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*Mathematics* *Subject* *Classification*

"... Abstract. It is shown that every non-compact hyperbolic manifold of finite volume has a finite cover admitting a geodesic ideal triangulation. Also, every hyperbolic manifold of finite volume with non-empty, totally geodesic boundary has a finite regular cover which has a geodesic partially truncat ..."

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Abstract. It is shown that every non-compact hyperbolic manifold of finite volume has a finite cover admitting a geodesic ideal triangulation. Also, every hyperbolic manifold of finite volume with non-empty, totally geodesic boundary has a finite regular cover which has a geodesic partially truncated triangulation. The proofs use an extension of a result due to Long and Niblo concerning the separability of peripheral subgroups. Epstein and Penner [2] used a convex hull construction in Lorentzian space to show that every non-compact hyperbolic manifold of finite volume has a canonical subdivision into convex geodesic polyhedra all of whose vertices lie on the sphere at infinity of hyperbolic space. In general, one cannot expect to further subdivide these polyhedra into ideal geodesic simplices such that the result is an ideal triangulation. That this is possible after lifting the cell decomposition to an appropriate finite cover is the first main result of this paper. A cell decomposition of a hyperbolic nmanifold into ideal geodesic n-simplices all of which are embedded will be referred to as an embedded geodesic ideal triangulation. Theorem 1. Any non-compact hyperbolic manifold of finite volume has a finite regular cover which admits an embedded geodesic ideal triangulation. The study of geodesic ideal triangulations of hyperbolic 3-manifolds goes back to Thurston

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*Mathematics* *Subject* *Classification*

"... Abstract. For any set S let seq 1-1 (S) denote the cardinality of the set of all finite one-to-one sequences that can be formed from S, and for positive integers a let a S denote the cardinality of all functions from S to a. Using a result from combinatorial number theory, Halbeisen and Shelah have ..."

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Abstract. For any set S let seq 1-1 (S) denote the cardinality of the set of all finite one-to-one sequences that can be formed from S, and for positive integers a let a S denote the cardinality of all functions from S to a. Using a result from combinatorial number theory, Halbeisen and Shelah have shown that even in the absence of the axiom of choice, for infinite sets S one always has seq 1-1 (S) = 2 S (but nothing more can be proved without the aid of the axiom of choice). Combining stronger number-theoretic results with the combinatorial proof for a = 2, it will be shown that for most positive integers a one can prove the inequality seq 1-1 (S) = a S without using any form of the axiom of choice. Moreover, it is shown that a very probable number-theoretic conjecture implies that this inequality holds for every positive integer a in any model of set theory. Motivation It was proved in [3, Theorem 4] that for any set S with more than one element, the cardinality seq 1-1 (S) of the set of all finite one-to-one sequences that can be formed from S can never be equal to the cardinality of the power set of S, denoted by 2 S . The proof does not make use of any form of the axiom of choice and hence, the result also holds in models of set theory where the axiom of choice fails. Moreover, in the absence of the axiom of choice, seq 1-1 (S) = 2 S is all one can prove about the relation between these two cardinalities. In other words, for each of the statements seq 1-1 (S) < 2 S , seq 1-1 (S) > 2 S , and seq 1-1 (S) incomparable to 2 S , there are models of Zermelo-Fraenkel's set theory without the axiom of choice in which the statement is true (cf. [4, §9]). However, in the presence of the axiom of choice, for any infinite set S we always have seq 1-1 (S) < 2 S . Now,

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*Mathematics* *Subject* *Classification*

"... Abstract Spatial point processes are stochastic models for point patterns, systems of points scattered in R d . A point process can be used as a generating stochastic mechanism for additional spatial random systems such as random tessellations, random fields and random graphs, which are collectivel ..."

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Abstract Spatial point processes are stochastic models for point patterns, systems of points scattered in R d . A point process can be used as a generating stochastic mechanism for additional spatial random systems such as random tessellations, random fields and random graphs, which are collectively called secondary structures of point processes. Secondary structures have a role in the statistical analysis of point processes, e.g. in the form of statistical summaries based on tessellations, and in a method called regionalisation which bridges point pattern statistics with geostatistics. In this study the objective is to use geometric graphs together with graph-based summaries in the statistical analysis of smallscale properties of point patterns. The functional summaries of this study are connectivity function, cumulative connectivity function and clustering function. The concepts and their estimators are given, their properties are discussed, and a simulation study is conducted. The simulation experiment gives evidence that the graph-theoretical summaries are able to detect differences between point patterns where the second-order statistics such as Ripley's K or pair-correlation function fail. An R library has been developed for the computation of the graph-based summaries.

