The strong Markov property for canonical Wiener processes

A noncommutative analog of the concept of Markov time is formulated, in association with a canonical Wiener process (P, Q) [4]. For such a Markov time T, the quantities P_T(t) = P(t + T) ? P(T), Q_T(t) = Q(t + T) ? Q(T) can be defined using spectral integrals with operator-valued integrands, and constitute a new canonical Wiener process (P_T, Q_T), which is independent of the analog of the σ-field of events generated by the process up to the Markov time.     

Weighted rational approximation in the complex plane

Given a triple (G, W, γ) of an open bounded set G in the complex plane, a weight function W(z) which is analytic and different from zero in G, and a number γ with 0 ≤ γ ≤ 1, we consider the problem of locally uniform rational approximation of any function ƒ(z), which is analytic in G, by weighted rational functions W^m_i+n_i(z)R_mi, n_i(z)_{i = 0}^∞, where R_mi, n_i(z) = P_mi(z)/Q_ni(z) with deg P_mi ≤ m_i and deg Q_ni ≤ n_i for all i ≥ 0 and where m_i + n_i → ∞ as i → ∞ such that lim m_i/[m_i + n_i] = γ. Our main result is a necessary and sufficient condition for such an approximation to be valid. Applications of the result to various classical weights are also included.     

The minimum increment of f-divergences given total variation distances

Let (P _i , Q _i ), i = 0, 1, be two pairs of probability measures defined on measurable spaces (Ω _i ,F _i ) respectively. Assume that the pair (P₁, Q₁) is more informative than (P₀,Q₀) for testing problems. This amounts to say that If (P₁,Q₁) ≥ If (P₀,Q₀), where If (·, ·) is an arbitrary f-divergence. We find a precise lower bound for the increment of f-divergences I_f(P₁,Q₁) ? I_f(P₀,Q₀) provided that the total variation distances ||Q₁ ? P₁|| and ||Q₀ ? P₀|| are given. This optimization problem can be reduced to the case where P₁ and Q₁ are defined on the space consisting of four points, and P₀ and Q₀ are obtained from P₁ and Q₁ respectively by merging two of these four points. The result includes the well-known lower and upper bounds for I_f(P,Q) given ||Q ? P||.     

Sufficient conditions for the existence of a left quotient ring of a ring decomposed into a direct sum of left ideals

Let R=P₁⊕P₂⊕…⊕P_n be a decomposition of a ring into a direct sum of indecomposable left ideals. Assume that these ideals possess the following properties: (1) any nonzero homomorphisms ϕ: P_i→P_j is a monomorphism; (2) if subideals Q₁, Q₂ of the ideal P_j are isomorphic to the ideal P_i, then there exists a subideal Q₃⊆Q₁⊎Q₂, which is also isomorphic to P_i. It is proved that, under these asumptions, a left quotient ring of the ring R exists. This left quotient ring inherits properties(1), (2) and satisfies condition (3): any nonzero homomorphism ϕ: P_i→P_i is an automorphism of the ideal P_i. Bibliography: 2 titles. Translated fromZapiski Nauchnyk Seminarov POMI, Vol. 227, 1995, pp. 9–14.     

A polynomial dual of partitions

Let P(x) = Σ_i=0ⁿa_ixⁱ be a nonnegative integral polynomial. The polynomial P(x) is m-graphical, and a multi-graph G a realization of P(x), provided there exists a multi-graph G containing exactly P(1) points where a_i of these points have degree i for 0≤i≤n. For multigraphs G, H having polynomials P(x), Q(x) and number-theoretic partitions (degree sequences) π, ?, the usual product P(x)Q(x) is shown to be the polynomial of the Cartesian product G × H, thus inducing a natural product π? which extends that of juxtaposing integral multiple copies of ?. Skeletal results are given on synthesizing a multi-graph G via a natural Cartesian product G₁ × … × G_k having the same polynomial (partition) as G. Other results include an elementary sufficient condition for arbitrary nonnegative integral polynomials to be graphical.     

On double bound graphs and forbidden subposets

For a poset P=(X,≤_P), the double bound graph (DB-graph) of P is the graph DB(P)=(X,E_DB(P)), where xy∈E_DB(P) if and only if x≠y and there exist n,m∈X such that n≤_Px,y≤_Pm. We obtain that for a subposet Q of a poset P,Q is an (n, m)-subposet of P if and only if DB(Q) is an induced subgraph DB(P). Using this result, we show some characterizations of split double bound graphs, threshold double bound graphs and difference double bound graphs in terms of (n, m)-subposets and double canonical posets.     

