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Axioms

Axioms is an international, peer-reviewed, open access journal of mathematics, mathematical logic and mathematical physics, published monthly online by MDPI. The European Society for Fuzzy Logic and Technology (EUSFLAT), International Fuzzy Systems Association (IFSA), Union of Slovak Mathematicians and Physicists (JSMF) are Eurekas Community are affiliated with Axioms and their members receive discounts on the article processing charges.

Open Access— free for readers, with article processing charges (APC) paid by authors or their institutions.
High visibility: indexed within SCIE (Web of Science), dblp, and other databases.
Journal Rank: JCR - Q1 (Mathematics, Applied)
Rapid Publication: manuscripts are peer-reviewed and a first decision is provided to authors approximately 22.8 days after submission; acceptance to publication is undertaken in 2.6 days (median values for papers published in this journal in the second half of 2024).
Recognition of Reviewers: reviewers who provide timely, thorough peer-review reports receive vouchers entitling them to a discount on the APC of their next publication in any MDPI journal, in appreciation of the work done.
Companion journal: Logics.

Impact Factor: 1.9 (2023); 5-Year Impact Factor: 1.7 (2023)

Imprint Information Journal Flyer Open Access ISSN: 2075-1680

Latest Articles

26 pages, 478 KiB

Open AccessArticle

Treatment Effect Estimation in Survival Analysis Using Copula-Based Deep Learning Models for Causal Inference

by Jong-Min Kim

Axioms 2025, 14(6), 458; https://doi.org/10.3390/axioms14060458 - 10 Jun 2025

This paper presents the use of Copula-based deep learning with Horvitz–Thompson (HT) weights and inverse probability of treatment weighting (IPTW) for estimating propensity scores in causal inference problems. This study compares the performance of the statistical methods—Copula-based deep learning with HT and IPTW [...] Read more.

This paper presents the use of Copula-based deep learning with Horvitz–Thompson (HT) weights and inverse probability of treatment weighting (IPTW) for estimating propensity scores in causal inference problems. This study compares the performance of the statistical methods—Copula-based deep learning with HT and IPTW weights, propensity score matching (PSM), and logistic regression—in estimating the treatment effect (ATE) using both simulated and real-world data. Our results show that the Copula-based recurrent neural network (RNN) with the method of HT weights provides the most precise and robust treatment effect estimate, with narrow confidence intervals indicating high confidence in the results. The PSM model yields the largest treatment effect estimate, but with greater uncertainty, suggesting sensitivity to data imbalances. In contrast, logistic regression and causal forests produce a substantially smaller estimate, potentially underestimating the treatment effect, particularly in structured datasets such as COMPAS scores. Overall, copula-based methods (HT and IPTW) tend to produce higher and more precise estimates, making them effective choices for treatment effect estimation in complex settings. Our findings emphasize the importance of method selection based on both the magnitude and precision of the treatment effect for accurate analysis. Full article

(This article belongs to the Special Issue New Perspectives in Mathematical Statistics, 2nd Edition)

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14 pages, 680 KiB

Open AccessArticle

Judgment Criteria for Reliability Comparison of Three-Terminal Graphs with High Edge Failure Probability

by Sun Xie, Haixing Zhao, Jun Yin and Jinyu Zou

Axioms 2025, 14(6), 457; https://doi.org/10.3390/axioms14060457 - 10 Jun 2025

A three-terminal graph is defined as a simple graph comprising three specified target vertices. The reliability of three-terminal graphs represents the probability that these three target vertices remain connected, given that each edge fails independently with a constant probability q. In this [...] Read more.

