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 flushed partitions and concave compositions

In this work, we give combinatorial proofs for generating functions of two problems, i.e., flushed partitions and concave compositions of even length. We also give combinatorial interpretation of one problem posed by Sylvester involving flushed partitions and then prove it. For these purposes, we first describe an involution and use it to prove core identities. Using this involution with modifications, we prove several problems of different nature, including Andrews’ partition identities involving initial repetitions and partition theoretical interpretations of three mock theta functions of third order f(q), ?(q) and ψ(q). An identity of Ramanujan is proved combinatorially. Several new identities are also established.     

New criteria for hyperbolicity based on periodic sets

We prove some criteria for uniform hyperbolicity based on the periodic points of the transformation. More precisely, if a mild hyperbolicity condition holds for the periodic points of any diffeomorphism in a residual subset of a C ¹-open set U then there exists an open and dense subset A ? U of Axiom A diffeomorphisms. Moreover, we also prove a noninvertible version of Ergodic Closing Lemma which we use to prove a counterpart of this result for local diffeomorphisms.     

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.     

Stable weakly shadowable volume-preserving systems are volume-hyperbolic

We prove that any C1-stable weakly shadowable volume-preserving diffeomorphism defined on a compact manifold displays a dominated splitting E ⊕ F. Moreover, both E and F are volume-hyperbolic. Finally, we prove the version of this result for divergence-free vector fields. As a consequence, in low dimensions, we obtain global hyperbolicity.     

Strongly graceful graphs

In this paper, we prove that the n-cube is graceful, thus answering a conjecture of J.-C. Bermond and Gangopadhyay and Rao Hebbare. To do that, we introduce a special kind of graceful numbering, particular case of α-valuation, called strongly graceful and we prove that if a graph G is strongly graceful, G + K₂ is also strongly graceful.     

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.     

Algebras generated by inner functions

Let A be a uniform algebra on a compact space X. An inner function is a function in A unimodular on X. For three algebras of type H^∞ we prove A is generated by its inner functions. Whenever A is generated by its inner functions we prove the unit ball of A is the closed convex ball of the inner functions.     

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.     

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.     

Mixed-norm estimates for a class of nonisotropic directional maximal operators and Hilbert transforms

For all d?2 and p∈(1,max(2,(d+1)/2)], we prove sharp Lp to Lp(Lq) estimates (modulo an endpoint) for a directional maximal operator associated to curves generated by the dilation matrices exp((logt)P), where P has real entries and eigenvalues with positive real part. For the corresponding Hilbert transform we prove an analogous result for all d?2 and p∈(1,2]. As corollaries, we prove Lp bounds for variable kernel singular integral operators and Nikodym-type maximal operators taking averages over certain families of curved sets in Rd.     

Harmonic functions on [IN] and central hypergroups

In this paper we introduce the notions of [I N] and [S I N]-hypergroups and prove a Choquet-Deny type theorem for [I N] and central hypergroups. More precisely, we prove a Liouville theorem for bounded harmonic functions on a class of [I N]-hypergroups. Further, we show that positive harmonic functions on [I N]-hypergroups are integrals of exponential functions. Similar results are proved for [S I N] and central hypergroups.     

Vertex-minor reductions can simulate edge contractions

We prove that local complementation and vertex deletion, operations from which vertex-minors are defined, can simulate edge contractions. As an application, we prove that the rank-width of a graph is linearly bounded in term of its tree-width.     

Weighted integral inequalities for conjugate A-harmonic tensors

We first prove a local weighted integral inequality for conjugate A-harmonic tensors. Then, as an application of our local result, we prove a global weighted integral inequality for conjugate A-harmonic tensors in L^s(μ)-averaging domains, which can be considered as a generalization of the classical result. Finally, we give applications of the above results to quasiregular mappings.     

Generalized Derivations of Lie Color Algebras

In this paper, we give some basic properties of the generalized derivation algebra GDer(L) of a Lie color algebra L. In particular, we prove that GDer(L) = QDer(L) + QC(L), the sum of the quasiderivation algebra and the quasicentroid. We also prove that QDer(L) can be embedded as derivations in a larger Lie color algebra.     

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.     

Three theorems on common splitting fields of central simple algebras

LetA ₁, …A _n be central simple algebras over a fieldF. Suppose that we possess information on the Schur indexes of some tensor products of (some tensor powers of) the algebras. What can be said (in general) about possible degrees of finite field extensions ofF splitting the algebras? In Part I, we prove a positive result of that kind. In Part II, we prove a negative result. In Part III, we develop a general approach.     

On v-adic periods of t-motives

In this paper, we prove the equality between the transcendental degree of the field generated by the v-adic periods of a t-motive M and the dimension of the Tannakian Galois group for M, where v is a “finite” place of the rational function field over a finite field. As an application, we prove the algebraic independence of certain “formal” polylogarithms.     

A nonlocal nonlinear diffusion equation in higher space dimensions

We study the initial-value problem for a nonlocal nonlinear diffusion operator which is analogous to the porous medium equation, in the whole RN, N?1, or in a bounded smooth domain with Neumann or Dirichlet boundary conditions. First, we prove the existence, uniqueness and the validity of a comparison principle for solutions of these problems. In RN we show that if initial data is bounded and compactly supported, then the solutions is compactly supported for all positive time t, this implies the existence of a free boundary. Concerning the Neumann problem, we prove that the asymptotic behavior of the solutions as t→∞, they converge to the mean value of the initial data. For the Dirichlet problem we prove that the asymptotic behavior of the solutions as t→∞, they converge to zero.     

Factorization of delta-monotone linear mappings

For a delta-monotone linear mapping we prove that the factors in the polar decomposition are delta-monotone. Also, we prove that every delta-monotone linear mapping can be factored into a product of (1-ε)-monotone mappings for any ε∈(0,1). As an application in nonlinear case, we give a new proof of the following fact: the quasiconformality constant K(δ,n) of a δ-monotone mapping can be chosen such that K(δ,n) tends to 1 as δ tends to 1.