Union-intersecting set systems

Three intersection theorems are proved. First, we determine the size of the largest set system, where the system of the pairwise unions is l-intersecting. Then we investigate set systems where the union of any s sets intersect the union of any t sets. The maximal size of such a set system is determined exactly if s+t4. Finally, we exactly determine the maximal size of a k-uniform set system that has the above described (s,t)-union-intersecting property, for large enough n.Comment: 9 page

Baldness/delocalization in intersecting brane systems

Marginally bound systems of two types of branes are considered, such as the prototypical case of Dp+4 branes and Dp branes. As the transverse separation between the two types of branes goes to zero, different behaviour occurs in the supergravity solutions depending on p; no-hair theorems result for p<=1 only. Within the framework of the AdS/CFT correspondence, these supergravity no-hair results are understood as dual manifestations of the Coleman-Mermin-Wagner theorem. Furthermore, the rates of delocalization for p<=1 are matched in a scaling analysis. Talk given at ``Strings '99''; based on hep-th/9903213 with D. Marolf.Comment: LaTeX, 10 pages, 1 figure; contribution to Strings'99 proceeding

Intersecting Surface Defects and Instanton Partition Functions

We analyze intersecting surface defects inserted in interacting four-dimensional N = 2 supersymmetric quantum field theories. We employ the realization of a class of such systems as the infrared fixed points of renormalization group flows from larger theories, triggered by perturbed Seiberg-Witten monopole-like configurations, to compute their partition functions. These results are cast into the form of a partition function of 4d/2d/0d coupled systems. Our computations provide concrete expressions for the instanton partition function in the presence of intersecting defects and we study the corresponding ADHM model.Comment: 66 pages; v2: minor typos correcte

Distributed Systems of Intersecting Branes at Arbitrary Angles

A `reduced' action formulation for a general class of the supergravity solutions, corresponding to the `marginally' bound `distributed' systems of various types of branes at arbitrary angles, is developed. It turns out that all the information regarding the classical features of such solutions is encoded in a first order Lagrangian (the `reduced' Lagrangian) corresponding to the desired geometry of branes. The marginal solution for a system of $N$ such distributions (for various distribution functions) span an $N$ dimensional submanifold of the fields' configuration (target) space, parametrised by a set of $N$ independent harmonic functions on the transverse space. This submanifold, which we call it as the `$H$-surface', is a null surface with respect to a metric on the configuration space, which is defined by the reduced Lagrangian. The equations of motion then transform to a set of equations describing the embedding of a null geodesic surface in this space, which is identified as the $H$-surface. Using these facts, we present a very simple derivation of the conventional orthogonal solutions together with their intersection rules. Then a new solution for a (distributed) pair of $p$-branes at SU(2) angles in $D$ dimensions is derived.Comment: Latex file, 58 pages, no figures, 5 tables, This revision contains some minor changes of the original version including those of the title, abstract and referrences. Some comments are adde

Cross-intersecting families and primitivity of symmetric systems

Let $X$ be a finite set and $\mathfrak p\subseteq 2^X$, the power set of $X$, satisfying three conditions: (a) $\mathfrak p$ is an ideal in $2^X$, that is, if $A\in \mathfrak p$ and $B\subset A$, then $B\in \mathfrak p$; (b) For $A\in 2^X$ with $|A|\geq 2$, $A\in \mathfrak p$ if $\{x,y\}\in \mathfrak p$ for any $x,y\in A$ with $x\neq y$; (c) $\{x\}\in \mathfrak p$ for every $x\in X$. The pair $(X,\mathfrak p)$ is called a symmetric system if there is a group $\Gamma$ transitively acting on $X$ and preserving the ideal $\mathfrak p$. A family $\{A_1,A_2,\ldots,A_m\}\subseteq 2^X$ is said to be a cross-$\mathfrak{p}$-family of $X$ if $\{a, b\}\in \mathfrak{p}$ for any $a\in A_i$ and $b\in A_j$ with $i\neq j$. We prove that if $(X,\mathfrak p)$ is a symmetric system and $\{A_1,A_2,\ldots,A_m\}\subseteq 2^X$ is a cross-$\mathfrak{p}$-family of $X$, then $$\sum_{i=1}^m|{A}_i|\leq\left\{ \begin{array}{cl} |X| & \hbox{if $m\leq \frac{|X|}{\alpha(X,\, \mathfrak p)}$,} \\ m\, \alpha(X,\, \mathfrak p) & \hbox{if $m\geq \frac{|X|}{\alpha{(X,\, \mathfrak p)}}$,} \end{array}\right.$$ where $\alpha(X,\, \mathfrak p)=\max\{|A|:A\in\mathfrak p\}$. This generalizes Hilton's theorem on cross-intersecting families of finite sets, and provides analogs for cross-$t$-intersecting families of finite sets, finite vector spaces and permutations, etc. Moreover, the primitivity of symmetric systems is introduced to characterize the optimal families.Comment: 15 page

Dynamics of intersecting brane systems -- Classification and their applications --

We present dynamical intersecting brane solutions in higher-dimensional gravitational theory coupled to dilaton and several forms. Assuming the forms of metric, form fields, and dilaton field, we give a complete classification of dynamical intersecting brane solutions with/without M-waves and Kaluza-Klein monopoles in eleven-dimensional supergravity. We apply these solutions to cosmology and black holes. It is shown that these give FRW cosmological solutions and in some cases Lorentz invariance is broken in our world. If we regard the bulk space as our universe, we may interpret them as black holes in the expanding universe. We also discuss lower-dimensional effective theories and point out naive effective theories may give us some solutions which are inconsistent with the higher-dimensional Einstein equations.Comment: 44 pages; v2: minor corrections, references adde