1
vote
0answers
12 views

A complete math programme

I want to correct my biggest mistake in life, which is a complete lack of interest in math when I was a kid. Now, after learning a lot of new things, knowing science and working as a programmer I ...
0
votes
0answers
4 views

Partial derivatives of all orders of linear map exist

If F is a linear map from R^n to R^m is it true that F is C^infinity, i.e. partial derivatives of all orders exist? My thought is that the answer should be "yes," because the derivative of F is just F ...
0
votes
1answer
3 views

Let $\{X_n; n\geq 0\}$ be a martingale with respect to $\{Y_n\}$. Proove for any set of integers $k\leq l<m$ that

Let $\{X_n; n\geq 0\}$ be a martingale with respect to $\{Y_n\}$. Proove for any set of integers $k\leq l<m$ that the difference $X_m-X_l$ is uncorrelated with $X_k$, that is, ...
0
votes
0answers
2 views

Value distribution into random variables

I want to distribute a large number into quantities of 100s, 50s, 25s, 5s, 1s Lets say, I'm selling apples in quantities of 100s, 50s .... 1. Based on the quantity there is different pricing. 1 apple ...
0
votes
1answer
9 views

Induction and trigonometry

Do you know how to prove that cos(X/2) + cos(3x/2)... + cos(2n-1)/2 = sin(nx)/(2sin1/2x) with induction? I have tried with n = 1 which gives cos x/2 = sin(nx)/(2sin1/2x) I am not sure on how to ...
0
votes
0answers
10 views

Solving Logical equivalence & propositional logic problems without truth tables

I have no particular "Logic question" in hand at the time being, but need help to understand a way that can be used to prove "Logical equivalence without using truth tables". moreover can we solve ...
0
votes
2answers
16 views

Does $\pi \left(\dfrac{x+y}{2}\right)=\pi\left(\sqrt{xy}\right)$ hold for infinitely many values of $x$ and $y$?

The problem is (assume $\pi(x)$ to be the prime-counting function, Does there exist infinitely many solutions to the equality $\pi \left(\dfrac{x+y}{2}\right)=\pi\left(\sqrt{xy}\right)$ with ...
1
vote
0answers
11 views

How can check that for a given norm, we can found an inner product?

Let $$\Bbb C^2=\{w=(z_1,z_2) : z_1,z_2\in\Bbb C\}$$ be the vector space of all ordered pairs of complex numbers. Can we obtain the norm defined on $\Bbb C^2$ by $$||w||=|z_1|+|z_2|$$ from an inner ...
0
votes
1answer
12 views

How to find right child in a pyramid number?

A pyramid number: 0 1 2 3 4 5 6 7 8 9 So is there any equation like: ...
1
vote
0answers
18 views

Surface area of a slightly deformed sphere

Consider the unit sphere, which can either be described by $x^2+y^2+z^2=1$ or by the equation $r(\theta,\phi)=1$, where $(r,\theta,\phi)$ are spherical polar coordinates. I define a deformed sphere ...
0
votes
0answers
20 views

Wich way to do this integral?

Let $\delta(\phi) = \phi(0)$ be the dirac delta. I would like to compute $\int_{\mathbb{R}} h(x) \delta(\lambda x) dx$ 1) Since $\delta$ is an unit mass on $0$ $$\int_{\mathbb{R}} h(x) ...
0
votes
1answer
22 views

How to calculate powers of a permutation in cyclic notation?

How do I calculate powers of an 8-cycle (1 2 3 4 5 6 7 8) ?
3
votes
0answers
25 views

Origin of the Integral (Theory Behind It - How it came about)?

How exactly was the integral derived? Like similarly to how the difference quotient explains where the derivative came from, what can we use to explain the origins of the integral? Like how does ...
2
votes
4answers
27 views

Am I using the chain rule correctly?

I'm supposed to find $y'$ and $y''$ of this function: $$y=e^{\alpha x}sin\beta x$$ This is what I have done so far: $$y'=e^{\alpha x}\sin\beta x\cdot \alpha x'\sin\beta x\cdot \sin'\beta x \cdot ...
0
votes
0answers
7 views

How to find estimation value of $u$ s.t minimize $\sum_{i=1}^n \sum_{j=1}^m \|x_i-c_j\|^2u_{ij}^p $

I have a dataset $x=\{ x_1,x_2...x_n\}$. I will divide the data into $m$ classes with cluster center (mean of each data in a cluster) is $c_j$. The absolute square distance between each data $x_i$ ...
1
vote
0answers
14 views

Characterizing the A-module ${M}/{S}$

I've been working through this for a little while, and I'm not 100% sure I understand what I'm supposed to be doing here, or maybe I'm not grasping correctly what they mean by "Characterize". ...
1
vote
0answers
25 views

The “argument” of a quaternion

My question is pretty simple. I've been trying to read a pretty introductory text on Clifford algebras, and I encountered how they define the "argument" of a quaternion as an ordered quadruple ...
2
votes
0answers
20 views

How can I calculate the formula of this fractal-like structure?

