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Let $\kappa$ be an uncountable cardinal with $\kappa=\kappa^{{\kappa^+$'' implies the statement "every closed subset of ${}^\kappa\kappa$ is a continuous image of ${}^\kappa(\kappa^+)$'' or its negation., Comment: 30 pages

Two given orbits of a minimal circle homeomorphism $f$ are said to be geometrically equivalent if there exists a quasisymmetric circle homeomorphism identifying both orbits and commuting with $f$. By a well-known theorem due to Herman and Yoccoz, if $f$ is a smooth diffeomorphism with Diophantine rotation number, then any two orbits are geometrically equivalent. As it follows from the a-priori bounds of Herman and Swiatek, the same holds if $f$ is a critical circle map with rotation number of bounded type. By contrast, we prove in the present paper that if $f$ is a critical circle map whose rotation number belongs to a certain full Lebesgue measure set in $(0,1)$, then the number of equivalence classes is uncountable (Theorem A). The proof of this result relies on the ergodicity of a two-dimensional skew product over the Gauss map. As a by-product of our techniques, we construct topological conjugacies between multicritical circle maps which are not quasisymmetric, and we show that this phenomenon is abundant, both from the topological and measure-theoretical viewpoints (Theorems B and C)., Comment: 38 pages, 5 figures. To appear in Ergodic Theory and Dynamical Systems

The purpose of this research is to describe the ability of the eleventh-grade students of SMA Deli Murni Bandar Baru on using Countable and Uncountable Nouns. The population of this research was the eleventh-grade students, with 58 students taken as sample. The instrument of collecting data is a test concerning Countable and Uncountable Nouns. The tryout test was done to know the validity, reliability, item difficulty of test items. The result showed that 5 students (15 %) belong to the high category, 24 students (73 %) to the moderate category, and 4 students (12 %) to the low category. The mean score was 61,39 it was only 24 % of the total students who can do the test well with 12 students who get a score above 75, it means that the eleventh-grade students of SMA Deli Murni Bandar Baru are not yet able to use Countable and Uncountable Nouns. Based on the total incorrect answers made by the students in using countable and uncountable was 502. The percentage of students’ mistakes made by students in uncountable multiple choice including indefinite and quantifier uncountable was 33 %, in countable multiple choice including singular, regular, irregular countable 34%, in the countable essay including regular and irregular countable was 33%. Based on the findings and conclusions, some suggestions are offered to English teachers, English students, and other researchers. Especially to English teachers, who teach in school, are advised to improve students' ability to use countable and uncountable nouns.

We study a family of finitely generated residually finite groups. These groups are doubles $F_2*_H F_2$ of a rank-$2$ free group $F_2$ along an infinitely generated subgroup $H$. Varying $H$ yields uncountably many groups up to isomorphism., Comment: 10 pages, 2 figures

Let D be a division ring with involution ⋆ and S the set of all symmetric elements of D. Assume that the center F of D is uncountable and K is a division subring of D containing F. The main aim of this note is to show that S is right algebraic over K if and only if so is D. This result allows us to construct an example of division rings K ⊂ D such that D is right algebraic but not left algebraic over K.

We study the Galvin property. We show that various square principles imply that the cofinality of the Galvin number is uncountable (or even greater than $$\aleph _1$$ ). We prove that the proper forcing axiom is consistent with a strong negation of the Glavin property.

We prove that it is relatively consistent with $\mathrm{ZFC}$ that every strong measure zero subset of the real line is meager-additive while there are uncountable strong measure zero sets (i.e., Borel's conjecture fails). This answers a long-standing question due to Bartoszynski and Judah.

We generalize the analysis of Andrews, Schweber and Sorbi of the first-order theory of the partial order of degrees of c.e. equivalence relations to higher computability theory, specifically to the setting of a regular cardinal.

Recently in Dikranjan et al. (Fund Math 249: 185–209, 2020) an uncountable Borel subgroup $$t^s_{(2^n)}({\mathbb T}) $$ (called statistically characterized subgroup) was constructed containing the Prufer group $${\mathbb Z}(2^\infty )$$ using the notion of statistical convergence. This note is based on the recent work (Bose et al. in Acta Math Hungar, 2020) which helps us to show that an uncountable chain of distinct Borel subgroups (each of size $$\mathfrak {c}$$ ) can be generated between $${\mathbb Z}(2^\infty )$$ and $$t^s_{(2^n)}({\mathbb T}) $$ , whereas their intersection actually strictly contains the Prufer group, with their union being strictly contained in $$t^s_{(2^n)}({\mathbb T})$$ .

