摘要：
Embryonic stem cells are conventionally differentiated by modulating specific growth factors in the cell culture media. Recently the effect of cellular mechanical microenvironment in inducing phenotype specific differentiation has attracted considerable attention. We have shown the possibility of inducing endoderm differentiation by culturing the stem cells on fibrin substrates of specific stiffness [1]. Here, we analyze the regulatory network involved in such mechanically induced endoderm differentiation under two different experimental configurations of 2-dimensional and 3-dimensional culture, respectively. Mouse embryonic stem cells are differentiated on an array of substrates of varying mechanical properties and analyzed for relevant endoderm markers. The experimental data set is further analyzed for identification of co-regulated transcription factors across different substrate conditions using the technique of bi-clustering. Overlapped bi-clusters are identified following an optimization formulation, which is solved using an evolutionary algorithm. While typically such analysis is performed at the mean value of expression data across experimental repeats, the variability of stem cell systems reduces the confidence on such analysis of mean data. Bootstrapping technique is thus integrated with the bi-clustering algorithm to determine sets of robust bi-clusters, which is found to differ significantly from corresponding bi-clusters at the mean data value. Analysis of robust bi-clusters reveals an overall similar network interaction as has been reported for chemically induced endoderm or endodermal organs but with differences in patterning between 2-dimensional and 3-dimensional culture. Such analysis sheds light on the pathway of stem cell differentiation indicating the prospect of the two culture configurations for further maturation.
期刊：
Journal of Differential Equations,2014年257(2):566-600 ISSN：0022-0396
通讯作者：
Li, Gongbao
作者机构：
[Li, Gongbao] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Li, Gongbao] C;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
关键词：
Kirchhoff equation;Ground state solutions;Pohoz̆aev type identity;Variational methods
摘要：
In this paper, we study the following nonlinear problem of Kirchhoff type with pure power nonlinearities: {-(a + b integral(R3) vertical bar Du vertical bar(2)) Delta u + V(x)u = vertical bar u vertical bar(p-1)u, x is an element of R-3, (0.1) u is an element of H-1 (R-3), u > 0, x is an element of R-3, where a, b > 0 are constants, 2 < p < 5 and V : R-3 -> R. Under certain assumptions on V. we prove that (0.1) has a positive ground state solution by using a monotonicity trick and a new version of global compactness lemma. Our main results especially solve problem (0.1) in the case where p is an element of (2, 3], which has been an open problem for Kirchhoff equations and can be viewed as a partial extension of a recent result of He and Zou in [14] concerning the existence of positive solutions to the nonlinear Kirchhoff problem {-(epsilon(2)a + epsilon b integral(R3) vertical bar Du vertical bar(2)) Delta u + V(x)u = f(u), x is an element of R-3, u is an element of H-1 (R-3), u > 0, x is an element of R-3, where epsilon > 0 is a parameter, V(x) is a positive continuous potential and f (u) similar to vertical bar u vertical bar(p-1)u with 3 < p < 5 and satisfies the Ambrosetti-Rabinowitz type condition. Our main results extend also the arguments used in [7,33], which deal with Schrodinger-Poisson system with pure power nonlinearities, to the Kirchhoff type problem. (C) 2014 Elsevier Inc. All rights reserved.
期刊：
Journal of Differential Equations,2015年259(4):1256-1274 ISSN：0022-0396
通讯作者：
Shuai, Wei
作者机构：
[Shuai, Wei; Shuai, W] Cent China Normal Univ, Dept Math, Wuhan 430079, Peoples R China.
通讯机构：
[Shuai, Wei] C;Cent China Normal Univ, Dept Math, Wuhan 430079, Peoples R China.
关键词：
Kirchhoff-type equations;Sign-changing solutions;Nonlocal term
摘要：
We are interested in the existence of least energy sign-changing solutions for a class of Kirchhoff-type problem in bounded domains. Because the so-called nonlocal term b(integral(Omega)vertical bar del u vertical bar(2)dx)Delta u is involving in the equation, the variational functional of the equation has totally different properties from the case of b = 0. Combining constraint variational method and quantitative deformation lemma, we prove that the problem possesses one least energy sign-changing solution u(b). Moreover, we show that the energy of u(b) is strictly larger than the ground state energy. Finally, we regard b as a parameter and give a convergence property of ub as b SE arrow 0. (C) 2015 Elsevier Inc. All rights reserved.
