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As a contribution to the study of Hartree-Fock theory we prove rigorously that the Hartree-Fock approximation to the ground state of the ddimensional Hubbard model leads to saturated ferromagnetism when the particle density (more precisely, the chemical potential µ) is small and the coupling constant U is large, but finite. This ferromagnetism contradicts the known fact that there is no magnetization at low density, for any U , and thus shows that HF theory is wrong in this case. As in the usual Hartree-Fock theory we restrict attention to Slater determinants that are eigenvectors of the z-component of the total spin, S z = x n x,↑ − n x,↓ , and we find that the choice 2S z = N = particle number gives the lowest energy at fixed 0 < µ < 4d.

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This paper has been motivated by the curiosity that the circulant matrix ${\rm Circ }(1/2, -1/4, 0, \dots, 0,-1/4)$ is the $n\times n$ positive semidefinite, tridiagonal matrix $A$ of smallest Euclidean norm having the property that $Ae = 0$ and $Af = f$, where $e$ and $f$ are, respectively, the vector of all $1$s and the vector of alternating $1$ and $-1$s. It then raises the following question (minimization problem): What should be the matrix $A$ if the tridiagonal restriction is replaced by a general bandwidth $2r + 1$ ($1\leq r \leq \tfrac{n}{2 } -1$)? It is first easily shown that the solution of this problem must still be a circulant matrix. Then the determination of the first row of this circulant matrix consists in solving a least-squares problem having $\tfrac{n}{2} \, - 1$ nonnegative variables (Nonnegative Orthant) subject to $\tfrac{n}{2} - r$ linear equations. Alternatively, this problem can be viewed as the minimization of the norm of an even function vanishing at the points $|i|>r$ of the set $\left\{-\tfrac{n}{2} + 1, \dots, -1, 0, 1, \dots ,\tfrac{n}{2} \right\}$, and whose Fourier-transform is nonnegative, vanishes at zero, and assumes the value one at $\tfrac{n}{2}$. Explicit solutions are given for the special cases of $r=\tfrac{n}{2}$, $r=\tfrac{n}{2} -1$, and $r=2$. The solution for the particular case of $r=2$ can be physically interpreted as the vibrational mode of a ring-like chain of masses and springs in which the springs link both the nearest neighbors (with positive stiffness) and the next-nearest neighbors (with negative stiffness). The paper ends wiih a numerical illustration of the six cases ($1\leq r \leq 6$)corresponding to $n=12$.

Abstract. Let C be a n × n symmetric matrix. For each integer 1 ≤ k < n we consider the minimization problem m(ε) := min X {Tr{C X } + ε f (X )}. Here the variable X is an n × n symmetric matrix, whose eigenvalues satisfythe number ε is a positive (perturbation) parameter and f is a Lipchitz-continuous function (in general nonlinear). It is well known that when ε = 0 the minimum value, m(0), is the sum of the smallest k eigenvalues of C. Assuming that the eigenvalues of C satisfywe establish the following upper and lower bounds for the minimum value m(ε):wheref is the minimum value of f over the solution set of unperturbed problem and L is the Lipschitz-constant of f . The above inequality shows that the error by replacing the upper bound (or the lower bound) by the exact value is at least quadratic in the perturbation parameter. We also treat the case that λ k+1 (C) = λ k (C). We compare the exact solution with the upper and lower bounds for some examples.Mathematical subject classification: 15A42, 15A18, 90C22.

We optimize the spring constants $k^{i,j}$ (stiffness) of circular spring-mass systems with nearest-neighbour (NN) and next-nearest-neighbour (NNN) springs only. In this optimization problem, such systems are also subjected to damping and periodic external forces. The function to be minimized is the average ratio of the square norm of the on-site internal forces (response) to the square norm of the external on-site forces (input). Under the average of this response/input ratio is meant the average over time and over all configurations of external forces. As main result, it is established that the optimum stiffness matrix converges to the discrete and periodic bi-Laplacian operator as the size $n$ of the system increases. Such a result is obtained under the following assumptions: (a) the system has the natural mode shape (eigenvector) of alternating $1$s and $-1$s; and (b) the (external) forcing frequency is at least $1.095$ times higher than the highest natural frequency. It is remarkable that this optimum stiffness matrix exhibits negative stiffness for the springs linking NNN point masses. More specifically, as $n$ increases, $0&gt; k^{i,i+2} \, \, = \, \, - \tfrac{1}{4} \, k^{i,i+1}$ is the relation between the optimum NNN spring constant and the optimum NN spring constant. Such systems illustrate that the introduction of negative stiffness springs in some specific positions does in fact reduce the average response/input ratio. Numerical tables illustrating the main result are given.