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1.
It has been known for a long time that the Deligne–Lusztig curves associated to the algebraic groups of type and defined over the finite field all have the maximum number of -rational points allowed by the Weil “explicit formulas”, and that these curves are -maximal curves over infinitely many algebraic extensions of . Serre showed that an -rational curve which is -covered by an -maximal curve is also -maximal. This has posed the problem of the existence of -maximal curves other than the Deligne–Lusztig curves and their -subcovers, see for instance Garcia (On curves with many rational points over finite fields. In: Finite Fields with Applications
to Coding Theory, Cryptography and Related Areas, pp. 152–163. Springer, Berlin, 2002) and Garcia and Stichtenoth (A maximal
curve which is not a Galois subcover of the Hermitan curve. Bull. Braz. Math. Soc. (N.S.) 37, 139–152, 2006). In this paper, a positive answer to this problem is obtained. For every q = n
3 with n = p
r
> 2, p ≥ 2 prime, we give a simple, explicit construction of an -maximal curve that is not -covered by any -maximal Deligne–Lusztig curve. Furthermore, the -automorphism group Aut has size n
3(n
3 + 1)(n
2 − 1)(n
2 − n + 1). Interestingly, has a very large -automorphism group with respect to its genus .
Research supported by the Italian Ministry MURST, Strutture geometriche, combinatoria e loro applicazioni, PRIN 2006–2007. 相似文献
2.
J. Bagherian Ilia Ponomarenko A. Rahnamai Barghi 《Journal of Algebraic Combinatorics》2008,27(2):173-185
We introduce a concept of cyclotomic association scheme over a finite near-field
. It is proved that any isomorphism of two such nontrivial schemes is induced by a suitable element of the group AGL(V), where V is the linear space associated with
. A sufficient condition on a cyclotomic scheme
that guarantee the inclusion
where
is a finite field with
elements, is given.
I. Ponomarenko partially supported by RFFI, grants 03-01-00349, NSH-2251.2003.1. 相似文献
3.
John P. Steinberger 《Results in Mathematics》2008,51(3-4):319-338
If is any ring or semi-ring (e.g., ) and G is a finite abelian group, two elements a, b of the group (semi-)ring are said to form a factorization of G if ab = rΣ
g∈G
g for some . A factorization is called quasiperiodic if there is some element g ∈ G of order m > 1 such that either a or b – say b – can be written as a sum b
0 + ... + b
m−1 of m elements of such that ab
h
= g
h
ab
0 for h = 0, ... , m − 1. Hajós [5] conjectured that all factorizations are quasiperiodic when and r = 1 but Sands [15] found a counterexample for the group . Here we show however that all factorizations of abelian groups are quasiperiodic when and that all factorizations of cyclic groups or of groups of the type are quasiperiodic when . We also give some new examples of non-quasiperiodic factorizations with for the smaller groups and .
Received: May 12, 2006. Revised: October 3, 2007. 相似文献
4.
In 1960 Hughes and Kleinfeld (Am J Math 82:389–392, 1960) constructed a finite semifield which is two-dimensional over a weak
nucleus, given an automorphism σ of a finite field and elements with the property that has no roots in . In 1965 Knuth (J Algebra 2:182–217, 1965) constructed a further three finite semifields which are also two-dimensional over
a weak nucleus, given the same parameter set . Moreover, in the same article, Knuth describes operations that allow one to obtain up to six semifields from a given semifield.
We show how these operations in fact relate these four finite semifields, for a fixed parameter set, and yield at most five
non-isotopic semifields out of a possible 24. These five semifields form two sets of semifields, one of which consists of
at most two non-isotopic semifields related by Knuth operations and the other of which consists of at most three non-isotopic
semifields.
相似文献
5.
Let be a field and q be a nonzero element of that is not a root of unity. We give a criterion for 〈0〉 to be a primitive ideal of the algebra of quantum matrices. Next, we describe all height one primes of ; these two problems are actually interlinked since it turns out that 〈0〉 is a primitive ideal of whenever has only finitely many height one primes. Finally, we compute the automorphism group of in the case where m ≠ n. In order to do this, we first study the action of this group on the prime spectrum of . Then, by using the preferred basis of and PBW bases, we prove that the automorphism group of is isomorphic to the torus when m ≠ n and (m,n) ≠ (1, 3),(3, 1).
This research was supported by a Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme
and by Leverhulme Research Interchange Grant F/00158/X. 相似文献
6.
Jean-Paul Bézivin 《manuscripta mathematica》2008,126(1):41-47
Résumé Let with |q| > 1, and a be a rational number such that a
2 is not equal to for . In this note, we prove that the sum is irrational. 相似文献
7.
Let (V, g) be a Riemannian manifold and let be the isometric immersion operator which, to a map , associates the induced metric on V, where denotes the Euclidean scalar product in . By Nash–Gromov implicit function theorem is infinitesimally invertible over the space of free maps. In this paper we study non-free isometric immersions . We show that the operator (where denotes the space of C
∞- smooth quadratic forms on ) is infinitesimally invertible over a non-empty open subset of and therefore is an open map in the respective fine topologies.
相似文献
8.
