0
votes
0answers
3 views

Hyperbolic Coxeter Systems

This question is regarding the proof of a proposition in the text book reflection groups and coxeter groups by Hymphreys section 6.8. Prop: Let $(W,S)$ be an irreducible coxeter system with graph ...
0
votes
0answers
4 views

sum of integral parts of real number and fraction

For any real x and positive integer n ,show that [x] + [x +1/n] + [x + 2/n] + .... + [x + n-1/n] = [nx] I have used the fact that x-1 < [x] <= x,for all terms and added,but not able to get ...
0
votes
0answers
12 views

$\frac{a+c}{a+b}+\frac{b+d}{b+c}+\frac{c+a}{c+d}+\frac{d+b}{d+a}\geq 4$

$a, b, c, d$ are positive reals. How would I prove the inequality $$\frac{a+c}{a+b}+\frac{b+d}{b+c}+\frac{c+a}{c+d}+\frac{d+b}{d+a} \geq 4$$ I have tried using the rearrangement inequality with ...
0
votes
0answers
6 views

How to Calculate Flat rate using Effective Rate?

Following calculations are made thriugh an Excel calculator in my company. I had to input EMI to calculate Flat Rate , but I want asolution to calculate Flat rate straight from Full/Effective rate ...
0
votes
0answers
12 views

Is there any large diffeomorphisms of $S^{n}\times S^1 $like Torus?

We know that a Torus is mapped onto itself in a special discontinuous transformation given by $PSL(2,\mathbb{Z})$. Thinking of torus as $S^{1}\times S^{1}$ and thus as a lattice, we can easily show ...
0
votes
1answer
11 views

find the values of a, b and c from sin graph

The diagram shows part of the graph of a function whose equation is $y = a\sin(bx^{\circ}) + c$ (a) Write down the values of a, b and c. (b) Determine the exact value of P ...
0
votes
3answers
9 views

Check if a given coordinate lies in path of a ray (coordinate geometry)

As shown in the image I have two known coordinate pair A and B and few other known coordinate pairs (RED blob) on the graph. I need to know if any of the other given coordinates fall in line of the ...
0
votes
1answer
7 views

Cost prices, selling prices

I have a question in maths regarding GST, cost prices and selling prices. (GST is government services tax, the amount added on to an amount for the government, so it is basically tax) There are two ...
0
votes
2answers
21 views

Inequality with a square root

If the inequality $ (x+2)^{\frac{1}{2}} > x $ is satisfied. what is the range of x ? My approach - I squared both the sides and proceeded on to solve the quadratic obtained in order to solve the ...
3
votes
0answers
11 views

Convex polyhedron with minimum faces

A convex polyhedron has at least three faces which are pentagons. What is the minimum number of faces the polyhedron might have? I have a polyhedron with seven faces but I don't know whether it is ...
0
votes
1answer
16 views

Let $w = \log(u^{2} + v^{2})$ where $u=e^{(x^{2}+y)}$ and $v= e^{(x+y^{2})}$

Then $\frac{\partial w}{\partial x}$ for $(x=0,y=0)$ is ? I got answer as 0 since on partial differentiation I got, ...
0
votes
0answers
15 views

What is the name of the group $\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$?

I know, that $\mathbb Z_2\times \mathbb Z_2$ is the Klein four-group. Is there a nice name for $\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$ too?
0
votes
0answers
7 views

Equivalent conditions for $\mathfrak{F}$ to be a differential ideal

Heres the question: Let $\mathfrak{F}$ be an ideal of forms on a manifold $M$ locally generated by $r$ independent $1$-forms. Say $\mathfrak{F}$ is generated by $\omega_1,\ldots, \omega_r$ on $U$. ...
0
votes
0answers
22 views

Why are is partitions counting technique wrong?

I recently heard about partitions. I tried to count them using the following technique: 1) Ways to write $5$ as a sum of five positive integers: $$1+1+1+1+1$$ 2) Number of ways to write $5$ a sum of ...
1
vote
1answer
19 views

Initial value problem $y'=e^{-y^{2}}-1.$

Let y be a solution of $y'=e^{-y^{2}}-1$ on $[0,1]$ which satisfy $y(0)=0$. Then how to prove that $y=0$ on $[0,1]$? According to me as the function $e^{-y^{2}}-1$ is Lipsctiz and continuos on $[0,1]$ ...
1
vote
0answers
12 views

Mass of lamina defined in y ≥ 0, with edges given by y = 0, y = (4-x^2)/3 and x = −y + 2y^2, and density is y.

I've been trying work this out, but I'm stuck on the the integral calculation. I've drawn a diagram, got all the points of intersection and relevant points, but I still can't get it. I had a go at ...
0
votes
0answers
9 views

Example of a “abrupt function”

I need example of a simple function to show that cubic spline gives better result than Lagrange's interpolation in case of some special functions. Thank you
0
votes
0answers
10 views

A problem on random q-colourings of a graph for randomly chosen vertex

Here is an exercise from Olle Haggstrom's "Finite Markov Chains and Algorithmic Applications" from the chapter "Fast Convergence of MCMC Algorithms". The exercise is based on random $q$-colorings of ...
1
vote
0answers
19 views

Is there a combinatorial identity for the following sum?

