Additive Maps Preserving Nilpotent Perturbation of Scalars

Let X be a Banach space over F(= R or C) with dimension greater than 2. Let N(X) be the set of all nilpotent operators and B_0(X) the set spanned by N(X). We give a structure result to the additive maps on FI + B_0(X) that preserve rank-1 perturbation of scalars in both directions. Based on it, a characterization of surjective additive maps on FI + B_0(X) that preserve nilpotent perturbation of scalars in both directions are obtained. Such a map Φ has the form either Φ(T) = cAT A~(-1)+ φ(T)I for all T ∈ FI + B_0(X) or Φ(T) = cAT*A~(-1)+ φ(T)I for all T ∈ FI + B_0(X), where c is a nonzero scalar,A is a τ-linear bijective transformation for some automorphism τ of F and φ is an additive functional.In addition, if dim X = ∞, then A is in fact a linear or conjugate linear invertible bounded operator.     

The kernels of irreducible representations of the general linear group

A derivation for the kernel of the irreducible representation T_(λ) of the general linear group GL_n(C) is given. This is then applied to the problem of determining necessary and sufficient conditions under which T_(λ)(A) = T_(λ)(B), where A and B are linear transformations, not necessarily invertible. Finally, conditions are obtained under which normality of T_(λ)(A) implies normality of A.     

考虑非周期检测的视情维修与备件订购联合策略优化

针对单部件系统/关键部件提出视情维修与备件订购联合策略,其中系统退化服从两阶段延迟时间过程且采用非周期检测策略,退化初期以检测间隔T₁检查系统状态,而在第一次识别缺陷状态时,缩短检测周期为T₂、订购备件且进行不完美维修;若系统在随后的退化中被识别处于缺陷状态,执行不完美维修直至超过阈值次数N_max并采取预防性更换,但若在检测周期内发生故障则进行更换。根据系统状态和备件状态分析各种可能更新事件及相应的联合决策,利用更新报酬理论构建最小化单位时间内期望成本的目标函数,优化T₁,T₂, N_max。与对比模型策略相比,算例结果表明所提出的联合策略能有效降低单位时间内的期望成本。     

Analytic modelling of solar radiation: driven temperature variations of outdoor surfaces

An analytic model is described for the variation of surface temperature T₀(t) ofa freely evaporating or totally dry surface exposed to complicated, non-analytic variations of meteorological conditions expressed in (N + 1)-point time series form. Its basis is a parameterized, analytic representation of T₀, inserted in linearized equations describing the surface energy balance and the concurrent underlying heat conduction. Applied to diurnal variations (with N = 24) over bare soil, the model reproduces observed temperatures with a typical standard deviation less than 1K, and the daily mean within 0.1 K.

The method should find general application in modelling of storage systems subject to complicated, discretely-sampled boundary conditions, where a linear analytic equation links the latter to the storage term.     

Strong equality of domination parameters in trees

We study the concept of strong equality of domination parameters. Let P₁ and P₂ be properties of vertex subsets of a graph, and assume that every subset of V(G) with property P₂ also has property P₁. Let ψ₁(G) and ψ₂(G), respectively, denote the minimum cardinalities of sets with properties P₁ and P₂, respectively. Then ψ₁(G)ψ₂(G). If ψ₁(G)=ψ₂(G) and every ψ₁(G)-set is also a ψ₂(G)-set, then we say ψ₁(G) strongly equals ψ₂(G), written ψ₁(G)≡ψ₂(G). We provide a constructive characterization of the trees T such that γ(T)≡i(T), where γ(T) and i(T) are the domination and independent domination numbers, respectively. A constructive characterization of the trees T for which γ(T)=γ_t(T), where γ_t(T) denotes the total domination number of T, is also presented.     

Ordinal notations and well-orderings in bounded arithmetic

This paper investigates provability and non-provability of well-foundedness of ordinal notations in weak theories of bounded arithmetic. We define a notion of well-foundedness on bounded domains. We show that T₂¹ and S₂² can prove the well-foundedness on bounded domains of the ordinal notations below ₀ and Γ₀. As a corollary, the class of polynomial local search problems, PLS, can be augmented with cost functions that take ordinal values below ₀ and Γ₀ without increasing the class PLS.     

