Search

The article explores the longest distance problem in mathematics, specifically focusing on the longest path between two points. It introduces a mathematical model and discusses the requirements for acceptable paths, such as continuity and differentiability. The article provides mathematical formulas and proofs to support its arguments. It also discusses the longest path problem with restricted slope and the optimization of path length. The findings are useful for solving optimization problems and mathematical analysis. [Extracted from the article]

In a recent Mathematical Intelligencer note, B. Chakraborty [A visual proof that $ \pi ^{e} π e < e π . Mathematical Intelligencer, 41(1), 56] provided a visual proof of the inequality $ \pi ^{e} π e < e π . The aim of this classroom capsule is to visualize the more general inequality $ b^a b a < a b , where $ e \leq a e ≤ a < b in an amazing way. [ABSTRACT FROM AUTHOR]

As we try to understand this tension and its evolution through the next century of European mathematics, we will find that mathematics was something slightly different for the sixteenth century than it is for us, and relearn important lessons about reading old books with modern eyes. Abstracts from past meetings History of mathematics and flight Saturday 2 July 2022 Concorde Centre, Manchester Airport A day of talks about the history of mathematics and flight. Black Heroes of Mathematics Tuesday 4 to Wednesday 5 October Online Repeating the success of the past two years, the BSHM, the International Centre for Mathematical Sciences, the Institute of Mathematics and its Applications, the Isaac Newton Institute, the London Mathematical Society, and the Mathematical Association held a two day conference on Black Heroes of Mathematics. It shall also use the arguments of recent mathematics historiography to establish a place for medieval European arithmetic in the canonical history of mathematics. [Extracted from the article]

Speakers and titles: Alastair Compton (Cambridge): I Thomas Willis (1621-1675) i Nigel Aston (Leicester): I The Theologians (1704-1782) i Jim Bennett (Oxford): I Thomas Hornsby (1733-1810) i Christopher Hollings (Oxford): I George Leigh Cooke (c.1779-1853) and Bartholomew Price (1818-1898) i June Barrow-Green (Open): I Augustus Love (1863-1940) i Peter Cargill (St Andrews): I Sydney Chapman (1888-1970) i Mark McCartney (Ulster): I George Temple (1901-1992) and Albert Green (1912-1999) i History of mathematics and flight Saturday 2 July 2022 Concorde Centre, Manchester Airport A day of talks about the history of mathematics and flight. This is somewhat surprising, considering that Somerville had explicitly depicted analytical mathematics as a fertile ground waiting to be farmed. The programme will include two talks by historians of mathematics and astronomy, and three by contemporary astronomers/cosmologists. [Extracted from the article]

This paper focuses on the convergence analysis for a patch structure Nicholson's blowflies system involving an oscillating death rate and multiple different time-varying delays. By using inequality techniques and concise mathematical analysis proof, some sufficient criteria are established to guarantee the global exponential convergence of the zero equilibrium point for the addressed system. Our results are novel and supplement some existing ones. Furthermore, the effectiveness and feasibility of the obtained results are demonstrated by some numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, an optimal robust adaptive fuzzy backstepping control is presented to the position control of the electro-hydraulic servo (EHS) system in the presence of structured and unstructured uncertainties. Initially, the robust control using the backstepping technique is presented to overcome the existing uncertainties in the dynamic equations. Mathematical proof demonstrates that the closed-loop system in the presence of uncertainties has a global asymptotic stability. Then, to overcome the chattering problem, a very simple fuzzy approximator is presented where it approximates the bounds of the uncertainties. Although the proposed robust fuzzy backstepping control has a desirable performance, it has no mathematical analysis to prove the stability of the closed-loop system. Therefore, to solve this problem, the proposed fuzzy approximator has been transformed into a one-law adaptive fuzzy approximator with a single-input single-output fuzzy rule. Mathematical analysis illustrates that the closed-loop system in the presence of uncertainties has a global asymptotic stability under the proposed robust adaptive fuzzy backstepping control. Furthermore, a novel modified harmony search algorithm (MHSA) has been developed, by using the original harmony search algorithm (OHSA) as an optimization technique, to achieve the optimal values of the membership functions and the control coefficients. Finally, a comparative study has been conducted between the proposed control scheme under the MHSA and the OHSA, and other existing advanced control approaches to verify the effectiveness of the proposed control. Results show that the proposed control scheme under the MHSA can suppress the chattering problem and reduce the disturbances effectively while ensuring that the performance is tracked. [ABSTRACT FROM AUTHOR]

