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1.
The paper studies the differential properties of functions of the form
$g(x) = \mathop {\max }\limits_{y \in Y} f(x,y),$
where xX (X is an open convex set from ? m ) and yY (Y is a compact from ? n ). Apart from the conventional smoothness conditions imposed on f(x, y), the condition of the concavity of g(x) on X is also imposed.
The differentiability of function g(x) on X is proved.The results of the study facilitate the derivation of the conditions ensuring the sufficiency of Pontryagin’s maximum principle.  相似文献   

2.
Let (P, ≤) be a finite poset (partially ordered set), where P has cardinality n. Consider linear extensions of P as permutations x1x2?xn in one-line notation. For distinct elements x, yP, we define ?(x ? y) to be the proportion of linear extensions of P in which x comes before y. For \(0\leq \alpha \leq \frac {1}{2}\), we say (x, y) is an α-balanced pair if α ≤ ?(x ? y) ≤?1 ? α. The 1/3–2/3 Conjecture states that every finite partially ordered set which is not a chain has a 1/3-balanced pair. We make progress on this conjecture by showing that it holds for certain families of posets. These include lattices such as the Boolean, set partition, and subspace lattices; partial orders that arise from a Young diagram; and some partial orders of dimension 2. We also consider various posets which satisfy the stronger condition of having a 1/2-balanced pair. For example, this happens when the poset has an automorphism with a cycle of length 2. Various questions for future research are posed.  相似文献   

3.
Let (X, μ) and (Y, ν) be standard measure spaces. A function \({\varphi\in L^\infty(X\times Y,\mu\times\nu)}\) is called a (measurable) Schur multiplier if the map S φ , defined on the space of Hilbert-Schmidt operators from L 2(X, μ) to L 2(Y, ν) by multiplying their integral kernels by φ, is bounded in the operator norm. The paper studies measurable functions φ for which S φ is closable in the norm topology or in the weak* topology. We obtain a characterisation of w*-closable multipliers and relate the question about norm closability to the theory of operator synthesis. We also study multipliers of two special types: if φ is of Toeplitz type, that is, if φ(x, y) = f(x ? y), \({x,y\in G}\), where G is a locally compact abelian group, then the closability of φ is related to the local inclusion of f in the Fourier algebra A(G) of G. If φ is a divided difference, that is, a function of the form (f(x) ? f(y))/(x ? y), then its closability is related to the “operator smoothness” of the function f. A number of examples of non-closable, norm closable and w*-closable multipliers are presented.  相似文献   

4.
In this paper, we essentially compute the set of x,y>0 such that the mapping \(z\longmapsto(1-r+re^{z})^{x}(\frac{\lambda}{\lambda-z})^{y}\) is a Laplace transform. If X and Y are two independent random variables which have respectively Bernoulli and Gamma distributions, we denote by μ the distribution of X+Y. The above problem is equivalent to finding the set of x>0 such that μ *x exists.  相似文献   

5.
Written in the evolutionary form, the multidimensional integrable dispersionless equations, exactly like the soliton equations in 2+1 dimensions, become nonlocal. In particular, the Pavlov equation is brought to the form vt = vxvy - ?x-1?y[vy + vx2], where the formal integral ?x?1 becomes the asymmetric integral \( - \int_x^\infty {dx'} \). We show that this result could be guessed using an apparently new integral geometry lemma. It states that the integral of a sufficiently general smooth function f(X, Y) over a parabola in the plane (X, Y) can be expressed in terms of the integrals of f(X, Y) over straight lines not intersecting the parabola. We expect that this result can have applications in two-dimensional linear tomography problems with an opaque parabolic obstacle.  相似文献   

6.
In this paper, the Fokas unified method is used to analyze the initial-boundary value for the Chen- Lee-Liu equation
$i{\partial _t}u + {\partial_{xx}u - i |u{|^2}{\partial _x}u = 0}$
on the half line (?∞, 0] with decaying initial value. Assuming that the solution u(x, t) exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter λ. The jump matrix has explicit (x, t) dependence and is given in terms of the spectral functions {a(λ), b(λ)} and {A(λ), B(λ)}, which are obtained from the initial data u0(x) = u(x, 0) and the boundary data g0(t) = u(0, t), g1(t) = ux(0, t), respectively. The spectral functions are not independent, but satisfy a so-called global relation.
  相似文献   

