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Niche segregation is a mechanism by which competition between coexisting species is reduced. The ecological niche is a multidimensional space shaped by the conditions and resources that enable the existence of species.We conducted comprehensive univariate and multidimensional analyses of phenotypic traits encompassing morphology, functional performance and ecophysiology, to investigate which phenotypic traits contribute to niche segregation and overlap in two coexisting green lizard species.Our analyses revealed that the main driver of niche segregation was body size. However, when considering size‐corrected phenotypic spaces, ecophysiological traits were still distinct, with little overlap between co‐occurring species. Such differentiation was linked mainly to preferred temperature variance and water loss, highlighting the importance of thermal and hydric niche segregation.Some particular traits such as limb length also contributed to niche segregation being maintained in the morphological space, even when the effect of size was accounted for. Instead, the phenotypic space of functional performance traits (i.e. bite force, locomotor performance) showed the greatest overlap between the two species, thus, less niche segregation was observed, once size effects were removed. Therefore, functional performance traits contribute in a minor proportion to the effective niche segregation between the two species. In light of our results, the most adequate perspective to understand niche segregation in coexisting species is through a multidimensional approach in differentiated phenotypic spaces. Our dismantling of phenotypic traits allowed us to identify niche areas in which trait overlap occurs and others that promote niche segregation, considering or not size effects.Our results suggest that differential use of structural and functional niche space promotes effective niche segregation, potentially reducing direct competition between species. We highlight the importance of studies that include the combination of several phenotypic traits that, as a whole, provide insights to better understand the mechanisms by which coexisting organisms exploit differentiated resources in multidimensional spaces. Read the free Plain Language Summary for this article on the Journal blog. [ABSTRACT FROM AUTHOR]

Background: Adaptive radiotherapy (ART) workflows have been increasingly adopted to achieve dose escalation and tissue sparing under shifting anatomic conditions, but the necessity of recontouring and the associated time burden hinders a real‐time or online ART workflow. In response to this challenge, approaches to auto‐segmentation involving deformable image registration, atlas‐based segmentation, and deep learning‐based segmentation (DLS) have been developed. Despite the particular promise shown by DLS methods, implementing these approaches in a clinical setting remains a challenge, namely due to the difficulty of curating a data set of sufficient size and quality so as to achieve generalizability in a trained model. Purpose: To address this challenge, we have developed an intentional deep overfit learning (IDOL) framework tailored to the auto‐segmentation task. However, certain limitations were identified, particularly the insufficiency of the personalized dataset to effectively overfit the model. In this study, we introduce a personalized hyperspace learning (PHL)‐IDOL segmentation framework capable of generating datasets that induce the model to overfit specific patient characteristics for medical image segmentation. Methods: The PHL‐IDOL model is trained in two stages. In the first, a conventional, general model is trained with a diverse set of patient data (n = 100 patients) consisting of CT images and clinical contours. Following this, the general model is tuned with a data set consisting of two components: (a) selection of a subset of the patient data (m < n) using the similarity metrics (mean square error (MSE), peak signal‐to‐noise ratio (PSNR), structural similarity index (SSIM), and the universal quality image index (UQI) values); (b) adjust the CT and the clinical contours using a deformed vector generated from the reference patient and the selected patients using (a). After training, the general model, the continual model, the conventional IDOL model, and the proposed PHL‐IDOL model were evaluated using the volumetric dice similarity coefficient (VDSC) and the Hausdorff distance 95% (HD95%) computed for 18 structures in 20 test patients. Results: Implementing the PHL‐IDOL framework resulted in improved segmentation performance for each patient. The Dice scores increased from 0.81±$ \pm $0.05 with the general model, 0.83±0.04$ \pm 0.04$ for the continual model, 0.83±0.04$ \pm 0.04$ for the conventional IDOL model to an average of 0.87±0.03$ \pm 0.03$ with the PHL‐IDOL model. Similarly, the Hausdorff distance decreased from 3.06±0.99$ \pm 0.99$ with the general model, 2.84±0.69$ \pm 0.69$ for the continual model, 2.79±0.79$ \pm 0.79$ for the conventional IDOL model and 2.36±0.52$ \pm 0.52$ for the PHL‐IDOL model. All the standard deviations were decreased by nearly half of the values comparing the general model and the PHL‐IDOL model. Conclusion: The PHL‐IDOL framework applied to the auto‐segmentation task achieves improved performance compared to the general DLS approach, demonstrating the promise of leveraging patient‐specific prior information in a task central to online ART workflows. [ABSTRACT FROM AUTHOR]