In 2011, Nathanson proposed several questions on minimal complements in a group or a semigroup. The notion of minimal complements and being a minimal complement leads to the notion of co-minimal pairs which was considered in a prior work of the authors. In this article, we study which type of subsets in the integers and free abelian groups of higher rank can be a part of a co-minimal pair. We show that a majority of lacunary sequences have this property. From the conditions established, one can show that any infinite subset of any finitely generated abelian group has uncountably many subsets which is a part of a co-minimal pair. Further, the uncountable collection of sets can be chosen so that they satisfy certain algebraic properties.

We show that the tree property, stationary reflection and the failure of approachability at $$\kappa ^{++}$$ are consistent with $$\mathfrak {u}(\kappa )= \kappa ^+ < 2^\kappa $$ , where $$\kappa $$ is a singular strong limit cardinal with the countable or uncountable cofinality. As a by-product, we show that if $$\lambda $$ is a regular cardinal, then stationary reflection at $$\lambda ^+$$ is indestructible under all $$\lambda $$ -cc forcings (out of general interest, we also state a related result for the preservation of club stationary reflection).

In this article, new generalised neutrosophic soft open known as neutrosophic soft ∗ b open set is introduced in neutrosophic soft topological spaces. Neutrosophic soft ∗ b open set is generated with the help of neutrosophic soft semiopen and neutrosophic soft preopen sets. Then, with the application of this new definition, some soft neutrosophical separation axioms, countability theorems, and countable space can be Hausdorff space under the subjection of neutrosophic soft sequence which is convergent, the cardinality of neutrosophic soft countable space, engagement of neutrosophic soft countable and uncountable spaces, neutrosophic soft topological features of the various spaces, soft neutrosophical continuity, the product of different soft neutrosophical spaces, and neutrosophic soft countably compact that has the characteristics of Bolzano Weierstrass Property (BVP) are studied. In addition to this, BVP shifting from one space to another through neutrosophic soft continuous functions, neutrosophic soft sequence convergence, and its marriage with neutrosophic soft compact space, sequentially compactness are addressed.

The main aim of this paper is proving the corresponding lattice version of James distortion theorem for $$c_0(\Gamma )$$ and $$\ell _1(\Gamma )$$ : if a Banach lattice contains lattice copy of $$c_0(\Gamma )$$ ( $$\ell _1(\Gamma )$$ respectively), then it contains lattice-almost isometric copies of $$c_0(\Gamma )$$ ( $$\ell _1(\Gamma )$$ respectively). We also show that the classical Lozanovskii and Meyer-Nieberg Theorem for $$c_0$$ is not longer valid for $$c_0(\Gamma )$$ whenever $$\Gamma $$ is uncountable.

Sustainable urban transformation increasingly relies upon technicities of computation and interoperability among variegated registers and domains. In contrast, the notion of an “urban majority,” first introduced by the authors nearly a decade ago, points to a different “mathematics” of combination. Here the ways in which different economic practices, demeanors, behavioral tactics, forms of social organization, territory, and mobility intersect and detach, coalesce into enduring cultures of inhabitation or proliferate as momentary occupancies of short-lived situations make up a kind of algorithmic process that continuously produces new functions and new values for individual and collective capacities, backgrounds, and ways of doing things. This capacity, albeit facing new vulnerabilities and recalibration, will become increasingly important in shaping urban change in a post-pandemic era.

