Compact Closure Research Articles

Compact subsets of C λ,u (X)

Abstract The famous Ascoli-Arzelà theorem served as a springboard for research into compactness in function spaces, particularly spaces of continuous functions. This paper investigates compact subsets of spaces of continuous functions endowed with topologies between the topology of pointwise convergence and the topology of uniform convergence. More precisely, this paper studies necessary and sufficient conditions for a subset to be compact in C λ,u (X) for a locally-λ space X when λ ⊇ 𝓕(X), for a hemi- λ λf -space X when λ ⊆ 𝓟 𝓢(X), and for a k-space X when λ ⊇ 𝓚(X). This paper also studies that every bounded subset of C λ,u (X) has compact closure for some classes of topological spaces X.

On spaces with a π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi $$\\end{document}-base with elements with compact closure

In this paper we show that if X is a T1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_1$$\\end{document}-space with a π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi $$\\end{document}-base whose elements have compact closure, then d(X)≤c(X)·2ψ(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d(X)\\le c(X)\\cdot 2^{\\psi (X)}$$\\end{document} and therefore, for such spaces we have d(X)ψ(X)=c(X)ψ(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d(X)^{\\psi (X)} = c(X)^{\\psi (X)}$$\\end{document}. This result allows us to restate several known upper bounds of the cardinality of a Hausdorff space X by replacing in them d(X) with c(X). In addition, we show that for such spaces X Šapirovskiĭ’s inequality d(X)≤πχ(X)c(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d(X)\\le \\pi \\chi (X)^{c(X)}$$\\end{document}, which is known to be true for regular Hausdorff spaces, is also valid. In the case when the space X is in addition sequential or radial, we show that |X|≤2c(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|X|\\le 2^{c(X)}$$\\end{document}. This result extends two theorems of Arhangel′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$'$$\\end{document}skiĭ to the class of Hausdorff spaces with a π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi $$\\end{document}-base whose elements have compact closures. We also show that spaces with a π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi $$\\end{document}-base with elements with compact closures are α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-favorable in the Banach–Mazur game, which implies such spaces are Baire. It was shown in Bella et al. (Quaest Math 46(4):745–760, 2023) that if a Hausdorff space X has a π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi $$\\end{document}-base consisting of elements with compact closure, then |X|≤2wL(X)t(X)ψc(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|X|\\le 2^{wL(X)t(X)\\psi _c(X)}$$\\end{document}. We give a variation of this result by showing |X|≤πχ(X)wL(X)ot(X)ψc(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|X|\\le \\pi \\chi (X)^{wL(X)\ extrm{ot}(X)\\psi _c(X)}$$\\end{document} for such a space X. Note that since wL(X)ot(X)≤c(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$wL(X)\ extrm{ot}(X)\\le c(X)$$\\end{document}, this result is at least as good as that given by Sun (Proc Am Math Soc 104:313–316, 1988). We also give a possible improvement of the bound in Bella et al. (2023) by showing that |X|≤2wL(X)wt(X)ψc(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|X|\\le 2^{wL(X)wt(X)\\psi _c(X)}$$\\end{document} for a Hausdorff space X with a π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi $$\\end{document}-base consisting of elements with compact closure. This uses the weak tightness wt(X) defined in Carlson (Topol Appl 249:103–111, 2018), which has the property ot(X)≤wt(X)≤t(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{ot}(X)\\le wt(X)\\le t(X)$$\\end{document}. We also show that if X is a Hausdorff homogeneous space with a π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi $$\\end{document}-base consisting of elements with compact closure (such spaces are locally compact), then |X|≤wL(X)wt(X)πχ(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|X|\\le wL(X)^{wt(X)\\pi \\chi (X)}$$\\end{document}. This generalizes the result in Bella and Carlson (Monatsh Math 192(1):39–48, 2020) that if X is a homogeneous compactum, then |X|≤2wt(X)πχ(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|X|\\le 2^{wt(X)\\pi \\chi (X)}$$\\end{document}.

