On complexity of three-dimensional hyperbolic manifolds with geodesic boundary

The nonintersecting classes ? _p,q are defined, with p, q ?? ? and p ?? q ?? 1, of orientable hyperbolic 3-manifolds with geodesic boundary. If M ?? ? _p,q , then the complexity c(M) and the Euler characteristic ??(M) of M are related by the formula c(M) = p???(M). The classes ? _q,q , q ?? 1, and ?_2,1 are known to contain infinite series of manifolds for each of which the exact values of complexity were found. There is given an infinite series of manifolds from ?_3,1 and obtained exact values of complexity for these manifolds. The method of proof is based on calculating the ?-invariants of manifolds.     

Constructing quasi-cyclic codes from linear algebra theory

Let F _q be a finite field of cardinality q, l and m be positive integers and M _l (F _q ) the F _q -algebra of all l × l matrices over F _q . We investigate the relationship between monic factors of X ^m ? 1 in the polynomial ring M _l (F _q )[X] and quasi-cyclic (QC) codes of length lm and index l over F _q . Then we consider the idea of constructing QC codes from monic factors of X ^m ? 1 in polynomial rings over F _q -subalgebras of M _l (F _q ). This idea includes ideas of constructing QC codes of length lm and index l over F _q from cyclic codes of length m over a finite field F _q l, the finite chain ring F _q  + uF _q  + · · · + u ^{l ? 1} F _q (u ^l  = 0) and other type of finite chain rings.     

A characterization of the minimalbasis of the torus

D. König asks the interesting question in [7] whether there are facts corresponding to the theorem of Kuratowski which apply to closed orientable or non-orientable surfaces of any genus. Since then this problem has been solved only for the projective plane ([2], [3], [8]). In order to demonstrate that König’s question can be affirmed we shall first prove, that every minimal graph of the minimal basis of all graphs which cannot be embedded into the orientable surface f of genusp has orientable genusp+1 and non-orientable genusq with 1≦q≦2p+2. Then let f be the torus. We shall derive a characterization of all minimal graphs of the minimal basis with the nonorientable genusq=1 which are not embeddable into the torus. There will be two very important graphs signed withX ₈ andX ₇ later. Furthermore 19 graphsG ₁,G ₂, ...,G ₁₉ of the minimal basisM(torus, >₄) will be specified. We shall prove that five of them have non-orientable genusq=1, ten of them have non-orientable genusq=2 and four of them non-orientable genusq=3. Then we shall point out a method of determining graphs of the minimal basisM(torus, >₄) which are embeddable into the projective plane. Using the possibilities of embedding into the projective plane the results of [2] and [3] are necessary. This method will be called saturation method. Using the minimal basisM(projective plane, >₄) of [3] we shall at last develop a method of determining all graphs ofM(torus, >₄) which have non-orientable genusq≧2. Applying this method we shall succeed in characterizing all minimal graphs which are not embeddable into the torus. The importance of the saturation method will be shown by determining another graphG ₂₀≠G ₁,G ₂, ...,G ₁₉ ofM(torus, >₄).     

Multipliers of a wandering subspace for a shift invariant subspace

For a shift-invariant subspace M of the two variable Hardy space H², we consider the associated wandering subspace M₀=M?zM. Then there exists a nonconstant function ? in H^∞ such that ?M₀⊆M₀ if and only if M=qH² for some inner function q.     

The t-singular homology of orbifolds

For an orbifold M we define a new homology group, called t-singular homology group t-Hq(M) by using singular simplicies intersecting ‘transversely’ with ΣM. The rightness of this homology group is ensured by the facts that the 1-dimensional homology group t-H₁(M) is isomorphic to the abelianization of the orbifold fundamental group π₁(M,x₀). If M is a manifold, t-Hq(M) coincides with the usual singular homology group. We prove that it is a ‘b-homotopy’ invariant of orbifolds and develop many algebraic tools for the calculations. Consequently we calculate the t-singular homology groups of several orbifolds.     

Lévy functional and jump process martingales

An arbitrary jump process is considered without any assumption about the jump times and allowing the jump times to have accumulation points of arbitrary order. Certain basic martingales q(t, A) and the related Lévy system describing the jumps are introduced, and a notion of quadratic integration with respect to the predictable quadratic variation <q, q> of the basic martingales is defined. If M_t is a (locally) square integrable martingale with respect to the family of σ-fields generated by the jump process it is shown that M_t can be represented as a stochastic integral with respect to the basic martingales and with an integrand (locally) square integrable with respect to <q, q>.     

