Let Q be a nite set of generalized quantiiers. By L k (Q) we denote the k-variable fragment of FO(Q), rst order logic extended with Q. We show that for each k, there is a PFP(Q)-deenable linear pre-order whose equivalence classes in any nite structure A are the L k (Q)-types in A. For some special classes of generalized quantiiers Q, we show that such an ordering of L k (Q)-types is already dee...

In 2008 Brändén proved a (p,q)-analogue of the γ-expansion formula for Eulerian polynomials and conjectured divisibility γ-coefficient γn,k(p,q) by (p+q)k. As follow-up, in 2012 Shin Zeng showed that fraction γn,k(p,q)/(p+q)k is polynomial N[p,q]. The aim this paper to give combinatorial interpretation latter terms André permutations, class objects first defined studied Foata, Schützenberger St...

In a recent Letter [1], Kondo and Morimatsu present a QCD sum rule calculation of nucleon-nucleon scattering lengths. They also relate the empirical scattering lengths to the nucleon mass shift δM in nuclear matter to cast doubt on the " linear density approximation. " In this Comment, we point out flaws in both parts of their analysis. It is useful to start with the (nonrelativistic) optical p...

In this paper we continue the investigation into the group of algebras with uniformly distributed invariants, U(k’), and its relation to the Schur subgroup S(K), undertaken in [6, 71. We maintain the notation of [6, 71. In the first section we investigate U(K) for a real quadratic field K. We calculate generators of U(K) explicitly. In the second section we investigate U(K) for other real field...

We demonstrate propagation rules of subsystem code constructions by extending, shortening and combining given subsystem codes. Given an [[n, k, r, d]]q subsystem code, we drive new subsystem codes with parameters [[n + 1, k, r,≥ d]]q, [[n − 1, k + 1, r,≥ d − 1]]q , [[n, k − 1, r + 1, d]]q . We present the short subsystem codes. The interested readers shall consult our companion papers for upper...

In this article, some structures in the projective plane of order q are found which allow us to construct small k regular balanced bipartite graphs of girth 6 for all k ≤ q . When k = q , the order of these q-regular graphs is 2(q2−1); andwhen k ≤ q−1, the order of these k -regular graphs is 2(qk − 2). Moreover, the incidence matrix of a k -regular balanced bipartite graph of girth 6 having 2(q...

Let $n_q(k,d)$ denote the smallest value of $n$ for which there exists a linear $[n,k,d]$-code over the Galois field $GF(q)$. An $[n,k,d]$-code whose length is equal to $n_q(k,d)$ is called {em optimal}. In this paper we present some matrix generators for the family of optimal $[n,3,d]$ codes over $GF(7)$ and $GF(11)$. Most of our given codes in $GF(7)$ are non-isomorphic with the codes pre...

and Applied Analysis 3 Theneuron activation functions,g p (y p (⋅)) (p = 1, . . . , n), are assumed to be nondecreasing, bounded, and globally Lipschitz; that is, l − p ≤ g p (ξ p ) − g p (ξ q ) ξ p − ξ q ≤ l + p , ∀ξ p , ξ q ∈ R, ξ p ̸ = ξ q , (5) where l− p and l+ p are constant values. For simplicity, in stability analysis of the network (1), the equilibrium point y∗ = [y∗ 1 , . . . , y ∗ n ]...

We prove that $L(SL_2(\textbf{k}))$ is a maximal Haagerup von Neumann subalgebra in $L(\textbf{k}^2\rtimes SL_2(\textbf{k}))$ for $\textbf{k}=\mathbb{Q}$. Then we show how to modify the proof handle $\textbf{k}=\mathbb{Z}$. The key step complete description of all intermediate subalgebras between and $L^{\infty}(Y)\rtimes SL_2(\textbf{k})$, where $SL_2(\textbf{k})\curvearrowright Y$ denotes quo...