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1.
We analyze polynomials P
n
that are biorthogonal to exponentials
, in the sense that
Here α>−1. We show that the zero distribution of P
n
as n→∞ is closely related to that of the associated exponent polynomial
More precisely, we show that the zero counting measures of {P
n
(−4nx)}
n=1∞ converge weakly if and only if the zero counting measures of {Q
n
}
n=1∞ converge weakly. A key step is relating the zero distribution of such a polynomial to that of the composite polynomial
under appropriate assumptions on {Δ
n,j
}.
相似文献
2.
Let H be an infinite-dimensional real Hilbert space equipped with the scalar product (⋅,⋅)
H
. Let us consider three linear bounded operators,
We define the functions
where a
i
∈H and α
i
∈ℝ. In this paper, we discuss the closure and the convexity of the sets Φ
H
⊂ℝ2 and F
H
⊂ℝ3 defined by
Our work can be considered as an extension of Polyak’s results concerning the finite-dimensional case. 相似文献
3.
Gaston Casanova 《Advances in Applied Clifford Algebras》2005,15(1):151-155
All the letters represent relative integers, except
and
and i = e1e2 in R(2, 0) oder e1 in R(1, 0).
We study the Fermat’s equation
a, b, c being prime two and two and by utilizing an elementary method. We use the Gauss’ formula
where n = 5, 7, 11, 17.
相似文献
| (1) |
| 1. |
If 2 is the p · g · c · d· of A and B we put
|
||
| 2. |
If βn = bn / (c − a) is not divisible by n, we write the expansion
|
||
| 3. | If one a, b oder c is divisible by n we prove the impossibility | ||
| 4. | In the case n = 3 the ring {a + bj} is euclidian which permits to conclude in favour of the impossibility. |
4.
Annalisa Malusa Luigi Orsina 《Calculus of Variations and Partial Differential Equations》2006,27(2):179-202
We study the limit as n goes to +∞ of the renormalized solutions u
n
to the nonlinear elliptic problems
where Ω is a bounded open set of ℝ
N
, N≥ 2, and μ is a Radon measure with bounded variation in Ω. Under the assumption of G-convergence of the operators , defined for , to the operator , we shall prove that the sequence (u
n
) admits a subsequence converging almost everywhere in Ω to a function u which is a renormalized solution to the problem
相似文献
5.
Let {X
n
,n ≥ 1} be a sequence of i.i.d. random variables. Let M
n
and m
n
denote the first and the second largest maxima. Assume that there are normalizing sequences a
n
> 0, b
n
and a nondegenerate limit distribution G, such that . Assume also that {d
k
,k ≥ 1} are positive weights obeying some mild conditions. Then for x > y we have
when G(y) > 0 (and to zero when G(y) = 0).
相似文献
6.
Katalin Gyarmati 《The Ramanujan Journal》2008,17(3):387-403
Let τ(n) be the number of positive divisors of an integer n, and for a polynomial P(X)∈ℤ[X], let
R. de la Bretèche studied the maximum values of τ
P
(n) in intervals. Here the following is proved: if P(X)∈ℤ[X] is not of the form a(X+b)
k
with a,b∈ℚ, and k∈ℕ then
This improves partially on La Bretèche’s results.
Research partially supported by Hungarian National Foundation for Scientific Research, Grants T043631, T043623 and T049693. 相似文献
7.
8.
Let X
1, X
2, ... be i.i.d. random variables. The sample range is R
n
= max {X
i
, 1 ≤ i ≤ n} − min {X
i
, 1 ≤ i ≤ n}. If for a non-degenerate distribution G and some sequences (α
k
), (β
k
) then we have
and
almost surely for any continuity point x of G and for any bounded Lipschitz function f: R → R.
相似文献
9.
