Bichromatic and Equichromatic Lines in ℂ2 and ℝ2

Let G and R each be a finite set of green and red points, respectively, such that |G|=n, |R|=n−k, G∩R=∅, and the points of G∪R are not all collinear. Let t be the total number of lines determined by G∪R. The number of equichromatic lines (a subset of bichromatic) is at least (t+2n+3−k(k+1))/4. A slightly weaker lower bound exists for bichromatic lines determined by points in ℂ². For sufficiently large point sets, a proof of a conjecture by Kleitman and Pinchasi is provided. A lower bound of (2t+14n−k(3k+7))/14 is demonstrated for bichromatic lines passing through at most six points. Lower bounds are also established for equichromatic lines passing through at most four, five, or six points.     

A note on the ranks of 2 × 2 × 2 and 2 × 2 × 2 × 2 tensors

Using rank-1 reduction formula and the vector space spanned by the real rank-1 matrices, we present a different way to show that the maximum possible rank of the 2?×?2?×?2 tensors over the real field is 3. Following, we obtain that the maximum rank of the 2?×?2?×?2?×?2 tensors over the real field is less than or equal to 5 and propose another way to show that the maximum rank of the 2?×?2?×?2?×?2 tensors over the complex field is 4, except one special case.     

Canonical forms of 2 × 2 × 2 and 2 × 2 × 2 × 2 arrays over 𝔽2 and 𝔽3

We consider arrays of size 2?×?2?×?2 and 2?×?2?×?2?×?2 over the fields with two and three elements. We use computer algebra to determine the canonical forms of these arrays with respect to the action of the semidirect product of general linear groups with the symmetric group. For each canonical form, we determine the size of its orbit and the rank of the arrays in its orbit.     

Tautness and Fatou Components in ?2

Hyperbolicity played an important role in the classification of Fatou components for rational functions in the Riemann sphere. In higher dimensions Fatou components are not nearly as well understood. We investigate the Kobayashi completeness and tautness of invariant Fatou components for holomorphic endomorphisms of ?² and for Hénon mappings. We show that basins of attraction and domains with an attracting Riemann surface, previously known to be taut, are also complete, which is strictly stronger. We also prove tautness for a class of Siegel domains.     

The 2×2 matrix Schlesinger system and the Belavin-Polyakov-Zamolodchikov system

We show that the Belavin-Polyakov-Zamolodchikov equation of the minimal model of conformal field theory with the central charge c = 1 for the Virasoro algebra is contained in a system of linear equations that generates the Schlesinger system with 2×2 tmatrices. This generalizes Suleimanov’s result on the Painlevé equations. We consider the properties of the solutions, which are expressible in terms of the Riemann theta function.     

Ordinary and ℤ2-Graded Cocharacters of UT 2(E)

Let E be the infinite dimensional Grassmann algebra over a field F of characteristic 0. In this article we consider the algebra R of 2 × 2 matrices with entries in E and its subalgebra G, which is one of the minimal algebras of polynominal identity (PI) exponent 2. We compute firstly the Hilbert series of G and, as a consequence, we compute its cocharacter sequence. Then we find the Hilbert series of R, using the tool of proper Hilbert series, and we compute its cocharacter sequence. Finally we describe explicitely the ?₂-graded cocharacters of R.     

T_2 Separation Subsets and T_2 Separateness in L-fuzzy Topological Spaces

5l.Introductionowingtothestratiformstructure,L-fuzzyseparfltenessarefarcomplicatedtocrisl)case,forexamp1eL-fuzzy7'2separateness,ithasarisentl1atL-fuzzy7'2sPacesandweak7',sPacespresentedbyLiuandLuo[lJ,wl1icl1arecalledbyWangF2jS7',sl7aceand,,I'2sPaces,ant1a-7',spacesandstratiformY'2spacesintroductedbyPengL3j'a[lda-7'2sPacesgivenbyYiL4jetc..Eachofthemhasitsownstrongpoints,butalltl1eyaredef1nedfortltewholeL-fuzzytopologicalspace-Inthepaper,anew7'2separateness,whichiscalle(lC7'2,isdefinedfo…     

The Gauss-Bonnet theorem and ζ(2)

Using a counting argument in hyperbolic geometry due to Siegel and the Gauss-Bonnet theorem, we compute that (2)=²/6.NATO Postdoctoral Fellow.     

Urey-Bradley force field—ketene and ketene-d 2

Urey-Bradley force constants of ketene and deuteroketene have been evaluated by the normal co-ordinate treatment. Using these force constants, the fundamental vibrational wave-numbers for ketene and its deuterium analogue have been calculated and compared with the observed bands.     

