Pantelis Sopasakis
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 Dec28 awarded Yearling Dec28 answered $f(x):S^{2n} \rightarrow S^{2n}$ continuous so that there is $x \in S^{2n}$ with $f(x)=x$ or $f(x)=-x$ Dec28 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 Dec28 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 Dec28 revised Prove: $abc\geqslant 162$ Math Format (I changed some * into \cdot) & some rephrasing Dec28 suggested approved edit on Prove: $abc\geqslant 162$ Dec15 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 $(-1)^2=1$). Dec12 revised fractional derivative of a heaviside function Format of equations (introduced $) Dec12 comment Reducing a system of differential equations It's just a thought...$\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})$. Dec12 suggested approved edit on fractional derivative of a heaviside function Dec12 comment Reducing a system of differential equations I guess you can do so if the higher-dimensional equation "doesn't need" the additional states. I guess that what you are looking for is not generally true (unless you impose additional conditions on$\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)$? Dec12 comment Solving$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}$and the solution follows easily. Dec12 revised Maximal Positive Invariant Set — Some fine print deleted 2 characters in body Dec12 comment Calculating$\frac{\partial^{1/2}}{\partial x^{1/2}}\left( e^{-\alpha x^2 + \beta x} \right) $It seems that the access to your paper is restricted. Can you provide the details of the paper? (authors, journal, etc.) Dec12 answered Kernel of Fractional Differential Operator Dec10 comment fractional derivaitve of logarithm function$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). Dec9 revised Solutions of fractional linear dynamical systems added 208 characters in body; edited title Dec9 revised Solutions of fractional linear dynamical systems added 42 characters in body Dec9 asked Solutions of fractional linear dynamical systems Dec9 awarded Caucus