Monotone Property Research Articles (Page 1)

Abstract We develop an augmented characteristic, first-order formulation of the field equations in f(R) gravity governing the global evolution of a (possibly) massive scalar field $\phi$ under spherical symmetry. This formulation is designed to isolate the genuine dynamical degrees of freedom while preserving the geometric structure of the theory. By treating the spacetime scalar curvature as an independent unknown, we obtain a closed first-order nonlocal system for the pair (\phi,R). This augmentation eliminates the higher-derivative character of the original equations at the level of the principal part. Our formulation allows us to pose the characteristic initial value problem and to establish several structural properties of solutions. More precisely, we work in generalized Bondi–Sachs coordinates and prescribe initial data on an asymptotically flat future light cone with vertex at the center of symmetry, and we identify the minimal regularity conditions required at the center. These regularity conditions are shown to be precisely those ensuring equivalence between the reduced system and the full f(R) equations. Extending Christodoulou’s method for the Einstein–scalar-field system, we recast the f(R) field equations as an integro-differential system of two coupled, first-order, nonlocal, nonlinear hyperbolic equations, whose principal unknowns are the scalar field and the spacetime scalar curvature. In deriving this reduced two-equation system, we impose natural assumptions on the scalar-field potential and on the function f(R) governing the gravitational Lagrangian density. These hypotheses correspond to standard viability and positivity conditions commonly imposed in the f(R) literature.
We prove several equivalence and monotonicity properties, including for the Hawking mass in this setting.The proposed formulation separates the essential null evolution from the radial constraint reconstruction (via explicit integral relations) on the future domain of dependence of the initial light cone. This structure makes the system amenable to characteristic energy estimates and to stable numerical implementation in spherical symmetry.

A bstract We demonstrate some very special features of the Dirac-Born-Infeld-like (DBI) gravitational counter-term in AdS 4 spacetime, in the context of holography with a sharp radial cut-off. We show that the three-sphere partition function is not only independent of a constant radial cut-off, but also remains unchanged under deformations of the cut-off surface. We also consider the renormalized holographic entanglement entropy for an equatorial Ryu-Takayanagi surface with a cut-off with an arbitrary shape and show that it can also be independent of the cut-off under a special condition. We also numerically study the behavior of the renormalized entropy with different counter-terms and relate the results to monotonicity properties under holographic renormalization group flow. The DBI counter-term is always seen to be associated with integrating out fewer degrees of freedom compared to other counter-terms.

In this paper, the concepts of compactness, almost compactness, and nearly compactness for ech fuzzy soft closure spaces are introduced and discussed. Their characterizations in terms of the finite intersection property are established, and their hereditary properties are investigated for compactness and almost compactness types. The relationships between the three types of compactness are investigated and illustrated with examples. The sufficient condition for the equivalence between compactness and almost compactness has been provided. In addition, new types of fuzzy soft mappings on ech fuzzy soft closure spaces, namely ech fuzzy soft strongly (respectively ech fuzzy soft θ, and ech fuzzy soft almost) continuous mapping, are introduced to study the behavior of the presented compactness types under fuzzy soft mappings.

In recent years, fuzzy soft topology has emerged as a powerful approach that contributes significantly to advancing practical applications in uncertain environments, especially in information systems design, predictive modeling, and decision‐making. The structure of fuzzy soft topology, defined over fuzzy soft sets as a universal set instead of a crisp universal set, was introduced and discussed. This article aims to develop and present a comprehensive study of this structure in more depth by investigating new concepts, properties, and related findings, supported by illustrative examples. We also explore several forms and instances of this type of fuzzy soft topologies. The focus of this article is the exploration of new classes of separation and regularity axioms in the context of these types of fuzzy soft topology. We study these classes, establish their key properties, and reveal the interconnections between them. The relationships between the presented separation axioms and those in the induced crisp and soft topologies are examined. Moreover, we demonstrate that the separation and regularity axioms have the harmonic and hereditary properties. The new concepts, examples, and systematic investigation of the fuzzy soft separation and regularity axioms with their interrelationships achieved in this study contribute to the development of the theoretical foundation for the structure of topology defined over fuzzy soft sets. Moreover, our results not only deepen the understanding of this framework but also pave the way for its improved applications, especially in dealing with uncertain cases in real‐life problems.

