The linking pairings of orientable Seifert manifolds

We compute the p-primary components of the linking pairings of orientable 3-manifolds admitting a fixed-point free S¹-action. Any linking pairing on a finite abelian group of odd order is realized by such a manifold. We find necessary and sufficient conditions for a pairing on an abelian 2-group to be the 2-primary component of such a linking pairing, and give simple examples which are not realizable by any Seifert fibred 3-manifold.     

Approximation Properties AP s and p-Nuclear Operators (the Case 0 < s ≤ 1)

We investigate Banach spaces possessing (or not possessing) the approximation properties AP _s, 0 < s ≤ 1, in connection with the following known question in the geometric theory of operators: under which conditions on Banach spaces X and Y and on positive numbers r and p does the p-nuclearity of the second adjoint of a continuous operator T from X to Y imply the p-nuclearity of T? Actually, we give necessary and sufficient conditions under which this question is answered affirmatively. In addition, the corresponding counterexamples are obtained in the maximally strong form. For instance, it is shown (and this statement is a significant strengthening of the previous results of that sort) that there exists a pair of separable Banach spaces Z and W such that the spaces Z ^** and W have Schauder bases, while for every p, 1 ≤ p < 2, there is a non-p-nuclear operator from W to Z with a p-nuclear second adjoint. Earlier, in similar examples, the corresponding spaces did not possess even the Grothendieck approximation property. The technique developed in this paper does not allow us to treat the case p > 2. That case will be studied in a forthcoming paper of the author. Bibliography: 11 titles.     

Relationship between strictly collapsible and perfect contingency tables

Whittemore (1978) conjectured that an N-dimensional contingency table p is strictly collapsible over each factor with respect to the set of remaining factors if and only if p has a certain factorization. I prove this conjecture for N = 3 and show by counterexamples that it is false for N > 3.     

Foliations and global injectivity in ℝ n

Let F: ? ⁿ → ? ⁿ be a polynomial local diffeomorphism and let S _F denote the set of not proper points of F. The Jelonek’s real Jacobian Conjecture states that if codim(S _F ) ≥ 2, then F is bijective. We prove a weak version of such conjecture establishing the sufficiency of a necessary condition for the bijectivity of F.     

Subnormal operators whose adjoints have rich point spectrum

A generalized version of the Glauber-Klauder basic formula of quantum optics is shown to be valid for any cyclic subnormal operator S whose adjoint has a rich point spectrum σp(S^∗) (in the sense that a semispectral measure of S vanishes on C?σp^∗(S^∗)). It is exhibited that such operators always have analytic models. The point spectrum of the adjoint of a subnormal operator which satisfies a generalized version of the Glauber-Klauder formula is proved to be rich (in the above sense).     

On highly potential words

We introduce a class of infinite words, called highly potential words because of their seemingly high potential of being a good supply of examples and counterexamples regarding various problems on words. We prove that they are all aperiodic words of finite positive defect, and having their set of factors closed under reversal, thus giving examples Brlek and Reutenauer were looking for. We prove that they indeed satisfy the Brlek–Reutenauer conjecture. We observe that each highly potential word is recurrent, but not uniformly recurrent. Considering a theorem from the paper of Balková, Pelantová and Starosta, later found to be incorrect, we show that highly potential words constitute an infinite family of counterexamples to that theorem. Finally, we construct a highly potential word which is a fixed point of a nonidentical morphism, thus showing that a stronger version of a conjecture by Blondin-Massé et al., as stated by Brlek and Reutenauer, is false.     

Chords and disjoint paths in matroids

A chord of a circuit C of a matroid M on E is a cell e ? S\C such that C spans e. Menger's theorem gives necessary and sufficient conditions for a cell of a graphic matroid to be a chord of some circuit. We extend this result to a large class of matroids and find all minimal counterexamples. The theorem is used to obtain results on disjoint paths and to characterize a class of matroid sums.     

The local regularity for strong solutions of the Hessian quotient equation

By means of the Reilly formula and the Alexandrov maximum principle, we obtain the local C1,1 estimates of the W2,p strong solutions to the Hessian quotient equations for p sufficiently large, and then prove that these solutions are smooth. There are counterexamples to show that the integral exponent p is optimal in some cases. We modify partially the known result in the Hessian case, and extend the regularity result in the special Lagrangian case to the Hessian quotient case.     

Spectrum of a network of Euler-Bernoulli beams

A network of N flexible beams connected by n vibrating point masses is considered. The spectrum of the spatial operator involved in this evolution problem is studied. If λ² is any real number outside a discrete set of values S and if λ is an eigenvalue, then it satisfies a characteristic equation which is given. The associated eigenvectors are also characterized. If λ² lies in S and if the N beams are identical (same mechanical properties), another characteristic equation is available. It is not the case for different beams: no general result can be stated. Some numerical examples and counterexamples are given to illustrate the impossibility of such a generalization. At last the asymptotic behaviour of the eigenvalues is investigated by proving the so-called Weyl's formula.     

Distribution of Points and Hardy Type Inequalities in Spaces of Homogeneous Type

In the setting of spaces of homogeneous type, we study some Hardy type inequalities, which notably appeared in the proofs of local T(b) theorems. We give some sufficient conditions ensuring their validity, related to the geometry and distribution of points in the homogeneous space. We study the relationships between these conditions and give some examples and counterexamples in the complex plane.     

Unbalanced bipartite factorizations of complete bipartite graphs

We construct a new infinite family of factorizations of complete bipartite graphs by factors all of whose components are copies of a (fixed) complete bipartite graph K_p,q. There are simple necessary conditions for such factorizations to exist. The family constructed here demonstrates sufficiency in many new cases. In particular, the conditions are always sufficient when q=p+1.     

