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The stochastic approximation (SA) method stands as the foundational mathematical tool for modern large-scale optimization and machine learning. Therefore, gaining a theoretical understanding of SA algorithms is of fundamental interest. In their paper titled "A Lyapunov Theory for Finite-Sample Guarantees of Markovian Stochastic Approximation," Chen et al. present a unified Lyapunov framework for the finite-sample analysis of a Markovian SA algorithm under a contractive operator with respect to an arbitrary norm. The key novelty lies in the construction of a smooth Lyapunov function called the generalized Moreau envelope. The authors demonstrate the effectiveness of their SA results in the context of reinforcement learning (RL), specifically through popular algorithms such as variants of temporal difference (TD) learning and Q-learning. As byproducts, the results provide theoretical insights into the efficiency of bootstrapping in TD learning with eligibility traces and the bias-variance tradeoff in off-policy learning. This paper develops a unified Lyapunov framework for finite-sample analysis of a Markovian stochastic approximation (SA) algorithm under a contraction operator with respect to an arbitrary norm. The main novelty lies in the construction of a valid Lyapunov function called the generalized Moreau envelope. The smoothness and an approximation property of the generalized Moreau envelope enable us to derive a one-step Lyapunov drift inequality, which is the key to establishing the finite-sample bounds. Our SA result has wide applications, especially in the context of reinforcement learning (RL). Specifically, we show that a large class of value-based RL algorithms can be modeled in the exact form of our Markovian SA algorithm. Therefore, our SA results immediately imply finite-sample guarantees for popular RL algorithms such as n-step temporal difference (TD) learning, TD (λ) , off-policy V-trace, and Q-learning. As byproducts, by analyzing the convergence bounds of n-step TD and TD (λ) , we provide theoretical insight into the problem about the efficiency of bootstrapping. Moreover, our finite-sample bounds of off-policy V-trace explicitly capture the tradeoff between the variance of the stochastic iterates and the bias in the limit. Funding: This work was supported by RTX, the National Science Foundation [Grants 2019844, 2107037, 211247, 2112533, 2144316, and 2240982], and the Machine Learning Laboratory at University of Texas at Austin. Supplemental Material: The online appendix is available at https://doi.org/10.1287/opre.2022.0249. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new mapping namely L-contraction for fixed function and we also prove a novel fixed function theorem using L-contraction mapping. Moreover, we apply our result to solve integral equation. [ABSTRACT FROM AUTHOR]

The unique solvability of the inverse problem of recovering the source function in a nonuniformly parabolic equation with many independent variables in a bounded domain is proved. As an additional condition, the integral observation condition is given. The unknown source function can be found by iteration from the operator equation with contraction operator. An example of an the inverse problem for which the results proved in this paper are applicable is presented. The results obtained for the inverse problem are based on the preliminary study of the unique solvability of the corresponding direct problem, which is also of interest in itself. [ABSTRACT FROM AUTHOR]

