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Search Results (28)

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16 pages, 304 KiB

Open AccessArticle

On the Characterizations of Some Strongly Bounded Operators on C(K, X) Spaces

by Ioana Ghenciu

Axioms 2025, 14(8), 558; https://doi.org/10.3390/axioms14080558 - 23 Jul 2025

Viewed by 117

Abstract

Suppose X and Y are Banach spaces, K is a compact Hausdorff space, and

C (K, X)

is the Banach space of all continuous X-valued functions (with the supremum norm). We will study some strongly bounded operators [...] Read more.

Suppose X and Y are Banach spaces, K is a compact Hausdorff space, and

C (K, X)

is the Banach space of all continuous X-valued functions (with the supremum norm). We will study some strongly bounded operators

T : C (K, X) \to Y

with representing measures

m : Σ \to L (X, Y)

, where

L (X, Y)

is the Banach space of all operators

T : X \to Y

and

Σ

is the

σ

-algebra of Borel subsets of K. The classes of operators that we will discuss are the Grothendieck, p-limited, p-compact, limited, operators with completely continuous, unconditionally converging, and p-converging adjoints, compact, and absolutely summing. We give a characterization of the limited operators (resp. operators with completely continuous, unconditionally converging, p-convergent adjoints) in terms of their representing measures. Full article

25 pages, 371 KiB

Open AccessArticle

Involutions of the Moduli Space of Principal E₆-Bundles over a Compact Riemann Surface

by Álvaro Antón-Sancho

Axioms 2025, 14(6), 423; https://doi.org/10.3390/axioms14060423 - 29 May 2025

Viewed by 275

Abstract

In this paper, the fixed points of involutions on the moduli space of principal

E_{6}

-bundles over a compact Riemann surface X are investigated. In particular, it is proved that the combined action of a representative

σ

of the outer involution of [...] Read more.

In this paper, the fixed points of involutions on the moduli space of principal

E_{6}

-bundles over a compact Riemann surface X are investigated. In particular, it is proved that the combined action of a representative

σ

of the outer involution of

E_{6}

with the pull-back action of a surface involution

τ

admits fixed points if and only if a specific topological obstruction in

H^{2} (X / τ, π_{0} (E_{6}^{σ}))

vanishes. For an involution

τ

with

2 k

fixed points, it is proved that the fixed point set is isomorphic to the moduli space of principal H-bundles over the quotient curve

X / τ

, where H is either

F_{4}

PSp (8, C)

and it consists of

2^{g - k + 1}

components. The complex dimensions of these components are computed, and their singular loci are determined as corresponding to H-bundles admitting non-trivial automorphisms. Furthermore, it is checked that the stability of fixed

E_{6}

-bundles implies the stability of their corresponding H-bundles over

X / τ

, and the behavior of characteristic classes is discussed under this correspondence. Finally, as an application of the above results, it is proved that the fixed points correspond to octonionic structures on

X / τ

, and an explicit construction of these octonionic structures is provided. Full article

(This article belongs to the Special Issue Trends in Differential Geometry and Algebraic Topology)

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24 pages, 379 KiB

Open AccessArticle

Involutive Symmetries and Langlands Duality in Moduli Spaces of Principal G-Bundles

by Álvaro Antón-Sancho

Symmetry 2025, 17(6), 819; https://doi.org/10.3390/sym17060819 - 24 May 2025

Viewed by 392

Abstract

Let X be a compact Riemann surface of genus

g \geq 2

, G be a complex semisimple Lie group, and

M_{G} (X)

be the moduli space of stable principal G-bundles. This paper studies the fixed point set of [...] Read more.