On squares in Lucas sequences

Let P and Q be non-zero integers. The Lucas sequence {Un(P,Q)} is defined by U₀=0, U₁=1, Un=PUn−1−QUn−2 (n?2). The question of when Un(P,Q) can be a perfect square has generated interest in the literature. We show that for n=2,…,7, Un is a square for infinitely many pairs (P,Q) with gcd(P,Q)=1; further, for n=8,…,12, the only non-degenerate sequences where gcd(P,Q)=1 and Un(P,Q)=□, are given by U₈(1,−4)=²¹², U₈(4,−17)=⁶²⁰², and U₁₂(1,−1)=¹²².     

The complexity of the fixed point property

It is NP-complete to determine whether a given ordered set has a fixed point free order-preserving self-map. On the way to this result, we establish the NP-completeness of a related problem: Given ordered sets P and Q with t-tuples (p ₁, ... , p _t) and (q ₁, ... , q _t) from P and Q respectively, is there an order-preserving map f: P→Q satisfying f(p _i)≥q _i for each i=1, ... , t?     

Valeurs aux entiers négatifs des séries de Dirichlet associées a un polynôme,I

In this paper, we are studying Dirichlet series Z(P,ξ,s) = Σ_n?$\text{N}$^1rP(n)^?s ξⁿ, where P ∈ $\text{R}$₊ [X₁,…,X_r] and ξⁿ = ξ₁ⁿ¹ … ξ_r^nr, with ξ_i ∈ $\text{C}$, such that |ξ_i| = 1 and ξ_i ≠ 1, 1 ≦ i ≦ r. We show that Z(P, ξ,·) can be continued holomorphically to the whole complex plane, and that the values Z(P, ξ, ?k) for all non negative integers, belong to the field generated over $\text{Q}$ by the ξ_i and the coefficients of P. If, there exists a number field K, containing the ξ_i, 1 ≦ i ≦ r, and the coefficients of P, then we study the denominators of Z(P, ξ, ?k) and we define a $\text{B}$-adic function Z$\text{B}$(P, ξ,·) which is equal, on class of negative integers, to Z(P, ξ, ?k).     

Quantum mechanical Wiener processes

Quantum mechanical analogues of Wiener processes are defined and their existence proved, in terms of which the field operators of extremal universally invariant representations of the canonical commutation relations are expressible as stochastic integrals. A noncommutative analogue of the Wiener transformations is constructed and shown to have properties analogous to the classical Wiener transformation.     

A note on bideterminants for Schur superalgebras

Let S(m|n,r)_Z be a Z-form of a Schur superalgebra S(m|n,r) generated by elements ξ_i,j. We solve a problem of Muir and describe a Z-form of a simple S(m|n,r)-module D_λ,Q over the field Q of rational numbers, under the action of S(m|n,r)_Z. This Z-form is the Z-span of modified bideterminants [T_?:T_i] defined in this work. We also prove that each [T_?:T_i] is a Z-linear combination of modified bideterminants corresponding to (m|n)-semistandard tableaux T_i.     

Classification of certain pairs of operators (P, Q) satisfying [P, Q] = −i Id

In a separable Hilbert space a certain class of pairs of operators (P, Q) satisfying the Born-Heisenberg commutation relation [P, Q] = ?i Id on a dense domain Ω is investigated. This class is essentially defined by requiring Q bounded and self-adjoint, P symmetric and $\text{PΩ ? Ω, QΩ ? Ω}$. We show that Q is absolutely continuous and that P can be thought of as a first order differential operator. The class considered contains the pair “angle ?” and “angular momentum L_z.” It is expected that the methods of this paper can be applied to more general classes of operators (P, Q) including the Schrödinger case.     

Spanning multi-paths in hypercubes

A set A of vertices of a hypercube is called balanced if . We prove that for every natural number n there exists a natural number π₁(n) such that for every hypercube Q with dim(Q)?π₁(n) there exists a family of pairwise vertex-disjoint paths P_i between A_i and B_i for i=1,2,…,n with if and only if {A_i,B_i∣i=1,2,…,n} is a balanced set.     

Local duality for equitable partitions of a Hamming space

The Hamming space Qn is the set of binary words of length n. A partition (C₁,C₂,…,Cr) of Qn with quotient matrix B=[bij]_r×r is equitable if for all i and j, any word in the cell Ci has exactly bij neighbors in the cell Cj. In this paper, we provide an explicit formula relating the local spectrum of cells in the face to that in the orthogonal face.     

