Exercise [2.1.5] Let > 0 and pick K = K( ) finite such that if k ≥ K then r(k) ≤ . Applying the Cauchy-Schwarz inequality for Xi −EXi and Xj −EXj we have that Cov(Xi, Xj) ≤ [Var(Xi)Var(Xj)] ≤ r(0) <∞ for all i, j. Thus, breaking the double sum in Var(Sn) = ∑n i,j=1 Cov(Xi, Xj) into {(i, j) : |i − j| < K} and {(i, j) : |i− j| ≥ K} gives the bound Var(Sn) ≤ 2Knr(0) + n . Dividing by n we see that...

sequence with minimized acquisition delays (left). The duration of each gradient is minimized individually. Thus the acquisition delay depends on the k-space position (right). Figure 2: Central slices of a phantom containing 0.9% saline solution measured with the normal 3D-CSI sequence (a), the 3D-TEMin CSI sequence with minimized acquisition delay (b),the 3D-AW-CSI sequence with acquisition we...

Abstract In this paper, we introduce the notions of regularly invariant convergence, strongly p -strongly $(\mathcal{I}_{\sigma },\mathcal{I}^{\sigma }_{2})$ ( I σ , 2 ) -co...

Let $G$ be a graph. Let $f:V(G)to{0,1,2, ldots, k-1}$ be a map where $k in mathbb{N}$ and $k>1$. For each edge $uv$, assign the label $left|f(u)-f(v)right|$. $f$ is called a $k$-total difference cordial labeling of $G$ if $left|t_{df}(i)-t_{df}(j)right|leq 1$, $i,j in {0,1,2, ldots, k-1}$ where $t_{df}(x)$ denotes the total number of vertices and the edges labeled with $x$.A graph with admits a...

Exercise [2.1.5] Let > 0 and pick K = K( ) finite such that if k ≥ K then r(k) ≤ . Applying the Cauchy-Schwarz inequality for Xi −EXi and Xj −EXj we have that Cov(Xi, Xj) ≤ [Var(Xi)Var(Xj)] ≤ r(0) <∞ for all i, j. Thus, breaking the double sum in Var(Sn) = ∑n i,j=1 Cov(Xi, Xj) into {(i, j) : |i − j| < K} and {(i, j) : |i− j| ≥ K} gives the bound Var(Sn) ≤ 2Knr(0) + n . Dividing by n we see that...

Yarn's [6] algorithm solves the problem of finding an optimal code tree, optimal in the sense of minimum average cost, when the code symbols are of unequal cost and the source symbols are equiprobable. He addresses both exhaustive codes, for which the code tree is a full tree, as well as nonexhaustive codes, but only the exhaustive case will be of concern here. In particular, for code symbol co...

The pressure dependences of sound velocities, refractive index and elastic constants in ice-VII phase have been determined up to 8 GPa at 300 K by high-pressure Brillouin spectroscopy developed for a single molecular crystal grown in a diamond anvil cell. The Cauchy relation of C12=C44 was found at each pressure. This result suggests that the interactions between the oxygen atoms can be describ...

This paper is the continuation of the work ”On an inverse problem for finite-difference operators of second order” ([1]). We consider the Cauchy problem for the Toda lattice in the case when the corresponding L-operator is a Jacobi matrix with bounded elements, whose spectrum of multiplicity 2 is separated from its simple spectrum and contains an interval of absolutely continuous spectrum. Usin...

This paper is the continuation of the work ”On an inverse problem for finite-difference operators of second order” ([1]). We consider the Cauchy problem for the Toda lattice in the case when the corresponding L-operator is a Jacobi matrix with bounded elements, whose spectrum of multiplicity 2 is separated from its simple spectrum and contains an interval of absolutely continuous spectrum. Usin...