Saugata Basu 《Discrete and Computational Geometry》2008,40(4):481-503
Let
be an o-minimal structure over ℝ,
a closed definable set, and
the projection maps as depicted below:
For any collection
of subsets of
, and
, let
denote the collection of subsets of
where
. We prove that there exists a constant C=C(T)>0 such that for any family
of definable sets, where each A
i
=π
1(T∩π
2−1(y
i
)), for some y
i
∈ℝ
ℓ
, the number of distinct stable homotopy types amongst the arrangements
is bounded by
while the number of distinct homotopy types is bounded by
This generalizes to the o-minimal setting, bounds of the same type proved in Basu and Vorobjov (J. Lond. Math. Soc. (2) 76(3):757–776,
2007) for semi-algebraic and semi-Pfaffian families. One technical tool used in the proof of the above results is a pair of topological
comparison theorems reminiscent of Helly’s theorem in convexity theory. These theorems might be of independent interest in
the quantitative study of arrangements.
The author was supported in part by NSF grant CCF-0634907. 相似文献
9.
Gregg Musiker 《Journal of Algebraic Combinatorics》2009,30(2):255-276
Let q be a power of a prime, and E be an elliptic curve defined over
. Such curves have a classical group structure, and one can form an infinite tower of groups by considering E over field extensions
for all k≥1. The critical group of a graph may be defined as the cokernel of L(G), the Laplacian matrix of G. In this paper, we compare elliptic curve groups with the critical groups of a certain family of graphs. This collection
of critical groups also decomposes into towers of subgroups, and we highlight additional comparisons by using the Frobenius
map of E over
.
This work was partially supported by the NSF, grant DMS-0500557 during the author’s graduate school at the University of California,
San Diego, and partially supported by an NSF Postdoctoral Fellowship. 相似文献
10.
In this paper we offer a computational approach to the spectral function for a finite family of commuting operators, and give
applications. Motivated by questions in wavelets and in signal processing, we study a problem about spectral concentration
of integral translations of functions in the Hilbert space . Our approach applies more generally to families of n arbitrary commuting unitary operators in a complex Hilbert space , or equivalent the spectral theory of a unitary representation U of the rank-n lattice in . Starting with a non-zero vector , we look for relations among the vectors in the cyclic subspace in generated by ψ. Since these vectors involve infinite “linear combinations,” the problem arises of giving geometric characterizations of these non-trivial linear
relations. A special case of the problem arose initially in work of Kolmogorov under the name L
2-independence. This refers to infinite linear combinations of integral translates of a fixed function with l
2-coefficients. While we were motivated by the study of translation operators arising in wavelet and frame theory, we stress
that our present results are general; our theorems are about spectral densities for general unitary operators, and for stochastic
integrals.
Work supported in part by the U.S. National Science Foundation. 相似文献
11.
Let be the classical kernel density estimator based on a kernel K and n independent random vectors X
i
each distributed according to an absolutely continuous law on . It is shown that the processes , , converge in law in the Banach space , for many interesting classes of functions or sets, some -Donsker, some just -pregaussian. The conditions allow for the classical bandwidths h
n
that simultaneously ensure optimal rates of convergence of the kernel density estimator in mean integrated squared error,
thus showing that, subject to some natural conditions, kernel density estimators are ‘plug-in’ estimators in the sense of
Bickel and Ritov (Ann Statist 31:1033–1053, 2003). Some new results on the uniform central limit theorem for smoothed empirical
processes, needed in the proofs, are also included.
相似文献
12.
In the study of the asymptotic behaviour of solutions of differential-difference equations the
-spectrum has been useful, where
and
implies Fourier transform
, with
given
, φ∈L
∞(ℝ,X), X a Banach space,
(half)line. Here we study
and related concepts, give relations between them, especially
weak Laplace half-line spectrum of φ, and thus ⊂ classical Beurling spectrum = Carleman spectrum =
; also
= Beurling spectrum of “φ modulo
” (Chill-Fasangova). If
satisfies a Loomis type condition (L
U
), then
countable and
uniformly continuous ∈U are shown to imply
; here (L
U
) usually means
, indefinite integral Pf of f in U imply Pf in
(the Bohl-Bohr theorem for
= almost periodic functions, U=bounded functions). This spectral characterization and other results are extended to unbounded functions via mean classes
, ℳ
m
U ((2.1) below) and even to distributions, generalizing various recent results for uniformly continuous bounded φ. Furthermore for solutions of convolution systems S*φ=b with
in some
we show
. With these above results, one gets generalizations of earlier results on the asymptotic behaviour of solutions of neutral
integro-differential-difference systems. Also many examples and special cases are discussed. 相似文献
13.
Emanuele Delucchi 《Journal of Algebraic Combinatorics》2007,26(4):477-494
Given a finite group G and a natural number n, we study the structure of the complex of nested sets of the associated Dowling lattice
(Proc. Internat. Sympos., 1971, pp. 101–115) and of its subposet of the G-symmetric partitions
which was recently introduced by Hultman (, 2006), together with the complex of G-symmetric phylogenetic trees
. Hultman shows that the complexes
and
are homotopy equivalent and Cohen–Macaulay, and determines the rank of their top homology.