A dynamical system with a plastic self-organising velocity vector field was introduced in Janson and Marsden (Sci Rep 7:17007, 2017) as a mathematical prototype of new explainable intelligent systems. Although inspired by the brain plasticity, it does not model or explain any specific brain mechanisms or processes, but instead expresses a hypothesised principle possibly implemented by the brain. The hypothesis states that, by means of its plastic architecture, the brain creates a plastic self-organising velocity vector field, which embodies self-organising rules governing neural activity and through that the behaviour of the whole body. The model is represented by a two-tier dynamical system, in which the observable behaviour obeys a velocity field, which is itself controlled by another dynamical system. Contrary to standard brain models, in the new model the sensory input affects the velocity field directly, rather than indirectly via neural activity. However, this model was postulated without sufficient explication or theoretical proof of its mathematical consistency. Here we provide a more rigorous mathematical formulation of this problem, make several simplifying assumptions about the form of the model and of the applied stimulus, and perform its mathematical analysis. Namely, we explore the existence, uniqueness, continuity and smoothness of both the plastic velocity vector field controlling the observable behaviour of the system, and the of the behaviour itself. We also analyse the existence of pullback attractors and of forward limit sets in such a non-autonomous system of a special form. Our results verify the consistency of the problem and pave the way to constructing more models with specific pre-defined cognitive functions. [ABSTRACT FROM AUTHOR]

Purpose: This paper aims to propose a general and rigorous study on the propagation property of invariant errors for the model conversion of state estimation problems with discrete group affine systems. Design/methodology/approach: The evolution and operation properties of error propagation model of discrete group affine physical systems are investigated in detail. The general expressions of the propagation properties are proposed together with the rigorous proof and analysis which provide a deeper insight and are beneficial to the control and estimation of discrete group affine systems. Findings: The investigation on the state independency and log-linearity of invariant errors for discrete group affine systems are presented in this work, and it is pivotal for the convergence and stability of estimation and control of physical systems in engineering practice. The general expressions of the propagation properties are proposed together with the rigorous proof and analysis. Practical implications: An example application to the attitude dynamics of a rigid body together with the attitude estimation problem is used to illustrate the theoretical results. Originality/value: The mathematical proof and analysis of the state independency and log-linearity property are the unique and original contributions of this work. [ABSTRACT FROM AUTHOR]

We investigate two notions of independence—(usual) independence and complete independence—applied to the Peano axioms for the sequence of natural numbers. We review the results that, although they are independent, the Peano axioms are not completely independent. The standard proof that the Peano axioms are not completely independent is algebraic, in the sense that it makes essential reference to the relationship between several mathematical structures that satisfy, or do not satisfy, these axioms. We then present an alternative logical proof, which makes no essential references to the relationship between mathematical structures. There is a completely independent set of axioms for the sequence of natural numbers, but it is based on primitives different from those originally adopted by Peano. Therefore, we present a new completely independent set of axioms based on the same set of primitives as the one originally adopted by Peano. [ABSTRACT FROM AUTHOR]

