In this note, we estimate the distance between two $q$-nomial coefficients $\left|{\left(\genfrac{}{}{0pt}{}{n}{k}\right)}_q-{\left(\genfrac{}{}{0pt}{}{n^{\text{'}}}{k^{\text{'}}}\right)}_q\right|$, where $(n,k)\ne (n^{\text{'}},k^{\text{'}})$ and $q\ge 2$ is an integer.

The surjectivity of the operator $D_2+i{x}_2^{2k}{D}_1$ from the Gevrey space ${\mathcal{E}}^{\left\{s\right\}}\left({\mathbb{R}}^2\right)$, $s\ge 1$, onto itself and its non-surjectivity from ${\mathcal{E}}^{\left\{s\right\}}\left({\mathbb{R}}^3\right)$ to ${\mathcal{E}}^{\left\{s\right\}}\left({\mathbb{R}}^3\right)$ is proved.

The scope of this note is a self-contained presentation of a mathematical method that enables us to give an absolute upper bound for the difference of the Gini coefficients $$\left|G({\sigma }_1,\cdots ,{\sigma }_n)-G({\gamma }_1,\cdots ,{\gamma }_n)\right|,$$ where $({\gamma }_1,\cdots ,{\gamma }_n)$ represents the vector of the gross wages and $({\sigma }_1,\cdots ,{\sigma }_n)$ represents the vector of the corresponding super-gross wages that is used in the Czech Republic for calculating the net wage. Since (as of June 2019) ${\sigma }_i=100·⌈1.34{\gamma }_i/100⌉$, the study of the above difference seems to be somewhat inaccessible for many economists. However, our estimate...

Let: $𝐘=\left({𝐘}_i\right)$, where ${𝐘}_i=\left(Y_{i,1},...,Y_{i,d}\right)$, $i=1,2,\cdots$, be a $d$-dimensional, identically distributed, stationary, centered process with uniform marginals and a joint cdf $F$, and $F_n\left(𝐱\right):=\frac{1}{n}{\sum }_{i=1}^n𝕀\left(Y_{i,1}\le x_1,\cdots ,Y_{i,d}\le x_d\right)$ denote the corresponding empirical cdf. In our work, we prove the almost sure central limit theorem for an empirical process $B_n=\sqrt{n}\left(F_n-F\right)$ under some weak dependence conditions due to Doukhan and Louhichi. Some application of the established result to copula processes is also presented.

Let $P\left(z\right)={\sum }_{\nu =0}^n{a}_{\nu }{z}^{\nu }$ be a polynomial of degree at most $n$ which does not vanish in the disk $\left|z\right|<1$, then for $1\le p<\infty$ and $R>1$, Boas and Rahman proved $${∥P\left(Rz\right)∥}_p\le ({∥R^n+z∥}_p/{∥1+z∥}_p){∥P∥}_p.$$ In this paper, we improve the above inequality for $0\le p<\infty$ by involving some of the coefficients of the polynomial $P\left(z\right)$. Analogous result for the class of polynomials $P\left(z\right)$ having no zero in $\left|z\right|>1$ is also given.

Let $N$ be a sufficiently large integer. We prove that almost all sufficiently large even integers $n\in [N-6U,N+6U]$ can be represented as $$n=p_1^2+p_2^3+p_3^3+p_4^3+p_5^3+p_6^3,\left|p_1^2-{\displaystyle \frac{N}{6}}\right|\le U,\left|p_i^3-{\displaystyle \frac{N}{6}}\right|\le U,i=2,3,4,5,6,$$ where $U=N^{1-\delta +\epsilon }$ with $\delta \le 8/225$.

The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an $L^2\left(ℝ\right)$ maximum principle, in the form of a new “log” conservation law which is satisfied by the equation (1) for the interface. Our second result is a proof of global existence for unique strong solutions if the initial data is smaller than an explicitly computable constant, for instance ${∥f∥}_1\le 1/5$. Previous results...

Let $\mathrm{\Omega }$ be a bounded open convex set of class $C^2$. Let $a\left(x,H\left(u\right)\right)$ be a non linear operator satisfying the condition (A) (elliptic) with constants $\alpha$, $\gamma$, $\delta$. We prove that a number $\lambda \ge 0$ is an eigenvalue for the operator $a\left(x,H\left(u\right)\right)$ if and only if the number $\alpha \lambda$ is an eigen-value for the operator $\mathrm{\Delta }u$. If $\lambda \ge 0$ , the two systems $a\left(x,H\left(u\right)\right)=\lambda u$ and $\mathrm{\Delta }u=\alpha \lambda u$ have the same solutions. In particular, also the eventual eigen-values of the operator $a\left(x,H\left(u\right)\right)$ should all be negative. Finally, we obtain a sufficient condition for the existence of solutions $u\in H^2\cap H_0^1\left(\mathrm{\Omega }\right)$...

We study the existence of solutions of the system $$\left\{\begin{array}{c}{\left({\phi }_1\left(u_1^{\text{'}}\left(t\right)\right)\right)}^{\text{'}}=f_1(t,u_1\left(t\right),u_2\left(t\right),u_1^{\text{'}}\left(t\right),u_2^{\text{'}}\left(t\right)),\text{a.e.}t\in [0,T],{\left({\phi }_2\left(u_2^{\text{'}}\left(t\right)\right)\right)}^{\text{'}}=f_2(t,u_1\left(t\right),u_2\left(t\right),u_1^{\text{'}}\left(t\right),u_2^{\text{'}}\left(t\right)),\text{a.e.}t\in [0,T],\hfill \end{array}\right.$$ submitted to nonlinear coupled boundary conditions on $[0,T]$ where ${\phi }_1,{\phi }_2:(-a,a)\to ℝ$, with $0<a<+\infty$, are two increasing homeomorphisms such that ${\phi }_1\left(0\right)={\phi }_2\left(0\right)=0$, and $f_i:[0,T]\times {ℝ}^4\to ℝ$, $i\in \{1,2\}$ are two $L^1$-Carathéodory functions. Using some new conditions and Schauder fixed point Theorem, we obtain solvability result.