Isotopisms of quadratic quasigroups

We establish the restricted sumset analog of the celebrated conjecture of Sárközy on additive decompositions of the set of nonzero squares over a finite field. More precisely, we show that if $q>13$ is an odd prime power, then the set of nonzero squares in $\mathbb {F}_q$ cannot be written as a restricted sumset $A \hat {+} A$ , extending a result of Shkredov. More generally, we study restricted sumsets in multiplicative subgroups over finite fields as well as restricted sumsets in perfect powers (over integers) motivated by a question of Erdős and Moser. We also prove an analog of van Lint–MacWilliams’ conjecture for restricted sumsets, which appears to be the first analogue of Erdős--Ko–Rado theorem in a family of Cayley sum graphs.

SUMMARY A general method for constructing quasi-complete Latin squares based on groups is given. This method leads to a relatively straightforward way of counting the number of inequivalent quasi-complete Latin squares of side at most 9. Randomization of such designs is discussed, and an explicit construction for valid randomization sets of quasi-complete Latin squares whose side is an odd prime power is given. It is shown that, contrary to common belief, randomization using a subset of all possible quasi-complete Latin squares may be valid while that using the whole set is not.

On the existence of small quasimultiples of affine and projective planes of arbitrary order

In parallel with the various generalizations of the Banach fixed point theorem in metric spaces, this theory is also transported to some different types of spaces including ultra metric spaces, fuzzy metric spaces, uniform spaces, partial metric spaces, $b$-metric spaces etc. In this context, first we define a binary normed operation on nonnegative real numbers and give some examples. Then we recall the concept of $T$-metric space and some important and fundamental properties of it. A $T$-metric space is a $3$-tuple $(X, T, \diamond)$, where $X$ is a nonempty set, $\diamond$ is a binary normed operation and $T$ is a $T$-metric on $X$. Since the triangular inequality of $T$-metric depends on a binary operation, which includes the sum as a special case, a $T$-metric space is a real generalization of ordinary metric space. As main results, we present three coupled fixed point theorems for bivariate mappings satisfying some certain contractive inequalities on a complete $T$-metric space. It is easily seen that not only existence but also uniqueness of coupled fixed point guaranteed in these theorems. Also, we provide some suitable examples that illustrate our results.

The starting point of this work is the following lemma in Hirschfeld and SZOnyi (submitted): For q an odd prime power, let a be an element of GF(q2)\GF(q). Then there are precisely (q - l)/2 elements u in GF(q) for which a - u is a non-zero square in GF(q2).

In this paper, we present several new constructions for k holey mutually orthogonal Latin squares (HMOLS) of type gn. We concentrate mainly on ke4; here, for all but two values of n, namely 6 and 15, only a finite number of unsolved cases remain. Some new sets of 5 and 6 HMOLS are also given, in particular 5 HMOLS(2q) for q≥63 or q an odd prime power between 6 and 62, plus 6 HMOLS(4q)for q an odd prime power between 8 and 60.

There has been much interest in mutually unbiased bases (MUBs) and their connections with various other discrete structures, such as projective planes, mutually orthogonal Latin squares (MOLS) etc. It has been conjectured by Saniga et al. (J Opt B Quantum Semiclass Opt 6:L19---L20, 2004) that the existence of a complete set of MUBs in ? d is linked to the existence of a complete set of MOLS of side length d. Since more is known about MOLS than about MUBs, most research has concentrated on constructing MUBs from MOLS (Roy and Scott, J Math Phys 48:072110, 2007; Paterek et al., Phys Rev A 70:012109, 2009). This paper gives a simple algebraic construction of MOLS from two known constructions of MUBs in the odd prime power case.

Let q be an odd prime power and suppose that $a,b\in \mathbb {F}_q$ are such that $ab$ and $(1{-}a)(1{-}b)$ are nonzero squares. Let $Q_{a,b} = (\mathbb {F}_q,*)$ be the quasigroup in which the operation is defined by $u*v=u+a(v{-}u)$ if $v-u$ is a square, and $u*v=u+b(v{-}u)$ if $v-u$ is a nonsquare. This quasigroup is called maximally nonassociative if it satisfies $x*(y*z) = (x*y)*z \Leftrightarrow x=y=z$ . Denote by $\sigma (q)$ the number of $(a,b)$ for which $Q_{a,b}$ is maximally nonassociative. We show that there exist constants $\alpha \approx 0.029\,08$ and $\beta \approx 0.012\,59$ such that if $q\equiv 1 \bmod 4$ , then $\lim \sigma (q)/q^2 = \alpha $ , and if $q \equiv 3 \bmod 4$ , then $\lim \sigma (q)/q^2 = \beta $ .

