michael, the Sorgenfrey, and the real line, it is shown that C(M)⊈M(R) but C(S)⊆M(R).

Magnetic Dipole Moments, Occupation Numbers and Magnetic Form Factors of Some Neutron Rich Nuclei in sdpf-shell

In the sdpf shell model, static and dynamic characteristics of certain neutron-rich nuclei are measured by using the SDPF interactions: Magnetic dipole moments (𝜇), occupation numbers (occ#) and magnetic electron scattering form factors for the ground state, they are presented for some Vanadium (23V) and Manganese (25Mn) isotopes using radial wave functions for the single-particle matrix elements of harmonic-oscillator potential (HO). These isotopes are 48V (4 1+), 49V (7/2- 3/2), 50V (6+ 2), 53Mn (7/2- 3/2), 54Mn (3+ 2) and 56Mn (3+ 3). The calculations introduced with model space (MS) and core-polarization CP effects are included through effective g factors. In model space and with core-polarization effects, the magnetic transition probability B(M1) is also calculated. The calculations of one-body density matrix OBDM together with the interaction SDPFK are carried out in the sdpf-model space using the OXBASH code. The theoretical results of the 𝜇 moments agree with recent experimental data.

Regularization of closed positive currents and extension theory

In this note we extend to any bidimension (p,p) the Demailly theorem of regularization of closed positive (1,1)-currents on a compact Kähler manifold X of dimension n. When the manifold X is projective, we get explicitly a closed regularization with bounded negative part, constructed by using the space Cp(X) of effective algebraic cycles of X of dimension p. This space can be injected in the space of divisors of Cn−p−1(X) and we arrive at an intrinsic construction of the Skoda potential associated with a closed positive current of X. On another hand, in the case of a divisor D of X, we give an explicit bound for the degree of an irreducible component of the singular locus Dsing, involving the geometry of X. Lastly when X is embedded in the projective space PN, we prove the existence of a closed current extending in the generalized sense a given closed current of X, by using here a space of cycles. As an application, we obtain a characterization of the cohomology classes contained in some algebraic hypersurface of X.

Linear equivalence of (pseudo) compact spaces

Given Tychonoff spaces X and Y, Uspenskij proved in [15] that if X is pseudocompact and Cp(X) is uniformly homeomorphic to Cp (Y), then Y is also pseudocompact. In particular, if Cp (X) is linearly homeomorphic to Cp (Y), then X is pseudocompact if and only if so is Y. This easily implies Arhangel’skii’s theorem [1] which states that, in the case when Cp (X) is linearly homeomorphic to Cp (Y the space X is compact if and only if Y is compact. We will establish that existence of a linear homeomorphism between the spaces Cp *(X) and Cp *(Y) implies that X is (pseudo)compact if and only if so is Y. We will also show that the methods of proof used by Arhangel’skii and Uspenskij do not work in our case.

Analysis of Solutions of Some Multi-Term Fractional Bessel Equations

We construct the existence theory for generalized fractional Bessel differential equations and find the solutions in the form of fractional or logarithmic fractional power series. We figure out the cases when the series solution is unique, non-unique, or does not exist. The uniqueness theorem in space Cp is proved for the corresponding initial value problem. We are concerned with the following homogeneous generalized fractional Bessel equation $$\sum\limits_{i = 1}^m {{d_i}{x^{{\alpha _i}}}} {D^{{\alpha _i}}}u\left( x \right) + \left( {{x^\beta } - {v^2}} \right)u\left( x \right) = 0,\,{\alpha _i} >0,\,\beta >0,$$ which includes the standard fractional and classical Bessel equations as particular cases. Mostly, we consider fractional derivatives in Caputo sense and construct the theory for positive coefficients di. Our theory leads to a threshold admissible value for ν2, which perfectly fits to the known results. Our findings are supported by several numerical examples and counterexamples that justify the necessity of the imposed conditions. The key point in the investigation is forming proper fractional power series leading to an algebraic characteristic equation. Depending on its roots and their multiplicity/complexity, we find the system of linearly independent solutions.

