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Given maps 
$f_1,\ldots ,f_n:X\to Y$ between (finite and connected) graphs, with 
$n\geq 3$ (the case 
$n=2$ is well known), we say that they are loose if they can be deformed by homotopy to coincidence free maps, and totally loose if they can be deformed by homotopy to maps which are two by two coincidence free. We prove that: (i) if Y is not homeomorphic to the circle, then any maps are totally loose; (ii) otherwise, any maps are loose and they are totally loose if and only if they are homotopic.
$L^p$-improving measures in 
${\Bbb F}_q^d$
The purpose of this paper is to introduce and study the following graph-theoretic paradigm. Let where 
$$ \begin{align*}T_Kf(x)=\int K(x,y) f(y) d\mu(y),\end{align*} $$
$f: X \to {\Bbb R}$, X a set, finite or infinite, and K and 
$\mu $ denote a suitable kernel and a measure, respectively. Given a connected ordered graph G on n vertices, consider the multi-linear form where 
$$ \begin{align*}\Lambda_G(f_1,f_2, \dots, f_n)=\int_{x^1, \dots, x^n \in X} \ \prod_{(i,j) \in {\mathcal E}(G)} K(x^i,x^j) \prod_{l=1}^n f_l(x^l) d\mu(x^l),\end{align*} $$
${\mathcal E}(G)$ is the edge set of G. Define 
$\Lambda _G(p_1, \ldots , p_n)$ as the smallest constant 
$C>0$ such that the inequality (0.1)holds for all nonnegative real-valued functions 
$$ \begin{align} \Lambda_G(f_1, \dots, f_n) \leq C \prod_{i=1}^n {||f_i||}_{L^{p_i}(X, \mu)} \end{align} $$
$f_i$, 
$1\le i\le n$, on X. The basic question is, how does the structure of G and the mapping properties of the operator 
$T_K$ influence the sharp exponents in (0.1). In this paper, this question is investigated mainly in the case 
$X={\Bbb F}_q^d$, the d-dimensional vector space over the field with q elements, 
$K(x^i,x^j)$ is the indicator function of the sphere evaluated at 
$x^i-x^j$, and connected graphs G with at most four vertices.
Let G be a complex classical group, and let V be its defining representation (possibly plus a copy of the dual). A foundational problem in classical invariant theory is to write down generators and relations for the ring of G-invariant polynomial functions on the space 
$\mathcal P^m(V)$ of degree-m homogeneous polynomial functions on V. In this paper, we replace 
$\mathcal P^m(V)$ with the full polynomial algebra 
$\mathcal P(V)$. As a result, the invariant ring is no longer finitely generated. Hence, instead of seeking generators, we aim to write down linear bases for bigraded components. Indeed, when G is of sufficiently high rank, we realize these bases as sets of graphs with prescribed number of vertices and edges. When the rank of G is small, there arise complicated linear dependencies among the graphs, but we remedy this setback via representation theory: in particular, we determine the dimension of an arbitrary component in terms of branching multiplicities from the general linear group to the symmetric group. We thereby obtain an expression for the bigraded Hilbert series of the ring of invariants on 
$\mathcal P(V)$. We conclude with examples using our graphical notation, several of which recover classical results.
