Optimal bounds for a colorful Tverberg–Vre?ica type problem

We prove the following optimal colorful Tverberg–Vre?ica type transversal theorem: For prime r and for any k+1 colored collections of points C? in Rd, , |C?|=(r−1)(d−k+1)+1, , ?=0,…,k, there are partitions of the collections C? into colorful sets such that there is a k-plane that meets all the convex hulls , under the assumption that r(d−k) is even or k=0.Along the proof we obtain three results of independent interest: We present two alternative proofs for the special case k=0 (our optimal colored Tverberg theorem (2009) [2]), calculate the cohomological index for joins of chessboard complexes, and establish a new Borsuk–Ulam type theorem for m(Zp)-equivariant bundles that generalizes results of Volovikov (1996) [17] and ?ivaljevi? (1999) [21].     

Quasi-symmetric functions and up–down compositions

Carlitz (1973) [5] and Rawlings (2000) [13] studied two different analogues of up–down permutations for compositions with parts in {1,…,n}. Cristea and Prodinger (2008/2009) [7] studied additional analogues for compositions with unbounded parts. We show that the results of Carlitz, Rawlings, and Cristea and Prodinger on up–down compositions are special cases of four different analogues of generalized Euler numbers for compositions. That is, for any s≥2, we consider classes of compositions that can be divided into an initial set of blocks of size s followed by a block of size j where 0≤j≤s−1. We then consider the classes of such compositions where all the blocks are strictly increasing (weakly increasing) and there are strict (weak) decreases between blocks. We show that the weight generating functions of such compositions w=w₁?w_m, where the weight of w is , are always the quotients of sums of quasi-symmetric functions. Moreover, we give a direct combinatorial proof of our results via simple involutions.     

Locating–dominating codes in paths

Bertrand, Charon, Hudry and Lobstein studied, in their paper in 2004 [1], r-locating–dominating codes in paths P_n. They conjectured that if r≥2 is a fixed integer, then the smallest cardinality of an r-locating–dominating code in P_n, denoted by , satisfies for infinitely many values of n. We prove that this conjecture holds. In fact, we show a stronger result saying that for any r≥3 we have for all n≥n_r when n_r is large enough. In addition, we solve a conjecture on location–domination with segments of even length in the infinite path.     

Functions of normal operators under perturbations

In Peller (1980) [27], Peller (1985) [28], Aleksandrov and Peller (2009) [2], Aleksandrov and Peller (2010) [3], and Aleksandrov and Peller (2010) [4] sharp estimates for f(A)−f(B) were obtained for self-adjoint operators A and B and for various classes of functions f on the real line R. In this paper we extend those results to the case of functions of normal operators. We show that if a function f belongs to the Hölder class Λα(R²), 0<α<1, of functions of two variables, and N₁ and N₂ are normal operators, then ‖f(N₁)−f(N₂)‖?const‖fΛα_‖‖N₁−N₂α^‖. We obtain a more general result for functions in the space for an arbitrary modulus of continuity ω. We prove that if f belongs to the Besov class , then it is operator Lipschitz, i.e., . We also study properties of f(N₁)−f(N₂) in the case when f∈Λα(R²) and N₁−N₂ belongs to the Schatten–von Neumann class Sp.     

Regularity of solutions for an integral system of Wolff type

We consider the fully nonlinear integral systems involving Wolff potentials:(1) whereThis system includes many known systems as special cases, in particular, when and γ=2, system (1) reduces to(2) The solutions (u,v) of (2) are critical points of the functional associated with the well-known Hardy–Littlewood–Sobolev inequality. We can show that (2) is equivalent to a system of semi-linear elliptic PDEs which comprises the well-known Lane–Emden system and Yamabe equation.We obtain integrability and regularity for the positive solutions to systems (1). A regularity lifting method by contracting operators is used in proving the integrability, and while deriving the Lipschitz continuity, a brand new idea – Lifting Regularity by Shrinking Operators is introduced. We hope to see many more applications of this new idea in lifting regularities of solutions for nonlinear problems.     

The quaternionic evolution operator

In the recent years, the notion of slice regular functions has allowed the introduction of a quaternionic functional calculus. In this paper, motivated also by the applications in quaternionic quantum mechanics, see Adler (1995) [1], we study the quaternionic semigroups and groups generated by a quaternionic (bounded or unbounded) linear operator T=T₀+iT₁+jT₂+kT₃. It is crucial to note that we consider operators with components T?(?=0,1,2,3) that do not necessarily commute. Among other results, we prove the quaternionic version of the classical Hille–Phillips–Yosida theorem. This result is based on the fact that the Laplace transform of the quaternionic semigroup etT is the S-resolvent operator , the quaternionic analogue of the classical resolvent operator. The noncommutative setting entails that the results we obtain are somewhat different from their analogues in the complex setting. In particular, we have four possible formulations according to the use of left or right slice regular functions for left or right linear operators.     

