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].     

Operator Hölder-Zygmund functions

It is well known that a Lipschitz function on the real line does not have to be operator Lipschitz. We show that the situation changes dramatically if we pass to Hölder classes. Namely, we prove that if f belongs to the Hölder class Λα(R) with 0<α<1, then for arbitrary self-adjoint operators A and B. We prove a similar result for functions f in the Zygmund class Λ₁(R): for arbitrary self-adjoint operators A and K we have . We also obtain analogs of this result for all Hölder-Zygmund classes Λα(R), α>0. Then we find a sharp estimate for ‖f(A)−f(B)‖ for functions f of class for an arbitrary modulus of continuity ω. In particular, we study moduli of continuity, for which for self-adjoint A and B, and for an arbitrary function f in Λω. We obtain similar estimates for commutators f(A)Q−Qf(A) and quasicommutators f(A)Q−Qf(B). Finally, we estimate the norms of finite differences for f in the class Λω,m that is defined in terms of finite differences and a modulus continuity ω of order m. We also obtain similar results for unitary operators and for contractions.     

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.     

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.     

On the Schur Test forL2-Boundedness of Positive Integral Operators with a Wiener–Hopf Example

The Schur sufficiency condition for boundedness of any integral operator with non-negative kernel betweenL₂-spaces is deduced from an observation, Proposition 1.2, about the central role played byL₂-spaces in the general theory of these operators. Suppose (Ω, , μ) is a measure space and thatK: Ω×Ω→[0, ∞) is an ×-measurable kernel. The special case of Proposition 1.2 for symmetrical kernels says that such a linear integral operator is bounded onanyreasonable normed linear spaceXof -measurable functions only if it is bounded onL₂(Ω, , μ) where its norm is no larger. The general form of Schur's condition (Halmos and Sunder “Bounded Integral Operators onL₂-Spaces,” Springer-Verlag, Berlin/New York, 1978) is a simple corollary which, in the symmetrical case, says that the existence of an -measurable (not necessarily square-integrable) functionh>0μ-almost-everywhere onΩwithimplies thatKis a bounded (self-adjoint) operator onL₂(Ω, , μ) of norm at mostΛ. When (Ω, , μ) isσ-finite, we show that Schur's condition is sharp: in the symmetrical case the boundedness of onL₂(Ω, , μ) implies, for anyΛ>‖‖₂, the existence of a functionh∈L₂(Ω, , μ) which is positiveμ-almost-everywhere and satisfies (*). Such functionshsatisfying (*), whether inL₂(Ω, , μ) or not, will be calledSchur test functions. They can be found explicitly in significant examples to yield best-possible estimates of the norms for classes of integral operators with non-negative kernels. In the general theory the operators are not required to be symmetrical (a theorem of Chisholm and Everitt (Proc. Roy. Soc. Edinburgh Sect. A69(14) (1970/1971), 199–204) on non-self-adjoint operators is derived in this way). They may even act between differentL₂-spaces. Section 2 is a rather substantial study of how this method yields the exact value of the norm of a particular operator between differentL₂-spaces which arises naturally in Wiener–Hopf theory and which has several puzzling features.     

Best Approximation of Functions like |x| exp(−A|x|)

We determine the exact order of best approximation by polynomials and entire functions of exponential type of functions like?_λ, α(x)=|x|^λ exp(−A|x|^−α). In particular, it is shown thatE(?_λ, α, _n, L_p(−1, 1))∼n^{−(2λp+αp+2)/2p(1+α)}×exp(−(1+α⁻¹)(Aα)^1/(1+α) cos απ/2(1+α) n^α/(1+α)), whereE(?_λ, α, _n, L_p(−1, 1)) denotes best polynomial approximation of?_λ, αinL_p(−1, 1),λ∈,α∈(0, 2],A>0, 1?p?∞. The problem, concerning the exact order of decrease ofE(?_0, 2, _n, L_∞(−1, 1)), has been posed by S. N. Bernstein.     

Kac–Moody groups and cluster algebras

Let Q be a finite quiver without oriented cycles, let Λ be the associated preprojective algebra, let g be the associated Kac–Moody Lie algebra with Weyl group W, and let n be the positive part of g. For each Weyl group element w, a subcategory Cw of mod(Λ) was introduced by Buan, Iyama, Reiten and Scott. It is known that Cw is a Frobenius category and that its stable category is a Calabi–Yau category of dimension two. We show that Cw yields a cluster algebra structure on the coordinate ring C[N(w)] of the unipotent group N(w):=N∩(w−1N₋w). Here N is the pro-unipotent pro-group with Lie algebra the completion of n. One can identify C[N(w)] with a subalgebra of , the graded dual of the universal enveloping algebra U(n) of n. Let S^? be the dual of Lusztig?s semicanonical basis S of U(n). We show that all cluster monomials of C[N(w)] belong to S^?, and that S^?∩C[N(w)] is a C-basis of C[N(w)]. Moreover, we show that the cluster algebra obtained from C[N(w)] by formally inverting the generators of the coefficient ring is isomorphic to the algebra C[Nw] of regular functions on the unipotent cell Nw of the Kac–Moody group with Lie algebra g. We obtain a corresponding dual semicanonical basis of C[Nw]. As one application we obtain a basis for each acyclic cluster algebra, which contains all cluster monomials in a natural way.     

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.     