Invariant record processes and applications to best choice modelling

Let X₁, X₂,…be identically distributed random variables from an unknown continuous distribution. Further let I_r(1), I_r(2),…be a sequence of indicator functions defined on X₁, X₂,…by I_r(k) = 0 if k < r, I_r(k) = 1 if X_k is a r-record AND = 0 otherwise. Suppose that we observe X₁, X₂,… at times T₁ < T₂ <… where the T_k's are realisations of some regular counting process (N(τ)) defined on the positive half-line. Having observed [0, τ], say, the problem is to predict the future behaviour of the counting processes (R_r(τ, s))_sτ = # r-records in [τ, s]. More specifically the objective of this paper is to show that these processes can be (inhomogeneous) Poisson processes even if (N(τ))_τ0 has dependent increments.

The strong link between optimal selection and optimal stopping of record sequences or record processes, perhaps not fully recognized so far, is pointed out in this paper. It is shown to lead to a unification of the treatment of problems which, at first sight, are rather different. Moreover the stopping of record processes in continuous time can lead to rigorous and elegant solutions in cases where dynamic programming is bound to fail. Several examples will be given to facilitate a comparison with other methods.     

The consistency of a non-linear least squares estimator from diffusion processes

Consider the following Itô stochastic differential equation dX(t) = ƒ(θ⁰, X(t)) dt + dW(t), where (W(t), t 0), is a standard Wiener process in R^N. On the basis of discrete data 0 = t₀ < t₁ < …<t_n = T; X(t₁),...,X(t_n) we would like to estimate the parameter θ⁰. We shall define the least squares estimator and show that under some regularity conditions, is strongly consistent.     

Weak convergence to the multiple Stratonovich integral

We have considered the problem of the weak convergence, as tends to zero, of the multiple integral processes

in the space , where fL²([0,T]ⁿ) is a given function, and {η(t)}_>0 is a family of stochastic processes with absolutely continuous paths that converges weakly to the Brownian motion. In view of the known results when n2 and f(t₁,…,t_n)=1_{t1<t2<<tn}, we cannot expect that these multiple integrals converge to the multiple Itô–Wiener integral of f, because the quadratic variations of the η are null. We have obtained the existence of the limit for any {η}, when f is given by a multimeasure, and under some conditions on {η} when f is a continuous function and when f(t₁,…,t_n)=f₁(t₁)f_n(t_n)1_{t1<t2<<tn}, with f_iL²([0,T]) for any i=1,…,n. In all these cases the limit process is the multiple Stratonovich integral of the function f.     

All trees are 1-embeddable and all except stars are 2-embeddable

A graph G with n vertices is said to be embeddable (in its complement) if there is an automorphism φ of K_n such that E(G) ∩ E(φ(G))=. It is known that all trees T with n (≥2) vertices and T K_1,n−1 are embeddable. We say that G is 1-embeddable if, for every edge e, there is an automorphism φ of K_n such that E(G) ∩ E(φ(G))={e};and that it is 2-embeddable if,for every pair e₁, e₂ of edges, there is an automorphism φ of K_n such that E(G) ∩ E(φ(G))={e₁, e₂}. We prove here that all trees with n (3) vertices are 1-embeddable; and that all trees T with n (4) vertices and T K_1,n−1 are 2-embeddable. In a certain sense, this result is sharp.     

First passage time for some stationary processes

For a 1-dependent stationary sequence {X_n} we first show that if u satisfies p₁=p₁(u)=P(X₁>u)0.025 and n>3 is such that 88np₁³1, then

P{max(X₁,…,X_n)u}=ν·μⁿ+O{p₁³(88n(1+124np₁³)+561)}, n>3,

where

ν=1−p₂+2p₃−3p₄+p₁²+6p₂²−6p₁p₂,μ=(1+p₁−p₂+p₃−p₄+2p₁²+3p₂²−5p₁p₂)⁻¹

with

p_k=p_k(u)=P{min(X₁,…,X_k)>u}, k1

and

|O(x)||x|.

From this result we deduce, for a stationary T-dependent process with a.s. continuous path {Y_s}, a similar, in terms of P{max_0skTY_s<u}, k=1,2 formula for P{max_0stY_su}, t>3T and apply this formula to the process Y_s=W(s+1)−W(s), s0, where {W(s)} is the Wiener process. We then obtain numerical estimations of the above probabilities.     

