Countable Research Articles

Bicolor Graph of Morse-Smale Cascades on Manifolds of Dimension Three

The purpose of this study is to single out a class of Morse-Smale cascades (diffeomorphisms) with a three-dimensional phase space that allow topological classification using combinatorial invariants. In the general case, an obstacle to such a classification is the possibility of wild embedding of separatrix closures in the ambient manifold, which leads to a countable set of topologically nonequivalent systems. To solve the problem, we study the orbit space of a cascade. The ambient manifold of a diffeomorphism can be represented as a union of three pairwise disjoint sets: a connected attractor and a repeller whose dimension does not exceed one, and their complement consisting of wandering points of a cascade called the characteristic set. It is known that the topology of the orbit space of the restriction of the Morse-Smale diffeomorphism to the characteristic set and the embedding of the projections of two-dimensional separatrices into it is a complete topological invariant for Morse-Smale cascades on three-dimensional manifolds. Moreover, a criterion for the inclusion of Morse-Smale cascades in the topological flow was obtained earlier.These results are used in this paper to show that the topological conjugacy classes of Morse-Smale cascades that are included in a topological flow and do not have heteroclinic curves admit a combinatorial description. More exactly, the class of Morse-Smale diffeomorphisms without heteroclinic intersections, defined on closed three-dimensional manifolds included in topological flows and not having heteroclinic curves, is considered. Each cascade from this class is associated with a two-color graph describing the mutual arrangement of two-dimensional separatrices of saddle periodic points. It is proved that the existence of an isomorphism of two-color graphs that preserves the color of edges is a necessary and sufficient condition for the topological conjugacy of cascades. It is shown that the speed of the algorithm that distinguishes two-color graphs depends polynomially on the number of its vertices. An algorithm for constructing a representative of each topological conjugacy class is described.

Spectra of algebras of analytic functions, generated by sequences of polynomials on Banach spaces, and operations on spectra

We consider the subalgebra of the Fréchet algebra of entire functions of bounded type, generated by a countable set of algebraically independent homogeneous polynomials on the complex Banach space $X.$ We investigate the spectrum of this subalgebra in the case $X = \ell_1.$ We also consider some shift type operations that can be performed on the spectrum of this subalgebra in the case $X = \ell_p$ with $p \geq 1$.

Echocardiographic markers in the diagnosis of cardiac masses

Abstract Funding Acknowledgements Type of funding sources: None. Background The echocardiographic parameters required for a comprehensive assessment of cardiac mass (CMs) are still largely unknown. Objectives To identify and integrate the echocardiographic features of CMs that can accurately predict malignancy. Methods Observational cohort study of 286 consecutive patients who underwent a standard echocardiographic assessment for suspected cardiac mass between 2004 and 2022. A definitive diagnosis was achieved by histological examination or, in case of cardiac thrombi, with radiological evidence of thrombus resolution after an appropriate anticoagulant treatment. Logistic and multivariable regression analysis was performed to confirm the ability of 6 echocardiographic parameters to discriminate malignant from benign masses. The unweighted count of these parameters was used as a numerical score, ranging from 0 to 6, with a cut-off of >3 balancing sensitivity and specificity with respect to the histological diagnosis of malignancy. Classification tree analysis (CTA) was used to determine the ability of echocardiographic parameters to discriminate sub-groups of patients with a differential risk of malignancy. Results Benign masses were more frequently pedunculated, mobile, and adhered to the interatrial septum (p<0.001). Malignant masses showed a greater diameter and exhibited a higher frequency of irregular margins, an inhomogeneous appearance, sessile implantation, polylobate shape, and pericardial effusion (p<0.001). Infiltration, moderate-severe pericardial effusion, non-left localization, sessile, polylobate, and inhomogeneity were confirmed to be independent predictors of malignancy in both univariate and multivariable models. The predictive ability of the unweighted count of >3 was very high (>0.90) and similar to that of the previously published weighted score. The CTA generated an algorithm in which infiltration was the best discriminator of malignancy, followed by non-left localization and sessile shape. The percentage correctly classified by the CTA as malignant was 87.5%. Agreement between observer readings and cardiac mass histology ranged between 85.1–91.5%. The presence of at least 3 echocardiographic parameters was associated with a lower survival. Conclusions In the approach to CM, some echocardiographic parameters can serve as markers to accurately predict malignancy, thereby informing the need for second-level investigations and minimizing the diagnostic delay in such a complex clinical scenario.

A simple characterization of the existence of upper semicontinuous order-preserving functions

We introduce an upper semicontinuity condition concerning a not necessarily total preorder on a topological space, namely strong upper semicontinuity, and in this way we extend to the nontotal case the famous Rader’s theorem, which guarantees the existence of an upper semicontinuous order-preserving function for an upper semicontinuous total preorder on a second countable topological space. We show that Rader’s theorem is not generalizable if we adopt weaker upper semicontinuity conditions already introduced in the literature. We characterize the existence of an upper semicontinuous order-preserving function for all strongly upper semicontinuous preorders on a metrizable topological space.

