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1.
norms of random sums Paul Alton Hagelstein 《Proceedings of the American Mathematical Society》2005,133(8):2327-2334
Let denote a sequence of measurable functions on , and let denote the weak norm. It is shown that
where is a sequence of independent random variables taking on values and with equal probability. Moreover, it is shown that
The paper concludes by providing an example indicating that, if , then the estimate
is the best possible.
where is a sequence of independent random variables taking on values and with equal probability. Moreover, it is shown that
The paper concludes by providing an example indicating that, if , then the estimate
is the best possible.
2.
Aurel Spataru 《Proceedings of the American Mathematical Society》2004,132(11):3387-3395
Let be i.i.d. random variables with , and set . We prove that, for
under the assumption that and Necessary and sufficient conditions for the convergence of the sum above were established by Lai (1974).
under the assumption that and Necessary and sufficient conditions for the convergence of the sum above were established by Lai (1974).
3.
By using Krasnoselskii's fixed point theorem, we prove that the following periodic species Lotka-Volterra competition system with multiple deviating arguments has at least one positive periodic solution provided that the corresponding system of linear equations has a positive solution, where and are periodic functions with Furthermore, when and , , are constants but , remain -periodic, we show that the condition on is also necessary for to have at least one positive periodic solution.
4.
Gangsong Leng Lin Si Qingsan Zhu 《Proceedings of the American Mathematical Society》2004,132(9):2655-2660
For let and denote the arithmetic mean and geometric mean of elements of , respectively. It is proved that if is an integer in , then
with equality if and only if . Furthermore, as a generalization of this inequality, a mixed power-mean inequality for subsets is established.
with equality if and only if . Furthermore, as a generalization of this inequality, a mixed power-mean inequality for subsets is established.
5.
Let be a sequence of centered iid random variables. Let be a strongly additive arithmetic function such that and put . If and satisfies a Lindeberg-type condition, we prove the following law of the iterated logarithm: We also prove the validity of the corresponding weighted strong law of large numbers in .
6.
Todd Cochrane Christopher Pinner 《Proceedings of the American Mathematical Society》2005,133(2):313-320
For a sparse polynomial , with and , we show that
thus improving upon a bound of Mordell. Analogous results are obtained for Laurent polynomials and for mixed exponential sums.
thus improving upon a bound of Mordell. Analogous results are obtained for Laurent polynomials and for mixed exponential sums.
7.
Philippe Poulin 《Proceedings of the American Mathematical Society》2007,135(1):77-85
It is well known that the Green function of the standard discrete Laplacian on , exhibits a pathological behavior in dimension . In particular, the estimate fails for . This fact complicates the study of the scattering theory of discrete Schrödinger operators. Molchanov and Vainberg suggested the following alternative to the standard discrete Laplacian, and conjectured that the estimate holds for all . In this paper we prove this conjecture.
8.
Peter Borwein Richard Lockhart 《Proceedings of the American Mathematical Society》2001,129(5):1463-1472
The results of this paper concern the expected norm of random polynomials on the boundary of the unit disc (equivalently of random trigonometric polynomials on the interval ). Specifically, for a random polynomial
let
Assume the random variables , are independent and identically distributed, have mean 0, variance equal to 1 and, if 2$">, a finite moment . Then
and
as .
In particular if the polynomials in question have coefficients in the set (a much studied class of polynomials), then we can compute the expected norms of the polynomials and their derivatives
and
This complements results of Fielding in the case, Newman and Byrnes in the case, and Littlewood et al. in the case.
9.
Nakao Hayashi Pavel I. Naumkin Yasuko Yamazaki 《Proceedings of the American Mathematical Society》2002,130(3):779-789
We consider the derivative nonlinear Schrödinger equations
where the coefficient satisfies the time growth condition
is a sufficiently small constant and the nonlinear interaction term consists of cubic nonlinearities of derivative type
where and . We suppose that the initial data satifsfy the exponential decay conditions. Then we prove the sharp decay estimate , for all , where . Furthermore we show that for there exist the usual scattering states, when and the modified scattering states, when
where the coefficient satisfies the time growth condition
is a sufficiently small constant and the nonlinear interaction term consists of cubic nonlinearities of derivative type
where and . We suppose that the initial data satifsfy the exponential decay conditions. Then we prove the sharp decay estimate , for all , where . Furthermore we show that for there exist the usual scattering states, when and the modified scattering states, when
10.
