\input abstract.mac \title \endtitle \author \endauthor \keywords \endkeywords \subjclass \endsubjclass \abstract \endabstract %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title About a special class of quasigroups \endtitle \author H.\ Zeitler, \rm Bayreuth, Germany \endauthor \keywords quasigroups, Steiner--systems of block size 5 \endkeywords \subjclass 05B05 \endsubjclass \abstract In the axiomatic approach to the investigation of Steiner--systems, e.g.\ $S(2,4,v)$, quasigroups can be important tools. In this paper systems $S(2,5,v)$ are treated in a similar way. \endabstract %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title On Symmetric Block Designs (40,13,4) with Automorphisms of Order 13 \endtitle \author Vladimir \'Cepuli\'c, \rm Zagreb, Croatia \endauthor \keywords symmetric block design, automorphism group, orbital strusture, indexing \endkeywords \subjclass 05B05 \endsubjclass \abstract All symmetric block designs (40,13,4) admitting an automorphism group of order 13 are classified. Also the generators and orders of their automorphism groups are determined. \endabstract %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title Biplanes (56,11,2) with involutory automorphism fixing 14 points \endtitle \author Vladimir \'Cepuli\'c and Mario Essert, \rm Zagreb, Croatia \endauthor \keywords biplane, involutory automorphism, automorphism group, orbital structure, indexing \endkeywords \subjclass 05B05 \endsubjclass \abstract A biplane (56,11,2) admitting an involutory automorphism fixing 14 points is isomorphic to one of the three known biplanes (56,11,2) with automorphism groups of order 80640, 288 and 64, respectively. \endabstract %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title Symmetric (144,66,30)-designs with Frobenius group of order 78 as full automorphism group \endtitle \author Mario--Osvin Pav\v cevi\'c, \rm Zagreb, Croatia \endauthor \keywords symmetric design, Frobenius group, orbit structure, indexing \endkeywords \subjclass 05B05 \endsubjclass \abstract In this paper we construct two nonisomorphic selfdual symmetric designs with parameters (144,66,30) on which the Frobenius group of order 78 is operating so that the cyclic group of order 6 has exactly 12 fixed points and blocks. We show that these two designs are the only ones admitting this kind of operation of $Frob_{78}$. Further, this group is the full automorphism group of them. \endabstract %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title Some new partial symmetric designs derived from symmetric design with $\lambda>1$ \endtitle \author Vlado Cigi\'c, \rm Mostar, BiH \endauthor \keywords symmetric design, tight (Baer) subdesign, partial symmetric design, automorphism \endkeywords \subjclass 05B25 \endsubjclass \abstract In this paper we derive partial symmetric designs (PSD) by rejection of any tight subdesigns in symmetric design for $\lambda>1$. Also, we prove the existence of a PSD as the fixed structure of Baer automorphism on the points --- hyperplanes design $PG_{2d}(2d+1,q)$. \endabstract %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title Some theorems on equality of $L$-functions of elliptic curves \endtitle \author Ivica Gusi\'c, \rm Zagreb, Croatia \endauthor \keywords Galois field, discrete valuation, elliptic curve, abelian variety, $l$-adic representation, Frobenius element, inertia subgroup, $L$-function, Euler factor, isogeny, simple abelian variety, Weierstrass equation, index of inertia, Dirichlet density, good reduction of elliptic curve, elliptic curve with complex multiplication \endkeywords \subjclass 11G40, 14G10, 14H52 \endsubjclass \abstract Problem of equality $L(A,K,s)=L(B,M,s)$ where $A$, $B$ are elliptic curves over Galois number fields $K$, $M$ respectively, is studied. The problem is solved in the case if $A$, $B$ are defined over $\bf Q$ and $K$, $M$ are not quadratic over $K\cap M$ (Theorem 1), as well as in the case if $K\cap M=\bf Q$ and $(6,n)=1$, where $n=(K:\bf Q)$ (Theorem 2). \endabstract %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title Topological N-groups on the Reals \endtitle \author K.\ D.\ Magill, Jr., \rm Buffalo, USA \endauthor \keywords Topological nearrings, topological N-groups, ideals \endkeywords \subjclass 16Y30, 16W80, 54H13 \endsubjclass \abstract Let $N$ be a topological nearring and let $G$ be a topological group. Suppose there exists a continuous map from $N\times G$ which sends $(n,g)$ to $ng\in G$ such that $(n_1+n_2)g = n_1g+n_2g$ and $(n_1n_2)g = n_1(n_2g)$ for all $n_1,n_2\in N$ and $g\in G$. Then $G$, together with this multiplication on the left by elements of $N$, is referred to as a topological $N$-group or topological nearmodule. In this paper, $N$ will be a topological nearring whose additive group is a Euclidean $N$-group and $G$ will be the additive group $R$ of real numbers. Within this class of nearrings, we completely determine those nearrings $N$ for which there exist nontrivial multiplications such that $R$ is a topological $N$-group and we determine precisely when two of these nearrings are isomorphic. \endabstract %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title Cayley theorem for monoidoids \endtitle \author Gheorghe Ivan, \rm Timi\c soara, Romania \endauthor \keywords monoidoid, monoid bundle, mappings monoidoid, quasipermutation groupoid \endkeywords \subjclass 20L05, 20L99 \endsubjclass \abstract In this paper we define the monoidoids and give a brief summary of their most important properties. The main result is the Cayley's theorem for monoidoids. \endabstract %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\nat{\bf N} \title{Free Abelian topological groups and $k$-spaces} \endtitle \author{Kohzo