Idempotence preserving maps between matrix spaces

Suppose 𝔽 is an arbitrary field of characteristic not 2 and 𝔽?≠?𝔽₃. Let M _n (𝔽) be the space of all n?×?n full matrices over 𝔽 and P _n (𝔽) the subset of M _n (𝔽) consisting of all n?×?n idempotent matrices and GL _n (𝔽) the subset of M _n (𝔽) consisting of all n?×?n invertible matrices. Let Φ_𝔽(n,?m) denote the set of all maps from M _n (𝔽) to M _m (𝔽) satisfying A???λB?∈?P _n (𝔽)???φ(A)???λφ(B)?∈?P _m (𝔽) for every A,?B?∈?M _n (𝔽) and λ?∈?𝔽, where m and n are integers with 3?≤?n?≤?m. It is shown that if φ?∈?Φ_𝔽(n,?m), then there exists T?∈?GL _m (𝔽) such that φ(A)?=?T?[A???I _p ?⊕?A ^t ???I _q ?⊕?0]T?^?1 for every A?∈?M _n (𝔽), where I ₀?=?0. This improves the results of some related references.     

Milnor K-Groups and Finite Field Extensions

Let E/F be a finite separable field extension and let m denote the integral part of log₂ [E : F]. David Leep recently showed that if char(F) 2, then for n m the nth power of the fundamental ideal in the Witt ring of E satisfies the equality I ⁿ E = I ^n–m F · I ^m E. The aim of this note is to prove the analogous equality for the Milnor K-groups, that is K _n E = K _n–m F · K _m E for n m. In either of these equalities one may not replace m by m – 1, as examples of certain m-quadratic extensions indicate.     

On the approximate behavior of the posterior distribution for an extreme multivariate observation

The behavior of the posterior for a large observation is considered. Two basic situations are discussed; location vectors and natural parameters.Let X = (X₁, X₂, …, X_n) be an observation from a multivariate exponential distribution with that natural parameter Θ = (Θ₁, Θ₂, …, Θ_n). Let θ_x^* be the posterior mode. Sufficient conditions are presented for the distribution of Θ − θ_x^* given X = x to converge to a multivariate normal with mean vector 0 as |x| tends to infinity. These same conditions imply that E(Θ | X = x) − θ_x^* converges to the zero vector as |x| tends to infinity.The posterior for an observation X = (X₁, X₂, …, X_n is considered for a location vector Θ = (Θ₁, Θ₂, …, Θ_n) as x gets large along a path, γ, in Rⁿ. Sufficient conditions are given for the distribution of γ(t) − Θ given X = γ(t) to converge in law as t → ∞. Slightly stronger conditions ensure that γ(t) − E(Θ | X = γ(t)) converges to the mean of the limiting distribution.These basic results about the posterior mean are extended to cover other estimators. Loss functions which are convex functions of absolute error are considered. Let δ be a Bayes estimator for a loss function of this type. Generally, if the distribution of Θ − E(Θ | X = γ(t)) given X = γ(t) converges in law to a symmetric distribution as t → ∞, it is shown that δ(γ(t)) − E(Θ | X = γ(t)) → 0 as t → ∞.     

Third-order asymptomic properties of a class of test statistics under a local alternative

Suppose that {X_i; I = 1, 2, …,} is a sequence of p-dimensional random vectors forming a stochastic process. Let p_n, θ(X_n), X_n ^np, be the probability density function of X_n = (X₁, …, X_n) depending on θ Θ, where Θ is an open set of ¹. We consider to test a simple hypothesis H : θ = θ₀ against the alternative A : θ ≠ θ₀. For this testing problem we introduce a class of tests , which contains the likelihood ratio, Wald, modified Wald, and Rao tests as special cases. Then we derive the third-order asymptotic expansion of the distribution of T under a sequence of local alternatives. Using this result we elucidate various third-order asymptotic properties of T (e.g., Bartlett's adjustments, third-order asymptotically most powerful properties). Our results are very general, and can be applied to the i.i.d. case, multivariate analysis, and time series analysis. Two concrete examples will be given. One is a Gaussian ARMA process (dependent case), and the other is a nonlinear regression model (non-identically distributed case).     

A central limit theorem for generalized multilinear forms

Let X₁, …, X_n be independent random variables and define for each finite subset I {1, …, n} the σ-algebra = σ{X_i : i ε I}. In this paper -measurable random variables W_I are considered, subject to the centering condition E(W_I ) = 0 a.s. unless I J. A central limit theorem is proven for d-homogeneous sums W(n) = Σ_{I = d}W_I, with var W(n) = 1, where the summation extends over all (_n^d) subsets I {1, …, n} of size I = d, under the condition that the normed fourth moment of W(n) tends to 3. Under some extra conditions the condition is also necessary.     

