0
votes
0answers
2 views

Covariance of m-fold integrated Wiener process

The problem I'm trying to perform a Bayesian approach to the Maximum Likelihood Estimation procedure of Wecker and Ansley (1983). To this end, I need to compute the full likelihood of the data given ...
0
votes
0answers
7 views

Prove that there are no integer solutions to a given equation

Prove that $ 4x = y^2 + 1 $ has no integers solutions for (x,y) By rules of divisibility: $$ a | b \implies \frac {b}{a} = n $$ for $a,b,n \in \mathbb{Z}$ So let, $ a=4x$, $b=y^2$,and $ c =1$ ...
0
votes
0answers
4 views

Thoughts on Mac Lane's Homology for learning basic cohomology theory related to algebraic topology

I apologize, this may be a duplicate question as many reference requests here are. For those who have read or have experience with the content covered in Mac Lane 's Homology, what are your thoughts ...
0
votes
0answers
7 views

Curvature of curve

$r(t) = (-3sint)i + (-3sint)j + (cost)k$ I got as far as:$$||r'(u)|| = sqrt{(18cos^2u + sin^2u)}$$ But I cannot evaluate $\int_0^t||r'(u)||dt$
1
vote
0answers
25 views

Calling all genius: $p_1^{e_1} p_2^{e_2}…p_k^{e_k}=e_1^{p_1} e_2^{p_2}…e_k^{p_k}$

Find all positive integer $e_i$ and prime number $p_1^{e_1} p_2^{e_2}...p_k^{e_k}=e_1^{p_1} e_2^{p_2}...e_k^{p_k}$ for $k\ge 1$. Is this impossible or what? I've been trying for at least a week. I've ...
1
vote
0answers
15 views

Find an approximation for $P\{N=1\}$

I have the following question, I know I am somehow supposed to use binomial to Poisson approximation but can't figure out the trick. $X_1, X_2, ..., X_N$ are iid with $$P\{X_k = i\} = 1/10^6 , 0 ...
2
votes
1answer
9 views

Equilateral Triangle from three complex points

I need some help proving this, I've seen it proven in the other direction (prove the formula if it is an equilateral) but cant figure out how to prove it this way around. Any help would be much ...
1
vote
3answers
21 views

Why do polynomial terms with an even exponent bounce off the x-axis?

Like if it's $f(x) = (x-5)^2(x+6)$ Why, at $x=5$, does the graph reflect off the x-axis?
0
votes
0answers
6 views

Suggestion: good book on probability theory with emphasis on applications to other areas of mathematics and physics

On this website, there are many questions about books on probability theory, but I would like to ask if you can select (from all the references available on this website and elsewhere) a book ...
0
votes
0answers
9 views

Get the critical points and find the máximum or mínimum of $f(x,y,z) = (x^{2} + 2y^{2} +1)\cos{z}$

I'm trying to solve this problem: Get the critical points and find the máximum or mínimum of $f(x,y,z) = (x^{2} + 2y^{2} +1)\cos{z}$ First, I founded the gradient: $\nabla f(x,y,z)= (2x\cos{z}, ...
2
votes
0answers
16 views

Reference Request to Prepare for Hatcher's “Algebraic Topology”

Hatcher himself has an excellent and always generously free set of notes on point- set topology: http://www.math.cornell.edu/~hatcher/Top/TopNotes.pdf It includes up to quotient spaces. It seems ...
1
vote
0answers
7 views

Probability of 6 teams winning an 8 team debating competition

My son is in a debating team that competes in an 8 team round robin competition. Each team every other team once, and the result is a win or a loss (no draws). This year 6 teams came equal first, ...
0
votes
1answer
10 views

Show that $\int_{X}u\, \mathrm{d}\mu\leq 4$ and $\int_{X}u\, \mathrm{d}\mu=1$.

Let $(X,\mathcal{A},\mu)$ be a measureable space. Let $u\in \mathcal{M}_{\mathbb{R}}^{+}(\mathcal{A})$ and $\lbrace u_{j}\rbrace_{j\geq 1}$ be a sequence of functions in ...
1
vote
3answers
17 views

GCD's and how they generate groups

I was reading something today an it was talking about $U_{15}$, all the integers relatively prime to 15, and how it was generated by the set {7,11}. I understood it all, but I thought that if the ...
0
votes
0answers
15 views

Help with Rads?