Let $a,b,c$ be integers. Is there by any chance a neat combinatorial identity for the following sum? $$ \sum_{j=0}^c{a + jb \choose j}. $$ Thanks!
0
votes
1answer
23 views

Elementary number theory with some arithmetic progression.

Let A={n∈N┤n is the sum of seven consecutive integers}. B={n∈N┤|n is the sum of eight consecutive integers}. C={n∈N┤|n is the sum of nine consecutive integers}. Find A∩B∩C. I tried ...
0
votes
0answers
3 views

Is there a name for differential operators with certain homogeneity

Is there a name for the ODEs of the following form ? $$\sum_{m_k+n_k= N, }a_{m_k,n_k}u^{m_k}u^{(n_k)}=0,$$ where $u^{m_k}$ denotes the $m_k-$th power of $u$ and $u^{(n_k)}$ denotes the derivative of ...
1
vote
0answers
4 views

Rewrite each isometry as the composition of at most three reflections

Write each of the following isometries as a composition of at most three reflections: $\rho_{(1,0),\frac{\pi}{6}}$ $\tau_{(1,0)+(0,1)}=\tau_{(1,1)}$ $\sigma_{l_{BC}} \rho_{(1,0),\frac{\pi}{6}} ...
0
votes
3answers
40 views

How to differentiate $\sqrt[5]{7x^2}$?

How to differentiate $\sqrt[5]{7x^2}$? I'm not sure how to do it. I used the power and chain rules and have $\frac{14x}{5} \frac{1}{\sqrt[5]{28x^8}}$. Is this correct? I don't understand what ...
1
vote
0answers
21 views

Everyone in a room can't each have more friends than the average number of friends their friends have. Is my proof correct?

I know of simpler proofs, but I am concerned with the validity of the following probabilistic approach. Let $a_i$ be the number of friends that person $i$ has. Assume for contradiction that all $N$ ...
0
votes
1answer
14 views

Does every partition of n correspond to some permutation of [1,2, … n]?

It is known that every permutation can be decomposed into disjoint cycles. The cycle type gives the length of each cycle. The sum of cycles length is n. I am wondering whether every partition of n ...
-2
votes
0answers
13 views

Quotient ring $(\mathbb{Z}_4 \times \mathbb{Z}_6)/S$

Consider the ring Z4xZ6 with +6, *6, and +4, *4 in appropriate coordinates and S={(0,0),(2,0),(0,3),(2,3)}. Would the elements of the quotient ring Z4 x Z6 / S be: S+0 (trivial set above), ...
0
votes
1answer
15 views

Finding minimum distance between a circle and curve

what is the minimum distance between $x^2+y^2=9$ and $2x^2+10y^2+6xy=1$ in Question there is a circle and a curve and we have to find the least distance between them
0
votes
0answers
8 views

What will draw a shape of $L = \left\{ {\lambda \in \mathbb{C}:{s_4}(\lambda ) = {s_3}(\lambda )} \right\}$

Let $P(\lambda ) = \left( {\begin{array}{*{20}{c}} {{\lambda ^2} - 1} & 0 \\ 0 & {{\lambda ^2} - 2\lambda } \\ \end{array}} \right)$ and $\lambda \in \mathbb{C}.$ we say $\lambda$ is ...
0
votes
0answers
3 views

Example for injective and surjective bounded and unbounded operator

I am looking for some bounded and unbounded densely defined operators on a real Hilbert space $H$, let say $A:D(A)(\subset H)\to H$, that are one-to-one but they are not onto. I am wondering whether ...
1
vote
1answer
25 views

Prove that if $x\mapsto -x$ is continuous then $\sigma$ is the discrete topology.

Let $\tau $ be the topology on $\Bbb R$ for which the intervals $[a,b)$ form a base.Let $\sigma$ be a topology on $\Bbb R$ such that $\sigma \supseteq \tau. $ Prove that if $x\mapsto -x$ is ...
0
votes
1answer
17 views

Is $\langle A,B\rangle =trace(AB^T)$ an inner product in $R^{nxm}$?

I don't understand why one should take transpose of tr($AB^T$) and why we use the fact that tr(M)=tr($M^T$) for any M that is a square matrix to solve the problem.
0
votes
1answer
3 views

How does multiple integral change into terms multiplying each other in convolution theorem of Laplace?

In the steps of the proofs highlighted below, how does a multiple integral changes in to multiplication of two integral. This is only possible if V is independent of u, but as it turns out V = t - ...
0
votes
0answers
5 views

Prove that $l$ and $l_{AB}$ are parallel if and only if $\sigma_B \sigma_l \sigma_B \sigma_A \sigma_l \sigma_A = id$

Prove that $l$ and $l_{AB}$ are parallel if and only if $\sigma_B \sigma_l \sigma_B \sigma_A \sigma_l \sigma_A = id$ I imagine that this proof has to be along the lines of a proof by contradiction, ...
2
votes
1answer
17 views

$xP(|X|>x) \rightarrow 0 $ implies $E|X|^{1-\epsilon} < \infty$

Given X is a random variable, and $\epsilon \in (0,1)$, prove that : $xP(|X|>x) \rightarrow 0 $ implies $E|X|^{1-\epsilon} < \infty$ I got a hint to use the Lemma : $E(Y^p) = \int_0^{\infty} ...
0
votes
0answers
10 views

Looking for a easy proof of the fact that $H^i_m(R)=0$ if $i<n $?

