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1.
Rikard Olofsson 《Annales Henri Poincare》2010,11(7):1285-1302
We study the eigenfunctions of the quantized cat map, desymmetrized by Hecke operators. In the papers (Olofsson in Ann Henri
Poincaré 10(6):1111–1139, 2009; Math Phys 286(3):1051–1072, 2009) it was observed that when the inverse of Planck’s constant is a prime exponent N = p
n
, with n > 2, half of these eigenfunctions become large at some points, and half remains small for all points. In this paper we study
the large eigenfunctions more carefully. In particular, we answer the question of for which q the L
q
norms remain bounded as N goes to infinity. The answer is q ≤ 4. 相似文献
2.
T. A. Mel’nyk 《Ukrainian Mathematical Journal》2006,58(2):220-243
A spectral boundary-value problem is considered in a plane thick two-level junction Ωε formed as the union of a domain Ω0 and a large number 2N of thin rods with thickness of order ε = O(N
−1). The thin rods are split into two levels depending on their length. In addition, the thin rods from the indicated levels
are ε-periodically alternating. The Fourier conditions are given on the lateral boundaries of the thin rods. The asymptotic
behavior of the eigenvalues and eigenfunctions is investigated as ε → 0, i.e., when the number of thin rods infinitely increases
and their thickness approaches zero. The Hausdorff convergence of the spectrum is proved as ε → 0, the leading terms of asymptotics
are constructed, and the corresponding asymptotic estimates are justified for the eigenvalues and eigenfunctions.
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 195–216, February, 2006. 相似文献
3.
4.
Dong-Hyun Kim 《Acta Mathematica Hungarica》2001,90(1-2):75-83
We consider the asymptotic behavior of the distribution functions defined by FN(z)=1―N{1≦ n ≦ N : f(n)≦ z (mod 1)} in the case when f is q-additive. We give necessary and sufficient conditions for a q-additive function to have a uniform distribution modulo 1 or to have a non-uniform distribution modulo 1.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
5.
A. Sankaranarayanan 《Lithuanian Mathematical Journal》2006,46(4):459-474
We improve the existing upper bound for the quantity |∑
n⩽x
a(n
2)|, where a(n
2) is the n
2th Hecke eigenvalue of a normalized holomorphic cusp form (Hecke eigenform) of the full modular group SL(2, ℤ), whenever the weight of the original holomorphic cusp form (Hecke eigenform) lies in a certain range.
Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 4, pp. 565–583, October–December, 2006. 相似文献
6.
David Carlton 《manuscripta mathematica》2001,105(2):201-234
We study the moduli surface for pairs of elliptic curves together with an isomorphism between their N-torsion groups. The Weil pairing gives a “determinant” map from this moduli surface to (Z/N
Z)*; its fibers are the components of the surface. We define spaces of modular forms on these components and Hecke correspondences
between them, and study how those spaces of modular forms behave as modules for the Hecke algebra. We discover that the component
with determinant −1 is somehow the “dominant” one; we characterize the difference between its spaces of modular forms and
the spaces of modular forms on the other components using forms with complex multiplication. In addition, we prove Atkin–Lehner-style
results about these spaces of modular forms. Finally, we show some simplifications that arise when N is prime, including a complete determination of such CM-forms, and give numerical examples.
Received: 20 September 2000 / Revised version: 7 February 2001 相似文献
7.
Igor E. Shparlinski 《Archiv der Mathematik》2005,85(6):508-513
We give lower bounds on the number of distinct values of the Ramanujan function τ(n), n ≦ x, and on the number of distinct residues of τ(n), n ≦ x, modulo a prime ℓ. We also show that for any prime ℓ the values τ(n), n ≦ ℓ4, form a finite additive basis modulo ℓ.
Received: 6 October 2004 相似文献
8.
We study the spectrum of the boundary-value problem for the Laplace operator in a thin domain Ω(ε) obtained by small perturbation
of the cylinder Ω(ε)=ω×(-ε/2.ε/2) ⊂ ℝ3in a neighborhood of the lateral surface. The Dirichlet condition is imposed on the bases of the cylinder, and the Dirichlet
condition or the Neumann condition is imposed on the remaining part of ∂Ω(ε). We construct and justify asymptotic formulas
(as ε→+0) for eigenvalues and eigenfunctions. In view of a special form of the lateral surface, there are eigenfunctions of
boundary-layer type that exponentially decrease far from the lateral surface. For the mixed boundary-value problem such a
localization is possible in neighborhoods of local maxima of the curvature of the contour ∂ω. This property of eigenfunctions
is a characteristic feature of the first points of the spectrum (in particular, the first eigenvalue) and, under the passage
from Ω(h)() to Ω(h), the spectrum itself has perturbation O(h−2). Bibliography: 29 titles.
Translated fromProblemy Matematicheskogo Analiza, No. 19, 1999, pp. 105–149. 相似文献
9.
