4
votes
4answers
231 views

Velleman - How to prove it - Do these two statements really mean the same thing?

Hello and thanks in advance for reading! In How to Prove it P29 Velleman writes: " In general, the statement y ∈ { x | P(x)} means the same thing as P(y), ... " In my understanding the first ...
0
votes
1answer
149 views

Differentiation optimisation

I have a question stating that an insulation strip is to be sealed completely around three edges of a rectangular solar panel. The length of this strip is 200cm. It is asking what dimension of the ...
1
vote
1answer
52 views

A closed set in $\mathbb A^2_k\times\mathbb P^1_k$

Let $k$ be an algebraically closed field and consider the Zariski topology on $\mathbb A^2_k$ and on $\mathbb P^1_k$. If $$X:=\left\{((x_0,x_1),(y_0:y_1))\in\mathbb A^2_k\times \mathbb P^1_k\,\bigg| ...
3
votes
3answers
115 views

Prove that $m=(x+y^2, y+x^2+2xy^2+y^4)$ is a maximal ideal of $\mathbb{C}[x,y]$.

Prove that $m=(x+y^2, y+x^2+2xy^2+y^4)$ is a maximal ideal of $\mathbb{C}[x,y]$. I can show that the ideal $(x,y)$ of $\mathbb{C}[x,y]$ contains $m$ and $(x,y)$ is a maximal ideal. Therefore to ...
1
vote
0answers
359 views

What is the modern use of $\bigodot$ sign?

I've seen $\bigodot$ used in various contexts. It's used for a special set theory operation by some authors (say, Saks) and as sign for Hadamard product by a couple other authors (say, Wiener) in the ...
0
votes
1answer
51 views

Why does $\lim_{n\to\infty}\left(a_1+\sum a_k(z_n-z_0)^{k-1}\right)=a_1$

I couldn't understand the red boxed portion i.e. why does $$\lim_{n\to\infty}(a_1+\sum a_k(z_n-z_0)^{k-1})=a_1$$ Please Help!
0
votes
1answer
151 views

Alternatives to Monte-Carlo simulation

Imagine I have a model of economy of a region, which consists of several companies, importers and population. Let's assume that all local companies in question produce food and agricultural ...
-1
votes
3answers
67 views

Theorem proof of this equation

How would you prove the theorem $(-a)\cdot (-x)=ax$? If you used multiplication and addition axioms.
0
votes
1answer
90 views

Finding the supremum and infimum of a specified set

Hi, I've been stuck on this for a few hours now. I need to find sup(A) and inf(A) of this set. If anyone can provide a proof that would be a huge help! thanks.
3
votes
1answer
77 views

Domain of a solution of a differential equation

let $(x(t), y(t))$ be a solution to the system $x'=y-x^3, y'=-x-y^3$. I want to prove that $(x(t), y(t))$ is bounded for $t>0$ What I did: from the equations comes $0\ge ...
0
votes
2answers
54 views

I'm not sure with one is correct…

At the class I wrote this at my notebook: $\frac{a}{d}\equiv\frac{b}{d}\pmod{n}\Leftrightarrow a\equiv b\pmod{n}$ Assume that $d\mid{a}\wedge d\mid{b}\wedge d\mid{n}.$ this is right? or It should be ...
1
vote
3answers
151 views

Differentiation problem solving

I have a question which I'm unable to provide much working for as I'm not sure how to start it. A metal sphere is placed in seawater to study the corrosive effect of seawater. If the surface area ...
0
votes
1answer
111 views

On the five-point set $X=\lbrace a,b,c,d,e \rbrace$, construct two topologies, one that is Hausdorff and one that is not Hausdorff

On the five-point set $X=\lbrace a,b,c,d,e \rbrace$, construct two topologies, one that is Hausdorff (other than the discrete topology) and one that is not Hausdorff (other than the trivial ...
2
votes
0answers
259 views

Prove that the intervals of the form $[a,b)$ are closed in the lower limit topology on $\mathbb{R}$.

