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-4
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0answers
59 views

Should I use chain rule for this? $y = \sqrt[3]{(x+1)\sqrt[3]{(x^2+1)\sqrt[3]{x^3+1}}}$ , find $y'(0)$ [on hold]

The question is: $y = \sqrt[3]{(x+1)\sqrt[3]{(x^2+1)\sqrt[3]{x^3+1}}}$ , find $y'(0)$. I know that I can move the exponent $\frac{1}{3}$ down and differentiate the rest and go on and on, I'm ...
0
votes
0answers
50 views

Integrate the product of Fresnel integrals

If $S(x)=\int_0^x {\sin t^2 }\, dt$ and $C(x)=\int_0^x {\cos t^2\, dt}$ then how to express the following integral $\int S(x)C(x)\, dx$ in terms of $S(x)$ and $C(x)$?
1
vote
3answers
93 views

Literature request: Function that depends on a linear optimization problem

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}, $ ...
0
votes
0answers
141 views

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
213 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
121 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 ...
0
votes
0answers
41 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 ...
16
votes
0answers
320 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 ...
0
votes
1answer
91 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
votes
1answer
157 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 ...
3
votes
2answers
438 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 $\...
1
vote
0answers
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 ...
1
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0answers
114 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
155 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 )^\...
12
votes
3answers
805 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 ...
0
votes
2answers
204 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} \...
0
votes
0answers
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 ...
28
votes
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 ...
1
vote
2answers
181 views

Uniqueness of tangent space given local injectivity of orthogonal projection onto it

Definition. Let $X\subset \mathbb R^n$ be a locally Euclidean subset. Say it has a tangent space at $p\in X$ if there exists a linear subspace $V\leq \mathbb R^n$ satisfying the following conditions. ...
4
votes
1answer
363 views

Is the space of tangents actually the tangent space?

This is a crosspost of this MSE question. Given a locally Euclidean (locally homeomorphic to some Euclidean space) topological subspace $X\subset\mathbb R^n$ and $p\in X$, let $\mathrm{T}_pX$ denote ...
2
votes
0answers
311 views

Is there Calculus for (Almost) Continuous functions?

So I asked a similar question to this on Math Stack Exchange a couple of weeks ago, did a bounty, and I didn't receive any answers. I am struggling a bit with a part of my research (on CS). Suppose ...
2
votes
0answers
117 views

Computing harmonic sum [closed]

I want to show the following equalities for harmonic sum $$\sum_{k=1}^{\infty}\frac{(-1)^k}{k^3}=\lim_{n\to\infty}\int_0^{2n}\frac{-3}{x^4}([x]-2[\frac{x}{2}])dx$$ Any idea?
2
votes
0answers
90 views

Solve 4th order ODE with variable coefficients

I am trying to solve a 4th order boundary value problem with variable coefficients, namely the problem of a rotating cantilever beam: $u'''' - \frac{((1-x^2)u')'}{2\eta} - \frac{\alpha}{\eta}((1-x)u')...
1
vote
1answer
497 views

Question about the validity of Michael Grossman, Robert Katz and Jane Grossman Non-Newtonian / Meta / Multiplicative Calculus [closed]

I am no mathematician by far, but I have studied diff. and integral calc and beyond, in my undergrad years. I recently came upon this book: "The First Systems of Weighted Differential and Integral ...
1
vote
0answers
97 views

Inequality involving joint distribution and marginal distributions [closed]

Let $f(x,y)$ be a joint density of $X$ and $Y$. Let $f_{X}(x)$ and $f_{Y}(y)$ be the marginal density functions. Is the following inequality true? $$ \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}...
0
votes
0answers
32 views

Equivalent definitions of $C^1-$boundary

I am studying PDE, and I have two definition of $C^1$ open set as follow: Definition 1. (Evans' PDE book) An open set $\Omega \subset \mathbb{R}^N$ is $C^1$ if for each point $x_0 \in \partial \...
4
votes
1answer
364 views

An inequality inspired by the isoperimetric inequality

Let us consider the simplest isoperimetric inequality. Consider a smooth simple closed curve given by $r=\rho(\theta)$ in polar coordinates, where $\rho(\theta)>0$ can be regarded as a smooth ...
1
vote
1answer
141 views

Given a continuous function of many variables, how do I know when this function is equals to zero on all the corners of a unit hypercube?