Cardinal inequalities involving the Hausdorff pseudocharacter

We establish several bounds on the cardinality of a topological space involving the Hausdorff pseudocharacter Hpsi (X). This invariant has the property psi _c(X)le Hpsi (X)le chi (X) for a Hausdorff space X. We show the cardinality of a Hausdorff space X is bounded by 2^{pwL_c(X)Hpsi (X)}, where pwL_c(X)le L(X) and pwL_c(X)le c(X). This generalizes results of Bella and Spadaro, as well as Hodel. We show additionally that if X is a Hausdorff linearly Lindelöf space such that Hpsi (X)=omega , then |X|le 2^omega , under the assumption that either 2^{<{mathfrak {c}}}={mathfrak {c}} or {mathfrak {c}}<aleph _omega . The following game-theoretic result is shown: if X is a regular space such that player two has a winning strategy in the game G^{kappa }_1({mathcal {O}}, {mathcal {O}}_D), H psi (X) < kappa and chi (X) le 2^{<kappa }, then |X| le 2^{<kappa }. This improves a result of Aurichi, Bella, and Spadaro. Generalizing a result for first-countable spaces, we demonstrate that if X is a Hausdorff almost discretely Lindelöf space satisfying Hpsi (X)=omega , then |X|le 2^omega under the assumption 2^{<{mathfrak {c}}}={mathfrak {c}}. Finally, we show |X|le 2^{wL(X)Hpsi (X)} if X is a Hausdorff space with a pi -base with elements with compact closures. This is a variation of a result of Bella, Carlson, and Gotchev.

On the sticky particle solutions to the multi-dimensional pressureless Euler equations

In this paper we consider the multi-dimensional pressureless Euler system and we tackle the problem of existence and uniqueness of sticky particle solutions for general measure-type initial data. Although explicit counterexamples to both existence and uniqueness are known since [6], the problem of whether one can still find sticky particle solutions for a large set of data and of how one can select them was up to our knowledge still completely open.In this paper we prove that for a comeager set of initial data in the weak topology the pressureless Euler system admits a unique sticky particle solution given by a free flow where trajectories are disjoint straight lines.Indeed, such an existence and uniqueness result holds for a broader class of solutions decreasing their kinetic energy, which we call dissipative solutions, and which turns out to be the compact weak closure of the classical sticky particle solutions. Therefore any scheme for which the energy is l.s.c. and is dissipated will converge, for a comeager set of data, to our solution, i.e. the free flow.

Sealing ability in vitro study and biocompatibility in vivo animal study of different bioceramic based sealers.

The effectiveness of root canal therapy in endodontic practice is largely determined by providing a compact fluid-tight closure at the apex of the root canal, which inhibits irritant entry and buildup, which leads to a biological breakdown of the attachment mechanism and failure. During obturation, along with gutta-percha, root canal sealers are employed to fill voids and seal root canals. Root canal sealers come in a variety of shapes and sizes, each with its own set. Evaluation of sealing ability in vitro study by using scanning electron microscopy(SEM) and biocompatibility in vivo animals study of BioRoot RCS and meta Biomed bio_ceramic sealer (CeraSeal RCS) and compared the findings with that of Zinc oxide eugenol (ZOE) sealer as control. This study utilized two bio_ceramic sealers (BioRoot RCS and meta Biomed bio_ceramic sealer (CeraSeal RCS) and compared the findings with that of ZOE sealer as control. Biocompatibility was determined by examining histopathological biopsy specimens collected from rabbits. Each rabbit had four dentin tubes implanted into the subcutaneous tissues, one for BioRoot RCS, one for CeraSeal RCS, and one for ZOE RCS, with the fourth tube being empty haematoxylin and eosin were used to stain histological sections, and a light microscope was used to evaluate them. Extracted human single canal premolars were used to evaluate the sealing ability. The root canals were divided into three sections (coronal, middle, and apical). SEM was used to assess the adhesion quality at the sealer-dentin interface. BioRoot and CeraSeal sealers have excellent sealing adaptation and biocompatibility, as well as rapid tissue recovery, while ZOE sealers have a slower recovery of inflammatory reaction results when compared to bio_root and ceraSeal sealers, as well as a less sealing adaptation than the two other bio_ceramic sealers. In general, the two bioceramic sealers tested were biocompatible and capable of sealing or adhesion. While ZOEhad less adherence ability and less biocompatibility.