A three-terminal graph is defined as a simple graph comprising three specified target vertices. The reliability of three-terminal graphs represents the probability that these three target vertices remain connected, given that each edge fails independently with a constant probability q. In this paper, we focus on exploring the characteristics of more reliable three-terminal graphs when the edge failure probability approaches 1. Three reliability comparison criteria are proposed to characterize the locally most reliable three-terminal graph progressively when the number of edges m is in the range of

[5, 4 n - 10]

and

[(\binom{n}{2}) - n + 4, (\binom{n}{2})]

. At the same time, the locally optimal structures in the range of the edge number m with

(4 n - 10, (\binom{n}{2}) - n + 4)

are restricted to six specific classes of graphs. Furthermore, based on these criteria, a method is introduced to search local optimal structures and offer a theoretical foundation for constructing optimal networks and repairing faulty ones. Full article

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26 pages, 331 KiB

Open AccessArticle

A Stochastic Nash Equilibrium Problem for Crisis Rescue

by Cunlin Li and Yiyan Li

Axioms 2025, 14(6), 456; https://doi.org/10.3390/axioms14060456 - 10 Jun 2025

This paper proposes a two-stage stochastic non-cooperative game model to solve relief supplies procurement and distribution optimization of multiple rescue organizations in crisis rescue. Rescue organizations with limited budgets minimize rescue costs through relief supply procurement, storage, and transportation in an uncertain environment. [...] Read more.

This paper proposes a two-stage stochastic non-cooperative game model to solve relief supplies procurement and distribution optimization of multiple rescue organizations in crisis rescue. Rescue organizations with limited budgets minimize rescue costs through relief supply procurement, storage, and transportation in an uncertain environment. Under a mild assumption, we establish the existence and uniqueness of the equilibrium point and derive the optimality conditions by using the duality theory, characterizing the saddle point in the Lagrange framework. The problem is further reformulated as a constraint system governed by Lagrange multipliers, and its optimality is characterized by the Karush–Kuhn–Tucker condition. The economic interpretation of the multipliers as shadow prices is elucidated. Numerical experiments verify the effectiveness of the model in cost optimization in crisis rescue scenarios. Full article

24 pages, 1700 KiB

Open AccessArticle

Pearson and Deviance Residual-Based Control Charts for the Inverse Gaussian Ridge Regression Process: Simulation and an Application to Air Quality Monitoring

by Muhammad Amin, Samra Rani and Sadiah M. A. Aljeddani

Axioms 2025, 14(6), 455; https://doi.org/10.3390/axioms14060455 - 9 Jun 2025

In manufacturing and service industries, monitoring processes with correlated input variables and inverse Gaussian (IG)-distributed quality characteristics is challenging due to the limitations of maximum likelihood estimator (MLE)-based control charts. When input variables exhibit multicollinearity, traditional MLE-based inverse Gaussian regression model (IGRM) control [...] Read more.

In manufacturing and service industries, monitoring processes with correlated input variables and inverse Gaussian (IG)-distributed quality characteristics is challenging due to the limitations of maximum likelihood estimator (MLE)-based control charts. When input variables exhibit multicollinearity, traditional MLE-based inverse Gaussian regression model (IGRM) control charts become unreliable. This study introduces novel Shewhart control charts using Pearson and deviance residuals based on the inverse Gaussian ridge regression (IGRR) model to address this issue. The proposed IGRR-based charts effectively handle multicollinearity, offering a robust alternative for process monitoring. Their performance is evaluated through Monte Carlo simulations using average run length (ARL) as the main criteria, demonstrating that Pearson residual-based IGRR charts outperform deviance residual-based charts and MLE-based methods, particularly under high multicollinearity. A real-world application to a Pakistan air quality dataset confirms their superior sensitivity in detecting pollution spikes, enabling timely environmental negotiations. These findings establish Pearson residual-based IGRR control charts as a practical and reliable tool for monitoring complex processes with correlated variables. Full article

(This article belongs to the Special Issue Recent Advances in Statistical Modeling and Simulations with Applications, 2nd Edition)