作者机构：
[Peng, Shuangjie; Deng, Yinbin; Shuai, Wei] Cent China Normal Univ, Dept Math, Wuhan 430079, Peoples R China.
通讯机构：
[Peng, Shuangjie] C;Cent China Normal Univ, Dept Math, Wuhan 430079, Peoples R China.
关键词：
Kirchhoff-type equations;Nodal solutions;Nonlocal term
摘要：
In this paper, we study the existence and asymptotic behavior of nodal solutions to the following Kirchhoff problem -(a+b integral(3)(R)vertical bar del u vertical bar(2)dx)Delta u + v(vertical bar x vertical bar)u = f(vertical bar x vertical bar, u), in R-3 , u is an element of H-1(R-3), where V(x) is a smooth function, a, b are positive constants. Because the so-called nonlocal term (integral(3)(R)vertical bar del u vertical bar(2)dx)Delta u)Au is involved in the equation, the variational functional of the equation has totally different properties from the case of b = 0. Under suitable construction conditions, we prove that, for any positive integer k, the problem has a sign-changing solution u(k)(b), which changes signs exactly k times. Moreover, the energy of u(k)(b) is strictly increasing in k, and for any sequence {b(n)}-> 0(+) (n -> + infinity), there is a subsequence {b(ns)}, such that u(k)(bs) converges in H-1 (R-3) to wk as s -> infinity, where wk also changes signs exactly k times and solves the following equation -a Delta u +V (|x|), u) = f(|x|, u), in R-3, u is an element of H-1 (R-3). (C) 2015 Elsevier Inc. All rights reserved.
关键词：
Coordinated charging;load variance;microgrid;plug-in hybrid electric vehicle (EV) (HEV) (PHEV);regulated charging;smart grid;vehicle to grid (V2G)
摘要：
With the advent of the plug-in hybrid electric vehicles (PHEVs), the vehicle-to-grid (V2G) technology is attracting increasing attention recently. It is believed that the V2G option can aid to improve the efficiency and reliability of the power grid, as well as reduce overall cost and carbon emission. In this paper, the possibility of smoothing out the load variance in a household microgrid by regulating the charging patterns of family PHEVs is investigated. First, the mathematic model of the problem is built up. Then, the case study is conducted, which demonstrates that, by regulating the charging profiles of the PHEVs, the variance of load power can be dramatically reduced. Third, the energy losses and the subsidy mechanism are discussed. Finally, the impacts of the requested net charging quantities and the battery capacity of PHEVs on the performance of the regulated charging are investigated.
作者机构：
[Li, Gongbao] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Li, Gongbao] C;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
摘要：
We study the concentration and multiplicity of weak solutions to the Kirchhoff type equation with critical Sobolev growth, -(epsilon(2)a + epsilon b integral(R3) vertical bar del u vertical bar(2))del u + V(z)u = f(u) + u(5) in R-3, u is an element of H-1(R-3), u > 0 in R-3, where e is a small positive parameter and a, b > 0 are constants, f is an element of C-1(R+, R) is subcritical, V : R-3 -> R is a locally Holder continuous function. We first prove that for epsilon(0) > 0 sufficiently small, the above problem has a weak solution u(epsilon) with exponential decay at infinity. Moreover, u(epsilon) concentrates around a local minimum point of V in Lambda as epsilon -> 0. With minimax theorems and Ljusternik-Schnirelmann theory, we also obtain multiple solutions by employing the topological construction of the set where the potential V(z) attains its minimum.
通讯机构：
[Xu, Ying] U;Univ Georgia, Dept Biochem & Mol Biol, Computat Syst Biol Lab, Athens, GA 30602 USA.