Takeshi Okano 《Proceedings of the American Mathematical Society》2002,130(6):1603-1605
11.
Chunlei Liu 《Proceedings of the American Mathematical Society》2002,130(7):1887-1892
Let be a nontrivial Dirichlet character modulo an odd prime . Write
We shall prove
and, for complex ,
where is a constant depending only on .
We shall prove
and, for complex ,
0, \end{displaymath}">
where is a constant depending only on .
12.
Dan Popovici Zoltá n Sebestyé n 《Proceedings of the American Mathematical Society》2004,132(4):1029-1035
Given a family of vectors in a Hilbert space we characterize the existence of a family of commuting contractions on having regular dilation and such that
The theorem is a multi-dimensional analogue for some well-known operator moment problems due to Sebestyén in case or, recently, to Gavruta and Paunescu in case .
13.
Wojciech Matysiak Pawel J. Szablowski 《Transactions of the American Mathematical Society》2008,360(7):3909-3919
Existence, -stationarity and linearity of conditional expectations of square integrable random sequences satisfying for a real sequence is examined. The analysis is reliant upon the use of Laurent and Toeplitz operator techniques.
14.
Steven D. Taliaferro Lei Zhang 《Proceedings of the American Mathematical Society》2003,131(9):2895-2902
We study the conformal scalar curvature problem
where is a continuous function. We show that a necessary and sufficient condition on for this problem to have positive solutions which are arbitrarily large at is that be less than 1 on a sequence of points in which tends to .
where is a continuous function. We show that a necessary and sufficient condition on for this problem to have positive solutions which are arbitrarily large at is that be less than 1 on a sequence of points in which tends to .
15.
Nakao Hayashi Elena I. Kaikina Pavel I. Naumkin 《Transactions of the American Mathematical Society》2006,358(3):1165-1185
We study large time asymptotics of small solutions to the Cauchy problem for nonlinear damped wave equations with a critical nonlinearity
where 0,$"> and space dimensions . Assume that the initial data
where \frac{n}{2},$"> weighted Sobolev spaces are
Also we suppose that
where
Then we prove that there exists a positive such that the Cauchy problem above has a unique global solution satisfying the time decay property
for all 0,$"> where
where 0,$"> and space dimensions . Assume that the initial data
where \frac{n}{2},$"> weighted Sobolev spaces are
Also we suppose that
0,\int u_{0}\left( x\right) dx>0, \end{displaymath}">
where
Then we prove that there exists a positive such that the Cauchy problem above has a unique global solution satisfying the time decay property
for all 0,$"> where
16.
We prove that for integers 1,m\geq 1$"> and positive rationals the series
is irrational. Furthermore, if all the positive rationals are less than then the series
is also irrational.
is irrational. Furthermore, if all the positive rationals are less than then the series
is also irrational.
17.
Wolfgang Kö nig Peter Mö rters 《Transactions of the American Mathematical Society》2006,358(3):1223-1255
Consider independent Brownian motions in , each running up to its first exit time from an open domain , and their intersection local time as a measure on . We give a sharp criterion for the finiteness of exponential moments,
where are nonnegative, bounded functions with compact support in . We also derive a law of large numbers for intersection local time conditioned to have large total mass.
where are nonnegative, bounded functions with compact support in . We also derive a law of large numbers for intersection local time conditioned to have large total mass.
18.
We study doubly oscillatory integrals
and prove a sharp maximal estimate which is an immediate consequence of a well-known conjecture in Fourier analysis on .
and prove a sharp maximal estimate which is an immediate consequence of a well-known conjecture in Fourier analysis on .
19.
Norbert Peyerimhoff Evangelia Samiou 《Proceedings of the American Mathematical Society》2004,132(5):1525-1529
We show that the Cheeger isoperimetric constant of a solvable simply connected Lie group with Lie algebra is
20.
D. S. Lubinsky 《Proceedings of the American Mathematical Society》2001,129(12):3519-3529
Let 1$">, and be a rational function with numerator, denominator of degree , respectively. In several applications, one needs to know the size of the set such that for ,
In an earlier paper, we showed that
where denotes linear Lebesgue measure. Here we obtain, for each , the sharp version of this inequality in terms of condenser capacity. In particular, we show that as ,
In an earlier paper, we showed that
where denotes linear Lebesgue measure. Here we obtain, for each , the sharp version of this inequality in terms of condenser capacity. In particular, we show that as ,