Multiple harmonic-like numbers are studied using the generating function approach. A closed form is stated for binomial sums involving these numbers and two additional parameters. Several corollaries and examples are presented which are immediate consequences of the main result. Finally, combinatorial identities involving harmonic-like numbers and other prominent sequences like hyperharmonic numbers and odd harmonic numbers are offered. [ABSTRACT FROM AUTHOR]

Mathematician and philosopher Charles Howard Hinton posited a plausible correlation between higher-dimensional spaces, also referred to as 'hyperspaces', and the allegorical concept articulated by the Ancient Greek philosopher Plato in his work, Republic, known as the 'Cave.' In Plato's allegory, individuals find themselves situated in an underground 'Cave', constrained by chains on their legs and neck, perceiving shadows and sound reflections from the 'real' world cast on the 'Cave' wall as their immediate reality. Hinton extended the interpretation of these 'shadows' through the induction method, asserting that, akin to a 3D object casting a 2D shadow, the 'shadow' of a 4D hyper-object would exhibit one dimension less, manifesting as a 3D object. Building upon this conceptual framework, the authors posit a correlation between the perceived acoustic space of the bounded individuals within the 'Cave' and the characteristics of a 4D acoustic space, a proposition substantiated mathematically by scientific inquiry. Furthermore, the authors introduce an interactive art application developed as a methodical approach to exploring the hypothetical 4D acoustic space within Plato's 'Cave', as perceived by the bounded individuals and someone liberated from his constraints navigating through the 'Cave.' [ABSTRACT FROM AUTHOR]

The symbol TD(X) denotes the hyperspace of all nonempty totally disconnected compact subsets of a Hausdorff space X. This hyperspace is endowed with the Vietoris topology. For a mapping between Hausdorff spaces f : X → Y, define the induced mapping TD(f) : TD(X) → TD(Y) by TD(f)(A) = f(A) (the image of A under f). In the current paper, we study the relationships between the condition f belongs to a class of mappings between Hausdorff spaces 필 and the condition TD(f) belongs to 필. [ABSTRACT FROM AUTHOR]

If f is a continuous selection for the Vietoris hyperspace F (X) of the nonempty closed subsets of a space X, then the point f (X) ∈ X is not as arbitrary as it might seem at first glance. In this paper, we will characterise these points by local properties at them. Briefly, we will show that p = f (X) is a strong butterfly point precisely when it has a countable clopen base in ⊂ for some open set U ⊂ X \ {p} with ⊂ = U ⊂ {p}. Moreover, the same is valid when X is totally disconnected at p = f (X) and p is only assumed to be a butterfly point. This gives the complete affirmative solution to a question raised previously by the author. Finally, when p = f (X) lacks the above local base-like property, we will show that F(X) has a continuous selection h with the stronger property that h(S) = p for every closed S ⊂ X with p ∈ S. [ABSTRACT FROM AUTHOR]

