677 results
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2. Remarks on a paper of Olver
- Author
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David B. Fairlie
- Subjects
Power series ,Algebra ,Development (topology) ,Congruence (geometry) ,General Mathematics ,Product (mathematics) ,Null (mathematics) ,Field (mathematics) ,Group theory ,Associative property ,Mathematics - Abstract
SynopsisSome disparate ideas in the literature are drawn together. The work of P. J. Olver and his associates on Lagrangians which vanish for arbitrary variations, the so-called null Lagrangians, is viewed as a parallel development to Witten's study of topological field theories. A theorem of Olver, that all hyperjacobians are expressible as divergences, and are thus candidates for the construction of null Lagrangians, is shown to follow directly from the observation that such entities appear in a power series development of the general associative product, and this technique facilitates the construction of multi-dimensional examples.
- Published
- 1991
3. Some remarks on a paper by W. N. Everitt
- Author
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K. Daho and Heinz Langer
- Subjects
symbols.namesake ,Weight function ,Pure mathematics ,General Mathematics ,Operator (physics) ,Hilbert space ,symbols ,Space (mathematics) ,Mathematics - Abstract
Everitt has shown [1[, that for α ∊ [0, π/2] the undernoted problem (1.1–2) with an indefinite weight function r can be represented by a selfadjoint operator in a suitable Hilbert space. This result is extended to arbitrary α ∊ [0, π), replacing the Hilbert space in some cases by a Pontrjagin space with index one. The problem is also treated in the Krein space generated by the weight function r.
- Published
- 1977
4. A note on a paper of Atkinson concerning the asymptotics of an eigenvalue problem with interior singularity
- Author
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B. J. Harris
- Subjects
Singularity ,Differential equation ,General Mathematics ,Mathematical analysis ,Second order equation ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Singular equation ,Eigenvalues and eigenvectors ,Linear equation ,Mathematics - Abstract
SynopsisIn [2] Atkinson considers the asymptotic form of the eigenvalues of the linear differentialequationwhere a < 0 < b and y satisfies appropriate conditions at a andb. In particular Atkinson considers where q is singular at 0. In this case q(x) = xK, his results cover the case l ≦ K < . We extend Atkinson's results to cover more singular q, in the power case 1 ≦K <
- Published
- 1988
5. 16.—A Remark on a Paper by J. F. Toland and some Applications to Unilateral Problems
- Author
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Jesús Hernández and João-Paulo Dias
- Subjects
General Mathematics ,Mathematics - Abstract
SYNOPSISWe extend a result of J. F. Toland concerning bifurcation from infinity and we made some applications to variational inequalities.
- Published
- 1976
6. A note on the paper 'Existence conditions for eigenvalue problems generated by compact multiparameter operators' by P. Binding, P. J. Browne and L. Turyn
- Author
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H. Volkmer and R. Schäfke
- Subjects
Pure mathematics ,General Mathematics ,Calculus ,Eigenvalues and eigenvectors ,Mathematics - Abstract
SynopsisWe give a direct and simple proof of a central result of the above paper.
- Published
- 1985
7. Regularity results for the 2D critical Oldroyd-B model in the corotational case
- Author
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Zhuan Ye
- Subjects
Logarithm ,Cauchy stress tensor ,Computer Science::Information Retrieval ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Dissipation ,Vorticity ,01 natural sciences ,010101 applied mathematics ,A priori and a posteriori ,Oldroyd-B model ,Gravitational singularity ,0101 mathematics ,Laplace operator ,Mathematics - Abstract
This paper studies the regularity results of classical solutions to the two-dimensional critical Oldroyd-B model in the corotational case. The critical case refers to the full Laplacian dissipation in the velocity or the full Laplacian dissipation in the non-Newtonian part of the stress tensor. Whether or not their classical solutions develop finite time singularities is a difficult problem and remains open. The object of this paper is two-fold. Firstly, we establish the global regularity result to the case when the critical case occurs in the velocity and a logarithmic dissipation occurs in the non-Newtonian part of the stress tensor. Secondly, when the critical case occurs in the non-Newtonian part of the stress tensor, we first present many interesting global a priori bounds, then establish a conditional global regularity in terms of the non-Newtonian part of the stress tensor. This criterion comes naturally from our approach to obtain a global L∞-bound for the vorticity ω.
- Published
- 2019
8. Positive periodic solutions for singular fourth-order differential equations with a deviating argument
- Author
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Fanchao Kong and Zaitao Liang
- Subjects
010101 applied mathematics ,Fourth order ,Singularity ,Differential equation ,Argument ,General Mathematics ,010102 general mathematics ,Applied mathematics ,Continuation theorem ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, we study the singular fourth-order differential equation with a deviating argument:By using Mawhin's continuation theorem and some analytic techniques, we establish some criteria to guarantee the existence of positive periodic solutions. The significance of this paper is that g has a strong singularity at x = 0 and satisfies a small force condition at x = ∞, which is different from the known ones in the literature.
- Published
- 2018
9. Affine focal sets of codimension-2 submanifolds contained in hypersurfaces
- Author
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Marcelo José Saia, Marcos Craizer, and Luis F. Sánchez
- Subjects
Pure mathematics ,General Mathematics ,020207 software engineering ,02 engineering and technology ,Codimension ,GEOMETRIA DIFERENCIAL CLÁSSICA ,01 natural sciences ,Darboux vector ,0104 chemical sciences ,010404 medicinal & biomolecular chemistry ,Hypersurface ,Hyperplane ,Affine focal set ,0202 electrical engineering, electronic engineering, information engineering ,Tangent space ,Affine sphere ,Affine transformation ,Mathematics - Abstract
In this paper we study the affine focal set, which is the bifurcation set of the affine distance to submanifolds Nn contained in hypersurfaces Mn+1 of the (n + 2)-space. We give conditions under which this affine focal set is a regular hypersurface and, for curves in 3-space, we describe its stable singularities. For a given Darboux vector field ξ of the immersion N ⊂ M, one can define the affine metric g and the affine normal plane bundle . We prove that the g-Laplacian of the position vector belongs to if and only if ξ is parallel. For umbilic and normally flat immersions, the affine focal set reduces to a single line. Submanifolds contained in hyperplanes or hyperquadrics are always normally flat. For N contained in a hyperplane L, we show that N ⊂ M is umbilic if and only if N ⊂ L is an affine sphere and the envelope of tangent spaces is a cone. For M hyperquadric, we prove that N ⊂ M is umbilic if and only if N is contained in a hyperplane. The main result of the paper is a general description of the umbilic and normally flat immersions: given a hypersurface f and a point O in the (n + 1)-space, the immersion (ν, ν · (f − O)), where ν is the co-normal of f, is umbilic and normally flat, and conversely, any umbilic and normally flat immersion is of this type.
