Search Results (327)

We study the strong S-property for Nagata and Serre conjecture rings through the framework of $(T,M,D)$ construction rings, providing a unified approach that streamlines and extends previous results. Our main contribution is a concise, conceptual proof showing that the strong S-property of $R\left(X\right)$ versus $R⟨X⟩$ depends solely on the transcendence degree of the residue field extension $K/k$, where k is the quotient field of D. This perspective yields new, transparent counterexamples to both the Malik–Mott conjecture and a question of Cahen et al., and provides a clear characterization of the catenarity of Serre conjecture rings, $R⟨n⟩$. The approach is based on pullback constructions and the geometric structure of prime ideals, replacing intricate case analyses with arguments driven by natural invariants. Full article

This paper develops several new generalizations of Gronwall–Bellman–Bihari-type integral inequalities. We establish three novel integral inequalities that extend classical results to more complex settings, including integrals with mixed linear and nonlinear terms, delayed (retarded) arguments, and general integral kernels. In the preliminaries, we review known Gronwall–Bellman–Bihari inequalities and useful lemmas. In the main results, we present at least three new theorems. The first theorem provides an explicit bound for solutions of an integral inequality involving a separable kernel function and a nonlinear (Bihari-type) term, significantly extending the classical Bihari inequality. The second theorem addresses integral inequalities with delayed arguments, showing that the delay does not enlarge the growth bound compared to the non-delay case. The third theorem handles inequalities with combined linear and nonlinear terms; using a monotone iterative technique, we prove the existence of a maximal solution that bounds any solution of the inequality. Rigorous proofs are given for all main results. In the Applications section, we illustrate how these inequalities can be applied to deduce qualitative properties of differential equations. As an example, we prove a uniqueness result for an initial value problem with a non-Lipschitz nonlinear term using our new inequalities. The paper concludes with a summary of results and a brief discussion of potential further generalizations. Our results provide powerful tools for researchers to obtain a priori bounds and uniqueness criteria for various differential, integral, and functional equations. It is important to note that the integral inequalities established in this work provide bounds on the solution under the assumption of its existence on the considered interval $[t_0,T]$. For nonlinear differential or integral equations where the nonlinearity $F$ fails to be Lipschitz continuous, solutions may develop movable singularities (blow-up) in finite time. The bounds derived from our Gronwall–Bellman–Bihari-type inequalities are valid only on the maximal interval of existence of the solution. Determining the region where solutions are guaranteed to be free of such singularities is a separate and profound problem, often requiring additional techniques such as the construction of Lyapunov functions or the use of differential comparison principles. The primary contribution of this paper is to provide sharp estimates and uniqueness criteria within the domain where a solution is known to exist a priori. Full article

Are your perceptions, memories and observations merely a statistical fluctuation arising from of the thermal equilibrium of the universe, bearing no correlation to the actual past state of the universe? Arguments are given in the literature for and against this “Boltzmann brain” hypothesis. Complicating these arguments have been the many subtle—and very often implicit—joint dependencies among these arguments and others that have been given for the past hypothesis, the second law, and even for Bayesian inference of the reliability of experimental data. These dependencies can easily lead to circular reasoning. To avoid this problem, since all of these arguments involve the stochastic properties of the dynamics of the universe’s entropy, we begin by formalizing that dynamics as a time-symmetric, time-translation invariant Markov process, which we call the entropy conjecture. Crucially, like all stochastic processes, the entropy conjecture does not specify any time(s) which it should be conditioned on in order to infer the stochastic dynamics of our universe’s entropy. Any such choice of conditioning times and associated entropy values must be introduced as an independent assumption. This observation allows us to disentangle the standard Boltzmann brain hypothesis, its “1000CE” variant, the past hypothesis, the second law, and the reliability of our experimental data, all in a fully formal manner. In particular, we show that these all make an arbitrary assumption that the dynamics of the universe’s entropy should be conditioned on a single event at a single moment in time, differing only in the details of their assumptions. In this aspect, the Boltzmann brain hypothesis and the second law are equally legitimate (or not). Full article

