On the solvability of a nonlinear boundary value problem arising in the theory of growing cell populations

In this paper we establish some results regarding the existence of solution on L₁ spaces to a nonlinear boundary value problem originally proposed by Lebowitz and Rubinow (J. Math. Biol. 1974; 1 :17–36) to model an age‐structured proliferating cell population. Our approach, based on topological methods, uses essentially the specific properties of weakly compact sets on L₁ spaces. Our results provide positive answers to the questions posed in Jeribi (Nonlinear Anal. Real World Appl. 2002; 3 :85–105). Copyright © 2005 John Wiley & Sons, Ltd.     

Beamlets are densely embedded in H −1

The insufficiency of using ordinary measurable functions to model complex natural images was first emphasized by David Mumford (Q Appl Math 59:85–111, 2001). The idea was later rediscovered by Yves Meyer (Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, vol. 22, University Lecture Series, AMS, Providence, 2001) who introduced proper texture models based on generalized functions or distributions. The simpler but effective Sobolev texture model of H ^???1 was subsequently explored by Osher et al. (Multiscale Model Simul 1:349–370, 2003) to facilitate practical computation. H ^???1 textures have also been further employed in the recent works of Daubechies and Teschke (Appl Comput Harmon Anal 19(1):1–16, 2005), Lieu and Vese (UCLA CAM Tech Report, 05–33, 2005), Shen (Appl Math Res Express 4:143–167, 2005), and many others, leading to a new generation of models for image processing and analysis. On the other hand, beamlets are the unconventional class of geometric wavelets invented by Donoho and Huo (Multiscale and Multiresolution Methods, Lect Notes Comput Sci Eng, vol. 20, pp. 149–196. Springer, Berlin, 2002) to efficiently represent and detect lower dimensional singular image features. In the current work, we make an intriguing connection between the above two realms by demonstrating that H ^???1 is the natural space (of generalized functions) that hosts beamlets, and in return can be completely described by them. Computational evidences existing in the literature also help confirm this newly discovered bond.     

Asymptotic behavior of the Korteweg-de Vries equation posed in a quarter plane

The purpose of this work is to study the exponential stabilization of the Korteweg-de Vries equation in the right half-line under the effect of a localized damping term. We follow the methods in [G.P. Menzala, C.F. Vasconcellos, E. Zuazua, Stabilization of the Korteweg-de Vries equation with localized damping, Quart. Appl. Math. LX (1) (2002) 111-129] which combine multiplier techniques and compactness arguments and reduce the problem to prove the unique continuation property of weak solutions. Here, the unique continuation is obtained in two steps: we first prove that solutions vanishing on the support of the damping function are necessarily smooth and then we apply the unique continuation results proved in [J.C. Saut, B. Scheurer, Unique continuation for some evolution equations, J. Differential Equations 66 (1987) 118-139]. In particular, we show that the exponential rate of decay is uniform in bounded sets of initial data.     

Data dependence for the amplitude equation of surface waves

We consider the amplitude equation for nonlinear surface wave solutions of hyperbolic conservation laws. This is an asymptotic nonlocal, Hamiltonian evolution equation with quadratic nonlinearity. For example, this equation describes the propagation of nonlinear Rayleigh waves (Hamilton et al. in J Acoust Soc Am 97:891–897, 1995), surface waves on current-vortex sheets in incompressible MHD (Alì and Hunter in Q Appl Math 61(3):451–474, 2003; Alì et al. in Stud Appl Math 108(3):305–321, 2002) and on the incompressible plasma–vacuum interface (Secchi in Q Appl Math 73(4):711–737, 2015). The local-in-time existence of smooth solutions to the Cauchy problem for the amplitude equation in noncanonical variables was shown in Hunter (J Hyperbolic Differ Equ 3(2):247–267, 2006), Secchi (Q Appl Math 73(4):711–737, 2015). In the present paper we prove the continuous dependence in strong norm of solutions on the initial data. This completes the proof of the well-posedness of the problem in the classical sense of Hadamard.     

