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14 pages, 296 KB

Open AccessArticle

A Rigidity Theorem on Spacelike Hypersurfaces in Generalized Robertson–Walker Spacetimes

by Ning Zhang

Axioms 2026, 15(3), 241; https://doi.org/10.3390/axioms15030241 - 23 Mar 2026

Viewed by 261

We study the properties of complete parabolic constant mean curvature spacelike hypersurfaces in generalized Robertson–Walker (GRW) spacetimes

I \times_{φ} P^{n}

whose warping function

φ

fulfills a certain convexity criterion such that

- φ

is convex, and whose Ricci curvature of the [...] Read more.

We study the properties of complete parabolic constant mean curvature spacelike hypersurfaces in generalized Robertson–Walker (GRW) spacetimes

I \times_{φ} P^{n}

whose warping function

φ

fulfills a certain convexity criterion such that

- φ

is convex, and whose Ricci curvature of the fiber

P^{n}

is non-negative. Our approach is based on calculating the Laplacian of an appropriate function. Under appropriate conditions on the constant mean curvature, by using the parabolicity, we obtain a rigidity theorem and some corollaries of spacelike hypersurfaces. As a consequence, we solve new corresponding Calabi–Bernstein-type problems. Full article

(This article belongs to the Special Issue Recent Developments in Differential Geometry and Its Applications)

16 pages, 1247 KB

Open AccessArticle

Sharp Coefficient Bounds for a Class of Analytic Functions Related to Exponential Function

by Adel Salim Tayyah, Sibel Yalçın and Hasan Bayram

Mathematics 2025, 13(23), 3878; https://doi.org/10.3390/math13233878 - 3 Dec 2025

Cited by 2 | Viewed by 708

Abstract

In this paper, we introduce a new class of analytic functions, denoted by

S (ν, φ_{ϑ, e})

, and provide illustrative examples to elucidate its properties. This class generalizes the starlike and convex functions previously defined by Khatter [...] Read more.

In this paper, we introduce a new class of analytic functions, denoted by

S (ν, φ_{ϑ, e})

, and provide illustrative examples to elucidate its properties. This class generalizes the starlike and convex functions previously defined by Khatter et al. in relation to the exponential function. A significant contribution of this work is the derivation of sharp bounds for various coefficient-related problems within this class. The computational challenges involved in deriving these bounds were effectively addressed using Mathematica^TM codes. Additionally, figures illustrating the geometric properties and essential computations have been incorporated into the paper. Full article

(This article belongs to the Special Issue Advanced Research in Complex Analysis Operators and Special Classes of Analytic Functions)

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20 pages, 386 KB

Open AccessArticle

Measure of Non-p-Convexity in p-Seminormed Spaces

by Ghadah Albeladi and Naseer Shahzad

Mathematics 2025, 13(23), 3807; https://doi.org/10.3390/math13233807 - 27 Nov 2025

Viewed by 483

Abstract

This paper discusses two measures of non-p-convexity and aims to develop them for applications in p-normed spaces. We also extend and generalize some important properties and well-known results. Full article

(This article belongs to the Section C: Mathematical Analysis)

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16 pages, 319 KB

Open AccessArticle

A φ-Contractivity and Associated Fractal Dimensions

by Nifeen H. Altaweel, Olayan Albalawi and Razan Albalawi

Fractal Fract. 2025, 9(10), 628; https://doi.org/10.3390/fractalfract9100628 - 26 Sep 2025

Viewed by 752

Abstract

In this paper, we extend the concept of dimension of sets to some general frameworks relative to a gauge function

φ

, where two simultaneous dimensions are introduced. Unlike the classical cases where one dimension function is introduced based on the diameter power [...] Read more.

