Countable Research Articles

Lifting theorems for completely positive maps

We prove lifting theorems for completely positive maps going out of exact C^* -algebras, where we remain in control of which ideals are mapped into which. A consequence is, that if \mathsf X is a second countable topological space, \mathfrak A and \mathfrak B are separable, nuclear C^* -algebras over \mathsf X , and the action of \mathsf X on \mathfrak A is continuous, then E(\mathsf X; \mathfrak A, \mathfrak B) \cong KK(\mathsf X; \mathfrak A, \mathfrak B) naturally. As an application, we show that a separable, nuclear, strongly purely infinite C^* -algebra \mathfrak A absorbs a strongly self-absorbing C^* -algebra \mathscr D if and only if \mathfrak I and \mathfrak I\otimes \mathscr D are KK -equivalent for every two-sided, closed ideal \mathfrak I in \mathfrak A . In particular, if \mathfrak A is separable, nuclear, and strongly purely infinite, then \mathfrak A \otimes \mathcal O_2 \cong \mathfrak A if and only if every two-sided, closed ideal in \mathfrak A is KK -equivalent to zero.

A Note on Hyperspaces by Closed Sets with Vietoris Topology

For a topological space X, let CL(X) be the set of all non-empty closed subset of X, and denote the set CL(X) with the Vietoris topology by $$(CL(X), {\mathbb {V}})$$ . In this paper, we mainly discuss the hyperspace $$(CL(X), {\mathbb {V}})$$ when X is an infinite countable discrete space. As an application, we first prove that the hyperspace with the Vietoris topology on an infinite countable discrete space contains a closed copy of nth power of Sorgenfrey line for each $$n\in {\mathbb {N}}$$ . Then we investigate the tightness of the hyperspace $$(CL(X), {\mathbb {V}})$$ and prove that the tightness of $$(CL(X), {\mathbb {V}})$$ is equal to the set-tightness of X. Moreover, we extend some results about the generalized metric properties on the hyperspace $$(CL(X), {\mathbb {V}})$$ . Finally, we give a characterization of X such that $$(CL(X), {\mathbb {V}})$$ is a $$\gamma $$ -space.

On the spectrum of Euler–Lagrange operator in the stability analysis of Bénard problem

In studying the stability of Bénard problem, we usually have to solve a variational problem to determine the critical Rayleigh number for linear or nonlinear stability. To solve the variational problem, one usually transforms it to an eigenvalue problem which is called Euler–Lagrange equations. An operator related to the Euler–Lagrange equations is usually referred to as Euler–Lagrange operator whose spectrum is investigated in this paper. We have shown that the operator possesses only the point spectrum consisting of real number, which forms a countable set. Moreover, it is found that the spectrum of the Euler–Lagrange operator depends on the thickness of the fluid layer.

Arens regularity for totally ordered semigroups

Let S be a semigroup. We shall consider the centres of the semigroup (beta ,S, ,Box ,) and of the algebra (M(beta ,S), ,Box ,), where M(beta ,S) is the bidual of the semigroup algebra (ell ^{,1}(S),,star ,), and whether the semigroup and the semigroup algebra are Arens regular, strongly Arens irregular, or neither. We shall also determine subsets of S^* and of M(S^*) that are ‘determining for the left topological centre’ (DLTC sets) of beta ,S and M(beta ,S). It is known that, when the semigroup S is cancellative, ell ^{,1}(S) is strongly Arens irregular and that there is a DLTC set consisting of two points of S^*. In contrast, there is little that has been published about the Arens regularity of ell ^{,1}(S) when S is not cancellative. Totally ordered, abelian semigroups, with the map (s,t)rightarrow s wedge t as the semigroup operation, provide examples which show that several possibilities can occur. We shall determine the centres of beta ,S and of M(beta ,S) for all such semigroups, and give several examples, showing that the minimum cardinality of DTC sets may be arbitrarily large, and, in particular, we shall give an example of a countable, totally ordered, abelian semigroup S with this operation for which there is no countable DTC set for beta S or for M(beta S). There was no previously-known example of an abelian semigroup S for which beta ,S or M(beta S) did not have a finite DTC set.

