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1.
Agraph G is a diameter graph in ?d if its vertex set is a finite subset in ?d of diameter 1 and edges join pairs of vertices a unit distance apart. It is shown that if a diameter graph G in ?4 contains the complete subgraph K on five vertices, then any triangle in G shares a vertex with K. The geometric interpretation of this statement is as follows. Given any regular unit simplex on five vertices and any regular unit triangle in ?4, then either the simplex and the triangle have a common vertex or the diameter of the union of their vertex sets is strictly greater than 1. 相似文献
3.
For the equation $$L[u]: = K(y)u_{xx} + u_{yy} + r(x,y)u = f(x,y)$$ ( K (y)?0 whenever y?0) in G, bounded by a piecewise smooth curve Γ 0 for y>0 which intersects the line y=0 at the points A(?1, 0) and B(1, 0) and for y<0 by a smooth curve Γ 1 through A which meets the characteristic of (1) through B at the point P, the uniqueness of the Frankl-Morawetz problem is proved without assuming that Γ 1 is monotone. 相似文献
5.
Summary We present here some new families of non conforming finite elements in 3. These two families of finite elements, built on tetrahedrons or on cubes are respectively conforming in the spaces H(curl) and H(div). We give some applications of these elements for the approximation of Maxwell's equations and equations of elasticity.First, we introduce some notations
K
is a tetrahedron or a cube, the volume of which is
- K
is its boundary
-
f
is a face of K, the surface of which is
-
a
is an edge, the length of which is
-
L
2
(K)
is the usual Hilbert space of square integrable functions defined on K
-
H
m
(K)
{ L
2( K); L
2( K); || m}, where =( 1, 2, 3) is a multi-index; ||= 1+ 2+ 3
- curl u
u, (defined by using the distributional derivative) for u=( u
1, u
2, u
3); u
iL
2
(K)
-
H(curl)
{ u( L
2
(K)) 3; curl u( L
2
(K))
3}
- div u
· u
-
H(div)
{ u( L
2
(K))
3; div uL
2
(K)}
-
D
k
u
is the k-th differential operator associated to u, which is a ( k+1)-multilinear operator acting on 3
-
k
is an index
-
k
is the linear space of polynomials, the degree of which is less or equal to k
-
k
is the group of all permutations of the set {1, 2, ..., k}
-
c or c
will stand for any constant depending possibly on 相似文献
6.
Let χ( S r n?1 )) be the minimum number of colours needed to colour the points of a sphere S r n?1 of radius $r \geqslant \tfrac{1} {2}$ in ? n so that any two points at the distance 1 apart receive different colours. In 1981 P. Erd?s conjectured that χ( S r n?1 )→∞ for all $r \geqslant \tfrac{1} {2}$ . This conjecture was proved in 1983 by L. Lovász who showed in [11] that χ( S r n?1 ) ≥ n. In the same paper, Lovász claimed that if $r < \sqrt {\frac{n} {{2n + 2}}}$ , then χ( S r n?1 ) ≤ n+1, and he conjectured that χ( S r n?1 ) grows exponentially, provided $r \geqslant \sqrt {\frac{n} {{2n + 2}}}$ . In this paper, we show that Lovász’ claim is wrong and his conjecture is true: actually we prove that the quantity χ( S r n?1 ) grows exponentially for any $r > \tfrac{1} {2}$ . 相似文献
7.
Heilbronn conjectured that given arbitrary
n points in the 2-dimensional
unit square [0, 1] 2, there must be
three points which form a triangle of area at most
O(1/ n2). This
conjecture was disproved by a nonconstructive argument of
Komlós, Pintz and Szemerédi [10] who showed that for every
n there is a configuration of
n points in the unit square
[0, 1] 2 where all triangles have area
at least (log
n/ n2).
Considering a generalization of this problem to dimensions
d3, Barequet [3] showed for
every n the existence of
n points in the
d-dimensional unit cube [0,
1] d such that the minimum
volume of every simplex spanned by any ( d+1) of these n points is at least
(1/ nd). We improve on this
lower bound by a logarithmic factor (log n). 相似文献
8.
Russian Mathematics - We construct a reduction of the three-dimensional Darboux system for the Christoffel symbols which describes conjugate curvilinear coordinate systems. The reduction is... 相似文献
9.
We prove that for any expansive n×n integral matrix A with |det A|=2, there exist A-dilation minimally supported frequency
(MSF) wavelets that are associated with a multiresolution analysis (MRA). The condition |det A|=2 was known to be necessary,
and we prove that it is sufficient. A wavelet set is the support set of the Fourier transform of an MSF wavelet. We give some
concrete examples of MRA wavelet sets in the plane. The same technique of proof is also applied to yield an existence result
for A-dilation MRA subspace wavelets. 相似文献
10.
