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.     

How does within‐host dynamics affect population‐level dynamics? Insights from an immuno‐epidemiological model of malaria

Malaria is one of the most common mosquito‐borne diseases widespread in the tropical and subtropical regions. Few models coupling the within‐host malaria dynamics with the between‐host mosquito‐human dynamics have been developed. In this paper, by adopting the nested approach, a malaria transmission model with immune response of the host is formulated. Applying age‐structured partial differential equations for the between‐host dynamics, we describe the asymptomatic and symptomatic infectious host population for malaria transmission. The basic reproduction numbers for the within‐host model and for the coupled system are derived, respectively. The existence and stability of the equilibria of the coupled model are analyzed. We show numerically that the within‐host model can exhibit complex dynamical behavior, possibly even chaos. In contrast, equilibria in the immuno‐epidemiological model are globally stable and their stabilities are determined by the reproduction number. Increasing the activation rate of the within‐host immune response “dampens” the sensitivity of the population level reproduction number and prevalence to the increase of the within‐host reproduction of the pathogen. From public health perspective this means that treatment in a population with higher immunity has less impact on the population‐level reproduction number and prevalence than in a population with less immunity.     

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.     

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     

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     

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.     

Are the eigenvalues of the B‐spline isogeometric analysis approximation of −Δu = λu known in almost closed form?

We consider the B‐spline isogeometric analysis approximation of the Laplacian eigenvalue problem −Δu = λu over the d‐dimensional hypercube (0,1)^d. By using tensor‐product arguments, we show that the eigenvalue–eigenvector structure of the resulting discretization matrix is completely determined by the eigenvalue–eigenvector structure of the matrix arising from the isogeometric analysis approximation based on B‐splines of degree p of the unidimensional problem . Here, n is the mesh fineness parameter, and the size of is N(n,p) = n + p − 2. In previous works, it was established that the normalized sequence enjoys an asymptotic spectral distribution described by a function e_p(θ), the so‐called spectral symbol. The contributions of this paper can be summarized as follows:

1. the eigenvalues of are arranged in ascending order, ;
2. is a sequence of functions from [0,π] to , which depends only on p;
3. h = 1/n and θ_j,n = jπh for j = 1,…,n; and
4. is the remainder, which satisfies for some constant depending only on α and p. We also provide a proof of this expansion for α = 0 and j = 1,…,N(n,p) −(4p − 2), where 4p − 2 represents a theoretical estimate of the number of outliers .
5. We show through numerical experiments that, for p ≥ 3 and k ≥ 1, there exists a point θ( p,k) ∈ (0,π) such that vanishes on [0,θ( p,k)]. Moreover, as it is suggested by the numerics of this paper, the infimum of θ(p,k) over all k ≥ 1, say y_p, is strictly positive, and the equation holds numerically whenever θ_j,n < θ( p), where θ( p) is a point in (0,y_p] which grows with p.
6. For p ≥ 3, based on the asymptotic expansion in the above item 3, we propose a parallel interpolation–extrapolation algorithm for computing the eigenvalues of , excluding the outliers. The performance of the algorithm is illustrated through numerical experiments. Note that, by the previous item 4, the algorithm is actually not necessary for computing the eigenvalues corresponding to points θ_j,n < θ( p).

     

“断块”与 “板块”

从“断块”学说的观点看,“板块”是断块的一种特殊类型,它是被巨大的深切整个岩石圈的断裂带所围限的断块,是最大一级的断块,称做“岩石圈断块”. 板块学说主要是在对海洋的研究与地球物理研究的基础上发展起来的,它提出了一些有价值的新论点.但大多数研究板块的人,对大陆地区的地质情况还缺乏深入研究,不能充分利用本世纪以来在大陆地质构造研究方面所获得的科学结论与思想方法,这就使板块学说的进一步发展受到了阻碍。 本文先对断块学说的历史发展与现阶段理论作一简单介绍,然后应用断块构造的力学分析与历史分析方法,对板块构造研究中出现的一些主要问题谈一些看法.作者认为板块边界的力学机制主要是受基底锯齿状断裂的活动方式控制的,板块内部的应力分布也主要与板块内部次一级强度不同的断块的复杂活动有关。关于板块运动的驱动力,不但应考虑地球内部的热运动,而且应考虑到重力作用和地球作为一个旋转天体所具有的各种动力学行为.     

Post‐financial meltdown: What do the services industries need from us now?

In 2008 the global economy was rocked by a crisis that began on Wall Street, but quickly spread to Main Street U.S.A., and then to side streets around the world. Statisticians working in the service sector are not immune, with many concerned about losing their jobs. Given this dramatic course of events, how should statisticians respond? What, if anything, can we do to help our struggling organizations survive this recession, in order to prosper in the future? This expository article describes some approaches that we feel can help service industries deal with aftereffects of the financial meltdown. Based on an understanding of current needs of the service industries, we emphasize three approaches in particular: a greater emphasis on statistical engineering relative to statistical science, ‘embedding’ statistical methods and principles into key business processes, and the reinvigoration of Lean Six Sigma to drive immediate, tangible business results. Copyright © 2009 John Wiley & Sons, Ltd.     

Virtual age,is it real? ‐ Discussing virtual age in reliability context

Several settings, where the notion of virtual age is employed, are discussed. We argue that the age reduction imperfect repair modeling is often not sufficiently justified in practice, as it is not possible to execute repair in most of real situations that conforms with this model. On the other hand, a shock‐based virtual age model is suggested and justified via the probabilities of failures on shocks. The new notion of virtual age for degrading items is also introduced. We discuss how to recalculate virtual age after switching from one regime to another. Several examples illustrating different notions of virtual age are presented.     

