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1.
Georg Stamou 《Journal of Geometry》1983,20(1):155-168
Let S be a congruence of straight lines in an isotropic space of degree one. i.e. a three dimensional real affine space with the metric ds2=dx2+dy2. The purpose of this paper is to discuss ruled surfaces of the congruence S for which the parameter of distribution has a constant value δ (δ — surfaces). We study the curves of the middle surface and the focal surfaces (if there exist) of S which are base curves of δ — surfaces.Moreover we consider two congruences S and S′ which are “M-equivalent” and we investigate a correspondence between δ — surfaces of S and S′. 相似文献
2.
Stylianos S. Stamatakis 《Journal of Geometry》1991,42(1-2):148-161
Stephanidis has introduced the third quadratic differential form III of a rectilinear congruence S in the real 3-dimensional Euclidean space [6]. In the present paper we prove that the form III can be expressed in terms of the first and the second quadratic form of the congruence S and of the second fundamental form of the middle enveloppe M(u,v) of S. Further, we study the congruences, for which the III-principle surfaces coincide with some well-known ruled surfaces of the congruence S.
Herrn Professor Dr. Walter Benz zum 60. Geburtstag gewidmet 相似文献
Herrn Professor Dr. Walter Benz zum 60. Geburtstag gewidmet 相似文献
3.
C. Mendes 《Algebra Universalis》2005,54(3):279-290
An Ockham algebra
that satisfies the identity
is called a Kn, m-algebra. Generalizing some results obtained in [2], J. Varlet and T. Blyth, in [3, Chapter 8], study congruences on K1, 1-algebras. In particular, they describe the complement (when it exists) of a principal congruence and characterize these congruences
that are complemented. In this paper we study the same question for Kn, m-algebras.
Received March 24, 2005; accepted in final form April 28, 2005. 相似文献
4.
The variety pOconsists of those algebras (L;?,?,f,*,0,1) where (L;?,?,f,0,1) is an Ockham algebra, (L;?,?,f,*,0,1) is a p-algebra, and the unary operations fand *. commute. For an algebra in pK ωwe show that the compact congruences form a dual Stone lattice and use this to determine necessary and sufficient conditions for a principal congruence to be complemented. We also describe the lattice of subvarieties of pK 1,1identifying therein the biggest subvariety in which every principal congruence is complemented, and the biggest subvariety in which the intersection of two principal congruences is principal. 相似文献
5.
《Differential Geometry and its Applications》1999,10(1):1-26
The space of lines in R3 can be viewed as a four dimensional homogeneous space of the group of Euclidean motions, E(3). Line congruences arise in the classical method of transforming one surface to another by lines. These transformations are particularly interesting if some geometric property of the original surface is preserved. Line congruences, then, are two parameter families of lines and can be studied as surfaces in the space of lines. In this paper, we use the method of moving frames to study line congruences. We calculate the first order invariants of line congruences for which there are two real focal surfaces, and give the geometric meaning of these invariants. We look specifically at the case where the two first order invariants are constant and give a simple proof of Bäcklund's Theorem which relates to the transformation of one pseudospherical surface, a surface of constant negative Gaussian curvature, to another. These transformations are of interest since pseudospherical surfaces correspond to solutions to the sine-Gordon equation. We also give a proof of Bianchi's permutability theorem for pseudospherical surfaces in this context. Finally, we use the results of these theorems to generate some pseudospherical surfaces. All of these concepts and results are understood in terms of the structure equations of the line congruence. 相似文献
6.
Georg Stamou 《manuscripta mathematica》1975,15(4):329-340
In this paper we investigate a class Σ* of congruences which have common spherical image and which are mainly derived by suitable linear transformations from a given congruence S. A necessary and sufficient condition is given so that a congruence S* of Σ* has a common middle envelope with S. Further we investigate the correspondence between special ruled surfaces of a congruence S* and special curves on the middle envelope of the congruence S. 相似文献
7.
Boris Odehnal 《Journal of Geometry》2004,81(1-2):126-138
The aim of this paper is to show a way to find an explicit parametrization of rational isotropic congruences of lines in Euclidean three-space
It will be shown that also the focal surfaces of these congruences admit a rational parametrization. Furthermore, the close relation of isotropic congruences of lines to minimal surfaces will be used to find the related polynomial minimal surfaces. 相似文献
8.