- Published
- 2018
10. Flows of measures generated by vector fields
- Author
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Emanuele Paolini and Eugene Stepanov
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Lipschitz continuity ,01 natural sciences ,Measure (mathematics) ,Integral curve ,Flow (mathematics) ,Ordinary differential equation ,0103 physical sciences ,Vector field ,010307 mathematical physics ,0101 mathematics ,Borel measure ,Smooth structure ,Mathematics - Abstract
The scope of the paper is twofold. We show that for a large class of measurable vector fields in the sense of Weaver (i.e. derivations over the algebra of Lipschitz functions), called in the paper laminated, the notion of integral curves may be naturally defined and characterized (when appropriate) by an ordinary differential equation. We further show that for such vector fields the notion of a flow of the given positive Borel measure similar to the classical one generated by a smooth vector field (in a space with smooth structure) may be defined in a reasonable way, so that the measure ‘flows along’ the appropriately understood integral curves of the given vector field and the classical continuity equation is satisfied in the weak sense.
- Published
- 2018
11. Four notions of conjugacy for abstract semigroups
- Author
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João Araújo, Michael Kinyon, António Malheiro, and Janusz Konieczny
- Subjects
Pure mathematics ,Endomorphism ,Mathematics::Operator Algebras ,General Mathematics ,010102 general mathematics ,Group Theory (math.GR) ,0102 computer and information sciences ,01 natural sciences ,Representation theory ,Automaton ,Conjugacy class ,Areas of mathematics ,010201 computation theory & mathematics ,FOS: Mathematics ,Special classes of semigroups ,0101 mathematics ,Mathematics - Group Theory ,Group theory ,Mathematics - Abstract
The action of any group on itself by conjugation and the corresponding conjugacy relation play an important role in group theory. There have been many attempts to find notions of conjugacy in semigroups that would be useful in special classes of semigroups occurring in various areas of mathematics, such as semigroups of matrices, operator and topological semigroups, free semigroups, transition monoids for automata, semigroups given by presentations with prescribed properties, monoids of graph endomorphisms, etc. In this paper we study four notions of conjugacy for semigroups, their interconnections, similarities and dissimilarities. They appeared originally in various different settings (automata, representation theory, presentations or transformation semigroups). Here we study them in maximum generality. The paper ends with a large list of open problems., Comment: The paper is now more focused on abstract semigroups and a fourth notion of conjugacy was introduced for its importance in representation theory and finite semigroups
- Published
- 2017
12. Generalized Lagrange multiplier rule for non-convex vector optimization problems
- Author
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Maria Bernadette Donato
- Subjects
021103 operations research ,Augmented Lagrangian method ,General Mathematics ,010102 general mathematics ,Tangent cone ,0211 other engineering and technologies ,Regular polygon ,02 engineering and technology ,01 natural sciences ,Constraint (information theory) ,symbols.namesake ,Constraint algorithm ,Vector optimization ,Lagrange multiplier rule, vector optimization problems, tangent cone ,Lagrange multiplier ,symbols ,Applied mathematics ,Differentiable function ,0101 mathematics ,Mathematics - Abstract
In this paper a non-convex vector optimization problem among infinite-dimensional spaces is presented. In particular, a generalized Lagrange multiplier rule is formulated as a necessary and sufficient optimality condition for weakly minimal solutions of a constrained vector optimization problem, without requiring that the ordering cone that defines the inequality constraints has non-empty interior. This paper extends the result of Donato (J. Funct. Analysis261 (2011), 2083–2093) to the general setting of vector optimization by introducing a constraint qualification assumption that involves the Fréchet differentiability of the maps and the tangent cone to the image set. Moreover, the constraint qualification is a necessary and sufficient condition for the Lagrange multiplier rule to hold.
- Published
- 2016
13. The stability index of hypersurfaces with constant scalar curvature in spheres
- Author
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Qing-Ming Cheng, Haizhong Li, and Guoxin Wei
- Subjects
Mean curvature flow ,Mean curvature ,Hypersurface ,Geodesic ,General Mathematics ,Prescribed scalar curvature problem ,Mathematical analysis ,Constant (mathematics) ,Curvature ,Scalar curvature ,Mathematics ,Mathematical physics - Abstract
The totally umbilical and non-totally geodesic hypersurfaces in the (n + 1)-dimensional spheres are characterized as the only hypersurfaces with weak stability index 0. In our 2010 paper we proved that the weak stability index of a compact hypersurface M with constant scalar curvature n(n − 1)r, r> 1, in an (n + 1)-dimensional sphere Sn + 1(1), which is not a totally umbilical hypersurface, is greater than or equal to n + 2 if the mean curvature H and H3 are constant. In this paper, we prove the same results, without the assumption that H3 is constant. In fact, we show that the weak stability index of a compact hypersurface M with constant scalar curvature n(n − 1)r, r> 1, in Sn + 1(1), which is not a totally umbilical hypersurface, is greater than or equal to n + 2 if the mean curvature H is constant.