Telecommunications companies are constantly chasing continuous technological advances with their management methods of Operations & Maintenance (O&M) that still struggle to turn their eye toward innovation and simplification of processes. In a future that aims at fully fiber-optic networks, the objective of the research is to propose guidelines and provide data to support the Wholesale Operator for the definition of the acceptable attenuation threshold on the Passive Optical Network Fiber-To-The-Home (PON FTTH) infrastructure to determine the Service Level Agreements (SLA) to be contracted with Retail Operators. Following exceeding the threshold, the Retail Operator has the right to open a Trouble Ticket to request the resolution of the anomaly. Consequently, the definition of the attenuation threshold strongly affects the number of Trouble Tickets to be managed and the related costs the Wholesale Operator bears. This paper analyzes a specific case of the PON FTTH network of the Italian wholesale operator Open Fiber. The studied infrastructure, with a length of 11 km, allows a maximum attenuation of 37 dB without degrading the service. An economic model has been proposed to assess the cost impact of moving the attenuation threshold in the SLA. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the optimal integrability, regularity and symmetry of integrable solutions for the following Wolff type integral systems: (1.1) { u (x) = R 1 (x) W β , γ (v q) (x) , u (x) > 0 , x ∈ R N , v (x) = R 2 (x) W β , γ (u p) (x) , v (x) > 0 , x ∈ R N , where γ > 2 , β > 0 , β γ < N , p , q > γ − 1 with γ − 1 p + γ − 1 + γ − 1 q + γ − 1 = N − β γ N , R 1 , R 2 are double bounded in R N and W β , γ (h) (x) : = ∫ 0 ∞ [ ∫ B t (x) h (y) d y t N − β γ ] 1 γ − 1 d t t. Firstly, we prove the optimal integrability and boundedness of solutions (u , v) ∈ L p + γ − 1 (R N) × L q + γ − 1 (R N) for system (1.1) , by constructing a nonlinear, contracting operator and applying the regularity lifting lemma of C. Ma, W. Chen and C. Li (2011) [21]. Moreover, we exploit the general regularity lifting theorem to derive the Lipschitz continuity of u and v when R 1 ≡ R 2 ≡ 1 in R N. These extend the above important results of C. Ma, W. Chen and C. Li to γ > 2. We also prove that u and v vanish at infinity. As the corollaries of the above results, we obtain the optimal integrability, boundedness and the property of vanishing at infinity of integrable solutions for the corresponding γ -Laplace and k -Hessian systems. Secondly, we use the method of moving planes in integral forms to prove the symmetry and monotonicity of solutions (u , v) ∈ L p + γ − 1 (R N) × L q + γ − 1 (R N) for system (1.1) when R 1 ≡ R 2 ≡ 1 in R N , which extends the useful result of W. Chen and C. Li (2011) [5] to γ > 2. In comparison with the above two papers, Minkowski's inequality is crucial in our proofs. We believe that our arguments can be used to prove similar results for other Wolff type integral systems when γ > 2. [ABSTRACT FROM AUTHOR]

The interval Newton method, which combines the Newton method with the interval arithmetic and branch-and-prune algorithm, has been applied to the problem of finding all direct current operating points of nonlinear circuits. The modified linear programming narrowing technique has also been proposed to improve the computational efficiency of the interval Newton method when applying to a circuit whose nonlinear elements are only diodes and bipolar junction transistors. In this paper, a contraction operator based on interval linearization has been applied to further improve the performances. In addition, the application of the proposed method to circuits containing MOSFET is presented. The numerical experimental results on some test nonlinear circuits indicate that the proposed method can further reduce the number of search boxes and improve the computation time. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an orthogonal L ⋆ -contraction map concept and prove the fixed-point theorem in an orthogonal complete Branciari metric space (OCBMS). We also provide illustrative examples to support our theorems. We demonstrated the existence of a uniqueness solution to the fourth-order differential equation using a more orthogonal L ⋆ contraction operator in OCBMS as an application of the main results. [ABSTRACT FROM AUTHOR]

In this paper, we prove the controllability of mild solutions of neutral functional evolution equations with state-dependent delay and nonlocal conditions. We establish the non local controllability of mild solutions under certain conditions by combining Avramescu's nonlinear alternative for the sum of compact and contraction operators in Fréchet spaces with semigroup theory. [ABSTRACT FROM AUTHOR]

In this paper, our focus is to acquaint with the Suzuki-type mappings to establish some fixed point results using the new w-interpolative approach. We present some results for interpolative contraction operators via the w-admissible maps which satisfy the Kannan, Ćirić–Reich–Rus, and Hardy–Rogers contractions in quasi-partial b-metric space. Further, the outcomes so obtained are affirmed with relevant examples. [ABSTRACT FROM AUTHOR]

The aim of this paper is to define generalized (α-η)-Θ contraction and to extend the results of Jleli and Samet [M. Jleli, B. Samet, J. Inequal. Appl., 2014 (2014), 8 pages] by applying a simple condition on the function Θ. We also deduce certain fixed and periodic point results for orbitally continuous generalized Θ-contractions and certain fixed point results for integral inequalities are derived. Finally, we provide an example to show the significance of the investigation of this paper. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some common fixed point theorems for interpolative contraction operators using Perov operator which satisfy Suzuki type mappings. Further, some results are given. These results generalize several new results present in the literature. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of impulsive stochastic partial integrodifferential equations (ISPIEs) with delays and Poisson jumps. First, using the resolvent operator technique and contraction mapping principle, we can directly prove the existence and uniqueness of the mild solution for the system mentioned above. Then we develop a new impulsive integral inequality to obtain the global, both pth moment exponential stability and almost surely exponential stability of the mild solution is established with sufficient conditions. Also, a numerical example is provided to validate the theoretical result. [ABSTRACT FROM AUTHOR]