Given a metric continuum X, we consider the collection of all regular subcontinua of X and the collection of all meager subcontinua of X, these hyperspaces are denoted by D(X) and M(X), respectively. It is known that D(X) is compact if and only if D(X) is finite. In this way, we find some conditions related about the cardinality of D(X) and we reduce the fact to count the elements of D(X) to a Graph Theory problem, as an application of this, we prove in particular that |D(X)| G {2, 3,4, 5, 8, 9} for any continuum X. Also, we prove that D(X) is never homeomorphic to N. On the other hand, given a point p G X, we consider the meager composant and the filament composant of p in X, denoted by Mpf and FcsX (p), respectively, and we study some relations between Mpx and FcsX (p) such as the equality of them as a subset of X. Also, we construct examples showing that the collection Fcs(X) = {FcsX (p): p G X} can be homeomorphic to: any finite discrete space, the harmonic sequence, the closure of the harmonic sequence and the Cantor set. Finally, we study the contractibility of M(X); we prove the arc of pseudo-arcs, which is a no contractible continuum, satisfies that its hyperspace of meager subcontinua is contractible, given a solution to Problem 3 of [10]. Most of the results shown in this paper are focus to answer problems and questions posed in [6], [9] and [10]. Also, we rise open problems. [ABSTRACT FROM AUTHOR]

The mathematics curriculum has become more dynamic, rigorous, and interdisciplinary in recent years. Therefore, complementing the academic concepts with practical and experiential learning is essential to optimize the learning outcomes. When daunting scientific principles and topics are implemented by traditional diagrams and equations, most students have trouble visualizing; restricting their ability to grasp the idea in detail. Augmented reality has recently become an engineering education method to teach abstract concepts as it enhances the visualization and understanding of technical concepts. In this paper, an augmented reality-based application, 'Hyperspace', was developed to enhance undergraduate students' spatial skills and conceptual knowledge in trigonometry. Hyperspace provides students with features such as augmenting three-dimensional trigonometric functions. These trigonometric functions are dynamically generated using procedural content generation algorithms and computer graphics in augmented reality. An experimental study was conducted to evaluate the effectiveness of Hyperspace on the spatial skills and conceptual knowledge of the students. In total, 127 first-year engineering students took part in the study, and they were randomly assigned to two groups. The students of one group were taught using an AR-based application, and students of the other group were taught using a traditional approach. The experimental outcomes indicate that the augmented reality-based application 'Hyperspace' has significantly enhanced the spatial skills and conceptual knowledge of students when learning about trigonometry. [ABSTRACT FROM AUTHOR]

Gromov introduced two distance functions, the box distance and the observable distance, on the space of isomorphism classes of metric measure spaces and developed the convergence theory of metric measure spaces. We investigate several topological properties on the space equipped with these distance functions toward a deep understanding of convergence theory. [ABSTRACT FROM AUTHOR]

For a Hausdorff non-compact space X, relationships between closure-type properties of the hyperspace (Λ,τ+Δ) and covering properties of that of X have been studied. We then investigate selective separability and some variations of this property. Finally supertightness of (Λ,τ+Δ) has been studied. [ABSTRACT FROM AUTHOR]

Given a continuum X, Cn(X) denotes the hyperspace of nonempty closed subsets of X with at most n components. A strong size level is a subset of the form σ−1(t), where σ is a strong size map for Cn(X) and t ∈ (0, 1]. In this paper, answering a question by Capulín-Pérez, Fuentes-Montes de Oca, Lara-Mejía and Orozco-Zitli, we prove that for each n ≥ 2, no strong size level for Cn(X) is irreducible. [ABSTRACT FROM AUTHOR]

Let X ⊂ P(a0,. . ., an) be a quasi-smooth weighted Fano hypersurface of degree d and index IX such that ai | d for all i. If IX = 1, we show that, under a suitable condition, the α-invariant of X is greater than or equal to dimX/(dimX +1) and X is K-stable. This can be applied in particular to any X as above such that dimX ⩽ 3. If X is general and IX < dimX, then we show that X is K-stable. We also give a sufficient condition for the finiteness of automorphism groups of quasi-smooth Fano weighted complete intersections. [ABSTRACT FROM AUTHOR]

In this paper, by means of the null set defined by Wu (J Math Anal Appl. 2019;472:1741–1761), we introduce a set-valued equilibrium problem based on the null set, where the objective mapping takes values in a hyperspace equipped with a convex cone. Moreover, we obtain new existence results for set-valued equilibrium problems defined on compact or noncompact sets. Some applications are given to set optimization problems, to a set-valued variational inequality, to saddle point theorems for set-valued mappings, and to a generalized noncooperative game [ABSTRACT FROM AUTHOR]

