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1.
This paper deals with two main topics related to Diophantine approximation. Firstly, we show that if a point on an algebraic variety is approximable by rational vectors to a sufficiently large degree, the approximating vectors must lie in the topological closure of the rational points on the variety. In many interesting cases, in particular if the set of rational points on the variety is finite, this closure does not exceed the set of rational points on the variety itself. This result enables easier proofs of several known results as special cases. The proof can be generalized in some way and encourages to define a new exponent of simultaneous approximation. The second part of the paper is devoted to the study of this exponent. 相似文献
2.
Yu. V. Matiyasevich 《Proceedings of the Steklov Institute of Mathematics》2011,275(1):118-132
This survey presents various theorems (obtained mainly by specialists in mathematical logic and computability theory) stating
the impossibility of algorithms for solving certain Diophantine problems. Often the technique developed for obtaining such
“negative” results also allows one to prove many “positive” theorems on the possibility of formulating Diophantine problems
with special properties. This survey also lists a number of questions that remain open. 相似文献
3.
4.
A coincidence point of a pair of mappings is an element, at which these mappings take on the same value. Coincidence points of mappings of partially ordered spaces were studied by A.V. Arutyunov, E.S. Zhukovskiy, and S.E. Zhukovskiy (see Topology and its Applications, 2015, V. 179, No. 1, p. 13–33); they proved, in particular, that an orderly covering mapping and a monotone one, both acting from a partially ordered space to a partially ordered one, possess a coincidence point. In this paper, we study the existence of a coincidence point of a pair of mappings that act from a partially ordered space to a set, where no binary relation is defined, and, consequently, it is impossible to define covering and monotonicity properties of mappings. In order to study the mentioned problem, we define the notion of a “quasi-coincidence” point. We understand it as an element, for which there exists another element, which does not exceed the initial one and is such that the value of the first mapping at it equals the value of the second mapping at the initial element. It turns out that the following condition is sufficient for the existence of a coincidence point: any chain of “quasi-coincidence” points is bounded and has a lower bound, which also is a “quasi-coincidence” point. We give an example of mappings that satisfy the above requirements and do not allow the application of results obtained for coincidence points of orderly covering and monotone mappings. In addition, we give an interpretation of the stability notion for coincidence points of mappings that act in partially ordered spaces with respect to their small perturbations and establish the corresponding stability conditions. 相似文献
5.
Yong-Geun Oh 《Journal of Geometric Analysis》1997,7(2):305-327
In this paper, we study the structure ofJ-holomorphic discs in relation to the Fredholm theory of pseudo-holomorphic discs with totally real boundary conditions in
almost complex manifolds (M, J). We prove that anyJ-holomorphic disc with totally real boundary condition that is injective in the interior except at a discrete set of points,
which we call a “normalized disc,” must either have some boundary point that is regular and has multiplicity one, or satisfy
that its image forms a smooth immersed compact surface (without boundary) with a finite number of self-intersections and a
finite number of branch points. In the course of proving this theorem, we also prove several theorems on the local structure
of boundary points ofJ-holomorphic discs, and as an application we give a complete treatment of the transverslity result for Floer’s pseudo-holomorphic
trajectories for Lagrangian intersections in symplectic geometry.
This paper is supported in part by NSF Grant DMS 9215011. 相似文献
6.
Jean Bourgain 《Israel Journal of Mathematics》1992,79(2-3):193-206
In this paper, new results are obtained concerning the uniform approximation property (UAP) inL
p-spaces (p≠2,1,∞). First, it is shown that the “uniform approximation function” does not allow a polynomial estimate. This fact is rather
surprising since it disproves the analogy between UAP-features and the presence of “large” euclidian subspaces in the space
and its dual. The examples are translation invariant spaces on the Cantor group and this extra structure permits one to replace
the problem with statements about the nonexistence of certain multipliers in harmonic analysis. Secondly, it is proved that
the UAP-function has an exponential upper estimate (this was known forp=1, ∞). The argument uses Schauder’s fix point theorem. Its precise behaviour is left unclarified here. It appears as a difficult
question, even in the translation invariant context. 相似文献
7.
Giuseppe Tomassini 《Milan Journal of Mathematics》2007,75(1):399-412
We deal with the general problem of extension of analytic objects in a complex space X. After a short presentation of the classical results we discuss some recent developments obtained when X is a semi-1-corona. Semi-1-coronae are domains C
+ whose boundary is the union of a Levi flat part, a 1-pseudoconvex part and a 1-pseudoconcave part. Using the main result
in [31], we prove a “bump lemma” for compact semi-1-coronae in and then, applying Andreotti-Grauert theory, we get a cohomology finiteness theorem for coherent sheaves whose depth is at
least 3. As an application we get an extension theorem for coherent sheaves and analytic subsets.