Reverse Martingales and Approximation Operators

Let {ξ_n, _n, n ≥ m ≥ 1} be a reverse martingale such that the distribution of ξ_n depends on x I R =(− ∞, ∞)x. for each n ≥ m, and ξ_n[formula] For a continuous bounded function f on R let L_n(f, x) = Ef(ξ_n) be the associated positive linear operator. The properties of ξ_n are used to obtain the convergence properties of L_n(f, x), and some more details are given when ξ_n is a reverse martingale sequence of -statistics. Lipschitz properties for a subclass of these operators resulting from an exponential Family of distributions are also given. It is further shown that this class of operators of convex functions preserves convexity also. An example of a reverse supermartingale related to the Bleimann-Butzer-Hahn operator is also discussed.     

On exactlym times covers

In this paper it is shown that if every integer is covered bya ₁+n ₁ℤ,…,a _k +n _k ℤ exactlym times then for eachn=1,…,m there exist at least ( _n ^m ) subsetsI of {1,…k} such that ∑ _{i ∈ I} 1/n _i equalsn. The bound ( _n ^m ) is best possible. Research supported by the National Nature Science Foundation of P.R. of China.     

The Lebesgue Constant for Higher Order Hermite-Fejér Interpolation on the Chebyshev Nodes

For a fixed integer m ≥ 0, and for n = 1, 2, 3, ..., let λ_{2m, n}(x) denote the Lebesgue function associated with (0, 1,..., 2m) Hermite-Fejér polynomial interpolation at the Chebyshev nodes {cos[(2k−1) π/(2n)]: k=1, 2, ..., n}. We examine the Lebesgue constant Λ_{2m, n} max{λ_{2m, n}(x): −1 ≤ x ≤ 1}, and show that Λ_{2m, n} = λ_{m, n}(1), thereby generalising a result of H. Ehlich and K. Zeller for Lagrange interpolation on the Chebyshev nodes. As well, the infinite term in the asymptotic expansion of Λ_{2m, n)} as n → ∞ is obtained, and this result is extended to give a complete asymptotic expansion for Λ_{2, n}.     

On sets of integers with prescribed gaps

For a fixed setI of positive integers we consider the set of paths (p _o,...,p _k ) of arbitrary length satisfyingp _l –p _l–1I forl=2,...,k andp ₀=1,p _k =n. Equipping it with the uniform distribution, the random path lengthT _n is studied. Asymptotic expansions of the moments ofT _n are derived and its asymptotic normality is proved. The step lengthsp _l –p _l–1 are seen to follow asymptotically a restricted geometrical distribution. Analogous results are given for the free boundary case in which the values ofp ₀ andp _k are not specified. In the special caseI={m+1,m+2,...} (for some fixed m) we derive the exact distribution of a random m-gap subset of {1,...,n} and exhibit some connections to the theory of representations of natural numbers. A simple mechanism for generating a randomm-gap subset is also presented.     

On the decomposable numerical range of λI-N

Let N be an nxn normal matrix. For 1≤m≤n we characterize the convexity of the mth decomposable numerical range of λI_n -N which is defined to be

{det(X?(λI_n ?N)X) [sdot] X?C _n×m X ^? X=I_m }.

A related problem on mixed decomposable numerical range of λI_n -N is also discussed.     

Strong limit theorems for quasi-orthogonal random fields

This is a systematic and unified treatment of a variety of seemingly different strong limit problems. The main emphasis is laid on the study of the a.s. behavior of the rectangular means ζ_mn = 1/(λ₁(m) λ₂(n)) Σ_i=1^m Σ_k=1ⁿ X_ik as either max{m, n} → ∞ or min{m, n} → ∞. Here {X_ik: i, k ≥ 1} is an orthogonal or merely quasi-orthogonal random field, whereas {λ₁(m): m ≥ 1} and {λ₂(n): n ≥ 1} are nondecreasing sequences of positive numbers subject to certain growth conditions. The method applied provides the rate of convergence, as well. The sufficient conditions obtained are shown to be the best possible in general. Results on double subsequences and 1-parameter limit theorems are also included.     

Ramsey functions involving with large

For fixed integers m,k2, it is shown that the k-color Ramsey number r_k(K_m,n) and the bipartite Ramsey number b_k(m,n) are both asymptotically equal to k^mn as n→∞, and that for any graph H on m vertices, the two-color Ramsey number is at most (1+o(1))n^m+1/(logn)^m-1. Moreover, the order of magnitude of is proved to be n^m+1/(logn)^m if H≠K_m as n→∞.     

A-acceptability of derivatives of rational approximations to EXP(Z)

The question of A-acceptability in regard to derivatives of R_m/n, the [m/n] Padé approximation to the exponential, is examined for a range of values of m and n. It is proven that R′_{n − 1/n}, R′_n/n, R′_{n + 1/n}and R″_n/n are A-acceptable and that numerous other choices of m and n lead to non-A-acceptability. The results seem to indicate that the A-acceptability pattern of R_m/n^(k) displays an intriguing generalization of the Wanner-Hairer-Nørsett theorem on the A-acceptability of R_m/n.     

