0
votes
0answers
3 views

Find this limit.

What is the limit as x->1 of (x^2)+2? I greatly appreciate your help and I will continue to type to satisfy the minimum length requirement please do not downvote this question as it will harm my ...
0
votes
0answers
2 views

Related Rates Shadow Problem

The question is as follows: A man 6 feet tall walks at a rate of 5 feet per second away from a light that is 15 feet above the ground. When he is 10 feet from the base of the light, (a) at what rate ...
0
votes
0answers
3 views

Calculating central elements of Universal Enveloping Algebras?

Simply put, how do I calculate (in general) the central elements of the UEA of some Lie algebra given some desired degree in the algebra generators? I know the so-called 'quadratic Casimir', of ...
0
votes
0answers
5 views

Multiplicity in Solutions to Trig Function Equations

This is a very simple problem, but I can't figure out where I am going wrong! Say you have the following: $a \sin\theta + b \cos\theta = c. \tag{1}$ Now, this for example can be rewritten using: $R ...
0
votes
1answer
6 views

Subset of null set (set of measure zero)

Is there a proper subset of a set of measure zero that is not measurable? Any examples? Thanks a lot! I suspect the answer is yes due to some careful phrasing in books, e.g. let F be a subset of a ...
0
votes
0answers
3 views

Residue of $g(z)g'(z)$

I know how to use the residue theorem and the winding number to find the residue of $f$, but I have no idea how to relate the residue of $g$ to that of $g\cdot g'$, especially without knowing ...
0
votes
0answers
4 views

For any linear operator $\phi$ on $V$, prove such an integer $m$ exists.

Suppose $V$ is an $n$-dimensional vector space over some infinite number field $K$, $\phi\in\mathcal L(V)$, prove there exists such a (positive) integer $m$ that $$Im \phi^m=Im ...
0
votes
0answers
6 views

Opposite of NCF axiom

Say that two ultrafilters in $\omega^*$ are nearly coherent if there is a finite-to-one mapping $\varphi:\omega\to\omega$ such that $\beta\varphi$ maps one ultrafilter to the other. NCF (near ...
0
votes
0answers
4 views

Error Propagation, and Variance

I've recently been reading about error propagation, and I am very confused as to what is being assumed. On the wolfram page here it seems to derive the error propagation for the product of two ...
0
votes
0answers
4 views

A certain zeta function; or, the determinant of the Laplacian plus a constant on the circle

I am interested in a certain "zeta function," a meromorphic function of $s \in \mathbb{C}$ that depends on a real parameter $\alpha \neq 0$. It's defined for the real part of $s$ large by $$ ...
3
votes
0answers
14 views

Sequence of equations

The sequence continues infinitely, why do the equations below work? $$1+2=3$$ $$4+5+6=7+8$$ $$9+10+11+12=13+14+15$$ So I've been trying to observe some patterns but none seem to help me. So I ...
1
vote
1answer
23 views

Proof that $(a,b)\subset\mathbb{R}$ is not countable. Does it use Axiom of Choice?

I used this proof to show $(a,b)$ is uncountable, but looking at it, I don't really see if it uses AC or not. Until recently I was thinking it does use AC (In the choice of the $a_n,b_n$), now I think ...
0
votes
1answer
6 views

Equalities of ideals in Q[x,y]

I'm trying to prove something about polinomyal ideals. So I have to use this proposition: Let $I \subset k[x_{1}, \ldots, x_{n}] $ be an ideal, and let $f_{1}, \ldots, f_{s} \in k[x_{1}, \ldots, ...
-1
votes
0answers
4 views

Determining North-South Line Via Non-digital Watch Method: Discussion on Background Theory

Read this recently (page 9). What is the underlying geometric/celestial reason this works? What is the error to true north given this method? Is there a more accurate method using a watch and the sun? ...
0
votes
2answers
11 views

confused on to leave in centimeters or convert to cubic centimeters

The volume $V$ of the cylinder is $65\pi \mathrm{cm}^3$. The height of the cylinder is $5$ centimeters. Use the formula $V = Bh$ to find the area of the base of the cylinder.
2
votes
0answers
9 views

Proof of Vandermonde's Identity using a different aproach.

Hi I'd like to know if the following proof of Vandermonde's Identity is correct: Let $m,n,r$ be natural numbers such that $r\le \min \{m,n\}$. Then $$ \binom{m+n}r = \sum_{k=0}^r \binom mk \binom ...
1
vote
1answer
13 views

Show that $Ax=0, Bx=0$ share the same solution space iff there is some invertible $P$ s.t. $B=PA$.