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*Mathematics* *Subject* *Classification*

"... Abstract. Let G be a classical complex Lie group, P any parabolic subgroup of G, and G/P the corresponding partial flag variety. We prove an explicit combinatorial Giambelli formula which expresses an arbitrary Schubert class in H * (G/P ) as a polynomial in certain special Schubert class generator ..."

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Abstract. Let G be a classical complex Lie group, P any parabolic subgroup of G, and G/P the corresponding partial flag variety. We prove an explicit combinatorial Giambelli formula which expresses an arbitrary Schubert class in H * (G/P ) as a polynomial in certain special Schubert class generators. Our formula extends to one that applies to the torus-equivariant cohomology ring of G/P and to the setting of symplectic and orthogonal degeneracy loci. Introduction The Giambelli formula [G] is one of the fundamental results concerning Schubert calculus in the cohomology ring of the Grassmannian X. The variety X has a decomposition into Schubert cells, which gives an additive basis of Schubert classes for the cohomology of X. On the other hand, the ring H * (X, Z) is generated by certain special Schubert classes, which are the Chern classes of the universal quotient bundle over X. The formula of Giambelli expresses a general Schubert class as a determinant of a Jacobi-Trudi matrix with entries given by special classes. One can show that this formula is equivalent to the Pieri rule [P]; see for instance [T4]. The Schubert calculus on X can be generalized to any homogeneous space G/P , where G is a complex reductive Lie group and P a parabolic subgroup of G. However, more than a century since the theorems of Pieri and Giambelli were discovered, no combinatorially explicit analogues of these results are known in this generality, unless the Lie group G is of type A. One reason for this is that there is no uniform way to extend the notion of a special Schubert class over all possible Lie types and parabolics. Another serious concern is the more difficult algebro-combinatorial questions that arise in the other Lie types, about which more below. When G is a classical Lie group, one can define special Schubert class generators for the cohomology ring H * (G/P ) uniformly, as follows. In this situation, the variety G/P parametrizes partial flags of subspaces of a vector space, which in types B, C, and D are required to be isotropic with respect to an orthogonal or symplectic form. First, the special Schubert varieties on any Grassmannian are defined as the locus of (isotropic) linear subspaces which meet a given (isotropic or coisotropic) linear subspace nontrivially, following [BKT1]. The special Schubert classes are the cohomology classes determined by these Schubert varieties. Finally, the special Schubert classes on a partial flag variety G/P are the pullbacks of special Schubert classes on Grassmannians, in agreement with the convention in type A. In most cases, these special classes are equal to the Chern classes of the universal quotient

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*Mathematics* *Subject* *Classifications* (2000):

"... The Berezin transform for complex multivariable domains D ⊂ Cn is important to harmonic analysis because of its covariance with respect to holomorphic transformations. It can be regarded as an analogue of the Poisson transform, replacing the boundary integration by integrating over the domain itself ..."

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The Berezin transform for complex multivariable domains D ⊂ Cn is important to harmonic analysis because of its covariance with respect to holomorphic transformations. It can be regarded as an analogue of the Poisson transform, replacing the boundary integration by integrating over the domain itself. This applies in particular

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*Mathematics* *Subject* *Classifications* (2000): References

"... Abstract. We analyze an adaptive finite element method (AFEM) which does not require an exact computation of the Galerkin approximation at each mesh. We propose to approximate the Galerkin solution Uk+1 at step k + 1 of the adaptive process up to a tolerance dictated by the difference between Ũk an ..."

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Abstract. We analyze an adaptive finite element method (AFEM) which does not require an exact computation of the Galerkin approximation at each mesh. We propose to approximate the Galerkin solution Uk+1 at step k + 1 of the adaptive process up to a tolerance dictated by the difference between Ũk and Ũk+1. Here Ũk denotes the computed approximate solution at step k of the adaptive iteration. The adaptive algorithm is guided by Dörfler’s marking strategy [2], using the residual-type a posteriori error estimators for the inexact (computed) solutions Ũk. We prove a quasi-optimality result analogous to those of Stevenson [3] and Cascón et. al. [1]. This turns out to be a theoretical result of optimality for a very practical and realistic adaptive method, which is computationally cheaper than the usual AFEM. Several numerical tests show that a few iterations of the iterative linear solver are needed to obtain the desired accuracy in each step of the adaptive loop.