For a Hausdorff zero-dimensional topological space X and a totally ordered field F with interval topology, let $$C_c(X,F)$$ be the ring of all F-valued continuous functions on X with countable range. It is proved that if F is either an uncountable field or countable subfield of $${\mathbb {R}}$$ , then the structure space of $$C_c(X,F)$$ is $$\beta _0X$$ , the Banaschewski Compactification of X. The ideals $$\{O^{p,F}_c:p\in \beta _0X\}$$ in $$C_c(X,F)$$ are introduced as modified countable analogue of the ideals $$\{O^p:p\in \beta X\}$$ in C(X). It is realized that $$C_c(X,F)\cap C_K(X,F)=\bigcap _{p\in \beta _0X{\setminus } X} O^{p,F}_c$$ , and this may be called a countable analogue of the well-known formula $$C_K(X)=\bigcap _{p\in \beta X{\setminus } X}O^p$$ in C(X). Furthermore, it is shown that the hypothesis $$C_c(X,F)$$ is a Von-Neumann regular ring is equivalent to amongst others the condition that X is a P-space.

We construct an uncountable family of transversely Cantor laminations of compact spaces defined by free minimal actions of solvable groups, which are not affable and whose orbits are not quasi-isometric to Cayley graphs., 12 pages, 3 figures, to appear in Proceedings of the American Mathematical Society

The paper deals with local discrepancies in the problem of the distribution of the sequence $$\{k\alpha\}$$ , i.e., with the remainder terms in asymptotic formulas for the number of points in this sequence lying in prescribed intervals. A construction of intervals for which local discrepancies tend to infinity slower than any given function is presented. Moreover, it is shown that there exists an uncountable set of such intervals. Previously, similar results were obtained only for irrationalities with bounded partial quotients of their continued fraction expansions.

We find new lower bounds on the torsion orders of very general Fano hypersurfaces over (uncountable) fields of arbitrary characteristic. Our results imply that unirational parametrizations of most Fano hypersurfaces need to have enormously large degree. Our results also hold in characteristic two, where they solve the rationality problem for hypersurfaces under a logarithmic degree bound, thereby extending a previous result of the author from characteristic different from two to arbitrary characteristic., 32 pages, final version, to appear in Algebra and Number Theory

We prove that if X is a regular space with no uncountable free sequences, then the tightness of its $$G_\delta $$ topology is at most the continuum and if X is, in addition, assumed to be Lindelof then its $$G_\delta $$ topology contains no free sequences of length larger then the continuum. We also show that, surprisingly, the higher cardinal generalization of our theorem does not hold, by constructing a regular space with no free sequences of length larger than $$\omega _1$$ , but whose $$G_\delta $$ topology can have arbitrarily large tightness.

Countability is a universal lexical category that provides a binary division of nouns into countable and uncountable nouns or is also called count and mass nouns. Usually, count nouns refer to things or objects which can be individuated and thus counted, while mass nouns refer to substances or stuff such as water,wine, blood, or mud for which it is less easy to identify what and how to count. This cognitive division leaves abstract nouns out. Abstract nouns neither refer to things or objects nor to substances or stuff. On the contrary, the reference of abstract nouns is rather heterogeneous comprising different kinds of nouns such as processes, states, events, measure and time terms, and alike. The aim of this paper is to present the challenges abstract nouns pose for theories of countability, and to reflect on possibilities to incorporate abstract nouns in contemporary theories of countability. The research discussed in this paper circles around English abstract nouns but we will also discuss the application of certain semantic phenomena onto Bosnian nouns.

If $$\kappa > \aleph_0$$ then $$\kappa \to(\kappa,\operatorname{Top}K_\kappa)^2$$ , i.e., every graph on $$\kappa$$ vertices contains either an independent set of $$\kappa$$ vertices, or a topological $$K_\kappa$$ , iff $$\kappa$$ is regular and there is no $$\kappa$$ -Suslin tree. Concerning the statement $$\omega_2\to(\operatorname{Top}K_{\omega_2})^2_\omega$$ , i.e., in every coloring of the edges of $$K_{\omega_2}$$ with countably many colors, there is a monochromatic topological $$K_{\omega_2}$$ , both the statement and its negation are consistent with the Generalized Continuum Hypothesis.

It is proved that if an uncountable cardinal k has an ineffable subset of weakly compact cardinals, then k is a weakly compact cardinal, and if k has an ineffable subset of Ramsey (Rowbottom, Jónsson, ineffable or subtle) cardinals, then k is a Ramsey (Rowbottom, Jónsson, ineffable or subtle) cardinal. Resumen: Se prueba que si un cardinal no contable k tiene un subconjunto casi inefable de cardinales débilmente compactos entonces k es un cardinal débilmente compacto. Y si k tiene un conjunto inefable de cardinales de Ramsey (Rowbottom, Jónsson, inefables o sutiles) entonces k es cardinal de Ramsey (Rowbottom, Jónsson, inefable o sutil).