I'm stumped on this problem... A pateints thyroid gland is to be exposed to an average of 5.5 µCi for 16 days as an ingested sample of Iodine-131 decays. If the energy of a β radiation is 9.7 x ...
0
votes
0answers
6 views

Properties of a $B^\ast$-algebra

Defining a complex Banach algebra as a $B^\ast$-algebra when it is equipped with an application $\ast:B\to B$ such that for any $x,y\in B$ $$(x+y)^\ast=x^\ast+y^\ast,\quad(xy)^\ast=x^\ast ...
1
vote
0answers
13 views

Solving a simple Distance Geometry problem

I'm trying to solve the following problem: Given the absolute positions of four points in 3D space, and the distances from these four points to a fifth point, find the position of the fifth point. ...
0
votes
1answer
16 views

dimension of direct products

Suppose $\{V_i\}_{i\in I}$ is a family of $k$ vector spaces. Is it possible to calculate $\dim\oplus_{i\in I} V_i$ and $\dim\prod_{i\in I}V_i$?
0
votes
0answers
6 views

Frobenius theorem for singular foliations , what are hypotheses impose over an endomorphism $P:TM\to TM$ that it span a singular foliation in M?

Let $M$ a smooth manifold. Given a morphism $P:TM\to TM$, i.e, $\pi\circ P\equiv Id$ and is linear over the fibers. If we suppose that $P^2=P$, then for each $x\in M$, $P_{x}:T_xM\to T_xM$ is a ...
0
votes
0answers
9 views

showing 2 separate basis for a Vector space

Suppose $V$ is a vector space over the field $F$ and $V$ consists of more than the zero vector Suppose $V$ is spanned by a set $S$ of $s$ vectors and $V$ contains a linearly independent set $B$. ...
0
votes
1answer
14 views

Geometry , gre problem

Smallest distance from a point P to any point on the circle C is 5 and the largest distance from the the point P to the circle is C is 11 . If point P is situated outside the circle C , then what is ...
0
votes
1answer
10 views

Problem with spanning set and matrices

Let $V=M_{2,2}(\mathbb R)$, the set of $2\times 2$ real matrices, and consider the subset $$S=\left\{\begin{pmatrix}1&1\\0&0\end{pmatrix},\begin{pmatrix} ...
0
votes
0answers
10 views

Change of Variable involve derivative

Let me just give the 1-D version of my problem. Let $u\in C_c^\infty(R)$ and define $u_r(x):=u(rx)$. Then I am trying to evaluate the integration $\int_R u_r'(x)dx$. Here is my steps: $$\int_R ...
1
vote
3answers
39 views

How to prove which number is bigger??

Prove which number is larger: a) $10^{100}$ or $10^{10^{100}}$ b) $e^\pi$ or $\pi^e$ I know we all know how to plug these into the calculator and check, but how someone mathematically prove which ...
0
votes
0answers
18 views

Very basic question about the definition of the derivative

Why is the definition of the derivative shown here as $\dfrac{\Delta x}{\Delta y}$ if immediately above the slope (derivative) is defined as $\dfrac{\Delta y}{\Delta x}$? Why is $\Delta x$ in the ...
-2
votes
1answer
26 views

I do not know how to slove it

If a child can run 10 meters while a car travels 30 meters, how many meters can the child run while the car travels 66 meters
-2
votes
2answers
29 views

Please verify this

Using Newton's binomial theorem to argue that: $n \ge 1$
0
votes
1answer
12 views

What is this function of 2 variables?

Can you tell me the function f(K,N) that has the following values? For my education, please also explain how you tackled the problem. ...
0
votes
2answers
22 views

Finding derivative for multiple chain rules in one problem

Find the derivative of $$y=\sqrt{e^{-3t^2}+5}$$ There seems to be several layers to this. I'm not quite sure how to go from one to the next.
0
votes
1answer
10 views

Maclaurin series accuracy

Find an $n_1$ such that the $n_1$th-order Taylor polynomial for $\sin(x)$ about $x=0$ gives an approximation of $\sin(x)$ with an error of less than $5\cdot 10^{-10}$, for all $x$ between $0$ and ...
0
votes
2answers
7 views

When a symmetric densely defined operator is an adjoint operator?

I am wondering if it is possible to say that if a symmetric differential operator is densely defined then the operator is self-adjoint indeed? More Precisely, Let $A:D(A)(\subset H)\to H$ a densely ...
1
vote
3answers
23 views

If $g(x) = \sqrt{10 − x}$ What is the Domain for $(g\circ g)(x) = g(g(x))$?

What is the domain for $10 − \sqrt{10 − x} ≥$ ?
0
votes
0answers
7 views

Least sum of power of distances

Let $n$ points in a $3$-dimensional space. Are there any general intuitive methods to tackle the problem of finding the point $X$ such that the sum of distances $A_1X^x+ A_2X^x + ... +A_nX^x $ (where ...
2
votes
2answers
18 views

How to prove a polynomials roots are integers?

I'm having trouble finding a direct way to prove questions like the following: $$ \exists x\in\mathbb{Z} |x^2+x=271 $$ Now, I know this is false, because $$x^2+x-271=0$$ $$x= ...
0
votes
0answers
3 views

solution to curve limits (concept question)?