Cryopreserved platelets washed with a dialysis machine for dimethyl sulphoxide removal.

Cryopreserved platelets (cPLTs) can be stored for years and are mainly used in military settings. However, the commonly used cryoprotectant dimethyl sulphoxide (DMSO) has toxic side effects when utilized in high quantities. We developed a novel method to aseptically remove DMSO from thawed cPLTs by dialysis. One unit of platelets (N = 6) was mixed with 75 mL of 27% DMSO within 4 days after collection and stored at -80°C for 1 week. The platelet counts, platelet distribution width, mean platelet volume (MPV), platelet activity, platelet release, platelet aggregation, platelet metabolism indicators and platelet ultrastructural features (determined by electron microscopy) of the samples at the pre-freeze, post-thaw wash (post-TW) and 24 h post-thaw wash (24-PTW) stages were determined and compared. The DMSO clearance rate from the post-TW platelets was 95.56 ± 1.3%, and the platelet recovery rate after washing was 74.66 ± 6.34%. The total count, activity, release factors, aggregation and thrombolytic ability of the post-TW platelets were lower, whereas the MPV and apoptosis rates were higher compared with those of the pre-freeze platelets. The lactic acid, glucose and potassium ions released from the platelets during washing were filtered away by the dialyser, which significantly reduced their concentration. However, 24-PTW platelets were metabolically active, resulting in a decrease in pH and glucose content and an increase in lactic acid content. The level of potassium ions remained low after 24 h of storage and washing. The pre-freeze platelets maintained their normal disc shape and exhibited an open canalicular system (OCS) and a dense tubular system. The cPLTs appeared irregular after washing, with protruding pseudopodia and an extensive OCS, which increased the release of their contents. We developed a novel dialysis method to effectively remove DMSO from cPLTs under aseptic conditions and maintain platelet quality. The clinical efficacy of our method remains to be determined. However, the function of the platelets declined 24 h after washing, making them unsuitable for transfusion.

An exploration to random walk

There is a lot of uncertainty in many things that happen in real life, which leads to many events occurring randomly, and the quantitative description of the relationship between this sequence of random events is the stochastic process. Stochastic process is an important part of Probability. The stochastic process is a set of random variables, which is {Xt} t belongs to the T, The variable t is the parameter known as time, The parameter in a stochastic process are uncertain, each t corresponding to a random variable. T is the parameter set, and is a discrete parameter process when T is a finite set or a countable set, otherwise it is a continuous parameter process. The stochastic process can be divided into discrete state and continuous state according to the state space. If the random process x(t) takes countable values, then it is a discrete state process, otherwise it is not countable which is a continuous state process. The values of random variables change according to the results of random trials, and the pattern of change is also known as probability distribution. Stochastic process, which belonged to the field of physics in the early days of research, have developed into a powerful tool for research in many fields such as natural sciences, engineering sciences, and social sciences. Stochastic process use formulas to derive the probability of future events, find the inner laws from the apparent chance of events and describe the laws through probability. It can also be used as a statistical model to predict and deal with the development of some complex things in nature.

Countable dense sets of dendrites

For each separable space X, we consider the family of all countable dense sets of X modulo an equivalence relation defined as follows: we say that M∼N if and only if there is a homeomorphism h:X→X such that h[M]=N. The countable dense homogeneity degree of X is defined as the cardinality of the set of the equivalence classes under the relation ∼. In case such degree equals 1, the space is said to be countable dense homogeneous. Countable dense homogeneity has become a classical concept and has been studied with interest in the last decades. In this paper we present a full characterization of the countable dense homogeneity degree for the class of dendrites.

Revisiting Quantum Contextuality in an Algebraic Framework.

Within the framework of quantum contextuality, we discuss the ideas of extracontextuality and extravalence, that allow one to relate Kochen-Specker’s and Gleason’s theorems. We emphasize that whereas Kochen-Specker’s is essentially a no-go theorem, Gleason’s provides a mathematical justification of Born’s rule. Our extracontextual approach requires however a way to describe the “Heisenberg cut”. Following an article by John von Neumann on infinite tensor products, this can be done by noticing that the usual formalism of quantum mechanics, associated with unitary equivalence of representations, stops working when countable infinities of particles (or degrees of freedom) are encountered. This is because the dimension of the corresponding Hilbert space becomes uncountably infinite, leading to the loss of unitary equivalence, and to sectorisation. Such an intrinsically contextual approach provides a unified mathematical model including both quantum and classical physics, that appear as required incommensurable facets in the description of nature.