On the conformal walk dimension: quasisymmetric uniformization for symmetric diffusions

We introduce the notion of conformal walk dimension, which serves as a bridge between elliptic and parabolic Harnack inequalities. The importance of this notion is due to the fact that, for a given strongly local, regular symmetric Dirichlet space in which every metric ball has compact closure (MMD space), the finiteness of the conformal walk dimension characterizes the conjunction of the metric doubling property and the elliptic Harnack inequality. Roughly speaking, the conformal walk dimension of an MMD space is defined as the infimum over all possible values of the walk dimension with which the parabolic Harnack inequality can be made to hold by a time change of the associated diffusion and by a quasisymmetric change of the metric. We show that the conformal walk dimension of any MMD space satisfying the metric doubling property and the elliptic Harnack inequality is two, and provide a necessary condition for a pair of such changes to attain the infimum defining the conformal walk dimension when it is attained by the original pair. We also prove a necessary condition for the existence of such a pair attaining the infimum in the setting of a self-similar Dirichlet form on a self-similar set, and apply it to show that the infimum fails to be attained for the Vicsek set and the N-dimensional Sierpiński gasket with Nge 3, in contrast to the attainment for the two-dimensional Sierpiński gasket due to Kigami (Math Ann 340(4):781–804, 2008).

The metric space of limit laws for $q$-hook formulas

Billey-Konvalinka-Swanson studied the asymptotic distribution of the coefficients of Stanley's \(q\)-hook length formula, or equivalently the major index on standard tableaux of straight shape and certain skew shapes. We extend those investigations to Stanley's \(q\)-hook-content formula related to semistandard tableaux and \(q\)-hook length formulas of Björner-Wachs related to linear extensions of labeled forests. We show that, while their coefficients are "generically" asymptotically normal, there are uncountably many non-normal limit laws. More precisely, we introduce and completely describe the compact closure of the metric space of distributions of these statistics in several regimes. The additional limit distributions involve generalized uniform sum distributions which are topologically parameterized by certain decreasing sequence spaces with bounded \(2\)-norm. The closure of these distributions in the Lévy metric gives rise to the space of DUSTPAN distributions. As an application, we completely classify the limiting distributions of the size statistic on plane partitions fitting in a box.Mathematics Subject Classifications: 05A16 (Primary), 60C05, 60F05 (Secondary)Keywords: Hook length, \(q\)-analogues, major index, semistandard tableaux, plane partitions, forests, asymptotic normality, limit laws, Irwin-Hall distribution

More on cardinality bounds involving the weak Lindelöf degree

We give several new bounds for the cardinality of a Hausdorff topological space X involving the weak Lindelöf degree ωL(X). In particular, we show that if X is extremally disconnected, then |X| ≤ 2 ωL (X)πχ(X)ψ(X), and if X is additionally power homogeneous, then |X| ≤2 ωL (X)πχ(X). We also prove that if X is a star-DCCC space with a Gδ -diagonal of rank 3, then |X| ≤ 2ℵ0 ; and if X is any normal star-DCCC space with a Gδ -diagonal of rank 2, then |X| ≤2ℵ0. Several improvements of results in [10] are also given. We show that if X is locally compact, then |X| ≤ ωL(X) ψ (X) and that |X| ≤ ωL(X) t (X) if X is additionally power homogeneous. We also prove that |X| ≤ 2 ψc (X)t(X) ωL(X) for any space with a π-base whose elements have compact closures and that the stronger inequality |X| ≤ωL(X) ψc (X)t(X) is true when X is locally H-closed or locally Lindelöf.

ITClamp-Mediated Wound Closure Speeds Control of Arterial Hemorrhage With or Without Additional Hemostatic Agents.

Exsanguination is the leading cause of preventable posttraumatic death, especially in the prehospital arena. Traditional hemorrhage control methods involve packing the wound with hemostatic agents, providing manual pressure, and then applying a pressure dressing to stabilize the treatment. This is a lengthy process that frequently destabilizes upon patient transport. Conversely, the iTClamp, a compact wound closure device, is designed to rapidly seal wound edges mechanically, expediting clot formation at the site of injury. To determine the efficacy of the iTClamp with and without wound packing in the control of a lethal junction hemorrhage. Given the limited available information regarding the efficacy of the iTClamp in conjunction with traditional hemostatic agents, this study used a swine model of severe junctional hemorrhage. The goal was to compare a multiagent strategy using the iTClamp in conjunction with XSTAT to the traditional method of Combat Gauze packing with pressure dressing application. Readouts include application time, blood loss, and rebleed occurrence. Mean application times of the iTClamp treatment alone or in conjunction with other hemostatic agents were at least 75% faster than the application time of Combat Gauze with pressure dressing. Percent blood loss was not significantly different between groups but trended the highest for Combat Gauze treated swine, followed by iTClamp plus XSTAT, iTClamp alone and finally iTClamp plus Combat Gauze. The results from this study demonstrate that the iTClamp can be effectively utilized in conjunction with hemostatic packing to control junctional hemorrhages.

