For a directed graph G without loops or parallel edges, let β(G) denote the size of the smallest feedback arc set, i.e., the smallest subset X ⊂ E(G) such that G ∖ X has no directed cycles. Let γ(G) be the number of unordered pairs of vertices of G which are not adjacent. We prove that every directed graph whose shortest directed cycle has length at least r ≥ 4 satisfies β(G) ≤ cγ(G)/r2, where c is an absolute constant. This is tight up to the constant factor and extends a result of Chudnovsky, Seymour and Sullivan.

This result can also be used to answer a question of Yuster concerning almost given length cycles in digraphs. We show that for any fixed 0 < θ < 1/2 and sufficiently large n, if G is a digraph with n vertices and β(G) ≥ θn2, then for any 0 ≤ m ≤ θn − o(n) it contains a directed cycle whose length is between m and m + 6θ−1/2. Moreover, there is a constant C such that either G contains directed cycles of every length between C and θn − o(n) or it is close to a digraph G′ with a simple structure: every strong component of G′ is periodic. These results are also tight up to the constant factors.

On a separable C*-algebra A every (completely) bounded map which preserves closed two-sided ideals can be approximated uniformly by elementary operators if and only if A is a finite direct sum of C*-algebras of continuous sections vanishing at ∞ of locally trivial C*-bundles of finite type.

An important inequality due to Wolff on plate decompositions of cone multipliers is known to have consequences for sharp Lp results on cone multipliers, local smoothing for the wave equation, convolutions with radial kernels, Bergman projections in tubes over cones, averages over finite-type curves in ℝ3 and associated maximal functions. We observe that the range of p in Wolff's inequality, for the conic and the spherical versions, can be improved by using bilinear restriction results. We also use this inequality to give some improved estimates on square functions associated to decompositions of cone multipliers in low dimensions. This gives a new L4 bound for the cone multiplier operator in ℝ3.

Let H be the generic Iwahori–Hecke algebra associated with a finite Coxeter group W. Recently, we have shown that H admits a natural cellular basis in the sense of Graham and Lehrer, provided that W is a Weyl group and all parameters of H are equal. The construction involves some data arising from the Kazhdan–Lusztig basis {Cw} of H and Lusztig's asymptotic ring J}. We attempt to study J and its representation theory from a new point of view. We show that J can be obtained in an entirely different fashion from the generic representations of H, without any reference to {Cw}. We then extend the construction of the cellular basis to the case where W is not crystallographic. Furthermore, if H is a multi-parameter algebra, we see that there always exists at least one cellular structure on H. Finally, the new construction of J may be extended to Hecke algebras associated with complex reflection groups.

Let u(n)=f(gn), where g > 1 is integer and f(X) ∈ ℤ[X] is non-constant and has no multiple roots. We use the theory of -unit equations as well as bounds for character sums to obtain a lower bound on the number of distinct fields among for n ∈ . Fields of this type include the Shanks fields and their generalizations.

We present conditions guaranteeing the existence of non-trivial unstable sets for compact invariant sets in semiflows with certain compactness conditions, and then establish the existence of such unstable sets for an unstable equilibrium or a minimal compact invariant set, not containing equilibria, in an essentially strongly order-preserving semiflow. By appealing to the limit-set dichotomy for essentially strongly order-preserving semiflows, we prove the existence of an orbit connection from an equilibrium to a minimal compact invariant set, not consisting of equilibria. As an application, we establish a new generic convergence principle for essentially strongly order-preserving semiflows with certain compactness conditions.

Inspired by the results of Adin, Postnikov and Roichman, we propose combinatorial Gelfand models for semigroup algebras of some finite semigroups, which include the symmetric inverse semigroup, the dual symmetric inverse semigroup, the maximal factorizable subsemigroup in the dual symmetric inverse semigroup and the factor power of the symmetric group. Furthermore, we extend the Gelfand model for the semigroup algebras of the symmetric inverse semigroup to a Gelfand model for the q-rook monoid algebra.