7.
A uniform, on ?, estimate for the increment of the spectral function θ(λ; x, y) at x = y is proved for the self-adjoint Schrödinger operator A defined on the entire axis ? by the differential operation (?d/dx)2 + q(x) with a potential-distribution q(x) that uniformly locally belongs to the space W 2 ?1. As a consequence, it is shown that for any function f(x) from the domain of power Aα of the operator with α > 1/4, the spectral expansion of the function that corresponds to the operator A is convergent absolutely and uniformly on the entire axis ?.  相似文献   

8.
Let G be a nonabelian group, and associate the noncommuting graph ?(G) with G as follows: the vertex set of ?(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. Let S 4(q) be the projective symplectic simple group, where q is a prime power. We prove that if G is a group with ?(G) ? ?(S 4(q)) then G ? S 4(q).  相似文献   

9.
Let T X denote the full transformation semigroup on a set X. For an equivalence E on X, let
$T_{\exists}(X)=\{\alpha\in T_X:\forall x,y\in X,(x\alpha,y\alpha)\in E\Rightarrow(x,y)\in E\}.$
Then T ?(X) is exactly the semigroup of mappings on the topological space X for which the collection of all E-classes is a basis. In this paper, we discuss regularity of elements and Green’s relations for T ?(X).
  相似文献   

10.
It is proved that if an entire function f: ? → ? satisfies an equation of the form α 1(x)β 1(y) + α 2(x)β 2(y) + α 3(x)β 3(y), x,y ∈ C, for some α j , β j : ? → ? and there exist no \({\widetilde \alpha _j}\) and ?\({\widetilde \beta _j}\) for which \(f\left( {x + y} \right)f\left( {x - y} \right) = {\overline \alpha _1}\left( x \right){\widetilde \beta _1}\left( y \right) + {\overline \alpha _2}\left( x \right){\widetilde \beta _2}\left( y \right)\), then f(z) = exp(Az 2 + Bz + C) ? σ Γ(z - z 1) ? σ Γ(z - z 2), where Γ is a lattice in ?; σ Γ is the Weierstrass sigma-function associated with Γ; A,B,C, z 1, z 2 ∈ ?; and \({z_1} - {z_2} \notin \left( {\frac{1}{2}\Gamma } \right)\backslash \Gamma \).  相似文献   

11.
Let g be a linear combination with quasipolynomial coefficients of shifts of the Jacobi theta function and its derivatives in the argument. All entire functions f: ? → ? satisfying f(x+y)g(x?y) = α1(x)β1(y)+· · ·+αr(x)βr(y) for some r ∈ ? and αj, βj: ? → ? are described.  相似文献   

12.
Let R be a prime ring of characteristic different from 2 and 3, Qr its right Martindale quotient ring, C its extended centroid, L a non-central Lie ideal of R and n ≥ 1 a fixed positive integer. Let α be an automorphism of the ring R. An additive map D: RR is called an α-derivation (or a skew derivation) on R if D(xy) = D(x)y + α(x)D(y) for all x, yR. An additive mapping F: RR is called a generalized α-derivation (or a generalized skew derivation) on R if there exists a skew derivation D on R such that F(xy) = F(x)y + α(x)D(y) for all x, yR.  相似文献   

13.
14.
For the number n s , β; X) of points (x 1 , x 2) in the two-dimensional Fibonacci quasilattices \( \mathcal{F}_m^2 \) of level m?=?0, 1, 2,… lying on the hyperbola x 1 2 ? ??αx 2 2 ?=?β and such that 0?≤?x 1? ≤?X, x 2? ?0, the asymptotic formula
$ {n_s}\left( {\alpha, \beta; X} \right)\sim {c_s}\left( {\alpha, \beta } \right)\ln X\,\,\,\,{\text{as}}\,\,\,\,X \to \infty $
is established, and the coefficient c s (α, β) is calculated exactly. Using this, we obtain the following result. Let F m be the Fibonacci numbers, A i \( \mathbb{N} \), i?=?1, 2, and let \( \overleftarrow {{A_i}} \) be the shift of A i in the Fibonacci numeral system. Then the number n s (X) of all solutions (A 1 , A 2) of the Diophantine system
$ \left\{ {\begin{array}{*{20}{c}} {A_1^2 + \overleftarrow {A_1^2} - 2{A_2}{{\overleftarrow A }_2} + \overleftarrow {A_2^2} = {F_{2s}},} \\ {\overleftarrow {A_1^2} - 2{A_1}{{\overleftarrow A }_1} + A_2^2 - 2{A_2}{{\overleftarrow A }_2} + 2\overleftarrow {A_2^2} = {F_{2s - 1}},} \\ \end{array} } \right. $
0?≤?A 1? ≤?X, A 2? ?0, satisfies the asymptotic formula
$ {n_s}(X)\sim \frac{{{c_s}}}{{{\text{ar}}\cosh \left( {{{1} \left/ {\tau } \right.}} \right)}}\ln X\,\,\,\,{\text{as}}\,\,\,\,X \to \infty . $
Here τ?=?(?1?+?5)/2 is the golden ratio, and c s ?=?1/2 or 1 for s?=?0 or s?≥?1, respectively.
  相似文献   

