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4 views

If a graph has a perfect matching, does that mean that there is $S$ where $|S|$ is equal to $o(G-S)$?

In the mathematical discipline of graph theory the Tutte theorem, named after William Thomas Tutte, is a characterization of graphs with perfect matchings. Let ${\displaystyle o(G-S)}$ be the number ...
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1answer
23 views

Definition of the structure sheaf on $\text{Spec} A$

In his book Algebraic Geometry, Hartshorne defines the structure sheaf of $\text{Spec} A$ to be the set of functions $s:U\to\coprod_{p\in U}A_p$ such that $s(p)\in A_p$ and $s$ is locally a quotient ...
3
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0answers
28 views

Solution set to non-linear equation of $3$ variables

I have the following trigonometric equation of $3$ variables: $$f(\theta,\lambda,\phi)=3 \cos (\theta ) \cos (\lambda ) \cos (\phi )-(\cos (\theta )+3) \sin (\lambda ) \sin (\phi )$$ $$-\sin (\...
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0answers
2 views

Prove that following graph $G$ has no Hamilton circuit exists.

Prove that following graph $G$ has no Hamilton circuit exists. $\underline{Attempt}$ First I assumed that the graph $G$ has a Hamilton circuit. then, Since graph is symmetric, deleted the edge $(j,h)...
0
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1answer
883 views

Cauchy's theorem on connected (not simply) domains

I was trying to show that the function $f(z) = \frac1{z(1-z^2)}$ does not have an indefinite integral on the annulus $\mathbb{A} = \{z \in \mathbb{C} : 0 < |z| < 1 \}$ Indefinite integrals must ...
1
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3answers
32 views

Claim: $3\mid n^4-n^2$ for all $n \in \mathbb{Z^+}$, $n\ge2$.

Claim: $3\mid n^4-n^2$ for all $n \in \mathbb{Z^+}$, $n\ge2$. What I've done so far: Base case: Let $n=2$. Then $\exists k \in \mathbb{Z}$ such that $12=3k$, namely, $k=4$. Thus $3 \mid 12$ and hence ...
-3
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0answers
15 views

How to show if T(h,l) is an linear functional ? Is it bounded? And find the norm ||T||

\begin{array}{l}T\left(h,l\right)=\frac{\partial f\left(a,b\right)}{\partial x}h+\frac{\partial f\left(a,b\right)}{\partial x}l\\ \end{array} I'm trying to work through some example questions from my ...
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4 views

I tried substituting some values of x. I also know that f(x)=x is a solution to the function but I am just not able to systematically prove it.

Solve the functional equation f(x+1)+f(x-1)=2f(x) for f(x) I tried substituting for x= x-1; x-2; x+1;x+2 But I can't seem to get to a systematic method to solve this question. I do know that f(x)=x ...
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0answers
2 views

Linear maps preserving a revolution cone

Let $C$ be the cone of $R^{n+1}$ defined by $$ C=\lbrace x=(x_1,..,x_{n+1})|\quad \sum_1^n x_i^2 >x_{n+1}^2 \rbrace.$$ I am interested in the set $\mathcal{C}$ of $n\times n$ matrices $M$ which ...
0
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0answers
5 views

Proof verification of limit of a sequence

I was studying the definition of limits of a sequence in $\mathbb R$ and tried to verify whether the sequence $X=(1,2,3,......)$ converges to a real number or not. Solution: Here, the series does not ...
1
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0answers
27 views

Prove $\lim_{a\to\infty}a{\int_{0}^{1}{x^af(x)dx}} = f(1)$

A few days back, I was surfing the website and I saw an interesting theorem: for any continuously differentiable $f$ on $[0,1]$ : $$\lim_{a\to\infty}{a\int_{0}^{1}{x^af(x)dx}}=f(1)$$ It was said ...
2
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3answers
28 views

Solving a polynomial divison problem with a trick

If the polynomial $x^{19} +x^{17} +x^{13} +x^{11} +x^{7} +x^{5} +x^{3}$ is divided by $(x^ 2 +1)$, then the remainder is: How Do I solve this question without the tedious long division? Using ...
13
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2answers
653 views

