Let $k\subseteq k'$ be a field extension. We give relations between the kernels of higher derivations on $k[X]$ and $k'[X]$, where $k[X]:=k[x_1,\dots ,x_n]$ denotes the polynomial ring in $n$ variables over the field $k$. More precisely, let $D=\{D_n\}_{n=0}^\infty $ a higher $k$-derivation on $k[X]$ and $D'=\{D_n'\}_{n=0}^\infty $ a higher $k'$-derivation on $k'[X]$ such that $D'_m(x_i)=D_m(x_i)$ for all $m\geq 0$ and $i=1,2,\dots ,n$. Then (1) $k[X]^D=k$ if and only if $k'[X]^{D'}=k'$; (2) $k[X]^D$ is a finitely generated $k$-algebra if and only if $k'[X]^{D'}$ is a finitely generated $k'$-algebra. Furthermore, we also show that the kernel $k[X]^D$ of a higher derivation $D$ of $k[X]$ can be generated by a set of closed polynomials.
The main purpose of this paper is to use the analytic method to study the calculating problem of the general Kloosterman sums, and give an exact calculating formula for it.
Let a \subseteq \mathbb{C} [x1, ..., xn] be a monomial ideal andJ(a^{c}) the multiplier ideal of a with coefficient c. Then J(a^{c}) is also a monomial ideal of \mathbb{C} [x1, ..., xn], and the equality J(a^{c}) = a implies that 0 < c < n + 1. We mainly discuss the problem when J (a) = a or J({a^{n = 1 - \varepsilon }}) = a for all 0 < ε < 1. It is proved that if J (a) = a then a is principal, and if J({a^{n = 1 - \varepsilon }}) = a holds for all 0 < ε < 1 then a = (x1, ..., xn). One global result is also obtained. Let ã be the ideal sheaf on \mathbb{P}^{n-1} associated with a. Then it is proved that the equality J (ã) = ã implies that ã is principal., Cheng Gong, Zhongming Tang., and Obsahuje seznam literatury
Let $D$ be a positive integer, and let $p$ be an odd prime with $p\nmid D$. In this paper we use a result on the rational approximation of quadratic irrationals due to M. Bauer, M. A. Bennett: Applications of the hypergeometric method to the generalized Ramanujan-Nagell equation. Ramanujan J. 6 (2002), 209–270, give a better upper bound for $N(D, p)$, and also prove that if the equation $U^2-DV^2=-1$ has integer solutions $(U, V)$, the least solution $(u_1, v_1)$ of the equation $u^2-pv^2=1$ satisfies $p\nmid v_1$, and $D>C(p)$, where $C(p)$ is an effectively computable constant only depending on $p$, then the equation $x^2-D=p^n$ has at most two positive integer solutions $(x, n)$. In particular, we have $C(3)=10^7$.
We deal with the optimal portfolio problem in discrete-time setting. Employing the discrete It\^o formula, which is developed by Fujita, we establish the discrete Hamilton-Jacobi-Bellman (d-HJB) equation for the value function. Simple examples of the d-HJB equation are also discussed.
We give a sufficient condition for the oscillation of linear homogeneous second order differential equation $y^{\prime \prime }+p(x)y^{\prime }+q(x)y=0$, where $p(x), q(x)\in C[\alpha ,\infty )$ and $\alpha $ is positive real number.
A condition for solvability of an integral equation which is connected with the first boundary value problem for the heat equation is investigated. It is shown that if this condition is fulfilled then the boundary considered is 1⁄2-Hölder. Further, some simple concrete examples are examined.
In this note we give a negative answer to Zem�nek's question (1994) of whether it always holds that a Cesàro bounded operator $T$ on a Hilbert space with a single spectrum satisfies $\lim _{n \rightarrow \infty } \|T^{n+1} - T^n\| = 0.$.
In this paper, we deal with strong stationarity conditions for mathematical programs with equilibrium constraints (MPEC). The main task in deriving these conditions consists in calculating the Fréchet normal cone to the graph of the solution mapping associated with the underlying generalized equation of the MPEC. We derive an inner approximation to this cone, which is exact under an additional assumption. Even if the latter fails to hold, the inner approximation can be used to check strong stationarity via the weaker (but easier to calculate) concept of M-stationarity.