In this paper, we define some combinatorial principles to characterize spaces X whose hyperspace satisfies some variation of some classical star selection principle. Specifically, the variations characterized are the selective and absolute versions of the star selection principles for the Menger and Rothberger cases; also, the hyperspaces considered in these characterizations are CL(X), 핂(X), 픽(X) and ℂ핊(X) in both cases, endowed with either the Fell topology or the Vietoris topology. [ABSTRACT FROM AUTHOR]

A topological space X is selectively highly divergent (SHD) if for every sequence of non-empty open sets {Un: n ε ω} of X, we can find points xn ε Un, for every n < ω such that the sequence {xn: n ∊ ω} has no convergent subsequences. In this note we answer four questions related to this notion that were asked by Jiménez-Flores, Ríos-Herrejón, Rojas-Sánchez and Tovar-Acosta. [ABSTRACT FROM AUTHOR]

We give a lower bound for the polynomial entropy of the induced map on an n -fold symmetric product of X , for a homeomorphism f with at least one wandering point, on a compact space X . Also, we compute some polynomial entropies using this result. [ABSTRACT FROM AUTHOR]

For the estimation of the state of charge of lithium‐ion batteries Kalman filters are the state of the art. To ensure precise and reliable estimations these filters use covariance matrices, which need to be tuned correctly by the developer. This process is time‐consuming and depends largely on the experience and skill of the developer. Hence, filter tuning is not reproducible and not optimal with regard to goals as accuracy and convergence speed. Herein a multiobjective optimization framework called hyperspace exploration is used for the first time to automate the filter tuning procedure for an extended Kalman filter and two versions of adaptive extended Kalman filters. Four key performance indicators, including the maximum error in the estimation of the state of charge and the according root mean square error, are used to describe, validate, and compare the filter performance. This automated process enables optimal usage of the degrees of freedom in filter tuning and no longer requires manual tuning while the whole hyperspace, including different use cases and validation scenarios, is considered in the optimization. Furthermore, the proposed approach yields a novel method for the evaluation of filter parameters and their influence on the estimation behavior. [ABSTRACT FROM AUTHOR]

Given a uniform space (X, U), we introduce, from the uniformity U, some uniformities on the set F(X) of all normal, upper semicontinuous with compact support fuzzy sets on X: the Skorokhod uniformity, the level-wise uniformity, the endograph uniformity and the sendograph uniformity. For metric spaces we prove that these uniformities coincide with the uniformities induced by the Skorokhod metric (the level-wise metric, the endograph metric and the sendograph metric, respectively). We study completeness of this class of uniform spaces. [ABSTRACT FROM AUTHOR]

• Standard reflectance hyperspace vs -log reflectance hyperspace. • Deep learning-based vs linear-based unmixing algorithms. • Unmixing performed in the SWIR range. • Combined metric for pigment identification. • Application to a painting with inscription "Boceto di Pablo Veronese" on the back. Hyperspectral imaging has recently consolidated as a useful technique for pigment mapping and identification, although it is commonly supported by additional non-invasive analytical methods. Since it is relatively rare to find pure pigments in aged paintings, spectral unmixing can be helpful in facilitating pigment identification if suitable mixing models and endmember extraction procedures are chosen. In this study, a subtractive mixing model is assumed, and two approaches are compared for endmember extraction: one based on a linear mixture model, and the other, nonlinear and Deep-Learning based. Two spectral hyperspaces are used: the spectral reflectance (R hyperspace) and the -log(R) hyperspace, for which the subtractive model becomes additive. The performance of unmixing is evaluated by the similarity of the estimated reflectance to the measured data, and pigment identification accuracy. Two spectral ranges (400 to 1000 nm and 900 to 1700 nm) and two objects (a laboratory sample and an aged painting, both on copper) are tested. The main conclusion is that unmixing in the -log(R) hyperspace with a linear mixing model is better than for the non-linear model in R hyperspace, and that pigment identification is generally better in R hyperspace, improving by merging the results in both spectral ranges. [ABSTRACT FROM AUTHOR]

