On the nonexistence of closed timelike geodesics in flat Lorentz 2-step nilmanifolds

The main purpose of this paper is to prove that there are no closed timelike geodesics in a (compact or noncompact) flat Lorentz 2-step nilmanifold where is a simply connected 2-step nilpotent Lie group with a flat left-invariant Lorentz metric, and a discrete subgroup of acting on by left translations. For this purpose, we shall first show that if is a 2-step nilpotent Lie group endowed with a flat left-invariant Lorentz metric then the restriction of to the center of is degenerate. We shall then determine all 2-step nilpotent Lie groups that can admit a flat left-invariant Lorentz metric. We show that they are trivial central extensions of the three-dimensional Heisenberg Lie group . If is one such group, we prove that no timelike geodesic in can be translated by an element of By the way, we rediscover that the Heisenberg Lie group admits a flat left-invariant Lorentz metric if and only if

     

Resolutions of ideals of quasiuniform fat point subschemes of

The notion of a quasiuniform fat point subscheme is introduced and conjectures for the Hilbert function and minimal free resolution of the ideal defining are put forward. In a large range of cases, it is shown that the Hilbert function conjecture implies the resolution conjecture. In addition, the main result gives the first determination of the resolution of the th symbolic power of an ideal defining general points of when both and are large (in particular, for infinitely many for each of infinitely many , and for infinitely many for every 2$">). Resolutions in other cases, such as ``fat points with tails', are also given. Except where an explicit exception is made, all results hold for an arbitrary algebraically closed field . As an incidental result, a bound for the regularity of is given which is often a significant improvement on previously known bounds.

     

The space for nondoubling measures in terms of a grand maximal operator

Let be a Radon measure on , which may be nondoubling. The only condition that must satisfy is the size condition , for some fixed . Recently, some spaces of type and were introduced by the author. These new spaces have properties similar to those of the classical spaces and defined for doubling measures, and they have proved to be useful for studying the boundedness of Calderón-Zygmund operators without assuming doubling conditions. In this paper a characterization of the new atomic Hardy space in terms of a maximal operator is given. It is shown that belongs to if and only if , and , as in the usual doubling situation.

     

On one-dimensional self-similar tilings and -tiles

Let be an integer base, a digit set and the set of radix expansions. It is well known that if has nonvoid interior, then can tile with some translation set ( is called a tile and a tile digit set). There are two fundamental questions studied in the literature: (i) describe the structure of ; (ii) for a given , characterize so that is a tile.

We show that for a given pair , there is a unique self-replicating translation set , and it has period for some . This completes some earlier work of Kenyon. Our main result for (ii) is to characterize the tile digit sets for when are distinct primes. The only other known characterization is for , due to Lagarias and Wang. The proof for the case depends on the techniques of Kenyon and De Bruijn on the cyclotomic polynomials, and also on an extension of the product-form digit set of Odlyzko.

     

On partitioning the orbitals of a transitive permutation group

Let be a permutation group on a set with a transitive normal subgroup . Then acts on the set of nontrivial -orbitals in the natural way, and here we are interested in the case where has a partition such that acts transitively on . The problem of characterising such tuples , called TODs, arises naturally in permutation group theory, and also occurs in number theory and combinatorics. The case where is a prime-power is important in algebraic number theory in the study of arithmetically exceptional rational polynomials. The case where exactly corresponds to self-complementary vertex-transitive graphs, while the general case corresponds to a type of isomorphic factorisation of complete graphs, called a homogeneous factorisation. Characterising homogeneous factorisations is an important problem in graph theory with applications to Ramsey theory. This paper develops a framework for the study of TODs, establishes some numerical relations between the parameters involved in TODs, gives some reduction results with respect to the -actions on and on , and gives some construction methods for TODs.

     

Systems of diagonal Diophantine inequalities

We treat systems of real diagonal forms of degree , in variables. We give a lower bound , which depends only on and , such that if holds, then, under certain conditions on the forms, and for any positive real number , there is a nonzero integral simultaneous solution of the system of Diophantine inequalities for . In particular, our result is one of the first to treat systems of inequalities of even degree. The result is an extension of earlier work by the author on quadratic forms. Also, a restriction in that work is removed, which enables us to now treat combined systems of Diophantine equations and inequalities.