Prove that the intervals of the form $[a,b)$ are closed in the lower limit topology on $\mathbb{R}$. I manage to prove the statement by considering its complement, which is $\mathbb{R} \backslash ...
0
votes
2answers
217 views

Volume of a triangular prism with non parallel bases

Consider an $\mathbf{(v_1,v_2,v_3)}$ triangle and its $\mathbf{\hat{n}}$ unit normal. Let $\mathbf{p_i}=\lambda_i\mathbf{\hat{n}} + \mathbf{v_i}$, $i=\overline{1,3}$. Is it possible to compute the ...
0
votes
4answers
115 views

Probability of Finding an Empty Chair

A similar question here: Probability of finding empty seats If I have a table with ten seats, seven being taken, what is the likelihood that when I arrive, there will be an available grouping of two ...
2
votes
1answer
279 views

$f : [a, ∞) → R$ is a continuous function. If $\lim_{x→∞} f (x) = L$, prove that $f$ is uniformly continuous on $[a, ∞)$. [duplicate]

Suppose that $f : [a, ∞) → R$ is a continuous function. If $\lim\limits_{x→∞} f (x) = L$, prove that $f$ is uniformly continuous on $[a, ∞)$. My attempt at the proof: Well since I have to use both ...
0
votes
1answer
160 views

Surface element for sphere, can't find what's wrong

I know that surface element for any surface is: $$dS=\sqrt{ 1+\left(\frac{\partial f}{\partial x}\right)^2 + \left(\frac{\partial f}{\partial y}\right)^2 }dxdy$$ Say if we want to find a surface ...
2
votes
2answers
91 views

Variant on Russell's paradox: show $B = \varnothing$

Let $X$ be a set and $R$ a relationship on $X$. Define $N = \{x \in X\mid(x, x) \notin R\}$. Let $$B =\{b \in X\mid(\forall n \in N)(b\,R\,n) \land (\forall n \notin N)(\neg b\,R\,n)\}\;.$$ ...
0
votes
1answer
65 views

professionally writing proofs

I am writing a proof for the Theorem (x-a)(x+a)=x^2-a^2 and directly proved it by manipulating the equation using multiplication and addition axioms. But I'm not sure what should be included in the ...
1
vote
0answers
56 views

Determinant of symmetric p-diagonal matrices

Is there a simple method to compute the determinant of the following matrix $A =\begin{bmatrix} d_1 & a & 0 & b & 0 &c &0 &0\\ a & d_2 & a & 0 & b & 0 ...
1
vote
2answers
53 views

Is $f$ is non-prime, Can we say $|f|$ is also non-prime ; in convolution algebra?

By Schwartz-inequality and Riesz–Fischer theorem, one can deduced that, $$L^{2}(\mathbb T) \ast L^{2}(\mathbb T) = A(\mathbb T)(:= \{f\in L^{1}(\mathbb T): \sum_{n\in \mathbb Z} |\hat{f}(n)| < ...
0
votes
1answer
31 views

$p>2$ & $p$ prime, $p-1=q_1^{c_1}\cdots q_g^{c_g}$ where the $q_i$ are distinct primes…

and the $c_i$ are positive integers. Let $a \in \{1,2,\dots,p-1\}$. Show that if an integer $x < p-1$ divides $p-1$ then $x$ must divide at least one of $d_1,d_2,\dots,d_g$, where $d_j = ...
1
vote
3answers
208 views

Every Banach space isomorphic to a subspace of C(X) (for some X)

The book of Douglas says on page 12: Theorem (Banach): Every Banach space $B$ is isometrically isomorphic to a closed subspace of $C(X)$ for some compact Hausdorff space X. Proof: Let X be $(B^*)_1$ ...
1
vote
1answer
80 views

Evaluating Artin symbol

Consider the field $K=\mathbb Q(\sqrt{2})$. Let $mp=\frac{(2+\sqrt{2})^{p}-1}{1+\sqrt{2}}$. For the field extension, $K(-1+2\sqrt{2})/K$, and $p\equiv 5\bmod{6}$ how can one show $ ...
0
votes
1answer
40 views

