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The aim of this paper is to discuss the main result in the paper by D.Y. Gao and X. Lu [On the extrema of a nonconvex functional with double-well potential in 1D, Z. Angew. Math. Phys. (2016) 67:62]. More precisely we provide a detailed study of the problem considered in that paper, pointing out the importance of the norm on the space $C^{1}[a,b]$; because no norm (topology) is mentioned on $C^{1}[a,b]$ we look at it as being a subspace of $W^{1,p}(a,b)$ for $p\in [1,\infty]$ endowed with its usual norm. We show that the objective function has not local extrema with the mentioned constraints for $p\in [1,4)$, and has (up to an additive constant) only a local maximizer for $p=\infty$, unlike the conclusion of the main result of the discussed paper where it is mentioned that there are (up to additive constants) two local minimizers and a local maximizer. We also show that the same conclusions are valid for the similar problem treated in the preprint by X. Lu and D.Y. Gao [On the extrema of a nonconvex functional with double-well potential in higher dimensions, arXiv:1607.03995]., 12 pages; in this version we added the forgotten condition $F(x) \ne 0$ for $x\in (a,b)$ on page 3

In this paper, we consider weak, directional and strong matrix majorizations. Namely, for square matrices A and B of the same size we say that A is weakly majorized by B if there is a row stochastic matrix X such that A = XB. Further, A is strongly majorized by B if there is a doubly stochastic matrix X such that A = XB. Finally, A is directionally majorized by B if Ax is majorized by Bx for any vector x where the usual vector majorization is used. We introduce the notion of majorization of matrix tuples which is defined as a natural generalization of matrix majorizations: for a chosen type of majorization we say that one tuple of matrices is majorized by another tuple of the same size if every matrix of the “smaller” tuple is majorized by a matrix in the same position in the “bigger” tuple. We say that a linear operator preserves majorization if it maps ordered pairs to ordered pairs and the image of the smaller element does not exceed the image of the bigger one. This paper contains a full characterization of linear operators that preserve weak, strong or directional majorization of tuples of matrices and linear operators that map tuples that are ordered with respect to strong majorization to tuples that are ordered with respect to directional majorization. We have shown that every such operator preserves respective majorization of each component. For all types of majorization we provide counterexamples that demonstrate that the inverse statement does not hold, that is if majorization of each component is preserved, majorization of tuples may not.

Let E∗ G be a crossed product of a division ring E and a locally indicable group G. Hughes showed that up to E∗ G-isomorphism, there exists at most one Hughes-free division E∗G-ring. However, the existence of a Hughes-free division E∗ G-ring DE∗G for an arbitrary locally indicable group G is still an open question. Nevertheless, DE∗G exists, for example, if G is amenable or G is bi-orderable. In this paper we study, whether DE∗G is the universal division ring of fractions in some of these cases. In particular, we show that if G is a residually-(locally indicable and amenable) group, then there exists DE[G] and it is universal. In Appendix we give a description of DE[G] when G is a RFRS group, This paper is partially supported by the Spanish Ministry of Science and Innovation through the grant MTM2017-82690-P and the “Severo Ochoa Programme for Centres of Excellence in R&D” (CEX2019-000904-S4). I would like to thank Dawid Kielak and an anonymous referee for useful suggestions and comments

In this paper, the compositional structure of a finite group G is investigated, which has the Sylow 2-subgroup that is permutable with some non p-nilpotent biprimary subgroups, which contain the Sylow р-subgroup of G for all odd simple divisors of the р order of the group G, and such biprimary subgroups are taken one by one for each odd р, and mark the set SB(G). In this work, the existence of the subset SB(G)* in SB(G) is proved, which consists of р-closed subgroups. The main result of this paper is as follows: if the Sylow 2-subgroup of the group G is permutable with all subgroups SB(G)*, then G may have simple non-abelian compositional factors only of L2(7) type, if p > 3, and additionally of L2(3f) type, f = 3a, a ≥ 1, if p = 3.

An algebraic parameter of a graph – a difference between its maximum degree and its spectral radius is considered in this paper. It is well known that this graph parameter is always nonnegative and represents some measure of deviation of a graph from its regularity. In the last two decades, many papers have been devoted to the study of this parameter. In particular, its lower bound depending on the graph order and diameter was obtained in 2007 by mathematician S. M. Cioabă. In 2017 when studying the upper and the lower bounds of this parameter, M. R. Oboudi made a conjecture that the lower bound of a given parameter for an arbitrary graph is the difference between a maximum degree and a spectral radius of a chain. This is very similar to the analogous statement for the spectral radius of an arbitrary graph whose lower boundary is also the spectral radius of a chain. In this paper, the above conjecture is confirmed for some graph classes.

We present a theorem in the present paper on an approximation to functions of a Holder class on a countable union of segments lying on a positive ray by entire functions of order 1/2 bounded on this ray. Problems related to the approximation of entire functions on subsets of the semiaxis by using entire functions of order 1/2 are closely related to problems of approximating functions on subsets of the whole axis using entire functions of exponential type but have their own specifics. We consider segments In in this paper with lengths of order n such that the distance between In and In + 1 is also of order n. Cases of the whole semiaxis or the union of finitely many segments and a ray were considered in previous papers. As for the problem of approximating functions of the Holder class on the union of a countable set of segments on the whole axis, it turns out that the approximation rate at neighborhoods of the segment endpoints as the type of the functions increases is higher than that in a neighborhood of their midpoints.

