pinecone/core-2020-05-10-deduplication huggingface dataset:
Sewing Ribbons on Graphs in Space

AbstractAn open ribbon is a square with one side called the seam. A closed ribbon is a cylinder with one boundary component called the seam. We sew an open (resp. closed) ribbon onto a graph by identifying the seam with an open (resp. closed) walk in the graph. A ribbon complex is a graph with a finite number of ribbons sewn on. We investigate when a ribbon complex embeds in 3-dimensional Euclidean space. We give several characterizations of such spatial complexes which lead to algorithms. We examine special cases where (1) each edge of the graph is incident with at most three ribbons, and (2) every ribbon is closed together with a connectivity condition
The Quadrangulation Conjecture for Orientable Surfaces

AbstractBy means of character theory and symmetric functions, D. M. Jackson and T. I. Visentin (1990, Trans. Amer. Math. Soc.322, (343–363)) proved the existence of certain bijections between the set of quadrangulations in orientable surfaces and decorated maps (with marked edges and coloured vertices) in orientable surfaces. The bijections preserve a weight function consisting of a pair (g,n) of integer parameters. For quadrangulations, g is the genus and n is the number of faces. For decorated maps, g is the genus plus half the number of white vertices and n is the number of edges. The Quadrangulation Conjecture concerns the problem of finding a natural bijection of this type. Tutte's medial construction is a solution in the special case g=0 of planar maps. We give a construction of a bijection Ξ&#x0303; which both extends Tutte's medial construction to non-planar maps and preserves the parameter n of the Quadrangulation Conjecture. (The parameter g is not generally preserved, except when g=0.) Non-orientable surfaces play an important part in the construction of Ξ&#x0303;. As part of the construction, we introduce a bijection between orientable rooted quadrangulations and locally orientable, bipartite, rooted quadrangulations
Nilpotent Families of Endomorphisms of (P(V)+,∪)

AbstractA directed graph G=(V,A) is k-nice if for every u,v∈V (allowing t=v), and for every orientation of the edges of an undirected path of length k, there exists a u−v walk of length k in G whose orientation coincides with that of the given path. A graph is nice if it is k-nice for some k. We generalize this notion using the notion of a nilpotent semigroup of endomorphisms of (P(V)+,∪), and consider two basic problems:&#x02022;find bounds for the nilpotency class of such semigroups in terms of their generators (in the language of graphs: provided that a graph on vertices is nice, find the smallest such that is -nice);&#x02022;find a way to demonstrate non-nilpotency of such semigroups (find as simple as possible characterization of non-nice graphs)
Circular Colouring and Orientation of Graphs

AbstractThis paper proves that if a graph G has an orientation D such that for each cycle C with d∣C∣ (mod k)∈{1,2,…,2d−1} we have ∣C∣/∣C+∣⩽k/d and ∣C∣/∣C−∣⩽k/d, then G has a (k,d)-colouring and hence χc(G)⩽k/d. This is a generalization of a result of Tuza (J. Combin. Theory Ser. B55 (1992), 236–243) concerning the vertex colouring of a graph, and is also a strengthening of a result of Goddyn et al. (J. Graph Theory28 (1998), 155–161) concerning the relation between orientation and circular chromatic number of a graph
On the Connectivity Function of a Binary Matroid

AbstractIn this paper, we shall consider the following problem: up to duality, is a connected matroid reconstructible from its connectivity function? Cunningham conjectured that this question has an affirmative answer, but Seymour gave a counter-example for it. In the same paper, Seymour proved that a connected binary matroid is always reconstructible from its connectivity function, up to duality, in the class of binary matroids. In this paper, we prove that when M is a connected binary matroid and N another matroid with the same connectivity function as N, then M=N or M=N*
On Matroids of Branch-Width Three

AbstractFor all positive integers k, the class Bk of matroids of branch-width at most k is minor-closed. When k is 1 or 2, the class Bk is, respectively, the class of direct sums of loops and coloops, and the class of direct sums of series–parallel networks. B3 is a much richer class as it contains infinite antichains of matroids and is thus not well-quasi-ordered under the minor order. In this paper, it is shown that, like B1 and B2, the class B3 can be characterized by a finite list of excluded minors
On a Conjecture of Woodall