Discuss the singularities including the points at $\infty$

Discuss the singularities including the points at $\infty$: $\lim_{z\to 0} {z\over \sin z}=1$, so removable singularity, I dont know about $\infty$ $z\cos 1/z= z-{1\over 2! z}+{1\over ...
1
vote
2answers
415 views

Multiplicative Inverses in Non-Commutative Rings

My abstract book defines inverses (units) as solutions to the equation $ax=1$ then stipulates in the definition that $xa=1=ax$, even in non-commutative rings. But I'm having trouble understanding why ...
1
vote
1answer
115 views

Connectedness of the the punctured plane and the right open half-plane

Show that any set obtained by removing a single point from $\mathbb{R}^2$ is still connected, where $\mathbb{R}$ is the real numbers. Then show that $\Bbb H = \{(x,y) : x>0\}$ is connected. By ...
0
votes
1answer
36 views

Find out how many balls after 4 hours algebra

Question In a box, each ball is being divided by a second for half hour. At 9 in the morning, there were 10,000 balls. A. How many balls will we have at 14 oo (2)? B. how many balls did we have ...
0
votes
0answers
114 views

Development of imaginary exponent without appealing to “ambiguity” between $i$ and $-i$

Is there a way to develop the definition of the imaginary exponent, $z^i$, for complex $z$, that does not appeal to the notion that $i$ and $-i$ are "qualitatively indistinct" and that does not rely ...
1
vote
1answer
41 views

Pairwise independence implies intependence of unions

Is the following statement true? Let $\mathcal{A}$, $\mathcal{B}_1$, $\mathcal{B}_2$ be $\sigma$-algebras such that $\mathcal{A}$ is independent from $\mathcal{B_1}$ and $\mathcal{B}_2$. Then ...
2
votes
0answers
52 views

Convergence of $\sum_n^\infty (-1)^n\frac{\sin^2 n}n$ , questions

Could anyone give a hint how to prove the convergence of the following sum? $$\sum_n^\infty (-1)^n\frac{\sin^2 n}n$$ I tried writing it like this instead: $$\begin{array}{lcl}\sum_{n=1}^N ...
3
votes
1answer
229 views

Finding modulus and argument of z³ - 4√3 + 4i = 0

I think I am messing up somewhere as the principle argument should be a nice number from the standard triangles such as $\fracπ4$, $\fracπ3$ or $\fracπ6$ or something close. (That's what we have ...
4
votes
2answers
138 views

How do mathematics define a point?

I have a serious doubt. How do mathematicians define a 'point' in a space or a plot? If we have a clear explanation for a 'point' , I think my doubt on infinitesimals and infinity will be clarified.
1
vote
1answer
46 views

We need to find the limit of $\sum_{n=0}^{\infty} (n+1)z^n$

We need to find the limit of $\sum_{n=0}^{\infty} (n+1)z^n$ for what values of $z$ does the series converges? for convergence we need $\limsup|(n+1)z^n|^{1\over n}<1$ i.e $|z|<{1\over ...
0
votes
1answer
28 views

Determine truth value

If $ x \in \bigcup\{A:A \in \mathscr{A}\} $, then $x \in A$ for some $A \in \mathscr{A}$. $\mathscr{A}$ is a nonempty collection of sets and I have to determine the truth value of the above statement. ...
0
votes
1answer
57 views

The dimension of $\mathbb P^n$ is $n$

I'm trying to prove that the dimension of the projective space $\mathbb P^n$ is $n$. I've seeing some books saying that since the $\{U_i\}$ ($U_i$ homeomorphic to $\mathbb A^n$) is an open cover of ...
1
vote
1answer
132 views

Prove that for each $n$, there exist $n$ consecutive integers, each of which is divisible by a perfect square larger than $1$.