We work throughout in the theory Z F \mathsf {ZF} with the axiom of determinacy, A D \mathsf {AD} . We introduce and prove some club uniformization principles under A D \mathsf {AD} and A D R \mathsf {AD}_\mathbb {R} . Using these principles, we establish continuity results for functions of the form Φ : [ ω 1 ] ω 1 → ω 1 \Phi \colon [{\omega _{1}}]^{\omega _{1}} \rightarrow {\omega _{1}} and Ψ : [ ω 1 ] ω 1 → ω 1 ω 1 \Psi \colon [{\omega _{1}}]^{\omega _{1}} \rightarrow {}^{\omega _{1}}{\omega _{1}} . Specifically, for every function Φ : [ ω 1 ] ω 1 → ω 1 \Phi \colon [\omega _1]^{\omega _1} \rightarrow \omega _1 , there is a club C ⊆ ω 1 C \subseteq \omega _1 so that Φ ↾ [ C ] ∗ ω 1 \Phi \upharpoonright [C]^{\omega _1}_* is a continuous function. This has several consequences such as establishing the cardinal relation | [ ω 1 ] > ω 1 | > | [ ω 1 ] ω 1 | |[{\omega _{1}}]^{>{\omega _{1}}}| > |[{\omega _{1}}]^{\omega _{1}}| and answering a question of Zapletal by showing that if ⟨ X α : α > ω 1 ⟩ \langle X_\alpha : \alpha > \omega _1\rangle is a collection of subsets of [ ω 1 ] ω 1 [\omega _1]^{\omega _1} with the property that ⋃ α > ω 1 X α = [ ω 1 ] ω 1 \bigcup _{\alpha > \omega _1}X_\alpha = [\omega _1]^{\omega _1} , then there is an α > ω 1 \alpha > \omega _1 so that X α X_\alpha and [ ω 1 ] ω 1 [\omega _1]^{\omega _1} are in bijection. We show that under A D R \mathsf {AD}_\mathbb {R} everywhere [ ω 1 ] > ω 1 [\omega _1]^{>\omega _1} -club uniformization holds which is the following statement: Let c l u b ω 1 \mathsf {club}_{\omega _1} denote the collection of club subsets of ω 1 \omega _1 . Suppose R ⊆ [ ω 1 ] > ω 1 × c l u b ω 1 R \subseteq [\omega _1]^{>\omega _1} \times \mathsf {club}_{\omega _1} is ⊆ \subseteq -downward closed in the sense that for all σ ∈ [ ω 1 ] > ω 1 \sigma \in [\omega _1]^{>\omega _1} , for all clubs C ⊆ D C \subseteq D , R ( σ , D ) R(\sigma ,D) implies R ( σ , C ) R(\sigma ,C) . Then there is a function F : d o m ( R ) → c l u b ω 1 F \colon {\mathrm {dom}}(R) \rightarrow \mathsf {club}_{\omega _1} so that for all σ ∈ d o m ( R ) \sigma \in {\mathrm {dom}}(R) , R ( σ , F ( σ ) ) R(\sigma ,F(\sigma )) . We show that under A D \mathsf {AD} almost everywhere [ ω 1 ] > ω 1 [{\omega _{1}}]^{>{\omega _{1}}} -club uniformization holds which is the statement that for every R ⊆ [ ω 1 ] > ω 1 × c l u b ω 1 R \subseteq [{\omega _{1}}]^{>{\omega _{1}}} \times \mathsf {club}_{\omega _{1}} which is ⊆ \subseteq -downward closed, there is a club C C and a function F : d o m ( R ) ∩ [ C ] ∗ > ω 1 → c l u b ω 1 F \colon {\mathrm {dom}}(R) \cap [C]^{>{\omega _{1}}}_* \rightarrow \mathrm {club}_{\omega _{1}} so that for all σ ∈ d o m ( R ) ∩ [ C ] ∗ > ω 1 \sigma \in {\mathrm {dom}}(R) \cap [C]^{>{\omega _{1}}}_* , R ( σ , F ( σ ) ) R(\sigma ,F(\sigma )) .