In this paper, we extend the concept of dimension of sets to some general frameworks relative to a gauge function

φ

, where two simultaneous dimensions are introduced. Unlike the classical cases where one dimension function is introduced based on the diameter power relative to the associated measure power, and where the gauge is a set-valued function or a measure in the majority of cases, we no longer assume this hypothesis. The introduced variant generalizes many existing cases, such as Haudorff, packing, Carathéodory, and Billingsley original variants. Many characteristics of the dimensions are investigated, such as bijectivity, convexity, monotony, asymptotic behavior, and fixed points. Full article

(This article belongs to the Section General Mathematics, Analysis)

14 pages, 303 KB

Open AccessArticle

Luxemburg Norm Characterizations of BLO Spaces in General Metric Measure Frameworks

by Liping Yang and Xin Jiang

Mathematics 2025, 13(17), 2891; https://doi.org/10.3390/math13172891 - 7 Sep 2025

Viewed by 1064

Abstract

This study provides new equivalent descriptions of the Bounded Lower Oscillation (

BLO

) space through Luxemburg-type

L^{φ}

integrability conditions, where

φ

is a nonnegative function with either convexity or concavity. The framework accommodates various representative forms of

φ

, such as [...] Read more.

This study provides new equivalent descriptions of the Bounded Lower Oscillation (

BLO

) space through Luxemburg-type

L^{φ}

integrability conditions, where

φ

is a nonnegative function with either convexity or concavity. The framework accommodates various representative forms of

φ

, such as the power function

φ (t) = t^{p}

, exponential-type functions

φ (t) = e^{p t} - 1

, and logarithmic functions

φ (t) = {log}_{+}^{k} t

, with parameters

p \in (0, \infty)

and

k \in N

. These results unify and extend existing characterizations of

BLO

by encompassing a broad class of generating functions. Full article

19 pages, 300 KB

Open AccessArticle

Certain Novel Best Proximity Theorems with Applications to Complex Function Theory and Integral Equations

by Moosa Gabeleh

Axioms 2025, 14(9), 657; https://doi.org/10.3390/axioms14090657 - 27 Aug 2025

Viewed by 860

Abstract

Let

E

and

F

be nonempty disjoint subsets of a metric space

(M, d)

. For a non-self-mapping

φ : E \to F

, which is fixed-point free, a point

ϰ \in E

is said to be a best proximity [...] Read more.

Let

E

and

F

be nonempty disjoint subsets of a metric space

(M, d)

. For a non-self-mapping

φ : E \to F

, which is fixed-point free, a point

ϰ \in E

is said to be a best proximity point for the mapping

φ

whenever the distance of the point

ϰ

to its image under

φ

is equal to the distance between the sets,

E

and

F

. In this article, we establish new best proximity point theorems and obtain real extensions of Edelstein’s fixed point theorem in metric spaces, Krasnoselskii’s fixed point theorem in strictly convex Banach spaces, Dhage’s fixed point theorem in strictly convex Banach algebras, and Sadovskii’s fixed point problem in strictly convex Banach spaces. We then present applications of these best proximity point results to complex function theory, as well as the existence of a solution of a nonlinear functional integral equation and the existence of a mutually nearest solution for a system of integral equations. Full article

(This article belongs to the Special Issue New Theory and Applications of Nonlinear Analysis, Fractional Calculus and Optimization, 2nd Edition)

29 pages, 569 KB

Open AccessArticle

Born’s Rule from Contextual Relative-Entropy Minimization

by Arash Zaghi

Entropy 2025, 27(9), 898; https://doi.org/10.3390/e27090898 - 25 Aug 2025

Viewed by 3070

Abstract

We give a variational characterization of the Born rule. For each measurement context, we project a quantum state ρ onto the corresponding abelian algebra by minimizing Umegaki relative entropy; Petz’s Pythagorean identity makes the dephased state the unique local minimizer, so the Born [...] Read more.