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11 pages, 220 KiB

Open AccessArticle

Remarks on an Identity of Anastase and Díaz-Barrero

by Horst Alzer and Robert Frontczak

Axioms 2025, 14(6), 454; https://doi.org/10.3390/axioms14060454 - 9 Jun 2025

We extend an algebraic identity of Anastase and Díaz-Barrero (2022) and apply our results to deduce various formulas for sums and series involving (among others) Fibonacci and Lucas numbers, Bernoulli polynomials, and the Riemann zeta function. Full article

15 pages, 314 KiB

Open AccessArticle

Notes on the Free Additive Convolution

by Shokrya S. Alshqaq, Raouf Fakhfakh and Fatimah Alshahrani

Axioms 2025, 14(6), 453; https://doi.org/10.3390/axioms14060453 - 9 Jun 2025

The investigation of free additive convolution is a key concept in free probability theory, offering a framework for studying the sum of freely independent random variables. This paper uses free additive convolution and measure dilations to investigate various aspects of Marchenko–Pastur and free [...] Read more.

The investigation of free additive convolution is a key concept in free probability theory, offering a framework for studying the sum of freely independent random variables. This paper uses free additive convolution and measure dilations to investigate various aspects of Marchenko–Pastur and free Gamma laws in the setting of Cauchy-Stieltjes Kernel (CSK) families. Our investigation reveals the essential links between analytic function theory and free probability, highlighting the usefulness of CSK families in developing the theoretical and computational aspects of free additive convolution. Full article

(This article belongs to the Section Mathematical Analysis)

10 pages, 445 KiB

Open AccessArticle

About Some Unsolved Problems in the Stability Theory of Stochastic Differential and Difference Equations

by Leonid Shaikhet

Axioms 2025, 14(6), 452; https://doi.org/10.3390/axioms14060452 - 9 Jun 2025

This paper continues a series of papers by the author devoted to unsolved problems in the theory of stability and optimal control for stochastic systems. A delay differential equation with stochastic perturbations of the white noise and Poisson’s jump types is considered. In [...] Read more.

This paper continues a series of papers by the author devoted to unsolved problems in the theory of stability and optimal control for stochastic systems. A delay differential equation with stochastic perturbations of the white noise and Poisson’s jump types is considered. In contrast with the known stability condition, in which it is assumed that stochastic perturbations fade on the infinity quickly enough, a new situation is studied, in which stochastic perturbations can either fade on the infinity slowly or not fade at all. Some unsolved problem in this connection is brought to readers’ attention. Additionally, some unsolved problems of stabilization for one stochastic delay differential equation and one stochastic difference equation are also proposed. Full article

(This article belongs to the Section Mathematical Analysis)

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14 pages, 286 KiB

Open AccessArticle

A Class of the Generalized Ramanujan Tau Numbers and Their Associated Partition Functions

by Aleksandar Petojević, Hari M. Srivastava and Sonja Orlić

Axioms 2025, 14(6), 451; https://doi.org/10.3390/axioms14060451 - 7 Jun 2025

In this paper, the authors derive some believed-to-be new recursion and explicit formulas for the generalized Ramanujan numbers

τ_{s} (n)

(s \in N \cup {- 1})

, where, as usual,

N

is the set of positive [...] Read more.

In this paper, the authors derive some believed-to-be new recursion and explicit formulas for the generalized Ramanujan numbers

τ_{s} (n)

(s \in N \cup {- 1})

, where, as usual,

N

is the set of positive integers. The authors consider the associated partition functions and derive connections of the Eisenstein series with the numbers

τ_{s} (n)

. Several related corollaries and consequences of each of the presented results are also given. The paper concludes by presenting an open problem that is related to one of these results. Full article

(This article belongs to the Section Mathematical Analysis)

16 pages, 793 KiB

Open AccessArticle

Note on Iterations of Nonlinear Rational Functions

by Michal Fečkan, Amira Khelifa, Yacine Halim and Ibraheem M. Alsulami

Axioms 2025, 14(6), 450; https://doi.org/10.3390/axioms14060450 - 7 Jun 2025

This paper investigates a class of nonlinear rational difference equations with delayed terms, which often arise in various mathematical models. We analyze the iterative behavior of these rational functions and show how their iterations can be represented through second-order linear recurrence relations. By [...] Read more.

