Three-dimensional Lie group actions on compact (4n+3)-dimensional geometric manifolds

The (4n+3)-dimensional sphere S⁴ⁿ⁺³ can be viewed as the boundary of the quaternionic hyperbolic space and the group PSp(n+1,1) of quaternionic hyperbolic isometries extends to a real analytic transitive action on S⁴ⁿ⁺³. We call the pair (PSp(n+1,1),S⁴ⁿ⁺³) a spherical Q C-C geometry. A manifold M locally modelled on this geometry is said to be a spherical Q C-C manifold. We shall classify all pairs (G,M) where G is a three-dimensional connected Lie group which acts smoothly and almost freely on a compact spherical Q C-C manifold M, preserving the geometric structure. As an application, we shall determine all compact 3-pseudo-Sasakian manifolds admitting spherical Q C-C structures.     

On certain spaces of lattice diagram determinants

The aim of this work is to study some lattice diagram determinants Δ_L(X,Y) as defined in (Adv. Math. 142 (1999) 244) and to extend results of Aval et al. (J. Combin. Theory Ser. A, to appear). We recall that M_L denotes the space of all partial derivatives of Δ_L. In this paper, we want to study the space M_i,j^k(X,Y) which is defined as the sum of M_L spaces where the lattice diagrams L are obtained by removing k cells from a given partition, these cells being in the “shadow” of a given cell (i,j) in a fixed Ferrers diagram. We obtain an upper bound for the dimension of the resulting space M_i,j^k(X,Y), that we conjecture to be optimal. This dimension is a multiple of n! and thus we obtain a generalization of the n! conjecture. Moreover, these upper bounds associated to nice properties of some special symmetric differential operators (the “shift” operators) allow us to construct explicit bases in the case of one set of variables, i.e. for the subspace M_i,j^k(X) consisting of elements of 0 Y-degree.     

Positively regular homeomorphisms of Euclidean spaces

A homeomorphism of Rⁿ onto itself is called positively regular (or EC⁺) iff its family of non-negative iterates is pointwise equicontinuous. For EC⁺ homeomorphism of Rⁿ such that some point of Rⁿ has bounded positive semi-orbit, the nucleus M is defined, and the following theorems are proved.Theorem 1. If such a homeomorphism h:Rⁿ→Rⁿ has compact nucleus M, then M is a fully invariant compact AR. Further, for n≠4,5,h:Rⁿ/M→Rⁿ/M is conjugate to a contraction on Rⁿ.Theorem 2. In Rⁿ,n≠4,5,M compact iff there existsa disk D such that h(D)?IntD.Theorem 3. In R², either M is a disk and h|M is a rotation, or h|M is periodic. The relationship between M and the irregular set of ? is also studied.     

Lower bounds on the minimum average distance of binary codes

Let β(n,M) denote the minimum average Hamming distance of a binary code of length n and cardinality M. In this paper we consider lower bounds on β(n,M). All the known lower bounds on β(n,M) are useful when M is at least of size about 2ⁿ⁻¹/n. We derive new lower bounds which give good estimations when size of M is about n. These bounds are obtained using a linear programming approach. In particular, it is proved that lim_n→∞β(n,2n)=5/2. We also give a new recursive inequality for β(n,M).     

Commutative fundamental (m,n)-modules

In this paper, we introduce the concept of fundamental relation θ¹ on an (m, n)-hypermodule M as the smallest equivalence relation such that M/θ¹ is a commutative (m, n)-module, and then some related properties are investigated.     

Levi-Flat Hypersurfaces Tangent to Projective Foliations

Let M?? ⁿ be a singular real-analytic Levi-flat hypersurface tangent to a codimension-one holomorphic foliation \(\mathcal{F}\) on ? ⁿ . For n≥3, we give sufficient conditions to guarantee the existence of degenerate singularities in M, (in the sense of Segre varieties) and as a consequence we prove that \(\mathcal{F}\) can be defined by a global closed meromorphic 1-form.     

Complexity of Finding Irreducible Components of a Semialgebraic Set

Let W ⊂ Rⁿ be a semialgebraic set defined by a quantifier-free formula with k atomic polynomials of the kind f ∈ Z[X₁, . . . , X_n] such that deg_{X1, . . . , Xn}(f) < d and the absolute values of coefficients of f are less than 2^M for some positive integers d, M. An algorithm is proposed for producing the complexification, Zariski closure, and also for finding all irreducible components of W. The running time of the algorithm is bounded from above by M^O(1)(kd)^nO(1). The procedure is applied to computing a Whitney system for a semialgebraic set and the real radical of a polynomial ideal.     