One of the central aims of the philosophical analysis of mathematical explanation is to determine how one can distinguish explanatory proofs from non-explanatory proofs. In this paper, I take a closer look at the current status of the debate, and what the challenges for the philosophical analysis of explanatory proofs are. In order to provide an answer to these challenges, I suggest we start from analysing the concept understanding. More precisely, I will defend four claims: (1) understanding is a condition for explanation, (2) unificatory understanding is a type of explanatory understanding, (3) unificatory understanding is valuable in mathematics, and (4) mathematical proofs can contribute to unificatory understanding. As a result, in a context where the epistemic aim is to unify mathematical results, I argue it is fruitful to make a distinction between proofs based on their explanatory value. [ABSTRACT FROM AUTHOR]

Here HT ht , so that HT ht . To bring this brief melange to a close, the final result we need is the classical summation formula 11 HT ht Graph where all series involved converge. [Extracted from the article]

The solution of nonlinear optimization is usually encountered in many fields of scientific researches and engineering applications, which spawns a large number of corresponding algorithms to cope with it. Besides, with developments of modern cybernetics technology, it imperatively requires some advanced numerical algorithms to solve online dynamic nonlinear optimization (ODNO). Nevertheless, the major existing algorithms are limited to the static nonlinear optimization models, few works considering the dynamic ones, let alone tolerating noise. For the abovementioned reasons, this paper proposes a modified Newton integration (MNI) algorithm for ODNO with strong robustness and high-accuracy computing solution, which can effectively suppress the influence caused by noise components. In addition, the correlative theoretical analyses and mathematical proofs on convergence and robustness of the MNI algorithm are carried out, which indicates that computing solutions of the proposed MNI algorithm can globally converge to relative small value in the presence of various noise or zero noise conditions. Finally, to illustrate the advantages and feasibilities of the proposed MNI algorithm for ODNO problems, four numerical simulation examples and an application to robot manipulator motion generation are performed. [ABSTRACT FROM AUTHOR]

The article presents a study of the efficiency of the intermittent inverse-square Lévy walk strategy in finding sparse targets in finite two-dimensional terrains. Topics discussed include the role of the optimality claim in the optimality of Cauchy walks as a foraging strategy, benefit of intermittent Cauchy walks in terms of quickly finding sparse targets of unpredictable sizes, and the use of the Bachmann-Landau classical mathematical asymptotic notation in the study.

Purpose: The purpose of this paper is to introduce an artificial bee colony-based Kalman filter algorithm along with an extended objective function to ensure the optimality of the estimator of the quadrotor in the presence of unknown measurement noise statistics. Design/methodology/approach: Six degree-of-freedom mathematical model of the quadrotor is derived. Position controller for the quadrotor is designed. Kalman filter-based estimation algorithm is implemented in the sensor feedback loop. Artificial bee colony-based hybrid algorithm is used as an optimization method to handle the unknown noise statistics. Existing objective function is extended with a penalty term. Mathematical proof of the extended objective function is derived. Results of the proposed algorithm is compared with de facto genetic algorithm-based Kalman filter. Findings: Artificial bee colony algorithm-based Kalman filter and extended objective function duo are able to optimize the measurement noise covariance matrix with an absolute error as low as 0.001 [m2]. Proposed method and function is capable of reducing the noise from 2 to 0.09 [m] for x-axis, 3.4 to 0.14 [m] for y-axis and 3.7 to 0.2 [m] for z-axis, respectively. Originality/value: The motivation behind this paper is to bring a novel optimization-based solution for the estimation problem of the quadrotor when the measurement noise statistics are unknown along with an extended objective function to prevent the infeasible solutions with mathematical convergence analysis. [ABSTRACT FROM AUTHOR]

