We deal with an optimal matching problem, that is, we want to transport two measures to a given place (the target set) where they will match, minimizing the total transport cost that in our case is given by the sum of two different multiples of the Euclidean distance that each measure is transported. We show that such a problem has a solution with an optimal matching measure supported in the target set. This result can be proved by an approximation procedure using a p-Laplacian system. We prove that any optimal matching measure for this problem is supported on the boundary of the target set when the two multiples that affect the Euclidean distances involved in the cost are different. Moreover, we present simple examples showing uniqueness or non-uniqueness of the optimal measure.
Prestel introduced a generalization of the notion of an ordering of a field, which is called a semiordering. Prestel’s axioms for a semiordered field differ from the usual (Artin-Schreier) postulates in requiring only the closedness of the domain of positivity under $x \rightarrow x a^2$ for nonzero $a$, instead of requiring that positive elements have a positive product. In this work, this type of ordering is studied in the case of a division ring. It is shown that it actually behaves the same as in the commutative case. Further, it is shown that the bounded subring associated with that ordering is a valuation ring which is preserved under conjugation, so one can associate a natural valuation to a semiordering.
We give a construction of $p$ orthogonal Latin $p$-dimensional cubes (or Latin hypercubes) of order $n$ for every natural number $n\ne 2,6$ and $p \ge 2$. Our result generalizes the well known result about orthogonal Latin squares published in 1960 by R. C. Bose, S. S. Shikhande and E. T. Parker.
In this paper, oscillation and asymptotic behaviour of solutions of
\[ y^{\prime \prime \prime} + a(t)y^{\prime \prime}+b(t)y^{\prime} + c(t)y=0 \] have been studied under suitable assumptions on the coefficient functions $a,b,c\in C([\sigma ,\infty),R)$, $ \sigma \in R$, such that $a(t)\ge 0$, $b(t) \le 0$ and $c(t) < 0$.
In this paper, necessary and sufficient conditions are obtained for every bounded solution of \[ [y (t) - p (t) y (t - \tau )]^{(n)} + Q (t) G \bigl (y (t - \sigma )\bigr ) = f (t), \quad t \ge 0, \qquad \mathrm{(*)}\] to oscillate or tend to zero as $t \rightarrow \infty $ for different ranges of $p (t)$. It is shown, under some stronger conditions, that every solution of $(*)$ oscillates or tends to zero as $t \rightarrow \infty $. Our results hold for linear, a class of superlinear and other nonlinear equations and answer a conjecture by Ladas and Sficas, Austral. Math. Soc. Ser. B 27 (1986), 502–511, and generalize some known results.
In this work, oscillatory behaviour of solutions of a class of fourth-order neutral functional difference equations of the form ∆ 2 (r(n)∆2 (y(n) + p(n)y(n − m))) + q(n)G(y(n − k)) = 0 is studied under the assumption ∑∞ n=0 n ⁄ r(n) < ∞. New oscillation criteria have been established which generalize some of the existing results in the literature.
The paper deals with the oscillation of a differential equation $L_4y+P(t)L_2y+Q(t)y\equiv 0$ as well as with the structure of its fundamental system of solutions.
et $p\in (0,1)$ be a real number and let $n\ge 2$ be an even integer. We determine the largest value $c_n(p)$ such that the inequality
\[ \sum ^n_{i=1} |a_i|^p \ge c_n(p) \] holds for all real numbers $a_1,\ldots ,a_n$ which are pairwise distinct and satisfy $\min _{i\ne j} |a_i-a_j| = 1$. Our theorem completes results of Ozeki, Mitrinović-Kalajdžić, and Russell, who found the optimal value $c_n(p)$ in the case $p>0$ and $n$ odd, and in the case $p\ge 1$ and $n$ even.