We give a variational characterization of the Born rule. For each measurement context, we project a quantum state ρ onto the corresponding abelian algebra by minimizing Umegaki relative entropy; Petz’s Pythagorean identity makes the dephased state the unique local minimizer, so the Born weights

p_{C} (i) = Tr (ρ P_{i})

arise as a consequence, not an assumption. Globally, we measure contextuality by the minimum classical Kullback–Leibler distance from the bundle

{p_{C} (ρ)}

to the noncontextual polytope, yielding a convex objective

Φ (ρ)

. Thus,

Φ (ρ) = 0

exactly when a sheaf-theoretic global section exists (noncontextuality), and

Φ (ρ) > 0

otherwise; the closest noncontextual model is the classical I-projection of the Born bundle. Assuming finite dimension, full-rank states, and rank-1 projective contexts, the construction is unique and non-circular; it extends to degenerate PVMs and POVMs (via Naimark dilation) without change to the statements. Conceptually, the work unifies information-geometric projection, the presheaf view of contextuality, and categorical classical structure into a single optimization principle. Compared with Gleason-type, decision-theoretic, or envariance approaches, our scope is narrower but more explicit about contextuality and the relational, context-dependent status of quantum probabilities. Full article

(This article belongs to the Special Issue Quantum Foundations: 100 Years of Born’s Rule)

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34 pages, 1302 KB

Open AccessArticle

Integrated Information in Relational Quantum Dynamics (RQD)

by Arash Zaghi

Appl. Sci. 2025, 15(13), 7521; https://doi.org/10.3390/app15137521 - 4 Jul 2025

Cited by 1 | Viewed by 2012

Abstract

We introduce a quantum integrated-information measure

Φ

for multipartite states within the Relational Quantum Dynamics (RQD) framework.

Φ (ρ)

is defined as the minimum quantum Jensen–Shannon distance between an n-partite density operator

ρ

and any product state over a bipartition of [...] Read more.

We introduce a quantum integrated-information measure

Φ

for multipartite states within the Relational Quantum Dynamics (RQD) framework.

Φ (ρ)

is defined as the minimum quantum Jensen–Shannon distance between an n-partite density operator

ρ

and any product state over a bipartition of its subsystems. We prove that its square root induces a genuine metric on state space and that

Φ

is monotonic under all completely positive trace-preserving maps. Restricting the search to bipartitions yields a unique optimal split and a unique closest product state. From this geometric picture, we derive a canonical entanglement witness directly tied to

Φ

and construct an integration dendrogram that reveals the full hierarchical correlation structure of

ρ

. We further show that there always exists an “optimal observer”—a channel or basis—that preserves

Φ

better than any alternative. Finally, we propose a quantum Markov blanket theorem: the boundary of the optimal bipartition isolates subsystems most effectively. Our framework unites categorical enrichment, convex-geometric methods, and operational tools, forging a concrete bridge between integrated information theory and quantum information science. Full article

(This article belongs to the Special Issue Quantum Communication and Quantum Information)

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15 pages, 283 KB

Open AccessArticle

On Jensen-Related Inequalities for Various Types of Convexity via a Unified Approach

by Shoshana Abramovich

Axioms 2025, 14(7), 501; https://doi.org/10.3390/axioms14070501 - 26 Jun 2025

Viewed by 1296

Abstract

The focus of this paper is on three types of convexity: generalized uniform convexity,

Φ

-convexity and superquadracity. The similar structures of these types of convexity are such that the same processes can be applied to each one of them to obtain further [...] Read more.

The focus of this paper is on three types of convexity: generalized uniform convexity,

Φ

-convexity and superquadracity. The similar structures of these types of convexity are such that the same processes can be applied to each one of them to obtain further refinements of known inequalities. Full article

(This article belongs to the Special Issue Theory of Functions and Applications, 3rd Edition)

10 pages, 261 KB

Open AccessArticle

Properties for Close-to-Convex and Quasi-Convex Functions Using q-Linear Operator

by Ekram E. Ali, Rabha M. El-Ashwah, Abeer M. Albalahi and Wael W. Mohammed

Mathematics 2025, 13(6), 900; https://doi.org/10.3390/math13060900 - 7 Mar 2025

Cited by 2 | Viewed by 1023

Abstract

In this work, we describe the

q

-analogue of a multiplier–Ruscheweyh operator of a specific family of linear operators

I_{q, ρ}^{s} (ν, τ)

, and we obtain findings related to geometric function theory (GFT) by utilizing approaches [...] Read more.