摘要：
We have recently developed a new version of the DOOR operon database, DOOR 2.0, which is available online at http://csbl.bmb.uga.edu/DOOR/ and will be updated on a regular basis. DOOR 2.0 contains genome-scale operons for 2072 prokaryotes with complete genomes, three times the number of genomes covered in the previous version published in 2009. DOOR 2.0 has a number of new features, compared with its previous version, including (i) more than 250 000 transcription units, experimentally validated or computationally predicted based on RNA-seq data, providing a dynamic functional view of the underlying operons; (ii) an integrated operon-centric data resource that provides not only operons for each covered genome but also their functional and regulatory information such as their cis-regulatory binding sites for transcription initiation and termination, gene expression levels estimated based on RNA-seq data and conservation information across multiple genomes; (iii) a high-performance web service for online operon prediction on user-provided genomic sequences; (iv) an intuitive genome browser to support visualization of user-selected data; and (v) a keyword-based Google-like search engine for finding the needed information intuitively and rapidly in this database.
期刊：
JOURNAL OF DIFFERENTIAL EQUATIONS,2014年257(6):1721-1752 ISSN：0022-0396
通讯作者：
Huang, Jicai
作者机构：
[Huang, Jicai; Song, Jing] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Ruan, Shigui] Univ Miami, Dept Math, Coral Gables, FL 33124 USA.
通讯机构：
[Huang, Jicai] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.
关键词：
Predator–prey system;Holling type-III functional response;Hopf bifurcation;Bogdanov–Takens bifurcation;Degenerate focus type BT bifurcation of codim 3
摘要：
We consider a predator-prey system of Leslie type with generalized Holling type III functional response p(x) = mx(2)/ax(2)+bx+1. By allowing b to be negative (b > -2 root a), p(x) is monotonic for b > 0 and nonmonotonic for b < 0 when x >= 0. The model has two non-hyperbolic positive equilibria (one is a multiple focus of multiplicity one and the other is a cusp of codimension 2) for some values of parameters and a degenerate Bogdanov-Takens singularity (focus or center case) of codimension 3 for other values of parameters. When there exist a multiple focus of multiplicity one and a cusp of codimension 2, we show that the model exhibits subcritical Hopf bifurcation and Bogdanov-Takens bifurcation simultaneously in the corresponding small neighborhoods of the two degenerate equilibria, respectively. Different phase portraits of the model are obtained by computer numerical simulations which demonstrate that the model can have: (i) a stable limit cycle enclosing two non-hyperbolic positive equilibria; (ii) a stable limit cycle enclosing an unstable homoclinic loop; (iii) two limit cycles enclosing a hyperbolic positive equilibrium; (iv) one stable limit cycle enclosing three hyperbolic positive equilibria; or (v) the coexistence of three stable states (two stable equilibria and a stable limit cycle). When the model has a Bogdanov-Takens singularity of codimension 3, we prove that the model exhibits degenerate focus type Bogdanov-Takens bifurcation of codimension 3. These results not only demonstrate that the dynamics of this model when b > -2 root a are much more complex and far richer than the case when b > 0 but also provide new bifurcation phenomena for predator-prey systems. (C) 2014 Elsevier Inc. All rights reserved.
期刊：
Calculus of Variations and Partial Differential Equations,2015年54(3):3067-3106 ISSN：0944-2669
通讯作者：
Li, Gongbao
作者机构：
[He, Yi; Li, Gongbao] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;[He, Yi; Li, Gongbao] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Li, Gongbao] C;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
关键词：
35J60;35J92;Primary 35J20
摘要：
We are concerned with the following Kirchhoff type equation with critical nonlinearity: {-(epsilon(2)a + epsilon b integral(R3) vertical bar del u vertical bar(2))Delta u + V(x)u = lambda vertical bar u vertical bar(p-2)u + vertical bar u vertical bar(4)u in R-3, u > 0, u is an element of H-1(R-3), where epsilon is a small positive parameter, a, b > 0, lambda > 0, 2 < p <= 4. Under certain assumptions on the potential V, we construct a family of positive solutions u(epsilon) is an element of H-1(R-3) which concentrates around a local minimum of V as epsilon -> 0. Although, critical growth Kirchhoff type problem {-(epsilon(2)a + epsilon b integral(R3) vertical bar del u vertical bar(2))Delta u + V(x)u = f (u) + u(5) in R-3, u > 0, u is an element of H-1(R-3) has been studied in e.g. He et al. [18], where the assumption for f(u) is that f(u) similar to vertical bar u vertical bar(p-2)u with 4 < p < 6 and satisfies the Ambrosetti-Rabinowitz condition which forces the boundedness of any Palais-Smale sequence of the corresponding energy functional of the equation. As g(u) := lambda vertical bar u vertical bar(p-2)u + vertical bar u vertical bar(4)u with 2 < p <= 4 does not satisfy the Ambrosetti-Rabinowitz condition (there exists mu > 4, 0 < mu integral(u)(0) g(s)ds <= g(u)u), the boundedness of Palais-Smale sequence becomes a major difficulty in proving the existence of a positive solution. Also, the fact that the function g(s)/s(3) is not increasing for s > 0 prevents us from using the Nehari manifold directly as usual. Our result extends the main result in He et al. [18] concerning the existence and concentration of positive solutions to the case where f(u) similar to vertical bar u vertical bar(p-2)u with 4 < p < 6.