Let X be a compact Riemann surface of genus

g \geq 2

, G be a complex semisimple Lie group, and

M_{G} (X)

be the moduli space of stable principal G-bundles. This paper studies the fixed point set of involutions on

M_{G} (X)

induced by an anti-holomorphic involution

τ

on X and a Cartan involution

θ

of G, producing an involution

σ = θ_{*} \circ τ_{*}

. These fixed points are shown to correspond to stable

G_{R}

-bundles over the real curve

(X_{τ}, τ)

, where

G_{R}

is the real form associated with

θ

. The fixed point set

M_{G} {(X)}^{σ}

consists of exactly

2^{r}

connected components, each a smooth complex manifold of dimension

\frac{(g - 1) \dim G}{2}

, where r is the rank of the fundamental group of the compact form of G. A cohomological obstruction in

H^{2} (X_{τ}, π_{1} (G_{R}))

characterizes which bundles are fixed. A key result establishes a derived equivalence between coherent sheaves on

M_{G} {(X)}^{σ}

and on the fixed point set of the dual involution on the moduli space of

G^{\lor}

-local systems, where

G^{\lor}

denotes the Langlands dual of G. This provides an extension of the Geometric Langlands Correspondence to settings with involutions. An application to the Chern–Simons theory on real curves interprets

M_{G} {(X)}^{σ}

as a

(B, B, B)

-brane, mirror to an

(A, A, A)

-brane in the Hitchin system, revealing new links between real structures, quantization, and mirror symmetry. Full article

(This article belongs to the Special Issue Symmetry in Integrable Systems: Topics and Advances)

13 pages, 283 KiB

Open AccessArticle

The Finite Coarse Shape Paths

by Ivan Jelić and Ivančica Mirošević

Mathematics 2025, 13(3), 439; https://doi.org/10.3390/math13030439 - 28 Jan 2025

Viewed by 519

Abstract

In this paper, we introduce the notions of finite coarse shape path and finite coarse shape path connectedness of a topological space. We prove that the solenoid

Σ_{(p_{n})}

, which is known to be coarse shape path connected but [...] Read more.

In this paper, we introduce the notions of finite coarse shape path and finite coarse shape path connectedness of a topological space. We prove that the solenoid

Σ_{(p_{n})}

, which is known to be coarse shape path connected but not shape path connected, is not finite coarse shape path connected either. Furthermore, we show that every finite coarse shape path induces an isomorphism between finite coarse shape groups of the topological space at different base points, with some interesting and useful properties. We also show that finite coarse shape groups of the same space, in general, depend on the choice of a base point. Hence, the pointed finite coarse shape type of

(X, x)

, in general, depends on the choice of the point x. Finally, we prove that if X is a finite coarse shape path connected paracompact locally compact space, then the pointed finite coarse shape type of

(X, x)

does not depend on the choice of the point x. Full article

(This article belongs to the Section A: Algebra and Logic)

12 pages, 1404 KiB

Open AccessArticle

Covalent vs. Dative Bonding in Carbon Monoxide and Other 10-Valence-Electron Diatomics

by Khadija Rizwan and John Morrison Galbraith

Molecules 2024, 29(22), 5396; https://doi.org/10.3390/molecules29225396 - 15 Nov 2024

Cited by 1 | Viewed by 1439

Abstract

Valence bond theory (VB) was used to determine the extent and driving forces for covalent vs. dative bonding in 10-valence-electron diatomic molecules N₂, CO, NO⁺, CN⁻, P₂, SiS, PS⁺, and SiP⁻. VBSCF calculations were performed at the CCSD(T)/cc-pVDZ optimized geometries. The full triply bonded system included 20 VB structures. A separation of the σ and π space allowed for a subdivision of the full 20 structure set into sets of 8 and 3 for the π and σ systems, respectively. The smaller structure sets allowed for a more focused look at each type of bond. In situ bond energies for σ bonds, individual π bonds, the π system, and triple bonds follow expected trends. Our data shows that N₂ and P₂ have three covalent bonds whereas CO and SiS contain two covalent and one dative bond, and charged species NO⁺, CN⁻, PS⁺, and SiP⁻ are a mixture of N₂ and CO type electronic arrangements, resulting in a nearly equal charge distribution. Dative bonds prefer to be in the π position due to enhanced σ covalency and π resonance. Both σ and π resonance energies depend on a balance of ionic strength, orbital compactness, σ constraints, and bond directionality. Resonance energy is a major contributor to bond strength, making up more than 50% of the π bonds in SiS and PS⁺ (charge-shift bonds), and is greater than charge transfer in dative bonds. Full article

(This article belongs to the Section Physical Chemistry)