This paper investigates a class of nonlinear rational difference equations with delayed terms, which often arise in various mathematical models. We analyze the iterative behavior of these rational functions and show how their iterations can be represented through second-order linear recurrence relations. By establishing a connection with generalized Balancing sequences, we derive explicit formulas that describe the system’s asymptotic behavior. Our main contribution is proving the existence of a unique globally asymptotically stable equilibrium point for all trajectories, regardless of initial conditions. We also provide analytical expressions for the solutions and support our findings with numerical examples. These results offer valuable insights into the dynamics of nonlinear rational systems and form a theoretical basis for further exploration in this area. Full article

(This article belongs to the Section Mathematical Analysis)

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35 pages, 382 KiB

Open AccessArticle

Generalized Pauli Fibonacci Polynomial Quaternions

by Bahadır Yılmaz, Nazmiye Gönül Bilgin and Yüksel Soykan

Axioms 2025, 14(6), 449; https://doi.org/10.3390/axioms14060449 - 6 Jun 2025

Since Hamilton proposed quaternions as a system of numbers that does not satisfy the ordinary commutative rule of multiplication, quaternion algebras have played an important role in many mathematical and physical studies. This paper introduces the generalized notion of Pauli Fibonacci polynomial quaternions, [...] Read more.

Since Hamilton proposed quaternions as a system of numbers that does not satisfy the ordinary commutative rule of multiplication, quaternion algebras have played an important role in many mathematical and physical studies. This paper introduces the generalized notion of Pauli Fibonacci polynomial quaternions, a definition that incorporates the advantages of the Fibonacci number system augmented by the Pauli matrix structure. With the concept presented in the study, it aims to provide material that can be used in a more in-depth understanding of the principles of coding theory and quantum physics, which contribute to the confidentiality needed by the digital world, with the help of quaternions. In this study, an approach has been developed by integrating the advantageous and consistent structure of quaternions used to solve the problem of system lock-up and unresponsiveness during rotational movements in robot programming, the mathematically compact and functional form of Pauli matrices, and a generalized version of the Fibonacci sequence, which is an application of aesthetic patterns in nature. Full article

(This article belongs to the Special Issue Advances in Applied Algebra and Related Topics)

22 pages, 375 KiB

Open AccessFeature PaperArticle

A Well-Posed Evolutionary Inclusion in Mechanics

by Rawane Mansour and Mircea Sofonea

Axioms 2025, 14(6), 448; https://doi.org/10.3390/axioms14060448 - 6 Jun 2025

We consider an evolutionary inclusion associated with a time-dependent convex in an abstract Hilbert space. We recall a unique solvability result obtained based on arguments of nonlinear equations with maximal monotone operators combined with a penalty method. Then, we state and prove two [...] Read more.

We consider an evolutionary inclusion associated with a time-dependent convex in an abstract Hilbert space. We recall a unique solvability result obtained based on arguments of nonlinear equations with maximal monotone operators combined with a penalty method. Then, we state and prove two well-posedness results. Next, we provide three examples of such inclusions that arise in mechanics. The first one concerns an elastic–perfectly plastic constitutive law, while the last two examples are mathematical models that describe the equilibrium of an elastic body and an elastic–perfectly plastic body, respectively, in frictional contact with an obstacle. The contact is bilateral and the friction is modeled with the Tresca friction law. We use our abstract results in the study of these examples to provide the convergence of the solution with respect to the data. Full article

(This article belongs to the Special Issue Recent Trends in Numerical Methods for Functional Equations)

16 pages, 323 KiB

Open AccessArticle

Scattering in the Energy Space for Solutions of the Damped Nonlinear Schrödinger Equation on

R^{d} \times T

by Taim Saker, Mirko Tarulli and George Venkov

Axioms 2025, 14(6), 447; https://doi.org/10.3390/axioms14060447 - 6 Jun 2025

We will show, in any space dimension

d \geq 3

, the decay and scattering in the energy space for the solution to the damped nonlinear Schrödinger equation posed on

R^{d} \times T

and initial data in [...] Read more.

