Complete CAT Research Articles

The Closest Point Theorem in CAT p (0) Metric Spaces

In this work, we investigate the closest point theorem in complete CAT p ( 0 ) metric spaces for p ≥ 2 . As an application, an extension of the Banach Contraction Principle for best proximity points is obtained.

Gh-convergence of CAT(0)-spaces: stability of the Euclidean factor

AbstractWe prove that if a sequence of geodesically complete CAT(0)-spaces $$X_j$$ X j with uniformly cocompact discrete groups of isometries converges in the Gromov-Hausdorff sense to $$X_\infty $$ X ∞ , then the dimension of the maximal Euclidean factor splitted off by $$X_\infty $$ X ∞ and $$X_j$$ X j is the same, for j big enough. In other words, no additional Euclidean factors can appear in the limit.

Mean ergodic theorems for a sequence of nonexpansive mappings in complete CAT(0) spaces and its applications

Abstract In this article, we prove ergodic convergence for a sequence of nonexpansive mappings in Hadamard (complete CAT ( 0 ) {\rm{CAT}}\left(0) ) spaces. Our result extends the nonlinear ergodic theorem by Baillon for nonexpansive mappings from Hilbert spaces to Hadamard spaces. We further establish the standard nonlinear ergodic theorem and apply our results to the problem of finding a common fixed point of a countable family of nonexpansive mappings. Finally, we propose some applications of our results to solve convex optimization problems in Hadamard spaces.

Efficacy of inhaled tiotropium add-on to budesonide/formoterol in patients with bronchiolitis obliterans developing after hematopoietic stem cell transplantation

BackgroundBronchiolitis obliterans syndrome (BOS) is the lung manifestation of chronic graft-versus-host disease after hematopoietic stem cell transplantation (HSCT). We assessed whether inhaled tiotropium add-on to the combination regimen including budesonide/formoterol improve pulmonary function and the chronic obstructive pulmonary disease assessment test (CAT) scores in patients with BOS. MethodsPost-HSCT patients diagnosed as BOS in Seoul St. Mary's Hospital were reviewed retrospectively. Patients defined as BOS and treated with budesonide/formoterol/tiotropium combination therapy after budesonide/formoterol therapy from January 2011 to June 2019 were enrolled. ResultsTotal of 86 patients were evaluated. After tiotropium add-on, the absolute FEV1 increased significantly from 1.47 ± 0.49 to 1.53 ± 0.57 L (p = 0.023) and the % predicted FEV1 from 45.0 ± 12.8 to 46.8 ± 14.5% (p = 0.031). The % predicted DLCO increased significantly after tiotropium add-on (from 61.6 ± 16.7 to 64.3 ± 16.3%, p = 0.028). Among 56 patients with complete CAT scores, no significant change was present in total CAT scores. In all, 30 of the 72 patients (41.7%) evidenced FEV1 increases > 100 mL, and 20 of 56 patients (35.7%) had CAT score decreases of ≥ 2 points. When the FEV1 and CAT scores were combined, the overall response rate to tiotropium add-on was 56.2% (41/73). The response group evidenced a significantly greater FVC increase, and a significant decrease in the RV/TLC ratio compared to the no-response group. ConclusionsInhaled tiotropium add-on to combination budesonide/formoterol significantly improved lung function, but not respiratory symptoms, in patients with post-HSCT BOS.

Semi-simple actions of the Higman-Thompson groups T_n on finite-dimensional CAT(0) spaces

In this paper, we study isometric actions on finite-dimensional CAT(0) spaces for the Higman–Thompson groups T_n, which are generalizations of Thompson’s group T. It is known that every semi-simple action of T on a complete CAT(0) space of finite covering dimension has a global fixed point. After this result, we show that every semi-simple action of T_n on a complete CAT(0) space of finite covering dimension has a global fixed point. In the proof, we regard T_n as ring groups of homeomorphisms of S^1 introduced by Kim, Koberda and Lodha, and use general facts on these groups.

On convergence of infinite products of convex combinations of mappings in CAT(0) spaces

We study the weak convergence of infinite products of convex combinations of operators in complete CAT(0) spaces. We provide a new approach to this problem by considering a constructive selection of convex combinations in CAT(0) spaces that does not depend on the order of the involved elements and retain continuity properties with respect to them.

A modified viscosity iterative method for implicit midpoint rule for optimization and fixed point problems in CAT(0) spaces

In this paper, we introduce a proximal point-type of viscosity iterative method with double implicit midpoint rule comprising of a nonexpansive mapping and the resolvents of a monotone operator and a bifunction. Furthermore, we establish that the sequence generated by our proposed algorithm converges strongly to an element in the intersection of the solution sets of monotone inclusion problem, equilibrium problem and fixed point problem for a nonexpansive mapping in complete CAT(0) spaces. In addition, we give a numerical example of our method each in a finite dimensional Euclidean space and a non-Hilbert space setting to show the applicability of our method . Our results complement many recent results in the literature.