- Published
- 2014
14. The Lax–Oleinik semi-group: a Hamiltonian point of view
- Author
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Patrick Bernard, Université Paris sciences et lettres (PSL), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), European Project: 307062,EC:FP7:ERC,ERC-2012-StG_20111012,SAW(2012), Université Paris Dauphine-PSL, École normale supérieure - Paris (ENS-PSL), and Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris)
- Subjects
Pure mathematics ,Kolmogorov–Arnold–Moser theorem ,General Mathematics ,[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] ,010102 general mathematics ,Fixed point ,Invariant (physics) ,01 natural sciences ,Convexity ,Hamiltonian system ,010101 applied mathematics ,symbols.namesake ,Compact space ,symbols ,Configuration space ,0101 mathematics ,Hamiltonian (quantum mechanics) ,Mathematics - Abstract
International audience; The weak KAM theory was developed by Fathi in order to study the dynamics of convex Hamiltonian systems. It somehow makes a bridge between viscosity solutions of the Hamilton–Jacobi equation and Mather invariant sets of Hamiltonian systems, although this was fully understood only a posteriori. These theories converge under the hypothesis of convexity, and the richness of applications mostly comes from this remarkable convergence. In this paper, we provide an elementary exposition of some of the basic concepts of weak KAM theory. In a companion paper, Albert Fathi exposed the aspects of his theory which are more directly related to viscosity solutions. Here, on the contrary, we focus on dynamical applications, even if we also discuss some viscosity aspects to underline the connections with Fathi's lecture. The fundamental reference on weak KAM theory is the still unpublished book Weak KAM theorem in Lagrangian dynamics by Albert Fathi. Although we do not offer new results, our exposition is original in several aspects. We only work with the Hamiltonian and do not rely on the Lagrangian, even if some proofs are directly inspired by the classical Lagrangian proofs. This approach is made easier by the choice of a somewhat specific setting. We work on R d and make uniform hypotheses on the Hamiltonian. This allows us to replace some compactness arguments by explicit estimates. For the most interesting dynamical applications, however, the compactness of the configuration space remains a useful hypothesis and we retrieve it by considering periodic (in space) Hamiltonians. Our exposition is centred on the Cauchy problem for the Hamilton–Jacobi equation and the Lax–Oleinik evolution operators associated to it. Dynamical applications are reached by considering fixed points of these evolution operators, the weak KAM solutions. The evolution operators can also be used for their regularizing properties; this opens an alternative route to dynamical applications. 1. The method of characteristics, existence and uniqueness of regular solutions We consider a C 2 Hamiltonian H(t, q, p) : R × R d × R d * → R and study the associated Hamiltonian system ˙ q(t) = ∂ p H(t, q(t), p(t)), ˙ p(t) = −∂ q H(t, q(t), p(t)), (HS) * This paper is a late addition to the papers surveying active areas in partial differential equations , published in issue 141.2, which were based on a series of mini-courses held in the International Centre for Mathematical Sciences (ICMS) in Edinburgh during 2010. and Hamilton–Jacobi equation ∂ t u + H(t, q, ∂ q u(t, q)) = 0. (HJ) We denote by X H (x) = X H (q, p) the Hamiltonian vector field X H = J dH, where J is the matrix J = 0 I −I 0. The Hamiltonian system can be written in condensed terms ˙ x(t) = X H (t, x(t)). We shall always assume that the solutions extend to R. We denote by ϕ t τ = (Q t τ , P t τ): R d
- Published
- 2012
15. On the polynomial vector fields on
- Author
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Jaume Llibre and Yulin Zhao
- Subjects
Combinatorics ,Polynomial ,Polynomial vector fields ,Degree (graph theory) ,General Mathematics ,Homogeneous polynomial ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Vector field ,Mathematics - Abstract
Let X be a polynomial vector field of degree n on M, M = ℝm. The dynamics and the algebraic-geometric properties of the vector fields X have been studied intensively, mainly for the case when M = ℝm, and especially when n = 2. Several papers have been dedicated to the study of the homogeneous polynomial vector field of degree n on $\mathbb{S}^2$, mainly for the case where n = 2 and M = $\mathbb{S}^2$. But there are very few results on the non-homogeneous polynomial vector fields of degree n on $\mathbb{S}^2$. This paper attempts to rectify this slightly.
- Published
- 2011
16. On travelling wavefronts of Nicholson's blowflies equation with diffusion
- Author
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Ming Mei and Chi-Kun Lin
- Subjects
Wavefront ,General Mathematics ,Mathematical analysis ,Reaction–diffusion system ,Perturbation (astronomy) ,Wave speed ,Mathematics - Abstract
This paper is devoted to the study of Nicholson's blowflies equation with diffusion: a kind of time-delayed reaction diffusion. For any travelling wavefront with speed c > c* (c* is the minimum wave speed), we prove that the wavefront is time-asymptotically stable when the delay-time is sufficiently small, and the initial perturbation around the wavefront decays to zero exponentially in space as x → −∞, but it can be large in other locations. The result develops and improves the previous wave stability obtained by Mei et al. in 2004. The new approach developed in this paper is the comparison principle combined with the technical weighted-energy method. Numerical simulations are also carried out to confirm our theoretical results.
- Published
- 2010
17. Canard cycles in the presence of slow dynamics with singularities
- Author
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Freddy Dumortier and P. De Maesschalck
- Subjects
General Mathematics ,Mathematical analysis ,Tangent ,Periodic orbits ,Perturbation (astronomy) ,Gravitational singularity ,Turning point ,Critical curve ,Eigenvalues and eigenvectors ,Mathematics ,Stability change - Abstract
We study the cyclicity of limit periodic sets that occur in families of vector flelds of slow-fast type. The limit periodic sets are formed by a fast orbit and a curve of singularities containing a unique turning point. At this turning point a stability change takes place: on one side of the turning point, the dynamics point strongly towards the curve of singularities, on the other side the dynamics point away from the curve of singularities. The presence of periodic orbits in a perturbation is related to the presence of canard orbits passing near this turning point, i.e. orbits that stay close to the curve of singularities despite the exponentially-strong repulsion near this curve. All existing results deal with a non-zero slow movement permitting to get a good estimate of the cyclicity by considering the slow divergence integral along the curve of singularities. In this paper we study what happens when the slow dynamics exhibit singularities. In particular our study includes the cyclicity of the slow-fast 2-saddle cycle, formed by a regular saddle-connection (the fast part) and a part of the curve of singularities (the slow part). We see that the relevant information is no longer merely contained in the slow divergence integral. This paper concerns the study of the cyclicity of limit periodic sets in a quite general class of slow-fast vector flelds on a 2-manifold M. We are interested in families of vector flelds X" (possibly depending on other parameters as well) where the unperturbed \fast" vector fleld X0 has a curve of singular points ∞, called a critical curve. We call a point p on ∞ normally attracting (resp. normally repelling) when DX0(p) has a strictly negative (resp. strictly positive) eigenvalue corresponding to an eigendirection not tangent to ∞. When ∞ has both normally attracting points and normally repelling points it may occur that X0 has orbits connecting two such points. Let F be such a fast orbit so that the !-limit and fi-limit lie on ∞. We study the limit periodic set LF formed by F and the piece of ∞ going from the !-limit of F to the fi-limit of F. Of course the nonzero eigenvalue of DX0(p) must bifurcate along this piece of ∞; assume that this happens in a unique point p⁄, called a turning point. (One also says normal hyperbolicity of X0 is lost in p⁄.) In such a situation it is possible that the limit periodic set perturbs into one or more isolated periodic orbits; such orbits are called canard cycles. ‡ @’
- Published
- 2008
18. Singular solutions for the Uehling–Uhlenbeck equation
- Author
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Juan J. L. Velázquez, Stéphane Mischler, and Miguel Escobedo
- Subjects
Physical point ,Regular singular point ,Singular function ,Semigroup ,Singular solution ,General Mathematics ,Mathematical analysis ,Mathematics - Abstract
In this paper we prove the existence of solutions of the Uehling–Uhlenbeck equation that behave like k −7/6 as k → 0. From the physical point of view, such solutions can be thought as particle distributions in the space of momentum having a sink (or a source) of particles with zero momentum. Our construction is based on the precise estimates of the semigroup for the linearized equation around the singular function k −7/6 that we obtained in an earlier paper.