Taking a cognitive perspective on emotions as generally exemplified by appraisal theories, I suggest that attempts to “define” emotions is a theoretical exercise whose goal should be to specify necessary and jointly sufficient conditions for something to be an emotion. To this end, I advance arguments in support of the proposal that genuine emotions have the five necessary characteristics of being (i) intentional (i.e., about something), (ii) personally significant, (iii) valenced, (iv) consciously experienced, and (v) insuppressible. Collectively, these properties distinguish emotions from other kinds of mental states. I also argue that attempts to define emotions should resist the temptation to incorporate into definitions characteristics of emotions that are not always present, even though, when they are present, those characteristics may be typical and highly salient. It is suggested that two characteristics that are routinely taken to be constitutive of emotions—bodily changes and facial expression—are just such characteristics; they are typical and salient but not in fact necessary as evidenced by the fact that many (especially low intensity) emotions occur without them. Full article

Autonomy is enabled by the close connection of traditional mechanical systems with information technology. Historically, both communities have built norms for validation and verification (V&V), but with very different properties for safety and associated legal liability. Thus, combining the two in the context of autonomy has exposed unresolved challenges for V&V, and without a clear V&V structure, demonstrating safety is very difficult. Today, both traditional mechanical safety and information technology rely heavily on process-oriented mechanisms to demonstrate safety. In contrast, a third community, the semiconductor industry, has achieved remarkable success by inserting design artifacts which enable formally defined mathematical abstractions. These abstractions combined with associated software tooling (Electronics Design Automation) provide critical properties for scaling the V&V task, and effectively make an inductive argument for system correctness from well-defined component compositions. This article reviews the current methods in the mechanical and IT spaces, the current limitations of cyber-physical V&V, identifies open research questions, and proposes three directions for progress inspired by semiconductors: (i) guardian-based safety architectures, (ii) functional decompositions that preserve physical constraints, and (iii) abstraction mechanisms that enable scalable virtual testing. These perspectives highlight how principles from semiconductor V&V can inform a more rigorous and scalable safety framework for autonomous systems. Full article

This paper applies a general framework to explore the existence of multiple positive solutions for the fractional integral boundary value problem of high-order Caputo and Hadamard fractional coupled Laplacian systems with delayed or advanced arguments. We first focus on a generalized fractional homomorphic coupled boundary value problem with Hilfer fractional derivatives. Then we present the Green’s function corresponding to this Hilfer fractional system and its important properties. On this basis, by constructing a positive cone and applying a generalized cone fixed point theorem, we have established some novel criteria to ensure that the generalized fractional system has at least three positive solutions. As applications, we also obtain the multiplicity of the positive solutions of the Caputo and Hadamard fractional-order coupled Laplacian systems under two special Hilfer derivatives, respectively. Finally, we provide several examples to inspect the applicability of the main results. Full article

The findings of this study are connected with geometric function theory and were acquired using subordination-based techniques in conjunction with the Hurwitz–Lerch Zeta function. We used the Hurwitz–Lerch Zeta function to investigate certain properties of multivalent meromorphic functions. The primary objective of this study is to provide an investigation on the argument properties of multivalent meromorphic functions in a punctured open unit disc and to obtain some results for its subclass. Full article

In this paper we will be concerned with zeta-symmetry—the functional equation for the (Riemann) zeta-function (equivalents to which are called modular relations)—and reveal the reason why so many results are intrinsic to PFE (Partial Fraction Expansion) for the cotangent function. The hidden reason is that the cotangent function (as a function in the upper half-plane, say) is the polylogarithm function of order 0 (with complex exponential argument), and therefore it shares properties intrinsic to the Lerch zeta-function of order 0. Here we view the Lerch zeta-function defined in the unit circle as a zeta-function in a wider sense, as a function defined in the upper and lower half-planes. As evidence, we give a plausibly most natural proof of Ramanujan’s formula, including the eta transformation formula as a consequence of the modular relation via the cotangent function, speculating the reason why Ramanujan had been led to such a formula. Other evidence includes the pre-Poisson summation formula as the pick-up principle (which in turn is a generalization of the argument principle). Full article