Symmetric states for C∗-Fermi systems

After introducing the infinite Fermi [Formula: see text]-tensor product of a single [Formula: see text]-graded [Formula: see text]-algebra as an inductive limit, we systematically study the structure of the so-called symmetric states, that is those which are invariant under the group consisting of all finite permutations of a countable set. Among the obtained results, we mention the extension of De Finetti theorem which asserts that a symmetric state is a “mixture” of product states, each of which is a product of a single even state. This result induces a canonical morphism of the simplexes made of the symmetric even states on the usual infinite [Formula: see text]-tensor product and the symmetric states on the infinite Fermi [Formula: see text]-tensor product. We then extend the so-called Klein transformation to the infinite Fermi [Formula: see text]-tensor product, available when the parity automorphism is inner. In such a situation, we investigate further properties of product states, the last being the extremal symmetric states on such an infinite Fermi [Formula: see text]-tensor product [Formula: see text]-algebra. This paper is complemented with a finite dimensional illustrative example for which the Klein transformation is not implementable, and then the Fermi tensor product might not generate a usual tensor product. Therefore, in general, the study of the symmetric states on the Fermi algebra cannot be easily reduced to that of the corresponding symmetric states on the usual infinite tensor product, even if both share many common properties.

Characterizing Interoceptive Differences in Autism: A Systematic Review and Meta-analysis of Case-control Studies.

Interoception, the body's perception of its own internal states, is thought to be altered in autism, though results of empirical studies have been inconsistent. The current study systematically reviewed and meta-analyzed the extant literature comparing interoceptive outcomes between autistic (AUT) and neurotypical (NT) individuals, determining which domains of interoception demonstrate robust between-group differences. A three-level Bayesian meta-analysis compared heartbeat counting performance, heartbeat discrimination performance, heartbeat counting confidence ratings, and self-reported interoceptive attention between AUT and NT groups (15 studies; nAUT = 467, nNT = 478). Autistic participants showed significantly reduced heartbeat counting performance [g = -0.333, CrI95% (-0.535, -0.138)] and higher confidence in their heartbeat counting abilities [g = 0.430, CrI95% (0.123, 0.750)], but groups were equivalent on other meta-analyzed outcomes. Implications for future interoception research in autism are discussed.

Universality of Nodal Count Distribution in Large Metric Graphs

An eigenfunction of the Laplacian on a metric (quantum) graph has an excess number of zeros due to the graph’s non-trivial topology. This number, called the nodal surplus, is an integer between 0 and the graph’s first Betti number β. We study the distribution of the nodal surplus values in the countably infinite set of the graph’s eigenfunctions. We conjecture that this distribution converges to Gaussian for any sequence of graphs of growing β. We prove this conjecture for several special graph sequences and test it numerically for a variety of well-known graph families. Accurate computation of the distribution is made possible by a formula expressing the nodal surplus distribution as an integral over a high-dimensional torus.

IV-GNN : interval valued data handling using graph neural network.

Interval-valued data is an effective way to represent complex information where uncertainty, inaccuracy etc. are involved in the data space and they are worthy of taking into account. Interval analysis together with neural network has proven to work well on Euclidean data. However, in real-life scenarios, data follows a much more complex structure and is often represented as graphs, which is non-Euclidean in nature. Graph Neural Network is a powerful tool to handle graph like data with countable feature space. So, there is a research gap between the interval-valued data handling approaches and existing GNN model. No model in GNN literature can handle a graph with interval-valued features and, on the other hand, Multi Layer Perceptron (MLP) based on interval mathematics can not process the same due to non-Euclidean structure behind the graph. This article proposes an Interval-Valued Graph Neural Network, a novel GNN model where, for the first time, we relax the restriction of the feature space being countable without compromising the time complexity of the best performing GNN model in the literature. Our model is much more general than existing models as any countable set is always a subset of the universal set , which is uncountable. Here, to deal with interval-valued feature vectors, we propose a new aggregation scheme of intervals and show its expressive power to capture different interval structures. We validate our theoretical findings about our model for graph classification task by comparing its performance with those of the state-of-the-art models on several benchmark and synthetic network datasets.