- Published
- 2008
19. Graphs of $W^{1,1}$-maps with values in $S^{1}$: relaxed energies, minimal connections and lifting
- Author
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Mariano Giaquinta and Domenico Mucci
- Subjects
Combinatorics ,Work (thermodynamics) ,General Mathematics ,Link (knot theory) ,Mathematics - Abstract
The aim of this paper is to link the analytic results of Brezis et al. , Demengel and Ignat relative to $W^{1,1}$ -mappings from $B^n$ into $S^1$ to the measure-theoretical geometric results in our previous work. The paper also contains a few remarks about mappings in $W^{1,p}$ , $p\geq2$ , with values in $S^2$ .
- Published
- 2007
20. Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems
- Author
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Messoud Efendiev, Alain Miranville, and Sergey Zelik
- Subjects
Projected dynamical system ,Dynamical systems theory ,General Mathematics ,Mathematical analysis ,Attractor ,Statistical physics ,Limit set ,Dynamical system ,Random dynamical system ,Mathematics ,Linear dynamical system ,Hamiltonian system - Abstract
We suggest in this paper a new explicit algorithm allowing us to construct exponential attractors which are uniformly Hölder continuous with respect to the variation of the dynamical system in some natural large class. Moreover, we extend this construction to non-autonomous dynamical systems (dynamical processes) treating in that case the exponential attractor as a uniformly exponentially attracting, finite-dimensional and time-dependent set in the phase space. In particular, this result shows that, for a wide class of non-autonomous equations of mathematical physics, the limit dynamics remains finite dimensional no matter how complicated the dependence of the external forces on time is. We illustrate the main results of this paper on the model example of a non-autonomous reaction–diffusion system in a bounded domain.
- Published
- 2005
21. Constructing the symplectic Evans matrix using maximally analytic individual vectors
- Author
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Gianne Derks and Thomas J. Bridges
- Subjects
Determinant ,Symplectic vector space ,Pure mathematics ,General Mathematics ,Mathematical analysis ,Symplectic representation ,Symplectomorphism ,Mathematics::Symplectic Geometry ,Moment map ,Symplectic matrix ,Mathematics ,Symplectic manifold ,Symplectic geometry - Abstract
For linear systems with a multi-symplectic structure, arising from the linearization of Hamiltonian partial differential equations about a solitary wave, the Evans function can be characterized as the determinant of a matrix, and each entry of this matrix is a restricted symplectic form. This variant of the Evans function is useful for a geometric analysis of the linear stability problem. But, in general, this matrix of two-forms may have branch points at isolated points, shrinking the natural region of analyticity. In this paper, a new construction of the symplectic Evans matrix is presented, which is based on individual vectors but is analytic at the branch points—indeed, maximally analytic. In fact, this result has greater generality than just the symplectic case; it solves the following open problem in the literature: can the Evans function be constructed in a maximally analytic way when individual vectors are used? Although the non-symplectic case will be discussed in passing, the paper will concentrate on the symplectic case, where there are geometric reasons for evaluating the Evans function on individual vectors. This result simplifies and generalizes the multi-symplectic framework for the stability analysis of solitary waves, and some of the implications are discussed.
- Published
- 2003
22. Variational characterizations of weighted Hardy spaces and weighted spaces
- Author
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Yongming Wen, Huoxiong Wu, Weichao Guo, and Dongyong Yang
- Subjects
symbols.namesake ,Pure mathematics ,General Mathematics ,symbols ,Hardy space ,Mathematics - Abstract
This paper obtains new characterizations of weighted Hardy spaces and certain weighted $BMO$ type spaces via the boundedness of variation operators associated with approximate identities and their commutators, respectively.
- Published
- 2021
23. Analysis of the PML equations in general convex geometry
- Author
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Erkki Somersalo and Matti Lassas
- Subjects
Convex geometry ,Scattering ,General Mathematics ,Mathematical analysis ,Boundary (topology) ,Geometry ,010103 numerical & computational mathematics ,01 natural sciences ,Domain (mathematical analysis) ,010101 applied mathematics ,symbols.namesake ,Perfectly matched layer ,Bounded function ,Dirichlet boundary condition ,symbols ,0101 mathematics ,Convex function ,Mathematics - Abstract
In this work, we study a mesh termination scheme in acoustic scattering, known as the perfectly matched layer (PML) method. The main result of the paper is the following. Assume that the scatterer is contained in a bounded and strictly convex artificial domain. We surround this domain by a PML of constant thickness. On the peripheral boundary of this layer, a homogenous Dirichlet condition is imposed. We show in this paper that the resulting boundary-value problem for the scattered field is uniquely solvable for all wavenumbers and the solution within the artificial domain converges exponentially fast toward the full-space scattering solution when the layer thickness is increased. The proof is based on the idea of interpreting the PML medium as a complex stretching of the coordinates in Rn and on the use of complexified layer potential techniques.
- Published
- 2001
24. The spectral boundary of complemented invariant subspaces in Lp(R)
- Author
-
Zoltán Buczolich and Alexander Olevskii
- Subjects
Discrete mathematics ,Pure mathematics ,Invariant polynomial ,General Mathematics ,Reflexive operator algebra ,Invariant (mathematics) ,Linear subspace ,Mathematics - Abstract
In this paper we construct a compact set K of zero Hausdorff dimension that satisfies certain ‘arithmetic-type’ thickness properties. The concept of ‘arithmetic thickness’ has its origins in applications to harmonic analysis, introduced in a paper by Lebedev and Olevskiĭ. For example, there are no spectral sets whose ‘essential boundary’ can contain the above set K.