In this work, we describe the

q

-analogue of a multiplier–Ruscheweyh operator of a specific family of linear operators

I_{q, ρ}^{s} (ν, τ)

, and we obtain findings related to geometric function theory (GFT) by utilizing approaches established through subordination and knowledge of

q

-calculus operators. By using this operator, we develop generalized classes of quasi-convex and close-to-convex functions in this paper. Additionally, the classes

K_{q, ρ}^{s} (ν, τ) (φ)

,

Q_{q, ρ}^{s} (ν, τ) (φ)

are introduced. The invariance of these recently formed classes under the

q

-Bernardi integral operator is investigated, along with a number of intriguing inclusion relationships between them. Additionally, several unique situations and the beneficial outcomes of these studies are taken into account. Full article

(This article belongs to the Special Issue Advanced Research in Complex Analysis Operators and Special Classes of Analytic Functions)

16 pages, 276 KB

Open AccessArticle

Monotonicities of Quasi-Normed Orlicz Spaces

by Dong Ji and Yunan Cui

Axioms 2024, 13(10), 696; https://doi.org/10.3390/axioms13100696 - 7 Oct 2024

Cited by 2 | Viewed by 1274

Abstract

In this paper, we introduce a new Orlicz function, namely a b-Orlicz function, which is not necessarily convex. The Orlicz spaces

L^{Φ}

generated by the b-Orlicz function

Φ

equipped with a Luxemburg quasi-norm contain both classical spaces [...] Read more.

In this paper, we introduce a new Orlicz function, namely a b-Orlicz function, which is not necessarily convex. The Orlicz spaces

L^{Φ}

generated by the b-Orlicz function

Φ

equipped with a Luxemburg quasi-norm contain both classical spaces

L^{p} (p \geq 1)

and

L^{p} (0 < p < 1)

. The Orlicz spaces

L^{Φ}

are quasi-Banach spaces. Some basic properties in quasi-normed Orlicz spaces are discussed, and the criteria that a quasi-normed Orlicz space is strictly monotonic and lower (upper) locally uniformly monotonic are given. Full article

8 pages, 228 KB

Open AccessArticle

A Unified Version of Weighted Weak-Type Inequalities for the One-Sided Hardy–Littlewood Maximal Function in Orlicz Classes

by Erxin Zhang

Mathematics 2024, 12(18), 2814; https://doi.org/10.3390/math12182814 - 11 Sep 2024

Viewed by 1289

Abstract

Let

M_{g}^{+} f

be the one-sided Hardy–Littlewood maximal function,

φ_{1}

be a nonnegative and nondecreasing function on

[0, \infty)

,

γ

be a positive and nondecreasing function defined on

[0, \infty)

; let [...] Read more.

Let

M_{g}^{+} f

be the one-sided Hardy–Littlewood maximal function,

φ_{1}

be a nonnegative and nondecreasing function on

[0, \infty)

,

γ

be a positive and nondecreasing function defined on

[0, \infty)

; let

φ_{2}

be a quasi-convex function and

u, v, w

be three weight functions. In this paper, we present necessary and sufficient conditions on weight functions

(u, v, w)

such that the inequality

φ_{1} (λ) \int_{{M_{g}^{+} f > λ}} u (x) g (x) d x \leq C \int_{- \infty}^{+ \infty} φ_{2} (C \frac{| f (x) | v (x)}{γ (λ)}) w (x) g (x) d x

holds. Then, we unify the weak and extra-weak-type one-sided Hardy–Littlewood maximal inequalities in the above inequality. Full article

(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)

