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 Yearling
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Dec
28
awarded  Yearling
Dec
28
answered $f(x):S^{2n} \rightarrow S^{2n}$ continuous so that there is $x \in S^{2n}$ with $f(x)=x$ or $f(x)=-x$
Dec
28
comment $f(x):S^{2n} \rightarrow S^{2n}$ continuous so that there is $x \in S^{2n}$ with $f(x)=x$ or $f(x)=-x$
It seems to me that this is a consequence of the Borsuk-Ulam theorem - See en.wikipedia.org/wiki/Borsuk%E2%80%93Ulam_theorem
Dec
28
comment $f(x):S^{2n} \rightarrow S^{2n}$ continuous so that there is $x \in S^{2n}$ with $f(x)=x$ or $f(x)=-x$
I saw you tagged your question as (algebraic-topology), so maybe $S$ is some particular space I am not aware of, but if it is compact you can use some standard fixed-point theorem such as en.wikipedia.org/wiki/Brouwer_fixed-point_theorem. Also in the title of your question you write $f(x)=\pm x$; what does $\pm$ stand for? Do you mean $f$ is either $f(x)=x$ or $f(x)=-x$, or $f$ is a multi-valued function? You may also want to take a look at en.wikipedia.org/wiki/Schauder_fixed_point_theorem
Dec
28
revised Prove: $abc\geqslant 162$
Math Format (I changed some * into \cdot) & some rephrasing
Dec
28
suggested approved edit on Prove: $abc\geqslant 162$
Dec
15
comment What did I do here? This can't be right… ($i = -1$)?
An equally interesting fallacy: $-1=(-1)^{2/2}=[(-1)^2]^{1/2}=\sqrt{1}=1$. As others have mentions in their answers the mistake here is at the step: $\sqrt{1}=1$ actually $\sqrt{1}$ is defined as the positive root of the equation $x^2=1$, but there is also a negative root here (because $(-1)^2=1$).
Dec
12
revised fractional derivative of a heaviside function
Format of equations (introduced $)
Dec
12
comment Reducing a system of differential equations
It's just a thought... $\mathbf{C}$ is a rectangular matrix and $\mathbf{C}^{-1}$ is the (multivalued) inverse mapping, so the dynamics is then described by a differential inclusion, however, you can pick a Lipschitz-continuous selection, say $c\in\mathbf{C}^{-1}$. Also, it is $\frac{\mathrm{d}\mathbf{v}(t)}{\mathrm{d} t}=\frac{\mathrm{d}\mathbf{Cu}(t)}{\mathrm{d}t}=\mathbf{C}'\frac{\mathrm{d} \mathbf{u} (t) }{\mathrm{d}t}=\mathbf{C}'\mathbf{F}(\mathbf{u})\in\mathbf{C}'\mathbf{F}(\mathbf‌​{C}^{-1}\mathbf{v})$.
Dec
12
suggested approved edit on fractional derivative of a heaviside function
Dec
12
comment Reducing a system of differential equations
I guess you can do so if the higher-dimensional equation "doesn't need" the additional states. I guess that what you are looking for is not generally true (unless you impose additional conditions on $\mathbf{F}$). Just a thought: let $\mathbf{v}(t) = \mathbf{C}\mathbf{u}(t)$, where $\mathbf{C}\in\mathbb{R}^{3\times n}$ and and notice that $\frac{\mathrm{d}\mathbf{v}(t)}{\mathrm{d}t}=\mathbf{C}'\mathbf{F}(\mathbf{u})$; can you write $\mathbf{u}(t)=\mathbf{C}^{-1}\mathbf{v}(t)$?
Dec
12
comment Solving $z^3=-1+i$
Write $z$ as $z=\rho e^{i\theta}$; you then need to determine $\rho$ and $\theta$. Then write the right-hand side complex number as $-1+i=\sqrt{2}e^{-i3\pi/4}$ and the solution follows easily.
Dec
12
revised Maximal Positive Invariant Set — Some fine print
deleted 2 characters in body
Dec
12
comment Calculating $\frac{\partial^{1/2}}{\partial x^{1/2}}\left( e^{-\alpha x^2 + \beta x} \right) $
It seems that the access to your paper is restricted. Can you provide the details of the paper? (authors, journal, etc.)
Dec
12
answered Kernel of Fractional Differential Operator
Dec
10
comment fractional derivaitve of logarithm function $x^ {a} log(x) $
Which fractional derivative are you considering? The main difficulty here is that the Leibniz rule doesn't hold for fractional-order derivatives. There are however certain extensions. For the Riemann-Liouville case, check out Section 2.7.2 in: I. Podlubny, "Fractional Differential Equations," Academic Press, 1999. For the Caputo derivative see Eq. (9) in K. Diethelm et al., "Algorithms for the fractional calculus: A selection of numerical methods," Comp. meth. appl. mech. eng. 194 (2005).
Dec
9
revised Solutions of fractional linear dynamical systems
added 208 characters in body; edited title
Dec
9
revised Solutions of fractional linear dynamical systems
added 42 characters in body
Dec
9
asked Solutions of fractional linear dynamical systems
Dec
9
awarded  Caucus