共查询到20条相似文献,搜索用时 31 毫秒
1.
Malkhaz Ashordia 《Czechoslovak Mathematical Journal》2017,67(3):579-608
A general theorem (principle of a priori boundedness) on solvability of the boundary value problem dx = dA(t) · f(t, x), h(x) = 0 is established, where f: [a, b]×R n → R n is a vector-function belonging to the Carathéodory class corresponding to the matrix-function A: [a, b] → R n×n with bounded total variation components, and h: BVs([a, b],R n ) → R n is a continuous operator. Basing on the mentioned principle of a priori boundedness, effective criteria are obtained for the solvability of the system under the condition x(t1(x)) = B(x) · x(t 2(x))+c 0, where t i: BVs([a, b],R n ) → [a, b] (i = 1, 2) and B: BVs([a, b], R n ) → R n are continuous operators, and c 0 ∈ R n . 相似文献
2.
Mikhlin’s ideas and results related to the theory of spaces L ρ p(·) with nonstandard growth are developed. These spaces are called Lebesgue spaces with variable index; they are used in mechanics, the theory of differential equations, and variational problems. The boundedness of Fourier multipliers and singular operators on the spaces L ρ p(·) are considered. All theorems are derived from an extrapolation theorem due to Rubio de Francia. The considerations essentially use theorems on the boundedness of operators and maximal Hardy-Littlewood functions on Lebesgue spaces with constant index. 相似文献
3.
Let R be a commutative ring with identity. Let Γ(R) denote the maximal graph corresponding to the non-unit elements of R, i.e., Γ(R) is a graph with vertices the non-unit elements of R, where two distinct vertices a and b are adjacent if and only if there is a maximal ideal of R containing both. In this paper, we have shown that, for any finite ring R which is not a field, Γ(R) is a Euler graph if and only if R has odd cardinality. Moreover, for any finite ring R ? R 1×R 2× · · · ×R n, where the R i is a local ring of cardinality p i αi for all i, and the p i’s are distinct primes, it is shown that Aut(Γ(R)) is isomorphic to a finite direct product of symmetric groups. We have also proved that clique(G(R)’) = χ(G(R)’) for any semi-local ring R, where G(R)’ denote the comaximal graph associated to R. 相似文献
4.
We consider a self-adjoint matrix elliptic operator A ε, ε > 0, on L 2(R d ;C n ) given by the differential expression b(D)*g(x/ε)b(D). The matrix-valued function g(x) is bounded, positive definite, and periodic with respect to some lattice; b(D) is an (m × n)-matrix first order differential operator such that m ≥ n and the symbol b(ξ) has maximal rank. We study the operator cosine cos(τA ε 1/2 ), where τ ∈ R. It is shown that, as ε → 0, the operator cos(τA ε 1/2 ) converges to cos(τ(A 0)1/2) in the norm of operators acting from the Sobolev space H s (R d ;C n ) (with a suitable s) to L 2(R d ;C n ). Here A 0 is the effective operator with constant coefficients. Sharp-order error estimates are obtained. The question about the sharpness of the result with respect to the type of the operator norm is studied. Similar results are obtained for more general operators. The results are applied to study the behavior of the solution of the Cauchy problem for the hyperbolic equation ? τ 2 u ε (x, τ) = ?A ε u ε (x, τ). 相似文献
5.
N. N. Senik 《Functional Analysis and Its Applications》2017,51(2):152-156
In this note we consider the homogenization problem for a matrix locally periodic elliptic operator on R d of the form A ε = ?divA(x, x/ε)?. The function A is assumed to be Hölder continuous with exponent s ∈ [0, 1] in the “slow” variable and bounded in the “fast” variable. We construct approximations for (A ε ? μ)?1, including one with a corrector, and for (?Δ) s/2(A ε ? μ)?1 in the operator norm on L 2(R d ) n . For s ≠ 0, we also give estimates of the rates of approximation. 相似文献
6.