Generally, the internal states of most engineering systems are inaccessible to sensors under operational conditions, permitting indirect measurements and failures to be observed discretely. For this type of system, a class of proportional hazard (PH) models with multistate process have been proposed for performing prognostics. However, existing studies use a constant degradation mechanism with a very limited number of states, which may fail to model the actual deterioration of the system. Therefore, this article presents a novel PH model that utilizes a nonconstant degradation mechanism with a large number of states to perform health prognostics. A matrix partition-based approximation method is developed to include all the possible state transitions and to estimate important health quantities of the system, such as conditional reliability (CR), mean remaining useful life (MRUL), and remaining useful life distribution (RULD). An online prognostic scheme based on this matrix approach is presented to provide an update with the latest health quantities of the system. The proposed approach provides appealing computational features and avoids difficulties in calculating uncountable transitions for PH model-based prognostics. The effectiveness of the proposed method is demonstrated via a case of helical gearboxes under indirect monitoring and random failure conditions.

We analyse the complexity of the class of (special) Aronszajn, Suslin and Kurepa trees in the projective hierarchy of the higher Baire-space $\omega_1^{\omega_1}$. First, we will show that none of these classes have the Baire property (unless they are empty). Moreover, under $(V=L)$, (a) the class of Aronszajn and Suslin trees is $\Pi_1^1$-complete, (b) the class of special Aronszajn trees is $\Sigma_1^1$-complete, and (c) the class of Kurepa trees is $\Pi^1_2$-complete. We achieve these results by finding nicely definable reductions that map subsets $X$ of $\omega_1$ to trees $T_X$ so that $T_X$ is in a given tree-class $\mathcal T$ if and only if $X$ is stationary/non-stationary (depending on the class $\mathcal T$). Finally, we present models of CH where these classes have lower projective complexity., Comment: 16 pages

We construct a mean Li-Yorke chaotic set along polynomial sequences (the degree of this polynomial is not less than three) with full Hausdorff dimension and full topological entropy for \begin{document}$ \beta $\end{document} -transformation. An uncountable subset \begin{document}$ C $\end{document} is said to be a mean Li-Yorke chaotic set along sequence \begin{document}$ \{a_n\} $\end{document} , if both \begin{document}$ \begin{equation*} \liminf\limits_{N\to \infty}\frac{1}{N}\sum\limits_{j = 1}^{N}d(f^{a_j}(x),f^{a_j}(y )) = 0 \text{ and } \limsup\limits_{N\to \infty}\frac{1}{N}\sum\limits_{j = 1}^{N}d(f^{a_j}(x),f^{a_j}(y ))>0 \end{equation*} $\end{document} hold for any two distinct points \begin{document}$ x $\end{document} and \begin{document}$ y $\end{document} in \begin{document}$ C $\end{document} .

For a compact space $K$ we consider the space $P(K)$, of probability regular Borel measures on $K$, equipped with the $weak^\ast$ topology inherited from $C(K)^\ast$. We discuss possible characterizations of those compact spaces $K$ for which $P(K)$ is $\aleph_0$-monolithic. The main result states that under $\diamondsuit$ there exists a nonseparable Corson compact space $K$ such that $P(K)$ is $\aleph_0$-monolithic but $K$ supports a measure of uncountable type., Comment: 12 pages; final version

