Can Parity‐Time‐Symmetric Potentials Support Families of Non‐Parity‐Time‐Symmetric Solitons?

For the one‐dimensional nonlinear Schrödinger equations with parity‐time (PT) symmetric potentials, it is shown that when a real symmetric potential is perturbed by weak PT‐symmetric perturbations, continuous families of asymmetric solitary waves in the real potential are destroyed. It is also shown that in the same model with a general PT‐symmetric potential, symmetry breaking of PT‐symmetric solitary waves does not occur. Based on these findings, it is conjectured that one‐dimensional PT‐symmetric potentials cannot support continuous families of non‐PT‐symmetric solitary waves.     

Mean field dynamics of fermions and the time-dependent Hartree–Fock equation

The time-dependent Hartree–Fock equations are derived from the N-body linear Schrödinger equation with the mean-field scaling in the limit N→+∞ and for initial data that are close to Slater determinants. Only the case of bounded, symmetric binary interaction potentials is treated in this work. We prove that, as N→+∞, the first partial trace of the N-body density operator approaches the solution of the time-dependent Hartree–Fock equations (in operator form) in the sense of the trace norm.     

The global dynamics of a class of vector fields in ℝ<Superscript>3</Superscript>

In this paper, we find a bridge connecting a class of vector fields in ℝ³ with the planar vector fields and give a criterion of the existence of closed orbits, heteroclinic orbits and homoclinic orbits of a class of vector fields in ℝ³. All the possible nonwandering sets of this class of vector fields fall into three classes: (i) singularities; (ii) closed orbits; (iii) graphs of unions of singularities and the trajectories connecting them. The necessary and sufficient conditions for the boundedness of the vector fields are also obtained.     

Does a ‘volume‐filling effect’ always prevent chemotactic collapse?

The parabolic–parabolic Keller–Segel system for chemotaxis phenomena, is considered under homogeneous Neumann boundary conditions in a smooth bounded domain Ω??ⁿ with n?2. It is proved that if ψ(u)/?(u) grows faster than u^2/n as u→∞ and some further technical conditions are fulfilled, then there exist solutions that blow up in either finite or infinite time. Here, the total mass ∫_Ωu(x, t)dx may attain arbitrarily small positive values. In particular, in the framework of chemotaxis models incorporating a volume‐filling effect in the sense of Painter and Hillen (Can. Appl. Math. Q. 2002; 10 (4):501–543), the results indicate how strongly the cellular movement must be inhibited at large cell densities in order to rule out chemotactic collapse. Copyright © 2009 John Wiley & Sons, Ltd.     

The global attractor of a non-local PDE model with delay for population dynamics in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper, we consider a non-local PDE model with delay for population dynamics in ℝ ⁿ . First, we prove the existence and uniqueness of weak solutions under some suitable decayed assumptions on non-local term at infinity. Then, we obtain the global attractor by proving ω-limit compactness property of the solution operator semigroup.     

How many graphs are unions of k‐cliques?

We study the number of n‐vertex graphs that can be written as the edge‐union of k‐vertex cliques. We obtain reasonably tight estimates for in the cases (i) k = n ? o(n) and (ii) k = o(n) but . We also show that exhibits a phase transition around . We leave open several potentially interesting cases, and raise some other questions of a similar nature. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 87–107, 2006     

Random dynamics and limiting behaviors for 3D globally modified Navier–Stokes equations driven by colored noise

This paper is mainly concerned with the long-term random dynamics for the nonautonomous 3D globally modified Navier–Stokes equations with nonlinear colored noise. We first prove the existence of random attractors of the nonautonomous random dynamical system generated by the solution operators of such equations. Then we establish the existence of invariant measures supported on the random attractors of the underlying system. Random Liouville-type theorem is also derived for such invariant measures. Moreover, we further investigate the limiting relationship of invariant measures between the above equations and the corresponding limiting equations when the noise intensity approaches to zero. In addition, we show the invariant measures of such equations with additive white noise can be approximated by those of the corresponding equations with additive colored noise as the correlation time of the colored noise goes to zero.     