AbstractDirac proved in 1952 that every 2-connected graph of order n and minimum degree k admits a cycle of length at least {n,2k}. As a possible improvement, Woodall conjectured in 1975 that if a 2-connected graph of order n has at least n2+k vertices of degree at least k, then it has a cycle of length at least 2k. This conjecture was one of the 50 unsolved problems in Bondy and Murty (“Graph Theory with Applications,” Macmillan Press, New York, 1976). Häggkvist and Jackson showed in 1985 that this conjecture is true if n⩽3k−2. Häggkvist and Li proved that this result is true if the graph is 3-connected. In this paper, we essentially verify Woodall's conjecture
Another Proof of the Map Color Theorem for Nonorientable Surfaces

AbstractThe simplest known proof of the Map Color Theorem for nonorientable surfaces (obtained by Youngs, Ringel et al. and given in Ringel's book “Map Color Theorem”) uses index one and three current graphs, and index two and three inductive constructions. We give another proof, still using current graphs, but simpler than Youngs' and Ringel's in several ways. Our proof uses only index one current graphs and no inductive constructions. For every h⩾8, h≠9, 14, we construct an index one current graph Γ(h) that yields a minimal nonorientable embedding ψ(h) of Kh. The current graphs Γ(h) have the property that Γ(n) and Γ(n+1) are not too different from each other and share common ladder-like fragments. As a result, the embeddings ψ(n) and ψ(n+1) have a large number of common faces: it is shown that, as n approaches infinity, for n≢3, 9 mod 12 (resp. n≡3,9 mod 12) no less than 5/8 (resp. 5/16) of all faces of ψ(n) appear in ψ(n+1)
Reconstruction of Finite Truncated Semi-Modular Lattices

AbstractIn this paper, we prove reconstruction results for truncated lattices. The main results are that truncated lattices that contain a 4-crown and truncated semi-modular lattices are reconstructible. Reconstruction of the truncated lattices not covered by this work appears challenging. Indeed, the remaining truncated lattices possess very little lattice-typical structure. This seems to indicate that further progress on the reconstruction of truncated lattices is closely correlated with progress on reconstructing ordered sets in general
Packing Odd Circuits in Eulerian Graphs

AbstractLet C be the clutter of odd circuits of a signed graph (G,Σ). For nonnegative integral edge-weights w, we are interested in the linear program min(wtx: x(C)⩾1, for C∈C, and x⩾0), which we denote by (P). The problem of solving the related integer program clearly contains the maximum cut problem, which is NP-hard. Guenin proved that (P) has an optimal solution that is integral so long as (G,Σ) does not contain a minor isomorphic to odd-K5. We generalize this by showing that if (G,Σ) does not contain a minor isomorphic to odd-K5 then (P) has an integral optimal solution and its dual has a half-integral optimal solution
Closure and Forbidden Pairs for Hamiltonicity

AbstractLet C be the claw K1,3 and N the net, i.e. the only connected graph with degree sequence 333111. It is known (Bedrossian, Thesis, Memphis State University, USA, 1991; Faudree and Gould, Discrete Math.173 (1997), 45–60) that if X,Y is a pair of connected graphs, then, for any 2-connected graph G, G being XY-free implies G is hamiltonian if and only if X is the claw C and Y belongs to a finite list of graphs, one of them being the net N. For any such pair X,Y we show that the closures of all 2-connected XY-free graphs form a subclass of the class of CN-free graphs, and we fully describe their structure
Topological Minors in Graphs of Large Girth

AbstractWe prove that every graph of minimum degree at least r and girth at least 186 contains a subdivision of Kr+1 and that for r⩾435 a girth of at least 15 suffices. This implies that the conjecture of Hajós that every graph of chromatic number at least r contains a subdivision of Kr (which is false in general) is true for graphs of girth at least 186 (or 15 if r⩾436). More generally, we show that for every graph H of maximum degree Δ(H)⩾2, every graph G of minimum degree at least {Δ(H),3} and girth at least 166log∣H∣logΔ(H) contains a subdivision of H. This bound on the girth of G is best possible up to the value of the constant and improves a result of Mader, who gave a bound linear in ∣H∣
Packing Cycles in Graphs

