For every oriented tree T there exists a graph DT (called the dual of T ) such that T 6→ G ⇔ G → DT holds for every G (an arrow denotes the existence of a homomorphism). An explicit construction of DT has been found recently. Although the DT constructed this way may have exponential number of vertices in |V (T )| = n, we will prove that its diameter is linear in n (and therefore DT is ”small” i...

$K$-frames as a generalization of frames were introduced by L. Gu{a}vruc{t}a to study atomic systems on Hilbert spaces which allows, in a stable way, to reconstruct elements from the range of the bounded linear operator $K$ in a Hilbert space. Recently some generalizations of this concept are introduced and some of its difference with ordinary frames are studied. In this paper, we give a new ge...

We study the rapid decay property and polynomial growth for duals of bicrossed products coming from a matched pair discrete group compact

In this paper, we study the duals of some finite dimensional pointed Hopf algebras working over an algebraically closed field k of characteristic 0. In particular, we study pointed Hopf algebras with coradical k[Γ] for Γ a finite abelian group, and with associated graded Hopf algebra of the form B(V )#k[Γ] where B(V ) is the Nichols algebra of V = ⊕iV χi gi ∈ k[Γ] k[Γ]YD. As a corollary to a ge...

A linear code with a complementary dual (or an LCD code) is defined to be a linear code C whose dual code C⊥ satisfies C ∩ C⊥ = {0}. The algebraic characterization of LCD codes is given, and it is shown that asymptotically good LCD codes exist. LCD codes are shown to provide an optimum linear coding solution for the two-user binary adder channel. The nearest-neighbor (or maximum-likelihood) dec...

We introduce the notion of the automorphic dual of a matrix algebraic group defined over Q . This is the part of the unitary dual that corresponds to arithmetic spectrum. Basic funcional properties of this set are derived and used both to deduce arithmetic vanishing theorems of "Ramanujan" type as well as to give a new construction of automorphic forms. Let G be a semisimple linear algebraic gr...

We describe a primal-dual framework for the design and analysis of online convex optimization algorithms for drifting regret. Existing literature shows (nearly) optimal drifting regret bounds only for the l2 and the l1-norms. Our work provides a connection between these algorithms and the Online Mirror Descent (OMD) updates; one key insight that results from our work is that in order for these ...