Mathematical Analysis generally, including Real Analysis, Harmonic Analysis, Complex Variable Theory, the Calculus of Variations, Measure Theory, and Non-Standard Analysis.
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0answers
16 views
Completely continuous operator
how to prove that an operator
$T:K\rightarrow C[0,1]$ such that $K=\lbrace u\in C[0,1], u(t)\geq a(t)||u|| \text{on} [0,1] \rbrace \subset C[0,1]$
is completly continuous
(Article: Positive ...
1
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1answer
10 views
Asymptotic Approximation and Sign Convention
When I write the asymptotic approximation of a function, does the sign convention matter? i.e. suppose I have (though the formula might not make sense) $$f_n(x)=x^2+\dots-O(n),$$
If my function is ...
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1answer
24 views
How to use the Mean Value Theorem to find the “Contraction Constant”
Show that the contraction $T(x)= (1+x)^{1/3} $ on the interval $I=[1,2]$ satisfies the definition of a contraction.
It's not just this problem-- on this site and others explanations will say "the ...
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0answers
16 views
Question about theorem 3.2 from Morse theory by Milnor
THe demonstration of the theorem 3.2 in the book Morse theory by Milnor
is given in the special case whene the manifold is the Torus ,
My question is : can i prove it in the case where the ...
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votes
1answer
43 views
Compute the contour integral $\int_\gamma\frac{\sin z}{z^4}dz$
Im getting really confused looking at past exam style questions evaluating contour integrals... Can anyone help me in the right direction to solve these..
(i) $\displaystyle\int_\gamma \frac{\sin ...
-1
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0answers
23 views
find a special element in group algebra
Let $G=\langle x, y, z| xyx^{-1}=zy, xzx^{-1}=z, yz=zy\rangle$, denote $l^1(G)^{\times}$ to be the set of units in $l^1(G)$, which we have considered as a ring with multiplication defined by the usual ...
1
vote
4answers
36 views
Ratio Test $\sum\limits_{n=0}^\infty\frac{(2n)^{2n}}{(2n)!}$
Im getting a little stuck on the following Ratio Test example... I haven't done it in so long, but it forms part of a question in a complex analysis past exam question
$\displaystyle\sum_{n=0}^\infty ...
2
votes
1answer
99 views
Infinite series involving $\sqrt{n}$
I am looking for examples of infinite series, whose sum is expressed as distributions or known functions, with a $\sqrt{n}$ in each term, such as:
$$ \sum_{n=0}^{\infty} \sqrt{n} z^n, \quad ...
1
vote
2answers
31 views
Klein Bottle discrete harmonics?
Studying discrete representations of a function, on a $2$ Dimensional compact surface, brings to the use of spherical harmonics for the 2-sphere, and discrete Fourier transformations for the 2-torus.
...
1
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1answer
25 views
prove this equation has at least two real solutions
I want to prove that $2^x=2-x^2$ has at least two real solutions. I am trying to use Bolzano theorem to prove this, that is if $f(x)=2^x-2+x^2$=$0$ then $f(x)$ is continuous on R will be negative ...
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2answers
25 views
Inner product convention for $\ell^p$?
So I'm reading through some analysis problems and one is discussing $\ell^p$ (the space of $p$-summable sequences $x: \mathbb Z^+ \to \mathbb C$ such that $\sum_{n \in \mathbb Z^+}|x_n|^p < ...
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votes
3answers
86 views
Is there a way to compute $\lim\limits_{x\to\pi/3}\frac{\sqrt{3+2\cos x}-2}{\ln(1+\sin3x)}$ without using L'hopital?
I can compute
$$\lim_{x\to\pi/3}\frac{\sqrt{3+2\cos x}-2}{\ln(1+\sin3x)}$$ using L'hopital and the limit equals $\frac{\sqrt{3}}{12}$, but is there another way to compute this limit without using ...
2
votes
2answers
52 views
$\psi:A\rightarrow A^{-1}$ is continuous
Define a map that takes a matrix to it's inverse. Give $A\in M(n\times n)$ over reals field, define:
$$\psi:A\rightarrow A^{-1}$$
Is it always continuous, where defined? How do I prove this?
Thanks.
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votes
1answer
24 views
How to sum arithmetic combinations of divergent zeta-values?
By means of Ramanujan Summation, one can sum divergent values of the Riemann-zeta function. For instance, we have:
$$ \zeta(-2k) = 1 + 2^{2k} + 3^{2k} + \dots = 0 ( \mathfrak{R} ) , $$ and
$$ ...
3
votes
2answers
84 views
Prove there exists $y\neq 0$ but $x\cdot y=0$
I would like to know if I'm missing something with my solution - as an earlier version was wrong and I think I've managed to patch it up - to this problem from Rudin's Principles of Mathematical ...