Constant-Sign Solutions for Systems of Fredholm and Volterra Integral Equations: The Singular Case 总被引：1，自引：0，他引：1
We consider the system of Fredholm integral equations
and also the system of Volterra integral equations
where T>0 is fixed and the nonlinearities h
i
(t,u
1,u
2,…,u
n
) can be singular at t=0 and u
j
=0 where j∈{1,2,…,n}. Criteria are offered for the existence of constant-sign solutions, i.e., θ
i
u
i
(t)≥0 for t∈[0,1] and 1≤i≤n, where θ
i
∈{1,−1} is fixed. We also include examples to illustrate the usefulness of the results obtained.
相似文献
10.
We investigate limiting behavior as γ tends to ∞ of the best polynomial approximations in the Sobolev-Laguerre space WN,2([0, ∞); e−x) and the Sobolev-Legendre space WN,2([−1, 1]) with respect to the Sobolev-Laguerre inner product
and with respect to the Sobolev-Legendre inner product
respectively, where a0 = 1, ak ≥0, 1 ≤k ≤N −1, γ > 0, and N ≥1 is an integer. 相似文献
11.
We give the exact closed form solution of the following ordinary differential equation:
which is a modified logistic one, wherein x(t) is the population of a homogeneous species x at time t. Other than integrating the above nonlinear differential equation by means of Mathieu functions of the first kind, we also
provide a condition of a couple of inequalities involving a, b, c, h and x
0 whose fulfillment is sufficient to ensure that a bounded solution for x(t) there exists.
相似文献
12.
Let A
0, ... , A
n−1 be operators on a separable complex Hilbert space , and let α0,..., α
n−1 be positive real numbers such that 1. We prove that for every unitarily invariant norm,
for 2 ≤ p < ∞, and the reverse inequality holds for 0 < p ≤ 2. Moreover, we prove that if ω0,..., ω
n−1 are the n roots of unity with ω
j
= e
2πij/n
, 0 ≤ j ≤ n − 1, then for every unitarily invariant norm,
for 2 ≤ p < ∞, and the reverse inequalities hold for 0 < p ≤ 2. These inequalities, which involve n-tuples of operators, lead to natural generalizations and refinements of some of the classical Clarkson inequalities in the
Schatten p-norms. Extensions of these inequalities to certain convex and concave functions, including the power functions, are olso
optained.
相似文献
13.
Let K be an infinite field with characteristic different from two and (K) the n-sphere over K. We show that ambient polynomial automorphisms of (K) preserve the quadratic form x
02 + ⋯ + x
n
2 and the group Aut ((K
n+1, (K)) of such automorphisms of (K) is isomorphic to the (n + 1)-orthogonal group O(n + 1, K) provided K is real.
Next, the restriction map Aut (K
3, (K)) → Aut ( (K)) yields a surjection provided K is an algebraically closed field as well. Furthermore, for any such a field K, there is an imbedding
.
The second author was partially supported by the Ministerio de Ciencia y Tecnologia grant MTM2007-60016. 相似文献
14.
V. A. Kondratiev 《Journal of Mathematical Sciences》2006,135(1):2666-2674
The equations under consideration have the following structure:
where 0 < x
n < ∞, (x
1, …, x
n−1) ∈ Ω, Ω is a bounded Lipschitz domain,
is a function that is continuous and monotonic with respect to u, and all coefficients are bounded measurable functions. Asymptotic formulas are established for solutions of such equations
as x
n → + ∞; the solutions are assumed to satisfy zero Dirichlet or Neumann boundary conditions on ∂Ω. Previously, such formulas
were obtained in the case of a
ij, ai depending only on (x
1, …, x
n−1).
__________
Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 25, pp. 98–111, 2005. 相似文献
15.
V. A. Galaktionov 《Computational Mathematics and Mathematical Physics》2008,48(10):1823-1856
The third-order nonlinear dispersion PDE, as the key model,
is studied. Two Riemann’s problems for (0.1) with the initial data S
∓(x) = ∓ sgn.x create shock (u(x, t) ≡ S
−(x)) and smooth rarefaction (for the data S
+) waves (see [16]). The concept of “δ-entropy” solutions and others are developed for establishing the existence and uniqueness
for (0.1) by using stable smooth δ-deformations of shock-type solutions. These are analogous to entropy theory for scalar
conservation laws such as u
t
+ uu
x
= 0, which were developed by Oleinik and Kruzhkov (in x ∊ ℝ
N
) in the 1950s–1960s. The Rosenau-Hyman K(2, 2) (compacton) equation
which has a special importance for applications, is studied. Compactons as compactly supported travelling wave solutions
are shown to be δ-entropy. Shock and rarefaction waves are discussed for other NDEs such as
.
