It is shown that the Jacobian Conjecture holds for all polynomial maps F : k → k of the form F = x + H , such that JH is nilpotent and symmetric, when n ≤ 4. If H is also homogeneous a similar result is proved for all n ≤ 5. Introduction Let F := (F1, . . . , Fn) : C → C be a polynomial map i.e. each Fi is a polynomial in n variables over C. Denote by JF := (i ∂xj )1≤i,j≤n, the Jacobian matrix ...

Wang, Li and König have recently compared the cluster-theoretic properties of bi-stochasticized symmetric data similarity (e. g. kernel) matrices, produced by minimizing two different forms of Bregman divergences. We extend their investigation to non-symmetric matrices, specifically studying the 1995-2000 U. S. 3, 107× 3, 107 intercounty migration matrix. A particular bi-stochastized form of it...

We consider the first-order Cauchy problem ∂zu + a(z, x,Dx)u = 0, 0 < z ≤ Z, u |z=0 = u0, with Z > 0 and a(z, x,Dx) a k×k matrix of pseudodifferential operators of order one, whose principal part a1 is assumed symmetrizable: there exists L(z, x, ξ) of order 0, invertible, such that a1(z, x, ξ) = L(z, x, ξ) (−iβ1(z, x, ξ) + γ1(z, x, ξ)) (L(z, x, ξ))−1, where β1 and γ1 are hermitian symmetric and...

(2) ^ © = A' X(t) + X\t) A + B, te(0,co) dt for continuous systems. Here A is a given stable matrix, i.e. all its eigenvalues a, satisfy |a,| < i or Re a,< 0, respectively, and B is a given symmetric matrix. This difference/differential equation is to be solved with a given initial condition X(0). Of special interest is a limit for / -» co; it is a solution of the stationary Lyapunov equation (...

We study the matrix equation X+A>X−1A = Q, where A is a complex square matrix and Q is complex symmetric. Special cases of this equation appear in Green’s function calculation in nano research and also in the vibration analysis of fast trains. In those applications, the existence of a unique complex symmetric stabilizing solution has been proved using advanced results on linear operators. The s...

Let {X ij }, i, j = · · · , be a double array of independent and identically distributed (i.i.d.) real random variables with EX 11 = µ, E|X 11 − µ| 2 = 1 and E|X 11 | 4 < ∞. Consider sample covariance matrices (with/without empirical centering) S = 1 n n j=1 (s j − ¯ s)(s j − ¯ s) T and S = 1 n n j=1 s j s T j , where ¯ s = 1 n n j=1 s j and s j = T 1/2 n (X 1j , · · · , X pj) T with (T 1/2 n) ...

Real inner product. Let V be a vector space over R. A (real) inner product is a function 〈−,−〉 : V × V → R such that • 〈x, y〉 = 〈y, x〉 for all x, y ∈ V, • 〈c1x1 + c2x2, y〉 = c1〈x1, y〉+ c2〈x2, y〉 for all x1, x2, y ∈ V, c1, c2 ∈ R, • 〈x, x〉 ≥ 0 with 〈x, x〉 = 0 iff x = 0. That is, the pairing is symmetric, linear in the first variable (and therefore bilinear, by symmetry), and positive definite. E...

We study the initial value problem for integrable nonlocal nonlinear Schr\"odinger (NNLS) equation \[ iq_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0 \] with symmetric boundary conditions: $q(x,t)\to Ae^{2iA^2t}$ as $x\to\pm\infty$, where $A>0$ is an arbitrary constant. describe asymptotic stage of modulation instability NNLS by computing large-time asymptotics solution $q(x,t)$ this proble...

One of the major drawbacks of the symmetric QR algorithm is that it is not parallelizable. Each orthogonal similarity transformation that is needed to reduce the original matrix A to diagonal form is dependent upon the previous one. In view of the evolution of parallel architectures, it is therefore worthwhile to consider whether there are alternative approaches to reducing an n × n symmetric m...