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1.
: Higher-order recurrences Clemens Fuchs Attila Petho Robert F. Tichy 《Transactions of the American Mathematical Society》2003,355(11):4657-4681
Let be a field of characteristic and let be a linear recurring sequence of degree in defined by the initial terms and by the difference equation
with . Finally, let be an element of . In this paper we are giving fairly general conditions depending only on on , and on under which the Diophantine equation
has only finitely many solutions . Moreover, we are giving an upper bound for the number of solutions, which depends only on . This paper is a continuation of the work of the authors on this equation in the case of second-order linear recurring sequences.
with . Finally, let be an element of . In this paper we are giving fairly general conditions depending only on on , and on under which the Diophantine equation
has only finitely many solutions . Moreover, we are giving an upper bound for the number of solutions, which depends only on . This paper is a continuation of the work of the authors on this equation in the case of second-order linear recurring sequences.
2.
-estimates for stable integrals with drift Vladimir Kurenok 《Transactions of the American Mathematical Society》2008,360(2):925-938
Let be of the form where is a symmetric stable process of index with . We obtain various -estimates for the process . In particular, for and any measurable, nonnegative function we derive the inequality As an application of the obtained estimates, we prove the existence of solutions for the stochastic equation for any initial value .
3.
Zinoviy Grinshpun 《Proceedings of the American Mathematical Society》2003,131(5):1591-1600
We prove the following theorem. Any isometric operator , that acts from the Hilbert space with nonnegative weight to the Hilbert space with nonnegative weight , allows for the integral representation
where the kernels and satisfy certain conditions that are necessary and sufficient for these kernels to generate the corresponding isometric operators.
where the kernels and satisfy certain conditions that are necessary and sufficient for these kernels to generate the corresponding isometric operators.
4.
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.
5.
José Marí a Martell Carlos Pé rez Rodrigo Trujillo-Gonzá lez 《Transactions of the American Mathematical Society》2005,357(1):385-396
We show that the classical Hörmander condition, or analogously the -Hörmander condition, for singular integral operators is not sufficient to derive Coifman's inequality
where , is the Hardy-Littlewood maximal operator, is any weight and is a constant depending upon and the constant of . This estimate is well known to hold when is a Calderón-Zygmund operator.
where and where is an arbitrary weight. However, by a recent result due to A. Lerner, this inequality is satisfied whenever is a Calderón-Zygmund operator.
where , is the Hardy-Littlewood maximal operator, is any weight and is a constant depending upon and the constant of . This estimate is well known to hold when is a Calderón-Zygmund operator.
As a consequence we deduce that the following estimate does not hold:
where and where is an arbitrary weight. However, by a recent result due to A. Lerner, this inequality is satisfied whenever is a Calderón-Zygmund operator.
One of the main ingredients of the proof is a very general extrapolation theorem for weights.
6.
Lorenzo D'Ambrosio Enzo Mitidieri Stanislav I. Pohozaev 《Transactions of the American Mathematical Society》2006,358(2):893-910
Let be a possibly degenerate second order differential operator and let be its fundamental solution at ; here is a suitable distance. In this paper we study necessary and sufficient conditions for the weak solutions of on to satisfy the representation formula
then (R) holds with no growth assumptions on .
We prove that (R) holds provided is superlinear, without any assumption on the behavior of at infinity. On the other hand, if satisfies the condition
then (R) holds with no growth assumptions on .
7.
Prabhu Janakiraman 《Transactions of the American Mathematical Society》2006,358(5):1937-1952
The following limit result holds for the weak-type (1,1) constant of dilation-commuting singular integral operator in : for , , For the maximal operator , the corresponding result is
8.
Let be the radius of the largest disk covered after steps of a simple random walk. We prove that almost surely where denotes 3 iterations of the function. This is motivated by a question of Erdos and Taylor. We also obtain the analogous result for the Wiener sausage, refining a result of Meyre and Werner.
9.