Understanding how systems respond to external perturbations is a fundamental challenge in physics, particularly for nonequilibrium and nonstationary processes. The fluctuation-dissipation theorem provides a complete framework for near-equilibrium systems and various bounds have recently been reported for specific nonequilibrium regimes. Here, we present a compact, trajectory-score-based formulation that synthesizes and generalizes linear fluctuation-response relations for arbitrary Markov processes, decomposing system response into spatial correlations of local dynamical events. This decomposition reveals that response properties are encoded in correlations between transitions and dwelling times across the network, providing a natural generalization of the fluctuation-dissipation theorem and recently developed nonequilibrium linear response relations. Our theory unifies existing response bounds, extends them to time-dependent processes, and reveals fundamental monotonicity properties of the tightness of multiparameter response inequalities. Beyond its theoretical significance, this framework enables efficient numerical evaluation of response properties from sampling unperturbed trajectories, offering significant advantages over traditional finite-difference approaches for estimating response properties of complex networks and biological systems far from equilibrium.

Relevance . Discusses the principles of inheritance in the Muslim legal system. In modern conditions, due to migration, the religious foundations of the family structure associated with large families, the percentage of the Muslim population is growing almost all over the world. In such circumstances, the topic of inheritance is of particular scientific and practical interest. The institute of inheritance in Muslim countries is distinguished by certain features and specifics. The purpose is to examine the characteristic features of inheritance according to Sharia and highlight the basic principles of inheritance in the Muslim legal system. Objectives : to determine the order of inheritance by will and by law, taking into account the rules of Sharia, to determine the rights and obligations of heirs, the rules for the distribution of mandatory shares, freedom of probate, the division of property between heirs. Methodology . The research methods are a systematic review of research on this topic, interpretation of legal norms, comparative analysis, as well as methods of induction and deduction. Results. The following features of inheritance in Muslim law are highlighted: according to Sharia, there is a category of heirs who claim a mandatory share in the inheritance, regardless of the will. When making a will, the testator has the right to dispose of only one third of the inherited property, the rest of the property is reserved for mandatory heirs. Hereditary property is subject to division only after paying off all debts, giving alms and other donations provided for by religious rituals. When inheriting property, it is not divided in equal shares, but taking into account the proximity of kinship and gender, men receive twice as much property as women, and the next of kin receive a double share of inheritance, unlike relatives with a distant degree of kinship. Conclusion . The following principles of inheritance in the Muslim legal system are highlighted: the principle of balanced justice; the principle of compulsory share in inheritance; the principle of heir replacement; restriction of freedom of probate; consideration of blood and derivative ties in inheritance by law; consideration of kinship and gender in the division of inheritance; prohibition of deprivation of compulsory share in inheritance.

Abstract In this article we investigate the property of complete monotonicity within a special family $$\mathcal {F}_s$$ F s of functions in s variables involving logarithms. The main result of this work provides a linear isomorphism between $$\mathcal {F}_s$$ F s and the space of real multivariate polynomials. This isomorphism identifies the cone of completely monotone functions with the cone of non-negative polynomials. We conclude that the cone of completely monotone functions in $$\mathcal {F}_s$$ F s is semi-algebraic. This gives a finite time algorithm to decide whether a function in $$\mathcal {F}_s$$ F s is completely monotone.

In this paper, we carry out a comprehensive research on the function D ν ( x ) = x − ν ( I ν ( x ) − L ν ( x ) ) for -1/2 $ ]]> ν > − 1 / 2 and x>0, which finds wide applications in various fields, where I ν ( x ) and L ν ( x ) are the modified Bessel function of the first kind and the modified Struve function, respectively. Specifically, we present some basic properties of the function x ↦ D ν ( x ) , along with several useful bounds. Moreover, we establish the monotonicity of the bounded functions x ↦ x D ν ( x ) , x ↦ D μ ( x ) / D ν ( x ) , and x ↦ ( D ν + 1 ( x ) + D ν − 1 ( x ) ) / D ν ( x ) . Finally, numerical computations show that treating D ν ( x ) as a whole can avoid cancellation problems arising from the subtraction of two large numbers when x is extremely large.