Even and Odd Self-Similar Solutions of the Diffusion Equation for Infinite Horizon

In the description of transport phenomena, diffusion represents an important aspect. In certain cases, the diffusion may appear together with convection. In this paper, we study the diffusion equation with the self-similar Ansatz. With an appropriate change of variables, we have found an original new type of solution of the diffusion equation for infinite horizon. We derive novel even solutions of diffusion equation for the boundary conditions presented. For completeness, the odd solutions are also mentioned as well, as part of the previous works. We have found a countable set of even and odd solutions, of which linear combinations also fulfill the diffusion equation. Finally, the diffusion equation with a constant source term is discussed, which also has even and odd solutions.

An introduction to continuous functions, metric space, manifolds, topological spaces and its properties

A n-dimensional topological manifold is defined as an n-dimensional local Euclidean space M that is also a second countable space and a Hausdorff space. If a topological space has a countable base, it is referred to as a second countable space. At the same time, continuous function and metric space can help people to understand topological space better. This article will express some basic ideas about continuous functions, metric space, topological space and its properties, and topological manifolds. Moreover, the paper shows some basic ideas of how topological manifolds, topological space and function could be recognized and proved. Through analysis, this paper demonstrates the connection between them, such as using properties of continuous function to prove the definition and properties of topological manifolds and spaces.

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On Szymański theorem on hereditary normality of $\beta\omega$

We discuss the following result of A. Szymański in ``Retracts and non-normality points" (2012), Corollary 3.5.: If $F$ is a closed subspace of $\omega ^{*}$ and the $\pi$-weight of $F$ is countable, then every nonisolated point of $F$ is a non-normality point of $\omega ^{*}$. We obtain stronger results for all types of points, excluding the limits of countable discrete sets considered in ``Some non-normal subspaces of the Čech--Stone compactification of a discrete space" (1980) by A. Błaszczyk and A. Szymański. Perhaps our proofs look ``more natural in this area".

Countable products and countable direct sums of compact metrizable spaces in the absence of the Axiom of Choice

The main aim of the article is to show, in the absence of the Axiom of Choice, relationships between the following, independent of ZF, statements: “Every countable product of compact metrizable spaces is separable (respectively, compact)” and “Every countable product of compact metrizable spaces is metrizable”. Statements related to the above-mentioned ones are also studied. Permutation models (among them two new ones) are shown in which a countable direct sum (also a countable product) of metrizable spaces need not be metrizable, countable unions of countable sets are countable and there is a countable family of non-empty sets of size at most 2ℵ0 which does not have a choice function. A new permutation model is constructed in which every uncountable compact metrizable space is of size at least 2ℵ0 but a denumerable family of denumerable sets need not have a multiple choice function.

Stone space partitions indexed by a poset

Stone space partitions $\{X_{p}\mid p\in P\}$ satisfying conditions like $\bar{X_{p}}=\bigcup_{q\leqslant p}X_{q}$ for all $p\in P$, where $P$ is a poset or PO system (poset with a distinguished subset), arise naturally in the study both of primitive Boolean algebras and of $\omega$-categorical structures. A key concept for studying such partitions is that of a $p$-trim open set which meets precisely those $X_{q}$ for which $q\geqslant p$; for Stone spaces, this is the topological equivalent of a pseudo-indecomposable set. This paper develops the theory of infinite partitions of Stone spaces indexed by a poset or PO system where the trim sets form a neighbourhood base for the topology. We study the interplay between order properties of the poset/PO system and topological properties of the partition, examine extensions and completions of such partitions, and derive necessary and sufficient conditions on the poset/PO system for the existence of the various types of partition studied. We also identify circumstances in which a second countable Stone space with a trim partition indexed by a given PO system is unique up to homeomorphism, subject to choices on the isolated point structure and boundedness of the partition elements. One corollary of our results is that there is a partition $\{X_{r}\mid r\in[0,1]\}$ of the Cantor set such that $\bar{X_{r}}=\bigcup_{s\leqslant r}X_{s}\text{ for all }r\in[0,1]$.

The Polish topology of the isometry group of the infinite dimensional hyperbolic space

We consider the isometry group of the infinite-dimensional separable hyperbolic space with its Polish topology. This topology is given by pointwise convergence. For non-locally compact Polish groups, some striking phenomena like automatic continuity or extreme amenability may happen. Our leading idea is to compare this topological group with usual Lie groups on one side and with non-Archimedean infinite-dimensional groups like \mathcal{S}_\infty , the group of all permutations of a countable set on the other side. Our main results are All along the text, we lead a parallel study with the sibling group \mathrm{Isom}(\mathcal{H}) , where \mathcal{H} is a separable Hilbert space.

