Tutte uniqueness of line graphs

We prove that if a graph H has the same Tutte polynomial as the line graph of a d-regular, d-edge-connected graph, then H is the line graph of a d-regular graph. Using this result, we prove that the line graph of a regular complete t-partite graph is uniquely determined by its Tutte polynomial. We prove the same result for the line graph of any complete bipartite graph.     

The homogeneous dirichlet problem for quasilinear anisotropic degenerate parabolic-hyperbolic equation with L initial value

The aim of this paper is to prove the well-posedness (existence and uniqueness) of the L^p entropy solution to the homogeneous Dirichlet problems for the anisotropic degenerate parabolic-hyperbolic equations with L^p initial value. We use the device of doubling variables and some technical analysis to prove the uniqueness result. Moreover we can prove that the L^p entropy solution can be obtained as the limit of solutions of the corresponding regularized equations of nondegenerate parabolic type.     

On the forbidden induced subgraph sandwich problem

We consider the sandwich problem, a generalization of the recognition problem introduced by Golumbic et al. (1995) [15], with respect to classes of graphs defined by excluding induced subgraphs. We prove that the sandwich problem corresponding to excluding a chordless cycle of fixed length k is NP-complete. We prove that the sandwich problem corresponding to excluding K_r?e for fixed r is polynomial. We prove that the sandwich problem corresponding to 3PC(⋅,⋅)-free graphs is NP-complete. These complexity results are related to the classification of a long-standing open problem: the sandwich problem corresponding to perfect graphs.     

Structural stability of C1 diffeomorphisms

In this paper we prove that if f is a C¹ diffeomorphism that satisfies Axiom A and the strong transversality condition then it is structurally stable. J. Robbin proved this theorem for C² diffeomorphisms. In addition to reducing the amount of differentiability necessary to prove the theorem, we also give a new proof combining the d_f metric of Robbin with the stable and unstable manifold proof of D. Anosov. We also prove structural stability in the neighborhood of a single hyperbolic basic set (independent of its being part of a diffeomorphism that satisfies Axiom A and the strong transversality condition). These proofs are adapted to prove the structural stability of C¹ flows in another paper.     

On complete intersection toric ideals of graphs

We study the graphs G for which their toric ideals I _G are complete intersections. In particular, we prove that for a connected graph G such that I _G is a complete intersection all of its blocks are bipartite except for at most two. We prove that toric ideals of graphs which are complete intersections are circuit ideals. In this case, the generators of the toric ideal correspond to even cycles of G except of at most one generator, which corresponds to two edge disjoint odd cycles joint at a vertex or with a path. We prove that the blocks of these graphs satisfy the odd cycle condition. Finally, we characterize all complete intersection toric ideals of graphs which are normal.     

Nonnegative functions as squares or sums of squares

We prove that, for n?4, there are C^∞ nonnegative functions f of n variables (and even flat ones for n?5) which are not a finite sum of squares of C² functions. For n=1, where a decomposition in a sum of two squares is always possible, we investigate the possibility of writing f=g². We prove that, in general, one cannot require a better regularity than g∈C¹. Assuming that f vanishes at all its local minima, we prove that it is possible to get g∈C² but that one cannot require any additional regularity.     

Distortion of Banach spaces and supermultiplicative operational quantities

We prove that certain operational quantities q which characterize upper-semi Fredholm operators are supermultiplicative, in the sense of that q(S)q(T)?q(ST). Based on the distortion of Banach spaces we show that another is not supermultiplicative. Moreover we introduce two supermultiplicative operational quantities which characterize also the upper-semi Fredholm operators and we prove that they are not equivalent to some operational quantities known.     

L p -pinching and the geometry of compact Riemannian manifolds

We prove a Harnack-type inequality inf|S|/sup|S|>1?ε(W, M, V) satisfied by the sections of a Riemannian vector bundleW lying in the kernel of a Schrödinger operator ∨*∨+V underL ^p -pinching assumptions on the potentialV and derive various topological and geometric consequences. For instance, we prove a fibration theorem which gives a classification of almost non-negatively curved compact manifolds by the first Betti number. In the case of almost non-positively curved compact manifolds, we prove that the minimal volume must vanish whenever the isometry group is not finite and give conditions implying that it is abelian.     

Ideals in Graph Algebras

We show that the graph construction used to prove that a gauge-invariant ideal of a graph C ^???-algebra is isomorphic to a graph C ^???-algebra, and also used to prove that a graded ideal of a Leavitt path algebra is isomorphic to a Leavitt path algebra, is incorrect as stated in the literature. We give a new graph construction to remedy this problem, and prove that it can be used to realize a gauge-invariant ideal (respectively, a graded ideal) as a graph C ^???-algebra (respectively, a Leavitt path algebra).     

The Weyl bundle

Let F be a symplectic vector bundle over a space X. We construct a bundle of elementary C¹-algebras over X, and prove that the Dixmier-Douady invariant of this bundle is zero. The underlying Hilbert bundles, with their associated module structures, determine a characteristic class: we prove that this class is the second Stiefel-Whitney class of F.     

A Boolean algebra and a Banach space obtained by push-out iteration

Under the assumption that c is a regular cardinal, we prove the existence and uniqueness of a Boolean algebra B of size c defined by sharing the main structural properties that P(ω)/fin has under CH and in the ℵ₂-Cohen model. We prove a similar result in the category of Banach spaces.     

