Prime links in some skew-polynomial and skew-laurent rings

Let R be a Noetherian commutative ring and a α₁,…,α_n commuting automorphisms of R. Define T = R[θ₁,…,θ_n;α₁,…,α_n] to be the skew-polynomial ring with θ_ir = α_i(r)θ_i and θ_iθ_j= θ_jθ_i, for all i,j ? (1,…,n) and r ? R, and let S = Rθ₁,θ₁:^-1,…,θ_n:,θ_n;^-1;α₁:,…,α_n] be the corresponding skew-Laurent ring. In this paper we show that S and T satisfy the strong second layer condition and characterize the links between prime ideals in these rings.     

Local asymptotic normality for progressively censored likelihood ratio statistics and applications

Let X_n,1 ≤ X_n,2 ≤ … ≤ X_n,n be the ordered variables corresponding to a random sample of size n with respect to a family of probability measures {P_θ:θ ∈ Θ} where Θ is an open subset of the real line. In many practical situations the X_n,i are the observables and experimentation must be curtailed prior to X_n,n. If τ_n is a stopping variable adapted to the σ-fields {σ(X_n,1,…,X_n,k): 1 ≤ k ≤ n} and P_n,θ the projection of P_θ onto σ(X_n,1,…,X_n,τn), the local asymptotic normality of the stopped progressively censored likelihood ratio statistics is established with θ, $\text{θ}{}_{\mathrm{n}}\text{= θ + un}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{∈ Θ}$ and θ, u held fixed, under certain conditions on the underlying distribution and on τ_n. Conditions are also given to ensure the local asymptotic normality of likelihood ratio statistics where the underlying observations are given in a series scheme.     

On a problem of P. Erdös

Let n be a positive integer and let A = {a₁,…, a_s}, B = {b₁,…, b_t} be two sets of positive integers such that the product set consists of st distinct numbers. Then, for a certain positive constant c, $\text{st ≤ c}\text{n}{}^2\text{log}\text{n}$, establishing a conjecture made by P. Erdös.     

Approximation of curves by polygons

Let A be a smooth curve in a Euclidean space E given by an arc length parametrization f: [0, 1] → E. Let π_n = {0 = t₀ ≤ t₁ ≤ … ≤ t_n = 1} be a partition of [0, 1] and let P_n be the polygon with vertices f(t₀), f(t₁),…, f(t_n). Let L(A) and L(P_n) denote the lengths of A and P_n, respectively. The paper investigates the behavior of n² |L(A) ? L(P_n)| when the partition π_n is induced by the sequence nθ(mod 1) for some irrational number θ. It turns out that this behavior depends on the partial quotients of the continued fraction expansion of θ.     

Third order efficiency of conditional tests for exponential families

Let P_η, η = (θ, γ) ∈ Θ × Γ ? $\text{R}$ × $\text{R}$^k, be a (k + 1)-dimensional exponential family. Let $\text{?}{}_{\mathrm{n}}{}^1$, n ∈ $\text{N}$, be an optimal similar test for the hypothesis {P_(θ,γ)ⁿ: γ ∈ Γ} (θ ∈ Θ fixed) against alternatives P_(θ1,γ1)ⁿ, θ₁ > θ, γ₁ ∈ Γ. It is shown that (?_n¹)_{n∈$\text{N}$} is third order efficient in the class of all test-sequences that are asymptotically similar of level α + o(n^?1) (locally uniformly in the nuisance parameter γ).     

An asymptotic formula for the maximum size of an h-family in products of partially ordered sets

An h-family of a partially ordered set P is a subset of P such that no h + 1 elements of the h-family lie on any single chain. Let S₁, S₂,… be a sequence of partially ordered sets which are not antichains and have cardinality less than a given finite value. Let P_n be the direct product of S₁,…, S_n. An asymptotic formula of the maximum size of an h-family in P_n is given, where $\text{h=o(}\text{n}\text{)}$ and n → ∞.     

On inverse multiplicative eigenvalue problems for matrices

Let A be an n×n complex-valued matrix, all of whose principal minors are distinct from zero. Then there exists a complex diagonal matrix D, such that the spectrum of AD is a given set σ = {λ₁,…,λ_n} in C. The number of different matrices D is at most n!.     