We continue in this paper to study one of the models of a discrete analog of the Renyi problem, also known as the parking problem. Suppose that n and i are integers satisfying n ≥ 0 and 0 ≤ i ≤ n – 1. We place an open interval (i, i + 1) in the segment [0, n] with i being a random variable taking values 0, 1, 2, …, n – 1 with equal probability for all n ≥ 2. If n < 2, then we say that the interval does not fit. After placing the first interval, two free segments [0, i] and [i + 1, n] are formed and independently filled with intervals of unit length according to the same rule, and so on. At the end of the process of filling the segment [0, n] with intervals of unit length, the distance between any two adjacent unit intervals does not exceed one. Suppose now that Xn is the number of unit intervals placed. In our earlier work published in 2018, we studied the asymptotic behavior of the first moments of random variable Xn. In contrast to the classical case, the exact expressions for the expectation, variance, and third central moment were obtained. The asymptotic behavior of all central moments of random variable Xn is investigated in this paper and the asymptotic normality for Xn is proved.

This is the first of a series of two papers in which we present a solution to Manin’s Real Multiplication program (Manin in: Laudal and Piene (eds) The Legacy of Niels Henrik Abel, Springer, Berlin, 2004) —an approach to Hilbert’s 12th problem for real quadratic extensions of $$\mathbb Q$$ —in positive characteristic, using quantum analogs of the modular invariant and the exponential function. In this first paper, we treat the problem of Hilbert class field generation. If $$k=\mathbb F_{q}(T)$$ and $$k_{\infty }$$ is the analytic completion of k, we introduce the quantum modular invariant $$\begin{aligned} j^\mathrm{qt}: k_{\infty }\multimap k_{\infty } \end{aligned}$$ as a multivalued, discontinuous modular invariant function. Then if $$K=k(f)\subset k_{\infty }$$ is a real quadratic extension of k and f is a fundamental unit, we show that the Hilbert class field $$H_{\mathcal {O}_{K}}$$ (associated to $$\mathcal {O}_{K}=$$ integral closure of $$\mathbb F_{q}[T]$$ in K) is generated over K by the product of the multivalues of $$j^\mathrm{qt}(f)$$ .

This paper introduces self-similarity inherent in planar Milich-Jennings centered flip graphs derived from the Narayana sequence. We show that self-similarity found in a Narayana sequence yields a connected spanning subgraph with a centered flip. This paper has several main results (1) Every Narayana sequence constructs a flip graph, (2) Every Narayana sequence is self-similar and (3) Every Pascal 3-triangle has a free group presentation.

The energy of a graph which is denoted by is defined to be the sum of the absolute values of the eigenvalues of its adjacency matrix. In this paper we present the concepts of conjugacy class graph of dihedral groups and introduce the general formula for the energy of the conjugacy class graph of dihedral groups. The energy of any dihedral group of order in different cases, depends on the parity of is proved in this paper. Also we introduce the general formula for the conjugacy class graph of generalized quaternion groups and quasidihedral groups.

Let N be a module in category $$\sigma [M]$$ . N is called generalized $$\delta$$ –M-small (briefly G– $$\delta$$ –M) if, $$N \subseteq \delta (L)$$ for some $$L \in \sigma [M]$$ . In this paper we characterize G– $$\delta$$ –M modules and get some suitable results related to this kind of modules. We will show that a module $$N \in \sigma [M]$$ is G– $$\delta$$ –M if and only if $$N \subseteq \delta (\hat{N})$$ , where $$\hat{N}$$ is the M-injective envelope of N in $$\sigma [M]$$ . Also we prove that if there is no non-zero G– $$\delta$$ –M module, then M is cosemisimple and by giving an example we show that the converse need not be true. The relation between G– $$\delta$$ –M modules and some other classes of modules would be investigated in this paper. The torsion theory cogenerated by this class of modules will be introduced and studied in this paper. For a module $$N \in \sigma [M]$$ we show that $$N = \mathrm{{Re}}_{GD[M]}(N)$$ iff for every nonzero homomorphism $$f : N \longrightarrow K$$ in $$\sigma [M],$$ $$\mathrm{{Im}}(f) \not \subseteq \delta (K)$$ iff $$\Delta _{\delta }(N,A) = 0,$$ for all $$A \in \sigma [M]$$ .

In this paper, we investigate the factor properties and gap sequence of the Tribonacci sequence, the fixed point of the substitution σ( a , b , c ) = ( ab , ac , a ). Let ω p be the p-th occurrence of ω and G p ( ω ) be the gap between ω p and ω p +1 . We introduce a notion of kernel for each factor ω , and then give the decomposition of the factor ω with respect to its kernel. Using the kernel and the decomposition, we prove the main result of this paper: for each factor ω , the gap sequence { G p ( ω )} p ≤1 is the Tribonacci sequence over the alphabet { G 1 ( ω ), G 2 ( ω ), G 4 ( ω )}, and the expressions of gaps are determined completely. As an application, for each factor ω and p ∈ ℕ , we determine the position of ω p . Finally we introduce a notion of spectrum for studying some typical combinatorial properties, such as power, overlap and separate of factors.

Complexity may be one of the most important measurements for analysing time series data; it covers or is at least closely related to different data characteristics within nonlinear system theory. This paper provides a comprehensive literature review examining the complexity testing techniques for time series data. According to different features, the complexity measurements for time series data can be divided into three primary groups, i.e., fractality (mono- or multi-fractality) for self-similarity (or system memorability or long-term persistence), methods derived from nonlinear dynamics (via attractor invariants or diagram descriptions) for attractor properties in phase-space, and entropy (structural or dynamical entropy) for the disorder state of a nonlinear system. These estimations analyse time series dynamics from different perspectives but are closely related to or even dependent on each other at the same time. In particular, a weaker self-similarity, a more complex structure of attractor, and a higher-level disorder state of a system consistently indicate that the observed time series data are at a higher level of complexity. Accordingly, this paper presents a historical tour of the important measures and works for each group, as well as ground-breaking and recent applications and future research directions.