I did the following fractal-like structure manually, and I was trying to convert it to a formula to compute some parts of the drawing, but I get lost due to the complexity of the structure. Is as ...
1
vote
0answers
20 views

G acts freely on X. G is paradoxical implies X is also paradoxical

I am proving the Banach-Tarski paradox using a series of small results. For definition of certain terms, see here. Group $G$ acts freely on $X$ i.e. $\operatorname{Stab}(x)=e, \ \forall \ x\in X$. ...
-1
votes
2answers
7 views

Is the sum of a unimodal and increasing function still unimodal?

There is no specific function. I would like to know if there is information on summation of a unimodal and increasing function? (Continuous functions)
1
vote
0answers
30 views

Cardinality of the set of all field automorphisms of $\mathbb C$ [duplicate]

Does $\mathbb C$ have infinitely many field automorphisms? Does it have uncountably many field automorphisms?
1
vote
1answer
32 views

Integral Test question

So this is the problem: http://postimg.org/image/5g815zgk5/ I am getting $\lim_{b\to\infty} 2\sec^{-1}(2b) - 2\sec^{-1}2$ Now what? What do I do with $\sec^{-1}(2b)$? What happens to a trig function ...
1
vote
0answers
10 views

Matrix $B \in M_n(S)$, for $S$ an $R$ algebra, with $R$ independent entries, $A \in GL_n(R)$. Are the entries of $AB$ $R$-independent?

Let $R$ be a field (or a domain, or a commutative ring), and $S$ an $R$-algebra. Let $B \in M_n(S)$ have $R$ independent entries. Let $A \in GL_n(R)$. Are the entries of $AB$ $R$-independent? I am ...
0
votes
0answers
17 views

example of multiplication of ordinals with infinite cardinality with larger value on right where we dont' take the max?

I recall reading about a rule for multiplying ordinals where at least one is infinite, and where the cardinality of the multiplier (on right) is larger than the multiplicand (on left). If I recall ...
5
votes
4answers
53 views

Is there an alternative intuition for solving the probability of having one ace card in every bridge player's hand?

I am trying to get to know probability a little better since it's a weak point for me and I was wondering what other ways there were to intuitively think about the problem of finding the probability ...
1
vote
2answers
20 views

Incommensurable line segments

I have an issue with a proof given in my lecture hopefully someone can help me with. It will be shown that the hypotenuse $c$ of a right-angled and isosceles triangle will be incommensurable to the ...
0
votes
0answers
15 views

How does one maximize “mixedness” among seating charts over time?

Background: My class has $10$ students and $3$ tables; naturally, the students are distributed with $3, 3,$ and $4$ seated at the individual tables. On the second day of class, students sat in the ...
-4
votes
3answers
42 views

Can I divide $50$ cars on$ 5$ days? any trick? [on hold]

Can I divide $50$ cars on $5$ days, on condition that the numbers should be odd numbers? is there any trick? I'm asking for $5$ 5 odd numbers whose sum is $50$
1
vote
0answers
12 views

Analysing Exact Sequence

I have the following exact sequence $\mathbb{Z}\xrightarrow{f}\mathbb{Z} \xrightarrow{g} K_0(\mathcal{T})\xrightarrow{h}\mathbb{Z}\xrightarrow{0}0$. From here I want to conclude that ...
1
vote
0answers
23 views

Proving one expressions is greater than the other using limits?

In general, is it sufficient to show that one of them increases faster than the other? $$1-P_{k,1}< or > (1-P_{k,2})(M+B(1-p))/(M+B))$$ where $P_{k,1}$ and $P_{k,2}$ are decreasing with M. ...
0
votes
0answers
19 views

Are these two compositions of two functions differentiable?

Assuming $U=\{x\in\mathbb{R}^2:x_1^2+x_2^2<1\}$ is the open unit circle in the plane and $f,g:U\rightarrow\mathbb{R}^2$ two functions with $f(0)=g(0)=0$. $f$ is Fréchet-differentiable in $0$, and ...
0
votes
0answers
7 views

Divergence and formal adjoint operators: are they bounded/continuous?