Given a continuous and deriviable function of many variables, how do I know when this function is equals to zero on all the corners (or vertices) of a unit hypercube, i.e. all points, where each ...
0
votes
1answer
77 views

Maximizing sum of a product of logs

I came across the following note in a paper I'm reading and don't understand how it was derived. $\max_{\alpha_\ell}\sum_\ell^L\beta_\ell\log\alpha_\ell$ such that $\sum_\ell^L\alpha_\ell=1$ and $\...
0
votes
0answers
50 views

Existence of local minimizer

For a $f\in C^3$ function, if there is a sufficiently small $\epsilon$ $$\| \nabla F(x) \| < \epsilon$$ and a sufficiently large $\alpha$ where $$\lambda_{\min}[\nabla^2 F(x)] \ge \alpha$$ Can ...
6
votes
3answers
652 views

“Universal” differential identities

(This is a cross-post from MSE). Let $f:\mathbb{R}^d \to \mathbb{R}$ be smooth. The mixed derivatives commute: $f_{xy}=f_{yx}$. This identity is "universal" in the sense that it holds for any smooth ...
0
votes
1answer
145 views

Fixed point iteration using derivatives

Let $g(x)= \mathrm{\lim_{n \rightarrow \infty}}\,f^{\circ n} (x) $, where $f^{\circ n} (x)$ is the repeated application of $f$ to $x$, $n$ times. Why doesn't the following procedure work for ...
0
votes
1answer
159 views

The limitation of derivation of modified Bessel function of second kind

The final result I draw is related to the integral of modified Bessel function of the second kind. But I can not solve it, and I need a explicit solution Are you willing to help me? Thank all $I = \...
29
votes
1answer
855 views

How quickly can the derivative of an everywhere differentiable function change sign?

Let $f : [a,b] \to \Bbb R$ be everywhere differentiable with $f'(a) = 1$ and $f'(b) =-1$. By Darboux theorem, we know that $f'([a,b])$ is an interval containing $[-1,1]$. In particular, the set $\{x \...
0
votes
0answers
151 views

Time derivative of a quaternion-valued function in exponential form

I was wondering when a quaternion valued function $q(t) = e^{if_1 (t)+jf_2 (t) + kf_3 (t)}$ were written in exponential form, if there was a nice formula to express the time derivative $q'(t)$ in ...
2
votes
1answer
84 views

Question about optimizing a given function by optimizing an approximation

Let $f$ be a real-valued function. Suppose I want to find a local maximum of $f$, but I decide to work with an ''approximation'' to $f$ --let us call it $g$. What is a suitable notion of ''...
1
vote
0answers
305 views

Symmetry of higher order mixed partial derivatives under weaker assumptions

Suppose $U$ is an open subset of $\mathbb{R}^n$, and $f:U\to \mathbb{R}$. When $f$ is $C^2$ we know that the mixed partial derivatives are symmetric, i.e. $\partial_i\partial_jf= \partial_j\...
11
votes
2answers
716 views

Derivative formula

While trying to understand a paper of Cayley, he left something unexplained, I managed to show that it is equivalent to the following formula, which I got stuck at: $k \cdot (f^k)^{(k-1)} = \sum_{j=0}...
1
vote
1answer
411 views