期刊：
European Physical Journal Plus,2016年131(12):1-16 ISSN：2190-5444
通讯作者：
Selima, Ehab S.
作者机构：
[Yao, Xiaohua; Selima, Ehab S.] Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.;[Selima, Ehab S.] Menoufia Univ, Fac Sci, Dept Math, Shibin Al Kawm 32511, Egypt.;[Seadawy, Aly R.] Taibah Univ, Fac Sci, Dept Math, Al Ula, Saudi Arabia.;[Seadawy, Aly R.] Beni Suef Univ, Fac Sci, Dept Math, Bani Suwayf, Egypt.
通讯机构：
[Selima, Ehab S.] C;[Selima, Ehab S.] M;Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.;Menoufia Univ, Fac Sci, Dept Math, Shibin Al Kawm 32511, Egypt.
期刊：
ADVANCES IN MATHEMATICS,2014年259:511-531 ISSN：0001-8708
通讯作者：
He, Xing-Gang
作者机构：
[Dai, Xin-Rong] Sun Yat Sen Univ, Sch Math & Computat Sci, Guangzhou 510275, Guangdong, Peoples R China.;[He, Xing-Gang] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Lau, Ka-Sing] Chinese Univ Hong Kong, Dept Math, Hong Kong, Hong Kong, Peoples R China.
通讯机构：
[He, Xing-Gang] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
关键词：
Bernoulli convolution;Fourier transform;Orthogonal;Self-similar;Spectral measure;Spectrum;Bi-zero set
摘要：
For 0 < rho < 1 and N > 1 an integer, let mu be the self-similar measure defined by mu(center dot) = Sigma(N-1)(i=0) 1/N mu(rho(-1)(center dot) - i). We prove that L-2(mu) has an exponential orthonormal basis if and only if rho = 1/q for some q > 0 and N divides q. The special case is the Cantor measure with rho = 1/2k and N = 2 [16], which was proved recently to be the only spectral measure among the Bernoulli convolutions with 0 < rho < 1 [4]. (C) 2014 Elsevier Inc. All rights reserved.
摘要：
This paper is concerned with the positive solutions for generalized quasilinear Schrodinger equations in R-N with critical growth which have appeared from plasma physics, as well as high-power ultrashort laser in matter. By using a change of variables and variational argument, we obtain the existence of positive solutions for the given problem. (C) 2014 Elsevier Inc. All rights reserved.
期刊：
Advances in Mathematics,2013年242:187-208 ISSN：0001-8708
通讯作者：
He, Xing-Gang
作者机构：
[Dai, Xin-Rong] Sun Yat Sen Univ, Sch Math & Computat Sci, Guangzhou 510275, Guangdong, Peoples R China.;[He, Xing-Gang] Cent China Normal Univ, Coll Math & Stat, Wuhan 430079, Peoples R China.;[Lai, Chun-Kit] McMaster Univ, Dept Math & Stat, Hamilton, ON L8S 4K1, Canada.
通讯机构：
[He, Xing-Gang] C;Cent China Normal Univ, Coll Math & Stat, Wuhan 430079, Peoples R China.