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14 pages, 271 KiB

Open AccessArticle

Hodge Decomposition of Conformal Vector Fields on a Riemannian Manifold and Its Applications

by Hanan Alohali, Sharief Deshmukh, Bang-Yen Chen and Hemangi Madhusudan Shah

Mathematics 2024, 12(17), 2628; https://doi.org/10.3390/math12172628 - 24 Aug 2024

Cited by 1 | Viewed by 1045

Abstract

For a compact Riemannian m-manifold

(M^{m}, g), m > 1,

endowed with a nontrivial conformal vector field

ζ

with a conformal factor

σ

, there is an associated skew-symmetric tensor

φ

called the associated tensor, and [...] Read more.

For a compact Riemannian m-manifold

(M^{m}, g), m > 1,

endowed with a nontrivial conformal vector field

ζ

with a conformal factor

σ

, there is an associated skew-symmetric tensor

φ

called the associated tensor, and also,

ζ

admits the Hodge decomposition

ζ = \bar{ζ} + \nabla ρ

, where

\bar{ζ}

satisfies

div \bar{ζ} = 0

, which is called the Hodge vector, and

ρ

is the Hodge potential of

ζ

. The main purpose of this article is to initiate a study on the impact of the Hodge vector and its potential on

M^{m}

. The first result of this article states that a compact Riemannian m-manifold

M^{m}

is an m-sphere

S^{m} (c)

if and only if (1) for a nonzero constant c, the function

- σ / c

is a solution of the Poisson equation

Δ ρ = m σ

, and (2) the Ricci curvature satisfies

R i c (\bar{ζ}, \bar{ζ}) \geq {∥φ∥}^{2}

. The second result states that if

M^{m}

has constant scalar curvature

τ = m (m - 1) c > 0

, then it is an

S^{m} (c)

if and only if the Ricci curvature satisfies

R i c (\bar{ζ}, \bar{ζ}) \geq {∥φ∥}^{2}

and the Hodge potential

ρ

satisfies a certain static perfect fluid equation. The third result provides another new characterization of

S^{m} (c)

using the affinity tensor of the Hodge vector

\bar{ζ}

of a conformal vector field

ζ

on a compact Riemannian manifold

M^{m}

with positive Ricci curvature. The last result states that a complete, connected Riemannian manifold

M^{m}

m > 2

, is a Euclidean m-space if and only if it admits a nontrivial conformal vector field

ζ

whose affinity tensor vanishes identically and

ζ

annihilates its associated tensor

φ

. Full article

(This article belongs to the Special Issue Differentiable Manifolds and Geometric Structures)

37 pages, 485 KiB

Open AccessArticle

Existence and Stability of Solutions for p-Proportional ω-Weighted κ-Hilfer Fractional Differential Inclusions in the Presence of Non-Instantaneous Impulses in Banach Spaces

by Feryal Aladsani and Ahmed Gamal Ibrahim

Fractal Fract. 2024, 8(8), 475; https://doi.org/10.3390/fractalfract8080475 - 14 Aug 2024

Cited by 1 | Viewed by 958

Abstract

In this work, we introduce a new definition for the fractional differential operator that generalizes several well-known fractional differential operators. In fact, we introduce the notion of the

p

-proportional

ω

-weighted

κ

-Hilfer derivative includes an exponential function, [...] Read more.

In this work, we introduce a new definition for the fractional differential operator that generalizes several well-known fractional differential operators. In fact, we introduce the notion of the

p

-proportional

ω

-weighted

κ

-Hilfer derivative includes an exponential function,

D_{a, λ}^{σ, ρ, p, κ, ω}

, and then we consider a non-instantaneous impulse differential inclusion containing

D_{a, λ}^{σ, ρ, p, κ, ω}

with order

σ \in (1, 2)

and of kind

ρ \in [0, 1]

in Banach spaces. We deduce the relevant relationship between any solution to the studied problem and the integral equation that corresponds to it, and then, by using an appropriate fixed-point theorem for multi-valued functions, we give two results for the existence of these solutions. In the first result, we show the compactness of the solution set. Next, we introduce the concept of the

(p, ω, κ)