THE EXTREME POINTS OF SEVERAL CLASSES OF ANALYTIC FUNCTIONS

1Intr0ducti0nLetAden0tethesetofallfunctionsanalyticinA={z:Izl<1}.LetB={W:WEAandIW(z)l51}.Aisalocallyconvexlineaztop0l0gicalspacewithrespecttothetopologyofuniformconvergenceon`c0mpact8ubsetsofA-LetTh(c1,'tc.-1)={p(z):p(z)EA,Rop(z)>0,p(z)=1 clz czzz ' c.-lz"-l 4z" ',wherecl,',cn-1areforedcomplexconstants}.LetTh,.(b,,-..,b,-,)={p(z):P(z)'EAwithReP(z)>Oandp(z)=1 blz ' b.-lz"-l 4z" '-,wherebl,-'-jbu-1areffeedrealconstantsanddkarerealnumbersf0rk=n,n 1,'--}-LetTu(l1,'i'tI.-1)={…     

On the sequence of partial maxima of some random sequences

Let {X_n, n ? 1} be a sequence of identically distributed random variables, Z_n = max {X₁,…, X_n} and {u_n, n ? 1 } an increasing sequence of real numbers. Under certain additional requirements, necessary and sufficient conditions are given to have, with probability one, an infinite number of crossings of {Z_n} with respect to {u_n}, in two cases: (1) The X_n's are independent, (2) {X_n} is stationary Gaussian and satisfies a mixing condition.     

On tilings of the binary vector space

Let n be a positive integer, L a subset of {0, 1,…,n}. We discuss the existence of partitions (or tilings) of the n-dimensional binary vector space F_n into L-spheres. By a L-sphere around an x in F_n we mean {y ? F_n, d(x, y) ? L}, d(x, y) being the Hamming distance betwe en x and y. These tilings are generalizations of perfect error correcting codes. We show that very few such tilings exist (Theorem 2) and characterize them all for any L ? {0, 1,…,[$\text{1}\text{2}$n]}.     

Asymptotic normality of H-L estimators based on dependent data

LetX ₁,X ₂, ... be a strictly stationary φ-mixing sequence of r.v.'s with a common continuous cdfF. Let θ be a location parameter ofF. We prove the asymptotic normality of a class of Hodges-Lehmann estimators of θ under various regularity conditions on the mixing number φ and the underlyingF. We also establish the asymptotic linearity of signed rank statistics in the parameter θ. Our results also enable us to study the effect of φ-dependence on the asymptotic power of signed rank tests for testingH ₀: θ=0 againstH _n :θ=θ ₀ n ^?1/2,θ ₀≠0. Finally these results are shown to remain valid for strongly mixing processes {X _i } also.     

A Theorem related to Marcinkiewicz-Salem Conjecture

Let S_n, n = 1,2… be the sequence of partial sums of independent Bernoulli random variables. We show, for the randomly sampled trigonometric system {e_S?,? in N}, the validity of the Marcinkiewicz-Salem Conjecture.     

Simultaneous congruence of convex compact sets of Hermitian matrices with constant rank

Let S be a compact convex set of n × n hermitian matrices (n ⩾ 2). Suppose every member of S is nonsingular and has exactly one negative eigenvalue. Let (ε₁,…,ε_n) be any ordered n-tuple from the set {- 1, 1}. One of our main results is that a nonsingular matrix X exists such that, for every A in S and every 1 ⩽ j ⩽ n, the (j, j) entry of X¹AX has sign ε_j. A similar result, with only negative ε_j allowed, is proved also for a compact convex set S of n × n hermitian matrices such that every member of S has the same rank and exactly one negative eigenvalue.     

A relationship between stochastic matrices and eigenvalues of products

Let {B₁,…,B_n} be a set of n rank one n×n row stochastic matrices. The next two statements are equivalent: (1) If A is an n×n nonnegative matrix, then 1 is an eigenvalue ofB_kA for each k=1,…,n if and only if A is row stochastic. (2) The n×n row stochastic matrix S whose kth row is a row of B_k for k=1,…,n is nonsingular. For any set {B₁, B₂,…, B_p} of fewer than n row stochastic matrices of order n×n and of any rank, there exists a nonnegative n×n matrix A which is not row stochastic such that 1 is eigenvalue of every B_kA, k=1,…,p.     

Solution of the Szili type conjugate problem on the roots of the laguerre polynomials

Let ?1<α≤0 and let $$L_n^{(\alpha )} (x) = \frac{1}{{n!}}x^{ - \alpha } e^x \frac{{d^n }}{{dx^n }}(x^{\alpha + n} e^{ - x} )$$ be the generalizednth Laguerre polynomial,n=1,2,… Letx ₁,x ₂,…,x _n andx*₁,x*₂,…,x* _n?1 denote the roots ofL _n ^(α) (x) andL _n ^(α)′ (x) respectively and putx*₀=0. In this paper we prove the following theorem: Ify ₀,y ₁,…,y _n ?1 andy ₁ ^′ ,…,y _n ^′ are two systems of arbitrary real numbers, then there exists a unique polynomialP(x) of degree 2n?1 satisfying the conditions $$\begin{gathered} P\left( {x_k^* } \right) = y_k (k = 0,...,n - 1) \hfill \\ P'\left( {x_k } \right) = y_k^\prime (k = 1,...,n). \hfill \\ \end{gathered} $$ .     