Interior gradient estimates for solutions to the linearized Monge–Ampère equation

Let ? be a convex function on a convex domain Ω⊂Rn, n?1. The corresponding linearized Monge–Ampère equation istrace(ΦD²u)=f, where is the matrix of cofactors of D²?. We establish interior Hölder estimates for derivatives of solutions to such equation when the function f on the right-hand side belongs to Lp(Ω) for some p>n. The function ? is assumed to be such that with ?=0 on ∂Ω and the Monge–Ampère measure is given by a density g∈C(Ω) which is bounded away from zero and infinity.     

A further refinement of a three critical points theorem

In this paper, we complete the refinement process, made by Ricceri (2009) [4], of a result established by Ricceri (2000) [1], which is one of the most applied abstract multiplicity theorems in the past decade. A sample of application of our new result is as follows.Let (n≥3) be a bounded domain with smooth boundary and let .Then, for each ?>0 small enough, there exists λ_?>0 such that, for every compact interval , there exists ρ>0 with the following property: for every λ∈[a,b] and every continuous function satisfying for some , there exists δ>0 such that, for each ν∈[0,δ], the problem has at least three weak solutions whose norms in are less than ρ.     

A diffusion-type process with a given joint law for the terminal level and supremum at an independent exponential time

We construct a weak solution to the stochastic functional differential equation , where M_t=sup_0≤s≤tX_s. Using the excursion theory, we then solve explicitly the following problem: for a natural class of joint density functions μ(y,b), we specify σ(.,.), so that X is a martingale, and the terminal level and supremum of X, when stopped at an independent exponential time ξ_λ, is distributed according to μ. We can view (X_t∧ξλ) as an alternate solution to the problem of finding a continuous local martingale with a given joint law for the maximum and the drawdown, which was originally solved by Rogers (1993) [21] using the excursion theory. This complements the recent work of Carr (2009) [5] and Cox et al. (2010) [7], who consider a standard one-dimensional diffusion evaluated at an independent exponential time.¹     

Spectra of lifted Ramanujan graphs

A random n-lift of a base-graph G is its cover graph H on the vertices [n]×V(G), where for each edge uv in G there is an independent uniform bijection π, and H has all edges of the form (i,u),(π(i),v). A main motivation for studying lifts is understanding Ramanujan graphs, and namely whether typical covers of such a graph are also Ramanujan.Let G be a graph with largest eigenvalue λ₁ and let ρ be the spectral radius of its universal cover. Friedman (2003) [12] proved that every “new” eigenvalue of a random lift of G is with high probability, and conjectured a bound of ρ+o(1), which would be tight by results of Lubotzky and Greenberg (1995) [15]. Linial and Puder (2010) [17] improved Friedman?s bound to . For d-regular graphs, where λ₁=d and , this translates to a bound of O(d2/3), compared to the conjectured .Here we analyze the spectrum of a random n-lift of a d-regular graph whose nontrivial eigenvalues are all at most λ in absolute value. We show that with high probability the absolute value of every nontrivial eigenvalue of the lift is . This result is tight up to a logarithmic factor, and for λ?d2/3−ε it substantially improves the above upper bounds of Friedman and of Linial and Puder. In particular, it implies that a typical n-lift of a Ramanujan graph is nearly Ramanujan.     

Moment vanishing problem and positivity: Some examples

We consider the problem of vanishing of the momentswith Ω a compact domain in Rn and P(x), q(x) complex polynomials in x∈Ω (MVP). The main stress is on relations of this general vanishing problem to the following conjecture which has been studied recently in Mathieu (1997) [22], Duistermaat and van der Kallen (1998) [17], Zhao (2010) [34] and [35] and in other publications in connection with the vanishing problem for differential operators and with the Jacobian conjecture:

Conjecture A. 

For positive μ ifmk(P,1)=0fork=1,2,… , thenmk(P,q)=0fork?1for any q.     