The index formula and the spectral shift function for relatively trace class perturbations

We compute the Fredholm index, index(DA), of the operator DA=(d/dt)+A on L²(R;H) associated with the operator path , where (Af)(t)=A(t)f(t) for a.e. t∈R, and appropriate f∈L²(R;H), via the spectral shift function ξ(⋅;A₊,A₋) associated with the pair (A₊,A₋) of asymptotic operators A_±=A(±∞) on the separable complex Hilbert space H in the case when A(t) is generally an unbounded (relatively trace class) perturbation of the unbounded self-adjoint operator A₋.We derive a formula (an extension of a formula due to Pushnitski) relating the spectral shift function ξ(⋅;A₊,A₋) for the pair (A₊,A₋), and the corresponding spectral shift function ξ(⋅;H₂,H₁) for the pair of operators in this relative trace class context,This formula is then used to identify the Fredholm index of DA with ξ(0;A₊,A₋). In addition, we prove that index(DA) coincides with the spectral flow of the family {A(t)}_t∈R and also relate it to the (Fredholm) perturbation determinant for the pair (A₊,A₋): with the choice of the branch of ln(detH(⋅)) on C₊ such thatWe also provide some applications in the context of supersymmetric quantum mechanics to zeta function and heat kernel regularized spectral asymmetries and the eta-invariant.     

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.     

Some Best Constants in the Landau Inequality on a Finite Interval

An additive form of the Landau inequality forf∈Wⁿ_∞[−1, 1],is proved for 0<c?(cos(π/2n))⁻², 1?m?n−1, with equality forf(x)=T_n(1+(x−1)/c), 1?c?(cos(π/2n))⁻², whereT_nis the Chebyshev polynomial. From this follows a sharp multiplicative inequality,for ‖f⁽ⁿ⁾‖?σ ‖f‖, 2ⁿ⁻¹n! cos²ⁿ(π/2n)?σ?2ⁿ⁻¹n!, 1?m?n−1. For these values ofσ, the result confirms Karlin's conjecture on the Landau inequality for intermediate derivatives on a finite interval. For the proof of the additive inequality a Duffin and Schaeffer-type inequality for polynomials is shown.     

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.     

Functions of operators under perturbations of class Sp

This is a continuation of our paper [2]. We prove that for functions f in the Hölder class Λα(R) and 1<p<∞, the operator f(A)−f(B) belongs to Sp/α, whenever A and B are self-adjoint operators with A−B∈Sp. We also obtain sharp estimates for the Schatten-von Neumann norms ‖f(A)−f(B)Sp/α_‖ in terms of ‖A−BSp_‖ and establish similar results for other operator ideals. We also estimate Schatten-von Neumann norms of higher order differences . We prove that analogous results hold for functions on the unit circle and unitary operators and for analytic functions in the unit disk and contractions. Then we find necessary conditions on f for f(A)−f(B) to belong to Sq under the assumption that A−B∈Sp. We also obtain Schatten-von Neumann estimates for quasicommutators f(A)R−Rf(B), and introduce a spectral shift function and find a trace formula for operators of the form f(A−K)−2f(A)+f(A+K).     

The Randi? index and the diameter of graphs

The Randi? indexR(G) of a graph G is defined as the sum of over all edges uv of G, where d_u and d_v are the degrees of vertices u and v, respectively. Let D(G) be the diameter of G when G is connected. Aouchiche et al. (2007) [1] conjectured that among all connected graphs G on n vertices the path P_n achieves the minimum values for both R(G)/D(G) and R(G)−D(G). We prove this conjecture completely. In fact, we prove a stronger theorem: If G is a connected graph, then , with equality if and only if G is a path with at least three vertices.     

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.     

Critical heights on the moduli space of polynomials

Let Md be the moduli space of one-dimensional, degree d?2, complex polynomial dynamical systems. The escape rates of the critical points determine a critical heights mapG:Md→Rd−1. For generic values of G, we show that each connected component of a fiber of G is the deformation space for twist deformations on the basin of infinity. We analyze the quotient space obtained by collapsing each connected component of a fiber of G to a point. The space is a parameter-space analog of the polynomial tree T(f) associated to a polynomial f:C→C, studied in DeMarco and McMullen (2008) [6], and there is a natural projection from to the space of trees Td. We show that the projectivization is compact and contractible; further, the shift locus in has a canonical locally finite simplicial structure. The top-dimensional simplices are in one-to-one correspondence with topological conjugacy classes of structurally stable polynomials in the shift locus.     

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 .     

The Baer-invariant of a semidirect product

In 1972 K.I. Tahara [7,2, Theorem 2.2.5], using cohomological methods, showed that if a finite group is the semidirect product of a normal subgroup N and a subgroup T, then M(T) is a direct factor of M(G), where M(G) is the Schur-multiplicator of G and in the finite case, is the second cohomology group of G. In 1977 W. Haebich [1, Theorem 1.7] gave another proof using a different method for an arbitrary group G.In this paper we generalize the above theorem. We will show that scN_cM(T) is a direct factor of _cM(G), where _c[3, p. 102] is the variety of nilpotent groups of class at most c ≥ 1 and _cM(G) is the Baer-invariant of the group G with respect to the variety _c [3, p. 107].     

Bounded Functions in Möbius Invariant Dirichlet Spaces

Forp∈(0, 1), letQ_p(Q_{p, 0}) be the space of analytic functionsfon the unit diskΔwith sup_w∈Δ ‖f°?_w‖_p<∞ (lim_|w|→1 ‖f°?_w‖_p=0), where ‖·‖_pmeans the weighted Dirichlet norm and?_wis the Möbius map ofΔonto itself with?_w(0)=w. In this paper, we prove the Corona theorem for the algebraQ_p∩H^∞(Q_{p, 0}∩H^∞); then we provide a Fefferman–Stein type decomposition forQ_p(Q_{p, 0}), and finally we describe the interpolating sequences forQ_p∩H^∞(Q_{p, 0}∩H^∞)).