An application of the theory of building sets and nested set complexes by Feichtner and Kozlov (Selecta Math. (N.S.)
10, 37–60, 2004) shows that in fact
is subdivided by the order complex of
. We introduce the complex of Dowling trees
and prove that it is subdivided by the order complex of
. Application of a theorem of Feichtner and Sturmfels (Port. Math. (N.S.)
62, 437–468, 2005) shows that, as a simplicial complex,
is in fact isomorphic to the Bergman complex of the associated Dowling geometry.
Topologically, we prove that
is obtained from
by successive coning over certain subcomplexes. It is well known that
is shellable, and of the same dimension as
. We explicitly and independently calculate how many homology spheres are added in passing from
to
. Comparison with work of Gottlieb and Wachs (Adv. Appl. Math.
24(4), 301–336, 2000) shows that
is intimely related to the representation theory of the top homology of
.
Research partially supported by the Swiss National Science Foundation, project PP002-106403/1. 相似文献
14.
We study cyclicity of operators on a separable Banach space which admit a bicyclic vector such that the norms of its images
under the iterates of the operator satisfy certain growth conditions. A simple consequence of our main result is that a bicyclic
unitary operator on a Banach space with separable dual is cyclic. Our results also imply that if is the shift operator acting on the weighted space of sequences , if the weight ω satisfies some regularity conditions and ω(n) = 1 for nonnegative n, then S is cyclic if . On the other hand one can see that S is not cyclic if the series diverges. We show that the question of Herrero whether either S or S* is cyclic on admits a positive answer when the series is convergent. We also prove completeness results for translates in certain Banach spaces of functions on . 相似文献
15.
16.
Ameer Athavale 《Complex Analysis and Operator Theory》2008,2(3):417-428
Let be a strictly pseudoconvex bounded domain in with C
2 boundary . If a subnormal m-tuple T of Hilbert space operators has the spectral measure of its minimal normal extension N supported on , then T is referred to as a -isometry. Using some non-trivial approximation theorems in the theory of several complex variables, we establish a commutant
lifting theorem for those -isometries whose (joint) Taylor spectra are contained in a special superdomain Ω of . Further, we provide a function-theoretic characterization of those subnormal tuples whose Taylor spectra are contained in
Ω and that are quasisimilar to a certain (fixed) -isometry T (of which the multiplication tuple on the Hardy space of the unit ball in is a rather special example).
Submitted: September 9, 2007. Revised: October 10, 2007. Accepted: October 24, 2007. 相似文献
17.
Euisung Park 《Mathematische Zeitschrift》2007,256(3):685-697
In this article we study nondegenerate projective curves of degree d which are not arithmetically Cohen-Macaulay. Note that for a rational normal curve and a point . Our main result is about the relation between the geometric properties of X and the position of P with respect to . We show that the graded Betti numbers of X are uniquely determined by the rank of P with respect to . In particular, X satisfies property N
2,p
if and only if . Therefore property N
2,p
of X is controlled by and conversely can be read off from the minimal free resolution of X. This result provides a non-linearly normal example for which the converse to Theorem 1.1 in (Eisenbud et al., Compositio
Math 141:1460–1478, 2005) holds. Also our result implies that for nondegenerate projective curves of degree d which are not arithmetically Cohen–Macaulay, there are exactly distinct Betti tables. 相似文献
18.
S. De Winter 《Journal of Algebraic Combinatorics》2006,24(3):285-297
Let be a proper partial geometry pg(s,t,2), and let G be an abelian group of automorphisms of acting regularly on the points of . Then either t≡2±od
s+1 or is a pg(5,5,2) isomorphic to the partial geometry of van Lint and Schrijver (Combinatorica 1 (1981), 63–73). This result is a new step towards the classification of partial geometries with an abelian Singer group and further provides an interesting characterization of the geometry of van Lint and Schrijver.The author is Postdoctoral Fellow of the Fund for Scientific Research Flanders (FWO-Vlaanderen). 相似文献
19.
Bent functions have many applications in the fields of coding theory, communications and cryptography. This paper studies
the constructions of bent functions having the form
for odd n and
for even n, over the finite field
of odd characteristic p, where
. Based on the irreducibility of some polynomials on
, we focus on characterizing the bent functions for n=p
v
q
r
and n=2p
v
q
r
, where
is an odd prime and p a primitive root modulo q
2. Moreover, the enumerations of those functions are also considered.
Partially supported by the NSF of China under Grants No. 60603012 and No. 60573053. 相似文献
20.
Let k 1 and
be a system of rational functions forming a strongly linearly independent set over a finite field
. Let
be arbitrarily prescribed elements. We prove that for all sufficiently large extensions
, there is an element
of prescribed order such that
is the relative trace map from
onto
We give some applications to BCH codes, finite field arithmetic and ordered orthogonal arrays. We also solve a question of Helleseth et~al. (Hypercubic 4 and 5-designs from Double-Error-Correcting codes, Des. Codes. Cryptgr. 28(2003). pp. 265–282) completely.classification 11T30, 11G20, 05B15 相似文献