- Published
- 2001
25. Ergodic properties and Weyl M-functions for random linear Hamiltonian systems
- Author
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Russell Johnson, Sylvia Novo, and Rafael Obaya
- Subjects
Floquet theory ,Computer Science::Information Retrieval ,General Mathematics ,Mathematical analysis ,Schrödinger equation ,Hamiltonian system ,symbols.namesake ,symbols ,Ergodic theory ,Covariant Hamiltonian field theory ,Superintegrable Hamiltonian system ,Invariant (mathematics) ,Hamiltonian (control theory) ,Mathematical physics ,Mathematics - Abstract
This paper provides a topological and ergodic analysis of random linear Hamiltonian systems. We consider a class of Hamiltonian equations presenting absolutely continuous dynamics and prove the existence of the radial limits of the Weyl M-functions in the L1-topology. The proof is based on previous ergodic relations obtained for the Floquet coefficient. The second part of the paper is devoted to the qualitative description of disconjugate linear Hamiltonian equations. We show that the principal solutions at ±∞ define singular ergodic measures, and determine an invariant region in the Lagrange bundle which concentrates the essential dynamical information. We apply this theory to the study of the n-dimensional Schrödinger equation at the first point of the spectrum.
- Published
- 2000
26. The HELP inequality for lim-p Hamiltonian systems
- Author
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Marco Marletta and Brian Malcolm Brown
- Subjects
Combinatorics ,Pure mathematics ,Singularity ,Hamiltonian lattice gauge theory ,Cover (topology) ,General Mathematics ,Ordinary differential equation ,Covariant Hamiltonian field theory ,Superintegrable Hamiltonian system ,Limit (mathematics) ,Hamiltonian system ,Mathematics - Abstract
In a recent paper, Brown, Evans and Marletta extended the HardyEverittLittlewoodPolya inequality from 2nth-order formally self-adjoint ordinary differential equations to a wide class of linear Hamiltonian systems in 2n variables. The paper considered only problems on semi-infinite intervals [a, ∞) with a limit-point type singularity at infinity. In this paper we extend the theory to cover all types of endpoint ( lim-p for n ≤ p ≤ 2n ).
- Published
- 2000
27. Free quotients of infinite rank of GL2 over Dedekind domains
- Author
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A. W. Mason
- Subjects
Combinatorics ,symbols.namesake ,General Mathematics ,Dedekind sum ,symbols ,Dedekind eta function ,Rank (graph theory) ,Dedekind cut ,Quotient ,Dedekind–MacNeille completion ,Mathematics - Abstract
This paper is concerned with integral domains R, for which the factor group SL2(R)/U2(R) has a non-trivial, free quotient, where U2(R) is the subgroup of GL2(R) generated by the unipotent matrices. Recently, Krstić and McCool have proved that SL2(P[x])/U2(P[x]) has a free quotient of infinite rank, where P is a domain which is not a field. This extends earlier results of Grunewald, Mennicke and Vaserstein.Any ring of the type P[x] has Krull dimension at least 2. The purpose of this paper is to show that result of Krstić and McCool extends to some domains of Krull dimension 1, in particular to certain Dedekind domains. This result, which represents a two-dimensional anomaly is the best possible in the following sense. It is well known that SL2(R) = U2(R), when R is a domain of Krull dimension zero, i.e. when R is a field. It is already known that for some arithmetic Dedekind domains A, the factor group SL2(A)/U2(A) has a free quotient of finite (and not infinite) rank.
- Published
- 1999
28. Global regularity criterion for the dissipative systems modelling electrohydrodynamics involving the middle eigenvalue of the strain tensor
- Author
-
Fan Wu
- Subjects
Physics::Fluid Dynamics ,General Mathematics ,Mathematical analysis ,Dissipative system ,Infinitesimal strain theory ,Electrohydrodynamics ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper, we study a dissipative systems modelling electrohydrodynamics in incompressible viscous fluids. The system consists of the Navier–Stokes equations coupled with a classical Poisson–Nernst–Planck equations. In the three-dimensional case, we establish a global regularity criteria in terms of the middle eigenvalue of the strain tensor in the framework of the anisotropic Lorentz spaces for local smooth solution. The proof relies on the identity for entropy growth introduced by Miller in the Arch. Ration. Mech. Anal. [16].
- Published
- 2021
29. The Fokker–Planck equation for the time-changed fractional Ornstein–Uhlenbeck stochastic process
- Author
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Yuliya Mishura, Enrica Pirozzi, Giacomo Ascione, Ascione, G., Mishura, Y., and Pirozzi, E.
- Subjects
Stochastic process ,General Mathematics ,generalized Fokker-Planck equation ,fractional Brownian motion ,subordinator ,Ornstein–Uhlenbeck process ,Fokker–Planck equation ,Statistical physics ,Caputo-type derivative ,Bernstein function ,Mathematics - Abstract
In this paper, we study some properties of the generalized Fokker–Planck equation induced by the time-changed fractional Ornstein–Uhlenbeck process. First of all, we exploit some sufficient conditions to show that a mild solution of such equation is actually a classical solution. Then, we discuss an isolation result for mild solutions. Finally, we prove the weak maximum principle for strong solutions of the aforementioned equation and then a uniqueness result.
- Published
- 2021
30. Qualitative properties of singular solutions to fractional elliptic equations
- Author
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Shuibo Huang, Zhitao Zhang, and Zhisu Liu
- Subjects
General Mathematics ,Mathematics - Abstract
In this paper, by the moving spheres method, Caffarelli-Silvestre extension formula and blow-up analysis, we study the local behaviour of nonnegative solutions to fractional elliptic equations \begin{align*} (-\Delta)^{\alpha} u =f(u),~~ x\in \Omega\backslash \Gamma, \end{align*} where $0, $\Omega = \mathbb {R}^{N}$ or $\Omega$ is a smooth bounded domain, $\Gamma$ is a singular subset of $\Omega$ with fractional capacity zero, $f(t)$ is locally bounded and positive for $t\in [0,\,\infty )$, and $f(t)/t^{({N+2\alpha })/({N-2\alpha })}$ is nonincreasing in $t$ for large $t$, rather than for every $t>0$. Our main result is that the solutions satisfy the estimate \begin{align*} f(u(x))/ u(x)\leq C d(x,\Gamma)^{{-}2\alpha}. \end{align*} This estimate is new even for $\Gamma =\{0\}$. As applications, we derive the spherical Harnack inequality, asymptotic symmetry, cylindrical symmetry of the solutions.