Based on the notions of both contractibility and local contractibility, many works were done in fixed point theory. The present paper concerns a relation between digital contractibility and the existence of fixed points of digitally continuous maps. In this paper, establishing a new digital homotopy named by a K-homotopy in the category of Khalimsky topological spaces, we prove that in digital topology, whereas contractibility implies local contractibility, the converse does not hold. Furthermore, we address the following problem, which remains open. Let X be a Khalimsky ( K- for short) topological space with K-contractibility. Then we may pose the following question: does the space X have the fixed point property ( FPP)? In this paper, we prove that not every K-topological space with K-contractibility has the FPP. [ABSTRACT FROM AUTHOR]

The first purpose of this paper is to prove an existence and uniqueness result for the multivariate fixed point of a contraction type mapping in complete metric spaces. The proof is based on the new idea of introducing a convenient metric space and an appropriate mapping. This method leads to the changing of the non-self-mapping setting to the self-mapping one. Then the main result of the paper will be applied to an initial-value problem related to a class of differential equations of first order. The second aim of this paper is to prove strong and weak convergence theorems for the multivariate fixed point of a N-variables nonexpansive mapping. The results of this paper improve several important works published recently in the literature. [ABSTRACT FROM AUTHOR]

This paper proves a new q-Hermite-Hadamard inequality for convex functions using quantum integrals. We also prove some new midpoint-type inequalities for q-differentiable convex functions. Moreover, we present some examples to illustrate our established results, supplemented with graphs. [ABSTRACT FROM AUTHOR]

In this paper, some sufficient conditions for the oscillation of all solutions of second order dynamic equations with a negative sub-linear neutral term are established. The obtained results provide a unified platform that adequately covers both discrete and continuous equations. Furthermore, it covers a wide range of equations by utilizing different time scales. Illustrative examples are provided. [ABSTRACT FROM AUTHOR]

In this research paper, we investigate some new identifies for Sarıkaya fractional integrals which introduced by Sarıkaya and Ertugral in [20]. The fractional integral operators also have been applied to Hermite-Hadamard type integral inequalities to provide their generalized properties. Furthermore, as special cases of our main results, we present several known inequalities such as Simpson, Bullen, trapezoid for convex functions. [ABSTRACT FROM AUTHOR]

Two interesting extensions of Banach contraction principle to mappings that do not to be continuous, are Kannan and Chatterjea's theorems. Before this, in the cyclical form, extensions of these two theorems and Banach contraction principle were produced. But so far, these theorems have not been studied in the noncyclical form. In this paper, we answer the question whether there are versions of these theorems for non-cyclic mappings, also we give generalizations of existing results. For this purpose, in the setting of metric spaces we introduce the notions of cyclic and noncyclic contraction of Fisher type. We establish the existence of fixed points for these mappings and iterative algorithms are furnished to determine such fixed points. As a result of our results we give new theorems for cyclic orbital contractions. [ABSTRACT FROM AUTHOR]

In this paper, the notion of fuzzy (θ,L)-weak contraction in G--metric space is introduced, and sufficient conditions for the existence of fuzzy fixed points for such mappings are investigated. Relevant illustrative examples are constructed to support the assumptions of our established theorems. It is observed that the principal ideas obtained herein extend and subsume some well-known results in the corresponding literature. A few of these special cases of our results are noted and discussed as corollaries. [ABSTRACT FROM AUTHOR]

As an important emission reduction source for the transportation industry, biofuel has received strong support from the Chinese government. However, the development of the biofuel industry is still struggling. The high degree of uncertainty makes the development of the industry face huge challenges. Kitchen waste, as a biodiesel raw material with a large yield, has good development prospects. Reuse of kitchen waste can solve public health and safety problems. This paper proposes a two-stage stochastic programming model under supply disturbance to optimize the supply chain from the perspective of contract. Then current three main flow directions of kitchen waste are analysed and the reasonable price for biodiesel operators to purchase is determined. By signing contracts with the biodiesel operators, restaurant is guaranteed and encouraged to provide a certain percentage of kitchen waste to meet the demand for biodiesel production. Using actual case in the Yangtze River Delta region, the performance of the stochastic programming model under disturbance was compared. Through the sensitivity analysis of different parameters, this paper determines the influence of its supply chain network design and expected total system cost. Through the optimization of the waste cooking oil (WCO) for biodiesel supply chain, this paper can effectively improve the efficiency of the supply chain, reduce system costs, increase the profits of biofuel operators, and promote the sustainable development of the biofuel industry. [ABSTRACT FROM AUTHOR]