The safety factor (SF) in riverbank stability problems for noncohesive soils is treated mathematically in this article to unravel its inherent bimodality, as stipulated by the cusp catastrophe technique in a hyperspace. The developed methodology may be contrasted with traditional approaches delineating the SF space to failure and operational states. The emerging three states are a significant shift from traditional treatments overlooking bimodality. The mathematical treatment presented in the technical note incorporates classic soil equations for noncohesive soils into the cusp catastrophe technique, with clear formulations at multiple levels of a potential function, including the energy level. The integrated mathematical expressions use soil properties to express lower order properties such as catastrophe flags, for example, bifurcation sets, hysteresis, and bimodality delineating sudden and gradual change in the state of a system. These equations show that even the SF of operational states depends on soil properties and gravity, and therefore a safe use of the SF requires a deep knowledge of the SF hyperspace. Practical applications of the new mathematical development presented in the technical note may be viewed in three steps. In Step 1, the tacit nature of the cusp catastrophe bimodality of safety factor in riverbank stability problems needs to be tested through experimental data. The laboratory tests and fieldwork can be designed to encompass the full range of cases comprising: (i) the failure region; (ii) the operational region; and (iii) their bimodal zone. In Step 2, the existence of the three cases is verified by various applications to underpin the dependency of the safety factor on soil parameters. In Step 3, this new knowledge is realized by wide applications to gain an insight into behaviors of safety factors in wide-ranging problems such as natural slopes, channels, embankments, riverbanks, and levees. [ABSTRACT FROM AUTHOR]

For robots in human environments, learning complex and demanding interaction skills from humans and responding quickly to human motions are highly desirable. A common challenge for interaction tasks is that the robot has to satisfy both the task space and the joint space constraints on its motion trajectories in real time. Few studies have addressed the issue of hyperspace constraints in human-robot interaction, whereas researchers have investigated it in robot imitation learning. In this work, we propose a method of dual-space feature fusion to enhance the accuracy of the inferred trajectories in both task space and joint space; then, we introduce a linear mapping operator (LMO) to map the inferred task space trajectory to a joint space trajectory. Finally, we combine the dual-space fusion, LMO, and phase estimation into a unified probabilistic framework. We evaluate our dual-space feature fusion capability and real-time performance in the task of a robot following a human-handheld object and a ball-hitting experiment. Our inference accuracy in both task space and joint space is superior to standard Interaction Primitives (IP) which only use single-space inference (by more than 33%); the inference accuracy of the second order LMO is comparable to the kinematic-based mapping method, and the computation time of our unified inference framework is reduced by 54.87% relative to the comparison method. [ABSTRACT FROM AUTHOR]

This paper presents an investigation into the mathematical concepts of neutrosophic folding and neuretraction on neutrosophic manifolds, specifically focusing on their application in hyperspace. Through the application of specific transformations on a neutrosophic manifold situated in hyperspace, we can obtain neutrosophic manifolds in lower dimensions. Based on our research, we can accurately establish the connection between neutrosophic folding and neuretraction on a neutrosophic manifold. Furthermore, we can determine the relationship between neuretraction and neutrosophic folding. [ABSTRACT FROM AUTHOR]

The need to study boundary value problems for elliptic parabolic equations is dictated by numerous practical applications in the theoretical study of the processes of hydrodynamics, electrostatics, mechanics, heat conduction, elasticity theory, quantum physics. Let H (dimH ≥ 1) -- a finite-dimensional real Hilbert space with inner product <·,·> and norm || · ||. We will study the equation of the following form u + L (u) = g ∈ H, where L(·) is a non-linear continuous transformation, g is an element of the space H, u is the required solution of the problem from H. In this paper, we obtain two theorems on a priori estimates for solutions of nonlinear equations in a finite-dimensional Hilbert space. The work consists of four items. The conditions of the theorems are such that they can be used in the study of a certain class of initial-boundary value problems to obtain strong a priori estimates. This is the meaning of these theorems. [ABSTRACT FROM AUTHOR]