I had a question about how the limits work in that 4pi would not give the correct circle distance. I understand that if it has a radius 1 that the distance would be farther but that is only for a ...
0
votes
0answers
14 views

Example of a Schauder basis for $C^0(\mathbb{R})$

Can someone please provide me with an example of a Schauder basis for $C^0(\mathbb{R})$. If there isn't one could you please explain why not. My understanding of basis for infinite dimensional vector ...
-1
votes
0answers
18 views

Continuity and differentiability of piecewise defined function

I am struggling to answer: Let $f$ be a function whose domain of definition is the ensemble of real’s strictly positive where: $$ f(x) = \begin{cases} a+x^b & \text{if } x>1\\ 2x & ...
3
votes
0answers
17 views

When do Leibniz-like rules lead to unique linear operators

Background Usually one defines differentiation in terms of limits, and then shows that differentiation satisfies the Leibniz (product) rule, $$\frac{d}{dx}(f \cdot g) = f\frac{dg}{dx} + g ...
0
votes
2answers
22 views

How is the discriminant is able to find coefficients in a quadratic equation?

I know how to solve for $k$ in $kx^2-30x+25=0$ using $b^2-4ac$, but i want to know how the discriminant does this. How are we able to just plug the coefficients into the discriminant and get the ...
0
votes
0answers
23 views

Modular arthimetic

What are the four distinct values which are congruent to 6 when the base is 7? I already have two of them which is 6, and 48 but i can not figure out the other two I have tried all other
0
votes
1answer
22 views

Limit of a quotient. Proof through the definition of absolute and relative errors.

I have difficulty with understanding V. Zorich's (Mathematical Analysis, p.85 of English edition) proof of a $$\lim\limits_{n\to\infty}\frac{x_n}{y_n}=\frac{A}{B}, \quad y_n\neq0,\quad B\neq0$$ Proof: ...
0
votes
0answers
15 views

Enlightening ideas and methods that change one's appoach to problems, theorems or mathematics as a whole

I would like to collect a "big-list" of ideas and methods from different fields (although I'm particularly interested in elementary number theory, algebra, analysis, linear algebra, geometry, physics, ...
0
votes
0answers
12 views

Criticise work with simple graphs & problem solving

So I'm studying graph theory at the moment and would like some constructive criticism or thoughts on my method. The problem can be formulated as follows. I'm looking for someone to verify my answer as ...
0
votes
2answers
24 views

Let $u(x,y) = x^2 + 2axy + by^2$, where $a$ and $b$ are real, when is $u$ the real part of an analytic function and what's the imaginary part?

Does my approach here seem agreeable? Attempt: If $u$ is the real portion of an analytic function in the complex plane, it must satisfy the Cauchy-Riemann equations $u_x = v_y$ and $u_y = -v_x$, we ...
0
votes
1answer
24 views

probabilit on rolling die

If a fair die is rolled 6 times, what is the probability that the number shown will be less than 4 exactly 2 times? I have tried to do this problem and I just can't get it. I am not even sure how to ...
-1
votes
2answers
29 views

Verify the Identity

$\binom{n}{k-1} + \binom{n}{k} = \binom{n+1}{k}$ So far I have gotten $\frac{n!}{(k-1)!\big(n-(k-1)\big)!} + \frac{n!}{ k! (n-k)!}$ But I quickly lose myself once I have to start making the ...
2
votes
0answers
15 views

Is every field between $F$ and $F(\alpha_1,\cdots,\alpha_n)$ of the form $F(\alpha_j,\cdots,\alpha_k)$

Say I have a field $F$, and an extension field $L = F(\alpha_1,\cdots,\alpha_n)$. Is it true that every $K$ such that $$ F \subset K \subset F $$ (all field extensions), $K = ...
0
votes
1answer
18 views

Why $( Z_3\rtimes Z_2)\times Z_2 \cong (Z_3\times Z_2)\rtimes Z_2$?

I got an explanation, it says as $Z_2$ is in the kernel of the homomorphism. But I can't understand from that. Also can you tell me why $Z_3\rtimes Z_2\cong S_3$ ? Thank you.
0
votes
0answers
20 views

Avoid losing the woods for the trees in daily study and lecture time

When facing to some new material in mathematics, it is quite easy to overwhelmed by lots of details with losing the woods for the trees. So is there some good strategy to study the materials ...
0
votes
1answer
26 views

How to write a formal proof of the statement: For all real numbers $x$, if $x \ge 1$ then $\frac{3|x-2|}{x} \le4$

For all real numbers $x$, if $x\ge1$ then $\frac{3|x-2|}{x} \le 4$ I understand that I must algebraically show how to build on $x\ge1$ to reach $\frac{3|x-2|}{x} \le4$, but cant for the life of me! I ...

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