Sensitivity, local stable/unstable sets and shadowing

In this paper we study local stable/unstable sets of sensitive homeomorphisms with the shadowing property defined on compact metric spaces. We prove that local stable/unstable sets always contain a compact and perfect subset of the space. As a corollary we generalize results in [6] and [11] proving that positively countably expansive homeomorphisms defined on compact metric spaces satisfying either transitivity and the shadowing property, or the L-shadowing property, can only be defined in countable spaces.

Electromagnetic Steklov eigenvalues: existence and distribution in the self-adjoint case

In previous works, it was suggested to use Steklov eigenvalues for Maxwell equations as target signature for nondestructive testing, and it was recognized that this eigenvalue problem cannot be reformulated as a standard eigenvalue problem for a compact operator. Consequently, a modified eigenvalue problem with the desired properties was proposed. We report that apart for a countable set of particular frequencies, the spectrum of the original self-adjoint eigenvalue problem consists of three disjoint parts: The essential spectrum consisting of the origin, an infinite sequence of positive eigenvalues which accumulate only at infinity and an infinite sequence of negative eigenvalues which accumulate only at zero. The analysis is based on a suitable topological decomposition, a representation of the operator as block operator and Schur-factorizations. For each Schur-complement, the existence of an infinite sequence of eigenvalues is proven via an intermediate value technique. The modified eigenvalue problem arises as limit of one Schur-complement.

Coarse computability, the density metric, Hausdorff distances between Turing degrees, perfect trees, and reverse mathematics

For [Formula: see text], the coarse similarity class of A, denoted by [Formula: see text], is the set of all [Formula: see text] such that the symmetric difference of A and B has asymptotic density 0. There is a natural metric [Formula: see text] on the space [Formula: see text] of coarse similarity classes defined by letting [Formula: see text] be the upper density of the symmetric difference of A and B. We study the metric space of coarse similarity classes under this metric, and show in particular that between any two distinct points in this space there are continuum many geodesic paths. We also study subspaces of the form [Formula: see text] where [Formula: see text] is closed under Turing equivalence, and show that there is a tight connection between topological properties of such a space and computability-theoretic properties of [Formula: see text]. We then define a distance between Turing degrees based on Hausdorff distance in the metric space [Formula: see text]. We adapt a proof of Monin to show that the Hausdorff distances between Turing degrees that occur are exactly 0, [Formula: see text], and 1, and study which of these values occur most frequently in the senses of Lebesgue measure and Baire category. We define a degree a to be attractive if the class of all degrees at distance [Formula: see text] from a has measure 1, and dispersive otherwise. In particular, we study the distribution of attractive and dispersive degrees. We also study some properties of the metric space of Turing degrees under this Hausdorff distance, in particular the question of which countable metric spaces are isometrically embeddable in it, giving a graph-theoretic sufficient condition for embeddability. Motivated by a couple of issues arising in the above work, we also study the computability-theoretic and reverse-mathematical aspects of a Ramsey-theoretic theorem due to Mycielski, which in particular implies that there is a perfect set whose elements are mutually 1-random, as well as a perfect set whose elements are mutually 1-generic. Finally, we study the completeness of [Formula: see text] from the perspectives of computability theory and reverse mathematics.

Investigating the association between various platelet indices and different clinical sub-groups of severe malaria.

To evaluate the ability of platelet count, mean platelet volume, platelet distribution width, and platelet mass index to predict the severity of malaria. This study was conducted as a retrospective cohort study at a tertiary hospital in Somali. Patients grouped as severe and non-severe malaria. We compared groups in terms of platelet count, mean platelet volume, platelet distribution width, and platelet mass index. A total of 131 patients were included in the final analysis. Of the patients, 77 (58.7%) had non-severe malaria, and 54 (41.3%) had severe malaria. The multivariate analysis revealed that there was no significant difference between the groups in terms of platelet count, mean platelet volume, platelet distribution width, and platelet mass index (p: 0.183, 0.323, 0.204, and 0.139, respectively). In the receiver operating characteristic analysis, the area under the curve values for platelet count, mean platelet volume, platelet distribution width, and platelet mass index were 0.699, 0.619, 0.504, and 0.675, respectively. Of the platelet indices, platelet count, mean platelet volume, platelet distribution width, and platelet mass index were not clinically significant markers that could be used to predict the severity of malaria.

Contextual Unification of Classical and Quantum Physics

Following an article by John von Neumann on infinite tensor products, we develop the idea that the usual formalism of quantum mechanics, associated with unitary equivalence of representations, stops working when countable infinities of particles (or degrees of freedom) are encountered. This is because the dimension of the corresponding Hilbert space becomes uncountably infinite, leading to the loss of unitary equivalence, and to sectorization. By interpreting physically this mathematical fact, we show that it provides a natural way to describe the "Heisenberg cut", as well as a unified mathematical model including both quantum and classical physics, appearing as required incommensurable facets in the description of nature.