Rikard Olofsson 《Annales Henri Poincare》2009,10(6):1111-1139
This paper continues the work done in Olofsson [Commun Math Phys 286(3):1051–1072, 2009] about the supremum norm of eigenfunctions of desymmetrized quantized cat maps. N will denote the inverse of Planck’s constant and we will see that the arithmetic properties of N play an important role. We prove the sharp estimate ||ψ||∞ = O(N 1/4) for all normalized eigenfunctions and all N outside of a small exceptional set. We are also able to calculate the value of the supremum norms for most of the so called newforms. For a given N = p n , with n > 2, the newforms can be divided in two parts (leaving out a small number of them in some cases), the first half all have supremum norm about ${2/\sqrt{1\pm 1/p}}This paper continues the work done in Olofsson [Commun Math Phys 286(3):1051–1072, 2009] about the supremum norm of eigenfunctions
of desymmetrized quantized cat maps. N will denote the inverse of Planck’s constant and we will see that the arithmetic properties of N play an important role. We prove the sharp estimate ||ψ||∞ = O(N
1/4) for all normalized eigenfunctions and all N outside of a small exceptional set. We are also able to calculate the value of the supremum norms for most of the so called
newforms. For a given N = p
n
, with n > 2, the newforms can be divided in two parts (leaving out a small number of them in some cases), the first half all have
supremum norm about 2/?{1±1/p}{2/\sqrt{1\pm 1/p}} and the supremum norm of the newforms in the second half have at most three different values, all of the order N
1/6. The only dependence of A is that the normalization factor is different if A has eigenvectors modulo p or not. We also calculate the joint value distribution of the absolute value of n different newforms. 相似文献
10.
Miguel Angel Javaloyes Levi Lopes de Lima Paolo Piccione 《Mathematische Zeitschrift》2008,260(2):277-303
Following the lines of Bott in (Commun Pure Appl Math 9:171–206, 1956), we study the Morse index of the iterates of a closed
geodesic in stationary Lorentzian manifolds, or, more generally, of a closed Lorentzian geodesic that admits a timelike periodic
Jacobi field. Given one such closed geodesic γ, we prove the existence of a locally constant integer valued map Λγ on the unit circle with the property that the Morse index of the iterated γ
N
is equal, up to a correction term εγ∈{0,1}, to the sum of the values of Λγ at the N-th roots of unity. The discontinuities of Λγ occur at a finite number of points of the unit circle, that are special eigenvalues of the linearized Poincaré map of γ.
We discuss some applications of the theory. 相似文献
11.
We present a short and direct proof (based on the Pontryagin-Thom construction) of the following Pontryagin-Steenrod-Wu theorem:
(a) LetM be a connected orientable closed smooth (n + 1)-manifold,n≥3. Define the degree map deg: π
n
(M) →H
n
(M; ℤ) by the formula degf =f*[S
n
], where [S
n
] εH
n
(M; ℤ) is the fundamental class. The degree map is bijective, if there existsβ εH
2(M, ℤ/2ℤ) such thatβ ·w
2(M) ε 0. If suchβ does not exist, then deg is a 2-1 map; and (b) LetM be an orientable closed smooth (n+2)-manifold,n≥3. An elementα lies in the image of the degree map if and only ifρ
2
α ·w
2(M)=0, whereρ
2: ℤ → ℤ/2ℤ is reduction modulo 2. 相似文献
12.
S. N. M. Ruijsenaars 《Theoretical and Mathematical Physics》2006,146(1):25-33
Letting Al(x) denote the commuting analytic difference operators of elliptic relativistic Calogero-Moser type, we present and study
zero-eigenvalue eigenfunctions for the operators Al(x) − Al(−y) (with l = 1, 2,..., N and x, y ∈ ℂ
N). The eigenfunctions are products of elliptic gamma functions. They are invariant under permutations of x1,..., xN and y1,..., yN and under interchange of the step-size parameters.
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 31–41, January, 2006. 相似文献
13.
We investigate the adjoints of linear fractional composition operators Cφ acting on classical Dirichlet space D(BN ) in the unit ball BN of CN , and characterize the normality and essential normality of Cφ on D(BN ) and the Dirichlet space modulo constant function D0(BN ), where φ is a linear fractional map of BN . In addition, we also show that for any non-elliptic linear fractional map φ of BN , the composition maps σ ο φ and φ ο σ are elliptic or parabolic linear fractional maps of BN . 相似文献
14.
Sebahattin Ikikardes Recep Sahin I. Naci Cangul 《Bulletin of the Brazilian Mathematical Society》2009,40(4):479-494
In this paper, first, we determine the quotient groups of the Hecke groups H(λ
q
), where q ≥ 7 is prime, by their principal congruence subgroups H
p
(λ
q
) oflevel p, where p is also prime. We deal with the case of q = 7 separately, because of its close relation with the Hurwitz groups. Then, using the obtained results, we find the principal
congruence subgroups of the extended Hecke groups $
\overline H
$
\overline H
(λ
q
) for q ≥ 5 prime. Finally, we show that some of the quotient groups of the Hecke group H(λ
q
) and the extended Hecke group $
\overline H
$
\overline H
(λ
q
), q ≥ 5 prime, by their principal congruence subgroups H
p
(λ
q
) are M*-groups. 相似文献
15.