Quasi-Nearly Subharmonic Functions and Quasiconformal Mappings

We prove that the composition of a quasi-nearly subharmonic function and a quasiregular mapping of bounded multiplicity is quasi-nearly subharmonic. Also, we prove that if u?°?f is quasi-nearly subharmonic for all quasi-nearly subharmonic u and f satisfying some additional conditions, then f is quasiconformal. Similar results are further established for the class of regularly oscillating functions.     

The Palais-Smale condition on contact type energy levels for convex Lagrangian systems

We prove that for a uniformly convex Lagrangian system L on a compact manifold M, almost all energy levels contain a periodic orbit. We also prove that below Mañé's critical value of the lift of the Lagrangian to the universal cover, c _u (L), almost all energy levels have conjugate points. We in addition prove that if an energy level is of contact type, projects onto M and $M\ne{\mathbb T}^2We prove that for a uniformly convex Lagrangian system L on a compact manifold M, almost all energy levels contain a periodic orbit. We also prove that below Ma?é's critical value of the lift of the Lagrangian to the universal cover, c _u (L), almost all energy levels have conjugate points.We in addition prove that if an energy level is of contact type, projects onto M and , then the free time action functional of L+k satisfies the Palais-Smale condition.Partially supported by Conacyt, Mexico, grant 36496-E.     

The bar complex of an E-infinity algebra

The standard reduced bar complex B(A) of a differential graded algebra A inherits a natural commutative algebra structure if A is a commutative algebra. We address an extension of this construction in the context of E-infinity algebras. We prove that the bar complex of any E-infinity algebra can be equipped with the structure of an E-infinity algebra so that the bar construction defines a functor from E-infinity algebras to E-infinity algebras. We prove the homotopy uniqueness of such natural E-infinity structures on the bar construction.We apply our construction to cochain complexes of topological spaces, which are instances of E-infinity algebras. We prove that the n-th iterated bar complexes of the cochain algebra of a space X is equivalent to the cochain complex of the n-fold iterated loop space of X, under reasonable connectedness, completeness and finiteness assumptions on X.     

Nonautonomous Ornstein-Uhlenbeck operators in weighted spaces of continuous functions

We consider the nonautonomous Ornstein-Uhlenbeck operator in some weighted spaces of continuous functions in $\mathbb{R}^{N}$ . We prove sharp uniform estimates for the spatial derivatives of the associated evolution operator P _s,t , which we use to prove optimal Schauder estimates for the solution to some nonhomogeneous parabolic Cauchy problems associated with the Ornstein-Uhlenbeck operator. We also prove that, for any t>s, the evolution operator P _s,t is compact in the previous weighted spaces.     

A new approach to induction and imprimitivity results

Given a closed quantum subgroup of a locally compact quantum group, we study induction of unitary corepresentations of the quantum subgroup to the ambient quantum group. More generally, we study induction given a coaction of the quantum subgroup on a C^*-algebra. We prove imprimitivity theorems that unify the existing theorems for actions and coactions of groups. This means that we define quantum homogeneous spaces as C^*-algebras and that we prove Morita equivalence of crossed products and homogeneous spaces. We essentially use von Neumann algebraic techniques to prove these Morita equivalences between C^*-algebras.     

Acute triangulations of polyhedra and ℝ N

We study the problem of acute triangulations of convex polyhedra and the space ? ⁿ . Here an acute triangulation is a triangulation into simplices whose dihedral angles are acute. We prove that acute triangulations of the n-cube do not exist for n≥4. Further, we prove that acute triangulations of the space ? ⁿ do not exist for n≥5. In the opposite direction, in ?³, we present a construction of an acute triangulation of the cube, the regular octahedron and a non-trivial acute triangulation of the regular tetrahedron. We also prove nonexistence of an acute triangulation of ?⁴ if all dihedral angles are bounded away from π/2.     

On the Ono invariants of imaginary quadratic number fields

J. Cohen, J. Sonn, F. Sairaiji and K. Shimizu proved that there are only finitely many imaginary quadratic number fields K whose Ono invariants OnoK are equal to their class numbers hK. Assuming a Restricted Riemann Hypothesis, namely that the Dedekind zeta functions of imaginary quadratic number fields K have no Siegel zeros, we determine all these K's. There are 114 such K's. We also prove that we are missing at most one such K. M. Ishibashi proved that if OnoK is large enough compared with hK, then the ideal class groups of K is cyclic. We give a short proof and a precision of Ishibashi's result. We prove that there are only finitely many imaginary quadratic number fields K satisfying Ishibashi's sufficient condition. Assuming our Restricted Riemann Hypothesis, we prove that the absolute values dK of their discriminants are less than 2.3⋅¹⁰⁹. We determine all these K's with dK?¹⁰⁶. There are 76 such K's. We prove that there is at most one such K with dK?1.8⋅¹⁰¹¹.     

Relative homological algebra in the category of quasi-coherent sheaves

In this paper, we prove the existence of a flat cover and of a cotorsion envelope for any quasi-coherent sheaf over a scheme (X,O_X). Indeed we prove something more general. We define what it is understood by the category of quasi-coherent R-modules, where R is a representation by rings of a quiver Q, and we prove the existence of a flat cover and a cotorsion envelope for quasi-coherent R-modules. Then we use the fact that the category of quasi-coherent sheaves on (X,O_X) is equivalent to the category of quasi-coherent R-modules for some Q and R to get our result.     

On finite groups with given maximal subgroups

We prove that a finite group whose every maximal subgroup is simple or nilpotent is a Schmidt group. A group whose every maximal subgroup is simple or supersoluble can be nonsoluble, and in this case we prove that its chief series has the form 1 ? K ? G, K }~ PSL ₂(p) for a suitable prime p, |G: K| ≤ 2.