The question is said in the title, suppose $A,B\in M_{m\times n}(K)$, where $K$ is some infinite number field. If we regard $A,B$ as linear maps from $K^n$ to $K^m$, then they share the same ...
1
vote
1answer
26 views

Show $X$ and $Y$ are independent if we assume that $E[XY] = E[X] E[Y] $

Assume that $$E[XY] = E[X]E[Y]$$ Let $X$ and $Y$ be random variables taking two different values $a,b \in \mathbb{R}$. Show that X and Y are independent. Note: I've spent a long time on this ...
-1
votes
1answer
7 views

For one year what is the average price per Are?

What is the average price per 100m2 of land per year? 1 Are = 10m x 10m I have purchased a total of 55 Are of land in Bali for different lengths of time. I have purchased 30 Are of land for ...
0
votes
0answers
5 views

What are the asymptotic considerations in the following?

The following is from this paper that discusses polynomials and classic number theory functions. The proof of theorem 1.3 has a final statement saying that $R$ must be null because equation (1), ...
0
votes
3answers
16 views

Pre-image of $f(x,y) = xy$

$f: \mathbb{R^2} \to \mathbb{R}$ is $f(x,y) = xy$. Find the pre-image $f^{-1}((a,b))$ of an open interval $(a, b) \in \mathbb{R}$, and show that this pre-image is open in $\mathbb{R^2}$. I can't ...
0
votes
0answers
11 views

Show $F: (0, \infty) \rightarrow \mathbb{R}$ is differentiable with respect to $t$

Hi just need a bit of help with a few parts of this practice question: Show $F: (0, \infty) \rightarrow \mathbb{R}$ is differentiable with respect to $t \in (0, \infty)$, where $$F(t) := ...
1
vote
0answers
7 views

A ring satisfying a certain property is unital and has $ab = 1$ if $ba = 1$

Let $R$ be a finite ring satisfying for any $x \in R$ there exists $y \in R$ with $xyx = x$. Show that $R$ is unital and that if $ab = 1$, then $ba = 1$. Thoughts so far: If I can show that the ...
0
votes
0answers
7 views

Why is Distribution Prioritized Over Combining?

In every algebra (or basic analysis) book that I've seen, three properties of real numbers are taken as axiomatic: commutativity, association, and distribution of multiplication over addition [a(b + ...
0
votes
0answers
4 views

Number of possible non crossing paths on a grid of $m$ by $n$ size?

Given two points on 2 dimensional m by n grid, moving in units of 1 in either direction, how many non intersecting paths exist between the two points? in other words, with taxi cab metric, on a m by ...
1
vote
0answers
13 views

Is it possible to define submanifold like this

Wikipedia offers the following definition for an (embedded) submanifold: An embedded submanifold (also called a regular submanifold), is an immersed submanifold for which the inclusion map is a ...
0
votes
0answers
11 views

An compute of Riemannian geometry

According to Einstein summation convention , $g_{ij}$ is metric tensor,and $f$ is a real function. Show that : $$ g_{il}g^{ij}\frac{\partial f}{\partial x^i}g^{kl}\frac{\partial f}{\partial ...
1
vote
0answers
7 views

A functional equation relating two harmonic sums.

Introduction. I computed two Mellin transforms while browsing / working on the problem at this MSE link. No solution was found, but some interesting auxiliary results appeared. I am writing to ask for ...
0
votes
1answer
8 views

Find $\overrightarrow{b}$ knowing $comp_{\overrightarrow{a}}\overrightarrow{b} = 5$ and $\overrightarrow{a}$

I am trying to solve the following question: Given that $\overrightarrow{a} = \langle3, 2, -1\rangle$, find a vector $\overrightarrow{b}$ such that $comp_{\overrightarrow{a}}\overrightarrow{b} = ...
0
votes
0answers
4 views

Solving a system of ODEs with 4 repeated eigenvalues

I'm working on problem which requires me to solve a system of ODEs with 7 equations. I've gotten as far as determining the eigenvalues and vectors of my coefficient matrix $A$, but 4 of the ...
0
votes
1answer
12 views

Prove that $\tan x < \frac{4}{\pi}x,\forall x\in \left( 0;\frac{\pi}{4} \right)$

Prove that $$\tan x < \frac{4}{\pi}x,\forall x\in \left( 0;\frac{\pi}{4} \right)$$ I have known the solution that uses convex function. But I'd like another solution don't use it. :D
0
votes
1answer
5 views

Higher the percentage, lower the value

I need to work out a grading based on a percentage. The higher the percentage, the lower the grading. There are 7 major grades in the score, 1 to 7 (100% = 1) & (0% = 7) I need to work out the ...
0
votes
0answers
10 views

Summation of the reciprocals of the product of consecutive integers

It is well known that there is a closed formula for: $\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \cdots + \frac{1}{(n)(n + 1)}$ And likewise for: $\frac{1}{1 \cdot 2 \cdot 3} + \frac{1}{2 \cdot 3 ...
0
votes
2answers
16 views

How do you show that f(z)=z conjugate isn't linear?

let $x_1= a+ib,x_2= c+id,k=$scalar $f(x_1,x_2)=f(x_1) + f(x_2)$ $f(a+ib + c + id)=(a+c)-i(b+d)$ $f(a+ib)+f(c+id)=(a+c) - i(b+d)$ $f(kx_1)=kf(x_1)$ $f(k(a+ib))= k(a-ib)$ $kf(x_1)=k(a-ib)$ Looks ...
0
votes
0answers
9 views

Image is not a manifold when considered as a subset: how is this possible?