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*Mathematics* *Subject* *Classification*: 37C20

"... Abstract Ecological vector fieldsẋ i = x i f i (x) on the non-negative cone R n + on R n are often used to describe the dynamics of n interacting species. These vector fields are called permanent (or uniformly persistent) if the boundary ∂R n + of the nonnegative cone is repelling. We construct an ..."

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Abstract Ecological vector fieldsẋ i = x i f i (x) on the non-negative cone R n + on R n are often used to describe the dynamics of n interacting species. These vector fields are called permanent (or uniformly persistent) if the boundary ∂R n + of the nonnegative cone is repelling. We construct an open set of ecological vector fields containing a dense subset of permanent vector fields and containing a dense subset of vector fields with attractors on ∂R n + . In particular, this construction implies that robustly permanent vector fields are not dense in the space of permanent vector fields. Hence, verifying robust permanence is important. We illustrate this result with ecological vector fields involving five species that admit a heteroclinic cycle between two equilibria and the Hastings-Powell teacup attractor.

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*Mathematics* *Subject* *Classification*: 37A10

"... Abstract. It is proved that some velocity changes in flows on the torus determined by quasi-periodic Hamiltonians on R 2 : Hðx þ m; y þ nÞ ¼ Hðx; yÞ þ m 1 þ n 2 ; where 1 = 2 is an irrational number with bounded partial quotients, lead to singular flows on T 2 with an ergodic component having a min ..."

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Abstract. It is proved that some velocity changes in flows on the torus determined by quasi-periodic Hamiltonians on R 2 : Hðx þ m; y þ nÞ ¼ Hðx; yÞ þ m 1 þ n 2 ; where 1 = 2 is an irrational number with bounded partial quotients, lead to singular flows on T 2 with an ergodic component having a minimal set of self-joinings.

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*Mathematics* *Subject* *Classification*. Primary 53 C 25

"... Abstract. We study geometric structures of W4-type in the sense of A. Gray on a Riemannian manifold. If the structure group G ⊂ SO(n) preserves a spinor or a non-degenerate differential form, its intrinsic torsion Γ is a closed 1-form (Proposition 2.1 and Theorem 2.1). Using a G-invariant spinor we ..."

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Abstract. We study geometric structures of W4-type in the sense of A. Gray on a Riemannian manifold. If the structure group G ⊂ SO(n) preserves a spinor or a non-degenerate differential form, its intrinsic torsion Γ is a closed 1-form (Proposition 2.1 and Theorem 2.1). Using a G-invariant spinor we prove a splitting theorem (Proposition 2.2). The latter result generalizes and unifies a recent result obtained in c . An interesting class of geometric structures generalizing Hopf structures are those with a ∇ c -parallel intrinsic torsion Γ. In this case, Γ induces a Killing vector field (Proposition 4.1) and for some special structure groups it is even parallel. Adapted connections of a geometric structure of vectorial type Fix a subgroup G ⊂ SO(n) of the special orthogonal group and decompose the Lie algebra so(n) = g ⊕ m into the Lie algebra g of G and its orthogonal complement m. The different geometric types of G-structures on a Riemannian manifold correspond to the irreducible G-components of the representation R n ⊗m. Indeed, consider an oriented Riemannian manifold (M n , g) and denote its Riemannian frame bundle by F(M n ). It is a principal SO(n)-bundle over M n . A G-structure is a reduction R ⊂ F(M n ) of the frame bundle to the subgroup G. The Levi-Civita connection is a 1-form Z on F(M n ) with values in the Lie algebra so(n). We restrict the Levi-Civita connection to R and decompose it with respect to the decomposition of the Lie algebra so(n), Then, Z * is a connection in the principal G-bundle R and Γ is a 1-form on M n with values in the associated bundle R × G m. This 1-form, or more precisely the G-components of the element Γ ∈ R n ⊗ m, characterizes the different types of non-integrable Gstructures (see e i ⊗ pr m (e i T) .

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*Mathematics* *Subject* *Classification*: Primary 46G20

"... Abstract. We study projections and injections between projective tensor products spaces or spaces of polynomials and we show that the example of a polynomial constructed in Since Ryan ..."

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Abstract. We study projections and injections between projective tensor products spaces or spaces of polynomials and we show that the example of a polynomial constructed in Since Ryan