     

The Dirichlet problem and nondivergence harmonic measure

We consider the Dirichlet problem

for two second-order elliptic operators , , in a bounded Lipschitz domain . The coefficients belong to the space of bounded mean oscillation with a suitable small modulus. We assume that is regular in for some , , that is, for all continuous boundary data . Here is the surface measure on and is the nontangential maximal operator. The aim of this paper is to establish sufficient conditions on the difference of the coefficients that will assure the perturbed operator to be regular in for some , .

     

Analytic models for commuting operator tuples on bounded symmetric domains

For a domain in and a Hilbert space of analytic functions on which satisfies certain conditions, we characterize the commuting -tuples of operators on a separable Hilbert space  such that is unitarily equivalent to the restriction of to an invariant subspace, where is the operator -tuple on the Hilbert space tensor product  . For the unit disc and the Hardy space , this reduces to a well-known theorem of Sz.-Nagy and Foias; for a reproducing kernel Hilbert space on such that the reciprocal of its reproducing kernel is a polynomial in and  , this is a recent result of Ambrozie, Müller and the second author. In this paper, we extend the latter result by treating spaces for which ceases to be a polynomial, or even has a pole: namely, the standard weighted Bergman spaces (or, rather, their analytic continuation) on a Cartan domain corresponding to the parameter in the continuous Wallach set, and reproducing kernel Hilbert spaces for which is a rational function. Further, we treat also the more general problem when the operator is replaced by ,  being a certain generalization of a unitary operator tuple. For the case of the spaces on Cartan domains, our results are based on an analysis of the homogeneous multiplication operators on , which seems to be of an independent interest.

     

The -module structure of -modules

Let be a regular ring, essentially of finite type over a perfect field . An -module is called a unit -module if it comes equipped with an isomorphism , where denotes the Frobenius map on , and is the associated pullback functor. It is well known that then carries a natural -module structure. In this paper we investigate the relation between the unit -structure and the induced -structure on . In particular, it is shown that if is algebraically closed and is a simple finitely generated unit -module, then it is also simple as a -module. An example showing the necessity of being algebraically closed is also given.

     

Non-solvability for a class of left-invariant second-order differential operators on the Heisenberg group

We study the question of local solvability for second-order, left-invariant differential operators on the Heisenberg group , of the form

where is a complex matrix. Such operators never satisfy a cone condition in the sense of Sjöstrand and Hörmander. We may assume that cannot be viewed as a differential operator on a lower-dimensional Heisenberg group. Under the mild condition that and their commutator are linearly independent, we show that is not locally solvable, even in the presence of lower-order terms, provided that . In the case we show that there are some operators of the form described above that are locally solvable. This result extends to the Heisenberg group a phenomenon first observed by Karadzhov and Müller in the case of It is interesting to notice that the analysis of the exceptional operators for the case turns out to be more elementary than in the case When the analysis of these operators seems to become quite complex, from a technical point of view, and it remains open at this time.

     

Couples contacto-symplectiques

We introduce a new geometric structure on differentiable manifolds. A contact-symplectic pair on a manifold is a pair where is a Pfaffian form of constant class and a -form of constant class such that is a volume form. Each form has a characteristic foliation whose leaves are symplectic and contact manifolds respectively. These foliations are transverse and complementary. Some other differential objects are associated to it. We give a local model and several existence theorems on nilpotent Lie groups, nilmanifolds and principal torus bundles. As a deep application of this theory, we give a negative answer to the famous Reeb's problem which asks if every vector field without closed 1-codimensional transversal on a manifold having contact forms is the Reeb vector field of a contact form.

     

Automorphisms of finite order on Gorenstein del Pezzo surfaces

In this paper we shall determine all actions of groups of prime order with on Gorenstein del Pezzo (singular) surfaces of Picard number 1. We show that every order- element in ( , being the minimal resolution of ) is lifted from a projective transformation of . We also determine when is finite in terms of , and the number of singular members in . In particular, we show that either for some , or for every prime , there is at least one element of order in (hence is infinite).

