Neural oscillators, which are a mathematical model of central pattern generator, have been used to investigate human gait and control mobile robots. A typical neural oscillator uses two neurons that mutually inhibit each other’s activity. The exact orbitally-stable conditions the activity oscillators without external signals reported. However, behavior with is unclear because their depends on s...

Problem 7.8. (a) Let ω ∈ Ω(V ). Show that there is a basis φ1, . . . , φn of V ∗ such that ω = (φ1 ∧ φ2) + · · ·+ (φ2r−1 ∧ φ2r). (b) Show that the r-fold wedge product ω∧· · ·∧ω is non-zero and decomposable, and that the (r + 1)-fold wedge product is 0. Thus r is well-determined; it is called the rank of ω. (c) If ω = ∑ i<j aijψi ∧ ψj, and A is the upper triangular matrix with Aij = aij for i <...

A zero sum game is a simultaneous move game between 2 players. Such a game is represented by a matrix A ∈ Rm×n. The strategies of the “row” (resp. “column”) player are the rows (resp. columns) of A. If the row player plays strategy i ∈ [m] and the column player plays j ∈ [n] then the outcome is Aij . 1 Interpret this as that the row player pays Aij amount of money to the column player, therefor...

OBJECTIVES The purpose of the presentin vitro study was to predict to what extent dietary fiber (DF) takes up heterocyclic aromatic amine (HAA) and how DF acts to intercept HAAsin vivo. METHODS The sorption isotherms of 2-amino-3-methyl-3H-imidazo[4,5-f]quinoline (IQ), 2-amino-3,4-dimethyl-3H-imidazo[4,5-f]quinoline (MeIQ), 3-amino-1,4-dimethyl-5H-pyrido[4,3-b]indole acetate (Trp-P-1), and 3-...

A crucial problem in a decision-making process is the determination of a scale of relative importance for a set X $x1, x2, . . . , xn % of alternatives either with respect to a criterion C or an expert E. A widely used tool in Multicriteria Decision Making is the pairwise comparison matrix A ~aij !, where aij is a positive number expressing how much the alternative xi is preferred to the altern...

For a matrix A = [aij ] m,n i,j=1 ∈ Fm×n, the transpose of A is the matrix A> = [aji] n,m j,i=1 ∈ Fn×m. A square matrix A ∈ Rn×n is called symmetric if aji = aij for all i, j ∈ {1, . . . , n} and is called skew-symmetric or anti-symmetric if aji = −aij for all i, j ∈ {1, . . . , n}. A basis will be denoted B = {u1,u2, . . . ,un} when the ordering of the basis vectors is not important and B = [u...

= −divax, t∇u + b⃗x, t ⋅ ∇u + cx, tux, t with aij, bj, c ∈ LUT and for a0 > 0, z⃗ ⋅ aijx, t z⃗ ≥ a0 | z⃗| ∀z⃗ ∀x, t ∈ UT. Then L is said to be uniformly elliptic on UT. We recall that the assumptions on the coefficients aij, bj, c, imply the existence of positive constants, a1,a0 and μ0 such that for all u,v ∈ H1U, and μ > μ0, |Bu,v, t| ≤ a1‖u‖1‖v‖1 a.e. t ∈ 0,T Bu,u, t +...

A polynomially parametrized family of continuous-time controllable linear systems is always stabilizable by polynomially parametrized feedback. 1. RESULTS Theorem 1. Let (Aλ,Bλ) be a pair of matrices, all whose entries are real polynomial functions of the parameter λ∈Rr. Assume that Aλ is n×n , Bλ is n×m, and that the pair (Aλ,Bλ) is controllable for each λ∈Rr. Then, there exists an m×n matrix ...

Let Bn be an aperiodic Brandt semigroup M0[G;n, n;P ] where G is the trivial group, n ∈ N and P = (aij)n×n with aij = 1G if i = j and aij = 0 otherwise. The maximum size of an independent set in Bn is known to be bn2/4c+n, where bn2/4c denotes the largest integer not greater than n2/4. We reprove this result using Turán’s famous graph theorem. Moreover, we give a characterization of all indepen...