Gradient-like and integrable vector fields on ℝ2

Recently, the authors have obtained criteria for the integral curves of a nonsingular smooth vector field X on a smooth manifold M to be timelike, null or spacelike geodesics for some Lorentzian metric g for M. In this paper, we show that for smoothly contractible subsets S of ² null geodesibility of a vector field X is equivalent to X being preHamiltonian on S and timelike, spacelike or Riemannian pregeodesibility of X are all equivalent to X being gradient-like. It turns out that null geodesibility is quite rare as we prove that even among real analytic vector fields on S there are many open sets of vector fields which fail to be preHamiltonian.Partially supported by a grant from the Weldon Springs endowment of the University of Missouri-Columbia     

CALCULATIONS OF BIVARIATE SPLINES——Ⅰ. Truncated Powers and BM-Splines in S_2~1, (?)2 and S_4~2,(?)_2

Determining the lowest degree, dimensions and basis functions of btp (bivariate truncated power(s)) spaces with given smoothness requirements and establishing the calculation formulae of btp on a three-and four-direction mesh, we give the necessary and sufficient condition for the existence of the bivariate BM-spline in S_4~2, and the bivariate BM-spline in S_2~1, in terms of linear combinations of btp. The so-called "revolving around" teahnique is mentioned.     

From the Icosahedron to Natural Triangulations of??P 2 and S 2��S 2

We present two constructions in this paper: (a) a 10-vertex triangulation $\mathbb{C}P^{2}_{10}$ of the complex projective plane ?P ² as a subcomplex of the join of the standard sphere ( $S^{2}_{4}$ ) and the standard real projective plane ( $\mathbb{R}P^{2}_{6}$ , the decahedron), its automorphism group is A ₄; (b) a 12-vertex triangulation (S ²×S ²)₁₂ of S ²×S ² with automorphism group 2S ₅, the Schur double cover of the symmetric group S ₅. It is obtained by generalized bistellar moves from a simplicial subdivision of the standard cell structure of S ²×S ². Both constructions have surprising and intimate relationships with the icosahedron. It is well known that ?P ² has S ²×S ² as a two-fold branched cover; we construct the triangulation $\mathbb{C}P^{2}_{10}$ of ?P ² by presenting a simplicial realization of this covering map S ²×S ²???P ². The domain of this simplicial map is a simplicial subdivision of the standard cell structure of S ²×S ², different from the triangulation alluded to in (b). This gives a new proof that Kühnel??s $\mathbb{C}P^{2}_{9}$ triangulates ?P ². It is also shown that $\mathbb{C}P^{2}_{10}$ and (S ²×S ²)₁₂ induce the standard piecewise linear structure on ?P ² and S ²×S ² respectively.     

The Pell Equations x^2—8y^2=1 and y2—Dz^2=1

ＯｎｔｈｅＰｅｌｌｅｑｕａｔｉｏｎｓｘ２－８ｙ２＝１,　ｙ２－Ｄｚ２＝１（１）ｗｈｅｒｅＤ＞０ｉｓａｓｑｕａｒｅ－ｆｒｅｅｉｎｔｅｇｅｒ．ＣａｏＺｈｅｎｆｕ［１］ｓｈｏｗｅｄｔｈａｔｉｆＤ＝∏ｓｉ＝１Ｐｉ≡１（ｍｏｄ４）ｏｒＤ＝２∏Ｐｉ,１≤ｓ≤４,ｔｈｅｎｔｈｅｅｑｕａｔｉｏｎ（１）ｈａｓｎｏｌｙｐｏｓｉｔｉｖｅｉｎｔｅｇｅｒｓｏｌｕｔｉｏｎｚ＝６（Ｄ＝２·１７）．ＣｈｅｎｇＪｉａｎｈｕａ［２］ｓｈｏｗｅｄｔｈａｔｉｓＤ＝∏ｓｉ＝１Ｐｉ　１≤ｓ≤２,ｔｈｅｎｔｈｅｅｑｕａｔｉｏｎ（１）ｈａｓｏ…     

The inverse and determinant of a 2 × 2 uniformly distributed random matrix

Formulae are derived for the density of the determinant and the elements of the inverse of a 2 × 2 matrix, with entries which are independent random variables uniformlly distributed on [0,1]. Graphs of the densities are presented, and the relevance of the results to interval matrices is discussed.     

On Simple ℤ2-Invariant and Corner Function Germs

Mathematical Notes - V. I. Arnold has classified simple (i.e., having no moduli for the classification) singularities (function germs), and also simple boundary singularities: function germs...     

Isodual Codes over ℤ2k and Isodual Lattices

A code is called isodual if it is equivalent to its dual code, and a lattice is called isodual if it is isometric to its dual lattice. In this note, we investigate isodual codes over _2k . These codes give rise to isodual lattices; in particular, we construct a 22-dimensional isodual lattice with minimum norm 3 and kissing number 2464.     

Branched Coverings over ℂ2 and the Jacobian Conjecture

The class of branched coverings over $\mathbb{C}^2$ traditionally called exotic arouses interest because of its connection with the Jacobian conjecture. In this paper, we construct a series of examples of such coverings; in particular, methods of construction of coverings with arbitrarily many sheets, as well as with unknotted branch curves, are described. In addition, some topological characteristics of these coverings are computed, which allows us to answer some questions about a possible counterexample to the Jacobian conjecture.