Solution of axisymmetric Maxwell equations

In this article, we study the static and time‐dependent Maxwell equations in axisymmetric geometry. Using the mathematical tools introduced in (Math. Meth. Appl. Sci. 2002; 25 : 49), we investigate the decoupled problems induced in a meridian half‐plane, and the splitting of the solution in a regular part and a singular part, the former being in the Sobolev space H¹ component‐wise. It is proven that the singular parts are related to singularities of Laplace‐like or wave‐like operators. We infer from these characterizations: (i) the finite dimension of the space of singular fields; (ii) global space and space–time regularity results for the electromagnetic field. This paper is the continuation of (Modél. Math. Anal. Numér. 1998; 32 : 359, Math. Meth. Appl. Sci. 2002; 25 : 49). Copyright © 2003 John Wiley & Sons, Ltd.     

On the solutions of the linear integral equations of Volterra type

Some boundaries about the solution of the linear Volterra integral equations of the form f(t)=1?K*f were obtained as |f(t)|?1, |f(t)|?2 and |f(t)|?4 in (J. Math. Anal. Appl. 1978; 64 :381–397; Int. J. Math. Math. Sci. 1982; 5 (1):123–131). The boundary of the solution function of an equation in this type was found as |f(t)|?2ⁿ in (Integr. Equ. Oper. Theory 2002; 43 :466–479), where t∈[0, ∞) and n is a natural number such that n?2. In (Math. Comp. 2006; 75 :1175–1199), it is shown that the boundary of the solution function of an equation in the same form can also be derived as that of (Integr. Equ. Oper. Theory 2002; 43 :466–479) under different conditions than those of (Integr. Equ. Oper. Theory 2002; 43 :466–479). In the present paper, the sufficient conditions for the boundedness of functions f, f′, f′′, …, f⁽ⁿ⁺³⁾, (n∈?) defined on the infinite interval [0, ∞) are given by our method, where f is the solution of the equation f(t)=1?K*f. Copyright © 2007 John Wiley & Sons, Ltd.     

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

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

The small weight codewords of the functional codes associated to non-singular Hermitian varieties

This article studies the small weight codewords of the functional code C _Herm (X), with X a non-singular Hermitian variety of PG(N, q ²). The main result of this article is that the small weight codewords correspond to the intersections of X with the singular Hermitian varieties of PG(N, q ²) consisting of q + 1 hyperplanes through a common (N ? 2)-dimensional space Π, forming a Baer subline in the quotient space of Π. The number of codewords having these small weights is also calculated. In this way, similar results are obtained to the functional codes C ₂(Q), Q a non-singular quadric (Edoukou et al., J. Pure Appl. Algebra 214:1729–1739, 2010), and C ₂(X), X a non-singular Hermitian variety (Hallez and Storme, Finite Fields Appl. 16:27–35, 2010).     

On Sommerfeld representation and uniqueness in scattering by wedges

We consider a non‐stationary scattering of plane waves by a wedge. We prove the Sommerfeld‐type representation and uniqueness of solution to the Cauchy problem in appropriate functional spaces developing the general method of complex characteristics (Math. USSR Sb. 1973; 21 (1):91–135, Moscow Univ. Math. Bull. 1974; 29 (2):140–145, Oper. Theory Adv. Appl. 1992; 57 :171–183, Am. Math. Soc. Transl. (2) 2002; 206 :125–159). Copyright © 2004 John Wiley & Sons, Ltd.     

Existence of (q, 7, 1) difference families with q a prime power

The existence of a (q,k, 1) difference family in GF(q) has been completely solved for k = 3,4,5,6. For k = 7 only partial results have been given. In this article, we continue the investigation and use Weil's theorem on character sums to show that the necessary condition for the existence of a (q,7,1) difference family in GF(q), i.e. q ≡ 1; (mod 42) is also sufficient except for q = 43 and possibly except for q = 127, q = 211, q = 31⁶ and primes q∈ [261239791, 1.236597 × 10¹³] such that in GF(q). © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 126–138, 2002; DOI 10.1002/jcd.998     

Global nonexistence of solutions with positive initial energy for a class of wave equations

In this study, we consider a class of wave equations with strong damping and source terms associated with initial and Dirichlet boundary conditions. We establish a blow up result for certain solutions with nonpositive initial energy as well as positive initial energy. This further improves the results by Yang (Math. Meth. Appl. Sci. 2002; 25 :825–833) and Messaudi and Houari (Math. Meth. Appl. Sci. 2004; 27 : 1687–1696). Copyright © 2008 John Wiley & Sons, Ltd.     

