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Proving linearity of the differential operator $a_n(x)D^n+a_{n-1}(x)D^{n-1}+\ldots+a_1(x)D+a_0(x)$.

I am taking a first course on ordinary differential equations(with background in linear algebra). I would like to know, if the proof to the below claim is correct (and that I am not using advanced facts to prove basic facts). Prove that every expression of the form: $$a_n(x)D^n+a_{n-1}(x)D^{n-1}+\ldots+a_1(x)D+a_0(x)$$ defines a linear transformation from $C^n[a,b]$ to […]

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Let X be a space and C a connected subset of X. Suppose $C \subset D\subset\bar C$. Then D is connected.

This is how i showed it works. Let X be a space and C a connected subset of X. Suppose, C $\subset$ D $\subset$ $\overline{C}$, Then D is connected.\ Let, C $\subset$ X be connected and suppose C $\subset$ D $\subset$ $\overline{C}$. Then, if U,V $\not=$ $\emptyset$ form a separation of D, then we have […]

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For non-negative reals such that $a+b+c\geq x+y+z$, $ab+bc+ca\geq xy+yz+zx$, and $abc\geq xyz$, show $a^k+b^k+c^k\geq x^k+y^k+z^k$ for $0

Let $a$, $b$, $c$, $x$, $y$, $z$ be non-negative real numbers such that $$a+b+c \geq x+y+z,$$ $$ab+bc+ca \geq xy+yz+zx,$$ $$ abc \geq xyz$$ Show that $$a^k+b^k+c^k \geq x^k+y^k+z^k, \quad 0 < k < 1$$ This is case $n=3$ of a problem posted by Ji Chen in the Art of Problem Solving forums, 2008. I have […]

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For non-negative reals such that $a+b+c\geq x+y+z$, $ab+bc+ca\geq xy+yz+zx$, and $abc\geq xyz$, show $a^k+b^k+c^k\geq x^k+y^k+z^k$ for $0

Let $a$, $b$, $c$, $x$, $y$, $z$ be non-negative real numbers such that $$a+b+c \geq x+y+z,$$ $$ab+bc+ca \geq xy+yz+zx,$$ $$ abc \geq xyz$$ Show that $$a^k+b^k+c^k \geq x^k+y^k+z^k, \quad 0 < k < 1$$ This is case $n=3$ of a problem posted by Ji Chen in the Art of Problem Solving forums, 2008. I have […]