Prove the following (algebra of polynomials)

Let $P_1=1$ and let $P_2=n+1$ define $$P_{i+1}=\frac{P_i^2-1}{P_{i-1}}$$ Prove that if $a \mid b$ then $ P_a \mid P_b $ I am working on this problem for a while but I could use some help here are the ...
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0answers
41 views

About Hall's Theorem

Let $G$ be a bipartite graph. In order to find a match in the $G$ diagram so that there are no unpaired elements in the set $A$, a necessary and sufficient condition is $|A-N(T)|\leq|B-T|$ that is ...
18
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3answers
557 views

Prove or disprove that the function $f(x)=x^{x^{x^{x}}}$ is convex on $(0,1)$

Let $0<x<1$ and $f(x)=x^{x^{x^{x}}}$ then we have : Claim : $$f''(x)\geq 0$$ My attempt as a sketch of partial proof : We introduce the function ($0<a<1$): $$g(x)=x^{x^{a^{a}}}$$ Second ...
0
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1answer
14 views

Question on the elements of a sigma-ideal of meagre sets

Let $A$ be the free $\sigma$-algebra on $\omega$ free generators and $X$ its Stone space. Then $A$ is isomorphic to the quotient algebra $Ba(X)/M$, where $Ba(X)$ is the $\sigma$-field of Baire subsets ...
2
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4answers
78 views

Create a function such that…

I'm reading Kevin Houston's book "How to Think Like a Mathematician" and I came across this stumper: Find an example of a non-polynomial function $f: \mathbb{R} \rightarrow \mathbb{R}$...
0
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1answer
42 views

Solutions of $ 5^x+5^{x^2}=4^x+6^{x^2} \quad \left(x\in \mathbb{R}\right)$

For equation $$ 5^x+5^{x^2}=4^x+6^{x^2} \quad \left(x\in \mathbb{R}\right)$$ is there any nontrivial solution? Easy to find $x=0,1$ are the trivial solutions, and also, easy to figure out that $ x>...
22
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0answers
346 views
+200

For which $n$ we can divide set $\{1,2,3,…,3n\}$ in to $n$ subsets each with $3$ elements such that in each subset $\{x,y,z\}$ we have $x+y=3z$?

For which $n\in \mathbb{N}$ can we divide set $\{1,2,3,...,3n\}$ in to $n$ subsets each with $3$ elements such that in each subset $\{x,y,z\}$ we have $x+y=3z$? Since $x_i+y_i=3z_i$ for each subset $...
0
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0answers
9 views

$f\in L^q \Longrightarrow f/|x|^{\alpha} \in L^{p}$ for a Schwartz function?

Let $\alpha>0$ and suppose $f$ is a Schwartz function on $\mathbb{R}$. Is it true or false that $$\left(\int_{\mathbb{R}}\frac{|f|^{p}}{|x|^{\alpha p}}\right)^{\frac{1}{p}}\leq c \left(\int_{\...
0
votes
1answer
31 views

Is Separable space iff compact space?

I'm reading chapter2 of PMA. In exercise 22~26, it uses separable space. According to exercises, compact$\implies$every infinite subset has a limit point$\implies$separable$\implies$countable base. ...
0
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1answer
41 views

Cosets of alternating group in symmetric group

Let $G=S_n$ and $N=A_n$. There are exactly two left cosets of $A_n$ in $S_n$: $1\cdot A_n$ and $\left(1,2\right) \cdot A_n$, the latter consisting precisely of all odd permutations. Thus, the quotient ...
0
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1answer
49 views

Does for any infinite set $M \subseteq \Bbb N$ hold $|M| =|\Bbb N|$?

In a proof I do a case split about a set $M \subseteq \Bbb N$. In one case $M$ is finite and it means that there’s some bijection only between n elements of $\Bbb N$ and all the elements of $M$, so $|...
0
votes
2answers
17 views

Is the conditional expectation of the conditional expectation equal to the conditional expectation

Does $\int E_{Y\lvert X}[Y\lvert X=x] p(y\lvert x)dy = E_{Y\lvert X}[Y\lvert X=x]$ In other words is the conditional expectation of the conditional expectation equal to the conditional expectation?
1
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0answers
9 views

Explanation of a part of a continued fraction proof : if $ \left|\dfrac{p}{q} - x \right| < \dfrac{1}{2q^2} $ then $\dfrac{p}{q}$ is a convergent.