In the ontological argument for the existence of God, Descartes famously argues that the idea of God is the idea of a perfect being. As such, the idea of God must combine all of the perfections. Now, as (necessary) existence is a perfection, God must exist. Leibniz criticized Descartes’ argument, pointing out that it rests upon the hidden assumption that God is possible. Leibniz argues, however, that God is really possible because realities cannot oppose one another, and so there could be no real opposition between the perfections. So, at least in the case of God, conceivability entails real possibility. Kant rejects this assumption and insists that the non-contradictoriness of an idea is not an adequate criterion for the real possibility of the object of the idea, for although predicates may be combined in thought to form a concept, this does not entail the properties they indicate may be so combined in reality. For this reason, Kant believes that it is impossible to prove the real possibility of God, and so the ontological argument is not sound. In this paper, I examine Kant’s reasons for reaching this conclusion. I pay particular attention to Kant’s argument in the Amphiboly, which deals with the concepts of agreement and opposition, and where Kant stresses the importance of the distinction between logical and real opposition. I will argue that this distinction plays a crucial role in Kant’s rejection of the ontological argument and rationalist Leibnizian–Wolffian metaphysics in general. I also show how Kant’s rejection of the possibility of what he calls the complete determination of a concept in the Ideal of Pure Reason, plays a role in his rejection of the conceivability entails real possibility principle. Full article

In intuitionistic fuzzy 2-normed spaces, there are numerous symmetries in the topological structures and mapping theorems. In this work, we present the concept of an intuitionistic fuzzy 2-normed space(IF2NS) and demonstrate its structural properties using illustrative examples. This approach unifies and broadens the scope of both classical 2-normed spaces and intuitionistic fuzzy normed spaces when specific conditions are met. We introduce the idea of fuzzy open balls and explore the convergence of sequences with respect to the topology derived from the intuitionistic fuzzy 2-norm. In addition, we define left and right N-Cauchy sequences relative to the topologies ${\tau }_N$ and ${\tau }_{N^{-1}}$ and analyze their convergence characteristics. Special attention is given to the inherent symmetry of the 2-norm, where the magnitude of a pair of vectors remains invariant under exchange of arguments, and to the balanced interaction between membership and non-membership functions in the intuitionistic fuzzy setting. This intrinsic symmetry is further reflected in the proofs of the open mapping and closed graph theorems, which naturally preserve the symmetric structure of the underlying space The paper culminates with the formulation and proof of the open mapping theorem that can be considered for its symmetric properties and the closed graph theorem in the context of IF2NS, thereby generalizing essential theorems of functional analysis to this fuzzy setting. Full article

Multifractality in time series analysis characterizes the presence of multiple scaling exponents, indicating heterogeneous temporal structures and complex dynamical behaviors beyond simple monofractal models. In the context of digital currency markets, multifractal properties arise due to the interplay of long-range temporal correlations and heavy-tailed distributions of returns, reflecting intricate market microstructure and trader interactions. Incorporating multifractal analysis into the modeling of cryptocurrency price dynamics enhances the understanding of market inefficiencies. It may also improve volatility forecasting and facilitate the detection of critical transitions or regime shifts. Based on the multifractal cross-correlation analysis (MFCCA) whose spacial case is the multifractal detrended fluctuation analysis (MFDFA), as the most commonly used practical tools for quantifying multifractality, we applied a recently proposed method of disentangling sources of multifractality in time series to the most representative instruments from the digital market. They include Bitcoin (BTC), Ethereum (ETH), decentralized exchanges (DEX) and non-fungible tokens (NFT). The results indicate the significant role of heavy tails in generating a broad multifractal spectrum. However, they also clearly demonstrate that the primary source of multifractality encompasses the temporal correlations in the series, and without them, multifractality fades out. It appears characteristic that these temporal correlations, to a large extent, do not depend on the thickness of the tails of the fluctuation distribution. These observations, made here in the context of the digital currency market, provide a further strong argument for the validity of the proposed methodology of disentangling sources of multifractality in time series. Full article

