Most of the density-to-potential inversion methods developed over years follow a general algorithm $ v_{xc}^{i+1}(\textbf{r}) = v_{xc}^{i}(\textbf{r}) + \Delta v_{xc}(\textbf{r})$, where $\Delta v_{xc}(\textbf{r}) \frac{\delta S[\rho]}{\delta \rho(\textbf{r})} \Big |_{ \rho_i(\textbf{r})} - \rho(\textbf{r})}\Big|_{ \rho_0(\textbf{r})}$ and $S[\rho]$ is an appropriately chosen density functional...

1. If p is a prime other than 2, half of the numbers 1, 2, ..., p-l are quadratic residues (modp) and the other half are quadratic non-residues. Various questions have been proposed concerning the distribution of the quadratic residues and non-residues for large p, but as yet only very incomplete answers to these questions are known. Many of the known results are deductions from the inequality ...

Freedman’s inequality is a martingale counterpart to Bernstein’s inequality. This result shows that the large-deviation behavior of a martingale is controlled by the predictable quadratic variation and a uniform upper bound for the martingale difference sequence. Oliveira has recently established a natural extension of Freedman’s inequality that provides tail bounds for the maximum singular val...

An optimal control problem for a semilinear heat equation with distributed is discussed, where two-sided pointwise box constraints on the and mixed control-state are given. The objective functional sum of standard quadratic tracking type part multiple L 1 -norm that accounts sparsity. Under certain structural condition almost active sets solution, existence integrable Lagrange multipliers prove...

In this paper, we establish global optimality conditions for quadratic optimization problems with quadratic equality and bivalent constraints. We first present a necessary and sufficient condition for a global minimizer of quadratic optimization problems with quadratic equality and bivalent constraints. Then, we examine situations where this optimality condition is equivalent to checking the po...

In this paper, we investigate the Hyers-Ulam stability for the system of additive, quadratic, cubicand quartic functional equations with constants coecients in the sense of dectic mappings in non-Archimedean normed spaces.

The celebrated S-lemma establishes a powerful equivalent condition for the nonnegativity of a quadratic function over a single quadratic inequality. However, this lemma fails without the technical condition, known as the Slater condition. In this paper, we first show that the Slater condition is indeed necessary for the S-lemma and then establishes a regularized form of the S-lemma in the absen...

Let χ be a primitive Dirichlet character to the modulus q. Let Sχ(M,N) = ∑ M<n≤N χ(n). The Pólya-Vinogradov inequality states that |Sχ(M,N)| √ q log q. The smoothed Pólya–Vinogradov inequality, recently introduced by Levin, Pomerance and Soundararajan, is a numerically useful version of the Pólya–Vinogradov inequality that saves a log q factor. The smoothed Pólya–Vinogradov inequality has been ...

No. There has never been good evidence for a pattern of rising inequality in low-income countries and falling inequality in higher income countries. The only evidence that appears to support the Kuznets hypothesis is the cross-sectional pattern of inequality levels across countries, although the Kuznets hypothesis is an assertion about the path of inequality within countries. Numerous cross-sec...