Isotoping Heegaard surfaces in neat positions with respect to critical distance Heegaard surfaces

Suppose a closed orientable 3-manifold M has a genus g Heegaard surface P with distance d(P)=2g. Let Q be another genus g Heegaard surface which is strongly irreducible. Then we show that there is a height function f:M→I induced from P such that by isotopy, Q is deformed into a position satisfying the following; (1) fQ_| has 2g+2 critical points p₀,p₁,…,p2g+1 with f(p₀)<f(p₁)<?<f(p2g+1) where p₀ is a minimum and p2g+1 is a maximum, and p₁,…,p2g are saddles, (2) if we take regular values ri (i=1,…,2g+1) such that f(pi−1)<ri<f(pi), then f−1(ri)∩Q consists of a circle if i is odd, and f−1(ri)∩Q consists of two circles if i is even.     

On the unlinking conjecture of independent polynomial functions

The celebrated U-conjecture states that under the N_n(0,I_n) distribution of the random vector X=(X₁,…,X_n) in Rⁿ, two polynomials P(X) and Q(X) are unlinkable if they are independent [see Kagan et al., Characterization Problems in Mathematical Statistics, Wiley, New York, 1973]. Some results have been established in this direction, although the original conjecture is yet to be proved in generality. Here, we demonstrate that the conjecture is true in an important special case of the above, where P and Q are convex nonnegative polynomials with P(0)=0.     

Optimal labelling of a product of two paths

If G is a graph with p vertices and at least one edge, we set φ (G) = m n max |f(u) ? f(v)|, where the maximum is taken over all edges uv and the minimum over all one-to-one mappings f : V(G) → {1, 2, …, p}: V(G) denotes the set of vertices of G.P_n will denote a path of length n whose vertices are integers 1, 2, …, n with i adjacent to j if and only if |i ? j| = 1. P_m × P_n will denote a graph whose vertices are elements of {1, 2, …, m} × {1, 2, …, n} and in which (i, j), (r, s) are adjacent whenever either i = r and |j ? s| = 1 or j = s and |i ? r| = 1.Theorem.If max(m, n) ? 2, thenφ(P_m × P_n) = min(m, n).     

Small divisors of Bernoulli sums

Let ?= {?_i,i ≥1} be a sequence of independent Bernoulli random variables (P{?_i = 0} = P{?_i = 1 } = 1/2) with basic probability space (Ω, A, P). Consider the sequence of partial sums B_n=?₁+...+?_n, n=1,2..... We obtain an asymptotic estimate for the probability P{P^-(B_n) > >} for >≤n^{e/log log n}, c a positive constant.     

Unitary similarity of algebras generated by pairs of orthoprojectors

It is shown that the unitary similarity of two matrix algebras generated by pairs of orthoprojectors { P₁, Q₁} and {P₂,Q₂} can be verified by comparing the traces of P₁, Q₁, and (P₁Q₁)ⁱ, i = 1, 2, …, n, with those of P₂, Q₂, and (P₂Q₂)ⁱ. The conditions of the unitary similarity of two matrices with quadratic minimal polynomials presented in [A. George and Kh. D. Ikramov, Unitary similarity of matrices with quadratic minimal polynomials, Linear Algebra Appl., 349, 11–16 (2002)] are refined. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 5–14.     

Some explicit formulae in simultaneous padé approximation

Using old results on the explicit calculation of determinants, formulae are given for the coefficients of P₀(z) and P₀(z)f_i(z) ? P_i(z), where P_i(z) are polynomials of degree σ ? ρ_i (i=0,1,…,n), P₀(z)f_i(z) ? P_i(z) are power series in which the terms with z^k, 0?k?σ, vanish (i=1,2,…,n), (ρ₀,ρ₁,…,ρ_n) is an (n+1)-tuple of nonnegative integers, σ=ρ₀+ρ₁+?+ρ_n, and {f_i}ⁿ_i=1 is the set of hypergeometric functions {₁F₁(1;c_i;z)}ⁿ_i=1$\text{(c}{}_{\mathrm{i}}\text{?}\text{Z}\text{z.drule;}\text{N}\text{, c}{}_{\mathrm{i}}\text{? c}{}_{\mathrm{j}}\text{?}\text{Z}\text{)}$ or {₂F₀(a_i,1;z)}ⁿ_i=1$\text{(a}{}_{\mathrm{i}}\text{?}\text{Z}\text{?}\text{N}\text{, a}{}_{\mathrm{i}}\text{? a}{}_{\mathrm{j}}\text{?}\text{Z}\text{)}$ under the condition ρ₀?ρ_i ? 1 (i=1,2,…,n).