In this article, a novel real-time acceleration-continuous path-constrained trajectory planning algorithm is proposed with an appealing built-in tradeoff mechanism between the cruise motion and time-optimal motion. Different from existing approaches, the proposed approach smoothens time-optimal trajectories with bang-bang input structures to generate acceleration-continuous trajectories while preserving the completeness property. More importantly, a novel built-in tradeoff mechanism is proposed and embedded into the trajectory planning framework so that the proportion of the cruise motion and time-optimal motion can be flexibly adjusted by changing a user-specified functional parameter. Thus, the user can easily apply the trajectory planning algorithm for various tasks with different requirements on motion efficiency and cruise proportion. Moreover, it is shown that feasible trajectories are computed more quickly than optimal trajectories. Rigorous mathematical analysis and proofs are presented for those aforementioned theoretical results. Comparative simulations and experimental results on an omnidirectional wheeled mobile robot demonstrate that flexible tunings between the cruise and time-optimal motions can be achieved in a higher computational efficiency manner by the proposed algorithm. Note to Practitioners—This article is motivated by the time-optimal and smooth motion planning problem for mobile robots along given paths. Existing approaches generally use the piecewise polynomial interpolations to smoothen and adjust feasible trajectories. This article proposes a novel path-constrained trajectory planning approach, which preserves properties of completeness and a high-efficient tradeoff mechanism between the optimal and cruise motions when achieving a globally optimal and acceleration-continuous trajectory. Comparative experimental results with other methods show the effectiveness of the proposed approach. In future research, we will attempt to integrate the proposed approach with typical path planning methods to achieve a complete and high-efficient motion planning framework. [ABSTRACT FROM AUTHOR]

In the article, we prove that α = 3, β = log 4=(π=2 - log 4) = 7:51371 ..., γ = 1=4 and δ = 1 + log 2 - π/2= 0.122351 ... are the best possible constants such that the double inequalities ... hold for all r 2 (0; 1), where r0 = p 1 - r2, and K(r) = R =2 0 p ... and ... are the complete elliptic integrals of the first and second kinds. [ABSTRACT FROM AUTHOR]

The location of the terminal hinge axis of the temporomandibular joint is still a very wide-spread procedure in dentistry in order to replicate the movement in various articulator devices. Especially pantographic methods are claimed to provide accurate measurements and, additionally, are seen to be able to separate a pure rotation of the joint from a movement with an arbitrary combined shift and rotation. In the latter application, these methods were used in a lot of studies as a reference standard. The aim of this study was to analyze, whether common pantographic methods in general are able to distinguish between a pure rotation and a movement with rotational and translational portions. The mathematical proof of this analysis was done with theoretical kinematic considerations and compared with computer simulations. The results show for the first time that there exist combinations of rotational and translational movements of the temporomandibular joint which cannot be separated from pure rotational movements using actual pantographic methods. Even more, the consequence is a shifted location of the (combined) finite center (axis) of rotation in comparison to the true center (axis) of rotation: in case of a translational portion of only 1 mm, this is a displacement of around ±6 mm and, in case of 2 mm translation, a displacement of ±12 mm. This finding necessitates a critical reinterpretation of former studies using pantographic methods as a reference standard. Further, under some circumstances it may also affect the applicability of articulator concepts and the interpretation of functional signs. [ABSTRACT FROM AUTHOR]

In general, a proportional function is obviously not antiperiodic, yet a very interesting fact in this paper shows that it is possible there is an antiperiodic solution for some proportional delayed dynamical systems. We deal with the issue of antiperiodic solutions for RNNs (recurrent neural networks) incorporating multiproportional delays. Employing Lyapunov method, inequality techniques and concise mathematical analysis proof, sufficient criteria on the existence of antiperiodic solutions including its uniqueness and exponential stability are built up. The obtained results provide us some lights for designing a stable RNNs and complement some earlier publications. In addition, simulations show that the theoretical antiperiodic dynamics are in excellent agreement with the numerically observed behavior. [ABSTRACT FROM AUTHOR]

We prove that a canonical curve C of genus ≥11 is bielliptic if and only if its second syzygy scheme Syz 2 (C) is different from C. [ABSTRACT FROM AUTHOR]

