C0-semigroups on a locally convex space

A necessary and sufficient condition that a densely defined linear operator A in a sequentially complete locally convex space X be the infinitesimal generator of a quasi-equicontinuous C₀-semigroup on X is that there exist a real number β ? 0 such that, for each λ > β, the resolvent (λI ? A)^?1 exists and the family {(λ ? β)^k(λI ? A)^?k; λ > β, k = 0, 1, 2,…} is equicontinuous. In this case all resolvents (λI ? A)^?1, λ > β, of the given operator A and all exponentials exp(tA), t ? 0, of the operator A belong to a Banach algebra $\text{B}{}_{\mathrm{г}}\text{(X)}$ which is a subspace of the space L(X) of all continuous linear operators on X, and, for each t ? 0 and for each x?X, one has lim_{k → z} (I ? k^?1tA)^?kx = exp(tA) x. A perturbation theorem for the infinitesimal generator of a quasi-equicontinuous C₀-semigroup by an operator which is an element of $\text{B}{}_{\mathrm{г}}\text{(X)}$ is obtained.     

On a continuous analogue of the stochastic difference equation Xn=ρXn-1+Bn

Let B₁, B₂, ... be a sequence of independent, identically distributed random variables, letX₀ be a random variable that is independent ofB_n forn?1, let ρ be a constant such that 0<ρ<1 and letX₁,X₂, ... be another sequence of random variables that are defined recursively by the relationshipsX_n=ρX_n-1+B_n. It can be shown that the sequence of random variablesX₁,X₂, ... converges in law to a random variableX if and only ifE[log⁺¦B₁¦]<∞. In this paper we let {B(t):0≦t<∞} be a stochastic process with independent, homogeneous increments and define another stochastic process {X(t):0?t<∞} that stands in the same relationship to the stochastic process {B(t):0?t<∞} as the sequence of random variablesX₁,X₂,...stands toB₁,B₂,.... It is shown thatX(t) converges in law to a random variableX ast →+∞ if and only ifE[log⁺¦B(1)¦]<∞ in which caseX has a distribution function of class L. Several other related results are obtained. The main analytical tool used to obtain these results is a theorem of Lukacs concerning characteristic functions of certain stochastic integrals.     

Asymptotic equilibrium of ordinary differential systems

A theorem on asymptotic equilibrium is proved for the solutions of the system(1)X ⁿ=f(t,X), x ^t ₀=x_o where f(t,x) is majorized by a funciton g(t,u) which is non-increasing in u. It is of interest to notice that the funcitons f(t,x) and g(t,u) need not be defined for x=0 and u=0 respectively. Such majorant functions occur in gravitational problems and therefore the result is of pracitcal interest.Using this, the asymptotic relatiohship between the solutions of(2)y=A(t)y, y ^t _o=y_oand its nonlinear perturbation(3) X=A(t)x+f(t,x), X^t _o is investigated. This last result includes as a special case two theorems of Hallam[2]     

Normal Forms for Nonautonomous Differential Equations

We extend Henry Poincarés normal form theory for autonomous differential equations x=f(x) to nonautonomous differential equations x=f(t, x). Poincarés nonresonance condition λ_j−∑ⁿ_i=1 ?_iλ_i≠0 for eigenvalues is generalized to the new nonresonance condition λ_j∩∑ⁿ_i=1 ?_iλ_i=∅ for spectral intervals.     

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.     

Note on forced oscillation of nth-order sublinear differential equations

In the case of oscillatory potentials, we establish an oscillation theorem for the forced sublinear differential equation x(n)+q(t)λ|x|sgnx=e(t), t∈[t₀,∞). No restriction is imposed on the forcing term e(t) to be the nth derivative of an oscillatory function. In particular, we show that all solutions of the equation x^″+tαsintλ|x|sgnx=mtβcost, t?0, 0<λ<1 are oscillatory for all m≠0 if β>(α+2)/(1−λ). This provides an analogue of a result of Nasr [Proc. Amer. Math. Soc. 126 (1998) 123] for the forced superlinear equation and answers a question raised in an earlier paper [J.S.W. Wong, SIAM J. Math. Anal. 19 (1988) 673].     

