Questions tagged [differential-calculus]

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A Bessel-like integral

I encounter the following integral when trying to find the inverse Fourier transform of the characteristic function of a certain sum of random variables. Here, $0\le\lambda\le1$, $p\ge0$, $q\ge0$ are ...
3
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0answers
70 views

Implicit function theorem for subdifferentiable convex functions

I am trying to find a method to apply the implicit function theorem for subdifferential convex functions. The original theorem provides an equation for the partial derivative of the implicit function ...
1
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3answers
215 views

Does the generalised directional derivative satisfy any version of the chain rule?

Is there a chain rule of any kind for the generalised directional derivative (of the Clarke type)? There is certainly a chain rule for the generalised gradient. The generalised directional derivative ...
-3
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1answer
59 views

Does having the derivative in the limit suffice to solve the function at the limit? [closed]

Suppose that I have a function $f(x, \epsilon)$ and I know that $$ \lim_{\epsilon \to 0} f'(x, \epsilon) = g'(x). $$ Now let $g(x)$ be the function whose derivative appears above. How can I ...
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0answers
27 views

Logarithmic integrals and derivates in terms of lower orders

We define $\partial^{k}\ln^{n}(x)$ like the k-order derivative of the function $\ln^{n}(x)$ for $n\in\mathbb{Z}_{+}$, and we define $\partial^{-k}ln^{n}(x)=\int^{k}\ln(x)dx...dx$ like the iterate k-...
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0answers
53 views

Density of integers related to the size of its order of appearance in the Fibonacci sequence

Let $z(n)=\min\{k>0 : n\mid F_k\}$. This function is known as the Fibonacci entry point (for example). A result of Sallé gives the sharpest upper bound for $z(n)$, namely, $z(n)\leq 2n$, for all $n$...
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0answers
100 views

Roots of a family of 4-parameter polynomials

Let $k, \ell, p$ and $q$ be positive integers, with $q>p>1$ and $\gcd(p,q)=1$. Let $f(x)$ the polynomial given by $$ f(x)=x^q-kx^{q-p}-\ell. $$ This polynomial is related to a family of two-...
3
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0answers
99 views

Hadamard lemma without integration

Let $I$ be the ideal of smooth germs vanishing at zero. Let $I^{k+1}$ be the ideal generated by $(k+1)$-fold product of such germs. Write $F_k$ for the ideal of $k$-flat germs at zero. By the product ...
2
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1answer
60 views

Dominant root of a family of odd degree polynomials

Let $q$ be an odd positive integer and denote by $f_q(x)=x^q-x^{q-2}-1$. This family of polynomials appears in some problems related to recurrence sequences. In order to try to use some standard ...
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1answer
85 views

What are the algebraic steps to transform this expression? (Part of differential calculus) [closed]

sorry if this seems to be too easy for you, but I have struggled a lot with this expression and I don't get it how they got there: How can $\sqrt{2x^2}$ become $4x^2$ ?
2
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2answers
111 views

How to show that this series of rational functions has a maximum at $x=0$ using the “Descartesschen Regel”?

I am reading an old German paper, and at one step they mention that the function \begin{equation*} f(x) := \sum_{k=2}^\infty \frac{(1+x)(k(k-1)^2 + (2+x)(1+x)^2)}{(k+x)^3 (k + 1 + x)^2} \end{equation*}...
22
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1answer
3k views

Was Jacobi the first to notice the ambiguity in the partial derivatives notation? And did anyone object to his fix?

In his 1841 article De determinantibus, Jacobi remarked that the notation $\frac{\partial z}{\partial x}$ for partial derivatives is ambiguous. He observed that when $z$ is a function of $x,y$ as well ...
5
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3answers
363 views

Mathematical Techniques to Reduce the Width of a Gaussian Peak

In the chemical analysis by instruments, the signals of several molecules are overlapped which makes it difficult to determine the true area of each peak, such as those shown in red. I simulated this ...
50
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22answers
6k views

Which high-degree derivatives play an essential role?

Q. Which high-degree derivatives play an essential role in applications, or in theorems? Of course the 1st derivative of distance w.r.t. time (velocity), the 2nd derivative (acceleration), and the ...
3
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1answer
72 views

Representation of finite differences of order k

We define recursively finite differences $ g_k (x) $ of order $ k $ of function $ f $ as follows: $g_0(x)=f(x)$, $g_n(x)=g_{n-1}(x+h_n)-g_{n-1}(x) (n\in\mathbb{N})$. It is known that all arguments of ...
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0answers
155 views

A vector calculus formula

Let me answer my own question, hoping to be forgiven for that. I asked unsuccessfully that question on Mathematics. Let $A, B$ be vector fields in $\mathbb R^3$. We have $$ \text{curl}\bigl((A\cdot \...
1
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1answer
162 views

About $\displaystyle\int_{\mathbb{R}_y^3}\int_{\mathbb{S}^2}e^{-\frac{1}{2}|x-[(x-y)\cdot\omega]\omega|^2}d\omega dy$

An integral has been pushed me over the edge for several weeks. It reads as: $$\displaystyle\int_{\mathbb{R}_y^3}\int_{\mathbb{S}^2}e^{-\frac{1}{2}|x-[(x-y)\cdot\omega]\omega|^2}d\omega dy$$ I tried ...
0
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1answer
76 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
190 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
58 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 $g$...
1
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3answers
124 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}, $ ...
4
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1answer
274 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 ...
3
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1answer
124 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 ...
16
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0answers
363 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
409 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
198 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
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2answers
455 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
86 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
129 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
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1answer
464 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
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3answers
1k 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
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2answers
218 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} \...
33
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2answers
2k 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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2answers
223 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
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1answer
411 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 ...
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0answers
317 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
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0answers
126 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
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0answers
134 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
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1answer
687 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
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0answers
156 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}...
5
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1answer
421 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
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1answer
148 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
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1answer
90 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
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0answers
59 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
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3answers
733 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
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1answer
312 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
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1answer
267 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
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1answer
1k 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 \...
2
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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 ''...
2
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0answers
568 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\...