scholarly journals Constructing Large Set Systems with Given Intersection Sizes Modulo Composite Numbers

2002 ◽  
Vol 11 (5) ◽  
pp. 475-486 ◽  
Author(s):  
SAMUEL KUTIN
Keyword(s):  
Prime Power ◽  
Large Set ◽  
Residue Classes ◽  
Set Systems ◽  
Pairwise Intersection ◽  
Technical Report ◽  
Original Argument

We consider k-uniform set systems over a universe of size n such that the size of each pairwise intersection of sets lies in one of s residue classes mod q, but k does not lie in any of these s classes. A celebrated theorem of Frankl and Wilson [8] states that any such set system has size at most (ns) when q is prime. In a remarkable recent paper, Grolmusz [9] constructed set systems of superpolynomial size Ω(exp(c log2n/log log n)) when q = 6. We give a new, simpler construction achieving a slightly improved bound. Our construction combines a technique of Frankl [6] of ‘applying polynomials to set systems’ with Grolmusz's idea of employing polynomials introduced by Barrington, Beigel and Rudich [5]. We also extend Frankl's original argument to arbitrary prime-power moduli: for any ε > 0, we construct systems of size ns+g(s), where g(s) = Ω(s1−ε). Our work overlaps with a very recent technical report by Grolmusz [10].

scholarly journals A Note on Set Systems with no Union of Cardinality 0 modulo m

2003 ◽  
Vol Vol. 6 no. 1 ◽  
Author(s):  
Vince Grolmusz
Keyword(s):  
Prime Power ◽  
Explicit Construction ◽  
Alternative Proof ◽  
Maximal Degree ◽  
Set Systems ◽  
International Audience

International audience \emphAlon, Kleitman, Lipton, Meshulam, Rabin and \emphSpencer (Graphs. Combin. 7 (1991), no. 2, 97-99) proved, that for any hypergraph \textbf\textitF=\F_1,F_2,\ldots, F_d(q-1)+1\, where q is a prime-power, and d denotes the maximal degree of the hypergraph, there exists an \textbf\textitF_0⊂ \textbf\textitF, such that |\bigcup_F∈\textbf\textitF_0F| ≡ 0 (q). We give a direct, alternative proof for this theorem, and we also show that an explicit construction exists for a hypergraph of degree d and size Ω (d^2) which does not contain a non-empty sub-hypergraph with a union of size 0 modulo 6, consequently, the theorem does not generalize for non-prime-power moduli.

scholarly journals Set-Systems with Restricted Multiple Intersections

10.37236/1625 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Vince Grolmusz
Keyword(s):  
Upper Bound ◽  
Explicit Construction ◽  
Partial Answer ◽  
Residue Classes ◽  
Set Systems ◽  
Element Set ◽  
Multiple Intersections

We give a generalization for the Deza-Frankl-Singhi Theorem in case of multiple intersections. More exactly, we prove, that if ${\cal H}$ is a set-system, which satisfies that for some $k$, the $k$-wise intersections occupy only $\ell$ residue-classes modulo a $p$ prime, while the sizes of the members of ${\cal H}$ are not in these residue classes, then the size of ${\cal H}$ is at most $$(k-1)\sum_{i=0}^{\ell}{n\choose i}$$ This result considerably strengthens an upper bound of Füredi (1983), and gives partial answer to a question of T. Sós (1976). As an application, we give a direct, explicit construction for coloring the $k$-subsets of an $n$ element set with $t$ colors, such that no monochromatic complete hypergraph on $$\exp{(c(\log m)^{1/t}(\log \log m)^{1/(t-1)})}$$ vertices exists.

scholarly journals Set Systems with Restricted $t$-wise Intersections Modulo Prime Powers

10.37236/255 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Rudy X. J. Liu
Keyword(s):  
Upper Bound ◽  
Prime Power ◽  
Set Systems ◽  
Composite Moduli ◽  
Element Set

We give a polynomial upper bound on the size of set systems with restricted $t$-wise intersections modulo prime powers. Let $t\geq 2$. Let $p$ be a prime and $q=p^{\alpha}$ be a prime power. Let ${\cal L}=\{l_1,l_2,\ldots,l_s\}$ be a subset of $\{0, 1, 2, \ldots, q-1\}$. If ${\cal F}$ is a family of subsets of an $n$ element set $X$ such that $|F_{1}\cap \cdots \cap F_{t}| \pmod{q} \in {\cal L}$ for any collection of $t$ distinct sets from ${\cal F}$ and $|F| \pmod{q} \notin {\cal L}$ for every $F\in {\cal F}$, then $$ |{\cal F}|\leq {t(t-1)\over2}\sum_{i=0}^{2^{s-1}}{n\choose i}. $$ Our result extends a theorem of Babai, Frankl, Kutin, and Štefankovič, who studied the $2$-wise case for prime power moduli, and also complements a result of Grolmusz that no polynomial upper bound holds for non-prime-power composite moduli.