Received: April 2007 相似文献
8.
V. Elser 《Discrete and Computational Geometry》2001,25(3):445-476
The level set of an elliptic function is a doubly periodic point set in ℂ. To obtain a wider spectrum of point sets, we consider,
more generally, a Riemann surface S immersed in ℂ2 and its sections (“cuts”) by ℂ More specifically, we consider surfaces S defined in terms of a fundamental surface element obtained as a conformai map of triangular domains in ℂ. The discrete group
of isometries of ℂ2 generated by reflections in the triangle edges leaves S invariant and generalizes double-periodicity. Our main result concerns the special case of maps of right triangles, with
the right angle being a regular point of the map. For this class of maps we show that only seven Riemann surfaces, when cut,
form point sets that are discrete in ℂ. Their isometry groups all have a rank 4 lattice subgroup, but only three of the corresponding
point sets are doubly periodic in ℂ. The remaining surfaces form quasiperiodic point sets closely related to the vertex sets
of quasiperiodic tilings. In fact, vertex sets of familiar tilings are recovered in all cases by applying the construction
to a piecewise flat approximation of the corresponding Riemann surface. The geometry of point sets formed by cuts of Riemann
surfaces is no less “rigid” than the geometry determined by a tiling, and has the distinct advantage in having a regular behavior
with respect to the complex parameter which specifies the cut. 相似文献
9.
Charles F. Osgood 《Journal of Fourier Analysis and Applications》2001,7(3):309-317
This article treats the problem of the approximation of an analytic function f on the unit disk by rational functions having
integral coefficients, with the goodness of each approximation being judged in terms of the maximum of the absolute values
of the coefficients of the rational function. This relates to the more usual approximation by a rational function in that
it could imply how many decimal places are needed when applying a particularly good rational function approximation having
non-integrad coefficients.
It is shown how to obtain “good” approximations of this type and it is also shown how under certain circumstances “very good”
bounds are not possible. As in diophantine approximation this means that many merely “good” approximations do exist, which
may be the preferable case. The existence or nonexistence of “very good” approximations is closely related to the diophantine
approximation of the first nonzero power series coefficient of at z=0. Nevanlinna theory methods are used in the proofs. 相似文献
10.
Marco Caldarelli Giovanni Catino Zindine Djadli Annibale Magni Carlo Mantegazza 《Geometriae Dedicata》2010,145(1):127-137
By means of a Kaluza–Klein type argument we show that the Perelman’s F{\mathcal{F}} -functional is the Einstein–Hilbert action in a space with extra “phantom” dimensions. In this way, we try to interpret some
remarks of Perelman in the introduction and at the end of the first section in his famous paper (Perelman in The entropy formula
for the Ricci flow and its geometric applications, 2002). As a consequence the Ricci flow (modified by a diffeomorphism and
a time-dependent factor) is the evolution of the “real” part of the metric under a constrained gradient flow of the Einstein–Hilbert
gravitational action in higher dimension. 相似文献
11.
Going beyond the linearized study has been a longstanding problem in the theory of Landau damping. In this paper we establish
exponential Landau damping in analytic regularity. The damping phenomenon is reinterpreted in terms of transfer of regularity
between kinetic and spatial variables, rather than exchanges of energy; phase mixing is the driving mechanism. The analysis
involves new families of analytic norms, measuring regularity by comparison with solutions of the free transport equation;
new functional inequalities; a control of non-linear echoes; sharp “deflection” estimates; and a Newton approximation scheme.
Our results hold for any potential no more singular than Coulomb or Newton interaction; the limit cases are included with
specific technical effort. As a side result, the stability of homogeneous equilibria of the non-linear Vlasov equation is
established under sharp assumptions. We point out the strong analogy with the KAM theory, and discuss physical implications.
Finally, we extend these results to some Gevrey (non-analytic) distribution functions. 相似文献
12.
David Glickenstein 《Discrete and Computational Geometry》2007,38(4):651-664
Given a triangulation of points in the plane and a function on the points, one may consider the Dirichlet energy, which is
related to the Dirichlet energy of a smooth function. In fact, the Dirichlet energy can be derived from a finite element approximation.