The existence is proved of two new families of sextic threefolds in , which are not quadratically normal. These threefolds arise naturally in the realm of first order congruences of lines as
focal loci and in the study of the completely exceptional Monge–Ampère equations. One of these families comes from a smooth
congruence of multidegree (1, 3, 3) which is a smooth Fano fourfold of index two and genus 9.
相似文献
9.
William J. Keith 《Discrete Mathematics》2009,309(16):4960-4968
The distribution of values of the full ranks of marked Durfee symbols is examined in prime and nonprime arithmetic progressions. The relative populations of different residues for the same modulus are determined: the primary result is that k-marked Durfee symbols of n equally populate the residue classes a and bmod2k+1 if gcd(a,2k+1)=gcd(b,2k+1). These are used to construct a few congruences. The general procedure is illustrated with a particular theorem on 4-marked symbols for multiples of 3. 相似文献
10.
Consider oriented surfaces immersed in
. Associated to them,
here are studied pairs of transversal foliations with
singularities, defined on the Elliptic region, where the
Gaussian curvature
, given
by the product of the principal curvatures
k
1,
k
2 is
positive. The leaves of the foliations are the
lines of harmonic mean
curvature, also called characteristic or
diagonal lines, along which
the normal curvature of the immersion is given by
, where
is the
arithmetic mean curvature. That is,
is the harmonic mean of the
principal curvatures k
1,
k
2 of
the immersion. The singularities of the foliations are the
umbilic points and
parabolic curves, where
k
1 =
k
2 and
, respectively.Here are determined the structurally stable patterns of
harmonic mean curvature lines
near the umbilic points, parabolic
curves and harmonic mean
curvature cycles, the periodic leaves of the
foliations. The genericity of these patterns is
established.This provides the three essential local ingredients to
establish sufficient conditions, likely to be also necessary,
for Harmonic Mean Curvature Structural
Stability of immersed surfaces. This study, outlined
towards the end of the paper, is a natural analog and complement
for that carried out previously by the authors for the
Arithmetic Mean Curvature and
the Asymptotic Structural
Stability of immersed surfaces, [13, 14, 17], and
also extended recently to the case of the
Geometric Mean Curvature
Configuration [15].The first author was partially supported by FUNAPE/UFG.
Both authors are fellows of CNPq.
This work was done under the project PRONEX/FINEP/MCT -
Conv. 76.97.1080.00 - Teoria Qualitativa das Equações Diferenciais
Ordinárias and CNPq - Grant 476886/2001-5. 相似文献
11.
Making use of deep results from group theory we prove that if a finite algebra has permutable congruences and its congruence lattice is Mn, then n-1 is a prime power. 相似文献
12.
Ernest X. W. Xia 《The Ramanujan Journal》2018,45(2):331-348
The notion of broken k-diamond partitions was introduced by Andrews and Paule. Let \(\Delta _k(n)\) denote the number of broken k-diamond partitions of n for a fixed positive integer k. Recently, a number of parity results satisfied by \(\Delta _k(n)\) for small values of k have been proved by Radu and Sellers and others. However, congruences modulo 4 for \(\Delta _k(n)\) are unknown. In this paper, we will prove five congruences modulo 4 for \(\Delta _5(n)\), four infinite families of congruences modulo 4 for \(\Delta _7(n)\) and one congruence modulo 4 for \(\Delta _{11}(n)\) by employing theta function identities. Furthermore, we will prove a new parity result for \(\Delta _2(n)\). 相似文献
13.
N. P. Koneva 《Ukrainian Mathematical Journal》1991,43(11):1442-1447
In a biaxial space of the hyperbolic type in the case when Laplace transformations of a normal congruence are normal congruences the following theorem is proved: a focal net of lines on all focal surfaces of a sequence of Laplace transformations is a net R and the point and tangential Darboux invariants on all focal surfaces of the sequence of Laplace transformations, respectively equal to one another.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 11, pp. 1551–1557, November, 1991. 相似文献
14.