For a convex curve in an even-dimensional affine space we introduce a series of convex domains (called Young hulls) describe their structure and give a formulas for the volume of the biggest of these domains.To our Teacher Vladimir Igorevich Arnold on the occasion of his 60-th birthday 相似文献
11.
The so-called first selection lemma states the following: given any set P of n points in ℝ
d
, there exists a point in ℝ
d
contained in at least c
d
n
d+1− O( n
d
) simplices spanned by P, where the constant c
d
depends on d. We present improved bounds on the first selection lemma in ℝ 3. In particular, we prove that c
3≥0.00227, improving the previous best result of c
3≥0.00162 by Wagner (On k-sets and applications. Ph.D. thesis, ETH Zurich, 2003). This makes progress, for the three-dimensional case, on the open problems of Bukh et al. (Stabbing simplices by points
and flats. Discrete Comput. Geom., 2010) (where it is proven that c
3≤1/4 4≈0.00390) and Boros and Füredi (The number of triangles covering the center of an n-set. Geom. Dedic. 17(1):69–77, 1984) (where the two-dimensional case was settled). 相似文献
12.
We study degenerate homogeneous affine surfaces in 3. It is proved that such a surface is either an open part of a plane, a cylinder on an ellipse, parabola or hyperbola or of the surface given by xz – 1/2 y
2=0. 相似文献
13.
In the short note of 1927, Urysohn constructed the metric space R that is nowhere locally separable. There is no publication with indications that R is a (noncomplete) ?-tree that has valency c at each point. The author in 1989, as well as Polterovich and Shnirelman in 1997, constructed ?-trees isometric to R unaware of the paper by Urysohn. In this paper the author considers various constructions of the ?-tree R and of the minimal complete ?-tree of valency c including R, as well as the characterizations of ?-trees, their properties, and connections with ultrametric spaces. 相似文献
14.
It is proved in this article that any generalized solution of a sufficiently general class of elliptic-type differential inequalities in
n
that is non-negative almost everywhere in
n
and vanishes almost everywhere on an open set
n
is trivial in
n
. 相似文献
15.
Biharmonic functions are defined on Euclidean spaces, Riemannian manifolds, infinite trees, and more generally on abstract harmonic spaces. In this note, we consider biharmonic functions b defined on annular sets Ω \ K and obtain Laurent-type decompositions for b in the Euclidean spaces and in infinite trees. Particular importance is given to the investigation when b extends as a distribution on Ω. 相似文献
16.
We prove some uniform and pointwise gradient estimates for the Dirichlet and the Neumann evolution operators \(G_{\mathcal {D}}(t,s)\) and \(G_{\mathcal {N}}(t,s)\) associated with a class of nonautonomous elliptic operators ( t) with unbounded coefficients defined in I× \(\mathbb{R}_{+}\) (where I is a right-halfline or I= ?). We also prove the existence and the uniqueness of a tight evolution system of measures \(\left \{\mu _{t}^{\mathcal {N}}\right \}_{t \in I}\) associated with \(G_{\mathcal {N}}(t,s)\) , which turns out to be sub-invariant for \(G_{\mathcal {D}}(t,s)\) , and we study the asymptotic behaviour of the evolution operators \(G_{\mathcal {D}}(t,s)\) and \(G_{\mathcal {N}}(t,s)\) in the L p -spaces related to the system \(\left \{\mu _{t}^{\mathcal {N}}\right \}_{t \in I}\) . 相似文献
17.
In this paper the problem of G
2 continuous interpolation of curves in
d
by polynomial splines of degree n is studied. The interpolation of the data points and two tangent directions at the boundary is considered. The case n = r + 2 = d, where r is the number of interior points interpolated by each segment of the spline curve, is studied in detail. It is shown that the problem is uniquely solvable asymptotically, e., when the data points are sampled regularly and sufficiently dense, and lie on a regular, convex parametric curve in
d
. In this case the optimal approximation order is also determined. 相似文献
18.
Doklady Mathematics - Suppose that an object t moves within a given corridor Y in the presence of a groups S of hostile observers S ∉ Y, each having a fixed visibility cone K(S). The problem... 相似文献
19.
A class of a posteriori parameter choice strategies for the operator version of Tiknonov regularization (including variants of Morozov's and Arcangeli's methods) is proposed and used in investigating the rate of convergence of the regularized solution for ill-posed nonlinear equation involving a monotone operator in Banach space. 相似文献
20.
OnaProblemofHaymanChenHuaihui(陈怀惠)FangMingliang(方明亮)(DepartmentofMathematics,NanjingNormalUniversity,Nanjing,Jiangsu,210024)C... 相似文献
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