We study ranked enumeration for Conjunctive Queries (CQs) where the answers are ordered by a given ranking function (e.g., an ORDER BY clause in SQL). We develop “ any-k ” algorithms, which, without knowing the number k of desired answers, push down the ranking into joins by carefully ordering the computation of intermediate tuples and avoiding materialization of join answers until they are needed. For this to be possible, the ranking function needs to obey a certain type of monotonicity. Supported ranking functions include the common sum-of-weights, where answers are compared by the sum of input-tuple weights, as well as any commutative selective dioid. Our results extend a well-known unranked-enumeration dichotomy, which states that only free-connex CQs are tractable (under certain hardness hypotheses and for CQs without self-joins). For this class of queries and appropriately monotone ranking functions, the data complexity of our ranked enumeration approach for the time to the k th CQ answer is \({\mathcal {O}}(n + k \log k)\) with n denoting the size of the input, which is only a logarithmic factor slower than the \({\mathcal {O}}(n + k)\) unranked-enumeration guarantee. A core insight of our work is that ranked enumeration for CQs is closely related to Dynamic Programming and the fundamental task of path enumeration in a weighted DAG . We uncover a previously unknown tradeoff, both for this problem and for CQs, under the lens of combined complexity where the query size is not considered a constant: one any- k algorithm has lower complexity when the number of returned answers is small, the other when their number is large. This tradeoff is eliminated under a stricter monotonicity property that we define and exploit for a novel algorithm that asymptotically dominates all previously known alternatives , including Eppstein’s algorithm for sum-of-weights path enumeration. We empirically demonstrate the findings of our theoretical analysis in an experimental study that highlights the superiority of our approach over the join-then-rank approach that existing database systems follow. The project web page at https://northeastern-datalab.github.io/anyk/ contains further information including code, slides, and videos.

The present paper deals with the hereditary property on different algebraic structures and give some example of algebraic structures which has hereditary property. I discuss the proof and definition of some algebraic structures whose substructure has the same character of the structure.

The workshop provided a venue for discussing several recent major developments in graph theory, with a primary focus on results concerning the notion of twin-width and hereditary properties of graphs. Both areas have seen a very rapid development recently, as reflected in the many results presented during the workshop. In addition to many interesting talks spanning the whole breadth of graph theory, the workshop also offered valuable collaboration opportunities, which also engaged early career researchers.

This manuscript presents a new weaker version of supra septarian axioms based on supra $\epsilon$-open sets, along with its essential features, which are called supra-$\epsilon$-$T_{j}$-space, $j=0,1,2$, in the framework of supra topological spaces (or STSs). We give comprehensive explanations of each type of them, backed up by several examples and counterexamples that highlight the significance of our original approaches. We also provide a diagram that outlines these relationships. Additionally, we present the supra $\epsilon$-symmetric property and supra difference property and study their effects on these version of supra-$\epsilon$-septarian axioms. In especial, we show that the two concepts of supra-$\epsilon$-$T_{0}$-space and supra-$\epsilon$-$T_{1}$-space are the same for any STS that fulfills supra $\epsilon$-symmetric property. Finally, we study the supra topological and supra hereditary properties for each of the previously discussed approaches. In particular, we show that the property of being a supra-$\epsilon$-$T_{j}$-space, where $j=0,1,2$, is a supra-hereditary (topological) property.

In this study, we establish new oscillation criteria for solutions of the fourth-order differential equation (aϕuu‴)+q(u∘h)=0, which is of a functional type with a delay. The oscillation behavior of solutions of fourth-order delay equations has been studied using many techniques, but previous results did not take into account the existence of the function ϕ except in second-order studies. The existence of ϕ increases the difficulty of obtaining monotonic and asymptotic properties of the solutions and also increases the possibility of applying the results to a larger area of special cases. We present two criteria to ensure the oscillation of the solutions of the studied equation for two different cases of ϕ. Our approach is based on using the comparison principle with equations of the first or second order to benefit from recent developments in studying the oscillation of these orders. We also provide several examples and compare our results with previous ones to illustrate the novelty and effectiveness.