This theorem seems important but I don't understand a piece of the proof and I can't find help anywhere. At the beginning it says : "We write $x= \dfrac{\omega p_n + p_{n-1}}{\omega q_n + q_{n-1}}...
2
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1answer
84 views

An interesting integral with a lot of well-known functions

Today I was surfing through YouTube and I cam across this unique and interesting integral on BriTheMathGuy YouTube channel. The integral is as follows : $\text{Q. Compute}$ $$\int_{0}^{1} (x^x)^{(x^x)^...
4
votes
5answers
11k views

Prove a cubic equation has at least one real root

Show that the cubic eq: $$x^3+ax^2+bx+c = 0 \quad a,b,c\in \mathbb{R}$$ has at least one real root. I know that the above equation can be broken down into $(x-a)(x-b)(x-c) = 0$ , but I have no ...
1
vote
3answers
344 views

Evaluate the improper integral: $\int_o^\infty 1.35 \times 10^{-7} e^{-0.03x}x^4 dx$

Evaluate the improper integral: $$\int_0^\infty 1.35 \times 10^{-7} e^{-0.03x}x^4 dx$$ (Needed for a solid processing example in chemical engineering). Now according to my textbook, this is simply $...
7
votes
5answers
3k views

How do we identify twin primes .

as known , each prime number greater than 3 is of the form $6k-1$ or $6k+1$ . twin primes are all sort of two adjacent primes of difference $= 2$ as: $$(11,13) ,(17,19),\ldots,(6k-1,6k+1)$$ -Is ...
1
vote
1answer
37 views

How many 2-digit integer solutions are there to the equation $x_1 + x_2 + x_3 = n$, $n \ge 100, n \in \mathbb{N} $?

While solving problems about in how many ways some numbers can be added up to another number, when they have a lower or upper bound, I've though of this problem. I'm looking for a general formula for ...
0
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0answers
13 views

A question on normal noncyclic abelian subgroups

Let $P$ be a nonabelian $p$-group, where $p$ is an odd prime. Let $Q$ be a nonabelian normal subgroup of $P$. Does there exist a normal noncyclic abelian subgroup of $P$ that is contained in $Q$? ...
5
votes
2answers
67 views

How does the Hopf fibration generalize to maps $S^{2n+1}\to \mathbb{CP}^n$?

I've been reading about Hopf fibrations. In the Wikipedia page, they state that the Hopf construction generalises to higher-dimensional projective spaces. More specifically, they write that The Hopf ...
4
votes
1answer
59 views

How do I show that the rank of a $\mathbb{Z}$-module is less than a certain number? [closed]

Hey guys I was wondering how I could use the properties of the $\def\Z{\mathbb{Z}}\Z$-module here. Let $A = (a_{i,j}) \in \Z^{m\times n}$. Let $m \leq n$. Let $L$ be the $\Z$-module spanned by the ...
0
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0answers
4 views

pullback of line bundle correspond to the fiber product?

Sorry for my bad English. Let $f:X\to Y$ be morphism of schemes, and $\mathscr{L}$ be inventible sheaf over $Y$. Now we consider $\mathscr{L}$ as line bundle $L\to Y$ as Hartshorne II.Ex.5.18. On ...
1
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0answers
9 views

Proj of Almost Same Graded Rings are Isomorphic (Exercise from Vakil's FOAG)

I'm trying to solve Exercise 6.4.F from Vakil's FOAG: First, I think what Vakil means when he says same finitely generated rings except in a finite number of nonzero degrees is that $R_{\bullet}$ and ...
0
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1answer
38 views

Continuous function between $[0,2\pi]$ and $S^{1}$

Here is the function between $[0,2\pi]$ and $S^1$ : for all $x\in [0, 2\pi]$ $f(x)=(\cos x,\sin x)$ . I know of that both $\sin x$ and $\cos x$ is continuous that is why $f$ is continuous, but is ...
2
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0answers
7 views

Explicit isometry for embedding hyperbolic space $H^d$ as a totally geodesic submanifold in the symmetric space of positive definite matrices $P(d+1)$

Crossposted on MathOverflow With $K>0$, let $H_K^d$ denote the (real) hyperbolic space of dimension $d$ with (sectional) curvature $-K$. Let $S(d+1, R)$ denote the vector space of real symmetric $(...
8
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1answer
53 views

When is a field an algebraic extension of a fixed subfield?

