arXiv subjects
Publications and source records attributed to Christopher Bouchard.
Let $k$ and $n$ be positive integers such that $2 \leq k \leq n+1$. We prove that the number of $k$-chains in a union-closed family with universe $[n]$ and size $m$ is minimized when its member sets are largest possible. We also show that, whenever the minimum is nonzero and $m>n$, there are no other minimizing families.
Let $\mathcal{A}$ be a union-closed family of sets with universe $\bigcup_{A \in \mathcal{A}}A = [n] = \{1,\cdots,n\}$ and length $\ell$. We prove that $|\mathcal{A}| \leq \sum_{i=0}^{\ell} \binom{n}{i}$, with equality if and only if $\mathcal{A} = \bigcup_{i=0}^{\ell}\binom{[n]}{n-i}$. Additionally, by showing that $|\mathcal{A}| \leq \frac{\ell^p-1}{\ell-1}+2^n(1-2^{-\ell})^p$ for any nonnegative integer $p$, we establish for all integers $1 \leq k \leq n$ that $\sum_{i=0}^k \binom{n}{i} \leq \frac{k^{\hat{p}}-1}{k-1}+2^n(1-2^{-k})^{\hat{p}}$, where $\hat{p}=\lfloor (n-k)/\log_2(\frac{k}{1-2^{-k}})\rfloor + 1$.
Let $\mathcal{A}$ be a union-closed family of sets with base set $b(\mathcal{A})=\bigcup_{A \in \mathcal{A}}A$ denoted by $[n]=\{1, \cdots, n\}$, and for any real $x>0$, let $\mathcal{A}_{<x} = \{A \in \mathcal{A} \ | \ |A| < x\}$. Also, denote by $\mathcal{B}$ any smallest irredundant subfamily of $\mathcal{A}_{<n/2}$ such that $b(\mathcal{B})=b(\mathcal{A}_{<n/2})$. We prove that if $\mathcal{A}$ is separating with height $h = 4 \leq n$ and $0 \leq |\mathcal{B}| \leq 2$, then the average size of a member set from $\mathcal{A}$ is at least $n/2$. We show that $h=4$ is greatest possible with respect to this result, and conclude by considering the remaining domain $3 \leq |\mathcal{B}| \leq 4$.
The union-closed sets conjecture, also known as Frankl's conjecture, is a well-studied problem with various formulations. In terms of lattices, the conjecture states that every finite lattice $L$ with more than one element contains a join-irreducible element that is less than or equal to at most half of the elements in $L$. In this work, we obtain several necessary conditions for any counterexample $\tilde{L}$ of minimum size.
A family of sets $\mathcal{A}$ is union-closed if it is finite and nonempty with member sets that are all finite and distinct (at least one of which is nonempty) and it satisfies the property $X, Y \in \mathcal{A} \implies X \cup Y \in \mathcal{A}$. Let $\binom{S}{k}$ be the set of all $k$-element subsets of a set $S$, and let $[n]=\{1,2,\cdots,n\}$ represent $\bigcup_{A \in \mathcal{A}}A$. Further, let $\mathcal{A}_B=\{A\in\mathcal{A} \ | \ A \cap B = B\}$ and $\mathcal{A}_{\underline{B}}=\{A\in\mathcal{A} \ | \ A \cap B = \emptyset\}$. We consider, for any union-closed family $\mathcal{A}$, the class of conjectures $\textrm{UC}_x \colon \ \exists B \in \binom{[n]}{n-x+1} \ | \ |\mathcal{A}_B| \geq |\mathcal{A}_{\underline{B}}|$, where $x \in [n]$. The extremal case $x=n$ is equivalent to the union-closed sets conjecture, also known as Frankl's conjecture, which states that there exists an element of $[n]$ that is in at least $\frac{|\mathcal{A}|}{2}$ member sets of $\mathcal{A}$. We prove $\textrm{UC}_x$ for $x \in [\lceil \frac{n}{3} \rceil + 1]$, and also investigate two strengthenings of the union-closed sets conjecture.
A summary of WG II of the CKM 2018 conference on semileptonic and leptonic $b$-hadron decays is presented. This includes discussions on the CKM matrix element magitudes $|V_{ub}|$ and $|V_{cb}|$, lepton universality tests such as $R(D^{*})$ and leptonic decays. As is usual for semileptonic and leptonic decays, much discussion is devoted towards the interplay between theoretical QCD calculations and the experimental measurements.
We investigate unification problems related to the Cipher Block Chaining (CBC) mode of encryption. We first model chaining in terms of a simple, convergent, rewrite system over a signature with two disjoint sorts: list and element. By interpreting a particular symbol of this signature suitably, the rewrite system can model several practical situations of interest. An inference procedure is presented for deciding the unification problem modulo this rewrite system. The procedure is modular in the following sense: any given problem is handled by a system of `list-inferences', and the set of equations thus derived between the element-terms of the problem is then handed over to any (`black-box') procedure which is complete for solving these element-equations. An example of application of this unification procedure is given, as attack detection on a Needham-Schroeder like protocol, employing the CBC encryption mode based on the associative-commutative (AC) operator XOR. The 2-sorted convergent rewrite system is then extended into one that fully captures a block chaining encryption-decryption mode at an abstract level, using no AC-symbols; and unification modulo this extended system is also shown to be decidable.