Questions tagged [multivalued-functions]

In Mathematics, set theory, a Multivalued function is defined as a left-total relation (that is, every input is associated with at least one output) in which at least one input is associated with multiple (two or more) outputs.

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Orbit of Multivalued Function

Is orbit of a multivalued function unique? Let $(X,d)$ any metric space, $x_0 \in X$, and $T\colon X \rightarrow CL(X)$ is multivalued function, then $O(T,x_0) =\{x_0,x_1,x_2,\cdots\}$ is said to be ...
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Transform equation assuming $a$ and $b$ as new variables.

The task is to transform the following equation $x \frac{dx}{dz}+y \frac{dz}{dy} =0$ into new one with independent variables taking $\varphi=\varphi(a,b)$ as a new function, for $a=\frac{y}{x}, b=y, \...
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Which branch of Multivalued Function $x^s$ to Take on Real Axis

Let $a, b, y$ be fixed real numbers satisfying $0<a<b<1<y$. I want to evaluate an integral of the form $\int_\gamma y^sp(s) ds$ where $p$ is a single-valued function of $s$ holomorphic on $...
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marginal expectation of joint probability density function (difference in order of integration)

I am trying to compute $E[X_2]$ of a joint pdf with random variables $X_1$ and $X_2$ shown below: $f_{{1},{2}}(x_1,x_2)=\frac{1}{4}(x_1-x_2)e^{-x_1}, 0<x_1<\infty, -x_1<x_2<x_1 $, and $...
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Rate of convergence for the law of large numbers if Multivariate Random variables

I am looking for a reference on rates of convergence for the law of large numbers on random variables on $R^d$ for $d>1$. I haven't been able to find (or find very little) about LLN of ...
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Conventions regarding multivalued function and Riemann sheets.

Good day all. Is the value of a composed multivalued function $F(g(z),h(z),k(z),..)$ on different Riemann sheets given only by its formula or, in addition to the formula one needs to provide the ...
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Branch Points and Cuts

I am having a lot of problems understanding what exactly branch points are and how they are computed for a function. There is this one problem that I just can't seem to get around to understanding, ...
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Proving the continuity of a function using the definition

Proving the continuity of a function using the definition. The function is: $f(x,y) = e^x + sen(y)$ I don’t know where to start. usually I have a point in where I have to find if the function is ...
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Showing that the sequence $\left\{x_\lambda\right\}_{\lambda > 0}$ is bounded.

(Part of the following can be found at page 36 of "Non-linear Differential Equations of Monotone Types in Banach Spaces" of Viorel Barbu.) Note for the following, that $A$ is a coercive and maximal ...
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How do you call a function that is the gradient of a function?

Let us consider $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ such that there exists a real-valued function $F: \mathbb{R}^n \rightarrow \mathbb{R}$ that satisfies $\nabla F(x) =f(x)$. How would you ...
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A Poincare-Miranda theorem for multivalued functions

The Poincare-Miranda theorem is an extension of the intermediate value theorem to multi-dimensional functions. It is considered to be equivalent to Brouwer's fixed-point theorem. Both these theorems ...
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Limits of multivariable functions hints

I have tried switching to polar coordinates to no avail, I have tried using a gazillion path tests to no avail, I tried sandwiching to no avail and now I am really desperate, I even attempted an $\...
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Reference for Measurable Selection

Let $F(t,a):T\times A \to \mathbb{R}$, continuous in $a$ for each $t$ and measurable in $t$ for each $a$. Is there a selection theorem for the case $$ I(t) = \{a\in A: F(t,a) = B(t)\} $$ I found a ...
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Behavior of Mulivalued Functions after Multiple Closed Circuits

I am studying Muskhelishvili's book on the plane theory of elasticity which uses complex-valued, holomorphic functions to evaluate boundary value problems in the plane. I am currently working through ...
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What if in certain algebra $w^{-n}$ is multivalued?

In complex numbers we have $z^{1/n}$ and $a^z$ having multiple values. But what would happen if the inverse function $1/z$ were multi-valued? For instance, there is such $w$ that $1/w=\pi^2/6$ but $...
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Distribution of X-Y for identical independent random variables

I have X and Y which are independent and both have an exponential distribution with density function $f(x) = e^{-x}$ if $x\gt0$ I want to find the distribution of X+Y and X-Y. Let U=X+Y, V=X-Y My ...
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Meaning of class of function

Let $f : (0,T)\times\mathbb{R}^{N}$ be a function in $C^{1,2}((0,T)\times\mathbb{R}^{N})$ At the very least, we know that $\partial_{t}f$ and $\partial_{x_{i}x_{j}}f$ exist and continuous. Now, what ...
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Quick question about constant function [closed]

Are all constant function, a multi-valued function? Since, when we considering the constant function on real numbers, it's not one-to-one?
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Multivariate limit