When does the K4‐free process stop?

The K₄‐free process starts with the empty graph on n vertices and at each step adds a new edge chosen uniformly at random from all remaining edges that do not complete a copy of K₄. Let G be the random maximal K₄‐free graph obtained at the end of the process. We show that for some positive constant C, with high probability as , the maximum degree in G is at most . This resolves a conjecture of Bohman and Keevash for the K₄‐free process and improves on previous bounds obtained by Bollobás and Riordan and by Osthus and Taraz. Combined with results of Bohman and Keevash this shows that with high probability G has edges and is ‘nearly regular’, i.e., every vertex has degree . This answers a question of Erd?s, Suen and Winkler for the K₄‐free process. We furthermore deduce an additional structural property: we show that whp the independence number of G is at least , which matches an upper bound obtained by Bohman up to a factor of . Our analysis of the K₄‐free process also yields a new result in Ramsey theory: for a special case of a well‐studied function introduced by Erd?s and Rogers we slightly improve the best known upper bound.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 355‐397, 2014     

Rigid–flexible interactive dynamics modelling approach

Dynamics modelling of multi-body systems composed of rigid and flexible elements is elaborated in this article. The control of such systems is highly complicated due to severe underactuated conditions caused by flexible elements and an inherent uneven non-linear dynamics. Therefore, developing a compact dynamics model with the requirement of limited computations is extremely useful for controller design, simulation studies for design improvement and also practical implementations. In this article, the rigid–flexible interactive dynamics modelling (RFIM) approach is proposed as a combination of Lagrange and Newton–Euler methods, in which the motion equations of rigid and flexible members are separately developed in an explicit closed form. These equations are then assembled and solved simultaneously at each time step by considering the mutual interaction and constraint forces. The proposed approach yields a compact model rather than a common accumulation approach that leads to a massive set of equations in which the dynamics of flexible elements is united with the dynamics equations of rigid members. The proposed RFIM approach is first detailed for multi-body systems with flexible joints, and then with flexible members. Then, to reveal the merits of this new approach, few case studies are presented. A flexible inverted pendulum is studied first as a simple template for lucid comparisons, and next a space free-flying robotic system that contains a rigid main body equipped with two manipulating arms and two flexible solar panels is considered. Modelling verification of this complicated system is vigorously performed using ANSYS and ADAMS programs. The obtained results reveal the outcome accuracy of the new proposed approach for explicit dynamics modelling of rigid–flexible multi-body systems such as mobile robotic systems, while its limited computations provide an efficient tool for controller design, simulation studies and also practical implementations of model-based algorithms.     

Eigenvalue dynamics of a <Emphasis Type="Italic">PT</Emphasis>-symmetric Sturm–Liouville operator and criteria for similarity to a self-adjoint or a normal operator

The dynamics of the eigenvalues of a family of Sturm–Liouville operators with complex integrable PT-symmetric potential on a finite interval is studied. In the model case of the complex Airy operator, a criterion for the similarity of operators in the family to self-adjoint and normal operators is stated and the exceptional parameter values corresponding to multiple eigenvalues are analytically calculated.     

Are You Convinced? Middle‐Grade Students' Evaluations of Mathematical Arguments

Students learn norms of proving by observing teachers generating proofs, engaging in proving, and generalizing features of proofs deemed convincing by an authority, such as a textbook. Students at all grade levels have difficulties generating valid proof; however, little research exists on students' understandings about what makes a mathematical argument convincing prior to more formal instruction in methods of proof. This study investigated middle‐school students' (ages 12–14) evaluations of arguments for a statement in number theory. Students evaluated both an empirical and a general argument in an interview setting. The results show that students tend to prefer empirical arguments because examples enhance an argument's power to show that the statement is true. However, interview responses also reveal that a significant number of students find arguments to be most convincing when examples are supported with an explanation that “tells why” the statement is true. The analysis also examined the alignment of students' reasons for choosing arguments as more convincing along with the strategies they employ to make arguments more convincing. Overall, the findings show middle‐school students' conceptions about what makes arguments convincing are more sophisticated than their performance in generating arguments suggests.     