An almost disjoint family $\mathcal A$ of subsets of $\mathbb N$ is said to be $\mathbb R$-embeddable if there is a function $f:\mathbb N\rightarrow \mathbb R$ such that the sets $f[A]$ are ranges of real sequences converging to distinct reals for distinct $A\in \mathcal A$. It is well known that almost disjoint families which have few separations, such as Luzin families, are not $\mathbb R$-embeddable. We study extraction principles related to $\mathbb R$-embeddability and separation properties of almost disjoint families of $\mathbb N$ as well as their limitations. An extraction principle whose consistency is our main result is: every almost disjoint family of size continuum contains an $\mathbb R$-embeddable subfamily of size continuum. It is true in the Sacks model. The Cohen model serves to show that the above principle does not follow from the fact that every almost disjoint family of size continuum has two separated subfamilies of size continuum. We also construct in ZFC an almost disjoint family, where no two uncountable subfamilies can be separated but always a countable subfamily can be separated from any disjoint subfamily. Using a refinement of the $\mathbb R$-embeddability property called a controlled $\mathbb R$-embedding property we obtain the following results concerning Akemann-Doner C*-algebras which are induced by uncountable almost disjoint families: a) In ZFC there are Akemann-Doner C*-algebras of density $\mathfrak c$ with no commutative subalgebras of density $\mathfrak c$, b) It is independent from ZFC whether there is an Akemann-Doner algebra of density $\mathfrak c$ with no nonseparable commutative subalgebra. This completes an earlier result that there is in ZFC an Akemann-Doner algebra of density $\omega_1$ with no nonseparable commutative subalgebra.

Do the topologies of each dimension have to be same of any space? I show that this is not necessary with amply soft topology produced by classical topologies. For example, an amply soft topology produced by classical topologies may have got any indiscrete topologies, discrete topologies or any topological spaces in each different dimension. The amply soft topology allows to write elements of different classical topologies in its each parameter sets. The classical topologies may be finite, infinite, countable or uncountable. This situation removes the boundary in soft topology and cause it to spread over larger areas. Amply soft topology produced by classical topologies is a special case of an amply soft topology. For this, I define a new soft topology it is called as an amply soft topology. I introduce amply soft open sets, amply soft closed sets, interior and closure of an amply soft set and subspace of any amply soft topological space. I gave parametric separation axioms which are different from Ti separation axioms. Ti questions the relationship between the elements of space itself while Pi questions the strength of the connection between their parameters. An amply soft topology is built on amply soft sets. Amply soft sets use any kind of universal parameter set or initial universe (such as finite or infinite, countable or uncountable). Also, subset, superset, equality, empty set, whole set on amply soft sets are defined. And operations such as union, intersection, difference of two amply soft sets and complement of an amply soft set are given. Then three different amply soft point such as amply soft whole point, amply soft point and monad point are defined. And also I give examples related taking a universal set as uncountable.

The aim of the research was to create such a calculation model for the air defense efficiency that will enable to determine the degree of implementation of the task by anti-aircraft defense forces in combat conditions. The innovative approach to the efficiency of air defense presented in the article focuses on the methods and algorithms enabling the assessment of the feasibility of the air defense task. In its general form, it is based on the determination of the probable number of air assault assets intended for the implementation of an air task (destruction, incapacitation, disorganization of the cover object) and the possibility of air defense sets (means) to repel an air attack. The research was conducted with the use of qualitative methods – when determining the elements of protection or tactical and technical data. The results of the presented research can be implemented in the command process in air defense.

This paper provides some model-theoretic analysis for probability (modal) logic ($PL$). It is known that this logic does not enjoy the compactness property. However, by passing into the sublogic of $PL$, namely basic probability logic ($BPL$), it is shown that this logic satisfies the compactness property. Furthermore, by drawing some special attention to some essential model-theoretic properties of $PL$, a version of Lindström characterization theorem is investigated. In fact, it is verified that probability logic has the maximal expressive power among those abstract logics extending $PL$ and satisfying both the filtration and disjoint unions properties. Finally, by alternating the semantics to the finitely additive probability models ($\mathcal{F}\mathcal{P}\mathcal{M}$) and introducing positive sublogic of $PL$ including $BPL$, it is proved that this sublogic possesses the compactness property with respect to $\mathcal{F}\mathcal{P}\mathcal{M}$.

If a real a is random over a model M and $$x\in M[a]$$ is another real then either (1) $$x\in M$$ , or (2) $$M[x]=M[a]$$ , or (3) M[x] is a random extension of M and M[a] is a random extension of M[x]. This result may belong to the old set theoretic folklore. It appeared as Exapmle 1.17 in Jech’s book “Multiple forcing” without the claim that M[x] is a random extension of M in (3), but, likely, it has never been published with a detailed proof. A corollary: $${{\varvec{\Sigma }}}^{1}_{n}$$ -Reduction holds for all $$n\ge 3$$ , in models extending the constructible universe $$\mathbf{L}$$ by $$\kappa $$ -many random reals, $$\kappa $$ being any uncountable cardinal in $$\mathbf{L}$$ .