Quasi-symmetric mappings and their generalizations on Riemannian manifolds

A relation between 𝜂-quasi-symmetric homomorphisms and K-quasiconformal mappings on n-dimensional smooth connected Riemannian manifolds has been studied. The main results of the research are presented in Theorems 2.6 and 2.7. Several conditions for the boundary behavior of 𝜂-quasi-symmetric homomorphisms between two arbitrary domains with weakly flat boundaries and compact closures, QED and uniform domains on the Riemannian manifolds, which satisfy the obtained results, were also formulated. In addition, quasiballs, c-locally connected domains, and the corresponding results were also considered.

FastLCD: A fast and compact loop closure detection approach using 3D point cloud for indoor mobile mapping

In simultaneous localization and mapping (SLAM), loop closure detection is a significant yet still open problem. It contributes to construct a globally consistent and accurate map. This paper proposes a fast and compact loop closure detection method (FastLCD) based on comprehensive descriptors and machine learning to achieve reliable and precise results using 3D point cloud for indoor LiDAR mobile mapping. Comprehensive descriptors proposed in this paper encode discriminative multimodality features to describe each scan of point clouds. The specific values of descriptors of point cloud scan pairs are fed into a machine learning model. We leverage the pre-trained learning model as a classifier to distinguish whether a pair of laser scans is a loop candidate. Then, to ensure the results’ precision, a novel double-deck loop candidate verification strategy is used to reject false positives. The algorithm is evaluated on datasets of some typical indoor environments. Compared with some state-of-the-art loop closure detection algorithms, the proposed FastLCD algorithm demonstrates superior performance in precision and recall rate. Moreover, the method proposed also exhibits high time efficiency, excellent generalization performance and insensitivity to threshold changes.

Some Applications of the Poincaré–Bendixson Theorem

AbstractWe consider a $${\mathcal {C}}^1$$ C 1 vector field X defined on an open subset U of the plane with compact closure. If X has no singular points and if U is simply connected, a weak version of the Poincaré–Bendixson theorem says that the limit sets of X in U are empty but that one can define non empty extended limit sets contained in the boundary of U. We give an elementary proof of this result, independent of the classical Poincaré–Bendixson theorem. A trapping triangle $${\mathcal {T}}$$ T based at p, for a $${\mathcal {C}}^1$$ C 1 vector field X defined on an open subset $${\mathcal {U}}$$ U of the plane, is a topological triangle with a corner at a point p located on the boundary $$\partial {\mathcal {U}}$$ ∂ U and a good control of the tranversality of X along the sides. The principal application of the weak Poincaré–Bendixson theorem is that a trapping triangle at p contains a separatrix converging toward the point p. This does not depend on the properties of X along $$\partial {\mathcal {U}}$$ ∂ U . For instance, X could be non differentiable at p, as in the example presented in the last section.

Quasi-symmetric mappings and their generalizations on Riemannian manifolds

A relation between $\eta$-quasi-symmetric homomorphisms and $K$-quasiconformal mappings on $n$-dimensional smooth connected Riemannian manifolds has been studied. The main results of the research are presented in Theorems 2.6 and 2.7. Several conditions for the boundary behavior of $\eta$-quasi-symmetric homomorphisms between two arbitrary domains with weakly flat boundaries and compact closures, QED and uniform domains on the Riemannian mani\-folds, which satisfy the obtained results, were also formulated. In addition, quasiballs, $c$-locally connected domains, and the corresponding results were also considered.

On a connection between some classes of mapping on Riemannian manifolds

In this paper we study the connection between $\eta$-quasisymmetric homomorphisms and $K$-quasi\-con\-for\-mal mappings on $n$-dimensional smooth connected Riemannian manifolds. The main result of our research is the Theorem 3.1. For its proof we use a partition of unity method, which subordinate to the locally finite atlas of the manifold. Several results on the boundary behavior of $\eta$-quasisymmetric homomorphisms between two arbitrary domains, QED (uniform) domains and domains with weakly flat boundaries and compact closures on the Riemannian manifolds are also obtained in view of the above relations. The obtained results can be applied to Finsler manifolds with the addition of some conditions, which will take into account the specific of the initial manifold.