S. Rippa showed that the Dirichlet energy (which he refers to as the “roughness”) is minimized by the Delaunay triangulation
by showing that each edge flip which makes an edge Delaunay decreases the energy. In this paper, we introduce a Dirichlet
energy on a weighted triangulation which is a generalization of the energy on unweighted triangulations and an analogue of
the smooth Dirichlet energy on a domain. We show that this Dirichlet energy has the property that each edge flip which makes
an edge weighted Delaunay decreases the energy. The proof is done by a direct calculation, and so gives an alternate proof
of Rippa’s result. 相似文献
13.
In this paper, we explore the virtual technique that is very useful in studying the moduli problem from a differential geometric point of view. We introduce a class of new objects "virtual manifolds/orbifolds', on which we develop the integration theory. In particular, the virtual localization formula is obtained. 相似文献
14.
A dual variant of Benson’s “outer approximation algorithm” for multiple objective linear programming
Outcome space methods construct the set of nondominated points in the objective (outcome) space of a multiple objective linear
programme. In this paper, we employ results from geometric duality theory for multiple objective linear programmes to derive
a dual variant of Benson’s “outer approximation algorithm” to solve multiobjective linear programmes in objective space. We
also suggest some improvements of the original version of the algorithm and prove that solving the dual provides a weight
set decomposition. We compare both algorithms on small illustrative and on practically relevant examples. 相似文献
15.
Pengzi Miao Luen-Fai Tam 《Calculus of Variations and Partial Differential Equations》2009,36(2):141-171
We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive
a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space forms, on
which the standard metrics are critical points, are geodesic balls. In the zero scalar curvature case, assuming the boundary
can be isometrically embedded in the Euclidean space as a compact strictly convex hypersurface, we show that the volume of
a critical point is always no less than the Euclidean volume bounded by the isometric embedding of the boundary, and the two
volumes are equal if and only if the critical point is isometric to a standard Euclidean ball. We also derive a second variation
formula and apply it to show that, on Euclidean balls and “small” hyperbolic and spherical balls in dimensions 3 ≤ n ≤ 5, the standard space form metrics are indeed saddle points for the volume functional. 相似文献
16.
F. B. Pakovich 《Journal of Mathematical Sciences》2009,158(1):148-154
In this paper, in the context of the “dessins d’enfants” theory, we give a combinatorial criterion for a plane tree to cover
a tree from the classes of “chains” or “stars.” We also discuss some applications of this result that are related to the arithmetical
theory of torsion on curves.
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 6, pp. 207–215, 2007. 相似文献
17.
S. Yu. Dobrokhotov 《Theoretical and Mathematical Physics》1997,112(1):827-843
According to Maslov’s idea, many two-dimensional, quasilinear hyperbolic systems of partial differential equations admit only
three types of singularities that are in general position and have the property of “structure self-similarity and stability.”
Those are: shock waves, “narrow” solitons, and “square-root” point singularities (solitary vortices). Their propagation is
described by an infinite chain of ordinary differential equations (ODE) that generalize the well-known Hugoniot conditions
for shock waves. After some reasonable closure of the chain for the case of solitary vortices in the “shallow water” equations,
we obtain a nonlinear system of sixteen ODE, which is exactly equivalent to the (linear) Hill equation with a periodic potential.
This means that, in some approximations, the trajectory of a solitary vortex can be described by the Hill equation. This result
can be used to predict the trajectory of the vortex center if we know its observable part.
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 1, pp. 47–66. 相似文献
18.
Shyūichi Izumiya 《manuscripta mathematica》1979,28(4):337-360
In his paper [2], Bierstone proves the equivariant Gromov theorem which is an integrability theorem for “open regularity condition”
of equivariant sections of a smooth G-fibre bundle under the assumption that all orbit bundles of base manifold are non-closed.
Here, we prove the result without his assumption under a nice “open regularity condition” which we call “G-extensible”.
One of the examples of “G-extensible condition” is given by notions of Thom-Boardman singularities. 相似文献
19.
In the first part of the present paper, we established estimates for the rate of approach of the integrals of a family of
“physical” white noises to a family of Wiener processes. We use this result to establish the estimate for the rate of approach
of a family of solutions of ordinary differential equations perturbed by some “physical” white noises to a family of solutions
of the corresponding It? equations. We consider both the case where the coefficient of random perturbation is separated from
zero and the case where it is not separated from zero. 相似文献
20.
In this paper we develop the inhomogeneous metric theory of simultaneous Diophantine approximation on planar curves. Our results
naturally extend the homogeneous Khintchine and Jarník type theorems established in Beresnevich et al. (Ann Math 166(2):367–426,
2007) and Vaughan and Velani (Invent Math 166:103–124, 2006) and are the first of their kind. The key lies in obtaining essentially
the best possible results regarding the distribution of ‘shifted’ rational points near planar curves. 相似文献