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1.
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.
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2.
Thomas Eckl 《Geometriae Dedicata》2008,137(1):149-162
Using Dumnicki’s approach to showing non-specialty of linear systems consisting of plane curves with prescribed multiplicities
in sufficiently general points on we develop an asymptotic method to determine lower bounds for Seshadri constants of general points on . With this method we prove the lower bound for 10 general points on .
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3.
It is classically known that a real cubic surface in cannot have more than one solitary point (or -singularity, locally given by x
2 + y
2 + z
2 = 0) whereas it can have up to four nodes (or -singularity, locally given by x
2 + y
2 − z
2 = 0). We show that on any surface of degree d ≥ 3 in the maximum possible number of solitary points is strictly smaller than the maximum possible number of nodes. Conversely,
we adapt a construction of Chmutov to obtain surfaces with many solitary points by using a refined version of Brusotti’s Theorem.
Combining lower and upper bounds, we deduce: , where denotes the maximum possible number of solitary points on a real surface of degree d in . Finally, we adapt this construction to get real algebraic surfaces in with many singular points of type for all k ≥ 1.
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4.
We introduce a new existence result for compact normal geodesic graphs with constant mean curvature and boundary in a class
of warped product spaces. In particular, our result includes that of normal geodesic graphs with constant mean curvature in
hyperbolic space over a bounded domain in a totally geodesic .
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5.
We present a new distance characterization of Aleksandrov spaces of non-positive curvature. By introducing a quasilinearization
for abstract metric spaces we draw an analogy between characterization of Aleksandrov spaces and inner product spaces; the
quasi-inner product is defined by means of the quadrilateral cosine—a metric substitute for the angular measure between two
directions at different points. Our main result states that a geodesically connected metric space is an Aleksandrov domain (also known as a CAT(0) space) if and only if the quadrilateral cosine does not exceed one for every two pairs of
distinct points in . We also observe that a geodesically connected metric space is an domain if and only if, for every quadruple of points in , the quadrilateral inequality (known as Euler’s inequality in ) holds. As a corollary of our main result we give necessary and sufficient conditions for a semimetric space to be an domain. Our results provide a complete solution to the Curvature Problem posed by Gromov in the context of metric spaces
of non-positive curvature.
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6.
We consider cocycles over certain hyperbolic actions, , and show rigidity properties for cocycles with values in a Lie group or a diffeomorphism group, which are close to identity
on a set of generators, and are sufficiently smooth. The actions we consider are Cartan actions of or , for , and Γ torsion free cocompact lattice. The results in this paper rely on a technique developed recently by D. Damjanović
and A. Katok.
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7.
Let be the absolute Galois group of , and let T be the complete rooted d-ary tree, where d ≥ 2. In this article, we study “arboreal” representations of into the automorphism group of T, particularly in the case d = 2. In doing so, we propose a parallel to the well-developed and powerful theory of linear p-adic representations of . We first give some methods of constructing arboreal representations and discuss a few results of other authors concerning
their size in certain special cases. We then discuss the analogy between arboreal and linear representations of . Finally, we present some new examples and conjectures, particularly relating to the question of which subgroups of Aut(T) can occur as the image of an arboreal representation of .
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8.
Christopher Kennedy 《Algebras and Representation Theory》2006,9(5):525-537
Deep matrix algebras based on a set over a ring are introduced and studied by McCrimmon when has infinite cardinality. Here, we construct a new -module that is faithful for of any cardinality. For a field of arbitrary characteristic and , is studied in depth. The algebra is shown to possess a unique proper non-zero ideal, isomorphic to . This leads to a new and interesting simple algebra, , the quotient of by its unique ideal. Both and the quotient algebra are shown to have centers isomorphic to . Via the new faithful representation, all automorphisms of are shown to be inner in the sense of Definition 18.Presented by D. Passman. 相似文献
9.
Let be the homogeneous tree with degree q + 1 ≥ 3 and a finitely generated group whose Cayley graph is . The associated lamplighter group is the wreath product , where is a finite group. For a large class of random walks on this group, we prove almost sure convergence to a natural geometric
boundary. If the probability law governing the random walk has finite first moment, then the probability space formed by this
geometric boundary together with the limit distribution of the random walk is proved to be maximal, that is, the Poisson boundary.
We also prove that the Dirichlet problem at infinity is solvable for continuous functions on the active part of the boundary,
if the lamplighter “operates at bounded range”.
Supported by ESF program RDSES and by Austrian Science Fund (FWF) P15577. 相似文献
10.
11.
We prove that the properties of acting metrically properly on some space with walls or some CAT(0) cube complex are closed
by taking the wreath product with . We also give a lower bound for the (equivariant) Hilbert space compression of in terms of the (equivariant) Hilbert space compression of H.