Local behavior of mappings of metric spaces with branching

The local behavior of mappings with the inverse Poletsky inequality between metric spaces is studied. The case where one of the spaces satisfies the condition of weak sphericalization, is similar to the Riemannian sphere (extended Euclidean space), and is locally linearly connected under a mapping is considered. It is proved that the equicontinuity of the corresponding families of mappings of two domains, one of which is a domain with a weakly flat boundary, and another one is a fixed domain with a compact closure, the corresponding weight in the main inequality being supposed to be integrable.

Bone stump formation against the background of post-amputation pain syndrome

Introduction: Bone is the basis of the stump. Aim: To study the impact of post-amputation pain syndrome on the nature of reparative processes in the bone residual limb. Material and methods: Three series of experiments were performed on 45 rabbits, 15 in each with mid-third thigh amputation and muscular plasty. In series 1, 2, a perineural catheter was attached to the sciatic nerve stump during amputation, and mechanical irritation of the nerve was performed daily for 20 minutes in series 1 for 20 days. In series 2, 0.3 mL of lidocaine (1%) was injected through the catheter into the circumference of the nerve twice daily for 20 days. Series 3 is a control. The follow-up periods were 1, 3, 6 months. The study method was histological with infusion of the vessels with ink-gelatin mixture. Results and discussion: In series 1, there was a sharp disturbance of the reparative process, which consisted in shape changes, resorption of the cortical diaphyseal plate, fractures, stump deformity, absence of bone closure plate formation, and microcirculatory disturbances. In the overall majority of experiments of series 2, the stumps retained the shape and structure characteristic of diaphysis with normalization of microcirculation. In series 3, the results of the residual limb formation were better than in series1, but worse than in series 2. Conclusions: When the pain syndrome is resolved within 20 days after amputation, a bone stump is formed with an organotypic shape and structure characteristic of the diaphysis, the formation of a compact closure plate of mature bone tissue, normalization of the medullary tissues and blood circulation. In post-amputation pain syndrome, organotypic formation of the residual limb does not occur.

Defect engineering for nontrivial multiferroic orders inSrTiO3

Although nanoscale multiferroics with coupled ferroelectricity and magnetism are of great technological and fundamental importance, the ferroic orders are inevitably destroyed in atomic-scale dimensions due to the intrinsic size effect or depolarization field. Here, we propose a strategy for atomic-size multiferroics by engineering the otherwise detrimental point defects in nonmagnetic paraelectric $\mathrm{SrTi}{\mathrm{O}}_{3}$. Systematic exploration using first-principles and molecular dynamics simulations demonstrates that the Ti antisite $({\mathrm{Ti}}_{\mathrm{Sr}})$ defects in $\mathrm{SrTi}{\mathrm{O}}_{3}$ have a tendency toward clustering with compact closure, which contributes to complex electrical polarization due to the large off-centering displacements of ${\mathrm{Ti}}_{\mathrm{Sr}}$ defects and the resulting separation between negative and positive charges. The ${\mathrm{Ti}}_{\mathrm{Sr}}$ defects also give rise to highly localized magnetic moments attributed to their unpaired spin-polarized electrons. The defect clustering and the associated multiferroic properties thus lead to atomic-size multiferroics with unusual ferroic orders.

Local behavior of mappings of metric spaces with branching

The local behavior of mappings with the inverse Poletsky inequality between metric spaces is studied. The case where one of the spaces satisfies the condition of weak sphericalization, is similar to the Riemannian sphere (extended Euclidean space), and is locally linearly connected under a mapping is considered. It is proved that the equicontinuity of the corresponding families of mappings of two domains, one of which is a domain with a weakly flat boundary, and another one is a fixed domain with a compact closure, the corresponding weight in the main inequality being supposed to be integrable.

On the Equicontinuity of Families of Inverse Mappings of Riemannian Manifolds

We study homeomorphisms of Riemannian manifolds with unbounded characteristic such that the inverse mappings satisfy the Poletsky-type inequality. It is established that their families are equicontinuous if the function Q which is related to the Poletsky inequality and is responsible for a distortion of the modulus, is integrable in the given domain, here the original manifold is connected and the domain of definition and the range of values of mappings have compact closures.

On the equicontinuity of families of inverse mappings of Riemannian manifolds

We study homeomorphisms of Riemannian manifolds with unbounded characteristic such that the inverse mappings satisfy the Poletsky-type inequality. It is established that their families are equicontinuous if the function Q which is related to the Poletsky inequality and is responsible for a distortion of the modulus, is integrable in the given domain, here the original manifold is connected and the domain of definition and the range of values of mappings have compact closures.