Chain sequences and symmetric generalized orthogonal polynomials

In this paper, we derive an explicit expression for the parameter sequences of a chain sequence in terms of the corresponding orthogonal polynomials and their associated polynomials. We use this to study the orthogonal polynomials K_n^(λ,M,k) associated with the probability measure dφ(λ,M,k;x), which is the Gegenbauer measure of parameter λ+1 with two additional mass points at ±k. When k=1 we obtain information on the polynomials K_n^(λ,M) which are the symmetric Koornwinder polynomials. Monotonicity properties of the zeros of K_n^(λ,M,k) in relation to M and k are also given.     

Approximation by light maps and parametric Lelek maps

The class of metrizable spaces M with the following approximation property is introduced and investigated: M∈AP(n,0) if for every ε>0 and a map g:In→M there exists a 0-dimensional map g^′:In→M which is ε-homotopic to g. It is shown that this class has very nice properties. For example, if Mi∈AP(ni,0), i=1,2, then M₁×M₂∈AP(n₁+n₂,0). Moreover, M∈AP(n,0) if and only if each point of M has a local base of neighborhoods U with U∈AP(n,0). Using the properties of AP(n,0)-spaces, we generalize some results of Levin and Kato-Matsuhashi concerning the existence of residual sets of n-dimensional Lelek maps.     

On conformally flat Lorentz parabolic manifolds

We introduce conformally flat Fefferman-Lorentz manifold of parabolic type as a special class of Lorentz parabolic manifolds. It is a smooth (2n+2)-manifold locally modeled on (Û(n+1, 1), S ^2n+1,1). As the terminology suggests, when a Fefferman-Lorentz manifold M is conformally flat, M is a Fefferman-Lorentz manifold of parabolic type. We shall discuss which compact manifolds occur as a conformally flat Fefferman-Lorentz manifold of parabolic type.     

Some Applications of the Hodge-de Rham Decomposition to Ricci Solitons

The aim of this paper is to present a link between the Perelman potential for a compact Ricci soliton M ⁿ and the Hodge-de Rham decomposition theorem, we shall use this result to present an integral formula which enables us to establish conditions under which the Ricci soliton is trivial. Moreover, given a Ricci soliton such that its associated vector field X is a conformal vector field we show that in the compact case X is a Killing vector field, while for the non-compact case, either the soliton is Gaussian or X is a Killing vector field.     

Vector cascade algorithms with infinitely supported masks in weighted L 2-spaces

In this paper, we shall study the solutions of functional equations of the form Φ =∑α∈Zsa(α)Φ(M·-α), where Φ = (φ1, . . . , φr)T is an r×1 column vector of functions on the s-dimensional Euclidean space, a:=(a(α))α∈Zs is an exponentially decaying sequence of r×r complex matrices called refinement mask and M is an s×s integer matrix such that limn→∞M-n=0. We are interested in the question, for a mask a with exponential decay, if there exists a solution Φ to the functional equation with each function φj,j=1, . . . ,r, belonging to L2(Rs) and having exponential decay in some sense? Our approach will be to consider the convergence of vector cascade algorithms in weighted L2 spaces. The vector cascade operator Qa,M associated with mask a and matrix M is defined by Qa,Mf:=∑α∈Zsa(α)f (M·-α),f= (f1, . . . , fr)T∈(L2,μ(Rs))r.The iterative scheme (Qan,Mf)n=1,2,... is called a vector cascade algorithm or a vector subdivision scheme. The purpose of this paper is to provide some conditions for the vector cascade algorithm to converge in (L2,μ(Rs))r , the weighted L2 space. Inspired by some ideas in [Jia,R.Q.,Li,S.: Refinable functions with exponential decay: An approach via cascade algorithms. J. Fourier Anal. Appl., 17, 1008-1034 (2011)], we prove that if the vector cascade algorithm associated with a and M converges in (L2(Rs))r, then its limit function belongs to (L2,μ(Rs))r for some μ0.     

Minimal tori with low nullity

The nullity of a minimal submanifold M⊂Sn is the dimension of the nullspace of the second variation of the area functional. That space contains as a subspace the effect of the group of rigid motions SO(n+1) of the ambient space, modulo those motions which preserve M, whose dimension is the Killing nullitykn(M) of M. In the case of 2-dimensional tori M in S³, there is an additional naturally-defined 2-dimensional subspace that contributes to the nullity; the dimension of the sum of the action of the rigid motions and this space is the natural nullitynnt(M). In this paper we will study minimal tori in S³ with natural nullity less than 8. We construct minimal immersions of the plane R² in S³ that contain all possible examples of tori with nnt(M)<8. We prove that the examples of Lawson and Hsiang with kn(M)=5 also have nnt(M)=5, and we prove that if the nnt(M)?6 then the group of isometries of M is not trivial.     