In this paper, we consider the inverse boundary value problem for a partial differential equation of third order with nonlocal boundary conditions, including an integral condition. Using analytical and operator-theoretic methods, as well as the Fourier method, the existence and uniqueness of the classical solution of this problem is proved. [ABSTRACT FROM AUTHOR]

In this paper, we prove coincidence and common fixed points results under nonlinear contractions on a metric space equipped with an arbitrary binary relation. Our results extend, generalize, modify and unify several known results especially those are contained in Berzig [J. Fixed Point Theory Appl. 12, 221-238 (2012))] and Alam and Imdad [To appear in Filomat (arXiv:1603.09159 (2016))]. Interestingly, a corollary to one of our main results under symmetric closure of a binary relation remains a sharpened version of a theorem due to Berzig. Finally, we use examples to highlight the accomplished improvements in the results of this paper. [ABSTRACT FROM AUTHOR]

An efficient solution of the three-dimensional Helmholtz equation is known to be crucial in many applications, especially geophysics. In this paper, we present and test two preconditioning approaches for the discrete problem resulting from the second order finite-difference discretization of this equation. The first approach combines shifted-Laplacian preconditioner with inversion of a separable matrix, corresponding to the horizontally-layered velocity model, using fast Fourier based transforms. The second approach is novel and involves a special transformation resulting in a preconditioner with a contraction operator (CO preconditioner). The two approaches have near the same arithmetical complexity; however, the second approach, developed in this paper, provides a faster convergence of an iterative solver as illustrated by numerical experiments and analysis of the spectral properties of the preconditioned matrices. Our numerical experiments involve parallel modeling of highly heterogeneous lossy and lossless media at different frequencies. We show that the CO-based solver can tackle problems with hundreds of millions of unknowns on a conventional cluster node. The CO preconditioned solver demonstrates a very moderate increase of iteration count with the frequency. We have conducted a comparison of the performance of the developed method versus open-source parallel sweeping preconditioner. The results indicate that, the CO solver is several times faster with respect to the wall-clock time and consumes substantially less memory than the code based on the sweeping preconditioner at least in the example we tested. [ABSTRACT FROM AUTHOR]

Motivated and inspired by the growing contribution with respect to iterative approximations from some researchers in the literature, we design and investigate two types of brand-new semi-implicit viscosity iterative approximation methods for finding the fixed points of nonexpansive operators associated with contraction operators in complete CAT (0) spaces and for solving related variational inequality problems. Under some suitable assumptions, strong convergence theorems of the sequences generated by the approximation iterative methods are devised, and a numerical example and some applications to related variational inequality problems are included to verify the effectiveness and practical utility of the convergence theorems. Our main results presented in this paper do not only improve, extend and refine some corresponding consequences in the literature, but also show that the additional variational inequalities, general variational inequality systems and equilibrium problems can be solved via approximation of the iterative sequences. Finally, we provide an open question for future research. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of square-mean piecewise almost automorphic function. By using the theory of semigroups of operators and the contraction mapping principle, the existence of square-mean piecewise almost automorphic mild solutions for linear and nonlinear impulsive stochastic evolution equations is investigated. In addition, the exponential stability of square-mean piecewise almost automorphic mild solutions for nonlinear impulsive stochastic evolution equations is obtained by the generalized Gronwall–Bellman inequality. Finally, we provide an illustrative example to justify the results. [ABSTRACT FROM AUTHOR]