For a finite group $G$, let $\H_{g,G,\xi}$ be the stack of admissible $G$-covers $C\to D$ of stable curves with ramification data $\xi$, $g(C)=g$ and $g(D)=g'$. There are source and target morphisms $\phi\colon \H_{g,G,\xi}\to \M_{g,r}$ and $\delta\colon \H_{g,G,\xi}\to \M_{g',b}$, remembering the curves $C$ and $D$ together with the ramification or branch points of the cover respectively. In this paper we study admissible cover cycles, i.e. cycles of the form $\phi_* [\H_{g,G,\xi}]$. Examples include the fundamental classes of the loci of hyperelliptic or bielliptic curves $C$ with marked ramification points. The two main results of this paper are as follows: Firstly, for the gluing morphism $\xi_A\colon \M_A\to \M_{g,r}$ associated to to a stable graph $A$ we give a combinatorial formula for the pullback $\xi^*_A \phi_*[\H_{g,G,\xi}]$ in terms of spaces of admissible $G$-covers and $\psi$ classes. This allows us to describe the intersection of the cycles $\phi_*[\H_{g,G,\xi}]$ with tautological classes. Secondly, the pull-push $\delta_*\phi^*$ sends tautological classes to tautological classes and we also give a combinatorial description of this map in terms of standard generators of the tautological rings. We show how to use the pullbacks to algorithmically compute tautological expressions for cycles of the form $\phi_* [\H_{g,G,\xi}]$. In particular, we compute the classes $[\Hyp_5]$ and $[\Hyp_6]$ of the hyperelliptic loci in $\M_5$ and $\M_6$ and the class $[\B_4]$ of the bielliptic locus in $\M_4$., Comment: 69 pages, comments very welcome

The main object in this paper is a certain rational convex polytope whose lattice points give a polyhedral realization of a highest weight crystal basis. This is also identical to a Newton-Okounkov body of a flag variety, and it gives a toric degeneration. In this paper, we prove that a specific class of this polytope is given by Kiritchenko's Demazure operators on polytopes. This implies that polytopes in this class are all lattice polytopes. As an application, we give a sufficient condition for the corresponding toric variety to be Gorenstein Fano., Comment: 21 pages

This paper studies the problem of finding optimal parameters for a Poincare section used for determining the type of behavior of a time series: a deterministic or stochastic one. To reach that goal optimization algorithms are coupled with the Poincare & Higuchi (P&H) method, which calculates the Higuchi dimension using points obtained by performing a Poincare section of a certain attractor. The P&H method generates distinctive patterns that can be used for determining if a given attractor is produced by a deterministic or a stochastic system, but this method is sensitive to the parameters of the Poincare section. Patterns generated by the P&H method can be characterized using numerical measures which in turn can be used for finding such parameters for the Poincare section for which the patterns produced by the P&H method are the most prominent. This paper studies several approaches to parameterization of the Poincare section. Proposed approaches are tested on twelve time series, six produced by deterministic chaotic systems and six generated randomly. The obtained results show, that finding good parameters of the Poincare section is important for determining the type of behavior of a time series. Among the tested methods the evolutionary algorithm was able to find the best Poincare sections for use with the P&H method.

The aim of the present paper is to introduce a generalized quantum cluster character, which assigns to each object V of a finitary Abelian category C over a finite field FF_q and any sequence ii of simple objects in C the element X_{V,ii} of the corresponding algebra P_{C,ii} of q-polynomials. We prove that if C was hereditary, then the assignments V-> X_{V,ii} define algebra homomorphisms from the (dual) Hall-Ringel algebra of C to the P_{C,ii}, which generalize the well-known Feigin homomorphisms from the upper half of a quantum group to q-polynomial algebras. If C is the representation category of an acyclic valued quiver (Q,d) and ii=(ii_0,ii_0), where ii_0 is a repetition-free source-adapted sequence, then we prove that the ii-character X_{V,ii} equals the quantum cluster character X_V introduced earlier by the second author in [29] and [30]. Using this identification, we deduce a quantum cluster structure on the quantum unipotent cell corresponding to the square of a Coxeter element. As a corollary, we prove a conjecture from the joint paper [5] of the first author with A. Zelevinsky for such quantum unipotent cells. As a byproduct, we construct the quantum twist and prove that it preserves the triangular basis introduced by A. Zelevinsky and the first author in [6]., AMS LaTeX, 46 pages, a reference added, to appear in Selecta Mathematica

This paper deals with an initial-boundary value problem for the chemotaxis system $$\left\{\begin{array}{ll} u_t = \nabla \cdot (D (u) \nabla u)- \nabla \cdot (u \nabla v), \quad & x\in \Omega, \quad t > 0, \\ v_t= \Delta v-uv, \quad & x \in \Omega, \quad t > 0, \end{array}\right.$$ under homogeneous Neumann boundary conditions in a convex smooth bounded domain \({\Omega\subset \mathbb{R}^n}\) with \({n\geq3}\), where the diffusion function D(u) satisfying $$\begin{array}{ll}D(u)\geq c_Du^{m-1}\quad\text{for all}\,\,u > 0 \end{array}$$ with some cD > 0 and m > 1. The main goal of this paper was to extend a previous result on global existence of solutions by Wang et al. (Z Angew Math Phys 65:1137–1152, 2014) under the condition that \({m > 2-\frac{2}{n}}\) can be relaxed to \({m > 2-\frac{6}{n+4}}\).

