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        Is it a weaker condition for the symmetry of mixed derivatives?

        Asked in MathStackExchange and I think it may be harder than usual problems. Conditions: $f:\mathbb{R^2}\mapsto\mathbb{R}$ has second-order derivatives on some neighborhood of the zero point. $f_{...
        4
        votes
        1answer
        207 views

        A Conjecture about the integral related to Chebyshev polynomial

        Recently, I am interested in the following integral related to the Chebychev polynomials $$I_{nm}:= \int_0^\pi \left(\frac {\sin nx}{\sin x}\right)^{m} dx,$$ where $n,m\in \mathbb{Z}^+.$ It is easy ...
        3
        votes
        1answer
        120 views

        A geometric property about certain polynomials in two variables

        Assume that $p(x,y)$ is a polynomial in $\mathbb{R}[x, y]$ in the form $$ p=p_{2n}+ p_{2n-1}+\ldots +p_1+p_0$$ where $p_i$ is a homogenous polynomial of degree $i$. Moreover we assume that the last ...
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        40 views

        Sensitivity of Lagrangian solution: implicit constraint

        just a question about a literature reference. I am writing a paper for engineers. Usually for the Lagrange multiplier problem ?f(x)+λ?g(x)=0 the sensitivity result that the multiplier λ gives the ...
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        318 views

        An integral in Gradshteyn and Ryzhik

        Section 3.248 of the 4th edition of the table of integrals by Gradshteyn and Ryzhik contains three entries. They are of elementary examples of the beta function. In the 5th edition there are two new ...
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        1answer
        86 views

        Properties of the argmin function (continuity, differentiability..) [closed]

        Given vectors $x_1,\ldots,x_N \in \mathbb{R}^d$ and a function, say $\psi \colon \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}$, one is interested in the properties of the function $$\Phi\colon (x_1,...
        3
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        1answer
        152 views

        If Derivative at each point $x \in \Bbb R^n$ is an orthogonal matrix then $f(x) = Ox +b $ [duplicate]

        Let $f : \Bbb R^n \to \Bbb R^n$ be a $C^1$ map such that it's Derivative at each point $x \in \Bbb R^n$ is an orthogonal matrix i.e. $Df_x \in O(n,\Bbb R) \text{ , } \forall x \in \Bbb R^n$ . Then ...
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        437 views

        Curl as a divergence… Is it possible? [closed]

        I want to know if it is possible to express the operation $$ \nabla \phi \times (\nabla \times \mathbf A) $$ as the divergence of second order tensor field $T$. Here $ \phi$ is a scalar field and $\...
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        72 views

        Reference for numerical solutions for differential equations like $f'(x)=f(x+1)+f(x-1)$

        One can solve a delay differential equation (like for example $f'(x)=f(x-1)$) if we have a function as a bounded condition (in my example we need to know $f$ on $[0,1)$) and then use a simple forward ...
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        113 views

        Determining a Closed Formula for the Positive Zeroes of the $n^{th}$ Derivatives of the Function $x?x^{-x}$

        The derivatives of the function $ f(x)=x^{-x}$ have interesting properties, especially when looking at their roots. I am interested in studying the behavior of the roots of the derivatives as the ...
        1
        vote
        1answer
        140 views

        integrate exponential function of quadratic form over unit norm vectors

        Let $A$ and $b$ be an $M\times M$ matrix and an $M\times 1$ vector, respectively. I need to solve $$\int_{\|x\|^2=1} \exp\left(-x^\ast Ax + 2\mathcal{R}\{x^\ast b\}\right) \, dx$$ where $(\cdot )^\...
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        796 views

        Differentiability of Eigenvalues - Perturbation Theory

        first, I have a general question. In perturbation theory, I saw perturbations in eigenvalues and eigenvectors of square, non-symmetric matrices and the calculations were all right but no one ever ...
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        2answers
        203 views

        exactness of a 1-form [closed]

        This may be a trivial question. If so, apologies in advance. Let a $1$-form $\omega$ be given. Suppose that for any line segments $C_1=[i,j]$, $C_2=[j,k]$, $C_3=[i,k]$, we have that $\int_{C_1} \...
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        68 views

        About Hessians of functions

        Say one is given a twice differentiable (at least) function $f : \mathbb{R}^n \rightarrow \mathbb{R}$. What are some conditions when one can put a lowerbound on the smallest eigenvalue of its ...
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        2answers
        1k views

        Is it always possible to calculate the limit of an elementary function?

        I already asked this question on https://math.stackexchange.com/questions/2691331/is-it-always-possible-to-calculate-the-limit-of-an-elementary-function but as I received no answer; maybe it is not as ...

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