Search results for "Geometry"
showing 10 items of 4487 documents
Periodic Orbits in the Isosceles Three-Body Problem
1991
The Saturn’s satellites Janus and Epimetheus are the first known bodies in the Solar System that has horseshoe orbits in a frame that rotates with uniform angular velocity. Both satellites have similar masses and orbital elements when they are far from one another. Moreover, their orbits are nearly symmetric. In fact, in the past, they have been identify as a unique satellite and afterwards, some mathematical theories about their orbits has been necessaries to understand why they do not collide. In particular, the interest in planar three-body problem with two small masses has increased6. We assume that the two small masses have similar symmetric initial conditions. The aim of this paper is…
“Ill-Conditioned” Vertices
1970
The round-off errors tend to increase particularly rapidly after pivoting at “ill-conditioned” vertices. Those vertices where two or more hyperplanes, each representing one constraint, intersect at a very slight angle are considered as “ill-conditioned”. An “ill-conditioned” vertex is for instance given by the intersection of the two constraints: $$\eqalign{ & 3\,{{\rm{x}}_{\rm{1}}}\, + \,{{\rm{x}}_{\rm{2}}}\, \le \,6 \cr & {{\rm{x}}_{\rm{1}}}\, + \,.354\,{{\rm{x}}_{\rm{2}}}\, \le \,2.001 \cr} $$
Lengths of radii under conformal maps of the unit disc
1999
If E f ( R ) E_{f}(R) is the set of endpoints of radii which have length greater than or equal to R > 0 R>0 under a conformal map f f of the unit disc, then cap E f ( R ) = O ( R − 1 / 2 ) \operatorname {cap} E_{f}(R)=O(R^{-1/2}) as R → ∞ R\to \infty for the logarithmic capacity of E f ( R ) E_{f}(R) . The exponent − 1 / 2 -1/2 is sharp.
Quasi-Modes in Higher Dimension
2019
Recall that if a(x, ξ) and b(x, ξ) are two C1-functions defined on some domain in \({\mathbf {R}}^{2n}_{x,\xi }\), then we can define the Poisson bracket to be the C0-function on the same domain given by $$\displaystyle \{ a,b\} =a^{\prime }_\xi \cdot b^{\prime }_x-a^{\prime }_x \cdot b^{\prime }_\xi =H_a(b). $$ Here \(H_a=a^{\prime }_\xi \cdot \partial _x-a^{\prime }_x\cdot \partial _\xi \) denotes the Hamilton vector field of a. The following result is due to Zworski, who obtained it via a semi-classical reduction from the above mentioned result of Hormander. A direct proof was given in Dencker et al. and here we give a variant. We will assume some familiarity with symplectic geometry.
h analogue of Newton's binomial formula
1998
In this letter, the $h$--analogue of Newton's binomial formula is obtained in the $h$--deformed quantum plane which does not have any $q$--analogue. For $h=0$, this is just the usual one as it should be. Furthermore, the binomial coefficients reduce to $\frac{n!}{(n-k)!}$ for $h=1$. \\ Some properties of the $h$--binomial coefficients are also given. \\ Finally, I hope that such results will contribute to an introduction of the $h$--analogue of the well--known functions, $h$--special functions and $h$--deformed analysis.
Historical Notes on Star Geometry in Mathematics, Art and Nature
2018
Gamma: “I can. Look at this Counterexample 3: a star-polyhedron I shall call it urchin. This consists of 12 star-pentagons. It has 12 vertices, 30 edges, and 12 pentagonal faces-you may check it if you like by counting. Thus the Descartes-Euler thesis is not true at all, since for this polyhedron \(V - E + F = - 6\)”. Delta: “Why do you think that your ‘urchin’ is a polyhedron?” Gamma: “Do you not see? This is a polyhedron, whose faces are the twelve star-pentagons”. Delta: “But then you do not even know what a polygon is! A star-pentagon is certainly not a polygon!”
Numerical bounds for semi-stable families of curves or of certain higher-dimensional manifolds
2005
Given an open subset U U of a projective curve Y Y and a smooth family f : V → U f:V\to U of curves, with semi-stable reduction over Y Y , we show that for a subvariation V \mathbb {V} of Hodge structures of R 1 f ∗ C V R^1f_*\mathbb {C}_V with rank ( V ) > 2 \textrm {rank} (\mathbb {V})>2 the Arakelov inequality must be strict. For families of n n -folds we prove a similar result under the assumption that the ( n , 0 ) (n,0) component of the Higgs bundle of V \mathbb {V} defines a birational map.
A note on coverings with special fibres and monodromy group $ S_{d}$
2012
We consider branched coverings of degree over with monodromy group , points of simple branching, special points and fixed branching data at the special points, where is a smooth connected complex projective curve of genus , and , are integers with . We prove that the corresponding Hurwitz spaces are irreducible if .
Der Satz von Tits für PGL2(R), R ein kommutativer Ring vom stabilen Rang 2
1996
Certain permutation groups on sets with distance relation are characterized as groups of projectivities PGL2(R) on the projective line over a commutative ring R of stable rank 2, thus generalizing a classical result of Tits where R is a field.
The Influence of H. Grassmann on Italian Projective N-Dimensional Geometry
1996
On May 29, 1883, Corrado Segre took his doctorate in Turin (Torino), under Enrico D’Ovidio’s guidance. His thesis (Segre 1884a,b) was published one year later in the Journal of the local Academy of Science, and after a short time it became a fundamental starting point for the development of Italian projective n-dimensional geometry.