This paper presents a novel numerical method for simulation controlled-source audio-magnetotellurics (CSAMT) and radio-magnetotellurics (CSRMT) data. These methods are widely used in mineral exploration. Interpretation of the CSAMT and CSRMT data collected over an area with the complex geology requires application of effective methods of numerical modeling capable to represent the geoelectrical model of a deposit well. In this paper, we considered an approach to 3D electromagnetic (EM) modeling based on new types of preconditioned iterative solvers for finite-difference (FD) EM simulation. The first preconditioner used fast direct inversion of the layered Earth FD matrix (Green's function preconditioner). The other combined the first with a contraction operator transformation. To illustrate the effectiveness of the developed numerical modeling methods, a 3D resistivity model of Aleksandrovka study area in Kaluga Region, Russia, was prepared based on drilling data, AMT, and a detailed CSRMT survey. We conducted parallel EM simulation of the full CSRMT survey. Our results indicated that the developed methods can be effectively used for modeling EM responses over a realistic complex geoelectrical model for a controlled source EM survey with hundreds of receiver stations. The contraction-operator preconditioner outperformed the Green's function preconditioner by factor of 7-10, both with respect to run-time and iteration count, and even more at higher frequencies. [ABSTRACT FROM AUTHOR]

The question if a polynomially bounded operator is similar to a con- traction was posed by Halmos and was answered in the negative by Pisier. His counterexample is an operator of infinite multiplicity, while all its restrictions on invariant subspaces of finite multiplicity are similar to contractions. In the paper, cyclic polynomially bounded operators which are not similar to contractions and are quasisimilar to C0-contractions or to isometries are constructed. The construction is based on a perturbation of the sequence of finite dimensional operators which is uniformly polynomially bounded, but is not uniformly completely polynomially bounded, constructed by Pisier. [ABSTRACT FROM AUTHOR]

The characteristics of construction projects-e.g., one-off, high complexity, information asymmetry, and asset specificity-have led to a very common phenomenon of opportunism. In this paper, a questionnaire based on a sample of contractors was employed to explore the effect of contract completeness on contractors' opportunistic behavior, as well as the moderating role of interdependence. The results show that term specificity, contingency adaptability, and contractual obligatoriness all have negative effects on contractors' opportunistic behavior (both weak and strong forms), while issue inclusiveness positively influences contractors' opportunistic behavior. In addition, except for the positive moderating role of interdependence magnitude on the relationship between contingency adaptability and contractors' opportunistic behavior (weak form), the other three significant moderating effects (magnitude on issue inclusiveness and strong form, magnitude on contractual obligatoriness and strong form, asymmetry on term specificity and strong form) are all found to be negative. These findings can give insight into the many ways that contracts affect opportunism, and thus help both sides strategically react to opportunistic disturbances and minimize transaction costs. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce the notions of Θ-type contractions and Θ-type Suzuki contractions and to establish some new fixed point theorems for these two kinds of mappings in the setting of complete metric spaces. The results presented in the paper are an extension of the Banach contraction principle, the Suzuki contraction theorem, and the Jleli and Samet fixed point theorem. As an application, we utilize our results to study the existence problem of solutions of nonlinear Hammerstein integral equations. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce the class of enriched interpolative Kannan type operators on Banach space that contains the classes of enriched Kannan operators, interpolative Kannan type contraction operators and some other classes of nonlinear operators. Some examples are presented to support the concepts introduced herein. A convergence theorem for the Krasnoselskii iteration method to approximate fixed point of the enriched interpolative Kannan type operators is proved. We study well-posedness, Ulam-Hyers stability and periodic point property of operators introduced herein. As an application of the main result, variational inequality problem is solved. [ABSTRACT FROM AUTHOR]

The aim of this paper is to improve some fixed point results given by I.A. Rus in (On a fixed point theorem of Maia, Studia Univ. Babes¸-Bolyai Math., 22(1977), 40-42), (Basic problem for Maia's theorem, Sem. on Fixed Point Theory, Preprint 3(1981), Babes¸-Bolyai Univ. Cluj- Napoca, 112-115) and (Data dependence of the fixed points in a set with two metrics, Fixed Point Theory, 8(2007), no. 1, 115-123), by applying, instead of contraction principle, some variants of Maia's fixed point theorem. 2010 Mathematics Subject Classification: 47H10, 47H09, 34K05, 34K12, 45D05, 45G10, [ABSTRACT FROM AUTHOR]