A note on countable compact spaces

A note on countable compact spaces

On 2-local automorphisms of symmetric groups

Let phi be a self-map of the symmetric group S_Omega acting on a countable set Omega . We show in an elementary fashion that if phi has the property that sigma tau is conjugate to phi (sigma )phi (tau ) for every choice of sigma ,tau in S_Omega , then it is necessarily an automorphism or an antiautomorphism of S_Omega . As a corollary to this result, the so-called 2-local automorphisms of S_Omega are also described.

Sunset similarity solution for a receding hydraulic fracture

This paper derives approximate ‘sunset’ similarity solutions for receding plane strain and radially symmetric hydraulic fractures in permeable elastic media close to the point of closure. Local analysis is used to show that a receding hydraulic fracture has a linear aperture asymptote$\hat {w}\sim \hat {s}$in the fracture tip, where$\hat {s}$is the distance from the fracture front. Due to the regularity of the linear asymptote, it is possible to determine similarity solutions in the form of power series expansions, which, for integers$N\ge 2$and values of the radius decay exponent$\gamma =1/N$, can be shown to terminate to yield polynomial solutions for the fracture aperture of degree$N$. Of this countable infinity of polynomial solutions, the final aperture profile as the fracture approaches closure is associated with the second-degree polynomial with$\gamma =1/2$called the sunset solution. For the reverse time$t^{\prime }$measured from closure, the sunset solution is characterized by$w\sim t^{\prime }$and$R\sim t^{\prime 1/2}$. Of all the admissible polynomial similarity solutions, the sunset solution is shown to form an attractor, as$t^{\prime }\rightarrow 0$, for receding hydraulic fractures associated with a wide variety of points in parametric space. Using the sunset solution, it is possible to estimate the duration of recession, assuming that the fracture aperture and radius at the start of recession are given, and determine how it scales with a dimensionless shut-in parameter. As the fracture approaches closure, the term responsible for coupling the elastic force balance and fluid conservation becomes subdominant to the other terms in the lubrication equation, which reduces to a local kinematic relation between the decaying fracture aperture and the leak-off velocity. This fundamental decoupling of dynamics from kinematics results in the sunset solution being dependent on only a single material parameter – namely the leak-off coefficient. This isolation of the leak-off coefficient by the sunset solution opens the possibility to determine this parameter from laboratory or field measurements.

Discrete Dirichlet forms as traces of Feller’s one-dimensional diffusions

Our primary purpose is to compute explicitly traces of the Dirichlet forms related to Feller's one-dimensional diffusions on countable sets via Fukushima's method. For discrete measures, the obtained trace form can be described as a Dirichlet form on the graph.

Comparison of CD146 +/− mesenchymal stem cells in improving premature ovarian failure