- Published
- 2021
31. Rank one plus a null-Lagrangian is an inherited property of two-dimensional compliance tensors under homogenisation
- Author
-
Graeme W. Milton and Yury Grabovsky
- Subjects
Shear modulus ,Algebra ,Rank (linear algebra) ,General Mathematics ,Mathematical analysis ,Isotropy ,Tensor ,Space (mathematics) ,Representation (mathematics) ,Null (physics) ,Mathematics ,Moduli - Abstract
Assume that the local compliance tensor of an elastic composite in two space dimensions is equal to a rank-one tensor plus a null-Lagrangian (there is only one symmetric one in two dimensions). The purpose of this paper is to prove that the effective compliance tensor has the same representation: rank-one plus the null-Lagrangian. This statement generalises the wellknown result of Hill that a composite of isotropic phases with a common shear modulus is necessarily elastically isotropic and shares the same shear modulus. It also generalises the surprising discovery of Avellaneda et al. that under a certain condition on the pure crystal moduli the shear modulus of an isotropic polycrystal is uniquely determined. The present paper sheds light on this effect by placing it in a more general framework and using some elliptic PDE theory rather than the translation method. Our results allow us to calculate the polycrystalline G-closure of the special class of crystals under consideration. Our analysis is contrasted with a two-dimensional model problem for shape-memory polycrystals. We show that the two problems can be thought of as ‘elastic percolation’ problems, one elliptic, one hyperbolic.
- Published
- 1998
32. A relationship between the periodic and the Dirichlet BVPs of singular differential equations
- Author
-
Meirong Zhang
- Subjects
Differential equation ,General Mathematics ,media_common.quotation_subject ,Mathematical analysis ,Infinity ,Dirichlet distribution ,symbols.namesake ,Singularity ,Ordinary differential equation ,Dirichlet's principle ,symbols ,Gravitational singularity ,Boundary value problem ,media_common ,Mathematics - Abstract
In this paper, a relationship between the periodic and the Dirichlet boundary value problems for second-order ordinary differential equations with singularities is established. This relationship may be useful in explaining the difference between the nonresonance of singular and nonsingular differential equations. Using this relationship, we give in this paper an existence result of positive periodic solutions to singular differential equations when the singular forces satisfy some strong force condition at the singularity 0 and some linear growth condition at infinity.
- Published
- 1998
33. Convergence of the viscosity solutions for weakly strictly hyperbolic conservation laws
- Author
-
Zhu Changjiang
- Subjects
Conservation law ,General Mathematics ,Viscosity (programming) ,Convergence (routing) ,Mathematical analysis ,Mathematics - Abstract
SynopsisThis paper is an extension of papers [14–16]. Using the theory of compensated compactness, we establish the convergence of the uniformly bounded approximate solution sequence for a class of ‘weakly strictly hyperbolic’ conservation laws.
- Published
- 1997
34. Regularity of the solutions for elliptic problems on nonsmooth domains in ℝ3. Part II: Regularity in neighbourhoods of edges
- Author
-
Ivo Babuška and Benqi Guo
- Subjects
Sobolev space ,Discrete mathematics ,Pure mathematics ,Partial differential equation ,Series (mathematics) ,General Mathematics ,Computation ,Convergence (routing) ,Poisson's equation ,Domain (mathematical analysis) ,Finite element method ,Mathematics - Abstract
This paper is the second in a series of three devoted to the analysis of the regularity of solutions of elliptic problems on nonsmooth domains in ℝ3. The present paper concentrates on the regularity of solutions of the Poisson equation in neighbourhoods of edges of a polyhedral domain in the framework of the weighted Sobolev spaces and countably normed spaces. These results can be generalised to elliptic problems arising from mechanics and engineering, for instance, the elasticity problem on polyhedral domains. Hence, the results are not only important to understand comprehensively the qualitative and quantitative aspects of the behaviours of the solution and its derivatives of all orders in neighbourhoods of edges, but also essential to design an effective computation and analyse the optimal convergence of the finite elements solutions for these problems.
- Published
- 1997
35. Periodic homogenisation of Hamilton–Jacobi equations: 2. Eikonal equations
- Author
-
Marie C. Concordel
- Subjects
symbols.namesake ,Integrable system ,Eikonal equation ,General Mathematics ,Calculus ,symbols ,Regular polygon ,Hamiltonian (quantum mechanics) ,Hamilton–Jacobi equation ,Convexity ,Mathematical physics ,Mathematics - Abstract
Homogenisation of the first-order Hamilton-Jacobi equations when H is periodic in the second variable, leads to an effective Hamiltonian H satisfying: uε converges, as ε → 0, to the solution u of ut + H(Du) = 0. In our first paper, we assumed that H is convex and we derived a variational formula giving H. In this second paper, we consider eikonal equations, i.e. H(p, x) = ½|p|2 – V(x). Using our variational formula, we compute explicitly the effective Hamiltonian in several cases, and we study precisely the lack of strict convexity for H (‘flat part’ around the origin).
- Published
- 1997
36. Regularity of the solutions for elliptic problems on nonsmooth domains in ℝ3, Part I: countably normed spaces on polyhedral domains
- Author
-
Benqi Guo and Ivo Babuška
- Subjects
Sobolev space ,Pure mathematics ,Continuous function ,General Mathematics ,Mathematical analysis ,Neighbourhood (graph theory) ,Structure (category theory) ,Piecewise ,Ellipse ,Mathematics ,Vector space ,Analytic function - Abstract
This is the first of a series of three papers devoted to the regularity of solutions of elliptic problems on nonsmooth domains in ℝ3. The present paper introduces various weighted spaces and countably weighted spaces in the neighbourhood of edges and vertices of polyhedral domains, and it concentrates on exploring the structure of these spaces, such as the embeddings of weighted Sobolev spaces, the relation between weighted Sobolev spaces and weighted continuous function spaces, and the relations between the weighted Sobolev spaces and countably weighted Sobolev spaces in Cartesian coordinates and in the spherical and cylindrical coordinates. These well-defined spaces are the foundation for the comprehensive study of the regularity theory of elliptic problems with piecewise analytic data in ℝ3, which are essential for the design of effective computation and the analysis of the h – p version of the finite element method for solving elliptic problems in three-dimensional nonsmooth domains arising from mechanics and engineering.