In this paper, we study the learning rate of generalized Bayes estimators in a general setting where the hypothesis class can be uncountable and have an irregular shape, the loss function can have heavy tails, and the optimal hypothesis may not be unique. We prove that under the multi-scale Bernstein’s condition, the generalized posterior distribution concentrates around the set of optimal hypotheses and the generalized Bayes estimator can achieve fast learning rate. Our results are applied to show that the standard Bayesian linear regression is robust to heavy-tailed distributions.

In this paper, we study E-semigroups over convex cones. We prove a structure theorem for E-semigroups which leave the algebra of compact operators invariant. Then we study in detail the CCR flows, E0-semigroups constructed from isometric representations, by describing their units and gauge groups. We exhibit an uncountable family of 2-parameter CCR flows, containing mutually non-cocycle-conjugate E0-semigroups.

This paper is dedicated to studying model-theoretic and complexity properties of the theory of regular languages with the Kleene star operation. We construct a ‘‘natural’’ infinite set of axioms for this theory, and we prove that it is not finitely axiomatizable. We establish that this theory is countably categorical, but it is not categorical in any uncountable cardinality. Also, we find the exact amount of non-isomorphic models of given cardinality. Finally, we prove that the theory of regular languages with the Kleene star is PSPACE-complete.

Urinary bladder stones are a common condition in elderly males, and they are usually related to infravesical obstruction secondary to prostate enlargement. Endoscopic management of bladder stone is the gold standard treatment; yet, in some cases, return to conventional open surgery is necessary. In our article, we reported the case of a 73-year-old male patient with accidentally discovered multiple urinary bladder stones. Cystolithotomy was the treatment of choice which went uneventfully with a smooth postoperative course.

We show that the Mallows measure on permutations of $1,\ldots,n$ arises as the law of the unique Gale-Shapley stable matching of the random bipartite graph conditioned to be perfect, where preferences arise from a total ordering of the vertices but are restricted to the (random) edges of the graph. We extend this correspondence to infinite intervals, for which the situation is more intricate. We prove that almost surely every stable matching of the random bipartite graph obtained by performing Bernoulli percolation on the complete bipartite graph $K_{\mathbb{Z},\mathbb{Z}}$ falls into one of two classes: a countable family $(\sigma_n)_{n\in\mathbb{Z}}$ of tame stable matchings, in which the length of the longest edge crossing $k$ is $O(\log |k|)$ as $k\to\pm \infty$, and an uncountable family of wild stable matchings, in which this length is $\exp \Omega(k)$ as $k\to +\infty$. The tame stable matching $\sigma_n$ has the law of the Mallows permutation of $\mathbb{Z}$ (as constructed by Gnedin and Olshanski) composed with the shift $k\mapsto k+n$. The permutation $\sigma_{n+1}$ dominates $\sigma_{n}$ pointwise, and the two permutations are related by a shift along a random strictly increasing sequence., Comment: 22 pages, 7 figures. V2: Very minor revisions

The idea of this paper is to explore the existence of canonical countably saturated models for different classes of structures. It is well-known that, under CH, there exists a unique countably saturated linear order of cardinality $$\mathfrak {c}$$ c . We provide some examples of pairwise non-isomorphic countably saturated linear orders of cardinality $$\mathfrak {c}$$ c , under different set-theoretic assumptions. We give a new proof of the old theorem of Harzheim, that the class of countably saturated linear orders has a uniquely determined one-element basis. From our proof it follows that this minimal linear order is a Fraïssé limit of certain Fraïssé class. In particular, it is homogeneous with respect to countable subsets. Next we prove the existence and uniqueness of the uncountable version of the random graph. This graph is isomorphic to $$(H(\omega _1),\in \cup \ni )$$ ( H ( ω 1 ) , ∈ ∪ ∋ ) , where $$H(\omega _1)$$ H ( ω 1 ) is the set of hereditarily countable sets, and two sets are connected if one of them is an element of the other. In the last section, an example of a prime countably saturated Boolean algebra is presented.