We will show, in any space dimension

d \geq 3

, the decay and scattering in the energy space for the solution to the damped nonlinear Schrödinger equation posed on

R^{d} \times T

and initial data in

H^{1} (R^{d} \times T)

. We will also derive new bilinear Morawetz identities and corresponding localized Morawetz estimates. Full article

26 pages, 513 KiB

Open AccessArticle

Stability of Weak Rescaled Pure Greedy Algorithms

by Wan Li, Man Lu, Peixin Ye and Wenhui Zhang

Axioms 2025, 14(6), 446; https://doi.org/10.3390/axioms14060446 - 6 Jun 2025

We study the stability of Weak Rescaled Pure Greedy Algorithms for convex optimization, WRPGA(co), in general Banach spaces. We obtain the convergence rates of WRPGA(co) with noise and errors under a weaker assumption for the modulus of smoothness of the objective function. The [...] Read more.

We study the stability of Weak Rescaled Pure Greedy Algorithms for convex optimization, WRPGA(co), in general Banach spaces. We obtain the convergence rates of WRPGA(co) with noise and errors under a weaker assumption for the modulus of smoothness of the objective function. The results show that the rate is almost the same as that of WRPGA(co) without noise and errors, which is optimal and independent of the spatial dimension. This makes WRPGA(co) more practically applicable and scalable for high-dimensional data. Furthermore, we apply WRPGA(co) with errors to the problem of m-term approximation and derive the optimal convergence rate. This indicates the flexibility of WRPGA(co) and its wide utility across machine learning and signal processing. Our numerical experiments verify the stability of WRPGA(co). Thus, WRPGA(co) is a desirable choice for practical implementation. Full article

(This article belongs to the Section Mathematical Analysis)

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17 pages, 698 KiB

Open AccessArticle

Numerical Method for Band Gap Structure and Dirac Point of Photonic Crystals Based on Recurrent Neural Network

by Yakun Wang and Jianhua Yuan

Axioms 2025, 14(6), 445; https://doi.org/10.3390/axioms14060445 - 6 Jun 2025

In this paper, we propose a recurrent neural network numerical method with the finite element method for partial differential equations to study the band gap structure and Dirac points in two-dimensional photonic crystals. Electromagnetic wave propagation is governed by Maxwell’s equations. We transform [...] Read more.

In this paper, we propose a recurrent neural network numerical method with the finite element method for partial differential equations to study the band gap structure and Dirac points in two-dimensional photonic crystals. Electromagnetic wave propagation is governed by Maxwell’s equations. We transform the partial differential equations into large-scale generalized eigenvalue problems by spatially discretising them using the finite element method. Compared with traditional numerical computation methods, neural networks can perform high-speed parallel computation. Existing neural network-based eigenvalue solvers are typically restricted to computing extremal eigenvalues of real symmetric matrix pairs. To overcome this limitation, we develop a novel RNN-based numerical scheme tailored for solving the band structure problem in photonic crystals. We validate our method by computing the dispersion relations of photonic crystals with periodic dielectric columns, achieving excellent agreement with the plane-wave expansion method. In addition, we calculate the Dirac points at the center of the Brillouin zone, which is crucial for understanding the unique optical properties of photonic crystals. We determine the precise filling ratios at which these Dirac points appear, thus providing insight into the relationship between geometrical and material parameters and the appearance of Dirac points. Full article

(This article belongs to the Topic Numerical Methods for Partial Differential Equations)

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14 pages, 3237 KiB

Open AccessArticle

Improving Stability and Reducing Vibrations of the Smooth and Discontinuous Oscillator Using a Proportional–Derivative Controller

by M. N. Abd El-Salam, Rageh K. Hussein and Sabry M. El-Shourbagy

Axioms 2025, 14(6), 444; https://doi.org/10.3390/axioms14060444 (registering DOI) - 5 Jun 2025

In this study, the Proportional and Derivative Controller (PD) is presented as a modified control that combines features of the Proportional Controller (P-Controller) and the Derivative Controller (D-Controller) to suppress the vibrations of the Smooth and Discontinuous Oscillator (SD). The investigated model has [...] Read more.

