1
vote
0answers
46 views

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 ...
1
vote
2answers
50 views

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 ...
1
vote
0answers
78 views

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 ...
0
votes
2answers
33 views

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 ...
0
votes
1answer
44 views

Solve equation of form $(d_B - 0.32)^{0.8} (d_B + 1.45)^{1.1} = exp(0.8)$ for the term $d_B$

I have the following equation: $$ \left(\frac{\sqrt{d_B}-\sqrt{d_{Beq}}}{\sqrt{d_{Bmin}}-\sqrt{d_{Beq}}} \right)^{1-\frac{c1}{c2}}\left(\frac{\sqrt{d_B}+\sqrt{c3}}{\sqrt{d_{Bmin}}+\sqrt{c3}} ...
0
votes
1answer
26 views

What will be the time complexity of insertion if a queue is implemented using two stacks?

A Queue could be implemented using two Stacks. So what will be the time complexity for insertion and deletion in this queue? Thanks in advance.
0
votes
1answer
23 views

Identifying some cyclic subgroup

Is there a fast way to argue that (for $a,b>1$ integers) the set of all $x\in\mathbf{Z}/b\mathbf{Z}$ with $ax=0$ is isomorphic to $\mathbf{Z}/{gcd(a,b)}\mathbf{Z}$? Maybe by counting the elements, ...
1
vote
3answers
111 views

How can efficiently derive $x$ and $y$ from $z$ where $z=2^x+3^y$.

How can efficiently derive $x$ and $y$ from $z$ where $z=2^x+3^y$. Note. $x$,$y$ and $z$ are integer values and $z$ is $4096$ bits integer or even more. For all $z>1$. And if equation be ...
1
vote
1answer
39 views

number of solutions of a system of linear equations

Consider a system $\Sigma (y)$ of $m$ linear equations in $n$ variables $x_1,\cdots,x_n$: $\sum_{j=1}^n a_{i,j}(y)\cdot x_j=b_i(y)$, $i=1,\cdots,m$, whose coefficients $a_{i,j}(y)$ and $b_i(y)$ are ...
1
vote
0answers
24 views

If X and Y are sequential tangents to a group G at the identity matrix, show that X+Y is also.

If X and Y are sequential tangents to a group G at the identity matrix, show that X+Y is also. Definition: X is a sequential tangent vector to G at the identity matrix I if there is a sequence ...
1
vote
1answer
79 views

When to rationalize numerator and/or denominator?

Sometimes, we have to rationalize either the numerator or the denominator, and sometimes we can still work the problem without rationalizing. So, in some cases, rationalizing can be done, although it ...
7
votes
4answers
249 views

Advanced Mathematics

I am a high school student and would like to pursue a career in mathematics and I am hoping to find a serious explanatory book on math (geometry, algebra, calculus, functions and trigonometry) for ...
1
vote
0answers
239 views

What are these physicists talking about? Dyadic green function?

I am interested in a mathematical explanation(in the sense that you say for example: is it a mapping from A to B) what a dyadic green function and the unit dyad actually is? I am reading this as ...
0
votes
0answers
80 views

Convert fundamental matrix Differential equations

I was wondering about the following: Assuming that I have found a fundamental matrix for a given ODE. How do I manage it, that the columns of the matrix $y_i$ fulfill the initial condition ...
0
votes
2answers
54 views

Proof: let $f:A \to B$ with $f$ bijective, then $f{\restriction_{C} }: C \to B$ is bijective

I need the proof of following: "let $f:A \to B$ with $f$ bijective, then $f{\restriction_{C} }: C \to B$ is bijective" Thanks in advance
-3
votes
6answers
488 views

Irrational roots don't exist [closed]

I'm going through Apostol's Calculus vol. 1. It is excellent. But I was surprised to see him "prove" each number has a square root. Inverse functions like divide usually introduce some invalid ...
3
votes
1answer
91 views

bilinear form $F(A, B) = n \cdot \text{tr}(AB) - \text{tr}(A)\cdot\text{tr}(B)$, find ortogonal subspaces, that satisfy…

Define $F$ as bilinear form $M_n(\mathbb{R}) \text{ x } M_n(\mathbb{R}) \rightarrow \mathbb{R}$ $F(A, B) = n \cdot \text{tr}(AB) - \text{tr}(A)\cdot\text{tr}(B)$ Prove, that $F$ is represented by ...
11
votes
2answers
408 views

When do equations represent the same curve?

Suppose we have two sets of parametric equations $\mathbf c_1(u) = (x_1(u), y_1(u))$ and $\mathbf c_2(v) = (x_2(v), y_2(v))$ representing two 2D planar curves. When I say "2D planar curves" I mean ...
10
votes
4answers
428 views

Is the theory of dual numbers strong enough to develop real analysis, and does it resemble Newton's historical method for doing calculus?

I've been interested in non-standard analysis recently. I was reading up on it and noticed the following interesting comment on the Wikipedia page about hyperreal numbers, right after giving an ...
0
votes
1answer
56 views

Simplifying equation from an aggregate one

I have got the 4th order coefficient of $\epsilon$ from the equation (9) from the paper. I got : $$\phi_4+ \ddot \phi_4+\omega_2 \ddot \phi_2 - \Delta \phi_2+g_2 \phi_2^2+2 \phi_1 \phi_3+ 3g_3 ...
0
votes
1answer
92 views

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.
6
votes
3answers
400 views

Can every real polynomial be factored up to quadratic factors?

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) ...
1
vote
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 ...
5
votes
1answer
269 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 ...
5
votes
3answers
141 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 ...
1
vote
0answers
107 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)= ...
2
votes
0answers
288 views

Preparing for reading Penrose's “Road to Reality”

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 ...
9
votes
3answers
367 views

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 ...
2
votes
1answer
104 views

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 ...
7
votes
2answers
429 views

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 ...
4
votes
10answers
770 views

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: ...