We experimentally investigate the structural, magnetic, transport, and electronic properties of two d⁵ iridate double perovskite materials La₂BIrO₆ (B = Mg, Zn). Notably, despite similar crystallographic structure, the two compounds show distinctly different magnetic behaviors. The M = Mg compound shows an antiferromagnetic-like linear field-dependent isothermal magnetization below its transition temperature, whereas the M = Zn counterpart displays a clear hysteresis loop followed by a noticeable coercive field, indicative of ferromagnetic components arising from a non-collinear Ir spin arrangement. The local structure studies authenticate perceptible M/Ir antisite disorder in both systems, which complicates the magnetic exchange interaction scenario by introducing Ir-O-Ir superexchange pathways in addition to the nominal Ir-O-B-O-Ir super-superexchange interactions expected for an ideally ordered structure. While spin–orbit coupling (SOC) plays a crucial role in establishing insulating behavior for both these compounds, the rotational and tilting distortions of the IrO₆ (and MO₆) octahedral units further lift the ideal cubic symmetry. Finally, by measuring the Ir-L₃ edge resonant inelastic X-ray scattering (RIXS) spectra for both the compounds, giving evidence of spin–orbit-derived low-energy inter-J-state (intra t_2g) transitions (below ~1 eV), the charge transfer (O 2p → Ir 5d), and the crystal field (Ir t_2g → e_g) excitations, we put forward a qualitative argument for the interplay among effective SOC, non-cubic crystal field, and intersite hopping in these two compounds. Full article

Fractional-order chaotic systems have emerged as powerful tools in secure communications and multimedia protection owing to their memory-dependent dynamics, large key spaces, and high sensitivity to initial conditions. However, most existing fractional-order image encryption schemes rely on fixed-order chaos and conventional solvers, which limit their complexity and reduce unpredictability, while also neglecting the potential of variable fractional-order (VFO) dynamics. Although similar phenomena have been reported in some fractional-order systems, the coexistence of hidden attractors and stable equilibria has not been extensively investigated within VFO frameworks. To address these gaps, this paper introduces a novel discrete variable fractional-order dark matter–dark energy (VFODM-DE) chaotic system. The system is discretized using the piecewise constant argument discretization (PWCAD) method, enabling chaos to emerge at significantly lower fractional orders than previously reported. A comprehensive dynamic analysis is performed, revealing rich behaviors such as multistability, symmetry properties, and hidden attractors coexisting with stable equilibria. Leveraging these enhanced chaotic features, a pseudorandom number generator (PRNG) is constructed from the VFODM-DE system and applied to grayscale image encryption through permutation–diffusion operations. Security evaluations demonstrate that the proposed scheme offers a substantially large key space (approximately $2^{249}$) and exceptional key sensitivity. The scheme generates ciphertexts with nearly uniform histograms, extremely low pixel correlation coefficients (less than 0.04), and high information entropy values (close to 8 bits). Moreover, it demonstrates strong resilience against differential attacks, achieving average NPCR and UACI values of about 99.6% and 33.46%, respectively, while maintaining robustness under data loss conditions. In addition, the proposed framework achieves a high encryption throughput, reaching an average speed of 647.56 Mbps. These results confirm that combining VFO dynamics with PWCAD enriches the chaotic complexity and provides a powerful framework for developing efficient and robust chaos-based image encryption algorithms. Full article

Nano-sanitizers, which exploit the unique physicochemical properties of nanomaterials, are being increasingly investigated as innovative tools to promote food safety. In this argumentative review, we compare and contrast nano-sanitizers with conventional sanitation methods by examining their underlying antimicrobial mechanisms, multifaceted benefits, inherent challenges, and wide-ranging public health implications. We evaluate regulatory conundrums and consumer perspectives alongside future outlooks for integration with advanced technologies such as artificial intelligence. Through selective synthesis of the published literature, our argumentative discussion demonstrates that nano-sanitizers not only promise superior performance in pathogen inactivation but could also contribute to overall food system sustainability, provided safety and regulatory concerns are adequately addressed. Full article

We investigate the solvability and stability properties of a class of nonlinear stochastic delay differential equations (SDDEs) driven by Wiener noise and incorporating discrete time delays. The equations are formulated within a Banach space of continuous, adapted sample paths. Under standard Lipschitz and linear growth conditions, we construct a solution operator and prove the existence and uniqueness of strong solutions using a fixed-point argument. Furthermore, we derive exponential mean-square stability via Lyapunov-type techniques and delay-dependent inequalities. This framework provides a unified and flexible approach to SDDE analysis that departs from traditional Hilbert space or semigroup-based methods. We explore a Banach space fixed-point approach to SDDEs with multiplicative noise and discrete delays, providing a novel functional-analytic framework for examining solvability and stability. Full article