-generalized Ulam-Hyeres stability of solutions, and, using the properties of the multi-valued weakly Picard operator, we present a result regarding the

(p, ω, κ)

-generalized Ulam-Rassias stability of the objective problem. Since many fractional differential operators are particular cases of the operator

D_{a, λ}^{σ, ρ, p, κ, ω}

, our work generalizes a number of recent findings. In addition, there are no past works on this kind of fractional differential inclusion, so this work is original and enjoyable. In the last section, we present examples to support our findings. Full article

(This article belongs to the Special Issue Boundary Value Problems for Nonlinear Fractional Differential Equations: Theory, Methods and Application)

33 pages, 442 KiB

Open AccessArticle

Antiperiodic Solutions for Impulsive ω-Weighted ϱ–Hilfer Fractional Differential Inclusions in Banach Spaces

by Zainab Alsheekhhussain, Ahmed Gamal Ibrahim, M. Mossa Al-Sawalha and Osama Yusuf Ababneh

Fractal Fract. 2024, 8(7), 376; https://doi.org/10.3390/fractalfract8070376 - 26 Jun 2024

Cited by 3 | Viewed by 1223

Abstract

In this article, we construct sufficient conditions that secure the non-emptiness and compactness of the set of antiperiodic solutions of an impulsive fractional differential inclusion involving an ω-weighted ϱ–Hilfer fractional derivative, [...] Read more.

D_{0, t}^{σ, v, ϱ, ω}

, of order

σ \in (1, 2),

in infinite-dimensional Banach spaces. First, we deduce the formula of antiperiodic solutions for the observed problem. Then, we give two theorems regarding the existence of these solutions. In the first, by using a fixed-point theorem for condensing multivalued functions, we show the non-emptiness and compactness of the set of antiperiodic solutions; and in the second, by applying a fixed-point theorem for contraction multivalued functions, we prove the non-emptiness of this set. Because many types of famous fractional differential operators are particular cases from the operator

D_{0, t}^{σ, v, ϱ, ω}

, our results generalize several recent results. Moreover, there are no previous studies on antiperiodic solutions for this type of fractional differential inclusion, so this work is novel and interesting. We provide two examples to illustrate and support our conclusions. Full article

22 pages, 409 KiB

Open AccessArticle

The Existence of Solutions for w-Weighted ψ-Hilfer Fractional Differential Inclusions of Order μ ∈ (1, 2) with Non-Instantaneous Impulses in Banach Spaces

by Zainab Alsheekhhussain, Ahmad Gamal Ibrahim, Mohammed Mossa Al-Sawalha and Yousef Jawarneh

Fractal Fract. 2024, 8(3), 144; https://doi.org/10.3390/fractalfract8030144 - 29 Feb 2024

Cited by 6 | Viewed by 1485

Abstract

In this research, we obtain the sufficient conditions that guarantee that the set of solutions for an impulsive fractional differential inclusion involving a w-weighted

ψ

-Hilfer fractional derivative,

D_{0, t}^{σ, v, ψ, w},

of order [...] Read more.

In this research, we obtain the sufficient conditions that guarantee that the set of solutions for an impulsive fractional differential inclusion involving a w-weighted

ψ

-Hilfer fractional derivative,

D_{0, t}^{σ, v, ψ, w},

of order

μ \in (1, 2),

in infinite dimensional Banach spaces that are not empty and compact. We demonstrate the exact relation between a differential equation involving

D_{0, t}^{σ, v, ψ, w}

of order

μ

\in (1, 2)

in the presence of non-instantaneous impulses and its corresponding fractional integral equation. Then, we derive the formula for the solution for the considered problem. The desired results are achieved using the properties of the w-weighted

ψ

-Hilfer fractional derivative and appropriate fixed-point theorems for multivalued functions. Since the operator

D_{0, t}^{σ, v, ψ, w}

includes many types of well-known fractional differential operators, our results generalize several results recently published in the literature. We give an example that illustrates and supports our theoretical results. Full article

7 pages, 247 KiB

Open AccessArticle

Compact Resolutions and Analyticity

by Salvador López-Alfonso, Manuel López-Pellicer and Santiago Moll-López

Mathematics 2024, 12(2), 318; https://doi.org/10.3390/math12020318 - 18 Jan 2024

Viewed by 1201

Abstract

We consider the large class

G

of locally convex spaces that includes, among others, the classes of

(D F)

-spaces and

(L F)

-spaces. For a space E in class

G

we have characterized that a subspace Y of [...] Read more.