I have troubles with such limit $$ \lim_{(x,y)\rightarrow(0,0)} \frac{x^3}{y^4+2\sin^2{x}}$$ Nothing works, as I approach on any line or curve I get limit equal to $0$. I try polar coordinates - ...
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If multi valued functions aren't functions how can they be differentiated and integrated

I recently learned that relations like $f(x)=\sqrt{x}$ and $f(x)=\arcsin(x)$ are not actually functions but multivalued functions, since they take multiple outputs for a single input. So how come we ...
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$(X,X+Y_1,X+Y_2,\dots,X+Y_n)$ a Gaussian vector

If $X$ and $Y_i$ for $\ 1\le i\le n$ are independent centered Gaussian r.v.'s then do we have $\overset{\rightarrow}X=(X,X+Y_1,X+Y_2,\dots,X+Y_n)$ a Gaussian vector. Sum of 2 independent Gaussian r....
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Residue theorem integration of a multivalued function

I have to evaluate the integral $$\int_{-\infty}^{+\infty}\frac{z^3e^{i\alpha z}e^{-\sqrt{z^2+a^2}\lvert b\rvert}dz}{(z^2-c^2)\sqrt{z^2+a^2}}$$ I know I have to use the residue theorem for that. The ...
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60 views

Directional derivative of a function $f(x,y)$ but direction in terms of $u$ and $v$.

I've been trying to solve this problem but I can't find the solution to it. The problem is as follows, Calculate the directional derivative of the function: $$ f(x,y) = 3xy^2+2x^2-5x $$ as ...
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calculating the center of mass of a set of vectors (during a proof of a Hilbert polynomial identity)

I was reading Hilbert's proof of the Hilbert-Waring theorem in this survey paper, and came across the following statement in page 25, which i rephrased a little, regarding the center of mass of a set ...
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Generalization of Kakutani's fixed point theorem.

I am studying some fixed point theory, and bumped into this Lemma from which Kakutani Fixed Point Theorem is claimed to follow trivially: Lemma Let $K\subset\mathbb{R}^m$ be compact and convex and let ...
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Integration of multi-valued function

Graph I have a set of discrete (x,y) points and need to find the area between the curve and the y-axis. However, the curve seems to be a multi-valued function (multiple x for the same y). Is such an ...
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Suppose there is a branch of the logarithm on a simply connected region $U$, when do we have the formula $\log z = \log r + i \theta$?

I am following Stein and Shakarchi on Complex analysis. In a previous thread, I learnt that if $U$ is a simply connected region with $1 \in U$ and $0 \not \in U$, then there is a branch of the ...
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If $z = re^{i\theta}$, how does a branch of the logarithm restrict the interval for $\theta$?

I am following Stein and Shakarchi on Complex analysis. On page 98 they define a branch of the logarithm to be a choice of domain for the logarithm. I suspect they mean a choice of interval for the ...
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Showing a multivariable function is constant under a condition

let $f : \mathbb{R}^3 \to \mathbb{R}$ be a $C^1$ function. and we have $\frac{\partial f}{\partial x }= \frac{\partial f}{\partial y} = 0$. Prove that function $g(x,y) = f(x,y,a)$ for fixed real ...
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Is every function a linear combination of separable functions?

Let $\mathcal{F}(\mathbb{R}^n\longrightarrow\mathbb{R})$ be the space of functions from $\Bbb R^n$ to $\Bbb R$. If I'm not mistaken, this space is canonically isomorph to the tensor product of $n$ ...
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Showing the equivalence of directional derivative with total derivative multiplied with direction in $\mathbb R^n$

Let $f:\mathbb R ^n \to \mathbb R^m, g:\mathbb R \to \mathbb R^n$, and $f$ totally differentiable. Then we have $$\frac{\partial }{\partial x} f(g(x)) = Df(g(x))\cdot Dg(x) \cdot x$$ and $$\frac{\...
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Show an inclusion for the range of a multi-valued dissipative operator

Let $Z$ be a $\mathbb R$-Banach space and $C$ be a multi-valued dissipative linear operator on $Z$, i.e. $C$ is a subspace of $Z\times Z$ with $$\forall\lambda>0:\forall(z,z')\in C:\left\|\lambda z-...
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Show that the semigroup given by the product of two semigroups is generated by the product of the generators

Let $X_n,Y$ be $\mathbb R$-Banach spaces for $n\in\mathbb N$, $A_n\subseteq X_n\times X_n$ and $B\subseteq Y\times Y$ be linear and dissipative with $\mathcal R(\lambda-A_n)=X_n$ and $\overline{\...
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Show that this multi-valued operator is surjective (Theorem 1.6.9 of Ethier and Kurtz)

Let $L_n,L$ be $\mathbb R$-Banach spaces for $n\in\mathbb N$, $A_n\subseteq L_n\times L_n$ and $A\subseteq L\times L$ be linear and dissipative with $\mathcal R(\lambda-A_n)=L_n$ and $\overline{\...
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If $A_0=\{(x,y)∈\overline A:y∈\overline{D(A)}\}$, then $R(λ-A_0)=\overline{D(A)}$ if and only if $R(λ-\overline A)⊇\overline{D(A)}$

Let $E$ be a $\mathbb R$-Banach space, $A$ be a multi-valued dissipative linear operator on $E$, $$A_0:=\left\{(x,y)\in\overline A:y\in\overline{\mathcal D(A)}\right\}$$ and $\lambda>0$. How can ...
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How do we call a multi-valued operator $L$ with $∃c>0:∀y\in Y:∃x\in X:(x,y)\in L\text{ and }\left\|x\right\|_X\le c\left\|y\right\|_Y$?