Yamada, \rm Shizuoka, Japan}\endauthor \keywords{free topological group, free Abelian topological group, metrizable space, $k$-space, locally compact space} \endkeywords \subjclass{22A05, 54D50, 54E45} \endsubjclass \abstract In this paper, we give a metrizable space $X$ such that $A_{n}(X)$ is a $k$-space for each $n \in\nat$, but $A(X)$ is not a $k$-space. Moreover, we shall show that for a metrizable space $X$, if each $F_{n}(X) \ (A_{n}(X))$ is a $k$-space, then either $X$ is locally compact or $X'$, the set of non-isolated points of $X$, is compact. On the other hand, we give a characterization of a metrizable space $X$ such that every $F_{n}(X)$ is locally compact. \endabstract %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title $K$-types of minimal representations ($p$-adic case) \endtitle \author Gordan Savin, \rm Salt Lake City, USA\endauthor \subjclass 22E35, 22E50, 11F70 \endsubjclass \abstract Let $F$ be a $p$-adic field. Let $G$ be a split simple simply connected group over $F$ of type $D_n$, $(n\geq 4),$ or $E_n$, $(n=6,7,8)$. Let $K$ be a hyperspecial maximal compact subgroup of $G$. In this article we describe $K$-types of the minimal representation of $G$. \endabstract %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title Generalization of some theorems of Steinhaus in locally compact groups \endtitle \author S.\ Basu, \rm Calcutta, India \endauthor \keywords Demi-sphere, density point, $\sigma$-finite outer measure, regularity of measure \endkeywords \subjclass 28A05 \endsubjclass \abstract The purpose of this paper is to generalize Theorems X and XI of Steinhaus [5] in locally compact Hausdorff topological groups in the light of transformations more general than those leading to the former. \endabstract %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title A generalization of Enestr\"om-Kakeya theorem \endtitle \author V.\ K.\ Jain, \rm Kharagpur, India \endauthor \keywords zeros, infinite series with complex coefficients \endkeywords \subjclass 30C15, 30C10 \endsubjclass \abstract A generalization of Enestr\"om-Kakeya theorem (well known in the theory of the distribution of zeros of polynomials) in relation to infinite series with complex coefficients has been obtained. The result is best possible. \endabstract %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title{Note on the oscillation of certain difference equations} \endtitle \author{B\l a\= zej Szmanda, \rm Pozna\'n, Poland} \endauthor \subjclass 39A10 \endsubjclass \keywords oscillation, nonoscillatory solution, difference equations \endkeywords \abstract The nonlinear difference equation of the form $ \Delta^{m}u(n)+a(n)f(u(r(n)))=0,\;\;m\geq 2,\;\; n \in {\bf N} $ is considered, where $\Delta^{m}$ is the $m$-th order forward difference operator, $a:{\bf N} \rightarrow [0,\infty)$, $r:{\bf N} \rightarrow \bf Z$, $f:{\bf R} \rightarrow {\bf R}$ and $uf(u)>0$ for $u \neq 0$. Some new criteria for the oscillatory and asymptotic behavior of solutions of the above equation are given. \endabstract %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title Qualitative behavior of Volterra difference systems \endtitle \author J.\ Morcha\l o, \rm Pozna\'n, Poland \endauthor \keywords Volterra difference systems, asymptotic behavior, $\rho$-admissible maps \endkeywords \subjclass 39A10, 39A12, 39A70 \endsubjclass \abstract In this paper we study the behavior of the solutions of Volterra difference system of the form $(N)$ treated as a perturbation of the linear system $(L)$ or $(E)$. We shown that $(L)$ inherits much of the behavior $(E)$ for an appropriate perturbation term $C(n)x$. \endabstract %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title Cauchy type equations related to some singular associative operations \endtitle \author Katarzyna Doma\'nska, \rm Cz\c estochowa, Poland \endauthor \keywords Functional equation, local solution, additive function, semigroup, associativity component of the set, Cauchy equation, Pexider equation with a restricted domain \endkeywords \subjclass 39B22, 39B52 \endsubjclass \abstract L.\ Losonczi [3] determined local solutions of the generalized Cauchy equation $f(F(x,y))=f(x)+f(y)$ on components of the domain of definition of a given associative operation $F$. In some cases local solutions were found on domains smaller than the components of $F$ as well. The aim of the present paper is to describe local solutions of the equation considered on components of the domain of definition of $F$ in cases which were omitted in [3]. \endabstract %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title Positive definite functions on separable function spaces \endtitle \author Hrvoje \v{S}iki\'{c}, \rm Zagreb, Croatia \endauthor \keywords positive definite functions, semigroups, Laplace transforms, superprocesses \endkeywords \subjclass 43A35, 60J99 \endsubjclass \abstract A notion of a separable function space is defined. It is proved that a positive definite function on the semigroup of the positive elements of a separable function space is the Laplace transform of a measure concentrated on the set of positive functionals. \endabstract %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title Pawlak and topological rough sets in terms of multifunctions \endtitle \author P.\ Maritz, \rm Stellenbosch, South Africa \endauthor \subjclass 54C60, 54A99, 04A99 \endsubjclass \keywords approximation space, multifunction, rough sets \endkeywords \abstract The purpose of this paper is to discuss approximation spaces in terms of surjective multifunctions. These multifunctions will be mainly semi-single-valued. The topologies associated with the upper and lower approximation operations will be studied. \endabstract