Since proportional delay is monotonically increasing, a neural network involving multi-proportional delays is obviously not anti-periodic, yet a very interesting fact in this paper shows that it is possible there is an anti-periodic solution for such systems. This paper aims to deal with the issue of anti-periodic solutions for SICNNs (Shunting Inhibitory Cellular Neural Networks) involving multi-proportional delays. With the help of Lyapunov method, inequality techniques and a concise mathematical analysis proof, sufficient criteria on the existence of anti-periodic solutions including its uniqueness and exponential stability are established. The obtained results provide us some lights for designing a stable SICNNs and complement some earlier publications. Furthermore, simulations show that the theoretical anti-periodic dynamics are in excellent agreement with the numerically observed behavior. [ABSTRACT FROM AUTHOR]

Abstract We establish the universal properties of the bicategory of polynomials, considering both cartesian and general morphisms between these polynomials. A direct proof of these universal properties would be impractical due to the complicated coherence conditions arising from polynomial composition; however, in this paper we avoid most of these coherence conditions using the properties of generic bicategories. In addition, we give a new proof of the universal properties of the bicategory of spans, and also establish the universal properties of the bicategory of spans with invertible 2-cells; showing how these properties may be used to describe the universal properties of polynomials. [ABSTRACT FROM AUTHOR]

Abstract In [3] well known results of Wall and Arnon on the monomial bases in the mod 2 Steenrod algebra (see [9,1]) were generalized to the subalgebra A ‾ p of the mod p Steenrod algebra, p > 2 , generated by the reduced powers. In the present paper we considered the case of the full Steenrod algebra A p. We constructed βX -, βZ -, βC -, ZA -, and XC -bases. We proved extremal properties of the βX -, βZ -, ZA -, and XC -bases. Also we constructed a new polynomial generators of the ring H ⁎ (K (Z / p , n) , Z / p) in terms of the βC -basis. [ABSTRACT FROM AUTHOR]

We determine the structure of all finite-dimensional selfinjective algebras over a field whose Auslander-Reiten quiver admits a hereditary stable slice. [ABSTRACT FROM AUTHOR]

Abstract The class of finite-dimensional unital right alternative superalgebras with semisimple strongly alternative even part over an algebraically closed field are studied. We introduce the concept of weak annihilator of a superalgebra and prove that it is a superideal, and the superalgebra itself splits into the direct sum of its weak annihilator and a superalgebra with a zero weak annihilator. We also classify superalgebras with zero weak annihilator in this class of superalgebras. [ABSTRACT FROM AUTHOR]

Abstract We show that bounding ramification at infinity bounds fierce ramification. This answers positively a question of Deligne posed to the first named author. [ABSTRACT FROM AUTHOR]

Abstract Syzygies or N p -property of an ample line bundles on abelian varieties are well known. In this paper, we study defining equations and syzygies among them of projective bundles over abelian varieties. We prove an analogue of Pareschi's theorem (or Lazarsfeld's conjecture) on abelian varieties, extended to projective bundles over an abelian variety. [ABSTRACT FROM AUTHOR]

Abstract We consider the following two problems: Let f and g be two polynomials over a field K. P1. When is { f + s , g + t } a Gröbner basis for all s , t ∈ K ? P2. When is { f λ , g σ } a Gröbner basis for all λ , σ ∈ N ? We provide a simpler proof of Hoon's criterion for P1 and a new criterion for P2. The two criteria are applied to the theory of Gröbner bases under composition. [ABSTRACT FROM AUTHOR]

Abstract In the article, we prove that the double inequality π 2 J (r ′) − 51 π − 160 160 r 16 < E (r) < π 2 J (r ′) − 5 π 3 × 2 31 r 16 holds for all r ∈ (0, 1), where E (r) = ∫ 0 π / 2 1 − r 2 sin 2 (t) d t is the complete elliptic integral of the second kind, r ′ = (1 − r 2) 1 / 2 and J (r) = 51 r 2 + 20 r r + 50 r + 20 r + 51 16 (5 r + 2 r + 5). [ABSTRACT FROM AUTHOR]