In this paper, the negative proportional dynamic feedback is designed in the right boundary of the wave component of the 1-d heat-wave system coupled at the interface and the long-time behavior of the system is discussed. The system is formulated into an abstract Cauchy problem on the energy space. The energy of the system does not increase because the semigroup generated by the system operator is contracted. In the meanwhile, the asymptotic stability of the system is derived in light of the spectral configuration of the system operator. Furthermore, the spectral expansions of the system operator are precisely investigated and the asymptotical stability is not exponential and is shown in view of the spectral expansions. [ABSTRACT FROM AUTHOR]

In this paper, we will introduce the concept of Suzuki type multivalued (θ,R)-contraction and we will prove some fixed point results in the setting of a metric space equipped with a binary relation. Our results generalize and extend various comparable results in the existing literature. Examples are provided to support the results proved here. As an application of our results, we obtain a homotopy result, proving the existence of a solution for a second-order differential equation and for a first-order fractional differential equation. [ABSTRACT FROM AUTHOR]

By means of the fixed point theory for a strict set contraction operator, this paper investigates the existence, nonexistence, and multiplicity of positive solutions for a nonlinear higher order boundary value problem with four point fractional integral boundary conditions in Banach spaces. In addition, an example is worked out to illustrate the main results. [ABSTRACT FROM AUTHOR]

Recall from [2], that a mapping T : X 7→ X is called a Reich mapping if it satisfies for all x; yϵ X, d(Tx; Ty) ≤ ad(x; Tx) + bd(y; Ty) + cd(x; y), where a,b,c are nonnegative and satisfy a + b + c < 1. Alternatively, one could define a Reich mapping as follows: T : X 7→ X is called a Reich mapping if there exists a nonnegative constant k with k < 1 3 such that d(Tx; Ty) ≤ k[d(x; Tx)+d(y; Ty)+d(x; y)]. In the present paper, we address the following: How do we characterize Theorem 3 [2], when T is a non-self map? We show such a characterization is given by Theorem 3.1 or Corollary 3.2 in this paper. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of non-oscillatory solutions to a kind of first-order neutral equations having both delay and advance terms. The new results are established using the Banach contraction principle. [ABSTRACT FROM AUTHOR]

European Institutions and scholars are reflecting upon the nature of the requirements that are or can be used to implement sustainability through Public Procurement. As far as public organisations are concerned, the Portuguese government's actions are in the same wavelength as the European guidelines on the matter. Public procurement involves economic operators. Thus, it is important to try to understand if the European Union proposal for a directive on "Corporate Sustainability Due Diligence" is an interesting tool in the discussion. Recently, the Portuguese Council of Ministers issued Resolution no. 132/2023, of 25 October, defining the ecological criteria applicable to certain kinds of contracts and products, goods and services concerning some administrative bodies, a Resolution which, nevertheless, is prone to criticism. May the shift of the European Union Public Procurement paradigm help implement Sustainable Public Procurement through mandatory requirements? Can the future Directive help economic operators align with contracting authorities to implement Sustainable Public Procurement? Might the obligation to include environmental factors help fulfil the Directive's due diligence obligations? And, lastly, has Portugal taken the right step with the issuance of Resolution no. 132/2023? [ABSTRACT FROM AUTHOR]