Triangles in 3-connected matroids

A collection F of 3-connected matroids is triangle-rounded if, whenever M is a 3-connected matroid having a minor in F, and T is a 3-element circuit of M, then M has a minor which uses T and is isomorphic to a member of F. An efficient theorem for testing a collection of matroids for this property is presented. This test is used to obtain several results including the following extension of a result of Asano, Nishizeki, and Seymour. Let T be a 3-element circuit of a 3-connected binary nonregular matroid M with at least eight elements. Then M has a minor using T that is isomorphic to S₈ or the generalized parallel connection across T of F₇ and M(K₄).     

Balanced partitions of two sets of points in the plane

We prove the following theorem. Let m≥2 and q≥1 be integers and let S and T be two disjoint sets of points in the plane such that no three points of ST are on the same line, |S|=2q and |T|=mq. Then ST can be partitioned into q disjoint subsets P₁,P₂,…,P_q satisfying the following two conditions: (i) conv(P_i)∩conv(P_j)=φ for all 1≤i<j≤q, where conv(P_i) denotes the convex hull of P_i; and (ii) |P_i∩S|=2 and |P_i∩T|=m for all 1≤i≤q.     

On the occupation time of an iterated process having no local time

We study the asymptotic behaviour of the occupation time process ∫^t₀ I_A(W₁(L₂(s)))ds, t 0, where W₁ is a standard Wiener process and L₂ is a Wiener local time process at zero that is independent from W₁. We prove limit laws, as well as almost sure upper and lower class theorems. Possible extensions of the obtained results are also discussed.     

On the fourth power moment of Fourier coefficients of cusp form

Let a(n)be the Fourier coefficients of a holomorphic cusp form of weightκ=2n≥12 for the full modular group and A(x)=∑_(n≤x)a(n).In this paper,we establish an asymptotic formula of the fourth power moment of A(x)and prove that ∫T1A~4(x)dx=3/(64κπ~4)s_4;2()T~(2κ)+O(T~(2κ-δ_4+ε))with δ_4=1/8,which improves the previous result.     

An integer sequence from a rational recursion

The sequence of numbers {a_i} defined by the recurrence for n>3 with initial values a₀, a₁, a₂, a₃=1 is shown to be integral. Other initial values are also considered.     

On the surface viscosity at the boundary between phases

The equations of motion of the interphase boundary are considered. It is shown that the conditions at the surface separating the phases obtained in /1, 2/ by different methods, are identical. The study of the dynamics of the fluid-fluid interface was initiated by Bussinesq /3/ who postulated a linear relationship between the surface stress tensor T_β and the strain rate tensor S_β, assigning two viscosity coefficients to the surface, the dilatation coefficient k (the analog of volume viscosity) and the two-dimensional shear viscosity . In the three-dimensional coordinate system two of whose axes u¹ and usu2 coincide with the axes of any coordinate system at the surface and whose third axis u³ is perpendicular to the surface, his results have the form Tβ = [γ + (k - )θ]a_β + S_β , θ = a^βS_β, V_{, β} = r_{. β} — v_sb_β,      

On a property specific to the tent map

Let a set {X_λ; λ  Λ} of subspaces of a topological space X be a cover of X. Mathematical conditions are proposed for each subspace X_λ to define a map g_Xλ:X_λ→X which has the following property specific to the tent map known in the baker’s transformation. Namely, for any infinite sequence ω₀, ω₁, ω₂, … of X_λ, λ  Λ, we can find an initial point x₀  ω₀ such that g_ω0(x₀)ω₁,g_ω1(g_ω0(x₀))ω₂,…. The conditions are successfully applied to a closed cover of a weak self-similar set.     

Combinatorial theorems on contractive mappings in power sets

We prove that to every positive integer n there exists a positive integer h such that the following holds: If S is a set of h elements and ƒ a mapping of the power set of S into such that ƒ(T)T for all T , then there exists a strictly increasing sequence T₁T_n of subsets of S such that one of the following three possibilities holds: (a) all sets ƒ(T_i), i= 1,…,n, are equal; (b) for all i=1,…, n, we have ƒ(T_i)=T_i; (c) T_i=ƒ(T_i+1) for all i= 1,…,n-1. This theorem generalizes theorems of the author, Rado, and Leeb. It has applications for subtrees in power sets.     

Mean residual life and increasing convex comparison of shock models

Two devices are subjected to shocks arriving according to a general counting process. Let M₁ and M₂ be the random number of shocks that cause the failure of the first and the second device, respectively. We find conditions on the counting process such that the mean residual life ordering, the increasing convex ordering and the expectation ordering between M₁ and M₂ are preserved in the random lifetimes of the two devices.