Unraveling the invisible demon: a study of the oxidative stress markers, antioxidant activities and inflammatory markers in patients admitted with complete heart block

Abstract Funding Acknowledgements Type of funding sources: None. Introduction Despite the recent advancements in the management of Complete Heart Block (CHB), the aetiology of CHB is still idiopathic in most of the cases. Our study explores this hitherto untouched aspect of complete heart block. Purpose We aimed to assess the aetiological profile of Complete Heart Block patients in our study. Methods The study population consisted of 60 patients with complete heart block aged between 30 to 80 years, attending as an inpatient in ER. Oxidative stress was measured by serum MDA, serum GSH, serum Catalase activity and serum SOD activity. Antioxidant activity was obtained by measuring the levels of serum total antioxidant capacity. Inflammatory stress was measured by IL-5 and TNF-alpha levels. These values were compared to 30 healthy controls with no prior history of smoking and diabetes mellitus. Results The mean age of the patient was 62.48 ± 7.98 years and the gender distribution was 37 males and 23 females out of 60 patients. The mean value of serum MDA (ng/mL) in cases is 1451.26 ± 206.32, and in controls, the mean value is 1197.98 ± 234.71 (p=<0.001). The mean value of serum GSH (mcg/mL) in cases is 46.982 ± 18.613, and in controls, the mean value is 54.155 ± 10.762 (p=0.027). The mean value of serum Catalase Activity (U/min/mg protein) in cases is 10.763 ± 4.038 and in controls, the mean value is 19.878 ± 7.787 (p=0.003). The mean value of serum SOD Activity (U/g) in cases is 24.950 ± 5.4565, and in controls, the mean value is 46.214 ± 14.6309 (p=0.891). The mean value of serum Total Antioxidant Capacity (U/mL) in cases is 5.546 ± 0.620 and in controls, the mean value is 8.346 ± 2.781 (p=0.025). The mean value of serum IL-5 (pg/mL) in cases is 481.442 ± 28.8995, and in controls, the mean value is 67.347 ± 20.445 (p<0.001). The mean value of serum TNF-ALFA (pg/mL) in cases is 196.741 ± 73.771, and in controls, the mean value is 144.530 ± 42.599 (p= 0.081). Conclusions During a complete heart block, SOD (p=0.891), CAT (p=0.003), GSH (p=0.027) and total antioxidant (TAOC) (p=0.025) were significantly decreased in cases, compared to healthy controls, thus suggesting that the elevated levels of oxidative free radicals causes endothelial dysfunctioning. The increase in ROS was observed by a highly significant increase of malondialdehyde (MDA) (p=<0.001) showing high ROS-mediated tissue damage. Besides damage by oxidative stress, our study suggests that there are certain inflammatory markers like TNF-α and IL-5 that actively participate in causing heart block. There was a significant increase in the concentration of IL-5 (p<0.001) in the cases as compared to the controls.

On Some Convergence Results for Hybrid Maximal Functions in Cat(0) Spaces

The study of CAT(0) spaces plays a crucial role not only in Reimann Geometry but also in Functional Analysis, Numerical Analysis, ect. The applications of Fixed Point Theory under some conditions in CAT(0) spaces have been on focus of many research papers. Their goals are in generalizing the expansive or contractive function, or giving new iteration which improves the convergent conditions for a function to its fixed points. The goal of this paper is the study of some convergence aspects of modified hybrid maximal functions in a complete CAT(0) space. There are given some sufficient conditions which guarantee the convergence of a sequence based on a new iteration for modified hybrid maximal function or hybrid partial maximal function using Fixed Point Theory.

Noncritical maps on geodesically complete spaces with curvature bounded above

We define and study the regularity of distance maps on geodesically complete spaces with curvature bounded above. We prove that such a regular map is locally a Hurewicz fibration. This regularity can be regarded as a dual concept of Perelman’s regularity in the geometry of Alexandrov spaces with curvature bounded below. As a corollary, we obtain a sphere theorem for geodesically complete CAT(1) spaces.

Convergence Theorems of Three-Step Iteration Algorithm in CAT (0) Spaces

In this paper, a modified three-step iteration algorithm for approximating a joint fixed point of non-expansive and contraction mapping is studied. Under appropriate conditions, several strong convergence theorems and Δ-convergence theorems are established in a complete CAT (0) space. a numerical example is introduced to show that this modified iteration algorithm is faster than other iteration algorithms. Finally, we prove that the modified iteration algorithm is stable. Therefore these results are extended and improved to a novel results that are stated by other researchers. Our results are also complement to many well-known theorems in the literature. This type of research can be played a vital role in computer programming, game theory and computational analysis.