The aim of this article is to introduce a new definition of domination number in graphs called hn-domination number denoted by . This paper presents some properties which show the concepts of connected and independent hn-domination. Furthermore, some bounds of these parameters are determined, specifically, the impact on hn-domination parameter is studied thoroughly in this paper when a graph is modified by deleting or adding a vertex or deleting an edge.

In this paper, we provide an analytical frame work for the following multi-valued parabolic variational inequality in a cylindrical domain \({Q = \Omega \times (0, \tau)}\) : Find \({{u \in K}}\) and an \({{\eta \in L^{p'}(Q)}}\) such that $$\eta \in f(\cdot,\cdot,u), \quad \langle u_t + Au, v - u\rangle + \int_Q \eta (v - u)\,{\rm d}x{\rm d}t \ge 0, \quad \forall \, v \in K,$$ where \({{K \subset X_0 = L^p(0,\tau;W_0^{1,p}(\Omega))}}\) is some closed and convex subset, A is a time-dependent quasilinear elliptic operator, and the multi-valued function \({{s \mapsto f(\cdot,\cdot,s)}}\) is assumed to be upper semicontinuous only, so that Clarke’s generalized gradient is included as a special case. Thus, parabolic variational–hemivariational inequalities are special cases of the problem considered here. The extension of parabolic variational–hemivariational inequalities to the general class of multi-valued problems considered in this paper is not only of disciplinary interest, but is motivated by the need in applications. The main goals are as follows. First, we provide an existence theory for the above-stated problem under coercivity assumptions. Second, in the noncoercive case, we establish an appropriate sub-supersolution method that allows us to get existence, comparison, and enclosure results. Third, the order structure of the solution set enclosed by sub-supersolutions is revealed. In particular, it is shown that the solution set within the sector of sub-supersolutions is a directed set. As an application, a multi-valued parabolic obstacle problem is treated.

The subject of this paper is to give the description, up to topological conjugacy, of possible minimal sets of triangular maps of the square of type 2 ∞ . In [4] , we give a general method allowing to embed any zero-dimensional almost 1–1 extension of the dyadic odometer (in particular any dyadic Toeplitz system) as a minimal set of a triangular map of this type. In this paper we present a method (a combination of that described in [4] with one introduced in [1] ) of similarly embedding a special class of zero-dimensional almost 2–1 extensions of the odometer. We conjecture that these two embedding theorems exhaust all possibilities for nonperiodic minimal sets. The paper was inspired by the last unsolved problem in the Sharkovski classification program of triangular maps: does there exist a triangular map with positive entropy attained on the set of uniformly recurrent points but with entropy zero on the set of regularly recurrent points. The paper answers this question positively, concluding the program.

In an earlier paper (arxiv:1108.4292) we introduced a new concept of dimension for metric spaces, the so called topological Hausdorff dimension. For a compact metric space $K$ let $\dim_{H}K$ and $\dim_{tH} K$ denote its Hausdorff and topological Hausdorff dimension, respectively. We proved that this new dimension describes the Hausdorff dimension of the level sets of the generic continuous function on $K$, namely $\sup{\dim_{H}f^{-1}(y) : y \in \mathbb{R}} = \dim_{tH} K - 1$ for the generic $f \in C(K)$, provided that $K$ is not totally disconnected, otherwise every non-empty level set is a singleton. We also proved that if $K$ is not totally disconnected and sufficiently homogeneous then $\dim_{H}f^{-1}(y) = \dim_{tH} K - 1$ for the generic $f \in C(K)$ and the generic $y \in f(K)$. The most important goal of this paper is to make these theorems more precise. As for the first result, we prove that the supremum is actually attained on the left hand side of the first equation above, and also show that there may only be a unique level set of maximal Hausdorff dimension. As for the second result, we characterize those compact metric spaces for which for the generic $f\in C(K)$ and the generic $y\in f(K)$ we have $\dim_{H} f^{-1}(y)=\dim_{tH}K-1$. We also generalize a result of B. Kirchheim by showing that if $K$ is self-similar then for the generic $f\in C(K)$ for every $y\in \inter f(K)$ we have $\dim_{H} f^{-1}(y)=\dim_{tH}K-1$. Finally, we prove that the graph of the generic $f\in C(K)$ has the same Hausdorff and topological Hausdorff dimension as $K$., 20 pages

As known, the method to obtain a sequence space by using convergence field of an infinite matrix is an old method in the theory of sequence spaces. However, the study of convergence field of an infinite matrix in the space of almost convergent sequences is so new (see [15]). The purpose of this paper is to introduce the new spaces and consisting of all sequences whose Cesaro transforms of order one are in the spaces f and f0, respectively. Also, in this paper, we show that and are linearly isomorphic to the spaces f and f0, respectively. The β- and γ-duals of the spaces and are computed. Furthermore, the classes ( : μ) and (μ: ) of infinite matrices are characterized for any given sequence space μ, and determined the necessary and sufficient conditions on a matrix A to satisfy BC-core(Ax) ⊆ K-core(x), K-core(Ax) ⊆ BC-core(x), BC-core(Ax) ⊆ BC-core(x), BC-core(Ax) ⊆ st-core(x) for all x ∈ l∈.

