Generating functions for Wilf equivalence under generalized factor order

Kitaev, Liese, Remmel, and Sagan recently defined generalized factor order on words comprised of letters from a partially ordered set $(P, \leq_P)$ by setting $u \leq_P w$ if there is a subword $v$ of $w$ of the same length as $u$ such that the $i$-th character of $v$ is greater than or equal to the $i$-th character of $u$ for all $i$. This subword $v$ is called an embedding of $u$ into $w$. For the case where $P$ is the positive integers with the usual ordering, they defined the weight of a word $w = w_1\ldots w_n$ to be $\text{wt}(w) = x^{\sum_{i=1}^n w_i} t^{n}$, and the corresponding weight generating function $F(u;t,x) = \sum_{w \geq_P u} \text{wt}(w)$. They then defined two words $u$ and $v$ to be Wilf equivalent, denoted $u \backsim v$, if and only if $F(u;t,x) = F(v;t,x)$. They also defined the related generating function $S(u;t,x) = \sum_{w \in \mathcal{S}(u)} \text{wt}(w)$ where $\mathcal{S}(u)$ is the set of all words $w$ such that the only embedding of $u$ into $w$ is a suffix of $w$, and showed that $u \backsim v$ if and only if $S(u;t,x) = S(v;t,x)$. We continue this study by giving an explicit formula for $S(u;t,x)$ if $u$ factors into a weakly increasing word followed by a weakly decreasing word. We use this formula as an aid to classify Wilf equivalence for all words of length 3. We also show that coefficients of related generating functions are well-known sequences in several special cases. Finally, we discuss a conjecture that if $u \backsim v$ then $u$ and $v$ must be rearrangements, and the stronger conjecture that there also must be a weight-preserving bijection $f: \mathcal{S}(u) \rightarrow \mathcal{S}(v)$ such that $f(u)$ is a rearrangement of $u$ for all $u$.Comment: 23 page

Elementary Deuring-Heilbronn Phenomenon

Adapting a technique of Pintz, we give an elementary demonstration of the Deuring phenomenon: a zero of \zeta(s) off the critical line gives a lower bound on L(1,\chi). The necessary tools are Dirichlet's 'method of the hyperbola', Euler summation, summation by parts, and the Polya-Vinogradov inequality.Comment: Minor revisions per referee's comments. To appear in Acta Arithmetic

Hypernetworks for reconstructing the dynamics of multilevel systems

Networks are fundamental for reconstructing the dynamics of many systems, but have the drawback that they are restricted to binary relations. Hypergraphs extend relational structure to multi-vertex edges, but are essentially set-theoretic and unable to represent essential structural properties. Hypernetworks are a natural multidimensional generalisation of networks, representing n-ary relations by simplices with n vertices. The assembly of vertices to make simplices is key for moving between levels in multilevel systems, and integrating dynamics between levels. It is argued that hypernetworks are necessary, if not sufficient, for reconstructing the dynamics of multilevel complex systems

A Quantitative Vainberg Method for Black Box Scattering

We give a quantitative version of Vainberg's method relating pole free regions to propagation of singularities for black box scatterers. In particular, we show that there is a logarithmic resonance free region near the real axis of size $\tau$ with polynomial bounds on the resolvent if and only if the wave propagator gains derivatives at rate $\tau$. Next we show that if there exist singularities in the wave trace at times tending to infinity which smooth at rate $\tau$, then there are resonances in logarithmic strips whose width is given by $\tau$. As our main application of these results, we give sharp bounds on the size of resonance free regions in scattering on geometrically nontrapping manifolds with conic points. Moreover, these bounds are generically optimal on exteriors of nontrapping polygonal domains.Comment: 22 pages, 1 figur

Enhancing Cohort Identity in Legal Education

This poster explores student 'cohort identity' and its link to student belonging and enhancement. It presents strategies for enhancing cohort identity and identifies practices that have worked in practice