Multi-valued solutions and branch point singularities for nonlinear hyperbolic or elliptic systems

Multi-valued solutions are constructed for 2 × 2 first-order systems using a generalization of the hodograph transformation. The solution is found as a complex analytic function on a complex Riemann surface for which the branch points move as part of the solution. The branch point singularities are envelopes for the characteristics and thus move at the characteristic speeds. We perform an analysis of stability of these singularities with respect to perturbations of the initial data. The generic singularity types are folds, cusps, and nondegenerate umbilic points with non-zero 3-jet. An isolated singularity is generically a square root branch point corresponding to a fold. Two types of collisions between singularities are generic: At a “tangential” collision between two singularities moving at the same characteristic speed, a cube root branch point is formed, corresponding to a cusp. A “non-tangential” collision, between two square root branch points moving at different characteristic speeds, remains a square root branch point at the collision and corresponds to a nondegenerate umbilic point. These results are also valid for a diagonalizable n-th order system for which there are exactly two speeds. © 1993 John Wiley & Sons, Inc.     

方阵的开方问题

运用Jordan标准形理论,完整解决了复数方阵、实数方阵能否开平方、开立方的问题.     

Colorings of plane graphs: A survey

After a brief historical account, a few simple structural theorems about plane graphs useful for coloring are stated, and two simple applications of discharging are given. Afterwards, the following types of proper colorings of plane graphs are discussed, both in their classical and choosability (list coloring) versions: simultaneous colorings of vertices, edges, and faces (in all possible combinations, including total coloring), edge-coloring, cyclic coloring (all vertices in any small face have different colors), 3-coloring, acyclic coloring (no 2-colored cycles), oriented coloring (homomorphism of directed graphs to small tournaments), a special case of circular coloring (the colors are points of a small cycle, and the colors of any two adjacent vertices must be nearly opposite on this cycle), 2-distance coloring (no 2-colored paths on three vertices), and star coloring (no 2-colored paths on four vertices). The only improper coloring discussed is injective coloring (any two vertices having a common neighbor should have distinct colors).     

Taking cube roots in Zm

We address the problem of taking cube roots modulo an integer. We generalize two of the fastest algorithms for computing square roots modulo a prime to algorithms for computing cube roots.     

Complex Hadamard matrices and equiangular tight frames

In this paper, we give a new construction of parametric families of complex Hadamard matrices of square orders, and connect them to equiangular tight frames. The results presented here generalize some of the recent ideas of Bodmann et al. [3] and extend the list of known equiangular tight frames. In particular, a (36, 21)-frame coming from a nontrivial cube root signature matrix is obtained for the first time.     

List 2-facial 5-colorability of plane graphs with girth at least 12

A proper vertex coloring of a plane graph is 2-facial if any two different vertices joined by a facial walk of length 2 are colored differently, and it is 2-distance if every two vertices at distance 2 from each other are colored differently. Note that any 2-facial coloring of a subcubic graph is 2-distance.It is known that every plane graph with girth at least 14 has a 2-facial 5-coloring [M. Montassier, A. Raspaud, A note on 2-facial coloring of plane graphs. Inform. Process. Lett. 98 (6) (2006) 235–241], and that every planar subcubic graph with girth at least 13 has a list 2-distance 5-coloring [F. Havet, Choosability of square of planar subcubic graphs with large girth, Discrete Math. 309 (2009) 3353–3563].We strengthen these results by proving the list 2-facial 5-colorability of plane graphs with girth at least 12.     

Discrepancy of line segments for general lattice checkerboards

In a series of papers recently “checkerboard discrepancy” has been introduced, where a black-and-white checkerboard background induces a coloring on any curve, and thus a discrepancy, i.e., the difference of the length of the curve colored white and the length colored black. Mainly straight lines and circles have been studied and the general situation is that, no matter what the background coloring, there is always a curve in the family studied whose discrepancy is at least of the order of the square root of the length of the curve.In this paper we generalize the shape of the background, keeping the lattice structure. Our background now consists of lattice copies of any bounded fundamental domain of the lattice, and not necessarily of squares, as was the case in the previous papers. As the decay properties of the Fourier transform of the indicator function of the square were strongly used before, we now have to use a quite different proof, in which the tiling and spectral properties of the fundamental domain play a role.     

A fast algorithm for balanced sampling

Summary  The cube method (Deville & Tillé 2004) is a large family of algorithms that allows selecting balanced samples with equal or unequal inclusion probabilities. In this paper, we propose a very fast implementation of the cube method. The execution time does not depend on the square of the population size anymore, but only on the population size. Balanced samples can thus be selected in very large populations of several hundreds of thousands of units.     

Generalization and Composition of Modal Squares of Oppositions

The first part of the paper aims at showing that the notion of an Aristotelian square may be seen as a special case of a variety of different more general notions: (1) the one of a subAristotelian square, (2) the one of a semiAristotelian square, (3) the one of an Aristotelian cube, which is a construction made up of six semiAristotelian squares, two of which are Aristotelian. Furthermore, if the standard Aristotelian square is seen as a special ordered 4-tuple of formulas, there are 4-tuples describing rotations of the original square which are non-standard Aristotelian squares. The second part of the paper focuses on the notion of a composition of squares. After a discussion of possible alternative definitions, a privileged notion of composition of squares is identified, thus opening the road to introducing and discussing the wider notion of composition of cubes.     