Thomas Keilen 《Transactions of the American Mathematical Society》2003,355(9):3485-3512
In 1985 Joe Harris proved the long-standing claim of Severi that equisingular families of plane nodal curves are irreducible whenever they are nonempty. For families with more complicated singularities this is no longer true. Given a divisor on a smooth projective surface it thus makes sense to look for conditions which ensure that the family of irreducible curves in the linear system with precisely singular points of types is irreducible. Considering different surfaces, including general surfaces in and products of curves, we produce a sufficient condition of the type
where is some constant and some zero-dimensional scheme associated to the singularity type. Our results carry the same asymptotics as the best known results in this direction in the plane case, even though the coefficient is worse. For most of the surfaces considered these are the only known results in that direction.
where is some constant and some zero-dimensional scheme associated to the singularity type. Our results carry the same asymptotics as the best known results in this direction in the plane case, even though the coefficient is worse. For most of the surfaces considered these are the only known results in that direction.
10.
Irene Fonseca Giovanni Leoni Jan Malý Roberto Paroni 《Transactions of the American Mathematical Society》2002,354(9):3723-3741
Lower semicontinuity properties of multiple integrals
are studied when may grow linearly with respect to the highest-order derivative, and admissible sequences converge strongly in It is shown that under certain continuity assumptions on convexity, -quasiconvexity or -polyconvexity of
ensures lower semicontinuity. The case where is -quasiconvex remains open except in some very particular cases, such as when
are studied when may grow linearly with respect to the highest-order derivative, and admissible sequences converge strongly in It is shown that under certain continuity assumptions on convexity, -quasiconvexity or -polyconvexity of
ensures lower semicontinuity. The case where is -quasiconvex remains open except in some very particular cases, such as when
11.
Paul A. Binding Patrick J. Browne Bruce A. Watson 《Transactions of the American Mathematical Society》2002,354(10):4043-4065
The Sturm-Liouville equation
is considered subject to the boundary conditions
We assume that is positive and that is piecewise continuous and changes sign at its discontinuities. We give asymptotic approximations up to for , or equivalently up to for , the eigenvalues of the above boundary value problem.
is considered subject to the boundary conditions
We assume that is positive and that is piecewise continuous and changes sign at its discontinuities. We give asymptotic approximations up to for , or equivalently up to for , the eigenvalues of the above boundary value problem.
12.
Jacques Boulanger Jean-Luc Chabert 《Transactions of the American Mathematical Society》2004,356(12):5071-5088
Let be a cyclotomic field with ring of integers and let be a polynomial whose values on belong to . If the ideal of generated by the values of on is itself, then every algebraic integer of may be written in the following form:
for some integer , where the 's are roots of unity of . Moreover, there are two effective constants and such that the least integer (for a fixed ) is less than , where
for some integer , where the 's are roots of unity of . Moreover, there are two effective constants and such that the least integer (for a fixed ) is less than , where
13.
Given a real polynomial in one variable such that , we consider the maximal operator in ,
We prove that is bounded on for 1$"> with bounds that only depend on the degree of .
0\,,\,i,j\in \mathbb{Z}}\frac{1... ...t f\big (x_{1}-2^{i}p(t),x_{2}-2^{j}p(t)\big )\big \vert\,dt . \end{displaymath}">
We prove that is bounded on for 1$"> with bounds that only depend on the degree of .
14.
Peter Borwein Kwok-Kwong Stephen Choi 《Transactions of the American Mathematical Society》2002,354(1):219-234
We give explicit formulas for the norm (or equivalently for the merit factors) of various sequences of polynomials related to the Fekete polynomials
where is the Legendre symbol. For example for an odd prime,
where is the class number of . Similar explicit formulas are given for various polynomials including an example of Turyn's that is constructed by cyclically permuting the first quarter of the coefficients of . This is the sequence that has the largest known asymptotic merit factor. Explicitly,
where denotes the nearest integer, satisfies
where
Indeed we derive a closed form for the norm of all shifted Fekete polynomials
Namely
and if .
where is the Legendre symbol. For example for an odd prime,
where is the class number of . Similar explicit formulas are given for various polynomials including an example of Turyn's that is constructed by cyclically permuting the first quarter of the coefficients of . This is the sequence that has the largest known asymptotic merit factor. Explicitly,
where denotes the nearest integer, satisfies
where
Indeed we derive a closed form for the norm of all shifted Fekete polynomials
Namely
and if .
15.