AbstractA graph G is called cycle Mengerian (CM) if for all nonnegative integral function w defined on V(G), the maximum number of cycles (repetition is allowed) in G such that each vertex v is used at most w(v) times is equal to the minimum of ∑{w(x):x∈X}, where the minimum is taken over all X⊆V(G) such that deleting X from G results in a forest. The purpose of this paper is to characterize all CM graphs in terms of forbidden structures. As a corollary, we prove that if the fractional version of the above minimization problem always have an integral optimal solution, then the fractional version of the maximization problem will always have an integral optimal solution as well
The Characterization of Reaction-Convection-Diffusion Processes by Travelling Waves

AbstractIt has long been known that the heat equation displays infinite speed of propagation. This is to say that if the initial data are nonnegative and have nonempty compact support, the solution of an initial-value problem is positive everywhere after any infinitesimal time. However, since the nineteen-fifties it has also been known that certain nonlinear diffusion equations of degenerate parabolic type do not display this phenomenon. For these equations, the (generalized) solution of an initial-value problem with compactly-supported initial data will have bounded support with respect to the spatial variable at all times. In this paper the necessary and sufficient criterion for finite speed of propagation for the general nonlinear reaction-convection-diffusion equation[formula]is determined. The assumptions on the coefficientsa,bandcare such that the classification unifies and generalizes previously-known results. The technique employed is comparision of an arbitrary solution of the equation with suitably-constructed travelling-wave solutions and subsolutions. Basically the central conclusion is that the equation exhibits finite speed of propagation if and only if it admits a travelling-wave solution with bounded support. Concurrently, the search for a travelling-wave solution with bounded support can be reduced to the study of a singular nonlinear integral equation whose solution must satisfy a certain constraint
Quasilinear Hyperbolic–Parabolic Equations of One-Dimensional Viscoelasticity

AbstractWe study the global existence of solutions of initial-boundary-value problems for a quasilinear hyperbolic-parabolic equation describing the longitudinal motion of a one-dimensional viscoelastic rod. We treat a variety of nonhomogeneous boundary conditions, requiring separate analyses, because they lead to distinctive physical effects. We employ a constitutive equation giving the stress as a general nonlinear function of the strain and the strain rate. All global analyses of this and related problems, except that of Dafermos (J. Differential Equations(1969), 71–86), have employed a stress that is merely affine in the strain rate. Dafermos's assumptions are far more appropriate for shearing motions than for longitudinal motions. Our constitutive equation satisfies the physically natural requirement that an infinite amount of compressive stress is needed to produce a total compression at any point of the rod. This requirement is the source of a severe singularity in the governing partial differential equations, which is particularly acute when time-dependent Dirichlet data are prescribed. The further novel, yet physically reasonable, restrictions we impose on the constitutive function yield estimates that preclude a total compression anywhere at any finite time. The resulting estimates are crucial for the global existence theory we obtain
Symmetry and the Chazy Equation

AbstractThere are three different actions of the unimodular Lie group SL(2, C) on a two-dimensional space. In every case, we show how an ordinary differential equation admitting SL(2) as a symmetry group can be reduced in order by three, and the solution recovered from that of the reduced equation via a pair of quadratures and the solution to a linear second order equation. A particular example is the Chazy equation, whose general solution can be expressed as a ratio of two solutions to a hypergeometric equation. The reduction method leads to an alternative formula in terms of solutions to the Lamé equation, resulting in a surprising transformation between the Lamé and hypergeometric equations. Finally, we discuss the Painlevé analysis of the singularities of solutions to the Chazy equation
The Principle Branch of Solutions of a Nonlinear Elliptic Eigenvalue Problem onRN

AbstractWe obtain existence, uniqueness and asymptotic decay properties of a semilinear elliptic eigenvalue problem inRN. The corresponding problem in dimensionN=1, which provided the motivation for this work, leads to bounds for the wavelengths and the power of guided waves propagating in a medium consisting of layers of dielectric material whose refractive index depends on the intensity of the electric field. In this paper we show the existence of a continuous branch of solutions (λ, u) bifurcating in the spacesR×C1(RN) orR×W2,p(RN) from the trivial solution atλ=Λ. HereΛ<0 is the lowest eigenvalue of the corresponding linear problem. We show that forplarger than a critical value depending onNthe branch is bounded inR×W2,p(RN), and for smallerpit is unbounded inR×Lp(RN). The unboundedness for smallpis demonstrated by comparison with a radially symmetric problem. Decay estimates are obtained from explicitly constructed supersolutions having known asymptotic decay rates. Subsolutions can be obtained as small multiples of the eigenfunction of the linear problem. Forλ=0 solutions do not decay exponentially, and we prove uniqueness only forλ<0. No assumptions are made concerning the growth of the nonlinearity at ∞
Phase Asymptotic Semiflows, Poincaré's Condition, and the Existence of Stable Limit Cycles

