Super-regular Steiner 2-designs

A design is additive under an abelian group $G$ (briefly, $G$-additive) if, up to isomorphism, its point set is contained in $G$ and the elements of each block sum up to zero. The only known Steiner 2-designs that are $G$-additive for some $G$ have block size which is either a prime power or a prime power plus one. Indeed they are the point-line designs of the affine spaces $AG(n,q)$, the point-line designs of the projective planes $PG(2,q)$, and the point-line designs of the projective spaces $PG(n,2)$. In the attempt to find new examples, possibly with a block size which is neither a prime power nor a prime power plus one, we look for Steiner 2-designs which are strictly $G$-additive (the point set is exactly $G$) and $G$-regular (any translate of any block is a block as well) at the same time. These designs will be called\break "$G$-super-regular". Our main result is that there are infinitely many values of $v$ for which there exists a super-regular, and therefore additive, $2$-$(v,k,1)$ design whenever $k$ is neither singly even nor of the form $2^n3\geq12$. The case $k\equiv2$ (mod 4) is a definite exception whereas $k=2^n3\geq12$ is at the moment a possible exception. We also find super-regular $2$-$(p^n,p,1)$ designs with $p\in\{5,7\}$ and $n\geq3$ which are not isomorphic to the point-line design of $AG(n,p)$.Comment: 31 page

Tight sets in finite classical polar spaces

We show that every i-tight set in the Hermitian variety H(2r + 1, q) is a union of pairwise disjoint (2r + 1)-dimensional Baer subgeometries PG(2r + 1, root q) and generators of H(2r + 1, q), if q >= 81 is an odd square and i < (q(2/3) - 1)/2. We also show that an i-tight set in the symplectic polar space W(2r + 1, q) is a union of pairwise disjoint generators of W(2r + 1, q), pairs of disjoint r-spaces {Delta,Delta(perpendicular to)}, and (2r + 1)-dimensional Baer subgeometries. For W(2r + 1, q) with r even, pairs of disjoint r-spaces {Delta,Delta(perpendicular to)} cannot occur. The (2r + 1)-dimensional Baer subgeometries in the i-tight set of W(2r + 1, q) are invariant under the symplectic polarity perpendicular to of W(2r + 1, q) or they arise in pairs of disjoint Baer subgeometries corresponding to each other under perpendicular to. This improves previous results where i < q(5/8)/root 2+ 1 was assumed. Generalizing known techniques and using recent results on blocking sets and minihypers, we present an alternative proof of this result and consequently improve the upper bound on i to (q(2/3) - 1)/2. We also apply our results on tight sets to improve a known result on maximal partial spreads in W(2r + 1, q)

Impact of technological synchronicity on prospects for CETI

For over 50 years, astronomers have searched the skies for evidence of electromagnetic signals from extraterrestrial civilizations that have reached or surpassed our level of technological development. Although often overlooked or given as granted, the parallel use of an equivalent communication technology is a necessary prerequisite for establishing contact in both leakage and deliberate messaging strategies. Civilization advancements, especially accelerating change and exponential growth, lessen the perspective for a simultaneous technological status of civilizations thus putting hard constraints on the likelihood of a dialogue. In this paper we consider the mathematical probability of technological synchronicity of our own and a number of other hypothetical extraterrestrial civilizations and explore the most likely scenarios for their concurrency. If SETI projects rely on a fortuitous detection of leaked interstellar signals (so called "eavesdropping") then without any prior assumptions N \geq 138-4991 Earth-like civilizations have to exist at this moment in the Galaxy for the technological usage synchronicity probability p \geq 0.95 in the next 20 years. We also show that since the emergence of complex life, coherent with the hypothesis of the Galactic habitable zone, N \geq 1497 extraterrestrial civilizations had to be created in the Galaxy in order to achieve the same estimated probability in the technological possession synchronicity which corresponds to the deliberate signaling scenario.Comment: 15 pages, 7 figures, 1 table, accepted for publication in the International Journal of Astrobiolog

Counting trees with a small number of vertices

Stablo je povezan jednostavan graf bez ciklusa. Prebrojavanje različitih stabala s n vrhova je težak kombinatorni problem. Za velike vrijednosti n, problem je još uvijek otvoren. U ovom će se članku prebrojiti i konstruirati sva stabla s najviše osam vrhova.A tree is a connected simple graph without cycles. Counting different trees with n vertices is a difficult combinatorial problem. For the large values of n, the problem is still open. In this paper we discuss the numbers and constructions of all trees with up to eight vertices

Prebrajanje razapinjućih stabala grafa

Ovaj se članak bavi tehnikama za prebrojavanje razapinjućih stabala grafa. Predstavljen je Kirchoffov matrični teorem o stablima koji povezuje broj razapinjućih stabala grafa i determinantu matrice čije vrijednosti ovise o grafu. Primjenom teorema izračunat je broj razapinjućih stabala od potpunog grafa Kn, potpunog bipartitnog grafa Krs i grafa kotača Wn

Counting trees with a small number of vertices

Stablo je povezan jednostavan graf bez ciklusa. Prebrojavanje različitih stabala s n vrhova je težak kombinatorni problem. Za velike vrijednosti n, problem je još uvijek otvoren. U ovom će se članku prebrojiti i konstruirati sva stabla s najviše osam vrhova.A tree is a connected simple graph without cycles. Counting different trees with n vertices is a difficult combinatorial problem. For the large values of n, the problem is still open. In this paper we discuss the numbers and constructions of all trees with up to eight vertices

q-analogs of group divisible designs

A well known class of objects in combinatorial design theory are {group divisible designs}. Here, we introduce the $q$-analogs of group divisible designs. It turns out that there are interesting connections to scattered subspaces, $q$-Steiner systems, design packings and $q^r$-divisible projective sets. We give necessary conditions for the existence of $q$-analogs of group divsible designs, construct an infinite series of examples, and provide further existence results with the help of a computer search. One example is a $(6,3,2,2)_2$ group divisible design over $\operatorname{GF}(2)$ which is a design packing consisting of $180$ blocks that such every $2$-dimensional subspace in $\operatorname{GF}(2)^6$ is covered at most twice.Comment: 18 pages, 3 tables, typos correcte

Problem klasifikacije grafova

Tema ovog članka je problem klasifikacije grafova. Predstavljeni su osnovni pojmovi teorije grafova, a zatim kombinatorne tehnike kojima se može pokazati da dva grafa nisu izomorfna te su klasificirani svi grafovi s 5 vrhova