A la mémoire de Michel Matthey 相似文献
12.
Let M be a four-holed sphere and Γ the mapping class group of M fixing the boundary ∂M. The group Γ acts on
which is the space of completely reducible SL (2,
-gauge equivalence classes of flat SL
-connections on M with fixed holonomy
on ∂M. Let
and
be the compact component of the real points of
. These points correspond to SU(2)-representations or SL(2,
-representations. The Γ-action preserves
and we study the topological dynamics of the Γ-action on
and show that for a dense set of holonomy
, the Γ-orbits are dense in
. We also produce a class of representations
such that the Γ-orbit of [ρ] is finite in the compact component of
, but
is dense in SL(2,
.Mathematics Subject Classiffications (2000). 57M05, 54H20, 11D99 相似文献
13.
A solution to the normalized Ricci flow is called non-singular if it exists for all time with uniformly bounded sectional
curvature. By using the techniques developed by the present authors [Ishida, The normalized Ricci flow on four-manifolds and
exotic smooth structures; Şuvaina, Einstein metrics and smooth structures on non-simply connected 4-manifolds] we prove that
for any finite cyclic group , where d > 1, there exist infinitely many compact topological 4-manifolds, with fundamental group , which admit at least one smooth structure for which non-singular solutions of the normalized Ricci flow exist, but also
admit infinitely many distinct smooth structures for which no non-singular solution of the normalized Ricci flow exists. We show that there are no non-singular -equivariant, d > 1, solutions to the normalized Ricci flow on appropriate connected sums of and . 相似文献
14.
J. S. Manhas 《Integral Equations and Operator Theory》2008,62(3):419-428
Let be the weighted Banach space of analytic functions with a topology generated by weighted sup-norm. In the present article,
we investigate the analytic mappings and which characterize the compactness of differences of two weighted composition operators on the space . As a consequence we characterize the compactness of differences of composition operators on weighted Bloch spaces.
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15.
Li-Ping Huang 《Geometriae Dedicata》2009,138(1):1-12
Let R, S be Bezout domains. Assume that n is an integer ≥ 3, 1 ≤ k ≤ n − 2. Denoted by the k-dimensional Grassmann space on . Let be a map. This paper proves the following are equivalent: (i) is an adjacency preserving bijection in both directions. (ii) is a diameter preserving bijection in both directions. Moreover, Chow’s theorem on Grassmann spaces over division rings is
extended to the case of Bezout domains: If is an adjacency preserving bijection in both directions, then is induced by either a collineation or the duality of a collineation.
Project 10671026 supported by National Natural Science Foundation of China. 相似文献
16.
We consider several kinds of partition relations on the set of real numbers and its powers, as well as their parameterizations with the set of all infinite sets of natural numbers, and show that they hold in some models of set theory. The proofs use generic absoluteness,
that is, absoluteness under the required forcing extensions. We show that Solovay models are absolute under those forcing
extensions, which yields, for instance, that in these models for every well ordered partition of there is a sequence of perfect sets whose product lies in one piece of the partition. Moreover, for every finite partition
of there is and a sequence of perfect sets such that the product lies in one piece of the partition, where is the set of all infinite subsets of X. The proofs yield the same results for Borel partitions in ZFC, and for more complex partitions in any model satisfying a certain degree of generic absoluteness.
This work was supported by the research projects MTM 2005-01025 of the Spanish Ministry of Science and Education and 2005SGR-00738
of the Generalitat de Catalunya. A substantial part of the work was carried out while the second-named author was ICREA Visiting
Professor at the Centre de Recerca Matemàtica in Bellaterra (Barcelona), and also during the first-named author’s stays at
the Instituto Venezolano de Investigaciones Científicas and the California Institute of Technology. The authors gratefully
acknowledge the support provided by these institutions. 相似文献
17.
Yu. I. Lyubich 《Designs, Codes and Cryptography》2009,51(1):21-31
It is shown that among all tight designs in , where is or , or (quaternions), only 5-designs in (Lyubich, Shatalora Geom Dedicata 86: 169–178, 2001) have irrational angle set. This is the only case of equal ranks of the
first and the last irreducible idempotent in the corresponding Bose-Mesner algebra.
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18.
Margherita Guida 《Ricerche di matematica》2008,57(1):159-167
In this paper we study the syzygy modules of a grid or a fat grid of . We compute the minimal free resolution for the ideal of a complete grid in , and we conjecture this resolution in . Moreover we compute the minimal free resolution for the ideal of an incomplete grid of . We also conjecture the minimal free resolution for the ideal of a fat complete grid in .
相似文献
19.
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
20.
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.
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