Quincunx multiresolution analysis for <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>(ℚ<Stack><Subscript>2</Subscript><Superscript>2</Superscript></Stack>)

With an eye on applications in quantum mechanics and other areas of science, much work has been done to generalize traditional analytic methods to p-adic systems. In 2002 the first paper on p-adic wavelets was published. Since then p-adic wavelet sets, multiresolution analyses, and wavelet frames have all been introduced. However, so far all constructions have involved dilations by p. This paper presents the first construction of a p-adic wavelet system with a more general matrix dilation, laying the foundation for further work in this direction.     

Approximation of the classes <Emphasis Type="Italic">C</Emphasis><Stack><Subscript>β</Subscript><Superscript>ψ</Superscript></Stack><Emphasis Type="Italic">H</Emphasis><Subscript>ω</Subscript> by generalized Zygmund sums

We obtain asymptotic equalities for the least upper bounds of approximations by Zygmund sums in the uniform metric on the classes of continuous 2π-periodic functions whose (ψ, β)-derivatives belong to the set H _ω in the case where the sequences ψ that generate the classes tend to zero not faster than a power function.     

传承&#183;探索&#183;创新

金风送爽,丹桂飘香。在这收获的季节里,中国珠算心算协会和世界珠算心算联合会相继在天津召开会员大会,组成新一届理事会和新的领导机构,这标志着我国和世界珠算心算事业将开启崭新的一页!     

The smooth structure set of <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> × <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>

We calculate \({\mathcal{S}^{{\it Diff}}(S^p \times S^q)}\), the smooth structure set of S ^p × S ^q , for p, q ≥ 2 and p + q ≥ 5. As a consequence we show that in general \({\mathcal{S}^{Diff}(S^{4j-1}\times S^{4k})}\) cannot admit a group structure such that the smooth surgery exact sequence is a long exact sequence of groups. We also show that the image of the forgetful map \({\mathcal{S}^{Diff}(S^{4j}\times S^{4k}) \rightarrow \mathcal{S}^{Top}(S^{4j}\times S^{4k})}\) is not in general a subgroup of the topological structure set.     

On the Crossing Number of <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">m</Emphasis></Subscript> □ <Emphasis Type="Italic">P</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

Crossing numbers of graphs are in general very difficult to compute. There are several known exact results on the crossing number of the Cartesian products of paths, cycles or stars with small graphs. In this paper we study cr(K_m □ P_n), the crossing number of the Cartesian product K_m □ P_n. We prove that for m ≥ 3,n ≥ 1 and cr(K_m □ P_n)≥ (n − 1)cr(K_m+2 − e) + 2cr(K_m+1). For m≤ 5, according to Klešč, Jendrol and Ščerbová, the equality holds. In this paper, we also prove that the equality holds for m = 6, i.e., cr(K₆ □ P_n) = 15n + 3. Research supported by NFSC (60373096, 60573022).     

Regular<Emphasis Type="Italic">n</Emphasis>-simplices in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> with vertices in ℤ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper theI andII regularn-simplices are introduced. We prove that the sufficient and necessary conditions for existence of anI regularn-simplex in ℝ ⁿ are that ifn is even thenn = 4m(m + 1), and ifn is odd thenn = 4m + 1 with thatn + 1 can be expressed as a sum of two integral squares orn = 4m - 1, and that the sufficient and necessary condition for existence of aII regularn-simplex in ℝ ⁿ isn = 2m ² - 1 orn = 4m(m + 1)(m ∈ ℕ). The connection between regularn-simplex in ℝ ⁿ and combinational design is given.     

The number of S-precomplete classes in the functional system <Emphasis Type="Italic">P</Emphasis><Stack><Subscript><Emphasis Type="Italic">κ</Emphasis></Subscript><Superscript><Emphasis Type="Italic">τ</Emphasis></Superscript></Stack>

The problem on the number of precomplete classes in the functional system P _κ ^τ is considered, elements of P _κ ^τ are deterministic S-functions defined on words of length τ composed from letters of an alphabet of cardinality κ. An asymptotics for the number of S-precomplete classes in P _κ ^τ is obtained for arbitrary fixed κ and τ tending to infinity.     

Equitable total coloring of <Emphasis Type="Italic">F</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> ⋁ <Emphasis Type="Italic">W</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

The minimum number of total independent partition sets of V ∪ E of graph G(V,E) is called the total chromatic number of G denoted by χ _t (G). If the difference of the numbers of any two total independent partition sets of V ∪ E is no more than one, then the minimum number of total independent partition sets of V ∪ E is called the equitable total chromatic number of G, denoted by χ _et (G). In this paper, we obtain the equitable total chromatic number of the join graph of fan and wheel with the same order. Supported by the National Natural Science Foundation of China (No. 10771091).     

On Dirichlet Type Spaces <Emphasis Type="Italic">A</Emphasis><Stack><Subscript><Emphasis Type="Italic">ω</Emphasis></Subscript><Superscript>2</Superscript></Stack> Over the Half–Plane

Some extensions of the results of the first author related with the Hilbert spaces A _ω,0 ² of functions holomorphic in the half–plane are proved. Some new Hilbert spaces A _ω ² of Dirichlet type are introduced, which are included in the Hardy space H2 over the half–plane. Several results on representations, boundary properties, isometry, interpolation, biorthogonal systems and bases are obtained for the spaces A _ω ² ? H₂.