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        Questions tagged [differential-calculus]

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        1answer
        72 views

        Existence of a certain type of function

        Trying to find functions with the given property: Given $M>0, K$ compact in $\mathbf{R^n}$,$f:U\rightarrow\mathbf{R}$ a $C^2$ function, where $U$ open in $\mathbf{R^n}$ and $K\subset U$such that $...
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        2answers
        150 views

        Dominant root of a family of polynomials

        Let $f(x)=x^5-x^4-x^3-x^2-x-c$, where $c>2$ is a real number. It is easy to prove that there exists a positive real root $\alpha>2$ of $f(x)$ and all the other roots are non real. Also, by ...
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        0answers
        52 views

        Looking for example of integral transformations that preserve number of zeros

        Let $f:\mathbb{R} \to \mathbb{R} $ have $n<\infty$ zeros. I am looking for non-trivial examples of integral transformation \begin{align} g(x)= \int f(t) h(t,x) dt \end{align} such that $f$ and $g$...
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        3answers
        109 views

        Literature request: Function that depends on a linear optimization problem [closed]

        my question is about functions of the following form: $$ f(t) = \max_{\mathbf{x}}~ \mathbf{c^T x} ~ {\rm s.t. \mathbf{Ax} +t \cdot \mathbf{a} \leq \mathbf{b}}, $$ where $\mathbf{x},\mathbf{b}, $ ...
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        1answer
        257 views

        A Conjecture about the integral related to Chebyshev polynomial

        I am interested in the following integral related to the Chebyshev polynomials $$I_{n,m}:= \int_0^\pi \left(\frac {\sin nx}{\sin x}\right)^{m} dx,$$ where $n,m\in \mathbb{Z}^+.$ It is easy to see ...
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        1answer
        123 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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        0answers
        327 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
        141 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,...
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        1answer
        161 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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        2answers
        443 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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        0answers
        77 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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        0answers
        118 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 ...
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        vote
        1answer
        212 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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        3answers
        880 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
        207 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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