Mathematics Stack Exchange
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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.

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. Reference: Wikipedia.

Synonyms: many-valued function, set-valued function, set-valued map, point-to-set map, multi-valued map, multimap, correspondence, carrier.

244 questions
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Minimizing the sum of the squares of the distance of points in $\mathbb{R}^3$

Let $P_{i} = (x_i,y_i,z_i)$ n points in $\mathbb{R}^3$. Show that the point $P=(x,y,z)$ that minimize the sum of the squares of the distances to the points $P_i$ is the gravity center $( x = \frac{1}{n} \sum x_{i}, y = \frac{1}{n} \sum y_{i}, z =…
user286485
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The existence of continuous approximate selection

I'm looking for a condition of the existence of continuous approximate selection for a (multi-valued) minimiser mapping. That is, let's say a continuous function $f: X \times U \to \mathbb{R}$ is given for $X \subset \mathbb{R}^{n}$ nonempty compact…
Jinrae Kim
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Is this function homogeneous on its domain?

I need help understanding this problem: $$f(x,y)=\begin{cases} \dfrac{y^4}{x}, & x>2, y>0 \\[6pt] y^3,& (x,y)\in E \end{cases}$$ $$D:=R^2 \smallsetminus \{{(x,y)\in R^2 : x=2, y\ge 0}\}$$ $$E:=D\smallsetminus \{{(x,y)\in R^2 : x>2, y> 0}\}$$ This…
Lamija37
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Given several values of x, y and z in the relation z=ax+by+c, determine a, b and c

I have inherited a cloud pricing mechanism that essentially uses two variables to derive prices. No-one in the organisation knows the original derivation, but I have about 16 sets of datapoints from which I am hoping to reverse-engineer the…
length
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How to solve equations involving multivalued functions in complex domain?

Does this equation have a solution in the complex domain? $$ \sqrt{x+3} = 3 + \sqrt{x} $$ Squaring both sides gives $\sqrt{x}=-1$, which suggests the solution to be $x=1$. But: How can the complex domain solution be a purely real number? Does this…
Mohammad Ali
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The definition of surjective multivalued

I am confused by the question of how to define surjectivity for multivalued mapping( if this definition has already existed).
Noura
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Quick question about constant function

Are all constant function, a multi-valued function? Since, when we considering the constant function on real numbers, it's not one-to-one?
user516076
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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). How would I write down this multi-valued function?…
zynga
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Analytic continuation of $a^{q}$

Although I have completed a Bachelor of Science degree, I have never learnt complex analysis in college. Let $F(a) = a^{q}$ with $q$ non integer. It is known that as $a$ passes over the negative real axis, we need to consider the situation in which…
Grown pains
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If $f:\mathbb{R}^2 \rightarrow \mathbb{R}, f(x,y) = (y-x^2)(y-2x^2)$, then $(0,0)$ is a min. point of $f$ restricted to any line through origin.

$f:\mathbb{R}^2 \rightarrow \mathbb{R}, f(x,y) = (y-x^2)(y-2x^2)$. Show that the origin is a minimum point for a restriction of $f$ to any line through origin, but isn't a minimum local point to $f$. Here's my attempt: $\partial_x f(x,y) =…
user286485