Let $(M,g)$ be a smooth Riemannian manifold. The divergence operator is the map \begin{align*} \delta_g:\Gamma^k(S^2M)&\rightarrow\Gamma^{k-1}(T^*M)\\ ...
0
votes
0answers
12 views

How to obtain the genus of the Riemann surface corresponding to an algebraic curve

I am trying to read about the genus of an algebraic curve. I have been told that there is a connection between topological genus and genus defined for an algebraic curve. Since an algebraic curve ...
2
votes
1answer
18 views

Ring homomorphism of tensor product of algebras

Let $B, C$ be two $A$-algebras, $f:A \to B, g: A\to C$ the corresponding ring homomorphisms. From this we can construct an $A$-algebra $B \otimes _A C$ and the mapping $ a \mapsto f(a) \otimes ...
2
votes
4answers
19 views

conditional probability of several events

I'm having a hard time understanding what this question wants: A person initially purchases either type A or type B. She will choose either type A or type B with an equal probability on her first ...
0
votes
1answer
5 views

Boolean algebra consensus theory

I want to simplify $wxy + x'z + y'z + wz = wxy + x'z + y'z$ but I can't seem to use the consensus theorem at the right place. I tried factoring cases for $x$ and $x'$ and $y$ and $y'$ but I don't ...
2
votes
0answers
12 views

Is this a correct understanding of what de Rham's Theorem is saying?

De Rham's Theorem states that $H_{dR}^k(M) \simeq H^k(M;\mathbf R)$ for all $k$. This is what it states. But I've been struggling to understanding what it's telling me. Here's my understanding so far: ...
0
votes
3answers
21 views

$A$ is a doubly stochastic matrix, how about $A^TA$

I am reading a paper with assumption that $A \in R^{n\times n}$ is a doubly stochastic matrix. However, the paper says $A^TA$ is symmetric and stochastic. Since $A^TA$ is symmetric, if $A^TA$ is ...
0
votes
0answers
7 views

non-linearity and non-convexity

I am taking a course on linear regression online and it talks about the sum of square difference cost function and one of the points it makes is that the cost function is always convex i.e. it has ...
2
votes
0answers
23 views

Polynomial-closed properties of rings

If $R$ is a ring with certain property, sometimes when we pass to the polynomial ring in one variable, the ring $R[x]$ still has the same property. For instance, it's a theorem that if $R$ is a UFD ...
0
votes
2answers
31 views

Calculating the Variance of a Dice Roll?

Here's my thinking: $$Var(X) = E(X^2) - E(X)^2$$ Assuming each roll is independent: $$E(X^2) = E(XX) = E(X) \cdot E(X) = E(X)^2$$ Thus: $$Var(X) = 0$$ However, this is not correct. Where did I ...
2
votes
1answer
26 views

Examples of irreducible representations

Which of the following representations are irreducible? 1) The tautological representation of $D_n$ on $\mathbb{R}^2$ 2) The action of $U(1)$ on $\mathbb{C}$ by multiplication 3) The ...
0
votes
3answers
54 views

Limit of $\{a_n\}$, where $a_{n+1} = \sqrt{2+a_n}$

I am struggling with this question: Let $\{a_n\}$ be defined recursively by $a_1=\sqrt2$, $a_{n+1}=\sqrt{2+a_n}$. Find $\lim\limits_{n\to\infty}a_n$. HINT: Let $L=\lim\limits_{n\to\infty}a_n$. ...
1
vote
3answers
27 views

Convert the equation to rectangular form $r = \frac {6}{1-\sinθ}$

Convert the equation to rectangular form $r = \frac {6}{1-\sinθ}$ The answer should be: $y = \frac{1}{12} x^2 -3$ But how to arrive at the answer? I tried replacing r with $\sqrt{x^2 + y^2}$, then ...
3
votes
1answer
36 views

Asymptotic behaviour of $\int_0^1 g(x)\exp(-nx)dx$ as $n\rightarrow\infty$

Let $g:(0,1]\rightarrow\mathbb{R}_+$ be an invertible monotonically non-increasing function that integrates to 1 and has g(1)=0, $g(0)=\infty$; eg. $g(x)=x^{-1/2}-1$ or $g(x)=ln(1/x)$. I believe it is ...
-1
votes
0answers
52 views

Epsilon delta proof of a limit

Show that the $\lim_{x \to 2} \frac{x-2}{x^2}=\frac{1}{2}$ using epsilon delta argument. To start with I have, not sure how to use the code so some brackets represent absolute value signs: ...
1
vote
0answers
18 views

clarification on eigendecomposition of a matrix

looking for some clarification on a couple things related to the eigendecomposition of a square matrix. Suppose we have a square n x n matrix, A, and we are interested in finding its eigenvectors and ...
1
vote
1answer
33 views

Analysis for Engineering : Practical Applications

I don't know much more about Analysis than what I've read about it on Wikipedia, although I have just begun reading Introduction to Calculus and Analysis I, by Richard Courant. My understanding is ...
1
vote
0answers
24 views

Intuitive reason for Fourier Series Convergence

I read that Fourier Series Converges to average of left side and right side limits at Jump Discontinuities. What is the intuitive explanation for it? Is it something regarding Energy minimization?
-2
votes
0answers
36 views

Finding all points on $y=x^2$ for which the normal line goes through the point $(0,3)$.

Find the coordinates of all points of the parabola $y=x^2$ for which the normal line goes through the point $(0,3)$. Give exact answers using radicals if necessary. No decimals.

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