On the cyclic homology of commutative algebras over arbitrary ground rings

Abstract

Let D(H) be the quantum double associated to a finite dimensional quasi-Hopf algebra H, as in Hausser and Nill ((Hausser, F., Nill, F. (1999a). Diagonal crossed products by duals of quasi-quantum groups. Rev. Math. Phys. 11:553–629) and (Hausser, F., Nill, F. (1999b). Doubles of quasi-quantum groups. Comm. Math. Phys. 199:547–589)). In this note, we first generalize a result of Majid (Majid, S. (1991). Doubles of quasitriangular Hopf algebras. Comm. Algebra 19:3061–3073) for Hopf algebras, and then prove that the quantum double of a finite dimensional quasitriangular quasi-Hopf algebra is a biproduct in the sense of Bulacu and Nauwelaerts (Bulacu, D., Nauwelaerts, E. (2002). Radford's biproduct for quasi-Hopf algebras and bosonization. J. Pure Appl. Algebra 179:1–42.).     

The phase transition in random horn satisfiability and its algorithmic implications

Let c > 0 be a constant, and Φ be a random Horn formula with n variables and m = c · 2ⁿ clauses, chosen uniformly at random (with repetition) from the set of all nonempty Horn clauses in the given variables. By analyzing PUR, a natural implementation of positive unit resolution, we show that lim_n→∞ Pr(Φ is satisfiable) = 1 ? F(e^?c), where F(x) = (1 ? x)(1 ? x²)(1 ? x⁴)(1 ? x⁸) …. Our method also yields as a byproduct an average‐case analysis of this algorithm. Published 2002 Wiley Periodicals, Inc. Random Struct. Alg., 20: 483–506, 2002     

Global C∞‐solutions to 1D compressible Navier–Stokes equations with density‐dependent viscosity

In this paper, we consider the existence of global smooth solutions to 1D compressible isentropic Navier–Stokes equations with density‐dependent viscosity and free boundaries. The initial density ρ₀∈W^1,2n is bounded below away from zero and the initial velocity u₀∈L²ⁿ. The viscosity coefficient µ is proportional to ρ^θ with 0<θ?1, where ρis the density. The existence and uniqueness of global solutions in Hⁱ([0,1])(i = 1,2,4) have been established in (J. Math. Phys. 2009; 50 :023101; Meth. Appl. Anal. 2005; 12 :239–252; J. Differ. Equations 2008; 245:3956–3973; Commun. Pure Appl. Anal. 2008; 7 :373–381). By mathematical induction method, we will establish the existence of global smooth solutions to 1D compressible isentropic Navier–Stokes equations with density‐dependent viscosity and free boundaries when the initial data ρ₀ and u₀ are smooth. Copyright © 2011 John Wiley & Sons, Ltd.     

Embeddability and construction of affine α-resolvable pairwise balanced designs

An affine α-resolvable PBD of index λ is a triple (V, B, R), where V is a set (of points), B is a collection of subsets of V (blocks), and R is a partition of B (resolution), satisfying the following conditions: (i) any two points occur together in λ blocks, (ii) any point occurs in α blocks of each resolution class, and (iii) |B| = |V| + |R| − 1. Those designs embeddable in symmetric designs are described and two infinite series of embeddable designs are constructed. The analog of the Bruck–Ryser–Chowla theorem for affine α-resolvable PBDs is obtained. © 1998 John Wiley & Sons, Inc. J Combin Designs 6:111–129, 1998     

The zeros of certain weakly holomorphic Drinfeld modular forms

Duke and Jenkins (Pure Appl Math Q 4(4):1327–1340, 2008) constructed a canonical basis for the space of weakly holomorphic modular forms for \({{\rm SL}_2(\mathbb{Z})}\) and investigated the zeros of the basis elements. In this paper we give an analogy in the Drinfeld setting of the result given by Duke and Jenkins (Pure Appl Math Q 4(4):1327–1340, 2008).     