This article was submitted by the author in English.
Dedicated to the memory of Professors O.A. Oleinik and S.N. Kruzhkov 相似文献
| ((0.1)) |
16.
A. I. Aptekarev J. S. Dehesa A. Martínez-Finkelshtein R. Yáñez 《Constructive Approximation》2009,30(1):93-119
Given a nontrivial Borel measure on ℝ, let p
n
be the corresponding orthonormal polynomial of degree n whose zeros are λ
j
(n), j=1,…,n. Then for each j=1,…,n,
with
defines a discrete probability distribution. The Shannon entropy of the sequence {p
n
} is consequently defined as
In the case of Chebyshev polynomials of the first and second kinds, an explicit and closed formula for
is obtained, revealing interesting connections with number theory. In addition, several results of numerical computations
exemplifying the behavior of
for other families are presented.
相似文献
17.
We analyse degenerate, second-order, elliptic operators H in divergence form on L
2(R
n
× R
m
). We assume the coefficients are real symmetric and a
1
H
δ
≥ H ≥ a
2
H
δ
for some a
1, a
2 > 0 where
Here x
1 ∈ R
n
, x
2 ∈ R
m
and are positive measurable functions such that behaves like as x → 0 and as with and . Our principal results state that the submarkovian semigroup is conservative and its kernel K
t
satisfies bounds
where |B(x; r)| denotes the volume of the ball B(x; r) centred at x with radius r measured with respect to the Riemannian distance associated with H. The proofs depend on detailed subelliptic estimations on H, a precise characterization of the Riemannian distance and the corresponding volumes and wave equation techniques which exploit
the finite speed of propagation. We discuss further implications of these bounds and give explicit examples that show the
kernel is not necessarily strictly positive, nor continuous. 相似文献
18.
Let (M, g, σ) be a compact Riemannian spin manifold of dimension ≥ 2. For any metric conformal to g, we denote by the first positive eigenvalue of the Dirac operator on . We show that
This inequality is a spinorial analogue of Aubin’s inequality, an important inequality in the solution of the Yamabe problem.
The inequality is already known in the case n ≥ 3 and in the case n = 2, ker D = {0}. Our proof also works in the remaining case n = 2, ker D ≠ {0}. With the same method we also prove that any conformal class on a Riemann surface contains a metric with , where denotes the first positive eigenvalue of the Laplace operator. 相似文献
19.
Simon Foucart 《Numerische Mathematik》2009,112(4):535-564
We describe the computations of some intrinsic constants associated to an n-dimensional normed space , namely the N-th “allometry” constants
These are related to Banach–Mazur distances and to several types of projection constants. We also present the results of our
computations for some low-dimensional spaces such as sequence spaces, polynomial spaces, and polygonal spaces. An eye is kept
on the optimal operators T and T′, or equivalently, in the case N = n, on the best conditioned bases. In particular, we uncover that the best conditioned bases of quadratic polynomials are not
symmetric, and that the Lagrange bases at equidistant nodes are best conditioned in the spaces of trigonometric polynomials
of degree at most one and two. 相似文献
20.
Yong Ge TIAN 《数学学报(英文版)》2006,22(1):289-300
For any element a in a generalized 2^n-dimensional Clifford algebra Lln （F） over an arbitrary field F of characteristic not equal to two, it is shown that there exits a universal invertible matrix Pn over Lln（F） such that Pn^-1DnPn= φ（α）∈F^2n×2n, where φ（a） is a matrix representation of α over and Dα is a diagonal matrix consisting of a or its conjugate. 相似文献