T. A. Suslina 《Functional Analysis and Its Applications》2016,50(3):241-246
We consider a self-adjoint elliptic operator Aε, ε> 0, on L2(Rd; Cn) given by the differential expression b(D)*g(x/ε)b(D). Here \(b(D) = \sum\nolimits_{j = 1}^d {b_j D_j }\) is a first-order matrix differential operator such that the symbol b(ξ) has maximal rank. The matrix-valued function g(x) is bounded, positive definite, and periodic with respect to some lattice. We study the operator exponential \({e^{ - i\tau {A_\varepsilon }}}\), where τ ∈ R. It is shown that, as ε → 0, the operator \({e^{ - i\tau {A_\varepsilon }}}\) converges to \({e^{ - i\tau {A^0}}}\) in the norm of operators acting from the Sobolev space Hs(Rd;Cn) (with suitable s) to L2(Rd;Cn). Here A0 is the effective operator with constant coefficients. Order-sharp error estimates are obtained. The question about the sharpness of the result with respect to the type of the operator norm is studied. Similar results are obtained for more general operators. The results are applied to study the behavior of the solution of the Cauchy problem for the Schrödinger-type equation i?τuε(x, τ) = Aεuε(x, τ). 相似文献
7.
Let d ≥ 1 and Z be a subordinate Brownian motion on R~d with infinitesimal generator ? + ψ(?),where ψ is the Laplace exponent of a one-dimensional non-decreasing L′evy process(called subordinator). We establish the existence and uniqueness of fundamental solution(also called heat kernel) pb(t, x, y) for non-local operator L~b= ? + ψ(?) + b ?, where Rb is an Rd-valued function in Kato class K_(d,1). We show that p~b(t, x, y)is jointly continuous and derive its sharp two-sided estimates. The kernel pb(t, x, y) determines a conservative Feller process X. We further show that the law of X is the unique solution of the martingale problem for(L~b, C_c~∞(R~d)) and X is a weak solution of Xt = X0+ Zt + integral from n=0 to t(b(Xs)ds, t ≥ 0).Moreover, we prove that the above stochastic differential equation has a unique weak solution. 相似文献
8.
As shown in Aldaz (Bull. Lond. Math. Soc. 39:203–208, 2007), the lowest constants appearing in the weak type (1, 1) inequalities satisfied by the centered Hardy-Littlewood maximal operator associated to certain finite radial measures, grow exponentially fast with the dimension. Here we extend this result to a wider class of radial measures and to some values of p > 1. Furthermore, we improve the previously known bounds for p = 1. Roughly speaking, whenever \({p\in (1, 1.03]}\), if μ is defined by a radial, radially decreasing density satisfying some mild growth conditions, then the best constants c p,d,μ in the weak type (p, p) inequalities satisfy c p,d,μ ≥ 1.005 d for all d sufficiently large. We also show that exponential increase of the best constants occurs for certain families of doubling measures, and for arbitrarily high values of p. 相似文献
9.
Matej Brešar 《Algebras and Representation Theory》2016,19(6):1437-1450
The fundamental theorem on functional identities states that a prime ring R with \(\deg (R)\ge d\) is a d-free subset of its maximal left ring of quotients Q m l (R). We consider the question whether the same conclusion holds for symmetric rings of quotients. This indeed turns out to be the case for the maximal symmetric ring of quotients Q m s (R), but not for the symmetric Martindale ring of quotients Q s (R). We show, however, that if the maps from the basic functional identities have their ranges in R, then the maps from their standard solutions have their ranges in Q s (R). We actually prove a more general theorem which implies both aforementioned results. Its proof is somewhat shorter and more compact than the standard proof used for establishing d-freeness in various situations. 相似文献
10.
We prove a theorem on the completeness of the system of root functions of the Schrödinger operator L = ?d 2/dx 2 + p(x) on the half-line R+ with a potential p for which L appears to be maximal sectorial. An application of this theorem to the complex Airy operator L c = ?d 2/dx 2 + cx, c = const, implies the completeness of the system of eigenfunctions of L c for the case in which |arg c| < 2π/3.We use subtler methods to prove a theorem stating that the system of eigenfunctions of this special operator remains complete under the condition that |arg c| < 5π/6. 相似文献
11.
Let L be a non-negative self-adjoint operator acting on L2(R n ) satisfying a pointwise Gaussian estimate for its heat kernel. Let w be an A r weight on R n × R n , 1 < r < ∞. In this article we obtain a weighted atomic decomposition for the weighted Hardy space H L,w p (R n ×R n ), 0 < p ≤ 1 associated to L. Based on the atomic decomposition, we show the dual relationship between H L,w 1 (R n × R n ) and BMOL,w(R n × R n ). 相似文献
12.