Thomas Keilen 《Transactions of the American Mathematical Society》2005,357(6):2467-2481
Francesco Severi (1921) showed that equisingular families of plane nodal curves are T-smooth, i.e. smooth of the expected dimension, whenever they are non-empty. For families with more complicated singularities this is no longer true. Given a divisor on a smooth projective surface it thus makes sense to look for conditions which ensure that the family of irreducible curves in the linear system with precisely singular points of types is T-smooth. Considering different surfaces including the projective plane, general surfaces in , products of curves and geometrically ruled surfaces, we produce a sufficient condition of the type
where is some invariant of the singularity type and is some constant. This generalises the results of Greuel, Lossen, and Shustin (2001) for the plane case, combining their methods and the method of Bogomolov instability. For many singularity types the -invariant leads to essentially better conditions than the invariants used by Greuel, Lossen, and Shustin (1997), and for most classes of geometrically ruled surfaces our results are the first known for T-smoothness at all.
where is some invariant of the singularity type and is some constant. This generalises the results of Greuel, Lossen, and Shustin (2001) for the plane case, combining their methods and the method of Bogomolov instability. For many singularity types the -invariant leads to essentially better conditions than the invariants used by Greuel, Lossen, and Shustin (1997), and for most classes of geometrically ruled surfaces our results are the first known for T-smoothness at all.
16.
Hichem Hajaiej 《Transactions of the American Mathematical Society》2005,357(12):4885-4896
We establish conditions under which the extended Hardy-Little- wood inequality
where each is non-negative and denotes its Schwarz symmetrization, holds. We also determine appropriate monotonicity assumptions on such that equality occurs in the above inequality if and only if each is Schwarz symmetric. We end this paper with some applications of our results in the calculus of variations and partial differential equations.
where each is non-negative and denotes its Schwarz symmetrization, holds. We also determine appropriate monotonicity assumptions on such that equality occurs in the above inequality if and only if each is Schwarz symmetric. We end this paper with some applications of our results in the calculus of variations and partial differential equations.
17.
W. R. Madych 《Transactions of the American Mathematical Society》2003,355(3):1109-1133
Given distinct real numbers and a positive approximation of the identity , which converges weakly to the Dirac delta measure as goes to zero, we investigate the polynomials which solve the interpolation problem
with prescribed data . More specifically, we are interested in the behavior of when the data is of the form for some prescribed function . One of our results asserts that if is sufficiently nice and has sufficiently well-behaved moments, then converges to a limit which can be completely characterized. As an application we identify the limits of certain fundamental interpolatory splines whose knot set is , where is an arbitrary finite subset of the integer lattice , as their degree goes to infinity.
with prescribed data . More specifically, we are interested in the behavior of when the data is of the form for some prescribed function . One of our results asserts that if is sufficiently nice and has sufficiently well-behaved moments, then converges to a limit which can be completely characterized. As an application we identify the limits of certain fundamental interpolatory splines whose knot set is , where is an arbitrary finite subset of the integer lattice , as their degree goes to infinity.
18.
-Laplacian with absorption Xinfu Chen Yuanwei Qi Mingxin Wang 《Transactions of the American Mathematical Society》2007,359(11):5653-5668
We consider, for and , the -Laplacian evolution equation with absorption We are interested in those solutions, which we call singular solutions, that are non-negative, non-trivial, continuous in , and satisfy for all . We prove the following:
- (i)
- When , there does not exist any such singular solution.
- (ii)
- When , there exists, for every , a unique singular solution that satisfies as .
Also, as , where is a singular solution that satisfies as .
Furthermore, any singular solution is either or for some finite positive .
19.
Monica Conti Susanna Terracini Gianmaria Verzini 《Transactions of the American Mathematical Society》2004,356(8):3283-3300
We present a new min-max approach to the search of multiple -periodic solutions to a class of fourth order equations
where is continuous, -periodic in and satisfies a superlinearity assumption when . For every , we prove the existence of a -periodic solution having exactly zeroes in .
where is continuous, -periodic in and satisfies a superlinearity assumption when . For every , we prove the existence of a -periodic solution having exactly zeroes in .
20.
Steven Dale Cutkosky Jü rgen Herzog Ana Reguera 《Transactions of the American Mathematical Society》2004,356(5):1833-1874
Let be a resolution of singularities of a normal surface singularity , with integral exceptional divisors . We consider the Poincaré series
where
We show that if has characteristic zero and is a semi-abelian variety, then the Poincaré series is rational. However, we give examples to show that this series can be irrational if either of these conditions fails.
where
We show that if has characteristic zero and is a semi-abelian variety, then the Poincaré series is rational. However, we give examples to show that this series can be irrational if either of these conditions fails.