L1-optimal estimates for a regression type function in R

Let X₁, X₂, …, X_n be random vectors that take values in a compact set in R^d, d ≥ 1. Let Y₁, Y₂, …, Y_n be random variables (“the responses”) which conditionally on X₁ = x₁, …, X_n = x_n are independent with densities f(y | x_i, θ(x_i)), i = 1, …, n. Assuming that θ lives in a sup-norm compact space Θ_q,d of real valued functions, an optimal L₁-consistent estimator of θ is constructed via empirical measures. The rate of convergence of the estimator to the true parameter θ depends on Kolmogorov's entropy of Θ_q,d.     

The Ramsey numbers for cycles versus wheels of odd order

For two given graphs G₁ and G₂, the Ramsey number R(G₁,G₂) is the smallest integer n such that for any graph G of order n, either G contains G₁ or the complement of G contains G₂. Let C_n denote a cycle of order n and W_m a wheel of order m+1. It is conjectured by Surahmat, E.T. Baskoro and I. Tomescu that R(C_n,W_m)=2n−1 for even m≥4, n≥m and (n,m)≠(4,4). In this paper, we confirm the conjecture for n≥3m/2+1.     

The Ramsey numbers for cycles versus wheels of even order

For two given graphs G₁ and G₂, the Ramsey number R(G₁,G₂) is the smallest integer n such that for any graph G of order n, either G contains G₁ or the complement of G contains G₂. Let C_n denote a cycle of order n and W_m a wheel of order m+1. Surahmat, Baskoro and Tomescu conjectured that R(C_n,W_m)=3n−2 for m odd, n≥m≥3 and (n,m)≠(3,3). In this paper, we confirm the conjecture for n≥20.     

On the sequence of powers of a stochastic matrix with large exponent

We consider the class of primitive stochastic n×n matrices A, whose exponent is at least (n²−2n+2)/2+2. It is known that for such an A, the associated directed graph has cycles of just two different lengths, say k and j with k>j, and that there is an α between 0 and 1 such that the characteristic polynomial of A is λⁿ−αλ^n−j−(1−α)λ^n−k. In this paper, we prove that for any mn, if α1/2, then A^m+k−A^m_∞A^m−1w^T_∞, where 1 is the all-ones vector and w^T is the left-Perron vector for A, normalized so that w^T1=1. We also prove that if jn/2, n31 and , then A^m+j−A^m_∞A^m−1w^T_∞ for all sufficiently large m. Both of these results lead to lower bounds on the rate of convergence of the sequence A^m.     

Limit distributions and one-parameter groups of linear operators on Banach spaces

Let = {U_t: t > 0} be a strongly continuous one-parameter group of operators on a Banach space X and Q be any subset of a set (X) of all probability measures on X. By (Q; ) we denote the class of all limit measures of {U_tn(μ₁ * μ₂*…*μ_n)*δ_xn}, where {μ_n}Q, {x_n}X and measures U_tnμ_j (j=1, 2,…, n; N=1, 2,…) form an infinitesimal triangular array. We define classes L_m( ) as follows: L₀( )= ( (X); ), L_m( )= (L_m−1( ); ) for m=1, 2,… and L_∞( )=_m=0^∞L_m( ). These classes are analogous to those defined earlier by Urbanik on the real line. Probability distributions from L_m( ), m=0, 1, 2,…, ∞, are described in terms of their characteristic functionals and their generalized Poisson exponents and Gaussian covariance operators.     

Lower bounds for the cardinality of minimal blocking sets in projective spaces

In this paper new lower bounds for the cardinality of minimal m-blocking sets are determined. Let r₂(q) be the number such that q+r₂(q)+1 is the cardinality of the smallest non-trivial line-blocking set in a plane of order q. If B is a minimal m-blocking set in PG(n,q) that contains at most q^m+q^m−1+…+q+1+r₂(q)·(∑_i=2m−n′^m−1qⁱ) points for an integer n′ satisfying mn′2m, then the dimension of B is at most n′. If the dimension of B is n′, then the following holds. The cardinality of B equals q^m+q^m−1+…+q+1+r₂(q)(∑_i=2m−n′^m−1qⁱ). For n′=m the set B is an m-dimensional subspace and for n′=m+1 the set B is a cone with an (m−2)-dimensional vertex over a non-trivial line-blocking set of cardinality q+r₂(q)+1 in a plane skew to the vertex. This result is due to Heim (Mitt. Math. Semin. Giessen 226 (1996), 4–82). For n′>m+1 and q not a prime the number q is a square and for q16 the set B is a Baer cone. If q is odd and |B|<q^m+q^m−1+…+q+1+r₂(q)(q^m−1+q^m−2), it follows from this result that the subspace generated by B has dimension at most m+1. Furthermore we prove that in this case, if , then B is an m-dimensional subspace or a cone with an (m−2)-dimensional vertex over a non-trivial line-blocking set of cardinality q+r₂(q)+1 in a plane skew to the vertex. For q=p^3h, p7 and q not a square we show this assertion for |B|q^m+q^m−1+…+q+1+q^2/3·(q^m−1+…+1).