BackgroundMesenchymal stem cells (MSCs) are a heterogeneous group of subpopulations with differentially expressed surface markers. CD146 + MSCs correlate with high therapeutic and secretory potency. However, their therapeutic efficacy and mechanisms in premature ovarian failure (POF) have not been explored.MethodsThe umbilical cord (UC)-derived CD146 +/− MSCs were sorted using magnetic beads. The proliferation of MSCs was assayed by dye670 staining and flow cytometry. A mouse POF model was established by injection of cyclophosphamide and busulfan, followed by treatment with CD146 +/− MSCs. The therapeutic effect of CD146 +/− MSCs was evaluated based on body weight, hormone levels, follicle count and reproductive ability. Differential gene expression was identified by mRNA sequencing and validated by RT-PCR. The lymphocyte percentage was detected by flow cytometry.ResultsCD146 +/− MSCs had similar morphology and surface marker expression. However, CD146 + MSCs exhibited a significantly stronger proliferation ability. Gene profiles revealed that CD146 + MSCs had a lower levels of immunoregulatory factor expression. CD146 + MSCs exhibited a stronger ability to inhibit T cell proliferation. CD146 +/− MSCs treatment markedly restored FSH and E2 hormone secretion level, reduced follicular atresia, and increased sinus follicle numbers in a mouse POF model. The recovery function of CD146 + MSCs in a reproductive assay was slightly improved than that of CD146 - MSCs. Ovary mRNA sequencing data indicated that UC-MSCs therapy improved ovarian endocrine locally, which was through PPAR and cholesterol metabolism pathways. The percentages of CD3, CD4, and CD8 lymphocytes were significantly reduced in the POF group compared to the control group. CD146 + MSCs treatment significantly reversed the changes in lymphocyte percentages. Meanwhile, CD146 - MSCs could not improve the decrease in CD4/8 ratio induced by chemotherapy.ConclusionUC-MSCs therapy improved premature ovarian failure significantly. CD146 +/− MSCs both had similar therapeutic effects in repairing reproductive ability. CD146 + MSCs had advantages in modulating immunology and cell proliferation characteristics.

Brief Pictorial Quizzes to Assess Carbohydrate Counting and Nutrition Knowledge in Youth With Type 1 Diabetes.

Available assessments of patient nutrition knowledge and carbohydrate counting ability are lengthy. This article reports on a study to implement and validate a series of brief nutrition quizzes of varying difficulty for use in pediatric type 1 diabetes. Among 129 youth with type 1 diabetes, participants completed an average of 2.4 ± 1 of the six quizzes, with a median score of 4.7 of 5. Higher quiz scores were associated with lower A1C (P <0.001), higher parental education (P = 0.02), and higher income (P = 0.01). Such quizzes can help to identify knowledge gaps and provide opportunities for education, which may improve glycemic outcomes in youth with type 1 diabetes.

THE STATES’ FINAL PROBABILITIES ANALYTICAL DESCRIPTION IN AN INCOMPLETELY ACCESSIBLE QUEUING SYSTEM WITH REFUSALS

Context. There is a problem of forecasting the efficiency of real queuing systems with refusals in the case of incomplete accessibility of service devices for the input flow of requirements. The solution of problem is necessary to create the possibility of more accurate design and control of such systems operation in real time. Objective. The aim of the research is to obtain an analytical description of the state’s final probabilities in a Markov queuing system with refusals and with incomplete accessibility of service devices for the input flow of requirements that is necessary to forecast the values of the queuing system performance indicators. Method. The probabilities of queuing systems’ states with refusals in the case of incomplete accessibility of service devices for the input flow of requirements are described by Kolmogorov differential equations. In a stationary state, these equations are transformed into a linearly dependent homogeneous system of algebraic equations. The number of equations is determined by the setdegree and for modern queuing and communication systems can be in the thousands, millions and more. Therefore, an attempt to predict the efficiency of a system is faced with the need to write down and numerically solve a countable set of algebraic equations systems that is quite difficult. The key idea of the proposed method for finding an analytical description of final probabilities for a given queuing system was the desire to move from the description of individual states (of 2n amount) to the description of groups of system states (of n+1 number) and to localize the influence of incomplete accessibility of service devices for the input flow of requirements in multiplicative functions of incomplete accessibility. Such functions allow obtaining the required analytical description and assessing the degree of the final probabilities transformation, in comparison with known systems, as well as assessing the forecasted values of the noted queuing system’s efficiency indicators when building a system and choosing the parameters for its controlling. Results. For the first time analytical expressions are obtained for the final probabilities of the queuing system states with refusals and with incomplete accessibility of service devices for the input flow of requirements, which makes it possible to evaluate as well as forecast values of all known system efficiency indicators. Conclusions. The resulting description turned out to be a general case for well-known type of Markov queuing systems with refusals. The results of the numerical experiment testify in favor of correctness the obtained analytical expressions for the final probabilities and in favor of possibility for their practical application in real queuing systems when solving problems of forecasting efficiency, as well as analyzing and synthesizing the parameters of real queuing systems.