Umeda et al. (Jpn J Appl Math 1:435–457, 1984) considered a rather general class of symmetric hyperbolic–parabolic systems: $$A^{0}z_{t}+\sum_{j=1}^{n}A^{j}z_{x_{j}}+Lz=\sum_{j,k=1}^{n}B^{jk}z_{x_{j}x_{k}}$$ and showed optimal decay rates with certain dissipative assumptions. In their results, the dissipation matrices $${L}$$ and $${B^{jk}(j,k=1,\ldots,n)}$$ are both assumed to be real symmetric. So far there are no general results in case that $${L}$$ and $${B^{jk}}$$ are not necessarily symmetric, which is left open now. In this paper, we investigate compressible Navier–Stokes–Maxwell (N–S–M) equations arising in plasmas physics, which is a concrete example of hyperbolic–parabolic composite systems with non-symmetric dissipation. It is observed that the Cauchy problem for N–S–M equations admits the dissipative mechanism of regularity-loss type. Consequently, extra higher regularity is usually needed to obtain the optimal decay rate of $${L^{1}({\mathbb{R}}^3)}$$ - $${L^2({\mathbb{R}}^3)}$$ type, in comparison with that for the global-in-time existence of smooth solutions. In this paper, we obtain the minimal decay regularity of global smooth solutions to N–S–M equations, with aid of $${L^p({\mathbb{R}}^n)}$$ - $${L^{q}({\mathbb{R}}^n)}$$ - $${L^{r}({\mathbb{R}}^n)}$$ estimates. It is worth noting that the relation between decay derivative orders and the regularity index of initial data is firstly found in the optimal decay estimates.

This paper is concerned with the asymptotic behavior on $${\partial\Omega}$$ of boundary blow-up solutions to semilinear elliptic equations $$\left\{\begin{array}{ll} \Delta u=b(x)f(u),~~ &x\in \Omega, \\ u(x)=\infty, ~~ &x\in\partial\Omega,\end{array} \right.$$ where b(x) is a nonnegative function on $${\Omega}$$ and may vanish on $${\partial\Omega}$$ at a very degenerate rate; f is nonnegative function on [0,∞) and normalized regularly varying or rapidly varying at infinity. The main feature of this paper is to establish a unified and explicit asymptotic formula when the function f is normalized regularly varying or grows faster than any power function at infinity. The effect of the mean curvature of the nearest point on the boundary in the second-order approximation of the boundary blow-up solution is also discussed. Our analysis relies on suitable upper and lower solutions and the Karamata regular variation theory.

Evolution in finite populations is often modelled using the classical Moran process. Over the last 10 years, this methodology has been extended to structured populations using evolutionary graph theory. An important question in any such population is whether a rare mutant has a higher or lower chance of fixating (the fixation probability) than the Moran probability, i.e. that from the original Moran model, which represents an unstructured population. As evolutionary graph theory has developed, different ways of considering the interactions between individuals through a graph and an associated matrix of weights have been considered, as have a number of important dynamics. In this paper, we revisit the original paper on evolutionary graph theory in light of these extensions to consider these developments in an integrated way. In particular, we find general criteria for when an evolutionary graph with general weights satisfies the Moran probability for the set of six common evolutionary dynamics.

Remarkably benign looking logistic transformations x n +1 = r x n (1 − x n ) for choosing x 0 between 0 and 1 and 0 r ⩽ 4 have found a celebrated place in chaos, fractals and discrete dynamics. The strong physical meaning of Mandelbrot and Julia sets is broadly accepted and nicely connected by Christian Beck [Beck C. Physical meaning for Mandelbrot and Julia sets. Physica D 1999;125(3–4):171–182. Zbl0988.37060] to the complex logistic maps, in the former case, and to the inverse complex logistic map, in the latter case. The purpose of this paper is to study the bounded behavior of the complex logistic map using superior iterates and generate fractals from the same. The analysis in this paper shows that many beautiful properties of the logistic map are extendable for a larger value of r .

This is the first paper in a general program to automate skein theoretic arguments. In this paper, we study skein theoretic invariants of planar trivalent graphs. Equivalently, we classify trivalent categories, which are nondegenerate pivotal tensor categories over $\mathbb C$ generated by a symmetric self-dual simple object $X$ and a rotationally invariant morphism $1 \rightarrow X \otimes X \otimes X$. Our main result is that the only trivalent categories with $\dim \operatorname{Hom}(1, X^{\otimes n})$ bounded by $1,0,1,1,4,11,40$ for $0 \leq n \leq 6$ are quantum $SO(3)$, quantum $G_2$, a one-parameter family of free products of certain Temperley-Lieb categories (which we call ABA categories), and the $H3$ Haagerup fusion category. We also prove similar results where the map $1 \rightarrow X^{\otimes 3}$ is not rotationally invariant, and we give a complete classification of nondegenerate braided trivalent categories with dimensions of invariant spaces bounded by $1,0,1,1,4$. Our main techniques are a new approach to finding skein relations which can be easily automated using Gr\"obner bases, and evaluation algorithms which use the discharging method developed in the proof of the $4$-color theorem., Comment: 50 pages; identical to published version

In this paper, we study the following Neumann–Robin boundary value problem - ( ϕ p ( u ′ ( x ) ) ) ′ = λ f ( u ( x ) ) , x ∈ ( 0 , 1 ) , u ′ ( 0 ) = 0 , u ′ ( 1 ) + α u ( 1 ) = 0 , whereα ∈ R, λ > 0 are parameters and p > 1, and p ′ = p p - 1 is the conjugate exponent of p and ϕp(x): = ∣x∣p−2x for all x ∈ R where (ϕp(u′))′ is the one dimensional p-Laplacian and f ∈ C2[0, ∞) such that f(0) 0, and also f is increasing and concave up. We shall investigate the existence and multiplicity of nonnegative solutions. Note that in this paper, we shall establish our existence results by using the quadrature method.

