Graphical models provide compact representations of complex probability distributions of many random variables through a collection of potential functions defined on small subsets of these variables. This representation is defined with respect to a graph in which nodes represent random variables and edges represent the interactions among those random variables. Graphical models provide a powerf...

In this lecture, I put forward conjectures asserting that, in all noncommutative field theories, (1) open Wilson lines and their descendants constitute a complete set of interpolating operators of ‘noncommutative dipoles’, obeying dipole relation, (2) infrared dynamics of the noncommutative dipoles is dual to ultraviolet dynamics of the elementary noncommutative fields, and (3) open string fiel...

Given an integer h, a graph G = (V,E) with arbitrary positive edge capacities and k pairs of vertices (s1, t1), (s2, t2), . . . , (sk, tk), called terminals, an h-route cut is a set F ⊆ E of edges such that after the removal of the edges in F no pair si−ti is connected by h edge-disjoint paths (i.e., the connectivity of every si− ti pair is at most h− 1 in (V,E\F )). The h-route cut is a natura...

Wei’s celebrated Duality Theorem is generalized in several ways, expressed as duality theorems for linear codes over division rings and, more generally, duality theorems for matroids. These results are further generalized, resulting in two Wei-type duality theorems for new combinatorial structures that are introduced and named demi-matroids. These generalize matroids and are the appropriate com...

For completeness, in this section we derive the dual (5) to the problem of computing proximal operator for the ERM objective (3).

We study (approximate) null-controllability of parabolic equations in $L_p(\mathbb{R}^d)$ and provide explicit bounds on the control cost. In particular we consider systems form $\dot{x}(t) = -A_p x(t) + \mathbf{1}_E u(t)$, $x(0) x_0\in L_p (\mathbb{R}^d)$, with interior a so-called thick set $E \subset \mathbb{R}^d$, where $p\in [1,\infty)$, $A$ is an elliptic operator order $m \in \mathbb{N}$...