     

A classification and examples of rank one chain domains

A chain order of a skew field is a subring of so that implies Such a ring has rank one if , the Jacobson radical of is its only nonzero completely prime ideal. We show that a rank one chain order of is either invariant, in which case corresponds to a real-valued valuation of or is nearly simple, in which case and are the only ideals of or is exceptional in which case contains a prime ideal that is not completely prime. We use the group of divisorial of with the subgroup of principal to characterize these cases. The exceptional case subdivides further into infinitely many cases depending on the index of in Using the covering group of and the result that the group ring is embeddable into a skew field for a skew field, examples of rank one chain orders are constructed for each possible exceptional case.

     

Degenerate fibres in the Stone-Cech compactification of the universal bundle of a finite group

Applied to a continuous surjection of completely regular Hausdorff spaces and , the Stone-Cech compactification functor yields a surjection . For an -fold covering map , we show that the fibres of , while never containing more than points, may degenerate to sets of cardinality properly dividing . In the special case of the universal bundle of a -group , we show more precisely that every possible type of -orbit occurs among the fibres of . To prove this, we use a weak form of the so-called generalized Sullivan conjecture.

     

The periodic Euler-Bernoulli equation

We continue the study of the Floquet (spectral) theory of the beam equation, namely the fourth-order eigenvalue problem

where the functions and are periodic and strictly positive. This equation models the transverse vibrations of a thin straight (periodic) beam whose physical characteristics are described by and . Here we develop a theory analogous to the theory of the Hill operator .

We first review some facts and notions from our previous works, including the concept of the pseudospectrum, or -spectrum.

Our new analysis begins with a detailed study of the zeros of the function , for any given ``quasimomentum' , where is the Floquet-Bloch variety of the beam equation (the Hill quantity corresponding to is , where is the discriminant and the period of ). We show that the multiplicity of any zero of can be one or two and (for some ) if and only if is also a zero of another entire function , independent of . Furthermore, we show that has exactly one zero in each gap of the spectrum and two zeros (counting multiplicities) in each -gap. If is a double zero of , it may happen that there is only one Floquet solution with quasimomentum ; thus, there are exceptional cases where the algebraic and geometric multiplicities do not agree.

Next we show that if is an open -gap of the pseudospectrum (i.e., ), then the Floquet matrix has a specific Jordan anomaly at and .

We then introduce a multipoint (Dirichlet-type) eigenvalue problem which is the analogue of the Dirichlet problem for the Hill equation. We denote by the eigenvalues of this multipoint problem and show that is also characterized as the set of values of for which there is a proper Floquet solution such that .

We also show (Theorem 7) that each gap of the -spectrum contains exactly one and each -gap of the pseudospectrum contains exactly two 's, counting multiplicities. Here when we say ``gap' or ``-gap' we also include the endpoints (so that when two consecutive bands or -bands touch, the in-between collapsed gap, or -gap, is a point). We believe that can be used to formulate the associated inverse spectral problem.

As an application of Theorem 7, we show that if is a collapsed (``closed') -gap, then the Floquet matrix is diagonalizable.

Some of the above results were conjectured in our previous works. However, our conjecture that if all the -gaps are closed, then the beam operator is the square of a second-order (Hill-type) operator, is still open.

     

A homotopy principle for maps with prescribed Thom-Boardman singularities

Let and be smooth manifolds of dimensions and ( ) respectively. Let denote an open subspace of which consists of all Boardman submanifolds of symbols with . An -regular map refers to a smooth map such that . We will prove what is called the homotopy principle for -regular maps on the existence level. Namely, a continuous section of over has an -regular map such that and are homotopic as sections.

     

Twisted sums with spaces

If is a separable Banach space, we consider the existence of non-trivial twisted sums , where or For the case we show that there exists a twisted sum whose quotient map is strictly singular if and only if contains no copy of . If we prove an analogue of a theorem of Johnson and Zippin (for ) by showing that all such twisted sums are trivial if is the dual of a space with summable Szlenk index (e.g., could be Tsirelson's space); a converse is established under the assumption that has an unconditional finite-dimensional decomposition. We also give conditions for the existence of a twisted sum with with strictly singular quotient map.

     

Stratified transversality by isotopy

For an abstract stratified set or a -regular stratification, hence for any -, - or -regular stratification, we prove that after stratified isotopy of , a stratified subspace of , or a stratified map , can be made transverse to a fixed stratified map .