Classification of the centers and their isochronicity for a class of polynomial differential systems of arbitrary degree

In this paper we classify the centers localized at the origin of coordinates, and their isochronicity for the polynomial differential systems in R² of degree d that in complex notation z=x+iy can be written as where j is either 0 or 1. If j=0 then d?5 is an odd integer and n is an even integer satisfying 2?n?(d+1)/2. If j=1 then d?3 is an integer and n is an integer with converse parity with d and satisfying 0<n?[(d+1)/3] where [⋅] denotes the integer part function. Furthermore λ∈R and A,B,C,D∈C. Note that if d=3 and j=0, we are obtaining the generalization of the polynomial differential systems with cubic homogeneous nonlinearities studied in K.E. Malkin (1964) [17], N.I. Vulpe and K.S. Sibirskii (1988) [25], J. Llibre and C. Valls (2009) [15], and if d=2, j=1 and C=0, we are also obtaining as a particular case the quadratic polynomial differential systems studied in N.N. Bautin (1952) [2], H. Zoladek (1994) [26]. So the class of polynomial differential systems here studied is very general having arbitrary degree and containing the two more relevant subclasses in the history of the center problem for polynomial differential equations.     

On Muckenhoupt–Wheeden Conjecture

Let M denote the dyadic Maximal Function. We show that there is a weight w, and Haar multiplier T for which the following weak-type inequality fails (With T replaced by M, this is a well-known fact.) This shows that a dyadic version of the so-called Muckenhoupt–Wheeden Conjecture is false. This accomplished by using current techniques in weighted inequalities to show that a particular L² consequence of the inequality above does not hold.     

Closure operation for even factors on claw-free graphs

Ryjá?ek (1997) [6] defined a powerful closure operation on claw-free graphs G. Very recently, Ryjá?ek et al. (2010) [8] have developed the closure operation on claw-free graphs which preserves the (non)-existence of a 2-factor. In this paper, we introduce a closure operation on claw-free graphs that generalizes the above two closure operations. The closure of a graph is unique determined and the closure turns a claw-free graph into the line graph of a graph containing no cycle of length at most 5 and no cycles of length 6 satisfying a certain condition and no induced subgraph being isomorphic to the unique tree with a degree sequence 111133. We show that these closure operations on claw-free graphs all preserve the minimum number of components of an even factor. In particular, we show that a claw-free graph G has an even factor with at most k components if and only if (, respectively) has an even factor with at most k components. However, the closure operation does not preserve the (non)-existence of a 2-factor.     

On a conjecture of Shanks

The conjecture in question concerns the function ?n related to the distribution of the zeroes of the Riemann zeta-function, γn, over the Gram points gn. It is the purpose of this article to show that for any α>0 the sum and this was conjectured, on numerical evidence, by Shanks (1961) [7] to be true for .     

Approximation of RNA multiple structural alignment

In the context of non-coding RNA (ncRNA) multiple structural alignment, Davydov and Batzoglou (2006) introduced in [7] the problem of finding the largest nested linear graph that occurs in a set G of linear graphs, the so-called Max-NLS problem. This problem generalizes both the longest common subsequence problem and the maximum common homeomorphic subtree problem for rooted ordered trees.In the present paper, we give a fast algorithm for finding the largest nested linear subgraph of a linear graph and a polynomial-time algorithm for a fixed number (k) of linear graphs. Also, we strongly strengthen the result of Davydov and Batzoglou (2006) [7] by proving that the problem is NP-complete even if G is composed of nested linear graphs of height at most 2, thereby precisely defining the borderline between tractable and intractable instances of the problem. Of particular importance, we improve the result of Davydov and Batzoglou (2006) [7] by showing that the Max-NLS problem is approximable within ratio in O(kn²) running time, where mopt is the size of an optimal solution. We also present O(1)-approximation of Max-NLS problem running in O(kn) time for restricted linear graphs. In particular, for ncRNA derived linear graphs, a -approximation is presented.     

Radially increasing minimizing surfaces or deformations under pointwise constraints on positions and gradients

In this paper, we prove existence of radially symmetric minimizersu_A(x)=U_A(|x|), having U_A(⋅)AC monotone and increasing, for the convex scalar multiple integral(∗ ) among those u(⋅) in the Sobolev space. Here, |∇u(x)| is the Euclidean norm of the gradient vector and B_R is the ball ; while A is the boundary data.Besides being e.g. superlinear (but no growth needed if (∗) is known to have minimum), our Lagrangian?^∗∗:R×R→[0,∞] is just convex lsc and and ?^∗∗(s,⋅) is even; while ρ₁(⋅) and ρ₂(⋅) are Borel bounded away from .Remarkably, (∗) may also be seen as the calculus of variations reformulation of a distributed-parameter scalar optimal control problem. Indeed, state and gradient pointwise constraints are, in a sense, built-in, since ?^∗∗(s,v)=∞ is freely allowed.