16 pages, 299 KB

Open AccessArticle

On New Generalized Hermite–Hadamard–Mercer-Type Inequalities for Raina Functions

by Zeynep Çiftci, Merve Coşkun, Çetin Yildiz, Luminiţa-Ioana Cotîrlă and Daniel Breaz

Fractal Fract. 2024, 8(8), 472; https://doi.org/10.3390/fractalfract8080472 - 13 Aug 2024

Cited by 5 | Viewed by 1681

Abstract

In this research, we demonstrate novel Hermite–Hadamard–Mercer fractional integral inequalities using a wide class of fractional integral operators (the Raina fractional operator). Moreover, a new lemma of this type is proved, and new identities are obtained using the definition of convex function. In [...] Read more.

In this research, we demonstrate novel Hermite–Hadamard–Mercer fractional integral inequalities using a wide class of fractional integral operators (the Raina fractional operator). Moreover, a new lemma of this type is proved, and new identities are obtained using the definition of convex function. In addition to a detailed derivation of a few special situations, certain known findings are summarized. We also point out that some results in this study, in some special cases, such as setting

α = 0 = φ, γ = 1,

and

w = 0, σ (0) = 1, λ = 1

, are more reasonable than those obtained. Finally, it is believed that the technique presented in this paper will encourage additional study in this field. Full article

(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications, 2nd Edition)

27 pages, 422 KB

Open AccessArticle

Multivalued Variational Inequalities with Generalized Fractional Φ-Laplacians

by Vy Khoi Le

Fractal Fract. 2024, 8(6), 324; https://doi.org/10.3390/fractalfract8060324 - 29 May 2024

Viewed by 1465

Abstract

In this article, we examine variational inequalities of the form

\{\begin{matrix} ⟨ A (u), v - u ⟩ + ⟨ F (u), v - u ⟩ \geq 0, \forall v \in K \\ u \in K, \end{matrix}

, [...] Read more.

In this article, we examine variational inequalities of the form

\{\begin{matrix} ⟨ A (u), v - u ⟩ + ⟨ F (u), v - u ⟩ \geq 0, \forall v \in K \\ u \in K, \end{matrix}

, where

A

is a generalized fractional

Φ

-Laplace operator, K is a closed convex set in a fractional Musielak–Orlicz–Sobolev space, and

F

is a multivalued integral operator. We consider a functional analytic framework for the above problem, including conditions on the multivalued lower order term

F

such that the problem can be properly formulated in a fractional Musielak–Orlicz–Sobolev space, and the involved mappings have certain useful monotonicity–continuity properties. Furthermore, we investigate the existence of solutions contingent upon certain coercivity conditions. Full article

(This article belongs to the Special Issue Feature Papers for Mathematical Physics Section)

16 pages, 318 KB

Open AccessArticle

Barrelled Weakly Köthe–Orlicz Summable Sequence Spaces

by Issam Aboutaib, Janusz Brzdęk and Lahbib Oubbi

Mathematics 2024, 12(1), 88; https://doi.org/10.3390/math12010088 - 26 Dec 2023

Viewed by 1394

Abstract

Let

E

be a Hausdorff locally convex space. We investigate the space

Λ_{φ} [E]

of weakly Köthe–Orlicz summable sequences in

E

with respect to an Orlicz function

φ

and a perfect sequence space

Λ

. We endow [...] Read more.

Let

E

be a Hausdorff locally convex space. We investigate the space

Λ_{φ} [E]

of weakly Köthe–Orlicz summable sequences in

E

with respect to an Orlicz function

φ

and a perfect sequence space

Λ

. We endow

Λ_{φ} [E]

with a Hausdorff locally convex topology and determine the continuous dual of the so-obtained space in terms of strongly Köthe–Orlicz summable sequences from the dual space

E^{'}

of

E

. Next, we give necessary and sufficient conditions for

Λ_{φ} [E]

to be barrelled or quasi-barrelled. This contributes to the understanding of different spaces of vector-valued sequences and their topological properties. Full article

(This article belongs to the Special Issue Topological Space and Its Applications)

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