Background: Tumor immune microenvironment (TiME) is prognostically instructive in Pancreatic adenocarcinoma (PAAD). However, the potential value of TiME-related genes in the individualized immunotherapy of PAAD has not been clarified. Methods: Correlation between Immune-Related Genes (IRGs) and immunerelated transcription factors (TFs) was performed to prove the immune correlation of selected genes. Immune-related molecular subtypes were identified by consensus clustering. The TiME-score, an immune microenvironment-related prognostic signature for PAAD, was constructed using minimum absolute contraction and selection operator regression (Lasso-Cox). The International Cancer Genome Consortium (ICGC) dataset validated the reliability of TiME-score as external validation. Single-cell samples from GSE197177 confirmed microenvironment differences of TiME-score hub genes between tumor and its paracancer tissues. Then, RARRES3, a hub gene in TiMEscore, was further analyzed about its upstream TP53 mutation and the specific immune landscape of itself in transcriptome and Single-cell level. Eventually, TiME-score were validated in different therapeutic cohorts of PAAD mice models. Results: A 14-genes PAAD immune-related risk signature, TiME-score, was constructed based on IRGs. The differences of TiME-score hub genes in single-cell samples of PAAD cancer tissues and adjacent tissues were consistent with the transcriptome. Single-cell samples of cancer tissues showed more pronounced immune cell infiltration. The upstream mutation factor TP53 of RARRES3 was significantly enriched in immune-related biological processes. High RARRES3 expression was correlated with a worse prognosis and high macrophages M1 infiltration. Additionally, the immunohistochemistry of hub genes AGT, DEFB1, GH1, IL20RB, and TRAF3 in different treatment cohorts of mice PAAD models were consistent with the predicted results. The combination of immunotherapy, chemotherapy and targeted therapy has shown significantly better therapeutic effects than single drug therapy in PAAD. Conclusion: TiME-score, as a prognostic signature related to PAAD-specific immune microenvironment constructed based on RARRES3, has predictive value for prognosis and the potential to guide individualized immunotherapy for PAAD patients. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish some fixed point results for cyclic contractions in the setting of dislocated quasi- b-metric spaces. We verify that some previous cyclic contraction results in dislocated quasi- b-metric spaces are just equivalent to the non-cyclic ones in the same spaces. Moreover, by using two examples, we highlight the superiority of the results obtained. [ABSTRACT FROM AUTHOR]

In the present paper we study stability of recurrence equations (which in particular case contain dynamics of rational functions) generated by contractive functions defined on an arbitrary non-Archimedean algebra. Moreover, multirecurrence equations are considered. We also investigate reverse recurrence equations which have application in the study of p -adic Gibbs measures. Note that our results also provide the existence of unique solutions of nonlinear functional equations. We should stress that the non-Archimedeanity of the algebra is essentially used, therefore, the methods applied in the present paper are not valid in the Archimedean setting. [ABSTRACT FROM AUTHOR]

Operators of type f → f ◦ ϕ acting on function spaces are called composition operators. We consider composition operators acting on the Hilbert Hardy space on the open unit disc or the right half-plane, study when they are similar to contractions, and obtain results interesting from the point of view of dilation theory of contractions and function theory. [ABSTRACT FROM AUTHOR]

The AGM framework is the benchmark approach in belief change. Since the framework assumes an underlying logic containing classical Propositional Logic, it can not be applied to systems with a logic weaker than Propositional Logic. To remedy this limitation, several researchers have studied AGM-style contraction and revision under the Horn fragment of Propositional Logic (i.e., Horn logic). In this paper, we contribute to this line of research by investigating the Horn version of the AGM entrenchment-based contraction. The study is challenging as the construction of entrenchment-based contraction refers to arbitrary disjunctions which are not expressible under Horn logic. In order to adapt the construction to Horn logic, we make use of a Horn approximation technique called Horn strengthening. We provide a representation theorem for the newly constructed contraction which we refer to as entrenchment-based Horn contraction. Ideally, contractions defined under Horn logic (i.e., Horn contractions) should be as rational as AGM contraction. We propose the notion of Horn equivalence which intuitively captures the equivalence between Horn contraction and AGM contraction. We show that, under this notion, entrenchment-based Horn contraction is equivalent to a restricted form of entrenchment-based contraction. [ABSTRACT FROM AUTHOR]

In this paper, we extend relation-theoretic contraction principle due to Alam and Im- dad to a nonlinear contraction using a relatively weaker class of continuous control functions employing a locally finitely T -transitive binary relation, which improves the corresponding fixed point theorems especially due to: Alam and Imdad (J. Fixed Point Theory Appl. 17 (2015) 693-702), Agarwal et al: (Applicable Analysis, 87 (1) (2008) 109-116), Berzig and Karapinar (Fixed Point Theory Appl. 2013:205 (2013) 18 pp), Berzig et al: (Abstr. Appl. Anal. 2014:259768 (2014) 12 pp) and Turinici (The Sci. World J. 2014:169358 (2014) 10 pp). [ABSTRACT FROM AUTHOR]

In this paper, we show that the contractive definition considered by Proinov [Fixed point theorems in metric spaces, Nonlinear Analysis 64 (2006) 546 - 557] is strong enough to generate a fixed point but does not force the mapping to be continuous at the fixed point. Thus we provide several answers to the open question posed by B.E. Rhoades in Contractive definitions and continuity, Contemporary Mathematics 72(1988), 233-245. [ABSTRACT FROM AUTHOR]