Geodesics on the regular tetrahedron and the cube

Consider all geodesics between two given points on a polyhedron. On the regular tetrahedron, we describe all the geodesics from a vertex to a point, which could be another vertex. Using the Stern–Brocot tree to explore the recursive structure of geodesics between vertices on a cube, we prove, in some precise sense, that there are twice as many geodesics between certain pairs of vertices than other pairs. We also obtain the fact that there are no geodesics that start and end at the same vertex on the regular tetrahedron or the cube.     

Maximal and Sylow Subgroups of Solvable Finite Groups

The structure of finite solvable groups in which any Sylow subgroup is the product of two cyclic subgroups is studied. In particular, it is proved that the nilpotent length of such a group is no greater than 4. It is also proved that the nilpotent length of a finite solvable group in which the index of any maximal subgroup is either a prime or the square of a prime or the cube of a prime does not exceed 5.     

Cube roots of integers. A conjecture about Heron's method in Metrika III. 20

Did Heron (or his teachers) use sequences of differences to find an approximate value of the cube root of an integer? I venture a conjecture of his heuristics and a couple of possible mathematical proofs of his method.     

Uniform spherical grids via equal area projection from the cube to the sphere

We construct an area preserving map from a cube to the unit sphere S², both centered at the origin. More precisely, each face F_i of the cube is first projected to a curved square S_i of the same area, and then each S_i is projected onto the sphere by inverse Lambert azimuthal equal area projection, with respect to the points situated at the intersection of the coordinate axes with S². This map is then used to construct uniform and refinable grids on a sphere, starting from any grid on a square.     

Vertex-distinguishing IE-total Colorings of Complete Bipartite Graphs K8,n

Let G be a simple graph. An IE-total coloring f of G is a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color. For each vertex x of G, let C(x) be the set of colors of vertex x and edges incident to x under f. For an IE-total coloring f of G using k colors, if C(u) 6= C(v) for any two different vertices u and v of G, then f is called a k-vertex-distinguishing IE-total-coloring of G or a k-VDIET coloring of G for short. The minimum number of colors required for a VDIET coloring of G is denoted by χievt(G) and is called vertex-distinguishing IE-total chromatic number or the VDIET chromatic number of G for short. The VDIET colorings of complete bipartite graphs K8,n are discussed in this paper. Particularly, the VDIET chromatic number of K8,n are obtained.     

Triangle colorings require at least seven colors

We show that if a coloring of the plane has the properties that any two points at distance one are colored differently and the plane is partitioned into uniformly colored triangles under certain conditions, then it requires at least seven colors. This is also true for a coloring using uniformly colored polygons if it has a point bordering at least four polygons.     

An Analogue of Gromov’s Waist Theorem for Coloring the Cube

It is proved that if we partition a d-dimensional cube into \(n^d\) small cubes and color the small cubes in \(m+1\) colors then there exists a monochromatic connected component consisting of at least \(f(d, m) n^{d-m}\) small cubes. Another proof of this result is given in Matdinov’s preprint (Size of components of a cube coloring, arXiv:1111.3911, 2011)     

Vertex-distinguishing E-total Coloring of Complete Bipartite Graph K_(7,n) when 7≤n≤95

Let G be a simple graph. A total coloring f of G is called an E-total coloring if no two adjacent vertices of G receive the same color, and no edge of G receives the same color as one of its endpoints. For an E-total coloring f of a graph G and any vertex x of G, let C(x) denote the set of colors of vertex x and of the edges incident with x, we call C(x) the color set of x. If C(u)≠ C(v) for any two different vertices u and v of V(G), then we say that f is a vertex-distinguishing E-total coloring of G or a VDET coloring of G for short. The minimum number of colors required for a VDET coloring of G is denoted by χ_(vt)~e(G) and is called the VDET chromatic number of G. The VDET coloring of complete bipartite graph K_(7,n)(7 ≤ n ≤ 95) is discussed in this paper and the VDET chromatic number of K_(7,n)(7 ≤ n ≤ 95) has been obtained.     

Remarks on Vertex-Distinguishing IE-Total Coloring of Complete Bipartite Graphs K4,n and Kn,n

Let G be a simple graph.An IE-total coloring f of G refers to a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color.Let C(u) be the set of colors of vertex u and edges incident to u under f.For an IE-total coloring f of G using k colors,if C(u)=C(v) for any two different vertices u and v of V(G),then f is called a k-vertex-distinguishing IE-total-coloring of G,or a k-VDIET coloring of G for short.The minimum number of colors required for a VDIET coloring of G is denoted by χ ie vt (G),and it is called the VDIET chromatic number of G.We will give VDIET chromatic numbers for complete bipartite graph K4,n (n≥4),K n,n (5≤ n ≤ 21) in this article.     

Pre-compatible almost endomorphisms and semigroups whose cube is a band

IThe structure of semigroups whose cube equals a given band L is determined modulo pre-compatible collections of mappings on L. This structure theorem is applied to determine the lattice of subvarieties of [xyz = xywyz] , the variety consisting precisely of every semigroup whose square is an inflation of its cube and whose cube is a rectangular band.