In this article, we investigate the notion of setwise betweenness, a concept introduced by P. Bankston as a generalisation of pointwise betweenness. In the context of continua, we say that a subset C of a continuum X is between distinct points a and b of X if every subcontinuum K of X containing both a and b intersects C. The notion of an interval [a,b] then arises naturally. Further interesting questions are derived from considering such intervals within an associated hyperspace on X. We explore these ideas within the context of the Vietoris topology and n-symmetric product hyperspaces on all nonempty closed subsets of a topological space X, CL(X). Moreover, an alternative pointwise interval, derived from setwise intervals, is introduced.

In this paper we continue to research relationships between closure-type properties of hyperspaces over a space X and covering properties of X. For a Hausdorff space X we denote by 2X the family of all closed subsets of X. We investigate selection properties of the bitopological space (2X, Δ1+ , Δ2+) where Δi+ is the upper Δi-topology for each i=1,2.

We consider the space Ψ(C) of equivalence classes of continuous open maps defined on the Cantor set C and endowed with the Vietoris topology. It is shown that Ψ(C) is homeomorphic to the space of irrational numbers. [ABSTRACT FROM AUTHOR]

Over the past few decades, physicists are seeking a unifying theory that could encapsulate the theories of general relativity and quantum mechanics, our understanding of the very big and the infinitesimal small, into one all-inclusive theory. This quest led to a renewed interest in the proposal of hyperspace, which states that the structure of space and time are folded into one another, creating multiple dimensions. The author believes that the biblical confession about the resurrected Christ could be beneficial to science and theology in this respect. Biblical testimony provides insight into the apparent natural and effortless movement of Christ between the different dimensions in nature. Astrotheology, as a nexus between the different disciplines, is well equipped to describe the meaning and implications of the resurrection with regard to the fabric of space-time. The author opines that it could facilitate as a natural interface between hyperspace and the Trinity. This proposal aims to accentuate, from a scriptural point of view, that reality indeed comprises more than four dimensions and that astrotheology could make a significant epistemological contribution in the dialogue about hyperspace and God's agency in creation. [ABSTRACT FROM AUTHOR]

Surface roughness parameters can be measured by variety of typical surface roughness testers available in the market. Parameters provide from the measuring devices are plethora, and, in turn, are used to represent the characteristic of the surfaces under investigation. In this particular research work, the authors seek their own initiative to utilize a multivariate statistical - based mean to classify the differences between major machining processes, namely, turning, milling, grinding, lapping, wire cut and EDM. Several multivariate statistical plotting techniques are employed in the machined process surface classification processes. In short, the technique may be employed to identify and/or differentiate some certain surface roughness parameter characteristics between typical machined surfaces. The techniques used are based mainly on the multivariate statistical approach in hyperspace. The relationship between conventional surface roughness parameters was studied utilizing the more fundamental exploratory statistical data analysis utilization of multivariate statistics in hyperspace i.e. Draftsman Plot, Principal Component Analysis and Andrews' Plot. Preliminary results for classification and differentiation of machined surfaces commonly used in industry are presented. It was found out that these techniques were useful for a machined surface profile roughness description in comparison with currently applied parameters, with respect to differentiating between machining processes. [ABSTRACT FROM AUTHOR]

We determine several classes of continua whose hyperspaces of subcontinua are infinite dimensional and homeomorphic to cones over (usually) other continuum. In particular, we obtain many Peano continua with such a property. [ABSTRACT FROM AUTHOR]