Relative symplectic caps, 4-genus and fibered knots

We prove relative versions of the symplectic capping theorem and sufficiency of Giroux’s criterion for Stein fillability and use these to study the 4-genus of knots. More precisely, suppose we have a symplectic 4-manifold X with convex boundary and a symplectic surface Σ in X such that ?Σ is a transverse knot in ?X. In this paper, we prove that there is a closed symplectic 4-manifold Y with a closed symplectic surface S such that (X,Σ) embeds into (Y,S) symplectically. As a consequence we obtain a relative version of the symplectic Thom conjecture. We also prove a relative version of the sufficiency part of Giroux’s criterion for Stein fillability, namely, we show that a fibered knot whose mondoromy is a product of positive Dehn twists bounds a symplectic surface in a Stein filling. We use this to study 4-genus of fibered knots in \(\mathbb {S}^{3} \). Further, we give a criterion for quasipositive fibered knots to be strongly quasipositive.     

The time-optimal control problem in Banach spaces

We show here that a variant of Pontryagin's maximum principle holds for time-optimal trajectories of the systemu′ = Au + f that begin and end at the domain ofA. HereA is the infinitesimal generator of a strongly continuous semigroupS(·) in the reflexive Banach spaceE. This partly generalizes a result of Balakrishnan ([1]) where no conditions are assumed on the trajectories but whereS(t) is supposed to beonto for allt ≥ 0. A weakened version of the maximum principle is then proved forall time-optimal trajectories whenA generates an analytic semigroup andE is a Hilbert space. Finally, the results are modified to handle two examples involving the heat equation where the spaceE is not reflexive.     

Sampling sets and sufficient sets for A

We give new characterizations of the subsets S of the unit disc of the complex plane such that the topology of the space A^−∞ of holomorphic functions of polynomial growth on coincides with the topology of space of the restrictions of the functions to the set S. These sets are called weakly sufficient sets for A^−∞. Our approach is based on a study of the so-called (p,q)-sampling sets which generalize the A^−p-sampling sets of Seip. A characterization of (p,q)-sampling and weakly sufficient rotation invariant sets is included. It permits us to obtain new examples and to solve an open question of Khôi and Thomas.     

Quadratic residues and the combinatorics of sign multiplication

If S is a nonempty finite set of positive integers, we find a criterion both necessary and sufficient for S to satisfy the following condition: if q is a fixed nonnegative integer, then there exists infinitely many primes p such that S contains exactly q quadratic residues of p. This result simultaneously generalizes two previous results of the author, and the criterion used is expressed by means of a purely combinatorial condition on the prime factors of the elements of S of odd multiplicity.     

Locally Lipschitz composition operators in spaces of functions of bounded variation

We give a necessary and sufficient condition on a function \({f:\mathbb{R}\to\mathbb{R}}\) under which the composition operator (Nemytskij operator) F defined by \({Fh=f\circ h}\) acts in the spaces \({BV_\varphi[a,b], HBV[a,b], {\rm and} \,RV_\varphi[a,b]}\) and satisfies a local Lipschitz condition. While the proof of sufficiency consists in a straightforward calculation, the proof of necessity builds on nontrivial arguments like Helly’s selection principle or the Arzelà–Ascoli compactness criterion.     

Nonnegative quadratic estimation and quadratic sufficiency in general linear models

Notions of linear sufficiency and quadratic sufficiency are of interest to some authors. In this paper, the problem of nonnegative quadratic estimation for β^′Hβ+hσ² is discussed in a general linear model and its transformed model. The notion of quadratic sufficiency is considered in the sense of generality, and the corresponding necessary and sufficient conditions for the transformation to be quadratically sufficient are investigated. As a direct consequence, the result on (ordinary) quadratic sufficiency is obtained. In addition, we pose a practical problem and extend a special situation to the multivariate case. Moreover, a simulated example is conducted, and applications to a model with compound symmetric covariance matrix are given. Finally, we derive a remark which indicates that our main results could be extended further to the quasi-normal case.     

Sewings of crumpled cubes revisited

This paper sets forth three mismatch properties, strictly ordered in strength, about sewings of crumpled n-cubes. The strongest is a sufficient but not a necessary condition for a sewing to yield Sn, and the weakest, a necessary but not sufficient one. We show that when both crumpled cubes satisfies the Disjoint Disks Property, then the weakest property implies the sewing yields Sn, and we also show that the intermediate property leads to the same conclusion when just one of the crumpled cubes possesses the Disjoint Disks Property. In addition, we develop examples that confirm sharpness of the relevant Disjoint Disks conditions.     

Counterexamples to isosystolic inequalities

We explore M. Gromov's counterexamples to systolic inequalities. Does the manifoldS ² ×S ² admit metrics of arbitrarily small volume such that every noncontractible surface inside it has at least unit area? This question is still open, but the answer is affirmative for its analogue in the case ofS ⁿ ×S ⁿ ,n ≥ 3. Our point of departure is M. Gromov's metric onS ¹ ×S ³, and more general examples, due to C. Pittet, of metrics onS ¹ ×S ⁿ with ‘voluminous’ homology. We take the metric product of these metrics with a sphereS ^n?1 of a suitable volume, and perform surgery to obtain the desired metrics onS ⁿ ×S ⁿ .     

Irreducible Specht modules for Hecke algebras of type A

Let F be a field, n a non-negative integer, λ a partition of n and S^λ the corresponding Specht module for the Iwahori-Hecke algebra H_F,q(S_n). James and Mathas conjecture a necessary and sufficient condition on λ for S^λ to be irreducible. We prove the sufficiency of this condition in the case where F has infinite characteristic and also in the case where q=1.