All Questions

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Questions about compact orientable manifold, need your help!

$13.$ Suppose $M$ is compact orientable $n$-manifold (with no boundary), and $\theta$ is an $(n-1)$-form on $M$. Show that $d\theta$ is $0$ at some point. This is another spivak's problem. ...
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Expressing the homomorphism condition with a commutative diagram.

I am asking myself how to express the homomorphic condition in commutative diagrams. For this let $(M, \cdot)$ and $(N, \cdot)$ be algebraic structures and let $\varphi : M \to N$ be a homomorphism, ...
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Objective questions on complex calculations.

how to solve these type of questions ? i have tried logarithm and inequalities but could not pin point the exact and correct method.
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lift of antiholomorphic involution of Riemann surface to its Jacobian's cohomology

Start from a connected closed Riemann surface $\Sigma_g.$ We have $2g$ one cycles $\delta_i$ and we can choose them such that $\delta_i$ intersects positively $\delta_{i+g}$ and no other (this gives ...
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Let $A$ and $B$ denote semigroups. Do such functions $A \times B \rightarrow B$ have a name?

Let $A$ and $B$ denote semigroups and let $f : A \times B \rightarrow B$ denote an action that is also "compatible" with the structure of $B.$ Meaning the following. $f(aa',b) = f(a,f(a',b))$ ...
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Tablet for reading textbooks and writing math

How good are tablets for the purposes of reading textbooks (maths) and writing mathematics? Whats the best way to write mathematics on a tablet: stylus or an app? In either cases, which one would you ...
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I need help with a few problems in my Algebra2 class. thanks!

This is the first one: Find the zeros of f(x) = -x^2 + 4x - 3 by using a table and a graph. Enter the missing numeral of the answer 1, _.
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Proof for convert $\left\{\int_a^tf(x)dx\right\}^{n}$ to multiplication of integrals

In a book, author use a relation like this: $\left\{\int_a^tf(x)dx\right\}^{n} = n!\int_a^tf(x_1)dx_1\int_a^{x_1}f(x_2)dx_2\dots\int_a^{x_{n-1}}f(x)dx_{n}$, How i can make a proof for this? which ...
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Probability of picking specific balls

Suppose I have $20$ red balls in one box and $20$ blue balls in another box. There $12$ red balls and $7$ blue balls have stars on them. I randomly take out one red ball and one blue ball at each ...
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Genus 2 surface equipped with hyperbolic metric is not a symmetric space.

Because the genus 2 surface with hyperbolic metric has constant sectional curvature, its Riemann tensor is covariantly zero. i.e. $\bigtriangledown R \equiv 0$ Therefore, it is locally symmetric ...
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Show the following is Local Martingale

$X_t$ bessel square process which satisfies $$\mathop{dx_t}= 2(a+1) \mathop{dt} +2 \sqrt{x_t} \mathop{dB_t}$$ and $u$ is a function which satisfies $x^2 u'' +x u' -u(a^2 + b x^{2p+2})= 0$. How can I ...
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extremas $\ln(x+y)- (x^2+y^2)/16$

Find max and min to $$\ln(x+y)-\frac1{16} (x^2+y^2)$$ where $2 \le x+y \le 8$ and $x,y \ge 0$. My attempt is very clearly written in the images. But I have no teacher available, so I need feedback if ...
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A criterion for complete lattice.

Is there an infinite partially ordered set $(X,\le)$, in which for each $A\subseteq X$, either $\inf A$ or $\sup A$ exists but for some $A\subseteq X$ either $\inf A$ or $\sup A$ does not exist.
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Prove thant if $a/b + c/d \in \mathbb Z, (a:b)= 1, (c:d) =1$ then $|b|=|d|$

Be $a,b,c,d \in \mathbb Z, b \ne 0, d \ne 0.$ Prove that if $a/b + c/d \in \mathbb Z, (a:b)= 1, (c:d) =1$ then $|b|=|d|$
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A polynomial in two variables with perfect square values

How can I find for which X and Y the values of the polynomial $100X^2+100Y^2+160X+80Y+81$ are perfect square?
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Probability of a player scoring multiple goals in a match

I'm struggling to work out this answer. Say Team A is estimated to score $1.6$ goals in a match and Team B is estimated to score $1.1$ goals. Team A's striker is expected to score 40% of his team's ...
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Help in this easy lemma about dimension in algebraic geometry