15.
In this paper, Let X, Y be two real Banach spaces and ε ≥ 0. A mapping f: XY is said to be a standard ε-isometry provided f(0) = 0 and
$$\parallel f\left( x \right) - f\left( y \right)\parallel - \parallel x - y\parallel | \leqslant \varepsilon $$
(1)
for all x, yX. If ε = 0, then it is simply called a standard isometry. We prove a sufficient and necessary condition for which {f(xn)}n≥1 is a basic sequence of Y equivalent to {xn}n≥1 whenever {xn}n≥1 is a basic sequence in X and f: XY is a nonlinear standard isometry. As a corollary we obtain the stability of basic sequences under the perturbation by nonlinear and non-surjective standard ε-isometries.
  相似文献   

16.
Let (X, d) be a locally compact separable ultrametric space. Let D be the set of all locally constant functions having compact support. Given a measure m and a symmetric function J(x, y) we consider the linear operator LJf(x) = ∫(f(x) ? f(y)) J(x, y)dm(y) defined on the set D. When J(x, y) is isotropic and satisfies certain conditions, the operator (?LJ, D) acts in L2(X,m), is essentially self-adjoint and extends as a self-adjoint Markov generator, its Markov semigroup admits a continuous heat kernel pJ (t, x, y). When J(x, y) is not isotropic but uniformly in x, y is comparable to isotropic function J(x, y) as above the operator (?LJ, D) extends in L2(X,m) as a self-adjointMarkov generator, its Markov semigroup admits a continuous heat kernel pJ(t, x, y), and the function pJ(t, x, y) is uniformly comparable in t, x, y to the function pJ(t, x, y), the heat kernel related to the operator (?LJ,D).  相似文献   

17.
Let d ? 3 be an integer, and set r = 2d?1 + 1 for 3 ? d ? 4, \(\tfrac{{17}}{{32}} \cdot 2^d + 1\) for 5 ? d ? 6, r = d2+d+1 for 7 ? d ? 8, and r = d2+d+2 for d ? 9, respectively. Suppose that Φ i (x, y) ∈ ?[x, y] (1 ? i ? r) are homogeneous and nondegenerate binary forms of degree d. Suppose further that λ1, λ2,..., λ r are nonzero real numbers with λ12 irrational, and λ1Φ1(x1, y1) + λ2Φ2(x2, y2) + · · · + λ r Φ r (x r , y r ) is indefinite. Then for any given real η and σ with 0 < σ < 22?d, it is proved that the inequality
$$\left| {\sum\limits_{i = 1}^r {{\lambda _i}\Phi {}_i\left( {{x_i},{y_i}} \right) + \eta } } \right| < {\left( {\mathop {\max \left\{ {\left| {{x_i}} \right|,\left| {{y_i}} \right|} \right\}}\limits_{1 \leqslant i \leqslant r} } \right)^{ - \sigma }}$$
has infinitely many solutions in integers x1, x2,..., x r , y1, y2,..., y r . This result constitutes an improvement upon that of B. Q. Xue.
  相似文献   

18.
We find the asymptotics as λ/? → ?∞ of the density of the spectral measure of the Sturm-Liouville operator in L 2(0,+∞) generated by the expression ?y″ + ?q(x)y, ? > 0, with the boundary condition y(0) cos α+y′(0) sinα = 0. The potential q(x) tends to ?∞ as x → +∞ and is assumed to satisfy the Sears condition and some additional regularity conditions.  相似文献   

19.
We prove that if X is a real Banach space, Y 1 ? Y 2 ? ... is a sequence of strictly embedded closed linear subspaces of X, and d 1d 2 ≥ ... is a nonincreasing sequence converging to zero, then there exists an element xX such that the distance ρ(x, Y n ) from x to Y n satisfies the inequalities d n ρ(x, Y n ) ≤ 8d n for n = 1, 2, ....  相似文献   

20.
Let T t : XX be a C 0-semigroup with generator A. We prove that if the abscissa of uniform boundedness of the resolvent s 0(A) is greater than zero then for each nondecreasing function h(s): ?+R + there are x′X′ and xX satisfying ∫ 0 h(|〈x′, T x x〉|)dt = ∞. If i? ∩ Sp(A) ≠ Ø then such x may be taken in D(A ).  相似文献   

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