An initial value problem for a singular parabolic equation of even order

At first Cauchy-problem for the equation: \(L[u(X,t)] \equiv \sum\limits_{i = 1}^n {\frac{{\partial ^2 u}}{{\partial x_1^2 }} + \frac{{2v}}{{\left| X \right|^2 }}} \sum\limits_{i = 1}^n {x_i \frac{{\partial u}}{{\partial x_i }} - \frac{{\partial u}}{{\partial t}} = 0} \) wheren≥1,v—an arbitrary constant,t>0,X=(x ₁, …, x_n)∈E ⁿ/{0}, |X|= =(x ₁ ² +…+x _n ² )^1/2, with 0 being a centre of coordinate system, is studied. Basing on the above, the solution of Cauchy-Nicolescu problem is given which consist in finding a solution of the equationL ^p [u (X, t)]=0, withp∈N subject the initial conditions \(\mathop {\lim }\limits_{t \to \infty } L^k [u(X,t)] = \varphi _k (X)\) ,k=0, 1,…,p?1 and ?_k(X) are given functions.     

On three-designs of small order

For positive integers t?k?v and λ we define a t-design, denoted B_i[k,λ;v], to be a pair (X,B) where X is a set of points and B is a family, (B_i:i?I), of subsets of X, called blocks, which satisfy the following conditions: (i) |X|=v, the order of the design, (ii) |B_i|=k for each i?I, and (iii) every t-subset of X is contained in precisely λ blocks. The purpose of this paper is to investigate the existence of 3-designs with 3?k?v?32 and λ>0.Wilson has shown that there exists a constant N(t, k, v) such that designs B_t[k,λ;v] exist provided λ>N(t,k,v) and λ satisfies the trivial necessary conditions. We show that N(3,k,v)=0 for most of the cases under consideration and we give a numerical upper bound on N(3, k, v) for all 3?k?v?32. We give explicit constructions for all the designs needed.     

On the eigenvalues of the fredholm operator

We prove that if ω(t, x, K ₂ ^(m) )?c(x)ω(t) for allxε[a, b] andx ε [0,b-a] wherec ∈L ¹(a, b) and ω is a modulus of continuity, then λ _n =O(n ^?m-1/2ω(1/n)) and this estimate is unimprovable.     

Some ratio inequalities for iterated stochastic integrals

Let X = (X_t, ?_t) be a continuous local martingale with quadratic variation 〈X〉 and X₀ = 0. Define iterated stochastic integrals I_n(X) = (I_n(t, X), ?_t), n ≥ 0, inductively by $$ I_{n} (t, X) = \int ^{t} _{0} I_{n-1} (s, X)dX_{s} $$ with I₀(t, X) = 1 and I₁(t, X) = X_t. Let (??^x_t(X)) be the local time of a continuous local martingale X at x ∈ ?. Denote ??*t(X) = sup_x∈? ??^x_t(X) and X* = sup_t≥0 |X_t|. In this paper, we shall establish various ratio inequalities for I_n(X). In particular, we show that the inequalities $$ c_{n,p} \, \left\Vert (G ( \langle X \rangle _{\infty} )) ^{n/2} \right\Vert _{p} \; \le \; \left\Vert {\mathop \sup \limits _{t \ge 0}} \; {\left\vert I_{n} (t, X) \right\vert \over {(1+ \langle X \rangle _{t} ) ^{n/2}}} \right\Vert _{p} \; \le C_{n, p} \, \left\Vert (G ( \langle X \rangle _{\infty} )) ^{n/2} \right\Vert _{p} $$ hold for 0 < p < ∞ with some positive constants c_n,p and C_n,p depending only on n and p, where G(t) = log(1+ log(1+ t)). Furthermore, we also show that for some γ ≥ 0 the inequality $$ E \left[ U ^{p}_{n} \exp \left( \gamma {U ^{1/n} _{n} \over {V}} \right) \right] \le C_{n, p, \gamma} E [V ^{n, p}] \quad (0 < p < \infty ) $$ holds with some positive constant C_n,p,γ depending only on n, p and γ, where U_n is one of 〈I_n(X)〉^1/2_∞ and I*_n(X), and V one of the three random variables X*, 〈X〉^1/2_∞ and ??*_∞(X). (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two-sided eigenvalue estimates for subordinate processes in domains

Let X={X_t,t?0} be a symmetric Markov process in a state space E and D an open set of E. Denote by X^D the subprocess of X killed upon leaving D. Let S={S_t,t?0} be a subordinator with Laplace exponent φ that is independent of X. The processes X^φ?{X_St,t?0} and are called the subordinate processes of X and X^D, respectively. Under some mild conditions, we show that, if {-μ_n,n?1} and {-λ_n,n?1} denote the eigenvalues of the generators of the subprocess of X^φ killed upon leaving D and of the process X^D respectively, then

     

Convexity of the bounds induced by Markov's inequality

X is a nonnegative random variable such that EX^t < ∞ for 0≤ t < λ ≤ ∞. The (l??) quantile of the distribution of X is bounded above by [?^?1 EX^t]^1?t. We show that there exist positive ?₁ ≥ ?₂ such that for all 0 <?≤?₁ the function g(t) = [?^-1EX^t]^1?t is log-convex in [0, c] and such that for all 0 < ? ≤ ?₂ the function log g(t) is nonincreasing in [0, c].     