Siegel’s Lemma and Sum-Distinct Sets

Let L(x)=a ₁ x ₁+a ₂ x ₂+???+a _n x _n , n≥2, be a linear form with integer coefficients a ₁,a ₂,…,a _n which are not all zero. A basic problem is to determine nonzero integer vectors x such that L(x)=0, and the maximum norm ‖x‖ is relatively small compared with the size of the coefficients a ₁,a ₂,…,a _n . The main result of this paper asserts that there exist linearly independent vectors x ₁,…,x _n?1∈? ⁿ such that L(x _i )=0, i=1,…,n?1, and $$\|{\mathbf{x}}_{1}\|\cdots\|{\mathbf{x}}_{n-1}\|<\frac{\|{\mathbf{a}}\|}{\sigma_{n}},$$ where a=(a ₁,a ₂,…,a _n ) and $$\sigma_{n}=\frac{2}{\pi}\int_{0}^{\infty}\left(\frac{\sin t}{t}\right)^{n}\,dt.$$ This result also implies a new lower bound on the greatest element of a sum-distinct set of positive integers (Erdös–Moser problem). The main tools are the Minkowski theorem on successive minima and the Busemann theorem from convex geometry.     

An almost sure central limit theorem for self-normalized weighted sums

Let X, X1 , X2 , ··· be a sequence of nondegenerate i.i.d. random variables with zero means, which is in the domain of attraction of the normal law. Let {a ni , 1≤i≤n, n≥1} be an array of real numbers with some suitable conditions. In this paper, we show that a central limit theorem for self-normalized weighted sums holds. We also deduce a version of ASCLT for self-normalized weighted sums.     

Monotonicity of permanents of certain doubly stochastic circulant matrices

Let p_k(A), k=2,…,n, denote the sum of the permanents of all k×k submatrices of the n×n matrix A. A conjecture of Ðokovi?, which is stronger than the famed van der Waerden permanent conjecture, asserts that the functions p_k((1?θ)J_n+;θA), k=2,…, n, are strictly increasing in the interval 0?θ?1 for every doubly stochastic matrix A. Here J_n is the n×n matrix all whose entries are equal $\text{1}\text{n}$. In the present paper it is proved that the conjecture holds true for the circulant matrices A=αI_n+ βP_n, α, β?0, α+;β=1, and $\text{A=}\text{(nJ}{}_{\mathrm{n}}\text{?I}{}_{\mathrm{n}}\text{?P}{}_{\mathrm{n}}\text{)}\text{(n?2)}$, where I_n and P_n are respectively the n×n identify matrix and the n×n permutation matrix with 1's in positions (1,2), (2,3),…, (n?1, n), (n, 1).     

Positive values of non-homogeneous indefinite quadratic forms of type (1,4)

Let Γ _r,n-r denote the infimum of all numbers Γ>0 such that for any real indefinite quadraticQ inn variables of type (r, n?r), determinantD≠0 and real numbersc ₁,…,c _n there exist (x ₁,…,x _n )≡(c ₁,…,c _n ) (mod 1) satisfying $$0< Q(x_1 ,...,x_n ) \leqslant (\Gamma \left| D \right|)^{1/n} .$$ . All the values of Γ _r,3 are known except Γ_1,4. It is shown that $$8 \leqslant \Gamma _{1,4} \leqslant 16.$$ .     

Multiplikatorsysteme der symplektischen Thetagruppe

Let Γ_θ be the subgroup of Siegel modular groupSp(n, ?) consisting of all matrices \(M = \left( {\begin{array}{*{20}c} {A B} \\ {C D} \\ \end{array} } \right)\) , such that the diagonal elements ofA ^t C andB ^t D are even. A multiplier system of weightr(∈?) is a system of complex numbers ν (M)≠0,M∈Γ_θ, such thatJ (M, Z)=ν(M) det(CZ+D) ^r is an automorphy factor (that isJ (M N, Z)=J (M, N Z) J (N, Z) forM, N ∈Sp(n,?) and $$Z \in S_n = \left\{ {Z = X + i Y \in M^{(n,n)} (\mathbb{C}); X = X^t , Y = Y^t > 0} \right\})$$ . We show that in casen≥2 such a multiplier system exists if and only if 2r∈?. A corollary of this fact is the following. From the cohomology theory of Siegel modular group we derive that in casen≥8 any Γ_θ-invariant divisor is the exact zero divisor of a modular form for Γ_θ. Therefore the zero divisor of classical theta function \(\theta (Z) = \sum\limits_{g \in \mathbb{Z}^n } {e^{\pi iZ[g]} } \) , a modular form of weight 1/2 is irreducible. In the second part of this paper we calculate the commutator factor group of Γ _{n, θ} forn≥2.