We find non-regular Hausdorff rU-spaces (connected or not). We also prove that a countable Hausdorff connected space cannot be an rU-space. [ABSTRACT FROM AUTHOR]

Let X ⊂ ℙ n be a general Fano complete intersection of type (d 1 , ... , d k). If at least one d i is greater than 2 , we show that X contains rational curves of degree e ≤ n with balanced normal bundle. If all d i are 2 and n ≥ 2 k + 1 , we show that X contains rational curves of degree e ≤ n − 1 with balanced normal bundle. As an application, we prove a stronger version of the theorem of Tian [27], Chen and Zhu [4] that X is separably rationally connected by exhibiting very free rational curves in X of optimal degrees. [ABSTRACT FROM AUTHOR]

In this paper, we consider the quasi-neutral limit of the compressible Navier-Stokes-Fourier-Poisson system in a periodic domain with the well-prepared initial data. We prove that the weak solution of the compressible Navier-Stokes-Fourier-Possion system converges to the strong solution of the compressible Navier-Stokes-Fourier system as long as the latter exists. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we establish a stronger version of Artemov's arithmetical completeness theorem of the Logic of Proofs LP 0. Moreover, we prove a version of the uniform arithmetical completeness theorem of LP 0. [ABSTRACT FROM AUTHOR]

We prove Alexander- and Markov-type theorems for virtual spatial trivalent graphs and virtual trivalent braids. We provide two versions for the Markov-type theorem: one uses an algebraic approach similar to the case of classical braids and the other one is based on L -moves. [ABSTRACT FROM AUTHOR]

For the Dunkl transforms associated with the weight functions Hk²(x) = ∏j=1d ..., k1, ⋯, kd ≥ 0 on ℝd, it is proved that if p ≥ 2 + 1/λk and λk := d-1/2 + ∑j=1dkj, the maximal Bochner-Riesz operator B*δ(hk²; f) order δ > 0 is bounded on the space Lp(ℝd; hk² dx) if and only if δ > δk(p) := max{(2λk +1)(1/2 -- 1/p ) -- 1/2, 0}. This extends a well known result of M. Christ for the classical Fourier transforms (Proc. Amer. Math. Soc. 95 (1985), 16-20). The proof relies on a new local restriction theorem for the Dunkl transforms, which is stronger than the corresponding global restriction theorem, but significantly more difficult to prove. [ABSTRACT FROM AUTHOR]

At the 1987 Ramanujan Centenary meeting Dyson asked for a coherent group-theoretical structure for Ramanujan's mock theta functions analogous to Hecke's theory of modular forms. Many of Ramanujan's mock theta functions can be written in terms of R(ζ, q), where R(z, q) is the two-variable generating function of Dyson's rank function and ζ is a root of unity. Building on earlier work of Watson, Zwegers, Gordon, and McIntosh, and motivated by Dyson's question, Bringmann, Ono, and Rhoades studied transformation properties of R(ζ, q). In this paper we strengthen and extend the results of Bringmann, Rhoades, and Ono, and the later work of Ahlgren and Treneer. As an application we give a new proof of Dyson's rank conjecture and show that Ramanujan's Dyson rank identity modulo 5 from the Lost Notebook has an analogue for all primes greater than 3. The proof of this analogue was inspired by recent work of Jennings-Shaffer on overpartition rank differences mod 7. [ABSTRACT FROM AUTHOR]

Abstract We introduce the notion of a generalized oscillating tableau and define a promotion operation on such tableaux that generalizes the classical promotion operation on standard Young tableaux. As our main application, we show that this promotion corresponds to rotation of the irreducible A 2 -webs of G. Kuperberg. [ABSTRACT FROM AUTHOR]

Abstract Let P be a poset of size 2 k that has a greatest and a least element. We prove that, for sufficiently large n , the Boolean lattice 2 [ n ] can be partitioned into copies of P. This resolves a conjecture of Lonc. [ABSTRACT FROM AUTHOR]