Nonvanishing cohomology and classes of Gorenstein rings

We give counterexamples to the following conjecture of Auslander: given a finitely generated module M over an Artin algebra Λ, there exists a positive integer n_M such that for all finitely generated Λ-modules N, if Ext_Λⁱ(M,N)=0 for all i?0, then Ext_Λⁱ(M,N)=0 for all i?n_M. Some of our examples moreover yield homologically defined classes of commutative local rings strictly between the class of local complete intersections and the class of local Gorenstein rings.     

Bloch constant of holomorphic mappings on the unit polydisk of ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper, we give a definition of Bloch mappings defined in the unit polydisk D ⁿ , which generalizes the concept of Bloch functions defined in the unit disk D. It is known that Bloch theorem fails unless we have some restrictive assumption on holomorphic mappings in several complex variables. We shall establish the corresponding distortion theorems for subfamilies β(K) and β _loc(K) of Bloch mappings defined in the polydisk D ⁿ , which extend the distortion theorems of Liu and Minda to higher dimensions. As an application, we obtain lower and upper bounds of Bloch constants for various subfamilies of Bloch mappings defined in D ⁿ . In particular, our results reduce to the classical results of Ahlfors and Landau when n = 1. This work was supported by the National Natural Science Foundation of China (Grant No. 10571164) and Specialized Research Fund for the Doctoral Program of Higher Education (SRFDP) (Grant No. 20050358052)     

A new class of multiset Wilf equivalent pairs

We say the pair of patterns (σ,τ) is multiset Wilf equivalent if, for any multiset M, the number of permutations of M that avoid σ is equal to the number of permutations of M that avoid τ. In this paper, we find a large new class of multiset Wilf equivalent pairs, namely, the pair (σ_n-2(n-1)n, σ_n-2n(n-1)), for n?3 and σ_n-2 a permutation of {1^x₁,2^x₂,…,(n-2)^x_n-2}. It is the most general multiset Wilf equivalence result to date.     

Geometric quantization on presymplectic manifolds

In this paper, we discuss the possibilities of adapting geometric quantization to presymplectic manifolds, i.e., differentiable manifoldsM ^2n+k (k>0) endowed with a closed 2-form ω of rank2n. We show that such an adaptation is possible in various manners, and that, as a general idea, it reduces the quantization onM to quantization on the symplectic quotientM/V, whereV is the foliation defined by the annihilator of ω.     

Über das Schwarzsche Lemma und Verwandte Sätze

Upper bounds for the Jacobian determinant by holomorphic mappings of bounded domainsD into itself were given first more then thirty years ago by Stefan Bergman by means of his theory of the kernel function ofD. In this paper a different method shall be developed and distortion theorems for holomorphic mappings of bounded domains of a Kähler manifoldM ⁿ into a Kähler manifoldM ₀ ⁿ shall be proved. The special casesM ⁿ =C _n (unit sphere of C ⁿ ) andM ⁿ =M ₀ ⁿ =|C _n shall also be considered. The proof depends essentially on the two Hermitian quadratic forms corresponding to the metric and to the Ricci tensor. The manifolds must be of negative Ricci curvature and fulfil two conditions given in section 4.     

On orthogonal systems of matrix algebras

In this work it is shown that certain interesting types of orthogonal system of subalgebras (whose existence cannot be ruled out by the trivial necessary conditions) cannot exist. In particular, it is proved that there is no orthogonal decomposition of M_n(C)⊗M_n(C)≡M_n²(C) into a number of maximal abelian subalgebras and factors isomorphic to M_n(C) in which the number of factors would be 1 or 3.In addition, some new tools are introduced, too: for example, a quantity c(A,B), which measures “how close” the subalgebras A,B⊂M_n(C) are to being orthogonal. It is shown that in the main cases of interest, c(A^′,B^′) - where A^′ and B^′ are the commutants of A and B, respectively - can be determined by c(A,B) and the dimensions of A and B. The corresponding formula is used to find some further obstructions regarding orthogonal systems.     

Freeness of orthogonal modules

A finitely generated module M over a commutative ring with unit R is said to be orthogonal stably free of type (n, m) if M is isomorphic to the solution space of a mxn matrix α such that αα^t=I_m. Geramita and Pullman have defined “generic” orthogonal stably free modules for each possible type and have obtained results on the freeness of these modules and on the supremum of the ranks of their free direct summands. We obtain further results of this type, concerning the generic modules of Geramita and Pullman as well as their sums with free modules and, in a few cases, their iterated sums. The last results are related to a theorem of T.Y. Lam stating that the iterated sum r · M of a stably free module M is free if r is greater than some lower bound. This lower bound is shown to be best possible in some cases.