Let $K$ be a field, $G\leq\mbox{Aut}(K)$ a group of field automorphisms of $K$. When is the extension $K/K^G$ algebraic? Recall that $K^G$ is the subfield of $K$ consisting of the elements of $K$ that ...
0
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0answers
24 views

Let G be the direct product of non nonabelian simple groups

Let $G$ be the direct product of non 2 nonabelian simple groups. Then $G$ has exactly four normal subgroups. I have answered the question but I believe it's wrong, I wanna know why this answer is not ...
-5
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0answers
25 views

How many subsets does B have?

Let $A$ be a set of five elements and $B$ is the set of all pairs $(x,y)$ such that $x,y\in A$. How many subsets does $B$ have?
0
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2answers
21 views

Identity in distributional sense, principal value $1/x$

How to prove that $$ T_{n}(\phi) =\lim_{r \to 0} \int_{r \le |x|} \frac{\cos(nx)}{x}\phi(x)\,\mathrm dx = \cos(nx)\ \operatorname {p.v.}\frac{1}{x}, $$ where $\phi$ belongs to $D(R)$ ($\phi$ is a test ...
1
vote
0answers
25 views

Let $f(x)=x^{4}-6x^{2}+5$. If $P(x_0,y_0)$ is a point such that $y_0>f(x_0)$ and there are exactly two distinct tangents through P

Let $f(x)=x^{4}-6x^{2}+5$. If $P(x_0,y_0)$ is a point such that $y_0>f(x_0)$ and there are exactly two distinct tangents through P drawn to the curve $y=f(x)$, then find the maximum possible value ...
0
votes
0answers
8 views

Can the transition function for Kalman Filter be a random normal distributed variable?

One simple question about Kalman Filters. It's told that you need a model of the system to estimate the next measurement, e.g state. The model is called a transition function. $$\dot x = f(x, u)$$ ...
0
votes
1answer
20 views

Besides circles, are there other curves for which $\vec{r}'(t)$ and $\vec{r}"(t)$ are always perpendicular, described by only elementary functions?

Provided that $\vec{r}_1$ and $\vec{r}_2$ are a quarter turn away from each other, a circle can be described using the equation $\vec{r}(t)=\left(\vec{r}_1-\vec{r}_c\right)cos(t)+\left(\vec{r}_2-\vec{...
0
votes
0answers
5 views

Orthogonality condition for a degenerate Matrix

Consider the following left and right eigenvalue problems: $$Lq = \lambda q$$ $$q^+ L = \mu q^+$$ One can derive the orthogonality condition: $q^+q(\lambda - \mu) = 0$. Which tells that if $\mu \neq \...
-7
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0answers
34 views

How do I solve this question [closed]

enter image description here Please how I can solve this question
1
vote
2answers
40 views

For a function $f = f(t,x)$, how is the space $C^0([0,T];(S'(\mathbb R^d))^n)$ defined?

I'm reading "Fourier Analysis and non-linear Partial Differential Equations" from H. Bahouri and they use a notation which is unknown to me. If $(t,x) \in \mathbb [0,T] \times \mathbb R^d$, $...
0
votes
2answers
35 views

Need 3 point-correspondences to determine a rotation

I read somewhere that you need three correspondences (pairs of points) to uniquely determine a rotation. How do you prove this? I'm only aware of making similar arguments when considering the solution ...
2
votes
0answers
28 views

Are there any subsets of $\ \mathbb{R}\ $ that have been investigated, but where it was not determined whether the set is countable or uncountable?

Are there any subsets of $\ \mathbb{R}\ $ that have been investigated, but where it was not determined whether the set is countable or uncountable? In particular, I am thinking of a subset of $\ \...

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