Prove that for each $n$, there exist $n$ consecutive integers, each of which is divisible by a perfect square larger than $1$. Can any one help with this? I was stuck because I know how to solve for a ...
0
votes
1answer
35 views

$f_n \rightarrow 0$ in $L^{1}$ implies $\exists N$ such that $\lim_{ k\rightarrow 0 }\frac{1}{2k}\int_{-1/k}^{1/k} f_N(x) dx = 0$?

Suppose $f_n \geq 0$ and $f_n \in L^1(\mathbb{R})$ for $n=1,2,\ldots.$ If $f_n \rightarrow 0$ in $L^{1}(\mathbb{R})$, must there be an $N$ such that $$ \lim_{ k\rightarrow 0 ...
1
vote
1answer
133 views

What line are inverse functions on the complex plane reflected over?

On the real plane (xy plane) inverse functions are reflections of their original functions over y=x. Is there such line for complex functions and their inverses?
3
votes
2answers
108 views

A theory of equation question from my exam paper

Consider The equation $x^3+3x^2+3x+3=0$ Then the sum of it's non-real roots is A) is equal to $0$ B)lies in $0$ and $1$ C)lies in $-1$ and $0$ D)Greter that $1$ Which one is correct , plz explain ...
2
votes
4answers
503 views

Show that the integers have infinite index in the additive group of rational numbers.

Show that the integers have infinite index in the additive group of rational numbers. Anybody in a good enough mood to tell me how this is done?
0
votes
1answer
435 views

Which of the following subsets of $R[x]$ are subrings of $R[x]$?

Which of the following subsets of $R[x]$ are subrings of $R[x]$? Prove or disprove. All polynomials with constant term $0_R$. The elements of this set have the form $r_nx^n + ...
1
vote
1answer
155 views

Normed vector space and uniform continuity

Let $V, V'$ be normed vector spaces and $f: V\to V'$ a linear transformation. a. Prove that if $f$ is continuous at one point it is continuous everywhere and in fact uniformly continuous. ...
3
votes
1answer
69 views

Is there a friendly pair $\{a, b\}$ where both $a$ and $b$ are odd?

Let $\sigma(x)$ denote the sum of the divisors of a positive integer $x$. $\{y, z\}$ is said to be a friendly pair if $$I(y) = I(z),$$ where $I(x) = \sigma(x)/x$ is the abundancy index of $x$. As ...
1
vote
1answer
311 views

“Problem of points (POP)” explanation

Quite recently, there was a question related to "Problem of Points" at MSE. I did some literature survey on POP in the internet and found explanations using an example of coin toss. I have been ...
0
votes
1answer
186 views

Show a subring contains certain elements.

Show that the set of all real numbers of the form $a_0 + a_1\pi + a_2\pi^2 +\cdots+ a_n\pi^n$ with $n≥0$ and $a_i ∈ \mathbb{Z}$ is a subring of $R$ that contains $\mathbb{Z}$ and $\pi$. ...
0
votes
1answer
86 views

how to solve $x^2 \equiv -1 \pmod{13}$

Knowing that $p$ is prime and if $p \equiv 1 \pmod 4$, then $\left(\left(\frac{p-1}{2}\right)!\right)^2 \equiv -1 \pmod p$; how do I solve $x^2 \equiv -1 \pmod{13}$?
1
vote
1answer
62 views

we need to find the points where $|f(z)|$ has maximum and minimum value

$f(z)=(z+1)^2$ and $R$ be the triangle with vertices $(0,0),(0,1),(2,0)$, we need to find the points where $|f(z)|$ has maximum and minimum value so here $z=2$ is the point of maximum and $z=0$ is ...
0
votes
1answer
79 views

the induced homomorphism $i_\#:\pi_1(P,x_0) \to \pi_1 (X,x_0)$ is an isomorphism.

If $P$ is a path component of $X$ and $X_0\in P$, then the inclusion map $i:P\to X$ can be regarded as a map of pointed spaces $P(X.x_0) \to (X,x_0)$. Prove that the induced homomorphism ...

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