Abstract We prove a regularity result for the weak solutions of H-systems in dimensions n ≥ 3. [ABSTRACT FROM AUTHOR]

The deviation of mathematical proof —proof in mathematical practice—from the ideal of formal proof —proof in formal logic—has led many philosophers of mathematics to reconsider the commonly accepted view according to which the notion of formal proof provides an accurate descriptive account of mathematical proof. This, in turn, has motivated a search for alternative accounts of mathematical proof purporting to be more faithful to the reality of mathematical practice. Yet, in order to develop and evaluate such alternative accounts, it appears as a necessary prerequisite to first possess a clear picture of what the deviation of mathematical proof from formal proof consists in. The present work aims to contribute building such a picture by investigating the relation between the elementary steps of deduction constituting the two types of proofs— mathematical inference and logical inference. Many claims have been made in the literature regarding the relation between mathematical inference and logical inference, most of them stating that the former is lacking properties that are constitutive of the latter. Such differentiating claims are, however, usually put forward without a clear conception of the properties occurring in them, and are generally considered to be immediately justified by our direct acquaintance, or phenomenological experience, with the two types of inferences. The present study purports to advance our understanding of the relation between mathematical inference and logical inference by developing a detailed philosophical analysis of the differentiating claims, that is, an analysis of the meaning of the differentiating claims—through the properties that occur in them—as well as the reasons that support them. To this end, we provide at the outset a representative list of the different properties of logical inference that have occurred in the differentiating claims, and we notice that they all boil down to the three properties of formality , generality , and mechanicality. For each one of these properties, our analysis proceeds in two steps: we first provide precise conceptual characterizations of the different ways logical inference has been said to be formal, general, and mechanical, in the philosophical and logical literature on formal proof; we then examine why mathematical inference does not appear to be formal, general, and mechanical, for the different variations of these notions identified. Our study results in a precise conceptual apparatus for expressing and discussing the properties differentiating mathematical inference from logical inference, and provides a first inventory of the various reasons supporting the observations of those differences. The differentiating claims constitute thus a set of data that any philosophical account of mathematical inference and proof purporting to be more faithful to mathematical practice ought to be able to accommodate and explain. [ABSTRACT FROM AUTHOR]

We consider a quantum system consisting of two subsystems (open system and environment). We have proved that the Wigner function of an open system is the integral of the Wigner function of a larger system with respect to coordinates of the phase space of the classical subsystem corresponding to the entourage. [ABSTRACT FROM AUTHOR]

Let σ be an equivalence on X and let E(X,σ) denote the semigroup (under composition) of all f:X→X, such that σ⊆ker(f). In this paper, we endow the semigroup E(X,σ) with a well-known natural partial order ≤ and provide a characterization for ≤ and prove necessary and sufficient conditions for ≤ to be both left and right compatible with the multiplication. We also describe the minimal and the maximal elements of E(X,σ) with respect to this order. [ABSTRACT FROM AUTHOR]

In this paper, we make several observations on cellular-Lindelöf spaces. We prove that in perfect spaces, the property of being cellular-Lindelöf is equivalent to the countable chain condition. Using this result, we prove that every cellular-Lindelöf first-countable perfect space has cardinality at most c and obtain a regular example of a weakly Lindelöf non-cellular-Lindelöf space. We also prove that if X is a cellular-Lindelöf space then every discrete family of non-empty open subsets of X is countable. Finally, we prove that if X is a cellular-Lindelöf space with a symmetric g-function such that ∩{g2(n,x):n∈ω}={x} for each x∈X then |X|≤2c. [ABSTRACT FROM AUTHOR]

We provide a maximum entropy derivation of a new family of BFGS-like methods. Similar results are then derived for block BFGS methods. This also yields an independent proof of a result of Fletcher 1991 and its generalization to the block case. [ABSTRACT FROM AUTHOR]