We study the existence of multiple solutions for a two-point boundary-value problem associated with a planar system of second-order ordinary differential equations by using a shooting technique. We consider asymptotically linear nonlinearities satisfying suitable sign conditions. Multiplicity is ensured by assumptions involving the Morse indices of the linearizations at zero and at infinity.

It is shown that if f is an analytic function of sufficiently small exponential type in the right half-plane, which takes integer values on a subset of the positive integers having positive lower density, then f is a polynomial.

Integrable operators arise in random matrix theory, where they describe the asymptotic eigenvalue distribution of large self-adjoint random matrices from the generalized unitary ensembles. We consider discrete Tracy–Widom operators and give sufficient conditions for a discrete integrable operator to be the square of a Hankel matrix. Examples include the discrete Bessel kernel and kernels arising from the almost Mathieu equation and the Fourier transform of Mathieu's equation.

Let ℳ be a von Neumann algebra acting on a Hilbert space and let be a von Neumann subalgebra of ℳ. If is singular in for every Hilbert space , is said to be completely singular in ℳ. We prove that if is a singular abelian von Neumann subalgebra or if is a singular subfactor of a type-II1 factor ℳ, then is completely singular in ℳ. is separable, we prove that is completely singular in ℳ if and only if, for every θ∈Aut(′) such that θ(X)=X for all X ∈ ℳ′, θ(Y)=Y for all Y∈′. As the first application, we prove that if ℳ is separable (with separable predual) and is completely singular in ℳ, then is completely singular in for every separable von Neumann algebra . As the second application, we prove that if 1 is a singular subfactor of a type-II1 factor ℳ1 and 2 is a completely singular von Neumann subalgebra of ℳ2, then is completely singular in .

We define a topological space over the p-adic numbers, in which Euler products and Dirichlet series converge. We then show how the classical Riemann zeta function has a (p-adic) Euler product structure at the negative integers. Finally, as a corollary of these results, we derive a new formula for the non-Archimedean Euler–Mascheroni constant.

The upper and lower solutions method and Leray–Schauder degree theory are employed to establish the existence result for a class of nonlinear third-order two-point boundary-value problems with a sign-type Nagumo condition.

We consider the existence of three non-trivial smooth solutions for nonlinear elliptic problems driven by the p-Laplacian. Using variational arguments, coupled with the method of upper and lower solutions, critical groups and suitable truncation techniques, we produce three non-trivial smooth solutions, two of which have constant sign. The hypotheses incorporate both coercive and non-coercive problems in our framework of analysis.

An analytic function f in the unit disc belongs to F(p,q,s), if

is uniformly bounded for all a ∈ . Here is the Green function of , and φa(z)=(a−z)/(1−āz). It is shown that for 0 < γ < ∞ and |w|=1 the singular inner function exp(γ(z+w)/(z−w)) belongs to F(p,q,s), 0<s≤1, if and only if . Moreover, it is proved that, if 0<s<1, then an inner function belongs to the Möbius invariant Besov-type space for some (equivalently for all) p > max{s,1−s} if and only if it is a Blaschke product whose zero sequence {zn} satisfies .

We study the membership of Blaschke products in Lipschitz spaces, especially for interpolating Blaschke products and for those whose zeros lie in a Stolz angle. We prove several theorems that complement or extend the earlier works of Ahern and the author.

This paper is concerned with a boundary-value problem on the half-line for nonlinear two-dimensional delay differential systems with positive delays. A theorem is established, which provides sufficient conditions for the existence of positive solutions. The application of this theorem to the special case of second-order nonlinear delay differential equations is given. Also, the application of the theorem to two-dimensional Emden–Fowler-type delay differential systems with constant delays is presented. Moreover, some general examples demonstrating the applicability of the theorem are included.

We establish the existence of positive solutions of the Sturm–Liouville problem

where

We assume g and to be non-negative, continuous functions, a(s) is a positive continuous function, c≥0, p>1, and the function h is sub-quadratic with respect to u′. We combine a priori estimates with a fixed-point result of Krasnosel'skii to obtain the existence of a positive solution.