Infinite Gray code has been introduced by Tsuiki [Ts02] as a redundancyfree representation of the reals. In applications the signed digit representation is mostly used which has maximal redundancy. Tsuiki presented a functional program converting signed digit code into infinite Gray code. Moreover, he showed that infinite Gray code can effectively be converted into signed digit code, but the program needs to have some non-deterministic features (see also [TS05]). Berger and Tsuiki [BT21a, BT21b] reproved the result in a system of formal first-order intuitionistic logic extended by inductive and co-inductive definitions, as well as some new logical connectives capturing concurrent behaviour. The programs extracted from the proofs are exactly the ones given by Tsuiki. In order to do so, co-inductive predicates S and G are defined and the inclusion S ⊆ G is derived. For the converse inclusion the new logical connectives are used to introduce a concurrent version S2 of S and G ⊆ S2 is shown. What one is looking for, however, is an equivalence proof of the involved concepts. One of the main aims of the present paper is to close the gap. A concurrent version G* of G and a modification S * of S2 are presented such that S * = G*. A crucial tool in [BT21a] is a formulation of the Archimedean property of the real numbers as an induction principle. We introduce a concurrent version of this principle which allows us to prove that S * and G* coincide. A further central contribution is the extension of the above results to the hyperspace of non-empty compact subsets of the reals. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a notion of shadowing property for a free semigroup action on a compact metric space, which is different of the notion of the shadowing property introduced by Bahabadi, called chain shadowing property. We study the relation between the shadowing property of a free semigroup action on a compact metric space X and the shadowing property of the induced free semigroup action on the hyperspace 2X. Specially, we not only theoretically prove that (F m + , F) ↷ X has the (chain) shadowing property if and only if (F m + , F) ↷ 2 X has the (chain) shadowing property, but also give examples to illustrate it. Finally, we compare the two notions of shadowing for free semigroup actions and obtain an interesting result that if (F m + , F) ↷ X has the shadowing property, then it has the chain shadowing property, but not vice versa. [ABSTRACT FROM AUTHOR]

In this paper, we study the relation between a space X satisfying certain generalized metric properties and the Pixley-Roy hyperspace F[X] over X satisfying the same properties. We prove that if X has a s-point-finite cn-network (resp., ck-network), then F[X] also has a s-point-finite cn-network (resp., ck-network). [ABSTRACT FROM AUTHOR]

Let X be a compact Hausdorff space. In this work we translate partial actions of X to partial actions on some hyperspaces determined by X, this gives an endofunctor 2- in the category of partial actions on compact Hausdorff spaces which generates a monad in this category. Moreover, structural relations between partial actions θ on X and partial determined by 2θ as well as their corresponding globalizations are established.

In this paper, we define the concept of pseudo-prime subhypermodules of hypermodules as a generalization of the prime hyperideal of commutative hyperrings. In particular, we examine the spectrum of the Zariski topology, which we built on the element of the pseudo-prime subhypermodules of a class of hypermodules. Moreover, we provide some relevant properties of the hypermodule in this topological hyperspace. [ABSTRACT FROM AUTHOR]

In this paper we point out an interesting geometric structure of nonnegative metric curvature emerging from the hyperspaces of decomposable, non-locally connected homogeneous continua, where "smooth" and "non-smooth" partitions live together, similarly to the macroscopic and the quantum realms of the Universe. [ABSTRACT FROM AUTHOR]

In this paper, we study the relationship between the multi-sensitivity and the topological maximal sequence entropy of dynamical systems for general group action. Furthermore, we also discuss the consistency of multi-sensitivity of a dynamical system (G ↷ X) and its hyperspace dynamical system G ↷ K(X). Moreover, we research the relationship between the multi-sensitivity of two dynamical systems and the multi-sensitivity of their product space dynamical system. Finally, we prove that if the topological sequence entropy of G ↷ X vanishes, then so does that of its induced system G ↷ ℳ (X) ; if the topological sequence entropy of G ↷ X is positive, then that of its induced system G ↷ ℳ (X) is infinity. [ABSTRACT FROM AUTHOR]