Abstract In this paper we study the multigrid algorithm for a quad-curl problem based on C 0 interior penalty method on bounded polygonal domains. It is shown that the contraction numbers of W -cycle multigrid algorithm for the C 0 interior penalty method converge uniformly with rate of m − α , where m is the number of pre-smoothing (and post-smoothing) steps, α is the index of elliptic regularity. The performance of the multigrid fast solver is illustrated by numerical results. [ABSTRACT FROM AUTHOR]

In this article, we study the dynamics of collective beliefs. As a first step, we formulate David Westlund's Principle of Collective Change (PCC)--a criterion that characterizes the evolution of collective knowledge--in the realm of belief revision. Thereafter, we establish a number of unsatisfiability results pointing out that the widely-accepted revision operators of Alchourrón, Gärdenfors and Makinson, combined with fundamental types of merging operations--including the ones proposed by Konieczny and Pino Pérez as well as Baral et al.--collide with the PCC. These impossibility results essentially extend in the context of belief revision the negative results established by Westlund for the operations of contraction and expansion. At the opposite of the impossibility results, we also establish a number of satisfiability results, proving that, under certain (rather strict) requirements, the PCC is indeed respected for specific merging operators. Overall, it is argued that the PCC is a rather unsuitable property for characterizing the process of collective change. Last but not least, mainly in response to the unsatisfactory situation related to the PCC, we explore some alternative criteria of collective change, and evaluate their compliance with belief revision and belief merging. [ABSTRACT FROM AUTHOR]

In this paper we study efficient iterative methods for real symmetric Toeplitz systems based on the trigonometric transformation splitting (TTS) of the real symmetric Toeplitz matrix A . Theoretical analyses show that if the generating function f of the n × n Toeplitz matrix A is a real positive even function, then the TTS iterative methods converge to the unique solution of the linear system of equations for sufficient large n . Moreover, we derive an upper bound of the contraction factor of the TTS iteration which is dependent solely on the spectra of the two TTS matrices involved. Different from the CSCS iterative method in Ng (2003) in which all operations counts concern complex operations when the DFTs are employed, even if the Toeplitz matrix A is real and symmetric, our method only involves real arithmetics when the DCTs and DSTs are used. The numerical experiments show that our method works better than CSCS iterative method and much better than the positive definite and skew-symmetric splitting (PSS) iterative method in Bai et al. (2005) and the symmetric Gauss–Seidel (SGS) iterative method. [ABSTRACT FROM AUTHOR]

We study asymptotic optimality of inference in a high-dimensional sparse normal means model using a broad class of one-group shrinkage priors. Assuming that the proportion of non-zero means is known, we show that the corresponding Bayes estimates asymptotically attain the minimax risk (up to a multiplicative constant) for estimation with squared error loss. The constant is shown to be 1 for the important sub-class of "horseshoe-type" priors proving exact asymptotic minimaxity property for these priors, a result hitherto unknown in the literature. An empirical Bayes version of the estimator is shown to achieve the minimax rate in case the level of sparsity is unknown. We prove that the resulting posterior distributions contract around the true mean vector at the minimax optimal rate and provide important insight about the possible rate of posterior contraction around the corresponding Bayes estimator. Our work shows that for rate optimality, a heavy tailed prior with sufficient mass around zero is enough, a pole at zero like the horseshoe prior is not necessary. This part of the work is inspired by van der Pas et al. (2014). We come up with novel unifying arguments to extend their results over the general class of priors. Next we focus on simultaneous hypothesis testing for the means under the additive 0 - 1 loss where the means are modeled through a two-groups mixture distribution. We study asymptotic risk properties of certain multiple testing procedures induced by the class of one-group priors under study, when applied in this set-up. Our key results show that the tests based on the "horseshoe-type" priors asymptotically achieve the risk of the optimal solution in this two-groups framework up to the correct constant and are thus asymptotically Bayes optimal under sparsity (ABOS). This is the first result showing that in a sparse problem a class of one-group priors can exactly mimic the performance of an optimal two-groups solution asymptotically. Our work shows an intrinsic technical connection between the theories of minimax estimation and simultaneous hypothesis testing for such one-group priors. [ABSTRACT FROM AUTHOR]