There is a need to decolonize English in order to reframe our relationships with fellow beings and our environment. English can frame water or oil as infinite, uncountable nouns , a tree as an inan...

We prove that Voiculescu’s noncommutative version of the Weyl-von Neumann Theorem can be extended to all unital, separably representable $\mathrm {C}^\ast $ -algebras whose density character is strictly smaller than the (uncountable) cardinal invariant $\mathfrak {p}$ . We show moreover that Voiculescu’s Theorem consistently fails for $\mathrm {C}^\ast $ -algebras of larger density character.

The basic goal of quantization for probability distribution is to reduce the number of values, which is typically uncountable, describing a probability distribution to some finite set and thus approximation of a continuous probability distribution by a discrete distribution. Mixtures of probability distributions, also known as mixed distributions, are an exciting new area for optimal quantization. In this paper, we investigate the optimal quantization for three different mixed distributions generated by uniform distributions associated with probability vectors., arXiv admin note: text overlap with arXiv:1902.03887

We examine two very different approaches to formalising real computation, commonly referred to as “Computable Analysis” and “the BSS approach”. The main models of computation underlying these approaches—bit computation (or Type-2 Effectivity) and BSS, respectively—have also been put forward as appropriate foundations for scientific computing. The two frameworks offer useful computability and complexity results about problems whose underlying domain is an uncountable space (such as $${\mathbb {R}}$$ or $${\mathbb {C}}$$). Since typically the problems dealt with in physical sciences, applied mathematics, economics, and engineering are also defined in uncountable domains, it is fitting that we choose between these two approaches a foundational framework for scientific computing. However, the models are incompatible as to their results. What is more, the BSS model is highly idealised and unrealistic; yet, it is the de facto implicit model in various areas of computational mathematics, with virtually no problems for the everyday practice. This paper serves three purposes. First, we attempt to delineate what the goal of developing foundations for scientific computing exactly is. We distinguish between two very different interpretations of that goal, and on the separate basis of each one, we put forward answers about the appropriateness of each framework. Second, we provide an account of the fruitfulness and wide use of BSS, despite its unrealistic assumptions. Third, according to one of our proposed interpretations of the scope of foundations, the target domain of both models is a certain mathematical structure (namely, floating-point arithmetic). In a clear sense, then, we are using idealised models to study a purely mathematical structure (actually a class of such structures). The third purpose is to point out and explain this intriguing (perhaps unique) phenomenon and attempt to make connections with the typical case of idealised models of empirical domains.

The interplay between ultrafilters and unbounded subsets of $${}^\omega \omega $$ with the order $$

Each series ∑n=1∞an of real strictly positive terms gives rise to a topology on ℕ={1,2,3,…} by declaring a proper subset A⊆ℕ to be closed if ∑n∈Aan

We study perturbations of a hybrid steepest descent method for locating common fixed points of an arbitrary pool {Tλ} of nonexpansive mappings. The difficulty of handling a possibly uncountable fam...

We present and analyze several definitions of Pareto optimality for multicriteria optimization or decision problems with uncertainty primarily in their objective function values. In comparison to related notions of Pareto robustness, we first provide a full characterization of an alternative efficient set hierarchy that is based on six different ordering relations both with respect to the multiple objectives and a possibly finite, countably infinite or uncountable number of scenarios. We then establish several scalarization results for the generation of the corresponding efficient points using generalized weighted-sum and epsilon-constraint techniques. Finally, we leverage these scalarization results to also derive more general conditions for the existence of efficient points in each of the corresponding optimality classes, under suitable assumptions.

Answering a question of Sakai (Arch Math Logic 52(1–2):29–45, 2013), we show that the existence of an $$\omega _1$$ -Erdős cardinal suffices to obtain the consistency of Chang’s Conjecture with $$\square _{\omega _1, 2}$$ . By a result of Donder (In: Set theory and model theory (Bonn, 1979), volume 872 of lecture notes in mathematics. Springer, Berlin, pp 55–97, 1981) this is best possible. We also give an answer to another question of Sakai relating to the incompatibility of $$\square _{\lambda , 2}$$ and $$(\lambda ^+, \lambda ) \twoheadrightarrow (\kappa ^+, \kappa )$$ for uncountable $$\kappa $$ .