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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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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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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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Some multivalued problems - Yosida approximation

I study multi-valued problems on the basis of M. Chipot's book and I have a seemingly simple problem. Set for a.e. $x \in \Omega$, $\lambda > 0$, $\forall t \in \mathbb{R}$: $$ J_\lambda (x,t) = ...
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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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137 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 ...
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Can raising a number to an irrational power have infinite solutions?

$a^{\frac{1}{2}}$ is generally considered to be the positive square root of $a$, but it also makes sense (depending on context) to consider it to be multivalued, returning all square roots of $a$ ...
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What does it mean for two [multivalued] complex functions to be equal?

For $f:X\subseteq\mathbb{R}\to\mathbb{R}$ and $g:X\subseteq\mathbb{R}\to\mathbb{R}$, we may say that $f$ and $g$ are equivalent if $\forall x\in{X}.f(x)=g(x)$. But for many complex functions $f:Z\...
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What's exactly meant if one is saying that the closure of a multivalued operator is the generator of a $C^0$-semigroup?

Let $E$ be $\mathbb R$-Banach space $A$ be a subspace of $E\times E$ and $$\mathcal D(A):=\left\{x\in E\mid\exists y\in E:(x,y)\in A\right\}$$ $(T(t))_{t\ge0}$ be a contractive $C^0$-semigroup on $\...
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Differentiabilty of $f(x,y)=\sqrt[3]{\lvert x^2-(y+1)^2 \lvert}\sin(\lvert x+y+1 \lvert))$ at $(0,-1)$ , $(1,0)$ and $(-1,0)$

I want to show that this function is differentiable at these points. At $(0,-1)$ i use the definition to check the differentiability: $\lim_{\substack{x\to 0 \\ y\to -1}} \frac{\sqrt[3]{\lvert x^2-(y+...
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My $ (i^i)^i $ result does not match Wolfram Alpha's result

See i^i on Wolfram Alpha. It provides a multivalued result that looks like this: $$ i^i = e^{-2 \pi n - \frac{\pi}{2}} \text{ for } n \in \mathbb{Z} $$ I was able to prove this result. From this ...
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Geometric Picture of z'(x, y(x))

When using the multivariable chain rule to compute $dz/dt$ of something with the form $z = f(x(t), y(t))$, the geometry involves moving from a number line ($t$) to a plane ($x$, $y$) to a new number ...
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“Counter-Stokes phenomenon” of $\sum_{n=0}^\infty \frac{(3)_n}{z^n}$

Let $$w\sim\sum_{n=0}^\infty \frac{(3)_n}{z^n}$$as $z\to\infty$, where $(3)_n$ is the Pochhammer symbol. One of the suitable choices of $w$ satisfies $zw'+(z-3)w-z=0.$ The solution of the differential ...
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How to show that limit value exists?

Here I have function $$f(x,y)=\frac{x^3-xy^2}{x^2+y^2}$$ I know that $\lim\limits_{(x,y) \to (0,0)}f(x,y)=0$ because if we put $y=mx$, then we have $$f(x,mx)=\frac{x(1-m^2)}{1+m^2}$$ $$\Rightarrow~\...
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Multivariate Recurrence Relation

$$A(n, k) = 2 \cdot \sum_{j=0}^{n-k}\sum_{i=1}^{k} (-1)^{i+1}A(n-j-i, k-i)\\$$ Some intial conditions that it holds are as follows: $$ n, k > 0\\ A(n, 0) = 1\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ A(n, ...
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180 views

What are meaning of these (P(x;y), P(x;y,z),P(x,y;z))?

I was reading a machine learning book that it uses probabilities like these: P(x;y), P(x;y,z),P(x,y;z) I couldn't find what they are and how can I read and ...
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How can I solve this problem ? regarding partial derivative ? satisfying laplace euation

Show that if $$w = f(u,v) $$ satisfies the so called Laplace equation $$w_{uu} + w_{vv} = 0 $$and if $$u = \frac{(x^2 −y^2)}{2} , v = xy ,$$ then $$ w = w(x,y)$$ also satisfies the Laplace ...
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3answers
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Does the limit exist ? and how to compute it?

$$\displaystyle\lim_{(x,y)\rightarrow (1,0)}\frac{x-1}{\sqrt{(x-1)^2+y^2}}$$ By direct substitution that's a ( $ \frac{0}{0}$ ) undefined so can I approach it by polar equation or by different ...
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writing down a multi valued function

Supposing a function can assume any value under a curve. As an example, we have a curve f(x). However, my actual function is that for a given value of x, the value y can be anything lower than f(x). ...
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What are examples of ORTHOGONAL function space for multivariable functions?

What are examples of orthogonal spaces of multivariable functions? I asked a question and learned that $$\langle f(x_1,x_2,\cdots x_n),g(x_1,x_2,\cdots x_n) \rangle = \idotsint_D f(x_1,x_2,\cdots x_n)...
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Confusion between function and multivalued function.