关键词：
Cantor measures;Spectral measures;Spectra;Trees
摘要：
We consider equally-weighted Cantor measures mu(q,b) arising from iterated function systems of the form b(-1)(x + i), i = 0, 1, ..., q - 1, where q < b. We classify the (q, b) so that they have infinitely many mutually orthogonal exponentials in L-2(mu(q,b)). In particular, if q divides b, the measures have a complete orthogonal exponential system and hence spectral measures. Improving the construction by Dutkay et al. (2009) [3], we characterize all the maximal orthogonal sets Lambda when q divides b via a maximal mapping on the q-adic tree in which all elements in Lambda are represented uniquely in finite b-adic expansions and we can separate the maximal orthogonal sets into two types: regular and irregular sets. For a regular maximal orthogonal set, we show that its completeness in L-2(mu(q,b)) is crucially determined by the certain growth rate of non-zero digits in the tail of the b-adic expansions of the elements. Furthermore, we exhibit complete orthogonal exponentials with zero Beurling dimensions. These examples show that the technical condition in Theorem 3.5 of Dutkay et al. (2011) [4] cannot be removed. For an irregular maximal orthogonal set, we show that under some conditions, its completeness is equivalent to that of the corresponding regularized mapping. (C) 2013 Elsevier Inc. All rights reserved.
作者机构：
[Peng, Shuangjie; Wang, Zhi-qiang] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;Nankai Univ, Chern Inst Math, Tianjin 300071, Peoples R China.;[Wang, Zhi-qiang] Utah State Univ, Dept Math & Stat, Logan, UT 84322 USA.
通讯机构：
[Wang, Zhi-qiang] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
摘要：
We consider the following nonlinear Schrodinger system in R-3 {-Delta u+P(vertical bar x vertical bar)u=mu u(3)+beta upsilon(2)u, x epsilon R-3, {-Delta upsilon+Q(vertical bar x vertical bar)upsilon=nu upsilon(3)+beta u(2)upsilon, x epsilon R-3, where P(r) and Q(r) are positive radial potentials, mu > 0, nu > 0 and beta epsilon R is a coupling constant. This type of system arises, in particular, in models in Bose-Einstein condensates theory. We examine the effect of nonlinear coupling on the solution structure. In the repulsive case, we construct an unbounded sequence of non-radial positive vector solutions of segregated type, and in the attractive case we construct an unbounded sequence of non-radial positive vector solutions of synchronized type. Depending upon the system being repulsive or attractive, our results exhibit distinct characteristic features of vector solutions.
摘要：
This paper is concerned with the solitary wave solutions for a generalized quasilinear Schrodinger equation in RN involving critical exponents, which have appeared from plasma physics, as well as high-power ultashort laser in matter. We find the related critical exponents for a generalized quasilinear Schrodinger equation and obtain its solitary wave solutions by using a change of variables and variational argument. (C) 2015 Elsevier Inc. All rights reserved.
摘要：
In this paper we study the effect of constant-yield predator harvesting on the dynamics of a Leslie-Gower type predator-prey model. It is shown that the model has a Bogdanov-Takens singularity (cusp case) of codimension 3 or a weak focus of multiplicity two for some parameter values, respectively. Saddle-node bifurcation, repelling and attracting Bogdanov-Takens bifurcations, supercritical and subcritical Hopf bifurcations, and degenerate Hopf bifurcation are shown as the values of parameters vary. Hence, there are different parameter values for which the model has a homoclinic loop or two limit cycles. It is also proven that there exists a critical harvesting value such that the predator specie goes extinct for all admissible initial densities of both species when the harvest rate is greater than the critical value. These results indicate that the dynamical behavior of the model is very sensitive to the constant-yield predator harvesting and the initial densities of both species and it requires careful management in the applied conservation and renewable resource contexts. Numerical simulations, including the repelling and attracting Bogdanov-Takens bifurcation diagrams and corresponding phase portraits, two limit cycles, the coexistence of a stable homoclinic loop and an unstable limit cycle, and a stable limit cycle enclosing an unstable multiple focus with multiplicity one, are presented which not only support the theoretical analysis but also indicate the existence of Bogdanov-Takens bifurcation (cusp case) of codimension 3. These results reveal far richer and much more complex dynamics compared to the model without harvesting or with only constant-yield prey harvesting.