On the [formula omitted]-equivalence of metric spaces

In this paper lp⁎-invariant properties of metric spaces are presented. These properties provide us with necessary and sufficient conditions for an isomorphic classification of function spaces Cp⁎(X), where X is any countable metric space of scattered height less than or equal to ω. Examples are presented to show that this isomorphic classification differs from the isomorphic classification of function spaces Cp(X).

The first Pontrjagin classes of homotopy complex projective spaces

Let M2n be an oriented closed smooth manifold homotopy equivalent to the complex projective space CP(n). The main purpose of this paper is to show that when n is even, the difference between the first Pontrjagin class of M2n and that of CP(n) is divisible by 16. As a geometric application of this result, we prove that the Kervaire sphere of dimension 4k+1 does not admit any free circle group action if k+1 is not a power of 2.

For every space X, either CpCp(X) or CpCpCp(X) is ψ-separable

We establish that, for any Tychonoff space X, at least one of the function spaces CpCp(X) and CpCpCp(X) must have a dense subspace of countable pseudocharacter. If GCH holds, then at least one of the spaces Cp(X) and CpCp(X) has a dense subspace of countable pseudocharacter. We also prove that all iterated function spaces Cp,n(K) have a uniformly dense subspace of countable pseudocharacter whenever K is a Corson compact space. Our results solve several published open questions.

On linear homeomorphisms of spaces of continuous functions on Hattori spaces

For a subset A of the real line R, the Hattori space H(A) is a topological space whose underlying point set is the real R and whose topology is defined as follows: points from A possess the usual Euclidean neighbourhoods while remaining points are given the neighbourhoods of the Sorgenfrey line. We consider the linear homeomorphisms between the spaces Cp(H(A)), Cp(H(B)) and Cp(S), where H(A) and H(B) are Hattori spaces and S is Sorgenfrey line.

REAL K-COHOMOLOGY OF COMPLEX PROJECTIVE SPACES

In this paper, we determine the structure of KO-cohomology of complex projective space P l and its product space P l × P m as algebras over the coefficient ring KO ∗ . We also give a description of the map KO ∗ ( P l+m ) → KO ∗ ( P l × P m ) induced by the map that classifies the tensor product of the canonical line bundles and show that its image is not contained in the image of the cross product KO ∗ ( P l ) ⊗KO∗ KO ∗ ( P m ) → KO ∗ ( P l × P m ) to see that non-existence of the formal group structure on KO ∗ ( P ∞ ). Introduction A commutative ring spectrum E is said to be complex oriented if an element x of the reduced E-cohomology of the infinite dimensional complex projective space CP ∞ is given such that x maps to a generator of the reduced E-cohomology of 1-dimensional complex projective space CP 1 ((2)). We call such an element x a complex orientation of E. On the other hand, if E-homology E∗E of E is a flat over the coefficient ring E∗, E∗E has a structure of a Hopf algebroid and E-homology theory takes values in the category of E∗E-comodule, in other words, the category of representations of the groupoid represented by the affine groupoid scheme represented by E∗E ((1)). If E is a complex oriented ring spectrum, the E-cohomology of the complex projective space is just a truncated polynomial algebra over E∗ and it is shown that E-homology E∗E of E is a flat over E∗. Moreover the product structure of CP ∞ gives a one dimensional formal group law over E ∗ ((5)) which closely relates with the structure of the Hopf algebroid ((2)). The complex K-theory is one of the most basic examples of complex oriented cohomology theories. However, KO-spectrum representing the real K-theory is one of a few well-known examples of spectra E without any complex orientation such that E∗E is flat over E∗ ((2), (7)). In fact, we see that KO-spectrum does not have any complex orientation by showing that the Atiyah-Hirzebruch spectral sequence converging to KO ∗ (CP l ) has a non-trivial