Motivated by Mandelbrot's [The Fractal Geometry of Nature, Freeman, San Francisco, 1983] idea of referring to lacunarity of Cantor sets in terms of departure from translation invariance, we study the properties of these translation sets and show how they can be used for a classification purpose. This first paper of a series of two will be devoted to set up the fundamental properties of Hausdorff measures of those intersection sets. Using the triadic expansion of the shifting number, we determine the fractal structure of intersection of triadic Cantor sets with their translates. We found that the Hausdorff measure of these sets forms a discrete spectrum whose non-zero values come only from those shifting numbers with a finite triadic expansion. We characterize this set of shifting numbers by giving a partition expression of it and the steps of its construction from a fundamental root set. Finally, we prove that intersection of Cantor sets with their translates verify a measure-conservation law with scales. The second paper will take advantage of the properties exposed here in order to utilize them in a classification context. Mainly, it will deal with the use of the discrete spectrum of measures to distinguish two Cantor-like sets of the same fractal dimension.

Let $$G=(V,E)$$ be a simple graph. A function $$f:V\longrightarrow \{-1,1\}$$ is a signed dominating function if for every vertex $$v \in V$$ , the closed neighborhood of v contains more vertices with function value 1 than with $$-1$$ . The weight of a function f is $$\omega (f)= \sum _{v \in V} f(v)$$ . The signed domination number of G, $$\gamma _s(G)$$ , is the minimum weight of a signed dominating function on G. A signed dominating function of weight $$\gamma _s(G)$$ is called $$\gamma _s(G)$$ -function. A $$\gamma _s(G)$$ -function can also be represented by a set of ordered pairs $$S_f=\{(v, f(v)) :v\in V\}$$ . A subset T of $$S_f$$ is called a forcing subset of $$S_f$$ if $$S_f$$ is the unique extension of T to a $$\gamma _s(G)$$ -function. The forcing signed domination number of $$S_f$$ $$f(S_f, \gamma _s) = {\rm min} \{|T|: T$$ is a forcing subset of $$S_f\}$$ and the forcing signed domination number of G, $$f(G,\gamma _s)$$ , is defined by $$f(G,\gamma _s) = {\rm min} \{f(S_f,\gamma _s): S_f$$ is a $$\gamma _s(G)$$ -function $$\}$$ . In this paper, we deal with the signed domination number of several classes of graph. Also the forcing signed domination number of some graphs are determined.

This paper concerns the reconstruction of a function f in the Hardy space of the unit disc $$\mathbb{D}$$ by using a sample value f (a) and certain n-intensity measurements $$\left| {\left\langle {f,{E_{{a_1} \cdots {a_n}}}} \right\rangle} \right|$$ , where a1, ⋯, $${a_n} \in \mathbb{D}$$ , and $${E_{{a_1} \cdots {a_n}}}$$ is the n-th term of the Gram-Schmidt orthogonalization of the Szego kernels $${k_{{a_1}}}, \cdots ,{k_{{a_n}}}$$ , or their multiple forms. Three schemes are presented. The first two schemes each directly obtain all the function values f (z). In the first one we use Nevanlinna’s inner and outer function factorization which merely requires the 1-intensity measurements equivalent to know the modulus |f (z)|. In the second scheme we do not use deep complex analysis, but require some 2- and 3-intensity measurements. The third scheme, as an application of AFD, gives sparse representation of f (z) converging quickly in the energy sense, depending on consecutively selected maximal n-intensity measurements $$\left| {\left\langle {f,{E_{{a_1} \cdots {a_n}}}} \right\rangle} \right|$$ .

TCLUST is a method in statistical clustering technique which is based on modification of trimmed k-means clustering algorithm. It is called “crisp” clustering approach because the observation is can be eliminated or assigned to a group. TCLUST strengthen the group assignment by putting constraint to the cluster scatter matrix. The emphasis in this paper is to restrict on the eigenvalues, λ of the scatter matrix. The idea of imposing constraints is to maximize the log-likelihood function of spurious-outlier model. A review of different robust clustering approach is presented as a comparison to TCLUST methods. This paper will discuss the nature of TCLUST algorithm and how to determine the number of cluster or group properly and measure the strength of group assignment. At the end of this paper, R-package on TCLUST implement the types of scatter restriction, making the algorithm to be more flexible for choosing the number of clusters and the trimming proportion.

DAHA-Jones polynomials of torus knots T(r, s) are studied systematically for reduced root systems and in the case of $$C^\vee C_1$$ . We prove the polynomiality and evaluation conjectures from the author’s previous paper on torus knots and extend the theory by the color exchange and further symmetries. The DAHA-Jones polynomials for $$C^\vee C_1$$ depend on five parameters. Their surprising connection to the DAHA-superpolynomials (type A) for the knots $$T(2p+1,2)$$ is obtained, a remarkable combination of the color exchange conditions and the author’s duality conjecture (justified by Gorsky and Negut). The uncolored DAHA-superpolynomials of torus knots are expected to coincide with the Khovanov–Rozansky stable polynomials and the superpolynomials defined via rational DAHA and/or in terms of certain Hilbert schemes. We end the paper with certain arithmetic counterparts of DAHA-Jones polynomials for the absolute Galois group in the case of $$C^\vee C_1$$ , developing the author’s previous results for $$A_1$$ .

This is a first of our papers devoted to "noncommutative topology and graph theory". Its origin is the paper math.QA/0002238 by I. Gelfand, V. Retakh, and R.L. Wilson where a new class of noncommutative algebras $Q_n$ was introduced. The algebra $Q_n$ is closely related to factorizations of a generic polynomial of degree $n$ over a division algebra into linear factors., Comment: 6 pages; amstex

Some definitions such as m-chain recurrent set, weakly gradient flow and generalized Morse decomposition for a flow defined on a topological space are introduced in this paper. Some conclusions, include the chain recurrent set contain the m-chain recurrent set; the m-chain recurrent set contain the non-wandering set, are proved. In some flows the non-wandering set is proper subset of the m-chain recurrent set; in the meantime the m-chain recurrent set is proper subset of the chain recurrent set. Moreover some criterions for the existence of trajectories joining singular points and a necessary and sufficient condition of the weakly gradient flow are also given here. At last the generalized Morse decomposition of the invariant set are discussed in the paper.