A graph is (k, d)-colorable if one can color the vertices with k colors such that no vertex is adjacent to more than d vertices of its same color. In this paper we investigate the existence of such colorings in surfaces and the complexity of coloring problems. It is shown that a toroidal graph is (3, 2)- and (5, 1)-colorable, and that a graph of genus γ is (χγ/(d + 1) + 4, d)-colorable, where χγ is the maximum chromatic number of a graph embeddable on the surface of genus γ. It is shown that the (2, k)-coloring, for k ≥ 1, and the (3, 1)-coloring problems are NP-complete even for planar graphs. In general graphs (k, d)-coloring is NP-complete for k ≥ 3, d ≥ 0. The tightness is considered. Also, generalizations to defects of several algorithms for approximate (proper) coloring are presented. © 1997 John Wiley & Sons, Inc. 相似文献
15.
A note on the congruence \left( {_{mp^k }^{np^k } } \right) \equiv \left( {_m^n } \right) (mod p
r
)
Romeo Me?trovi? 《Czechoslovak Mathematical Journal》2012,62(1):59-65
In the paper we discuss the following type congruences: $$\left( {_{mp^k }^{np^k } } \right) \equiv \left( {_m^n } \right)(\bmod p^r ),$$ where p is a prime, n, m, k and r are various positive integers with n ? m ? 1, k ? 1 and r ? 1. Given positive integers k and r, denote by W(k, r) the set of all primes p such that the above congruence holds for every pair of integers n ? m ? 1. Using Ljunggren’s and Jacobsthal’s type congruences, we establish several characterizations of sets W(k, r) and inclusion relations between them for various values k and r. In particular, we prove that W(k + i, r) = W(k ? 1, r) for all k ? 2, i ? 0 and 3 ? r ? 3k, and W(k, r) = W(1, r) for all 3 ? r ? 6 and k ? 2. We also noticed that some of these properties may be used for computational purposes related to congruences given above. 相似文献
16.
By the method of synthetic geometry, we define a seemingly new transformation of a three-dimensional projective space where
the corresponding points lie on the rays of the first order, nth class congruence C
n
1 and are conjugate with respect to a proper quadric Ψ. We prove that this transformation maps a straight line onto an n + 2 order space curve and a plane onto an n + 2 order surface which contains an n-ple (i.e. n-multiple) straight line. It is shown that in the Euclidean space the pedal surfaces of the congruences C
n
1 can be obtained by this transformation. The analytical approach enables new visualizations of the resulting curves and surfaces
with the program Mathematica. They are shown in four examples.
相似文献
17.
LetC be the normalization of an integral plane curve of degreed with δ ordinary nodes or cusps as its singularities. If δ=0, then Namba proved that there is no linear seriesg
d
−2/1
and that everyg
d
−1/1
is cut out by a pencil of lines passing through a point onC. The main purpose of this paper is to generalize his result to the case δ>0. A typical one is as follows: Ifd≥2(k+1), and δ<kd−(k+1)2+3 for somek>0, thenC has no linear seriesg
d
−3/1
. We also show that ifd≥2k+3 and δ<kd−(k+1)2+2, then each linear seriesg
d
−2/1
onC is cut out by a pencil of lines. We have similar results forg
d
−1/1
andg
2d
−9/1
. Furthermore, we also show that all of our theorems are sharp. 相似文献
18.
Andrzej Walendziak 《Czechoslovak Mathematical Journal》2002,52(1):117-127
Here we consider the weak congruence lattice
of an algebra
with the congruence extension property (the CEP for short) and the weak congruence intersection property (briefly the WCIP). In the first section we give necessary and sufficient conditions for the semimodularity of that lattice. In the second part we characterize algebras whose weak congruences form complemented lattices. 相似文献
19.
20.
Morris Newman 《Linear and Multilinear Algebra》2013,61(4):363-366
Let Rbe a principal ideal ringRn the ring of n× nmatrices over R, and dk (A) the kth determinantal divisor of Afor 1 ? k? n, where Ais any element of Rn , It is shown that if A,BεRn , det(A) det(B:) ≠ 0, then dk (AB) ≡ 0 mod dk (A) dk (B). If in addition (det(A), det(B)) = 1, then it is also shown that dk (AB) = dk (A) dk (B). This provides a new proof of the multiplicativity of the Smith normal form for matrices with relatively prime determinants. 相似文献