Generally, in any human field, a Smarandache Structure on a set A means a weak structure W on A such that there exists a proper subset B contained in A which is embedded with a stronger structure S. These types of structures occur in our everyday's life, that's why we study them in this book. Thus, as a particular case: A Near-ring is a non-empty set N together with two binary operations '+' and '.' such that (N, +) is a group (not necessarily abelian), (N, .) is a semigroup. For all a, b, c belonging to N we have (a + b) . c = a . c + b . c A Near-field is a non-empty set P together with two binary operations '+' and '.' such that (P, +) is a group (not-necessarily abelian), {P\{0}, .) is a group. For all a, b, c belonging to P we have (a + b) . c = a . c + b . c A Smarandache Near-ring is a near-ring N which has a proper subset P contained in N, where P is a near-field (with respect to the same binary operations on N).

A Latin square of order is an matrix of symbols, such that each symbol occurs exactly once in each row and column. For an odd prime power let denote the finite field of order . A quadratic Latin square is a Latin square defined byfor some such that and are quadratic residues in . Quadratic Latin squares have previously been used to construct perfect 1‐factorisations, mutually orthogonal Latin squares and atomic Latin squares. We first characterise quadratic Latin squares which are devoid of Latin subsquares. Let be a graph and a 1‐factorisation of . If the union of every pair of 1‐factors in induces a Hamiltonian cycle in then is called perfect, and if there is no pair of 1‐factors in which induce a Hamiltonian cycle in then is called antiperfect. We use quadratic Latin squares to construct new examples of antiperfect 1‐factorisations of complete graphs and complete bipartite graphs. We also demonstrate that for each odd prime , there are only finitely many orders , which are powers of , such that quadratic Latin squares of order could be used to construct perfect 1‐factorisations of complete graphs or complete bipartite graphs.

Resolvable bibd and sols

On nonsingular regular magic squares of odd order

Ring is a study of algebraic structures, which is defined as a non-empty set containing two binary operations. Regarding the first binary operation, the set is a group, and the second binary operation is a semigroup, and both operations fulfill the left distributive and right distributive properties. The generalized ring concept is an extension of the ring concept, namely that for the first binary operation, each element has an identity element that is not necessarily the same. This research aims to prove the elementary properties of generalized rings and the properties of generalized rings associated with the G-ring structure. Furthermore, this research also proves the properties of subsets related to identity elements in generalized rings. The results of this research are that the fundamental properties of the generalized ring are valid, which are analogous to the fundamental properties of the ring, and sufficient conditions for a generalized ring to be a G-ring are obtained. Furthermore, if the generalized ring has a unit element, it forms an abelian group with all elements having the same identity, and the generalized ring contains all identity elements.

The set of autotopisms of partial Latin squares

A binary operation on a nonempty set A is a function from A × A to A. A nonempty set F on which we have defined two binary operations, (a, b) ↦ a + b (called addition), and (a, b) ↦ ab (called multiplication), is a field if and only if the following conditions are satisfied: (1) (associativity of addition and multiplication) a + (b + c) = (a + b) + c and a(bc) = (ab)c for all a,b,c ∈ F. (2) (commutativity of addition and multiplication) a + b = b + a and ab = ba for all a, b ∈ F. (3) (existence of neutral elements with respect to addition and multiplication) There exist distinct elements of 0f ≠ 1f in F having the property that a + 0f = a and a1F = a for all a ∈ F. (4) (existence of additive and multiplicative inverses) For each a ∈ F there exists an element -a ∈ F satisfying a + (-a) = 0F and for each 0 ≠ a ∈ F there exists an element a-1 ∈ F satisfying aa-1 = 1f. (5) (distributivity of multiplication over addition) a(b + c) = ab + ac for all a, b, c ∈ F. The abstract theory of fields is due to Heinrich Weber at the end of the 19th century, based on previous work done by Leopold Kronecker and Richard Dedekind.