It is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from thed-dimensional Hardy spaceH p (R×···×R) toL p (R d ) (1/2<p<∞) and is of weak type (H 1 ?i ,L 1) (i=1,…,d), where the Hardy spaceH 1 ?i is defined by a hybrid maximal function. As a consequence, we obtain that the Fejér means of a functionf ∈H 1 ?i ?L(logL) d?1 converge a.e. to the function in question. Moreover, we prove that the Fejér means are uniformly bounded onH p (R×···×R) whenever 1/2<p<∞. Thus, in casef ∈H p (R×···×R) the Fejér means converge tof inH p (R×···×R) norm. The same results are proved for the conjugate Fejér means, too. 相似文献
13.
Let R k,s(n) denote the number of solutions of the equation \({n= x^2 + y_1^k + y_2^k + \cdots + y_s^k}\) in natural numbers x, y 1, . . . , y s . By a straightforward application of the circle method, an asymptotic formula for R k,s(n) is obtained when k ≥ 3 and s ≥ 2k–1 + 2. When k ≥ 6, work of Heath-Brown and Boklan is applied to establish the asymptotic formula under the milder constraint s ≥ 7 · 2k–4 + 3. Although the principal conclusions provided by Heath-Brown and Boklan are not available for smaller values of k, some of the underlying ideas are still applicable for k = 5, and the main objective of this article is to establish an asymptotic formula for R 5,17(n) by this strategy. 相似文献
14.
We obtain sharp bounds for the modulus of continuity of the uncentered maximal function in terms of the modulus of continuity of the given function, via integral formulas. Some of the results deduced from these formulas are the following: The best constants for Lipschitz and Hölder functions on proper subintervals of ? are Lip? α (Mf)≤(1+α)?1Lip? α (f), α∈(0,1]. On ?, the best bound for Lipschitz functions is \(\operatorname{Lip} ( Mf) \le (\sqrt{2} -1)\operatorname{Lip}( f)\). In higher dimensions, we determine the asymptotic behavior, as d→∞, of the norm of the maximal operator associated with cross-polytopes, Euclidean balls, and cubes, that is, ? p balls for p=1,2,∞. We do this for arbitrary moduli of continuity. In the specific case of Lipschitz and Hölder functions, the operator norm of the maximal operator is uniformly bounded by 2?α/q , where q is the conjugate exponent of p=1,2, and as d→∞ the norms approach this bound. When p=∞, best constants are the same as when p=1. 相似文献
15.
Every automaton (a letter-to-letter transducer) A whose both input and output alphabets are F p = {0, 1,..., p - 1} produces a 1-Lipschitz map f A from the space Z p of p-adic integers to Z p . The map fA can naturally be plotted in a unit real square I2 ? R2: To an m-letter non-empty word v = γ m-1γ m-2... γ0 there corresponds a number 0.v ∈ R with base-p expansion 0.γ m-1γ m-2... γ0; so to every m-letter input word w = α m-1α m-2 ··· α0 of A and to the respective m-letter output word a(w) = β m-1β m-2 ··· β0 of A there corresponds a point (0.w; 0.a(w)) ∈ R2. Denote P(A) a closure of the point set (0.w; 0.a(w)) where w ranges over all non-empty words.We prove that once some points of P(A) constitute a C 2-smooth curve in R2, the curve is a segment of a straight line with a rational slope. Moreover, when identifying P(A) with a subset of a 2-dimensional torus T2 ∈ R3, the smooth curves from P(A) constitute a collection of torus windings which can be ascribed to complex-valued functions ψ(x, t) = e i(Ax-2πBt) (x, t ∈ R), i.e., to matter waves. As automata are causal discrete systems, the main result may serve a mathematical reasoning why wave phenomena are inherent in quantum systems: This is just because of causality principle and discreteness of matter. 相似文献
16.