Existence of Quasiconformal Maps with Maximal Stretching on Any Given Countable Set

Quasiconformal maps are homeomorphisms with useful local distortion inequalities; infinitesimally, they map balls to ellipsoids with bounded eccentricity. This leads to a number of useful regularity properties, including quantitative Hölder continuity estimates; on the other hand, one can use the radial stretches to characterize the extremizers for Hölder continuity. In this work, given any bounded countable set in $$\mathbb {R}^d$$ , we will construct an example of a K-quasiconformal map which exhibits the maximum stretching at each point of the set. This will provide an example of a quasiconformal map that exhibits the worst-case regularity on a surprisingly large set, and generalizes constructions from the planar setting into $$\mathbb {R}^d$$ .

Bi-utility representable orderings on a countable set

An elementary characterization of the existence of a bi-utility representation for a partial order on a countable set is presented. The approach followed allows us to develop an algorithm for obtaining all such kind of representations in the finite case.

A new adaptive output-feedback controller against unknown control directions and intrinsic growth

ABSTRACT This paper addresses global output-feedback stabilisation in the context of unknown control directions and intrinsic unmeasurable-states-dependent growth, and focuses on rectifying the substantial deficiencies in the related works and developing a new adaptive output-feedback controller essentially reducing conservativeness. First, a varying parameter which takes value 1 or , instead of in an unbounded countable set, is specialised to capture the unknown control direction and accordingly a supervisory mechanism is delicately constructed to decide when to switch the parameter from 1 to or from to 1. A dynamic high gain, rather than a switching one, is appointed to compensate the serious system uncertainties. Then, the wanted controller is constructed, for which the design functions and design parameters are recursively generated by backstepping method (instead of analytically defined owing to the unknown control coefficients). Based on the designations and supervisory mechanism, it turns out that no Zeno phenomenon occurs, and global boundedness and convergence (to zero) hold, both for the resulting closed-loop system. An example is given to show the effectiveness and the advantage of the proposed scheme.

On spectral structure and spectral eigenvalue problems for a class of self similar spectral measure with product form

Let μ be a Borel probability measure with compact support on . We say that μ is a spectral measure if there exists , called a spectrum of μ, such that forms an orthonormal basis for L 2(μ). In this paper, we study the structure of spectra for a class of self-similar spectral measure μ R,B with product form on . We first give a partially characterize for E Λ to be a maximal orthogonal family in L 2(μ R,B ) by using the notion of maximal tree mapping. Based on this, we give a sufficient condition for a maximal orthogonal family E Λ (which corresponds to a maximal tree mapping) to be an orthonormal basis of L 2(μ R,B ). Moreover, we completely settle two types of spectral eigenvalue problems for μ R,B . Precisely, on the first case, for the model spectrum (simplest spectrum) of μ R,B , we characterize all possible real numbers t such that tΛ is also a spectrum of μ R,B . On the other case, we characterize all possible real numbers t such that there exists a countable set Λ such that Λ and tΛ are both spectra of μ R,B .

Entropy pair realization

AbstractWe show that the complete positive entropy (CPE) class $\alpha $ of Barbieri and García-Ramos contains a one-dimensional subshift for all countable ordinals $\alpha $ , that is, the process of alternating topological and transitive closure on the entropy pairs relation of a subshift can end on an arbitrary ordinal. This is the composition of three constructions. We first realize every ordinal as the length of an abstract ‘close-up’ process on a countable compact space. Next, we realize any abstract process on a compact zero-dimensional metrizable space as the process started from a shift-invariant relation on a subshift, the crucial construction being the implementation of every compact metrizable zero-dimensional space as an open invariant quotient of a subshift. Finally, we realize any shift-invariant relation E on a subshift X as the entropy pair relation of a supershift $Y \supset X$ , and under strong technical assumptions, we can make the CPE process on Y end on the same ordinal as the close-up process of E.