[x]i = (x (0) 0 , . . . , x (k−1) 0 , x (0) 1 , . . . , x (k−1) 1 , . . . , x (0) i−1, . . . , x (k−1) i−1 ), [y]i = (y (0) 0 , . . . , y (n−1) 0 , y (0) 1 , . . . , y (n−1) 1 , . . . , y (0) i−1, . . . , y (n−1) i−1 ). The column distance function (CDF) di is the minimum Hamming distance between all pairs of output sequences truncated at length i given that the input sequences differ in the fi...

This note provides new quantitative bounds for the recursive equation yn+1 = A + yn yn−k , n = 0, 1, . . . , where y−k, y−k+1, . . . , y−1, y0, A ∈ (0,∞) and k ∈ {2, 3, 4, . . .}. Issues regarding exponential convergence of solutions are also considered. In particular, it is shown that exponential convergence holds for all (A, k) for which global asymptotic stability was proven in R. M. Abu-Sar...

A reciprocal translocation between the long arm of the Y chromosome and the long arm of chromosome 1 was observed in an infertile man with non-obstructive azoospermia. The study was performed using a combination of techniques: immunocytogenetic analysis, which allows the detection of synaptonemal complexes (SCs) and recombination sites (MLH1) simultaneously, and fluorescence in-situ hybridizati...

1 Notion We introduce some notions used in this supplementary material. For regression task, we define y max = max y |y|. We further denote the set S as S = B 0, y max λ −1/2 if L2 is used and λ ≤1 R D otherwise where B 0, y max λ −1/2 = w ∈ R D : w ≤ y max λ −1/2 and R D specifies the whole feature space. We introduce five types of loss functions that can be used in our proposed algorithm, nam...

In Section 2 replace the definition of ∗◦Q in Definition 1 by x∗◦Q y = inf{u ∗◦ v | u > x, v > y}. It is defined only if the infimum exists. Proposition 1 remains unchanged. Theorem 0. Let (X, ∗◦,→∗◦,≤) be a commutative residuated semigroup on a complete chain equipped with the order topology. Let a, b, c ∈ X be such that a = b→∗◦ c. Let (x, y) ∈ X × X be such that 1. neither x nor y equals the...

and Applied Analysis 3 2. Some Identities of Bernoulli and Euler Polynomials By 1.4 , we get Ek ( x y ) k ∑ j 0 ( k j ) yk−jEj x , for ∈ Z . 2.1 From 2.1 , we note that Ek ( x y ) k ∑ j 0 ( k j ) yk−jEj x y k ∑ j 1 k j ( k − 1 j − 1 ) yk−jEj x . 2.2 Thus, we have k−1 ∑ j 0 ( k − 1 j ) yk−1−j Ej 1 x j 1 Ek ( x y ) − y k . 2.3 Replacing k by k 1 in 2.3 , we obtain the following proposition. Propo...

‎Let $G$ be a non-abelian group and let $Gamma(G)$ be the non-commuting graph of $G$‎. ‎In this paper we define an equivalence relation $sim$ on the set of $V(Gamma(G))=Gsetminus Z(G)$ by taking $xsim y$ if and only if $N(x)=N(y)$‎, ‎where $ N(x)={uin G | x textrm{ and } u textrm{ are adjacent in }Gamma(G)}$ is the open neighborhood of $x$ in $Gamma(G)$‎. ‎We introduce a new graph determined ...

Hölder’s inequality states that ‖x‖p ‖y‖q − 〈x, y〉 ≥ 0 for any (x, y) ∈ Lp(Ω) × Lq(Ω) with 1/p + 1/q = 1. In the same situation we prove the following stronger chains of inequalities, where z = y|y|q−2: ‖x‖p ‖y‖q − 〈x, y〉 ≥ (1/p) [( ‖x‖p + ‖z‖p )p − ‖x + z‖p ] ≥ 0 if p ∈ (1, 2], 0 ≤ ‖x‖p ‖y‖q − 〈x, y〉 ≤ (1/p) [( ‖x‖p + ‖z‖p )p − ‖x + z‖p ] if p ≥ 2. A similar result holds for complex valued fun...

We first prove that if y * optimizes (16), then |y * j | ≤ M, 1 ≤ j ≤ n. (EC.1) Suppose that in (16), given y and D, z * (y; D) optimizes ϕ(y; D). Then the objective function is C(y) = h · E[y − Az * (y; D)] + b · E[D − z * (y; D)]. i:a ji >0 a ji b i a ji E[D i − z * i ] ≥ ζ j A j · E[D − z * ] ≥ ζ j (A j · E[D] − y j), and C(y) ≥ h · E[y − Az * ] ≥ h j (y j − A j · E[z * ]) ≥ h j (y j − A j ·...

We suggest two grand unified models where the fifth dimensional coordinate is compactified on an S1/(Z2 × Z ′ 2) orbifold. Before showing the models, let us see the compactification of the fifth dimension. We denote the five dimensional coordinate as y, which is compactified on an S1/(Z2 × Z ′ 2) orbifold. Under the parity transformation of Z2 and Z ′ 2, which transform y → −y and y ′ → −y (y =...