The system where v = C*x is an output, u = S*y is a control, A(·) ∈ R n × n , B(·) ∈ R n × (n–p), C ∈ R n × (n–p), and D ∈ R n × (n–p), is considered. The elements αij(·) and βij(·) of the matrices A(·) and B(·) are arbitrary functionals satisfying the conditions It is assumed that A(·) ∈ Z 1 ∪ Z 3 and A*(·) ∈ Z 1 ∪ Z 3, where Z 1 is the class of matrices in which the first p elements of the kth superdiagonal are sign-definite and the elements above them are sufficiently small. The class Z 3 differs from Z t1 in that the elements between this superdiagonal and the (k + 1)th row are sufficiently small. If k > p, then the elements of the p × p square in the upper left corner of the matrix are sufficiently small as well. By using special quadratic Lyapunov functions, a matrix D for which y(t)–x(t) → 0 exponentially as t → ∞ is first found, and then a matrix S for which the vectors x(t) and y(t) have the same property is constructed.
相似文献
$$\frac{{dx}}{{dt}} = A\left( \cdot \right)x + B\left( \cdot \right)u,{\kern 1pt} \frac{{dy}}{{dt}} = A\left( \cdot \right)y + B\left( \cdot \right)u + D\left( {C*y - v} \right)$$
$$\mathop {\sup }\limits_{\left( \cdot \right)} |{\alpha _{ij}}\left( \cdot \right)| < \infty \left( {i,j \in 1,n} \right),\mathop {\sup }\limits_{\left( \cdot \right)} |{\beta _{ij}}\left( \cdot \right)| < \infty \left( {i \in 1,n,j \in 1,n - p} \right).$$
17.
Let R be a prime ring of char R ≠ 2, let d be a nonzero derivation of R, and let ρ be a nonzero right ideal of R such that [[d(x)x n , d(y)] m , [y, x] s ] t = 0 for all x, y ? ρ, where n ≥ 1, m ≥ 0, s ≥ 0, and t ≥ 1 are fixed integers. If [ρ, ρ]ρ ≠ 0 then d(ρ)ρ = 0. 相似文献
18.
Uri Yechiali 《The Journal of the Operational Research Society》1977,28(2):325-337
The expected steady-state waiting time, Wq(s), in a GI/M/s system with interarrival-time distribution H(·) is compared with the mean waiting time, Wq, in an "equivalent" system comprised of s separate GI/M/1 queues each fed by an interarrival-time distribution G(·) with mean arrival rate equal to 1/s times that of H(·). For H(·) assumed to be Exponential, Gamma or Deterministic three possible relationships between H(·) and G(·) are considered: G(·) can be of the "same type" as H(·); G(·) can be derived from H(·) by assigning new arrivals to the individual channels in a cyclic order; and G(·) may be obtained from H(·) by assigning customers probabilistically to the different queues. The limiting behaviour of the ratio R = Wq/Wq(s) is studied for the extreme values (1 and 0) of the common traffic intensity, ρ. Closed form results, which depend on the forms of H(·) and G(·) and on the relationships between them, are derived. It is shown that Wq is greater than Wq(s) by a factor of at least (s + 1)/2 when ρ approaches one, and that R is at least s(s!) when ρ tends to zero. In the latter case, however, R goes to infinity (!) in most cases treated. The results may be used to evaluate the effect on the waiting times when, for certain (non-queueing) reasons, it is needed to partition a group of s servers into several small groups. 相似文献
19.
We investigate the non-homogeneous modular Dirichlet problem Δ p (·)u(x) = f (x) (where Δ p (·)u(x) = div(|?u|p(x-2)?u(x)) from the functional analytic point of view and we prove the stability of the solutions \({\left( {{u_{{p_i}}}} \right)_i}\) of the equation \({\Delta _{{p_i}\left( \cdot \right)}}{u_{{p_i}\left( \cdot \right)}} = f\) as p i (·) → q(·) via Gamma-convergence of sequence of appropriate functionals. 相似文献
20.
For semicontinuous summation methods generated by Λ = {λn(h)} (n = 0, 1, 2,...; h > 0) of Fourier series in eigenfunctions of a discrete Sturm–Liouville operator of class B, some results on the uniform a.e. behavior of Λ-means are obtained. The results are based on strong- and weak-type estimates of maximal functions. As a consequence, some statements on the behavior of the summation methods generated by the exponential means λn(h) = exp(?uα(n)h) are obtained. An application to a generalized heat equation is given. 相似文献