- Published
- 1997
37. Sharpness of embeddings in logarithmic Bessel-potential spaces
- Author
-
Petr Gurka, Bohumír Opic, and David E. Edmunds
- Subjects
Pure mathematics ,Continuation ,Logarithm ,General Mathematics ,Double exponential function ,Bessel potential ,Embedding ,Sense (electronics) ,Type (model theory) ,Mathematics - Abstract
This paper is a continuation of [4], where embeddings of certain logarithmic Bessel-potential spaces (modelled upon generalised Lorentz-Zygmund spaces) in appropriate Orlicz spaces (with Young functions of single and double exponential type) were derived. The aim of this paper is to show that these embedding results are sharp in the sense of [8].
- Published
- 1996
38. Positivity of solutions of elliptic equations with nonlocal terms
- Author
-
A. Barabanova and W. Allegretto
- Subjects
Nonlinear system ,Partial differential equation ,Distribution (number theory) ,General Mathematics ,Mathematical analysis ,Value (mathematics) ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper we study a nonlocal problem for a second-order partial differential equation which depends on a parametern. We prove the existence of critical values 0
such that for all≦ɳ≦and for all non-negative right-hand sides, our problem has nonnegative solutions. We obtain a formula forɳ0, the maximal possible value of, and find the exact value of ɳ for spherical ɳ. We also study the corresponding eigenvalue problem. At the end of the paper, we consider the application of our results to the nonlinear system describing the distribution of temperature and potential in a microsensor. - Published
- 1996
39. L1-approximation of Fourier series of complex-valued functions
- Author
-
T. F. Xie and S. P. Zhou
- Subjects
symbols.namesake ,Fourier transform ,Fourier analysis ,Discrete-time Fourier transform ,General Mathematics ,Discrete Fourier series ,Mathematical analysis ,Conjugate Fourier series ,Fourier sine and cosine series ,symbols ,Fourier series ,Mathematics ,Parseval's theorem - Abstract
V. B. Stanojevic suggested in her recent paper that it would be of interest to prove a corresponding L1-convergence theorem for Fourier series with complex O-regularly varying quasimonotonc coefficients. The present paper will discuss this question and establish L1-convergence and. furthermore. L1-approximation theorems for complex-valued integrable functions.
- Published
- 1996
40. An exotic totally real minimal immersion of S3 in ℂP3 and its characterisation
- Author
-
Luc Vrancken, Bang-Yen Chen, Franki Dillen, and Leopold Verstraelen
- Subjects
Combinatorics ,Mean curvature ,Computer Science::Information Retrieval ,General Mathematics ,Complex projective space ,Space form ,Mathematics::Differential Geometry ,Sectional curvature ,Complex dimension ,Riemannian manifold ,Submanifold ,Scalar curvature ,Mathematics - Abstract
In a previous paper, B.-Y. Chen defined a Riemannian invariant δ by subtracting from the scalar curvature at every point of a Riemannian manifold the smallest sectional curvature at that point, and proved, for a submanifold of a real space form, a sharp inequality between δ and the mean curvature function. In this paper, we extend this inequality to totally real submanifolds of a complex space form. As a consequence, we obtain a metric obstruction for a Riemannian manifold Mn to admit a minimal totally real (i.e. Lagrangian) immersion into a complex space form of complex dimension n. Next we investigate three-dimensional submanifolds of the complex projective space ℂP3 which realise the equality in the inequality mentioned above. In particular, we construct and characterise a totally real minimal immersion of S3 in ℂP3.
- Published
- 1996
41. Lifting of recollements and gluing of partial silting sets
- Author
-
Alexandra Zvonareva and Manuel Saorín
- Subjects
Noetherian ,Pure mathematics ,General Mathematics ,01 natural sciences ,Lift (mathematics) ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,0103 physical sciences ,FOS: Mathematics ,Category Theory (math.CT) ,16E35, 18E30 ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics::Representation Theory ,Equivalence (measure theory) ,Mathematics ,Mathematics::Rings and Algebras ,010102 general mathematics ,Coproduct ,Mathematics - Category Theory ,Mathematics - Rings and Algebras ,Rings and Algebras (math.RA) ,Bounded function ,Torsion (algebra) ,010307 mathematical physics ,Mathematics - Representation Theory - Abstract
This paper focuses on recollements and silting theory in triangulated categories. It consists of two main parts. In the first part a criterion for a recollement of triangulated subcategories to lift to a torsion torsion-free triple (TTF triple) of ambient triangulated categories with coproducts is proved. As a consequence, lifting of TTF triples is possible for recollements of stable categories of repetitive algebras or self-injective finite length algebras and recollements of bounded derived categories of separated Noetherian schemes. When, in addition, the outer subcategories in the recollement are derived categories of small linear categories the conditions from the criterion are sufficient to lift the recollement to a recollement of ambient triangulated categories up to equivalence. In the second part we use these results to study the problem of constructing silting sets in the central category of a recollement generating the t-structure glued from the silting t-structures in the outer categories. In the case of a recollement of bounded derived categories of Artin algebras we provide an explicit construction for gluing classical silting objects.
- Published
- 2021
42. Unbalanced optimal total variation transport problems and generalized Wasserstein barycenters
- Author
-
Nhan-Phu Chung and Thanh-Son Trinh
- Subjects
010101 applied mathematics ,Variation (linguistics) ,General Mathematics ,010102 general mathematics ,Applied mathematics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, we establish a Kantorovich duality for unbalanced optimal total variation transport problems. As consequences, we recover a version of duality formula for partial optimal transports established by Caffarelli and McCann; and we also get another proof of Kantorovich–Rubinstein theorem for generalized Wasserstein distance$\widetilde {W}_1^{a,b}$proved before by Piccoli and Rossi. Then we apply our duality formula to study generalized Wasserstein barycenters. We show the existence of these barycenters for measures with compact supports. Finally, we prove the consistency of our barycenters.