In this article, we study the space Bμ(BX,Y) of Y-valued Bloch-type functions on the unit ball BX of an infinite dimensional Hilbert space X with μ is a normal weight on BX and Y is a Banach space. We also investigate the characterizations of the space WBμ(BX) of Y-valued, locally bounded, weakly holomorphic functions associated with the Bloch-type space Bμ(BX) of scalar-valued functions in the sense that f ∈ WBμ(BX) if w ∘ f ∈ Bμ(BX) for every w ∈ W, a separating subspace of the dual Y′ of Y. [ABSTRACT FROM AUTHOR]

Let M be a semifinite von Neumann algebra on a Hilbert space H equipped with a faithful normal semifinite trace τ, S(M, τ) be the *-algebra of all τ-measurable operators. Let S0 (M, τ) be the *-algebra of all τ-compact operators and T(M, τ) = S0 (M, τ) + CI be the *-algebra of all operators X = A + λI with A ∈ S0 (M, τ) and λ ∈ C. It is proved that every operator of T (M, τ) that is left-invertible in T(M, τ) is in fact invertible in T(M, τ). It is a generalization of Sterling Berberian theorem (1982) on the subalgebra of thin operators in B(H). For the singular value function μ(t; Q) of Q = Q2 ∈ S(M, τ), the inclusion μ(t; Q) ∈ {0}⋃[1, + ∞) holds for all t > 0. It gives the positive answer to the question posed by Daniyar Mushtari in 2010. [ABSTRACT FROM AUTHOR]

Let f : M → M be a Morse–Smale diffeomorphism defined on a compact and connected manifold without boundary. Let C(M) denote the hyperspace of all subcontinua of M endowed with the Hausdorff metric and C (f) : C (M) → C (M) denote the induced homeomorphism of f. We show in this paper that if M is the unit circle S 1 then the induced map C(f) has not the shadowing property. Also we show that the topological entropy of C(f) has only two possible values: 0 or ∞ . In particular, we show that the entropy of C(f) is 0 when M is the unit circle S 1 and it is ∞ if the dimension of the manifold M is greater than two. Furthermore, we study the recurrence of the induced maps 2 f and C(f) and sufficient conditions to obtain infinite topological entropy in the hyperspace. [ABSTRACT FROM AUTHOR]

In this paper, we study the connection between topological entropies of general tree maps and their extended maps on the spaces of fuzzy numbers. First, we show some aspects of the structures of ω -limit sets of the extended (and induced) topological tree maps. Next, we prove that topological entropies of a crisp tree map and its extended map all agree. [ABSTRACT FROM AUTHOR]

In this paper, we present translation hypersurfaces of semi-Euclidean spaces with zero scalar curvature. In addition, we prove that translation hypersurfaces with constant scalar curvature must have zero scalar curvature in the semi-Euclidean space ... . [ABSTRACT FROM AUTHOR]

The Vietoris space of a Stone space plays an important role in the coalgebraic approach to modal logic. When generalizing this to positive modal logic, there is a variety of relevant hyperspace constructions based on various topologies on a Priestley space and mechanisms to topologize the hyperspace of closed sets. A number of authors considered hyperspaces of Priestley spaces and their application to the coalgebraic approach to positive modal logic. A mixture of techniques from category theory, pointfree topology, and Priestley duality have been employed. Our aim is to provide a unifying approach to this area of research relying only on a basic familiarity with Priestley duality and related free constructions of distributive lattices. [ABSTRACT FROM AUTHOR]

In this article, we investigate the notion of setwise betweenness, a concept introduced by P. Bankston as a generalisation of pointwise betweenness. In the context of continua, we say that a subset C of a continuum X is between distinct points a and b of X if every subcontinuum K of X containing both a and b intersects C. The notion of an interval [a, b] then arises naturally. Further interesting questions are derived from considering such intervals within an associated hyperspace on X. We explore these ideas within the context of the Vietoris topology and n-fold symmetric product hyperspaces on all nonempty closed subsets of a topological space X, CL(X). Moreover, an alternative pointwise interval, derived from setwise intervals, is introduced. [ABSTRACT FROM AUTHOR]