The lemniscatic functions and their inverses play a vital role in various fields of applied science. In classical mechanics, they appear in the exact solutions of the nonlinear pendulum equation and describe the trajectories of particles in central force fields. In electromagnetism, they are involved in the modeling of electric and magnetic field distributions with elliptic symmetry. Moreover, in engineering mathematics and optical system design, lemniscatic functions arise in problems involving wave propagation, signal transmission, and conformal mappings. Their mathematical properties make them valuable tools in both theoretical analysis and computational modeling. Lemniscatic functions, such as Gauss’ arc lemniscate sine and the hyperbolic arc lemniscate sine, arise from the incompletely symmetric elliptic integral of the first kind. In 2007, Neuman introduced additional lemniscatic functions including the arc lemniscate tangent and its hyperbolic counterpart. These functions have attracted considerable attention due to their deep connections with elliptic integrals and their applications in inequality theory. In this paper, we build upon recent work by Wei, He, and Wang (2020), who established Shafer–Fink type inequalities for these lemniscatic functions. Employing methods from real analysis, we present new and sharper refinements of these inequalities. Specifically, we establish double inequalities for Gauss’ arc lemniscate sine, the hyperbolic arc lemniscate sine, arc lemniscate tangent and its hyperbolic counterpart bounded by rational expressions involving radical terms. These results not only improve the existing Shafer–Fink type inequalities but also provide further insight into the monotonicity and convexity properties of the involved lemniscatic functions. Our approach offers a unified framework for deriving tight bounds for all four Neuman lemniscatic functions, contributing to the broader understanding of special function inequalities. These results represent significant refinements over previous known estimates and open avenues for further generalizations.

In the present paper, we present certain geometric properties, such as starlikeness, convexity of order η(0≤η<1), and close-to-convexity, in an open unit disk of the normalized form of Ramanujan-type entire functions. As a consequence, a specific range of parameters is derived such that this function belongs to Hardy spaces H∞ and Hr. Finally, as an application, we present the monotonicity property of the Ramanujan-type entire function using the method of subordination factor sequences.

Abstract Hierarchical Clustering is a popular tool for understanding the hereditary properties of a data set. Such a clustering is actually a sequence of clusterings that starts with the trivial clustering in which every data point forms its own cluster and then successively merges two existing clusters until all points are in the same cluster. A hierarchical clustering achieves an approximation factor of $$\alpha $$ if the costs of each k-clustering in the hierarchy are at most $$\alpha $$ times the costs of an optimal k-clustering. We study as cost functions the maximum (discrete) radius of any cluster (k-center problem) and the maximum diameter of any cluster (k-diameter problem). In general, the optimal clusterings do not form a hierarchy and hence an approximation factor of 1 cannot be achieved. We call the smallest approximation factor that can be achieved for any instance the price of hierarchy. For the k-diameter problem we improve the upper bound on the price of hierarchy to $$3+2\sqrt{2}\approx 5.83$$ . Moreover we significantly improve the lower bounds for k-center and k-diameter, proving a price of hierarchy of exactly 4 and $$3+2\sqrt{2}$$ , respectively.

Abstract In the framework of data envelopment analysis, we review directional models and show that they are inadequate when inputs and outputs are improved simultaneously under constant returns to scale. Conversely, we introduce a new family of quadratically constrained models with generalized orientation and demonstrate that these models overcome this limitation. Furthermore, we extend the Farrell measure of technical efficiency using these new models. Additionally, we prove that the family of generalized‐oriented models satisfies some desired monotonicity properties. Finally, we show that the new models, although quadratically constrained, can be solved through linear programs in a fundamental particular case.

Abstract First, we solve a crucial problem under which conditions increasing uniform ‐monotonicity is equivalent to lower local uniform ‐monotonicity. Next, we investigate properties of substochastic operators on with applications. Namely, we show that a countable infinite combination of substochastic operators is also substochastic. Using ‐monotonicity properties, we prove several theorems devoted to the convergence of the sequence of substochastic operators in the norm of a symmetric space under addition assumption on . In our final discussion, we focus on compactness of admissible operators for Banach couples under additional assumption.

Monotonicity properties of classical functions and their q-analogues

After investigating several types of geodesic ball packing in S2×R space, in this paper we study the locally optimal geodesic of simply transitive ball packings with equal balls to the space groups generated by rotations in the H2×R geometry. These groups can be derived by direct product of the isometries in the hyperbolic plane H2 and the real line R. Moreover, we develop a procedure to determine the densities of the above locally densest geodesic ball packing configurations. Additionally, we examine the monotonicity properties of the densities within infinite series of the considered space groups. E. Molnár showed, that the homogeneous 3-spaces have a unified interpretation in the projective 3-sphere PS3(V4,V4,R). In our work, we use this projective model of H2×R to visualize the locally optimal ball arrangements.