In this study, the Proportional and Derivative Controller (PD) is presented as a modified control that combines features of the Proportional Controller (P-Controller) and the Derivative Controller (D-Controller) to suppress the vibrations of the Smooth and Discontinuous Oscillator (SD). The investigated model has been derived as a one-degree-of-freedom. The frequency response equation of the controlled system has been obtained using the perturbation technique up to a second approximation. The influence of the P-Controller, the D-Controller, and the PD-Controller on the SD-Oscillator amplitude has been studied by plotting the time histories. The numerical and approximate simulation established that the PD-Controller can inhibit the system vibration. Finally, there is high closeness between the numerical solutions (from time histories) and the approximate solutions (from perturbation analysis). Full article

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21 pages, 696 KiB

Open AccessArticle

Analyzing Dynamical Behaviors of a Stochastic Competitive Model with a Holling Type-II Functional Response Under Diffusion and the Ornstein–Uhlenbeck Process

by Di Yang, Chun Lu and Xiangcun Meng

Axioms 2025, 14(6), 443; https://doi.org/10.3390/axioms14060443 - 5 Jun 2025

Recognizing the crucial impacts of dispersal and noise intensity in ecosystems, this article explores a two-species stochastic competitive model with a Holling Type-II functional response, in which the intrinsic growth rates are driven by the Ornstein–Uhlenbeck process. Firstly, we demonstrate the existence and [...] Read more.

Recognizing the crucial impacts of dispersal and noise intensity in ecosystems, this article explores a two-species stochastic competitive model with a Holling Type-II functional response, in which the intrinsic growth rates are driven by the Ornstein–Uhlenbeck process. Firstly, we demonstrate the existence and uniqueness of the global solution to the model, as well as confirming the boundedness of the moment. Secondly, we proceed to derive sufficient conditions to guarantee the asymptotic stability of the model’s positive equilibrium point and acquire the value of constant b that will affect this property. This indicates that the weaker the noise intensity, the closer the stochastic model approaches the positive equilibrium of the corresponding deterministic model in the mean sense. Furthermore, we build the model by introducing a proper Lyapunov function and provide sufficient conditions under which a stationary distribution exists. Finally, through several numerical simulations, we yield results indicating that weaker noise can ensure the existence and uniqueness of a stationary distribution. Furthermore, this article extends the existing ones. Full article

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28 pages, 13036 KiB

Open AccessArticle

Statistical Analysis of a Generalized Variant of the Weibull Model Under Unified Hybrid Censoring with Applications to Cancer Data

by Mazen Nassar, Refah Alotaibi and Ahmed Elshahhat

Axioms 2025, 14(6), 442; https://doi.org/10.3390/axioms14060442 - 5 Jun 2025

This paper investigates an understudied generalization of the classical exponential, Rayleigh, and Weibull distributions, known as the power generalized Weibull distribution, particularly in the context of censored data. Characterized by one scale parameter and two shape parameters, the proposed model offers enhanced flexibility [...] Read more.