V. A. Bykovskii 《Journal of Mathematical Sciences》1998,89(1):915-932
A trace formula expressing the mean values of the form (k=2,3,...)
via certain arithmetic means on the group Г0(N1) is proved. Here the sum is taken over a normalized orthogonal basis in the space of holomorphic cusp forms of weight 2k
with respect to Г0(N1). By H
f
(x)
(s) we denote the Hecke series of the form f, twisted with the primitive character χ (mod N2), and λf(d), (d, N1N2)=1, are the eigenvalues of the Hecke operators
. The trace formula is used for obtaining the estimate
for the newform f for all ε>0, l=0,1,2,.... This improves the known result (Duke-Friedlander-Iwaniec, 1993) with upper bound
(1+|t|)2N
2
1/2−1/22+ε
on the right-hand side. As a corollary, we obtain the estimate
for the Fourier coefficients of holomorphic cusp forms of weight k+1/2, which improves Iwaniec' result (1987) with exponent
1/4–1/28+ε. Bibliography: 25 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 14–36. 相似文献
16.
The asymptotics of the solution to the Neumann spectral problem in a domain of the “dense-comb” type
Convergence theorems and asymptotic estimates (as ε → 0) are proved for the eigenvalues and the eigenfunctions of the Neumann
problem in a dense singular junction Ω
ɛ
of a domain Ω0 and a large number N of thin cylinders with thickness of order ε=lN−1, where l is the total length of common boundaries for Ω0 and the cylinders in question. Bibliography: 27 titles.
We dedicate the present paper to Olga Arsenievna Oleinik, as a symbol of our deep respect and gratitude
Translated from Trudy Seminara imeni I G. Petrovskogo, No. 19. pp. 000-000. 0000. 相似文献
17.
We consider residue fields of primes in the well-known fragment of arithmeticIΔ0+Ω1. We prove that each such residue field has exactly one extension of each degree. The standard proofs use counting and the
Frobenius map. Since little is known about these topics in fragments, we looked for, and found, another proof using permutation
groups and the elements of Galois cohomology. This proof fits nicely intoIΔ0 + Ω1 using, instead of exponentiation, exponentiation modulo a prime. 相似文献
18.
T. Christiansen 《Journal d'Analyse Mathématique》1998,76(1):1-43
We study the operatorH = -c
2
x,y)Μx,y)∇ · Μ
-1
(x,y)∇, wherec andΜ are perturbations of functionsc
0(y) andΜ
0(y) which depend only on the one-dimensional variabley. In particular, we study the spatial asymptotics of limε↺0(H - (λ +iε)2)-1 applied to functions which have compact support or are otherwise well-behaved at infinity and relate the scattering matrix
to the asymptotics of the generalized eigenfunctions. We then prove a trace formula for the operatorH in terms of the scattering phase, and, in a very special situation, use the trace formula to find spectral asymptotics forH.
Partially supported by an NSF Postdoctoral Fellowship and the University of Missouri Research Board. 相似文献
19.
Nearrings here are right nearrings. LetN be a nearring and fix an element α εN. Form another nearring Nα by taking addition to be the same as the addition inN but define the productx ⊗y of two elementsx, y ε Nα byx ⊗y =xay. The nearring Nα is referred to as a laminated nearring ofN andN is referred to as the base nearring. The element α is called the laminating element or the laminator. An elementx of a nearingN is a left zero ifxy =x for ally εN. A homomorphismϕ from a nearringN
1 into a nearringN
2 is a left zero covering homomorphism if for each left zeroy εN
2,ϕ(x) =y for somex εN
1. The left zero covering homomorphisms from one laminated nearring into another are investigated where the base nearring is
the nearring of all continuous selfmaps of the Euclidean group ℝ2 under pointwise addition and composition and the laminators are complex polynomials. Finally, it is shown that one can determine
whether or not two such laminated nearrings are isomorphic simply by inspecting the coefficients of the two laminating polynomials. 相似文献
20.
We consider a new way of establishing Navier wall laws. Considering a bounded domain Ω of R
N
, N=2,3, surrounded by a thin layer Σ
ε
, along a part Γ2 of its boundary ∂Ω, we consider a Navier-Stokes flow in Ω∪∂Ω∪Σ
ε
with Reynolds’ number of order 1/ε in Σ
ε
. Using Γ-convergence arguments, we describe the asymptotic behaviour of the solution of this problem and get a general Navier
law involving a matrix of Borel measures having the same support contained in the interface Γ2. We then consider two special cases where we characterize this matrix of measures. As a further application, we consider
an optimal control problem within this context. 相似文献