We consider the large class

G

of locally convex spaces that includes, among others, the classes of

(D F)

-spaces and

(L F)

-spaces. For a space E in class

G

we have characterized that a subspace Y of

(E, σ (E, E^{'}))

, endowed with the induced topology, is analytic if and only if Y has a

σ (E, E^{'})

-compact resolution and is contained in a

σ (E, E^{'})

-separable subset of E. This result is applied to reprove a known important result (due to Cascales and Orihuela) about weak metrizability of weakly compact sets in spaces of class

G

. The mentioned characterization follows from the following analogous result: The space

C (X)

of continuous real-valued functions on a completely regular Hausdorff space X endowed with a topology

ξ

stronger or equal than the pointwise topology

τ_{p}

C (X)

is analytic iff

(C (X), ξ)

is separable and is covered by a compact resolution. Full article

(This article belongs to the Special Issue Advances in Higher-Order Linear Equations and Linear Polynomial Differential Operators)

5 pages, 263 KiB

Open AccessArticle

Two Velichko-like Theorems for C(X)

by Salvador López-Alfonso, Manuel López-Pellicer and Santiago Moll-López

Mathematics 2023, 11(24), 4930; https://doi.org/10.3390/math11244930 - 12 Dec 2023

Viewed by 882

Abstract

This paper provides two new Velichko-like theorems for the weak counterpart of the locally convex space

C_{k} (X)

of all real-valued functions defined on a Tychonoff space X equipped with the compact-open topology

τ_{k}

. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Models and Applications)

16 pages, 345 KiB

Open AccessArticle

Property (h) of Banach Lattice and Order-to-Norm Continuous Operators

by Fu Zhang, Hanhan Shen and Zili Chen

Mathematics 2023, 11(12), 2747; https://doi.org/10.3390/math11122747 - 17 Jun 2023

Cited by 1 | Viewed by 1600

Abstract

In this paper, we introduce the property

(h)

on Banach lattices and present its characterization in terms of disjoint sequences. Then, an example is given to show that an order-to-norm continuous operator may not be

σ

-order continuous. Suppose [...] Read more.

In this paper, we introduce the property

(h)

on Banach lattices and present its characterization in terms of disjoint sequences. Then, an example is given to show that an order-to-norm continuous operator may not be

σ

-order continuous. Suppose

T : E \to F

is an order-bounded operator from Dedekind

σ

-complete Banach lattice E into Dedekind complete Banach lattice F. We prove that T is

σ

-order-to-norm continuous if and only if T is both order weakly compact and

σ

-order continuous. In addition, if E can be represented as an ideal of

L_{0} (μ)

, where

(Ω, Σ, μ)

is a

σ

-finite measure space, then T is

σ

-order-to-norm continuous if and only if T is order-to-norm continuous. As applications, we extend Wickstead’s results on the order continuity of norms on E and

E^{'}

. Full article

(This article belongs to the Special Issue Ordered Topological Vector Spaces, Operators Acting on Them and Related Applications)

12 pages, 322 KiB

Open AccessFeature PaperArticle

Generalized Universality for Compositions of the Riemann Zeta-Function in Short Intervals

by Antanas Laurinčikas and Renata Macaitienė

Mathematics 2023, 11(11), 2436; https://doi.org/10.3390/math11112436 - 24 May 2023

Cited by 1 | Viewed by 1807

Abstract

In the paper, the approximation of analytic functions on compact sets of the strip

{s = σ + i t \in C ∣ 1 / 2 < σ < 1}

by shifts [...] Read more.