- Published
- 2021
43. On the absolutely continuous subspaces of Floquet operators
- Author
-
Min-Jei Huang
- Subjects
Floquet theory ,Algebra ,Operator (computer programming) ,General Mathematics ,Mathematical analysis ,Absolute continuity ,Linear subspace ,Subspace topology ,Mathematics - Abstract
The purpose of this paper is to describe various subspaces that are closely related to the absolutely continuous subspace of a Floquet operator. This paper generalises and extends several known results.
- Published
- 1994
44. On the analysis and control of hyperbolic systems associated with vibrating networks
- Author
-
Günter Leugering, J. E. Lagnese, and E. J. P. G. Schmidt
- Subjects
Timoshenko beam theory ,Controllability ,General linear model ,Control theory ,General Mathematics ,Mathematical analysis ,Vibration control ,Control (linguistics) ,Closed loop ,Hyperbolic partial differential equation ,Hyperbolic systems ,Mathematics - Abstract
In this paper a general linear model for vibrating networks of one-dimensional elements is derived. This is applied to various situations including nonplanar networks of beams modelled by a three-dimensional variant on the Timoshenko beam, described for the first time in this paper. The existence and regularity of solutions is established for all the networks under consideration. The methods of first-order hyperbolic systems are used to obtain estimates from which exact controllability follows for networks containing no closed loops.
- Published
- 1994
45. Hyperasymptotics and the Stokes' phenomenon
- Author
-
A. B. Olde Daalhuis
- Subjects
symbols.namesake ,Integral representation ,General Mathematics ,Phenomenon ,Confluence ,Mathematical analysis ,symbols ,Hypergeometric function ,Series expansion ,Bessel function ,Mathematics - Abstract
SynopsisHyperasymptotic expansions were recently introduced by Berry and Howls, and yield refined information by expanding remainders in asymptotic expansions. In a recent paper of Olde Daalhuis, a method was given for obtaining hyperasymptotic expansions of integrals that represent the confluent hypergeometric U-function. This paper gives an extension of that method to neighbourhoods of the so-called Stokes lines. At each level, the remainder is exponentially small compared with the previous remainders. Two numerical illustrations confirm these exponential improvements.
- Published
- 1993
46. On the commutativity of certain quasi-differential expressions II
- Author
-
H. Frentzen, D. Race, and Anton Zettl
- Subjects
Pure mathematics ,Computer Science::Information Retrieval ,General Mathematics ,Mathematical analysis ,Commutative property ,Differential (mathematics) ,Mathematics - Abstract
SynopsisWe consider the question: when do two ordinary, linear, quasi-differential expressions commute? For classical differential expressions, answers to this question are well known. The set of all expressions which commute with a given such expression form a commutative ring. For quasi-differential expressions less is known and such an algebraiastructure can no longer be exploited. Using the theory of very general quasi-differential expressions with matrix-valued coefficients, we prove some general results concerning commutativity of such expressions. We show how, when specialised to scalar expressions, these results include a proof of the conjecture that if a 2nth-order scalar, J-symmetric (or real symmetric) quasi-differential expression commutes with a second order expression having the same properties, then the former must be an nth-order polynomial in the latter. This result was conjectured in a paper by Race and Zettl, to which this paper is a sequel.
- Published
- 1993
47. Symmetry sets of piecewise-circular curves
- Author
-
Thomas Banchoff and Peter Giblin
- Subjects
Pure mathematics ,General Mathematics ,Piecewise ,Geometry ,Symmetry (geometry) ,Mathematics - Abstract
SynopsisPiecewise-circular (PC) curves are made up of circular arcs and segments of straight lines, joined so that the (undirected) tangent line turns continuously. PC curves have arisen in various applications where they are used to approximate smooth curves. In a previous paper, the authors introduced some of their geometrical properties. In this paper they investigate the ‘symmetry sets’ of PC curves and one-parameter families of such curves. The symmetry set has also arisen in applications (this time to shape recognition) and its mathematical properties for smooth curves have been investigated by Bruce, Giblin and Gibson. It turns out that the symmetry sets of general one-parameter families of plane curves are mirrored remarkably faithfully by the symmetry sets arising from the much simpler class of PC curves.
- Published
- 1993
48. Solutions in Lebesgue spaces of the Navier-Stokes equations with dynamic boundary conditions
- Author
-
Niko Sauer and Marié Grobbelaar-Van Dalsen
- Subjects
General Mathematics ,Mathematical analysis ,Rigid body ,Lebesgue integration ,Lebesgue–Stieltjes integration ,Physics::Fluid Dynamics ,symbols.namesake ,Compressibility ,symbols ,Boundary value problem ,Lp space ,Navier–Stokes equations ,Rotation (mathematics) ,Mathematics - Abstract
SynopsisThis paper, although self-contained, is a continuation of the work done in [8], where the motion of a viscous, incompressible fluid is considered in conjunction with the rotation of a rigid body which is immersed in the fluid. The resulting mathematical model is a Navier-Stokes problem with dynamic boundary conditions. In [8] a uniqueL2,3solution is constructed under certain regularity assumptions on the initial states. In this paper we consider the Navier-Stokes problem with dynamic boundary conditions in the Lebesgue spacesLr,3(3
- Published
- 1993
49. The effects of diffusion on the principal eigenvalue for age-structured models with random diffusion
- Author
-
Hao Kang
- Subjects
010101 applied mathematics ,Random diffusion ,General Mathematics ,010102 general mathematics ,Principal (computer security) ,Statistical physics ,Mathematics::Spectral Theory ,0101 mathematics ,Diffusion (business) ,01 natural sciences ,Age structured ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper, we study the principal spectral theory of age-structured models with random diffusion. First, we provide an equivalent characteristic for the principal eigenvalue, the strong maximum principle and a positive strict super-solution. Then, we use the result to investigate the effects of diffusion rate on the principal eigenvalue. Finally, we study how the principal eigenvalue affects the global dynamics of the KPP model and verify that the principal eigenvalue being zero is a critical value.
- Published
- 2021
50. Existence of solution for elliptic equations with supercritical Trudinger–Moser growth
- Author
-
Luiz F. O. Faria and Marcelo Montenegro
- Subjects
010101 applied mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,0101 mathematics ,01 natural sciences ,Supercritical fluid ,Mathematics - Abstract
This paper is concerned with the existence of solutions for a class of elliptic equations on the unit ball with zero Dirichlet boundary condition. The nonlinearity is supercritical in the sense of Trudinger–Moser. Using a suitable approximating scheme we obtain the existence of at least one positive solution.
- Published
- 2021
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