作者机构：
[Cao, Daomin] Chinese Acad Sci, Inst Appl Math, AMSS, Beijing 100190, Peoples R China.;[Peng, Shuangjie] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Cao, Daomin] Chinese Acad Sci, Key Lab Random Complex Struct & Data Sci, Beijing 100190, Peoples R China.;[Yan, Shusen] Univ New England, Dept Math, Armidale, NSW 2351, Australia.
通讯机构：
[Peng, Shuangjie] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
关键词：
Critical Sobolev growth;Infinitely many solutions;p-Laplacian equations
摘要：
In this paper, we will prove the existence of infinitely many solutions for the following elliptic problem with critical Sobolev growth: -Delta(p)u = vertical bar u vertical bar(p)*(-2)u + mu vertical bar u vertical bar(p-2)u in Omega, u =0 on partial derivative Omega, provided N > p(2) + p, where Delta(p) is the p-Laplacian operator, 1 < p < N, p* = pN/N-p, mu > 0 and Omega is an open bounded domain in R-N. (C) 2012 Elsevier Inc. All rights reserved.
摘要：
In the paper, we establish a blow-up criterion in terms of the integrability of the density for strong solutions to the Cauchy problem of compressible isentropic Navier-Stokes equations in R-3 with vacuum, under the assumptions on the coefficients of viscosity: 29 mu/3 > lambda. This extends the corresponding results in Huang et al. (2011), Sun et al. (2011) [20,36] where a blow-up criterion in terms of the upper bound of the density was obtained under the condition 7 mu > lambda. As a byproduct, the restriction 7 mu > lambda. in Fan et al. (2010), Sun et al. (2011) [12,37] is relaxed to 29 mu/3 > lambda for the full compressible Navier-Stokes equations by giving a new proof of Lemma 3.1. Besides, we get a blow-up criterion in terms of the upper bound of the density and the temperature for strong solutions to the Cauchy problem of the full compressible Navier-Stokes equations in R-3. The appearance of vacuum could be allowed. This extends the corresponding results in Sun et al. (2011) [37] where a blow-up criterion in terms of the upper bound of (rho, 1/rho, theta) was obtained without vacuum. The effective viscous flux plays a very important role in the proofs. (C) 2013 Elsevier Inc. All rights reserved.
期刊：
Bulletin of the Malaysian Mathematical Sciences Society,2019年42(1):67-78 ISSN：0126-6705
通讯作者：
Wang, Shaohui
作者机构：
[Liu, Jia-Bao] Anhui Jianzhu Univ, Sch Math & Phys, Hefei 230601, Anhui, Peoples R China.;[Wang, Chunxiang] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Wang, Shaohui] Adelphi Univ, Dept Math & Comp Sci, Garden City, NY 11530 USA.;[Wei, Bing] Univ Mississippi, Dept Math, University, MS 38677 USA.
通讯机构：
[Wang, Shaohui] A;Adelphi Univ, Dept Math & Comp Sci, Garden City, NY 11530 USA.
关键词：
Eulerian graphs;Extremal bounds;Multiplicative Zagreb index;Zagreb index
期刊：
Journal of Functional Analysis,2014年266(1):343-354 ISSN：0022-1236
通讯作者：
He, Xing-Gang
作者机构：
[An, Li-Xiang; He, Xing-Gang] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[He, Xing-Gang] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
关键词：
Compatible pair;Moran iterated function systems;Moran measures;Spectral measures
摘要：
Let (D-k)(k=1)(infinity) be a sequence of digit sets in N and let {b(k)}(k=1)(infinity) be a sequence of integer numbers bigger than 1. We call the family {f(k),D-k (x) = b(k)(-1) (x + d): d is an element of D-k, k >= 1} a Moran iterated function system (IFS), which is a natural generalization of an IFS. We prove, under certain conditions in terms of (bk,Dk), that the associated Moran measure p, is a spectral measure, i.e., there exists a countable set Lambda subset of N such that {e(2 pi i lambda x): lambda is an element of Lambda} is an orthonormal basis for L-2(mu). (C) 2013 Elsevier Inc. All rights reserved.