This paper is concerned with anisotropic invariants of N symmetric second–order tensors relative to crystal classes and quasi–crystal classes as subgroups of the cylindrical group D α h . Simple irreducible functional bases for the latter are derived in unified forms by means of a simple procedure based upon recent development in isotropic extension of anisotropic functions. As a particular case, a new, simple derivation is available for the corresponding results on anisotropic invariants of a single symmetric second–order tensor presented in a paper by Xiao.

Algebraic numbers are the roots of integer polynomials. Each algebraic number α is characterized by its minimal polynomial Pα that is a polynomial of minimal positive degree with integer coprime coefficients, α being its root. The degree of α is the degree of this polynomial, and the height of α is the maximum of the absolute values of the coefficients of this polynomial. In this paper we consider the distribution of algebraic numbers α whose degree is fixed and height bounded by a growing parameter Q, and the minimal polynomial Pα is such that the absolute value of its derivative P'α (α) is bounded by a given parameter X. We show that if this bounding parameter X is from a certain range, then as Q → +∞ these algebraic numbers are distributed uniformly in the segment [-1+√2/3.1-√2/3]

This paper deals with two- and threefold weavings on Platonic polyhedral surfaces. Depending on the skewness of the weaving pattern with respect to the edges of the polyhedra, different numbers of closed strands are necessary in a complete weaving. The problem is present in basketry but can be addressed from the aspect of pure geometry (geodesics), graph theory (central circuits of 4-valent graphs) and even structural engineering (fastenings on a closed surface). Numbers of these strands are found to have a periodicity and symmetry, and, in some cases, this number can be predicted directly from the skewness of weaving. In this paper (i) a simple recursive method using symmetry operations is given to find the number of strands of cubic, octahedral and icosahedral weavings for cases where generic symmetry arguments fail; (ii) another simple method is presented to decide whether or not a single closed strand can run along the underlying Platonic without a turn (i.e. the linking number of the two edges of a strand is zero, and so the loop can be stretched to a circle without being twisted); and (iii) the linking number of individual strands in an alternate ‘check’ weaving pattern is determined.

The binary and ternary niobium oxides structurally related to H—Nb 2 O 5 have structures made up of columns of ReO 3 -type linked by boundary crystallographic shear planes. They are known as block structures in view of their appearance in projection. In this paper, the structures have been treated as two-dimensional shift lattices in which the ReO 3 -like columns are regarded as lattice fragments. The paper shows how the shift-lattice parameters may be derived and these are used to illustrate how the structures can be described in a systematic fashion and how hypothetical or new structures can also be incorporated into the framework of known block structures. It is also shown that many Wadsley defects occurring in block structures can be explained as simple manifestations of the operation of shift-lattice principles and in such instances the term ‘defects’ is inappropriate. The paper concludes with a discussion of the way in which the shift-lattice approach avoids difficulties associated with semicommensurate and incommensurate diffraction patterns which can occur in block-structure systems.

In this paper, two subclasses of close-to-star functions and two subclasses of close-to-convex functions are studied. By using the coefficient estimates of the known Caratheodory function classes, the upper bounds of the Zalcman functional $${J}_{3,4}$$ , Hankel determinant $${{H}_{2}}(3)$$ and Toeplitz determinant of given function classes are estimated. Some sharp results are obtained.

The skewfield K(∂) of rational pseudodifferential operators over a differential field K is the skewfield of fractions of the algebra of differential operators K[∂]. In our previous paper, we showed that any H ∈ K(∂) has a minimal fractional decomposition H = AB[superscript −1] , where A,B ∈ K[∂], B ≠ 0, and any common right divisor of A and B is a non-zero element of K . Moreover, any right fractional decomposition of H is obtained by multiplying A and B on the right by the same non-zero element of K[∂] . In the present paper, we study the ring M[subscript n](K(∂)) of n × n matrices over the skewfield K(∂). We show that similarly, any H ∈ M[subscript n](K(∂)) has a minimal fractional decomposition H = AB[superscript −1], where A,B ∈ M[subscript n](K[∂]), B is non-degenerate, and any common right divisor of A and B is an invertible element of the ring M[subscript n](K[∂]). Moreover, any right fractional decomposition of H is obtained by multiplying A and B on the right by the same non-degenerate element of M[subscript n](K[∂]). We give several equivalent definitions of the minimal fractional decomposition. These results are applied to the study of maximal isotropicity property, used in the theory of Dirac structures., National Science Foundation (U.S.)

The Moran process models the spread of genetic mutations through a population. A mutant with relative fitness $r$ is introduced into a population and the system evolves, either reaching fixation (in which every individual is a mutant) or extinction (in which none is). In a widely cited paper (Nature, 2005), Lieberman, Hauert and Nowak generalize the model to populations on the vertices of graphs. They describe a class of graphs (called "superstars"), with a parameter $k$. Superstars are designed to have an increasing fixation probability as $k$ increases. They state that the probability of fixation tends to $1-r^{-k}$ as graphs get larger but we show that this claim is untrue as stated. Specifically, for $k=5$, we show that the true fixation probability (in the limit, as graphs get larger) is at most $1-1/j(r)$ where $j(r)=\Theta(r^4)$, contrary to the claimed result. We do believe that the qualitative claim of Lieberman et al.\ --- that the fixation probability of superstars tends to 1 as $k$ increases --- is correct, and that it can probably be proved along the lines of their sketch. We were able to run larger computer simulations than the ones presented in their paper. However, simulations on graphs of around 40,000 vertices do not support their claim. Perhaps these graphs are too small to exhibit the limiting behaviour., Comment: Technical content remains the same but we have revised our interpretation of the experimental data and changed the title accordingly. We have included, as ancillary files, our c code (superstar.c) and also documentation demonstrating its correctness (superstarfixprob-ancillary.pdf): download the paper source package to view these