I'm studying dimension of quasiprojective varieties which is defined as $$\dim(X)=trdeg(k(X)|k)$$ if $X$ is a quasiprojective variety. I didn't understood this lemma: If $f:X\to Y$ is a finite ...
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$f(x)=\cos x$ is the unique continuous function such that $f(x)f(y)-f(x+y)=\sin x\sin y$

I need to find all solution of functional equation $f(x)f(y)-f(x+y)=\sin x\sin y$ for all $x,y\in\mathbb R$. Here $f:\mathbb R\to\mathbb R$ is a continuous function. I could find prove that ...
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The number system

Simple question(or maybe not...): Its quite fascinating that with only 10 symbols (for base 10) we can represent any possible natural number. In particular, it seems that no matter how you arrange any ...
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How can Busy beaver($10 \uparrow \uparrow 10$) have no provable upper bound?

This wikipedia article claims that the number of steps for a $10 \uparrow \uparrow 10$ state (halting) Turing Machine to halt has no provable upper bound: "... in the context of ordinary ...
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How to find P(X=r) from probability generating function of X?

I have a probability generating function G_X(s) = (p+ps)/(1-s+p+ps) and I need to find P(X=r) How do I get this from the probability generating function? I was thinking about finding the ...
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Formula for coordinates (lat, lon) on a small circle

I would like to know how to find a set of coordinates on a small circle some distance from another point. For example, lets say A = (39.73, -104.98) #Denver B = (39.83, -106.06) #point ...
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Resolution of direct image functor

Let $i: X \to Y$ be an embedding of compact complex manifolds (not necessarily projective) and $E\to X$ a holomorphic vector bundle. I've seen it stated that the direct image sheaf $i_* E$ has a ...
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Are there groups $G$, whose order is uncountable, such that the order of $G/[G,G]$ is countable?

Are there groups $G$, whose order is uncountable, such that the order of $G/[G,G]$ is countable? I am mostly concerned with looking at the groups in terms of generators and relations, so this can be ...
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How many complete theory extend theory T are there?

Т - theory of signature { <= }, defined by the axioms А1-А5: А1: х (х=х) А2: х и у ((х<=y & y<=x) -> x=y) A3: x,y и z((x<=y & y<=z) -> x<=z) A4: х и у(x<=y or ...
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a problem about the equivament of two vector groups $(I)$ and $(II)$

Today,I have a problem about the equivament of two vector groups $(I)$ and $(II)$ $(I),\alpha_1,\alpha_2,\alpha_3,\alpha_4$, and they are linear independence ...
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First eigenfunction on closed manifold

Is it true that $f^{-1}(0,\infty)$ is connected, where $f$ is the eigenfunction for the first eigenvalue of the Laplacian on a manifold (say, compact without boundary)? It seems intuitively clear to ...
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Evaluating the definite integral $\int^{\pi/2}_0 \frac{x+\sin x}{1+\cos x}\,\mathrm dx$

Problem : $$\int^{\pi/2}_0 \frac{x+\sin x}{1+\cos x}\,\mathrm dx \tag{i}$$ My approach : $$\frac{x+\sin x}{1+\cos x}\,\mathrm dx= \left ( \frac{x}{2\cos^2(x/2)} + \tan(x/2) \right )\,\mathrm dx$$ ...
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Exchanging the limit and Lebesgue integral for every set in the measure space

Let $(X,\mathcal{M},\mu)$ be a measure space and let a sequence of $\mathcal{M}$-measurable functions $\{f\}_k$, where $f_k: X\rightarrow [0,\infty]$. Assume that $f_k\rightarrow f$ pointwise and ...
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Value minimizing mean absolute percentage error

What value for $c$ would minimize the formula: $$\frac{1}{n}\;\sum^{n}_{i=1}\left | \frac{y_i-c}{y_i}\right|$$ given the values $y_1, ..., y_n$. For example in the mean squared error we have the ...
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Application of mathematical induction

If $p (x)= a_0 + a_1x^2+....a_nx^n$ in $\Bbb R [x]$ and $a \in \Bbb R$, then $p (x)$ can be written as $b_0+b_1(x-a)+.....b_n(x-a)^n$ ,Where $b_i \in\Bbb R \forall i\in\{0,1,2\ldots n\}$. ...
Here is a funny exercise $$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$ (If you prove it don't publish it here please). Do you have similar examples?
Consider the linear space $\mathcal{L}_\infty$ and let $x\in\mathcal{L}_\infty$ where $x=(a_1,a_2,...,a_n)$ and taking the norm of $x$ to be $\sup x_i$. My questions are: 1). When defining $x$ in ...