In the paper, the approximation of analytic functions on compact sets of the strip

{s = σ + i t \in C ∣ 1 / 2 < σ < 1}

by shifts

F (ζ (s + i u_{1} (τ)), \dots, ζ (s + i u_{r} (τ)))

, where

ζ (s)

is the Riemann zeta-function,

u_{1}, \dots, u_{r}

are certain differentiable increasing functions, and F is a certain continuous operator in the space of analytic functions, is considered. It is obtained that the set of the above shifts in the interval

[T, T + H]

with

H = o (T)

T \to \infty

, has a positive lower density. Additionally, the positivity of a density with a certain exceptional condition is discussed. Examples of considered operators F are given. Full article

(This article belongs to the Special Issue Analytic Methods in Number Theory and Allied Fields)

19 pages, 394 KiB

Open AccessArticle

A Characterization of Multipliers of the Herz Algebra

by Hans G. Feichtinger

Axioms 2023, 12(5), 482; https://doi.org/10.3390/axioms12050482 - 16 May 2023

Cited by 4 | Viewed by 1441

Abstract

For the characterization of multipliers of

L^{p} (R^{d})

or more generally, of

L^{p} (G)

for some locally compact Abelian group G, the so-called Figa-Talamanca–Herz algebra

A_{p} (G)

plays an important role. Following [...] Read more.

For the characterization of multipliers of

L^{p} (R^{d})

or more generally, of

L^{p} (G)

for some locally compact Abelian group G, the so-called Figa-Talamanca–Herz algebra

A_{p} (G)

plays an important role. Following Larsen’s book, we describe multipliers as bounded linear operators that commute with translations. The main result of this paper is the characterization of the multipliers of

A_{p} (G)

. In fact, we demonstrate that it coincides with the space of multipliers of

(L^{p} (G), {∥ \cdot ∥}_{p})

. Given a multiplier T of

(A_{p} (G), ∥ \cdot ∥_{A_{p} (G)})

and using the embedding

(A_{p} (G) {, ∥ \cdot ∥}_{A_{p} (G)}) ↪ (C_{0} (G), {∥ \cdot ∥}_{\infty})

, the linear functional

f \mapsto [T (f) (0)]

is bounded, and T can be written as a moving average for some element in the dual

P M_{p} (G)

(A_{p} (G), ∥ \cdot ∥_{A_{p} (G)})

. A key step for this identification is another elementary fact: showing that the multipliers from

(L^{p} (G), {∥ \cdot ∥}_{p})

(C_{0} (G), {∥ \cdot ∥}_{\infty})

are exactly the convolution operators with kernels in

(L^{q} (G), {∥ \cdot ∥}_{q})

for

1 < p < \infty

and

1 / p + 1 / q = 1

. The proofs make use of the space of mild distributions, which is the dual of the Segal algebra

(S_{0} (G), {∥ \cdot ∥}_{S_{0}})

, and the fact that multipliers T from

S_{0} (G)

S_{0}^{'} (G)

are convolution operators of the form

T : f \mapsto σ * f

for some uniquely determined

σ \in S_{0}^{'}

. This setting also allows us to switch from the description of these multipliers as convolution operators (by suitable pseudomeasures) to their description as Fourier multipliers, using the extended Fourier transform in the setting of

(S_{0}^{'} (G), {∥ \cdot ∥}_{S_{0}^{'}})

. The approach presented here extends to other function spaces, but a more detailed discussion is left to future publications. Full article

(This article belongs to the Special Issue Time-Frequency Analysis, Distributions, and Operators)

9 pages, 273 KiB

Open AccessArticle

Topological Structure and Existence of Solutions Set for q-Fractional Differential Inclusion in Banach Space

by Ali Rezaiguia and Taher S. Hassan

Mathematics 2023, 11(3), 683; https://doi.org/10.3390/math11030683 - 29 Jan 2023

Cited by 1 | Viewed by 1444

Abstract

In this work, we concentrate on the existence of the solutions set of the following problem [...] Read more.

In this work, we concentrate on the existence of the solutions set of the following problem

^{c} D_{q}^{α} σ (t) \in F (t, σ (t),^{c} D_{q}^{α} σ (t)), t \in I = [0, T] σ (0) = σ_{0} \in E

, as well as its topological structure in Banach space E. By transforming the problem posed into a fixed point problem, we provide the necessary conditions for the existence and compactness of solutions set. Finally, we present an example as an illustration of main results. Full article

(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications II)

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