Appendices A. Derivation of the CTC-EM Equations In the maximization step, the objective is to obtain the derivative of Q τ (w|x, z, w (n)) with respect to the input of the softmax layer, a t k , at time t. We first differentiate Q τ with respect to y t k at w = w (n) : ∂Q τ (w|x, z, w (n)) ∂y t k w=w (n) = |z| m=0 p(z 1:m |Z, x 1:τ , w (n)) ∂ ln p(z 1:m |x 1:τ , w (n)) ∂y t k. (1) With Bayes' ...

Ž . X Given a series-parallel network network, for short N, its dual network N is given by interchanging the series connection and the parallel connection of network N. We usually use a series-parallel graph to represent a network. Let w x w X x X G N and G N be graph representations of N and N , respectively. A sequence Ž w x w X x. of edges e , e , . . . , e is said to form a common trail on ...

Let $D$ be a finite simple digraph with vertex set $V(D)$ and arcset $A(D)$. A twin signed total Roman dominating function (TSTRDF) on thedigraph $D$ is a function $f:V(D)rightarrow{-1,1,2}$ satisfyingthe conditions that (i) $sum_{xin N^-(v)}f(x)ge 1$ and$sum_{xin N^+(v)}f(x)ge 1$ for each $vin V(D)$, where $N^-(v)$(resp. $N^+(v)$) consists of all in-neighbors (resp.out-neighbors) of $v$, and (...

Let N N be the normalizer of a maximal torus T"> T encoding="application/x-tex">T in split reductive ...

= zu2(i + k) for all i E I,. If I = Z we call w a Z-word or a bi-infinite word. If I is finite we call x finite and its length Ixl is defined as The usual notation for a finite word of length n is x = xi ... zn where all xi E {a, &}. We write {a, b}* for the collection of finite words and {a, b~+ for the nonempty finite words. (The empty word will be denoted throughout by e). The concatena...

Goals of this lecture-series: • Discuss existence of standing waves for the nonlinear Schrödinger equation (NLS) (1) i ∂w ∂t = −∆w + ˜ V (x)w − Γ(x)|w| p−1 w, x ∈ R n , t ∈ R for functions w : R n × R → C and for p > 1. • Discuss aspects of symmetry breaking and bifurcation. 1. Standing waves and their variational formulation A solution w(x, t) = u(x)e −iωt with u decaying to 0 at infinity is c...

This is a supplement to the paper " Kernel estimators of asymp-totic variance for adaptive Markov Chain Monte Carlo " and contains the proofs to Theorems 4.1-4.3. For improved readability, we recall the theorems and their assumptions. 1. Statement of the theorems. A1 For each θ ∈ Θ, P θ is phi-irreducible, aperiodic with invariant distribution π. There exists a measurable function V : X → [1, ∞...

Abstract The n-th order Wiener index of a molecular graph G was put forward by Estrada et al. [New J. Chem. 22 (1998) 819] as ( ) 1 ( , ) n n x W H G x where ( , ) H G x is the Hosoya polynomial. Recently Brückler et al. [Chem. Phys. Lett. 503 (2011) 336] considered a related graph invariant, ( ) 1 1 (1/ !) ( ( , )) / n n n n x W n d x H G x d x . For n=1, both W and W reduce to the ordinary W...