Wikipedia offers two definitions of a submanifold: One is that it is the image of an immersion. But I can't make sense of the remark that " in general this image will not be a submanifold as a ...
0
votes
0answers
3 views

Minimum of restricted linear combinations.

Let $\{N_0, ... , N_m\}$ be a set of natural numbers, then the minimum $(\geq 1)$ of all their linear combinations is their GCD. Is there a way to calculate that minimum if some $N$s can only be ...
1
vote
0answers
3 views

Right triangle and Sine function?

Given two angles and the hypotenuse of a right triangle, when trying to find the length of the side opposite the given angle, why and how does it's angle and supplementary angle yield the same answer? ...
0
votes
0answers
2 views

Lyapunov equation for discrete systems - LQR

I have following dicrete-time system $$ x(k+1) = A.x(k) + B.u(k) \\ y(k) = C.x(k) $$ I am supposed to find controller in form of $$ u^* = -K.x $$so that following performance index is minimized $$ ...
0
votes
0answers
7 views

Smooth section of Hodge bundle ($F^pH^k$) can be viewed as a smooth form of type$F^pH^k(X,C)$ over$ X$,$ X--->B$ is an analytic family.

I think it is due to Kodaira. could someone explain the idea that Kodaira come up with this. maybe I shouldn't say"can be viewed as". I really mean the smooth form restrict on each fibre is just the ...
0
votes
0answers
7 views

What are the loci of points z which satisfy the following relations?

a) |Z-Z1|=|Z-Z2| b) 0< Re(iZ)<1 c) |z|=ReZ+1 d) Im((Z-Z1)/(Z-Z2))=0 The professor did not explain loci in class and the text does not have any examples, so I am completely lost.
-9
votes
0answers
26 views

93388•9546985339 [on hold]

This girl purchased 3 pounds of fruit. Her sister Evelyn purchased 1/10 the amount how much did each girl purchase?
-6
votes
1answer
26 views

An RMO level question

Find all the integer solutions to the equation: $x^3 +y^4 =7$
0
votes
0answers
17 views

What is the physical significance of arithmetic operations?

Here is an example of what I mean by physical significance: When we use some geometric or trigonometric identity, let us say Pythagoras' theorem to calculate the length of the diagonal of a field, ...
1
vote
1answer
10 views

Count the number of strings of length 8 over A = {w, x, y, z} that begins with either w or y and have at least one x

Count the number of strings of length $8$ over $A = \{w, x, y, z\}$ that begins with either $w$ or $y$ and have at least one $x$ So here is what I came up with..Can someone check my work? $A = ...
0
votes
1answer
12 views

Show that f is a density and find the corresponding cdf

$f(x) = \frac{(1+\alpha x)}{2} $ for $-1 \leq x \leq 1$ and $f(x) = 0$ otherwise, where $-1\leq \alpha \leq 1$. Show that $f$ is a density and find the cdf. I am mainly having trouble with finding ...
3
votes
2answers
37 views

$f: X \to Y$ continuous if and only if $f: X \to f(X)$ continuous

Let the image of $f$, which is $f(X)$, be a subspace topology of $Y$. Prove that $f: X \to Y$ continuous if and only if $f: X \to f(X)$ continuous. 1) If $f: X \to Y$ continuous, then $f^{-1}(U)$ ...
0
votes
2answers
17 views

How do I complete composite functions that use cos(x) and a polynomial?

Let $f(x)= \cos(x)$ and $g(x)= 3x^3+4x^2-7$. Find $g(f(x))$ and $f(g(x))$
0
votes
0answers
7 views

Given the tetrahedron $OABC$, find a condition on $a OA+ b OB + c OC$ such that this is always inside $ABC$.

I did the following: Taking the tetrahedron $OABC$, one can decompose it in: $OA,OB,OC, AB,BC$. And then, writing: $$x(BC-AB)+AB\quad x\in[0,1]$$ We obtain all the points in the line segment from ...
-4
votes
2answers
62 views

What is the most complex mathematical topic? [on hold]

I'm a simple man living my simple life and often I like to learn more about math and science. Today my daughter asked me about how many numbers are there and I explained that there are infinite ...
0
votes
0answers
13 views

Example where probability theory fails without $\sigma$-algebra

I have just started reading theory of probability in a measured theory based approach and was wondering if someone could give an example where probability fails without using $\sigma$-algebra (or ...

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