This paper investigates an understudied generalization of the classical exponential, Rayleigh, and Weibull distributions, known as the power generalized Weibull distribution, particularly in the context of censored data. Characterized by one scale parameter and two shape parameters, the proposed model offers enhanced flexibility for modeling diverse lifetime data patterns and hazard rate behaviors. Notably, its hazard rate function can exhibit five distinct shapes, including upside-down bathtub and bathtub shapes. The study focuses on classical and Bayesian estimation frameworks for the model parameters and associated reliability metrics under a unified hybrid censoring scheme. Methodologies include both point estimation (maximum likelihood and posterior mean estimators) and interval estimation (approximate confidence intervals and Bayesian credible intervals). To evaluate the performance of these estimators, a comprehensive simulation study is conducted under varied experimental conditions. Furthermore, two empirical applications on real-world cancer datasets underscore the efficacy of the proposed estimation methods and the practical viability and flexibility of the explored model compared to eleven other existing lifespan models. Full article

(This article belongs to the Special Issue Methods and Applications of Advanced Statistical Analysis, 2nd Edition)

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38 pages, 678 KiB

Open AccessReview

Very Special Relativity Models: Infrared Regularization and Loop Corrections

by Jorge Alfaro

Axioms 2025, 14(6), 441; https://doi.org/10.3390/axioms14060441 - 4 Jun 2025

We review the

S i m (2)

invariant infrared regularization of Very Special Relativity models that we have proposed recently and apply it to compute loop corrections in quantum electrodynamics with VSR masses for neutrino and photon. Then, we compute the [...] Read more.

We review the

S i m (2)

invariant infrared regularization of Very Special Relativity models that we have proposed recently and apply it to compute loop corrections in quantum electrodynamics with VSR masses for neutrino and photon. Then, we compute the axial anomaly. Finally, we study the Gross–Neveu model with a VSR mass in the large N limit uncovering a new phase of the model. Full article

(This article belongs to the Special Issue Complex Variables in Quantum Gravity)

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18 pages, 251 KiB

Open AccessArticle

Complex Riemannian Spacetime: Removal of Black Hole Singularities and Black Hole Paradoxes

by John W. Moffat

Axioms 2025, 14(6), 440; https://doi.org/10.3390/axioms14060440 - 4 Jun 2025

An approach is presented to resolve key paradoxes in black hole physics through the application of complex Riemannian spacetime. We extend the Schwarzschild metric into the complex domain, employing contour integration techniques to remove singularities while preserving the essential features of the original [...] Read more.

An approach is presented to resolve key paradoxes in black hole physics through the application of complex Riemannian spacetime. We extend the Schwarzschild metric into the complex domain, employing contour integration techniques to remove singularities while preserving the essential features of the original solution. A new regularized radial coordinate is introduced, leading to a singularity-free description of black hole interiors. Crucially, we demonstrate how this complex extension resolves the long-standing paradox of event horizon formation occurring only in the infinite future of distant observers. By analyzing trajectories in complex spacetime, we show that the horizon can form in finite complex time, reconciling the apparent contradiction between proper and coordinate time descriptions. This approach also provides a framework for the analytic continuation of information across event horizons, resolving the Hawking information paradox. We explore the physical interpretation of the complex extension versus its projection onto real spacetime. The gravitational collapse of a dust sphere with negligible dust is explored in the complex spacetime extension. The approach offers a mathematically rigorous framework for exploring quantum gravity effects within the context of classical general relativity. Full article

(This article belongs to the Special Issue Complex Variables in Quantum Gravity)

18 pages, 297 KiB

Open AccessArticle

New Bounds for the Davis–Wielandt Radius via the Moore–Penrose Inverse of Bounded Linear Operators

by Xiaomei Dong, Yuzhen Guo and Deyu Wu

Axioms 2025, 14(6), 439; https://doi.org/10.3390/axioms14060439 - 4 Jun 2025

In this paper, we obtain some new upper bounds involving powers of the Davis–Wielandt radius of bounded linear operators with closed ranges by using the Moore–Penrose inverse. Moreover, by providing some examples, we show that the upper bounds obtained here are better than [...] Read more.

In this paper, we obtain some new upper bounds involving powers of the Davis–Wielandt radius of bounded linear operators with closed ranges by using the Moore–Penrose inverse. Moreover, by providing some examples, we show that the upper bounds obtained here are better than the existing ones in some situations. Full article

(This article belongs to the Special Issue Recent Advances in Functional Analysis and Operator Theory)

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