A note on the kernels of higher derivations

Let k ? k′ be a field extension. We give relations between the kernels of higher derivations on k[X] and k′[X], where k[X]:= k[x ₁,…, x _n ] denotes the polynomial ring in n variables over the field k. More precisely, let D = {D _n } _n=0 ^∞ a higher k-derivation on k[X] and D′ = {D′ _n } _n=0 ^∞ a higher k′-derivation on k′[X] such that D′ _m (x _i ) = D _m (x _i ) for all m ? 0 and i = 1, 2,…, n. Then (1) k[X] ^D = k if and only if k′[X] ^D′ = k′; (2) k[X] ^D is a finitely generated k-algebra if and only if k′[X] ^D′ is a finitely generated k′-algebra. Furthermore, we also show that the kernel k[X] ^D of a higher derivation D of k[X] can be generated by a set of closed polynomials.     

Outer Γ-Convexity in Vector Spaces

A subset S of some vector space X is said to be outer Γ-convex w.r.t. some given balanced subset Γ ? X if for all x ₀, x ₁ ? S there exists a closed subset Λ ? [0,1] such that {x _λ | λ ? Λ} ? S and [x ₀, x ₁] ? {x _λ | λ ? Λ} + 0.5 Γ, where x _λ: = (1 ? λ)x ₀ + λ x ₁. A real-valued function f:D → ? defined on some convex D ? X is called outer Γ-convex if for all x ₀, x ₁ ? D there exists a closed subset Λ ? [0,1] such that [x ₀, x ₁] ? {x _λ | λ ? Λ} + 0.5 Γ and f(x _λ) ≤ (1 ? λ)f(x ₀) + λ f(x ₁) holds for all λ ? Λ. Outer Γ-convex functions possess some similar optimization properties as these of convex functions, e.g., lower level sets of outer Γ-convex functions are outer Γ-convex and Γ-local minimizers are global minimizers. Some properties of outer Γ-convex sets and functions are presented, among others a simplex property of outer Γ-convex sets, which is applied for establishing a separation theorem and for proving the existence of modified subgradients of outer Γ-convex functions.     

Weak real integral characterizations for exponential stability of semigroups in reflexive spaces

Let (t _n ) be a sequence of nonnegative real numbers tending to ∞, such that 1≤t _n+1?t _n ≤α for all natural numbers n and some positive α. We prove that a strongly continuous semigroup {T(t)} _t≥0, acting on a Hilbert space H, is uniformly exponentially stable if $$\sum_{n=0}^\infty\varphi\bigl(\bigl|\bigl\langle T(t_n)x, y\bigr\rangle\bigr|\bigr)<\infty, $$ for all unit vectors x, y in H. We obtain the same conclusion under the assumption that the inequality $$\sum_{n=0}^\infty\varphi\bigl(\bigl|\bigl\langle T(t_n)x, x^\ast\bigr\rangle\bigr|\bigr)<\infty, $$ is fulfilled for all unit vectors x∈X and x ^?∈X ^?, X being a reflexive Banach space. These results are stated for functions φ belonging to a special class of functions, such as defined in the second section of this paper. We conclude our paper with a Rolewicz’s type result in the continuous case on Hilbert spaces.     

Functional inclusion and controllability of nonlinear neutral functional differential systems

Consider the nonlinear neutral functional differential inclusion (i) $$(d/dt)D(t, x_t ) \in R(t, x_t )$$ , whereD is a continuous operator onIXC, linear inx _t , indeed of the form (4) below, with kernelD(t, ·)={0}, and atomic at 0, andR is nonempty, closed, and convex. Here,I≡[t ₀,t _I ] andC=C([-h,0],E ⁿ ). In (i), the derivative is specified in terms of the state at timet as well as the state and the derivative of the state for values oft precedingt. We use the Fan fixed-point theorem to prove the existence of a solution of (i) which satisfies two-point boundary values \(x_\omega = \phi _0 ,x_{t_1 } = \phi _1\) , where φ₀, φ₁ belong toC. We next apply this existence result to study the exact function space controllability of the neutral functional differential system (ii) $$(d/dt)D(t, x_t ) = f(t, x_t , u), u(t) \in \Omega (t, x_t )$$ . We present sufficient conditions onf and Ω which imply exact controllability between two fixed functions inC.     