Good equidistant codes constructed from certain combinatorial designs

An (n,M,d;q) code is called equidistant code if the Hamming distance between any two codewords is d. It was proved that for any equidistant (n,M,d;q) code, d?nM(q-1)/(M-1)q(=d_opt, say). A necessary condition for the existence of an optimal equidistant code is that d_opt be an integer. If d_opt is not an integer, i.e. the equidistant code is not optimal, then the code with d=⌊d_opt⌋ is called good equidistant code, which is obviously the best possible one among equidistant codes with parameters n,M and q. In this paper, some constructions of good equidistant codes from balanced arrays and nested BIB designs are described.     

Embedding of a pseudo-point residual design into a Möbius plane

Let $\text{U}$ be a class of subsets of a finite set X. Elements of $\text{U}$ are called blocks. Let υ, t, λ and k be nonnegative integers such that υ?k?t?0. A pair (X, $\text{U}$) is called a (υ, k, λ) t-design, denoted by S_λ(t, k, υ), if (1) |X| = υ, (2) every t-subset of X is contained in exactly λ blocks and (3) for every block A in $\text{U}$, |A| = k. A Möbius plane M is an S₁(3, q+1, q²+1) where q is a positive integer. Let ∞ be a fixed point in M. If ∞ is deleted from M, together with all the blocks containing ∞, then we obtain a point-residual design M^*. It can be easily checked that M^* is an S_q(2, q+1, q²). Any S_q(2, q+1, q²) is called a pseudo-point-residual design of order q, abbreviated by PPRD(q). Let A and B be two blocks in a PPRD(q)M^*. A and B are said to be tangent to each other at z if and only if A∩B={z}. M^* is said to have the Tangency Property if for any block A in M^*, and points x and y such that x?A and y?A, there exists at most one block containing y and tangent to A at x. This paper proves that any PPRD(q)M^* is uniquely embeddable into a Möbius plane if and only if M^* satisfies the Tangency Property.     

Pseudo-Riemannian manifolds modelled on symmetric spaces

We construct irreducible pseudo-Riemannian manifolds (M, g) of arbitrary signature (p, q) with the same curvature tensor as a pseudo-Riemannian symmetric space which is a direct product of a two-dimensional Riemannian space form M ₂(c) and a pseudo-Euclidean space with the signature (p, q ? 2), or (p ? 2, q), respectively.     

Model-theoretic aspects of the Gurarij operator system

We establish some of the basic model theoretic facts about the Gurarij operator system GS recently constructed by the second-named author. In particular, we show: (1) GS is the unique separable 1-exact existentially closed operator system; (2) GS is the unique separable nuclear model of its theory; (3) every embedding of GS into its ultrapower is elementary; (4) GS is the prime model of its theory; and (5) GS does not have quantifier-elimination, whence the theory of operator systems does not have a model companion. We also show that, for any q ∈ ?, the theories of M_q-spaces and M_q-systems do have a model companion, namely the Fra¨?ssé limit of the class of finite-dimensional M_q-spaces and M_q-systems respectively; moreover, we show that the model companion is separably categorical. We conclude the paper by showing that no C* algebra can be existentially closed as an operator system.     

Sur les zéros des séries d'Eisenstein dans le demi-plan de Drinfeld

Eisenstein series for GL₂(F_q[T]) of weight q^k − 1 have zeroes in the Drinfeld upper half-plane. Let F be a fundamental domain for the GL₂(A)-action. We determine the number of zeroes in F of these series. Our method is essentially based on an assocíation between Eisenstein series and some functions defined on the edges of the Bruhat-Tits tree.     

An existence theorem for the complementarity problem

LetC be a pointed, solid, closed and convex cone in then-dimensional Euclidean spaceE ⁿ ,C* its polar cone,M:C→E ⁿ a map, andq a vector inE ⁿ . The complementarity problem (q|M) overC is that of finding a solution to the system $$(q|M) x \varepsilon C, M(x) + q \varepsilon C{^*} , \left\langle {x, M(x) + q} \right\rangle = 0.$$ It is shown that, ifM is continuous and positively homogeneous of some degree onC, and if (q|M) has a unique solution (namely,x=0) forq=0 and for someq=q ₀ ∈ intC*, then it has a solution for allq ∈E ⁿ .     

A Combinatorial Formula for the Schur Coefficients of the Integral Form of the Macdonald Polynomials in the Two Column and Certain Hook Cases

In this paper, we prove that the integral form of Macdonald polynomials J _?? [X; q, t] has the property that J _?? [X; q, t]/(1?q) ⁿ has Schur expansion with positive polynomial coefficient. Our proof proceeds by constructing constructing combinatorial formula for the Schur coefficients when??? is either a two column shape or a certain type of hook shape.     

