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Register a new userDifferentiability of the $n$-Variable Function Deduced by the Differentiability of the $n-1$-Variable Function
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In this paper, some sufficient conditions for the differentiability of the $n$-variable real-valued function are obtained, which are given based on the differentiability of the $n-1$-variable real-valued function and are weaker than classical conditions.
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Two articles published by Information Science discuss the derivatives of interval functions, in the sense of Svetoslav Markov. The authors of these articles tried to characterize for which functions and points such derivatives exist. Unfortunately, their characterization is inaccurate. This article describes this inaccuracy and explains how it can be corrected.
Let $ xgeq 1 $ be a large number, let $ [x]=x-{x} $ be the largest integer function, and let $ varphi(n)$ be the Euler totient function. The result $ sum_{nleq x}varphi([x/n])=(6/pi^2)xlog x+Oleft ( x(log x)^{2/3}(loglog x)^{1/3}right ) $ was proved very recently. This note presents a short elementary proof, and sharpen the error term to $ sum_{nleq x}varphi([x/n])=(6/pi^2)xlog x+O(x) $. In addition, the first proofs of the asymptotics formulas for the finite sums $ sum_{nleq x}psi([x/n])=(15/pi^2)xlog x+O(xlog log x) $, and $ sum_{nleq x}sigma([x/n])=(pi^2/6)xlog x+O(x log log x) $ are also evaluated here.
Let $ xgeq 1 $ be a large number, let $ [x]=x-{x} $ be the largest integer function, and let $ sigma(n)$ be the sum of divisors function. This note presents the first proof of the asymptotic formula for the average order $ sum_{pleq x}sigma([x/p])=c_0xlog log x+O(x) $ over the primes, where $c_0>0$ is a constant. More generally, $ sum_{pleq x}sigma([x/(p+a)])=c_0xlog log x+O(x) $ for any fixed integer $a$.
Let $R = k[w, x_1,..., x_n]/I$ be a graded Gorenstein Artin algebra . Then $I = ann F$ for some $F$ in the divided power algebra $k_{DP}[W, X_1,..., X_n]$. If $RI_2$ is a height one idealgenerated by $n$ quadrics, then $I_2 subset (w)$ after a possible change of variables. Let $J = I cap k[x_1,..., x_n]$. Then $mu(I) le mu(J)+n+1$ and $I$ is said to be generic if $mu(I) = mu(J) + n+1$. In this article we prove necessary conditions, in terms of $F$, for an ideal to be generic. With some extra assumptions on the exponents of terms of $F$, we obtain a characterization for $I = ann F$ to be generic in codimension four.
There is error in (2.1). I am very sorry for inconvenience.
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