Asymptotical Formulas For Solutions Of Linear Differential Systems Of The First Order

In this paper a system of differential equations y′ ? A(·,λ)y = 0 is considered on the finite interval [a,b] where λ ∈ C, A(·, λ):= λ A₁+ A ₀ +λ _?1A_?1(·,λ) and A ₁,A ₀, A _{? 1} are n × n matrix-functions. The main assumptions: A ₁ is absolutely continuous on the interval [a, b], A ₀ and A _{- 1}(·,λ) are summable on the same interval when ¦λ¦ is sufficiently large; the roots φ₁(x),…,φ_n (x) of the characteristic equation det (φ E — A ₁) = 0 are different for all x ∈ [a,b] and do not vanish; there exists some unlimited set Ω ? C on which the inequalities Re(λφ₁(x)) ≤ … ≤ Re (λφ_n(x)) are fulfilled for all x ∈ [a,b] and for some numeration of the functions φ_j(x). The asymptotic formula of the exponential type for a fundamental matrix of solutions of the system is obtained for sufficiently large ¦λ¦. The remainder term of this formula has a new type dependence on properties of the coefficients A ₁ (x), A _o (x) and A _{- 1} (x).     

A simple formula for an analogue of conditional wiener integrals and its applications II

Let C[0, T] denote the space of real-valued continuous functions on the interval [0, T] with an analogue w _ϕ of Wiener measure and for a partition 0 = t ₀ < t ₁ < ... < t _n < t _n+1 = T of [0, T], let X _n : C[0, T] → ℝ ⁿ⁺¹ and X ⁿ⁺¹: C[0, T] → ℝ ⁿ⁺² be given by X _n (x) = (x(t ₀), x(t ₁), ..., x(t _n )) and X _n+1(x) = (x(t ₀), x(t ₁), ..., x(t _n+1)), respectively. In this paper, using a simple formula for the conditional w _ϕ-integral of functions on C[0, T] with the conditioning function X _n+1, we derive a simple formula for the conditional w _ϕ-integral of the functions with the conditioning function X _n . As applications of the formula with the function X _n , we evaluate the conditional w _ϕ-integral of the functions of the form F _m (x) = ∫₀ ^T (x(t)) ^m for x ∈ C[0, T] and for any positive integer m. Moreover, with the conditioning X _n , we evaluate the conditional w _ϕ-integral of the functions in a Banach algebra which is an analogue of the Cameron and Storvick’s Banach algebra . Finally, we derive the conditional analytic Feynman w _ϕ-integrals of the functions in .      

Моносплайны и наилуч шие квадратурные фор мулы для некоторых классо в непериодических фу нкций

The following quadrature formulae are considered: $$\int\limits_0^1 {f(x)dx = \mathop \sum \limits_{k = 1}^n a_k f(x_k ) + \mathop \sum \limits_{i = 1}^l } b_i f^{(\alpha _i )} (0) + \mathop \sum \limits_{j = 1}^m c_j f^{(\beta _j )} (1) + R(f),$$ where 0≦x₁2<...n≦1 0≦α _i , β_j≦r?1;l, m, n, andr are positive integers. The problem of existence and uniqueness of the best quadrature formula is solved for the classesW ^r L _p (r=1, 2, ...; 1<p≦∞), obtaining the characteristic properties of its nodesx _k and weightsa _k ,b _i , andc _j .     

Generalized Christoffel functions for Jacobi-exponential weights

Assume that W=e ^?Q where I:=(a,b), ?∞≦a<0<b≦∞, and Q:?I→[0,∞) is continuous and increasing. Let 0<p<∞, a<t _r <t _r?1<?<t ₁<b, p _i >?1/p, i=1,2,…,r, and $U(x)=\prod_{i=1}^{r} {|x-t_{i}|}^{p_{i}}$ . We give the L _p Christoffel functions for the Jacobi-exponential weight WU. In addition, we obtain restricted range inequalities.