AbstractA concept of phase asymptotic semiflow is defined. It is shown that any Lagrange stable orbit at which the semiflow is phase asymptotic limits to a stable periodic orbit. A Lagrange stable solution of aC1differential equation is considered. When the second compound of the variational equation with respect to this solution is uniformly asymptotically stable and the omega limit set contains no equilibrium, then the semiflow is phase asymptotic at the orbit of the solution and the omega limit set is a stable periodic orbit. Analogous results are obtained for discrete semiflows and periodic differential equations
Evolutionary Semigroups and Dichotomy of Linear Skew-Product Flows on Locally Compact Spaces with Banach Fibers

AbstractWe study evolutionary semigroups generated by a strongly continuous semi-cocycle over a locally compact metric space acting on Banach fibers. This setting simultaneously covers evolutionary semigroups arising from non-autonomous abstract Cauchy problems andC0-semigroups, and linear skew-product flows. The spectral mapping theorem for these semigroups is proved. The hyperbolicity of the semigroup is related to the exponential dichotomy of the corresponding linear skew-product flow. To this end a Banach algebra of weighted composition operators is studied. The results are applied in the study of: “roughness” of the dichotomy, dichotomy and solutions of nonhomogeneous equations, Green's function for a linear skew-product flow, “pointwise” dichotomy versus “global” dichotomy, and evolutionary semigroups along trajectories of the flow
Behavior of Solutions of Model Equations for Incompressible Fluid Flow

AbstractWe study the behavior of solutions of model equations of inviscid incompressible fluid flow proposed by Constantin, Lax and Majda together with a viscous version studied by Schochet. A condition is found on initial data to guarantee that solution of viscous equation remains smooth when the inviscid solution blows up. We prove that global smooth solution of viscous model equation exists for a class of initial data, for which the explicit solution is not known. Singularities of solutions previously obtained are characterized here as distributions
Almost Planar Homoclinic Loops in R3

AbstractIn this paper we study homoclinic loops of vector fields in 3-dimensional space when the two principal eigenvalues are real of opposite sign, which we call almost planar. We are interested to have a theory for higher codimension bifurcations. Almost planar homoclinic loop bifurcations generically occur in two versions “non-twisted” and “twisted” loops. We consider high codimension homoclinic loop bifurcations under generic conditions. The generic condition forces the existence of a 2-dimensional topological invariant ring (non necessarily unique), which is a topological cylinder in the “non-twisted” case and a topological Möbius band in the “twisted” case. If the third eigenvalue is negative (resp. positive) theω-limit set (resp.α-limit set) of trajectories starting in a neighborhood of the loop is contained in the invariant ring. The dynamics is then given by fixed points or periodic points of period 2 of a 1-dimensional map which admits a nice asymptotic expansion, allowing to treat homoclinic loop bifurcations of arbitrarily high codimension in the non-twisted case
CkSmoothness of Invariant Curves in a Global Saddle-Node Bifurcation

AbstractThe birth ofCk-smooth invariant curves from a saddle-node bifurcation in a family ofCkdiffeomorphisms on a Banach manifold (possibly infinite dimensional) is constructed in the case that the fixed point is a stable node along hyperbolic directions, and has a smooth noncritical curve of homoclinic orbits. This ensures that the map restricted to the resulting curve is equivalent to aCkmap of the circle. In particular, for aC2family of diffeomorphisms the resulting curve isC2, and the “Denjoy example” cannot occur. Included is a new smoothness result for the foliation transversal to the center subspace, for the finite and infinite dimensional cases. Specifically,Ck-smoothness with respect to all variables of invariant foliations of the center-stable and center-unstable manifolds of a partially hyperbolic fixed point is proved in all cases
Counting Real Zeros of Analytic Functions Satisfying Linear Ordinary Differential Equations