"What is a function?" can be answered as "Single-valued relations are called functions". But how can "What are the multi-valued function?" be answered? Will someone clarify my doubt why multi-valued ...
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Extending Area Formulas to (Well-Behaved) Relations

According to my understanding, the formula to calculate the signed area between two functions (or axes, as when finding "area under a curve") $f(x)$ and $g(x)$ from $a$ to $e$ is $\int_a^e f(x) - g(x)\...
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3answers
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Where's my error in this partial derivatives problem? [duplicate]

Let $u(x, y)=x+y$. What is $\displaystyle\frac{\partial u}{\partial x}$ and $\displaystyle\frac{\partial u}{\partial y}$? My answers are $1$ and $1$. Suppose I now told you that $y=x$, so that $u=2x$....
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2answers
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$\arg(\overline{z}), \arg(z^2)$ [closed]

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 3.42,43 Exer 3.42 multiple-valued: NO in general. $$\arg(1+i)=\frac \pi 4 + 2k \pi$$ $$...
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Tricks for finding good “close enough” solutions to multivariate recursive relations

As part of an undergraduate project I am attempting to find a solution to this set of recursive inequalities: $$g(n,l,k) \leq g\left(\frac{n-1}{2},l,k-1\right)$$ $$k \cdot g(n,k,l)+\log_2 (n-2^l +1) ...
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Do any authors take the sheaf-theoretic viewpoint on multivalued functions and/or indefinite integrals?

It seems to me that multivalued functions and/or indefinite integrals can be thought of as sheaves. For example: The real square-root function can be viewed as the sheaf $\mathcal{F}$ defined on the ...
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Nonmonotone functions in compact convex space

I am asking for more explaining about : If we have a function $ F\mathrm{(}x\mathrm{)} $ which is not monotone in $ x $ And then we can say that the function : $ \overline{F}{\mathrm{(}}{x}{\mathrm{...
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1answer
47 views

fixed point of a multivalued map

Suppose I have a map $S:X \to Y$ between Banach spaces which is multivalued, so that $S(x)$ is a set. I have shown that $S$ takes a closed ball of radius $R$ to itself, and it also is such that if $...
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40 views

Prove a function is multivalued

I want to show that the function ((z-1)/(z+1))^(1/4) is multi valued only on the interval [-1,1]. Here's what I'm thinking: pick a point in [-1,1] and show that there exist another point in [-1,1] ...
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27 views

Two-valued change of variables for the given function

Consider the integral $$ \int \limits_{-1}^{1}d(\cos(\theta)) = 2 $$ I want to change the variable by the rule $$ \tag 1 \cos(\theta') = \frac{E'_{N}(\theta)p_{B} - m_{B}E_{N}}{p_{B}\sqrt{(E'_{N}(\...
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Does the two variable function f have only one root?

‎Alzer (2000) obtained the following new sharp upper bounds for Bernoulli numbers‎ ‎\begin{equation}‎ ‎|B_{2n}|\leq\frac{2(2n)!}{(2\pi)^{2n}}‎ ‎\cdot \frac{1}{1-2^{\beta-2n}}, n\geq1\qquad‎ ‎\end{...
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Quadratic Taylor approximation of an integral

I'm trying to find the quadratic Taylor approximation of $$f(x,y)=\int_0^{x+y^2}e^{-t^2}dt$$about the point $(0,0)$. I really don't know how to solve this though since I don't know even if it's ...
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1answer
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Continuity of multivariable function at point

I have to determine whether $$f(x,y)=\frac {x^4-y^4}{x^2+y^2}$$ is continous (or can be made continous) at $(x_0,y_0)=(0,0)$ or not, as far as it can be determined, using $y=mx$ only. Normally I'm ok ...
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1answer
82 views

Evaluate $ \int_0^\infty x^{a-1}\frac{\sin(\frac{1}{2}a \pi-bx)}{x^2+r^2}\, r\,dx $

Evaluate $$ \int_0^\infty x^{a-1}\frac{\sin(\frac{1}{2}a \pi-bx)}{x^2+r^2}\, r\,dx $$ with $0<a<2$, $b>0$, $r>0$, using methods of complex analysis. I can't find a proper contour ...
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1answer
130 views

How to choose a branch when there are multiple branch points?

$\sqrt[n]{1-z^2}$ =$\sqrt[n]{(1-z)(1+z)}$ two branch points at -1,and 1. $\sqrt[n]{z}$ hast n branches and only one branch point at z=0. My understanding: When we want to integrate on different ...
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20 views

Continuity of the maximal element of a multi-valued function

Let $(X ,d)$ be a complete metric space. $A :X \rightarrow 2^{\mathbb{R}}$ is a multivalued function (i.e., $A(x) \subseteq \mathbb{R}$ and $A(x)$ is non-empty for all $x \in X$). Suppose that $A(x)$...
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47 views

Relation between hemi-continuity of correspondences and semi-continuity of functions?

I have just been introduced to the concept of "hemi-continuity" (the "h" is not a typo) of correspondences. If I understand the concept correctly then the following conjecture should be true: A ...
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1answer
481 views

What are some examples of upper semicontinuous set valued functions?

I know the definition of an upper semicontinuous set valued function; A function $f:X\rightarrow 2^Y$ is upper semicontinuous at a point $x \in X$ provided that if $V$ is an open set in Y containing $...
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2answers
66 views

find the coefficient of the multivariate normal distribution?

We know that the multivariate normal distribution is given by $$f(x)=\frac 1 {c} e^{-\frac 1 2(x-\mu)^T\Sigma (x-\mu)}$$ Where $c =\sqrt {\det(\Sigma)2\pi}$ How do we derive this value for $c$? EDIT:...
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1answer
33 views

Can I get a differential of a multivariable vector field?

Say I got $\vec r=(xy,y^2,xz)$, can I have the differential $d\vec r$ ? What would be the generalization of it? I can't find in on internet, so I don't know if it's possible to do.
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67 views

A set valued intermediate value theorem.

Does anyone know of an intermediate value theorem for set valued maps which are upper semicontinuous? Specifically, I'm looking for a theorem which says that, for example, if we are in $R^3$ and we ...