Strong Convergence Theorem for Finding a Common Solution of Convex Minimization and Fixed Point Problems in CAT(0) Spaces

In this paper, we introduce a proximal point algorithm for approximating a common solution of finite family of convex minimization problems and fixed point problems fork-demicontractive mappings in complete CAT(0) spaces. We prove a strong convergence result and obtain other consequence results which generalize and extend some recent results in the literature. We further provide a numerical example to illustrate the convergence behaviour of the sequence generated by our algorithm.

"A modified Halpern extragradient method for equilibrium and fixed point problems in CAT(0) space"

"In this paper, we introduce a modified Halpern extragradient-type algorithm for approximating an element in the intersection of the set of common solutions of equilibrium problems and common fixed points of family of nonexpansive mappings in a complete CAT(0) space. We establish strong convergence theorem that improve and generalize recently announced results in the literature."

Minimization problem of proximal point algorithm in complete CAT(0) spaces

In this paper a proximal point algorithm for a convex function is considered in complete CAT(0) spaces. We introduce a necessary and sufficient condition for the set of minimizers of the function to be nonempty, and by showing that in this case, this iterative sequence converges strongly to the metric projection of some point onto the set of minimizers of the function.

Common Risk Factors in the Cross Section of Catastrophe Bond Returns

Historically, cat bonds have provided high single-digit average annual returns, paired with a low volatility and little correlation to other asset classes. While there is an extensive literature that explains (ex-ante) cat bonds spreads, there is no factor model in the academic literature that explains this (ex-post) realized return puzzle. Based on monthly quoted prices for the complete cat bond market from 2001 to 2020, we provide insights into relevant risk factors in the cross-section of cat bond returns. After investigating a battery of possible cat bond return factors in bivariate and multivariate portfolio sorts as well as Fama-MacBeth regressions, we propose a four-factor cat bond model. Its factors are the seasonality adjusted probability of first loss, a separate seasonality adjustment factor and the two corporate bond factors TERM and DEF from Fama & French. This novel four-factor model predicts 60% of the time series variation of the historical cat bond market returns - as opposed to 4% for the Fama & French three- or five-factor model - and substantially reduces the observable alpha of the cat bond market.

A Halpern Type Iterative Scheme for a Finite Number of Mappings in Complete Geodesic Spaces with Curvature Bounded above

In this paper, we show a convergence theorem to a common fixed point for finitely many mappings by the Halpern’s iterative scheme using balanced mappings on an admissible complete CAT(1) space.

Closed subsets of a CAT(0) 2–complex are intrinsically CAT(0)

Let k be at most 0, and let X be a locally-finite CAT(k) polyhedral 2-complex X, each face with constant curvature k. Let E be a closed, rectifiably-connected subset of X with trivial first singular homology. We show that E, under the induced path metric, is a complete CAT(k) space.

MODIFIED PROXIMAL POINT ALGORITHM FOR MINIMIZATION AND FIXED POINT PROBLEM IN CAT(0) SPACES

In this paper, we study modified-type proximal point algorithm for approximating a common solution of a lower semi-continuous mapping and fixed point of total asymptotically nonexpansive mapping in complete CAT(0) spaces. Under suitable conditions, some strong convergence theorems of the proposed algorithms to such a common solution are proved.

Existence and approximation of fixed points of λ-hybrid mappings in complete CAT(0) spaces

Existence and approximation of fixed points of λ-hybrid mappings in complete CAT(0) spaces

Monotone Vector Fields and Generation of Nonexpansive Semigroups in Complete CAT(0) Spaces

In this paper, we discuss about monotone vector fields, which is a typical extension to the theory of convex functions, by exploiting the tangent space structure. This new approach to monotonicity in spaces stands in opposed to the monotonicity defined earlier in spaces by Khatibzadeh and Ranjbar [J. Aust. Math. Soc. 103(1), 70–90 (2017).] and Chaipunya and Kumam [Optimization 66(10), 1647–1665 (2017).]. In particular, this new concept extends the theory from both Hilbert spaces and Hadamard manifolds, while the known concept barely has any obvious relationship to the theory in Hadamard manifolds. We also study the corresponding resolvents and Yosida approximations of a given monotone vector field and derive many of their important properties. Finally, we prove a generation theorem by showing convergence of an exponential formula applied to resolvents of a monotone vector field. Our findings improve several known results in the literature including generation theorems of Jost [AMS/IP Stud. Adv. Math., vol. 8, pp. 1–47. Amer. Math. Soc., Providence, RI (1998)], Mayer [Comm. Anal. Geom. 6(2), 199–253 (1998).], Stojkovic [Adv. Calc. Var. 5(1), 77–126 (2012).], and Bačák [Convex analysis and optimization in Hadamard spaces, De Gruyter Series in Nonlinear Analysis and Applications, vol. 22. De Gruyter, Berlin (2014).] for proper, convex, lower semicontinuous functions in the context of complete spaces, and also by Iwamiya and Okochi [Nonlinear Anal. 54(2), 205–214 (2003).] for monotone vector fields in the context of Hadamard manifolds.