Let $R=k[x_1,x_2,...,x_n]$ and $m=(x_1,...,x_n)$ be an ideal of $R$. Is there any easy (without any higher machinery ) proof of the fact that $H^i_m(R)=0$ if $i<n$ ? I know that the above ...
0
votes
0answers
4 views

What can you say about the $k$-th cohomology group of a closed orientable $n$-manifold?

What can you say about the $k$-th cohomology group of a closed orientable $n$-manifold for: (1) k=n, and (2) k=n-1 Poincare Duality tells us that for $M$ a closed $R$-orientable $n$-manifold, ...
1
vote
1answer
9 views

Isomorphic Galois groups imply isomorphic field extensions?

Suppose we have two field extensions $K/k$ and $L/k$. I am able to show that if these field extensions are isomorphic, then their corresponding Galois groups Aut$(K/k)$ and Aut$(L/k)$ are also ...
0
votes
1answer
18 views

Prove that for the matrix $A$, $A^TA$=$A$ and for the projection matrix $P^2=P$

I'm not sure how under what conditions the proofs are possible. I'm begging for help. This is not a homework question.
0
votes
1answer
8 views

Reducing a Boolean function

I have the following boolean function: f(x,y,z) = xyz + xyz' + xy'z + x'yz + xy'z' I could reduce it to the following: f(x,y,z) = xy + xy'z + x'yz + xy'z Im not sure what to do next, i know it can ...
1
vote
2answers
26 views

Is there a formula that finds middle between two angles

I am sorry in advance. If I am shooting a laser in front of me. Assume 90 degrees. And then I have a second laser that I shoot from that has a point 130 degrees. The midpoint between the two is not ...
0
votes
0answers
9 views

Upper bound on the norm of the inverse of matrices with zero limit

Let $\{L(\sigma)\}_{\sigma}$ be a family of matrices indexed by the parameter $\sigma$ so that the operator norm $||.||$ of $\{L(\sigma)\}_{\sigma}$ satisfies $Ae^{-a/\sigma}\leq ||L(\sigma)|| \leq ...
1
vote
1answer
11 views

Find a basis of the space V of all symmetric 3x3 matricies, and thus determine the dimension of V

I need help finding the general element or matrix of $V$. Do I need to find the basis of the nullspace and basis of the image to solve this problem?
0
votes
0answers
5 views

Non-Inductive formula for subdivision operator

This problem is from hatcher 2.1.25. Find an explicit, noninductive formula for the barycentric subdivision operator. I have no idea how to get that formula. The only way I see it geometrically is ...
0
votes
0answers
12 views

Cartesian equation from the complex equation

Find the Cartesian equation for the curve corresponding to the equation $|{z+8\over 16j-z}|=3$ Describe what curve is represented by the equation. Does my answer look correct? $|z+8|=3|16j-z|$ set ...
0
votes
1answer
23 views

Linear Algebra True or False Questions.

I have some problems determining why these statements are true would someone if they be willing help me with some of these? Now I know the definition of a column space it is the set col A of all ...
0
votes
0answers
12 views

Banach Theorem on Metric Space for Integral Equations

My instructor said that Banach doesn't apply in this case: f(x) = sin(x) + $\int_0^x$$f^2$(z)dz f(0) = 0; f'(0) = 1 > 0 f'(x) = cos(x) +$f(x)^2$, which is positive on (0,$\pi$) so f is positive ...
0
votes
0answers
11 views

Identifying Symmetric Matricies

If $A$ and $B$ are arbitrary $nxn$ matrices, which of the matrices must be symmetric? $something^T$ means the transpose of "something". $A^T$$A$ $A-A^T$ $A^TB^TBA$
0
votes
0answers
5 views

Is it a change of variable?

Hi everyone: In a book I am reading, they make a sort of "substitution" like this: let $B(0,R)$ be a ball in $\mathbb{R}^{N}$ $(N\geq2)$ and $f$ a locally integrable function. Let $\mu$ be a finite ...
0
votes
2answers
14 views

A coin is tossed 6 times. What is the probability that the no. of heads in the first 3 throws

A coin is tossed 6 times. What is the probability that the no. of heads in the first 3 throws is the same as the number number in the last three throws? To be honest, I don't know how to tackle this ...
0
votes
1answer
24 views

How do I get this integral? [duplicate]

This is from a list of problems from a local university's Integration Bee. I have no idea how to do it, but I thought maybe someone here can explain it to me. $$\int_{0}^{1} \frac{\ln(1+x)}{1+x^2} ...

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