scholarly journals Enumeration of groups of prime-power order

2007 ◽  
Vol 76 (3) ◽  
pp. 479-480 ◽  
Author(s):  
Brett E. Witty
Keyword(s):  
Finite Group ◽  
Finite Order ◽  
Prime Power ◽  
Prime Power Order ◽  
Preliminary Step ◽  
Residue Classes ◽  
Counting Functions

Finite group theorists have been interested in counting groups of prime-power order, as a preliminary step to counting groups of any finite order and to assist in explicitly listing such groups. In 1960, G. Higman considered when the functions giving the number of groups of prime-power order pn, for fixed n and varying p, is of a particular form, called polynomial on residue classes (PORC). The suggestion that such counting functions are PORC is known as Higman's PORC conjecture. In his 1960 paper [4] he proved that a certain class of groups of prime-power order, now called exponent-p class two groups, have counting functions that are PORC, but did not furnish explicit PORC functions.

Ab initio band theory approach to electron energy loss near edge structures

1988 ◽  
Vol 46 ◽  
pp. 506-507
Author(s):  
Xudong Weng ◽  
O.F. Sankey ◽  
Peter Rez
Keyword(s):  
Ab Initio ◽  
Local Density ◽  
Interpolation Method ◽  
Atomic Orbital ◽  
Plane Waves ◽  
Large Set ◽  
Theory Approach ◽  
Band Theory ◽  
Atomic Orbitals ◽  
Pseudopotential Calculations

Single electron band structure techniques have been applied successfully to the interpretation of the near edge structures of metals and other materials. Among various band theories, the linear combination of atomic orbital (LCAO) method is especially simple and interpretable. The commonly used empirical LCAO method is mainly an interpolation method, where the energies and wave functions of atomic orbitals are adjusted in order to fit experimental or more accurately determined electron states. To achieve better accuracy, the size of calculation has to be expanded, for example, to include excited states and more-distant-neighboring atoms. This tends to sacrifice the simplicity and interpretability of the method.In this paper. we adopt an ab initio scheme which incorporates the conceptual advantage of the LCAO method with the accuracy of ab initio pseudopotential calculations. The so called pscudo-atomic-orbitals (PAO's), computed from a free atom within the local-density approximation and the pseudopotential approximation, are used as the basis of expansion, replacing the usually very large set of plane waves in the conventional pseudopotential method. These PAO's however, do not consist of a rigorously complete set of orthonormal states.

Molecular Views of Ice-Embedded Lumbricus Terrestris Erythrocruorin Obtained By Invariant Classification

1990 ◽  
Vol 48 (1) ◽  
pp. 450-451
Author(s):  
Michael schatz ◽  
Joachim Jäger ◽  
Marin van Heel
Keyword(s):  
Image Interpretation ◽  
Lumbricus Terrestris ◽  
Reference Image ◽  
Large Set ◽  
Multivariate Statistical ◽  
Data Set ◽  
Low Contrast ◽  
Negative Stain ◽  
Reference Images ◽  
Statistical Resolution

Lumbricus terrestris erythrocruorin is a giant oxygen-transporting macromolecule in the blood of the common earth worm (worm "hemoglobin"). In our current study, we use specimens (kindly provided by Drs W.E. Royer and W.A. Hendrickson) embedded in vitreous ice (1) to avoid artefacts encountered with the negative stain preparation technigue used in previous studies (2-4).Although the molecular structure is well preserved in vitreous ice, the low contrast and high noise level in the micrographs represent a serious problem in image interpretation. Moreover, the molecules can exhibit many different orientations relative to the object plane of the microscope in this type of preparation. Existing techniques of analysis requiring alignment of the molecular views relative to one or more reference images often thus yield unsatisfactory results.We use a new method in which first rotation-, translation- and mirror invariant functions (5) are derived from the large set of input images, which functions are subsequently classified automatically using multivariate statistical techniques (6). The different molecular views in the data set can therewith be found unbiasedly (5). Within each class, all images are aligned relative to that member of the class which contributes least to the classes′ internal variance (6). This reference image is thus the most typical member of the class. Finally the aligned images from each class are averaged resulting in molecular views with enhanced statistical resolution.

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