Let $X,Y$ be $\mathbb R$-Banach spaces and $L$ be a multi-valued linear operator from $X$ to $Y$ with $$\exists c>0:\forall y\in Y:\exists x\in X:(x,y)\in L\text{ and }\left\|x\right\|_X\le c\left\|...
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Question regarding Inequation with multivariable functions

In order to simplify the notation consider: $$ x=[x_1,x_2,...x_n] $$ Consider the following inequation: $$ s(x)(u(x)+A(x))<0 $$ My goal is to choose the function u(x) such that the inequation ...
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Chain rule for direction derivative with multivariate and vector-valued functions

Given $f:\mathbb{R}^m\to \mathbb{R}^n$, $g:\mathbb{R}^n\to\mathbb{R}^q$, is the following statement for the directional derivative ($\mathbf{v}\in\mathbb{R}^m$) correct? $$\partial_{\mathbf{v}}(\...
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60 views

Local Maximum Point; Global Maximum Point

Given is the function: $f(x,y)=cos(x)+cos(y)$ Which of the following statements is correct? 1. The function has a local maximum point in $P (0, 0)$ This is correct, because the first order ...
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Why can't these two mappings be bijective?

Let $\phi : \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a continuously differentiable function and define the mapping $\mathbf{F} : \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ by $$\mathbf{F}(x, y) = (\...
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Proving that the composition of a harmonic function and a Cauchy-Riemann mapping is harmonic

Let $\mathcal{O}$ be an open subset of the plane $\mathbb{R}^{2}$ and let the mapping $F : \mathcal{O} \rightarrow \mathbb{R}^{2}$ be represented by $F(x, y) = (u(x, y), v(x, y))$ for $(x, y)$ in ...
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145 views

Logarithm being a multivalued function

This question is probably going to have a duplicate, but my main doubt is specifically on a particular question, so please consider reading the whole thing. First of all the function $y= \ln x$ seems ...
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Michael's selection theorem

Michael's selection theorem states that a lower hemicontinuous multivalued map with nonempty convex closed values $\displaystyle F\colon X\rightrightarrows E$ from a paracompact space $X$ to a Banach ...
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Is $(-1)^x$ multivalued?

$(-1)^\pi=(e^{i\pi})^\pi=e^{i\pi^2}=\cos(\pi^2)+i\sin(\pi^2)$. Wolfram Alpha lists this as the only answer. However it started with $e^{i\pi}=1$, although $e^{i\pi(2n+1)}=-1$ also for any integer n. ...
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Help with Multivariable Delta-Epsilon Proof for $(x,y)\to (0,\pi/2)$ of $\sin(x+y)=1$ [closed]

Help with Multivariable Delta-Epsilon Proof for $$\lim_{(x,y)\to (0,\frac{\pi}{2})}\sin(x+y)=1.$$
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Is $\ln|z|$ harmonic in the punctured disk [closed]

1. how can i show that $\ln|z|$ is harmonic in punctured disk ? also $\ln|z|$ has no harmonic conjugate in $\Bbb C\setminus\{0 \}$ but has in $\Bbb C\setminus[0, \infty)$.
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Is a limit point of branch points a branch point?

I have come into a discussion with my friends over a complex analysis question: Is $\infty$ a branch point of $\log(\cos z)$? I can't get a clear answer to this from the definition of branch points. ...
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$\frac1{x ^ 2 + y^2}$ is uniformly continuous in your domain? [closed]

They let me see if the function$$\frac1{x ^ 2 + y^2}$$ is uniformly continuous in their domain but I have not been able to solve the problem, will anyone have any suggestions on how to solve the ...
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411 views

Integration of a multivalued function

The integral is: $$I=\int_1^2\frac{\sqrt{(x-1)(2-x)}}{x^2}dx$$ To solve this problem I integrate over a path $C$ that surrounds clockwise the branch cut, so the integral becomes: $$I=\frac{1}{2}\oint_{...
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What about the solutions of $z^{1/3} +1 = 0$?

I'm trying to find the zeros of the equation $$z^{1/3} +1 = 0.$$ My professor said that the solutions are the third roots of unity multiplied by $-1$. My problem is that when I calculate the cubic ...