AbstractWe suggest an explicit procedure to establish upper bounds for the number of real zeros of analytic functions satisfying linear ordinary differential equations with meromorphic coefficients. If the equation[formula]has no singular points in a small neighborhoodUof a real segmentK, all the coefficientsaj(t) have absolute value ⩽AonUanda0(t)≡1, then any solution of this equation may have no more thanβ(A+ν) zeros onK, whereβ=β(U, K) is a geometric constant depending only onKandU. If the principal coefficienta0(t) is nonconstant, but its modulus is at leasta>0 somewhere onK, then the number of real zeros onKof any solution analytic inU, does not exceed (A/a+ν)μwith someμ=μ(U, K)
The Existence of Many Periodic Non-travelling Solutions to the Boussinesq Equation

AbstractThis paper uses variational methods—in particular, a generalization of the Mountain Pass Lemma of Rabinowitz—together with an invariance argument to demonstrate the existence of (weak Sobolev) periodic, non-travelling solutions to the Boussinesq equation[formula]In certain cases, the author demonstrates the existence of infinitely-many such solutions. The assumptions on the non-linear terms are that they are odd and roughly polynomial, and that they satisfy ∇G=(g1, −g2) for someG∈C2(R2; R)
Free Boundary Problems and Density Perimeter

AbstractWe introduce a new concept “the density perimeter,” which substitutes the measure of length of the perimeter of a set in free boundary problems. We deduce some links between the Hausdorff convergence and the char convergence for a family of domains with bounded density perimeter. If this perimeter is considered as a penalty term, we give existence results for the variational problem which appears in computer vision and in a Bernoulli-like free boundary problem. We also make some considerations concerning aΓ-convergence property of the density perimete
Structurally Stable Riemann Solutions

AbstractWe study the structure of solutions of Riemann problems for systems of two conservation laws. Such a solution comprises a sequence of elementary waves,viz., rarefaction and shock waves of various types; shock waves are required to have viscous profiles. We construct a Riemann solution by solving a system of equations characterizing its component waves. A Riemann solution is “structurally stable” if the number and types of its component waves are preserved when the initial data and the flux function are perturbed. Under the assumption that rarefaction waves and shock states lie in the stricly hyperbolic region, we characterize Riemann solutions for which the definition equations have maximal rank and we prove that such solutions are structurally stable. Structurally stable Riemann solutions cannot contain overcompressive shock waves, but they can contain transitional shock waves, including doubly sonic transitional shock waves, including doubly sonic transitional shock waves that have not been observed before
Dependence of Eigenvalues of Sturm–Liouville Problems on the Boundary

AbstractThe eigenvalues of Sturm–Liouville (SL) problems depend not only continuously but smoothly on boundary points. The derivative of thenth eigenvalue as a function of an endpoint satisfies a first order differential equation. This for arbitrary (separated or coupled) self-adjoint regular boundary conditions. In addition, as the length of the interval shrinks to zero all higher eigenvalues march off to plus infinity. This is also true for the first (i.e., lowest) Dirichlet eigenvalue but not for the lowest Neumann eigenvalue. The latter has a finite limit
Versal Deformations and Normal Forms for Reversible and Hamiltonian Linear Systems

AbstractThe problem of this article is the characterization of equivalence classes and their versal deformations for reversible and reversible Hamiltonian matrices. In both cases the admissible transformations form a subgroup G of Gl(m). Therefore the Gl(m)-orbits of a given matrix maysplitinto several G-orbits. These orbits are characterized bysigns. For each sign we have a normal form and a corresponding versal deformation. The main tool in the characterization is reduction to the semi simple case
Blow-Up for Quasilinear Heat Equations Described by Means of Nonlinear Hamilton–Jacobi Equations

AbstractIt is well-known that the nonnegative solutions of the semilinear heat equation[formula]blow up in a finite timeT(depending on the initial data, assumed to be large enough). This equation is interesting because it exhibits in differentβ-ranges the three most typical blow-up behaviours for solutions of nonlinear parabolic equations. Indeed, we consider radialy symmetric solutions and show that forβ>2 single-point blow-up occurs, forβ<2 blow-up is global, and forβ=2 we have regional blow-up. Moreover, the analysis shows that the precise asymptotic behaviour is described by a nonconstant self-similar blow-up solution of the first-order Hamilton–Jacobi equation[formula]This means that both equations are asymptotically equivalent near blow-up. This type of asymptotic “degeneracy” of a parabolic equation into a first-order equation is actually proved for a more general class of quasilinear heat equations