Computing homotopic shortest paths in the plane

We address the problem of computing homotopic shortest paths in the presence of obstacles in the plane. Problems on homotopy of paths received attention very recently [Cabello et al., in: Proc. 18th Annu. ACM Sympos. Comput. Geom., 2002, pp. 160–169; Efrat et al., in: Proc. 10th Annu. European Sympos. Algorithms, 2002, pp. 411–423]. We present two output-sensitive algorithms, for simple paths and non-simple paths. The algorithm for simple paths improves the previous algorithm [Efrat et al., in: Proc. 10th Annu. European Sympos. Algorithms, 2002, pp. 411–423]. The algorithm for non-simple paths achieves O(log²n) time per output vertex which is an improvement by a factor of O(n/log²n) of the previous algorithm [Hershberger, Snoeyink, Comput. Geom. Theory Appl. 4 (1994) 63–98], where n is the number of obstacles. The running time has an overhead O(n²⁺) for any positive constant . In the case k<n²⁺, where k is the total size of the input and output, we improve the running to O((n+k+(nk)^2/3)log^O(1)n).     

Local energy decay for linear wave equations with non‐compactly supported initial data

A local energy decay problem is studied to a typical linear wave equation in an exterior domain. For this purpose, we do not assume any compactness of the support on the initial data. This generalizes a previous famous result due to Morawetz (Comm. Pure Appl. Math. 1961; 14 :561–568). In order to prove local energy decay we mainly apply two types of new ideas due to Ikehata–Matsuyama (Sci. Math. Japon. 2002; 55 :33–42) and Todorova–Yordanov (J. Differential Equations 2001; 174 :464). Copyright © 2004 John Wiley & Sons, Ltd.     

A New Characterization of PSL(p,q)

Abstract

Order components of a finite group are introduced in Chen [Chen, G. Y. (1996c) On Thompson's conjecture. J. Algebra 185:184–193]. It was proved that PSL(3, q), where q is an odd prime power, is uniquely determined by its order components [Iranmanesh, A., Alavi, S. H., Khosravi, B. (2002a). A characterization of PSL(3, q) where q is an odd prime power. J. Pure Appl. Algebra 170(2–3): 243–254]. Also in Iranmanesh et al. [Iranmanesh, A., Alavi, S. H., Khosravi, B. (2002b). A characterization of PSL(3, q) where q = 2 ⁿ . Acta Math. Sinica, English Ser. 18(3):463–472] and [Iranmanesh, A., Alavi, S. H. (2002). A characterization of simple groups PSL(5, q). Bull. Austral. Math. Soc. 65:211–222] it was proved that PSL(3, q) for q = 2 ⁿ and PSL(5, q) are uniquely determined by their order components. In this paper we prove that PSL(p, q) can be uniquely determined by its order components, where p is an odd prime number. A main consequence of our results is the validity of Thompson's conjecture for the groups under consideration.     

The Navier–Stokes limit of the Boltzmann equation for bounded collision kernels

The present work establishes a Navier–Stokes limit for the Boltzmann equation considered over the infinite spatial domain R ³. Appropriately scaled families of DiPerna-Lions renormalized solutions are shown to have fluctuations whose limit points (in the w-L ¹ topology) are governed by Leray solutions of the limiting Navier–Stokes equations. This completes the arguments in Bardos-Golse-Levermore [Commun. Pure Appl. Math. 46(5), 667–753 (1993)] for the steady case, and in Lions-Masmoudi [Arch. Ration. Mech. Anal. 158(3), 173–193 (2001)] for the time-dependent case.Mathematics Subject Classification (2000) 35Q35, 35Q30, 82C40