The level sets of a differentiable function is a manifold

I have seen stronger propositions that imply this, but both their statement and their proofs require more advanced tools than I'd like to use in my text, which is aimed at a general scientific ...
3
votes
1answer
249 views

nth derivative of error function

Let $f(x)=erfi(a+x)$ and $g(x)=e^{cx}$ with \begin{align*} f^{(n)}(x)=\frac{2}{\sqrt{\pi}}e^{(a+x)^2}\sum_{m=0}^{n-1}\sum_{j=0}^{m}\frac{\binom{m}{j}(-1)^j(a+x)^{2m-n+1}}{m!}\\ \prod_{p=1}^{n-1}(2m-2j-...
4
votes
0answers
116 views

Basic calculus on topological fields

Let $K$ be a a topological field (I am mainly interested in the cases when K is either an ordered field or a valued field, e.g. $K = \mathbb Q$ or $ \mathbb Q_p$). 1) Let $f: K^n \to K$ be a ...
1
vote
0answers
359 views

The derivative of an integral function with indicator and max function as integrand

I encounter the following type of problem: \begin{equation} F(x) = \int_a^b \mathbf{1}_{\{v+x-h(v)\geq 0\}}\max\{h(v)-y-x,0\}dv \end{equation} where $\mathbf{1}_{\{z\geq 0\}}=1$ if event $z\geq 0$ ...
4
votes
1answer
406 views

Derivative is Zero on a dense G_delta set

I have the following question: I have a function $f: \mathbb R \to \mathbb R$ which is differentiable everywhere. I also have a set $G\subset\mathbb R$ which is dense in $\mathbb R$ and a $G_\delta$-...
0
votes
0answers
109 views

The Lipschitz continuity of the maximizer of a Liptschitz function

Let $$\phi\left(t,b\right)=-\tilde{c}\left(b\right)\alpha e^{r(T-t)}-\lambda\int_{0}^{+\infty}\exp\left\{\alpha yb e^{r(T-t)}\right\} d G(y),$$ where $G(y)$ is the distribution of a positive random ...
2
votes
2answers
467 views

Commuting of exterior derivative and contraction (vector-valued forms)

$\newcommand{\sig}{\sigma}$ $\newcommand{\tr}{\operatorname{tr}_{\eta}}$ $\newcommand{\al}{\alpha}$ $\newcommand{\be}{\beta}$ $\newcommand{\til}{\tilde}$ Let $E$ be a smooth vector bundle over a ...
2
votes
0answers
62 views

Combinatorial identity of Derivatives of super-Gaussian function

An asymptotic expansion I stumbled upon has real numbers $c^\alpha_{ij}$ as coefficients, where $i, j \in \mathbb{N}_0$ are non-negative integers and $\alpha \in \mathbb{N}_0^n$ is a multi-index. They ...
1
vote
0answers
73 views

Developing a functional equation for log-integral of theta function

I'm studying $u(z,q):=\exp(\left[\tfrac{\pi q}{12}-\ln2-\sum_{k\ge1}\ln(1-e^{-2k\pi q})\right]z+\int_1^z\ln\varphi(x,q)\,dx)$ , with $$\varphi(z,q)=2e^{-\pi qz^2-\pi q/4}\sinh \pi qz\prod_{k\ge1}(1-e^{...
2
votes
0answers
119 views

Universal chord theorem for curves

Let $\mathrm{\gamma} : [0,1] \to \mathbb{R}^2$ be a piecewise smooth, simple plane curve. Assume $\gamma(0) = (0,0)$, $\gamma(1) = (1,0)$ and that the slope of the tangent is not $0$ wherever it's ...
2
votes
0answers
86 views

Approximating a $C^1$ function in $Lip$ norm with piecewise linear

For a continuous function $f:[a,b]\to R$ there is a natural and obvious procedure to approximate it with a sequence of continuous, piecewise linear functions: take $N$ equally spaced points in $[a,b]$ ...
27
votes
1answer
2k views

Is there an explicit formula for the hessian of “Determinant”?

Let $f: G= \mbox{GL}(n,\mathbb{R}) \to \mathbb{R}$ be the determinant function. Then $\mbox{Hess} (f)$ is a two linear map on $M_{n}(\mathbb{R})\simeq T_{e}(G)$ where $e$ is the neutral element of $G$,...