Critical potentials of the eigenvalues and eigenvalue gaps of Schrödinger operators

Let M be a compact Riemannian manifold with or without boundary, and let −Δ be its Laplace-Beltrami operator. For any bounded scalar potential q, we denote by λi(q) the ith eigenvalue of the Schrödinger type operator −Δ+q acting on functions with Dirichlet or Neumann boundary conditions in case ∂M≠∅. We investigate critical potentials of the eigenvalues λi and the eigenvalue gaps Gij=λj−λi considered as functionals on the set of bounded potentials having a given mean value on M. We give necessary and sufficient conditions for a potential q to be critical or to be a local minimizer or a local maximizer of these functionals. For instance, we prove that a potential q∈L^∞(M) is critical for the functional λ₂ if and only if q is smooth, λ₂(q)=λ₃(q) and there exist second eigenfunctions f₁,…,fk of −Δ+q such that . In particular, λ₂ (as well as any λi) admits no critical potentials under Dirichlet boundary conditions. Moreover, the functional λ₂ never admits locally minimizing potentials.     

The Cut Loci, Conjugate Loci and Poles in a Complete Riemannian Manifold

Let M be a complete Riemannian manifold. We first prove that there exist at least two geodesics connecting p and every point in M if the tangent cut locus of ${p \in M}$ is not empty and does not meet its tangent conjugate locus. It follows from this that if M admits a pole and ${p \in M}$ is not a pole, then the tangent conjugate and tangent cut loci of p have a point in common. Here we say that a point q in M is a pole if the exponential map from the tangent space T _q M at q onto M is a diffeomorphism. Using this result, we estimate the size of the set of all poles in M having a pole whose sectional curvature is pinched by those of two von Mangoldt surfaces of revolution, meaning that their Gaussian curvatures are monotone and nonincreasing with respect to the distances to their vertices.     

Total palindrome complexity of finite words

The palindrome complexity function pal_w of a word w attaches to each n∈N the number of palindromes (factors equal to their mirror images) of length n contained in w. The number of all the nonempty palindromes in a finite word is called the total palindrome complexity of that word. We present exact bounds for the total palindrome complexity and construct words which have any palindrome complexity between these bounds, for binary alphabets as well as for alphabets with the cardinal greater than 2. Denoting by M_q(n) the average number of palindromes in all words of length n over an alphabet with q letters, we present an upper bound for M_q(n) and prove that the limit of M_q(n)/n is 0. A more elaborate estimation leads to .     

Multivariate model with a Kronecker product covariance structure: S. N. Roy method of estimation

In this paper we consider a (p × q)-matrix X = (X ₁, ..., X _q ), where a pq-vector vec (X) = (X ₁ ^T , ...,X _q ^T ) ^T is assumed to be distributed normally with mean vector vec (M) = (M ₁ ^T , ...,M _q ^T ) ^T and a positive definite covariance matrix Λ. Suppose that Λ follows a Kronecker product covariance structure, that is Λ = Φ?Σ, where Φ = (? _ij ) is a (q × q)-matrix and Σ = (σ _ij ) is a (p × p)-matrix and the matrices Φ, Σ are positive definite. Such a model is considered in [4], where the maximum likelihood estimates of the parameters M, Φ, Σ are obtained. Using S. N. Roy’s technique (see, e.g., [3]) of the multivariate statistical analysis, we obtain consistent and unbiased estimates of M, Φ, Σ as in [4], but with less calculations.     

Boundedness of the fractional maximal operator in generalized Morrey space on the Heisenberg group

In this paper we study the fractional maximal operator M _α , 0 ≤ α < Q on the Heisenberg group ? _n in the generalized Morrey spaces M _{p, ?}(? _n ), where Q = 2n + 2 is the homogeneous dimension of ? _n . We find the conditions on the pair (? ₁, ? ₂) which ensures the boundedness of the operator M _α from one generalized Morrey space M _{p, ?1}(? _n ) to another M _{q, ?2}(? _n ), 1 < p < q < ∞, 1/p?1/q = α/Q, and from the space M _{1, ?1}(? _n ) to the weak space WM _{q, ?2}(? _n ), 1 < q < ∞, 1 ? 1/q = α/Q. We also find conditions on the φ which ensure the Adams type boundedness of M _α from $M_{p,\phi ^{\tfrac{1} {p}} } \left( {\mathbb{H}_n } \right)$ to $M_{q,\phi ^{\tfrac{1} {q}} } \left( {\mathbb{H}_n } \right)$ for 1 < p < q < ∞ and from M _{1, ?}(? _n ) to $WM_{q,\phi ^{\tfrac{1} {q}} } \left( {\mathbb{H}_n } \right)$ for 1 < q < ∞. As applications we establish the boundedness of some Schrödinger type operators on generalized Morrey spaces related to certain nonnegative potentials V belonging to the reverse Hölder class B _∞(” _n ).