We prove that there are 14 distinct ways to tile the unit square (modulo the symmetries of the square) with 5 triangles such that the 5-tiling is not a subdivision of a tiling using fewer triangles. We then demonstrate how to construct infinitely many rational tilings in each of the 14 configurations. This stands in contrast to a long standing inability to find rational 4-tilings of the unit sq...

In this paper we establish the existence of S-idempotents in case of loop rings ZtLn(m) for a special class of loops Ln(m); over the ring of modulo integers Zt for a specific value of t. These loops satisfy the conditions g i for every gi ∈ Ln(m). We prove ZtLn(m) has an S-idempotent when t is a perfect number or when t is of the form 2p or 3p (where p is an odd prime) or in general when t = p1...

Rainbow vertex-connection number is the minimum k-coloring on vertex graph G and denoted by rvc(G). Besides, rainbow-vertex connection can be applied to some special graphs, such as prism path graph. Graph operation a method used create new combining two graphs. Therefore, this research uses corona product form at resulting from of (Pm,2 P3) (P3 Pm,2). The results study obtain that theorem rain...

imply the solvability of the equation a s-av = b' ; the notion of sum intersector set is defined similarly. The authors prove two general theorems saying that if a set {b, , b 2 BLOCKINbi} is well distributed simultaneously among and within all residue classes of small moduli then it must be both difference and sum intersector set. They apply these theorems to investigate the solvability of the...

1. INTRODUCTION. In this article we show that group representations permit simple, visual explanations of semidirect products and related concepts. Along the way, we prove an extension of Cayley's theorem, define blocks and tails for the resulting representations, and investigate the title question: Do normal subgroups have straight tails? We consider representations of finite groups over the f...

For a real abelian number field F and for a prime p we study the relation between the p-parts of the class groups and of the quotients of global units modulo cyclotomic units along the cyclotomic Zp-extension of F . Assuming Greenberg’s conjecture about the vanishing of the λ-invariant of the extension, a map between these groups has been constructed by several authors, and shown to be an isomo...

2 The Construction Fix n = m2 for a natural m and let An =Zm Zm,Zm being the group of integers modulo m. An may be thought of a combinatorial torus. Consider the following 5 bijections on An: 1. σ0 : (x;y) 7! (x;y), 2. σ1 : (x;y) 7! (x;x+ y), 3. σ2 : (x;y) 7! (x;x+ y+1), 4. σ3 : (x;y) 7! (x+ y;y), and 5. σ4 : (x;y) 7! (x+ y+1;y), addition modulo m. Now define Gn = (Un;Vn; En) as follows: Un = V...

1 K and K0 of schemes Let X be a quasi-projective variety. We define K(X) as follows. Let Fvb(X) be the free abelian group on all isomorphism classes of vector bundles (of finite rank) over X and consider the equivalence relation Rvb(X) generated by the following. If 0 → E1 → E2 → E3 → 0 is an exact sequence of vector bundles, then [E2] = [E1] + [E3]. The quotient Fvb(X)/Rvb(X) is called K(X). ...

In this paper, an exact bitwise MAP (Maximum A Posteriori) estimation algorithm for group testing problems is presented. We assume a simplest non-adaptive group testing scenario including N -objects with binary status and M disjunctive tests. If a group contains a positive object, the test result for the group is assumed to be one; otherwise, the test result becomes zero. Our inference problem ...

Let Uq be the quantum group associated to a Lie algebra g of rank n. The negative part U− of U has a canonical basis B with favourable properties (see Kashiwara [6] and Lusztig [11, §14.4.6]). The approaches of Lusztig and Kashiwara lead to a set of alternative parametrizations of the canonical basis, one for each reduced expression for the longest word in the Weyl group of g. We show that if g...