Conjugacy and dynamics in Thompson’s groups

We give a unified solution to the conjugacy problem for Thompson’s groups \(F, \,T\), and \(V\). The solution uses “strand diagrams”, which are similar in spirit to braids and generalize tree-pair diagrams for elements of Thompson’s groups. Strand diagrams are closely related to piecewise-linear functions for elements of Thompson’s groups, and we use this correspondence to investigate the dynamics of elements of \(F\). Though many of the results in this paper are known, our approach is new, and it yields elegant proofs of several old results.     

Mimetic trust and intra‐organizational network dynamics

Dynamic network models based on the homophily principle are criticized for neglecting organizational context conditions and the impact of role structures on the evolution of intra‐organizational trust networks. Using a neo‐institutional framework it is argued that individuals in competitive environments will attempt to reduce uncertainty about the trustworthiness of potential trustees by imitating the sociometric choice behavior of persons in similar network positions. Three hypotheses are developed. The positional trust hypothesis predicts that individuals tend to trust other actors who occupy a similar network position as themselves. The mimetic trust hypothesis argues that individuals trust actors who are trusted by persons in their own network position. Finally, the advisory trust hypothesis claims that individuals prefer to maintain trust relations to persons occupying a position of third party intermediary than to persons in other positions. An exploratory empirical test of the hypotheses is carried out by reanalyzing a longitudinal network study of the relationships among 25 salesmen in the furniture department of a North American retail sales store during the 1950s. Blockmodelling procedures are used to identify structural positions in the networks, and log‐linear analysis is applied to determine stability of choices within and between structural positions. The results support the mimetic trust and the advisory trust hypothesis.     

Elucidating strategic network dynamics through computational modeling

In this study we describe a comprehensive computational model of network dynamics (COM-NeD) and demonstrate how it may help us better understand and theorize the dynamics of strategic networks. Specifically, we model a population of firms characterized by idiosyncratic resource needs and productive capacities, while having to respond to the demands of external events by establishing ties and receiving needed resources from other firms. Through COM-NeD we investigate a set of established theoretical perspectives that represent distinct strategies for seeking and establishing interfirm ties. Rigorous computational experiments demonstrate the expected behavior of such a system under a broad range of assumptions. The results shed light on the complexity of strategic network dynamics, demonstrating novel interactions of firms’ internal resource capacity, relational search approaches, and external resource growth.

Large deviations for Glauber dynamics of continuous gas

This paper is devoted to the large deviation principles of the Glauber-type dynamics of finite or infinite volume continuous particle systems.We prove that the level-2 empirical process satisfies the large deviation principles in the weak convergence topology,while it does not satisfy the large deviation principles in the T-topology.     

Soliton dynamics for fractional Schrödinger equations

We investigate the soliton dynamics for the fractional nonlinear Schrödinger equation by a suitable modulational inequality. In the semiclassical limit, the solution concentrates along a trajectory determined by a Newtonian equation depending of the fractional diffusion parameter.     

On dynamics of the Sierpiński carpet

We prove that the Sierpiński curve admits a homeomorphism with strong mixing properties. We also prove that the constructed example does not have Bowen's specification property.     

Global dynamics of nonautonomous Hindmarsh–Rose equations

Global dynamics of nonautonomous diffusive Hindmarsh–Rose equations on a three-dimensional bounded domain in neurodynamics is investigated. The existence of a pullback attractor is proved through uniform estimates showing the pullback dissipative property and the pullback asymptotical compactness. Then the existence of pullback exponential attractor is also established by proving the smoothing Lipschitz continuity in a long run of the solution process.     

Global dynamics of the Benoît system

In this paper, we work with a two-degree polynomial differential system in \(\mathbb R ^3\) related with the canard phenomena. We show that this system is completely integrable, and we provide its global phase portrait in the Poincaré ball using the Poincaré–Lyapunov compactification.