# Tagged Questions

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### Continuous, differentiablee and continuous isomorphism, homomorphism

I have very big confusions on Continuous, differentiable and continuous isomorphism, homomorphism of a FUNCTION. I am seriously looking a single example, which can explain the following clearly. ...
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### Subset Sum represented as a perfect number

Can we form a set of $29$ distinct integer elements such that every subset of elements possible has a sum which is a perfect power? A perfect power is a positive integer that can be represented a p^q ...
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### Can operations on arbitrarily differentiable functions form a group?

Consider the the set $S=${All uniary operations (all operations that perform on one function and give you another function, e.g $(x^2)'=2x$) on all arbitrarily differentiable functions}, does it form ...
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### Transfer vector space properties to dual space

I am curious about this here (Actually, I don't know if my assumptions are true or not) a) Let $X$ be a Banach space that is isomorphic to $Y$, then $X^*$ is also isomorphic to $Y^*$. I sketched a ...
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### What is the difference between diffeomorphism and isomorphism?

What is the difference between diffeomorphisms and isomorphisms? I know isomorphisms already from my abstract algebra/group theory course, and now I'm studying analysis on (sub)manifolds, where this ...
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### Solving an 8th degree polynomial

I know that through the Abel Ruffini Theorem the general solution to a polynomial of degree five or more cannot be found explicitly. But are there are any other ways to find the roots of such a ...
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### Notion of Derivative in Discrete Space

We are accustomed with the notion of Calculus in $\mathbb{R}$. Can we define the notion of derivative/non-derivative in $\mathbb{N}$. I was thinking something as follows: Let $f$ be a function from ...
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### I want to find a closed form of the sum $\sum_{i=1}^n x^{i^2}$

Is there any closed form of the following sum? $\sum_{i=1}^n x^{i^2}$, where $x$ is a variable. Calculating the sum for a first few $n$ does not give any pattern.
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I know that you can't factor a real polynomial into $\Pi_{i=1}^N(x-a_i)$ in general. But is it possible to factor every real finite polynomial into this form: $(\Pi_{i=1}^N a_ix^2 + b_ix + c_i) ... 2answers 122 views ### What are the minimum constraints on the domain and range of a function in order for it to be differentiable? For example if$S$is a set with two elements$a$and$b$, then you could define:$f : \mathbb{R} \to S, s.t. f(x) = a, \forall x \in \mathbb{R}$Intuitively, since f is a constant function, it ... 1answer 275 views ### Antiderivative of Polynomials I really like how differentiation is introduced for polynomials: Let$P(t) \in A[t]$: $$D_P(t,s) = \frac{P(t) - P(s)}{t-s} \;\; \in A[t,s]$$ and the derivative of$P$is $$P'(t) = D_P(t,t).$$ It ... 3answers 143 views ### Can every continuous function that is curved be expressed as a formula? By "curved", I mean that there is no portion of a graph of a function that is a straight line. (Let us first limit the case into 2-dimensional cases.. and if anyone can explain the cases in more than ... 0answers 111 views ### length of orbits and fixed point Given a dynamical system$ \frac{dx}{dt}= F(x(t))$Then is there a relationship between the Cardinal of the fixed point of the classical system$ |\operatorname{Fix}(f^{m})| $with$ f^{m}(x)= ...
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I am reading Road to Reality by Roger Penrose and I although I know about calculus, complex analysis, differential equations I do not know about manifolds, Riemann surfaces and so on. Which books can ...
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### Why adjoining non-Archimedean element doesn't work as calculus foundation?

Consider the smallest ordered field that contains R and does not satisfy the Archimedean property. I assume this is a much simpler construction than ultrafilters and other big caliber artillery used ...
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### How to prove if P is a prepositive cone?

I am working through a Calculus book and I found an exercise, which I am not able to solve: Let $K$ be a field of rational functions over $\mathbb{R}$ and let $a,b$ be arbitrary in $\mathbb{Q}$ but ...
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### Algebraization of integral calculus

It is well known that the differential calculus has a nice algebraization in terms of the differential rings but what about integral calculus? Of course, one sometimes defines an integral in a ...
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### Polynomial satisfying $p(x)=3^{x}$ for $x \in \mathbb{N}$

Let $p(x)$ be a polynomial with integer coefficients, which is not constant. Then is this condition possible: $$p(x)=3^{x}$$ whenever $x \in \mathbb{N}$. Motivation: ...