In two 2006 papers, Kostant and Wallach constructed a complexified Gelfand-Zeitlin integrable system for the Lie algebra $\fgl(n+1,\C)$ and introduced the strongly regular elements, which are the points where the Gelfand-Zeitlin flow is Lagrangian. Later Colarusso studied the nilfibre, which consists of strongly regular elements such that each $i\times i$ submatrix in the upper left corner is nilpotent. In this paper, we prove that every Borel subalgebra contains strongly regular elements and determine the Borel subalgebras containing elements of the nilfibre by using the theory of $K_{i}=GL(i-1,\C) \times GL(1,\C)$-orbits on the flag variety for $\fgl(i,\C)$ for $2\leq i\leq n+1$. As a consequence, we obtain a more precise description of the nilfibre. The $K_{i}$-orbits contributing to the nilfibre are closely related to holomorphic and anti-holomorphic discrete series for the real Lie groups $U(i,1)$, with $i \le n$., 19 pages

Let G be a graph such that the color of its vertices is white or black. A dynamic vertex coloring for G is defined as follows. One starts with a certain set of black vertices. Then, at each time step, a black vertex with exactly one white neighbor forces its white neighbor to become black. The initial set of black vertices is called a zero forcing set if by iterating this process, all of the vertices of G become black. The zero forcing number of G (denoted by Z(G)) is the minimum cardinality of a zero forcing set in G. In this paper, we study the zero forcing number of trees. Let T be a tree with at least two vertices. We show that $$\Delta (T)-1\le Z(T)\le r(T)-1$$ , where $$\Delta (T)$$ and r(T) are the maximum degree and the number of pendant vertices of T, respectively. As a consequence, we obtain that $$Z(L(T))\ge Z(T)$$ , where L(T) is the line graph of T. We characterize all trees T such that $$Z(T)=\Delta (T)-1$$ . Finally, we study trees T with $$Z(T)= r(T)-1$$ .

A convex hull generated by a sample uniformly distributed on the plane is considered in the case when the support of a distribution is a convex polygon. A central limit theorem is proved for the joint distribution of the number of vertices and the area of a convex hull using the Poisson approximation of binomial point processes near the boundary of the support of distribution. Here we apply the results on the joint distribution of the number of vertices and the area of convex hulls generated by the Poisson distribution given in [6]. From the result obtained in the present paper, in particular, follow the results given in [3, 7], when the support is a convex polygon and the convex hull is generated by a homogeneous Poisson point process

Let G be a connected unipotent group over a finite field F_q with q elements. In this article we propose a definition of L-packets of complex irreducible representations of the finite group G(F_q) and give an explicit description of L-packets in terms of the so-called "admissible pairs" for G. We then apply our results to show that if the centralizer of every geometric point of G is connected, then the dimension of every complex irreducible representation of G(F_q) is a power of q, confirming a conjecture of V. Drinfeld. This paper is the first in a series of three papers exploring the relationship between representations of a group of the form G(F_q) (where G is a unipotent algebraic group over F_q), the geometry of G, and the theory of character sheaves., Version 4, 81 pages, LaTeX. Main change compared to the previous version: the term "$L$-packet" has been replaced with "$\mathbb{L}$-packet", which is short for "Lusztig packet" (to distinguish it from Langlands' notion of an $L$-packet)

The concept of (κ,τ)-regular vertex set appeared in 2004. It was proved that the existence of many classical combinatorial structures in a graph like perfect matchings, Hamiltonian cycles, effective dominating sets, etc., can be characterized by (κ,τ)-regular sets the definition whereof is equivalent to the determination of these classical combinatorial structures. On the other hand, the determination of (κ,τ)-regular sets is closely related to the properties of the main spectrum of a graph. This paper generalizes the well-known properties of (κ,κ)-regular sets of a graph to arbitrary (κ,τ)-regular sets of graphs with an emphasis on their connection with classical combinatorial structures. We also present a recognition algorithm for the Hamiltonicity of the graph that becomes polynomial in some classes of graphs, for example, in the class of graphs with a fixed cyclomatic number.

The theory of increasing and convex-along-rays (ICAR) functions defined on a convex cone in a real locally convex topological vector space X was already well developed. In this paper, we first examine abstract convexity of increasing plus-convex-along-rays (IPCAR) functions defined on a real normed linear space X. We also study, for this class of functions, some concepts of abstract convexity, such as support sets and subdifferentials. Finally, as an application, we characterize the maximal elements of the support set of strictly IPCAR functions and give optimality conditions for the global minimum of the difference between two IPCAR functions.

A collection of resistors with two possible resistivities is considered. This paper investigates the overall or macroscopic behaviour of a square two–dimensional lattice of such resistors when both types coexist in fixed proportions in the lattice. The macroscopic behaviour is that of an anisotropic conductor at the continuum level and the goal of the paper is to describe the set of all possible such conductors. This is thus a problem of bounds, following in the footsteps of an abundant literature on the topic in the continuum case. The originality of the paper is that the investigation focuses on the interplay between homogenization and the passage from a discrete network to a continuum. A set of bounds is proposed and its optimality is shown when the proportion of each resistor on the discrete lattice is½. We conjecture that the derived bounds are optimal for all proportions.