Homework questions are welcome as long as they are asked honestly, explain the problem, and show sufficient effort. Please do not use this as the only tag for a question. For the answers on homework questions, helpful hints or instructions are preferred to a complete solution. Please do not add ...

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1answer
19 views

Descartes rule of signs extension

Let $V(\text{sequence})$ be the number of sign changes in the sequence, e.g. $V(-3,0,-2,9,0,1)=1$. Show that $V(a_0,a_1,...,a_n)\ge V(a_0,a_0+a_1,a_0+a_1+a_2,...)$. Furthermore, prove that if ...
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1answer
38 views

Invertible Matrices Proof

Given that B is an invertible matrix and $B^3 + B^4 + B^7 = I$, find an expression for $B^{-1}$ in terms of only $B$. (where $I$ is an identity matrix) $B$ is a matrix that is $n \times n$.
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4answers
128 views

Real Analysis: Showing $f: \Bbb Q \to \Bbb Q$ is continuous

The following is all working in $\mathbb{Q}$, not $\mathbb{R}$. I am working with the function $f: \mathbb{Q} \to \mathbb{Q}$ defined piece-wise by $f(x)=-1$ if $x^2<2$ $f(x)=1$ if otherwise I ...
2
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1answer
273 views

Prove that if Triangles ABC = DEF in a metric geometry, then line AB contains exactly two of the points D, E, and F.

Prove that if Triangles ABC = DEF in a metric geometry, then line AB contains exactly two of the points D, E, and F. We are not allowed to use the facts: In a metric geometry, if triangles ABC=DEF, ...
2
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1answer
545 views

Minimizing L1 Regularization

I have given a high dimensional input $x \in \mathbb{R}^m$ where $m$ is a big number. Linear regression can be applied, but in generel it is expected, that a lot of these dimensions are actually ...
1
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4answers
112 views

Proof that arithmetic mean is greater than geometric mean? [duplicate]

I have to prove that $\frac{x + y}{2}> \sqrt{xy}$ algebraically for any $x,y \in \mathbb{R}$ such that $x,y \ge 0$ and $x\ne y.$ I'm fairly confused as to how to solve this problem algebraically, ...
2
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3answers
69 views

How to Prove This Trigonometry Identity?

I have to prove that: $$\tan^2\theta \sin^2\theta = \tan^2\theta - \sin^2 \theta$$ Here is what I have tried $$\tan^2\theta \sin^2\theta$$ ...
4
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4answers
366 views

Probablity that 3 husbands sit next to their wives round a circular table

There are 3 couples sitting randomly round a 6-seater circular table. What is the probability that all the husbands and wives sit next to each other? My attempt: First wife, say, takes any of the ...
2
votes
2answers
141 views

Inverse Laplace transform of $\frac{s}{\sqrt{(s+a)^3}}$

Trying to find the inverse Laplace transform of $\frac{s}{\sqrt{(s+a)^3}}$. So solving $\oint_B dz \: \frac{z}{\sqrt{(z+a)^3}} e^{z t}$ (Bromwich contour). I tried doing a u-substitution with $u=z+a$ ...
2
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2answers
43 views

Binary relation, reflexive, symmetric and transitive

I have a question regarding an image. I'm currently studying binary relations and the following image confused me: What got me confused is that the page from which I got the link ...
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1answer
30 views

Homomorphism from a commutative group?

I came across this question in a practice exercise and can't quite understand it. If f is a homomorphism from a commutative group $(S,*)$ to another group $(T,*')$, then prove that $(T,*') is also ...
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2answers
103 views

Omitting $i$ in calculations

Is it possible in various calculations related to the complex plane which also include analytic geometry , calculating distances etc, to omit $i$ and treat the imaginary axis as simply the cartesian ...
3
votes
3answers
168 views

Prove $\frac{\sec{A}+\csc{A}}{\tan{A} + \cot{A}} = \sin{A} + \cos{A}$ and $\cot{A} + \frac{\sin{A}}{1 + \cos{A}} = \csc{A}$

Can anyone help me solve the following trig equations. $$\frac{\sec{A}+\csc{A}}{\tan{A} + \cot{A}} = \sin{A} + \cos{A}$$ My work thus far ...
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1answer
9 views

Finding the conditions of (x,y,z,t) for them to belong to the span of a set of vectors

So I got this math exercise, and I don't know how to go about it: In $\mathbb{R}^4$, $S$ is the subspace spanned by the following set of vectors: $(1, 1, 1, 0) , (1, 2, 1, 1) , (2, 0, 1, 1) , (3, 0, ...
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1answer
56 views

Help finishing this exercise!

Given the following functions: $$ F(t)= \int_0^\infty e^{-tx}\dfrac{\sin{x}}{x}\,dx, \quad t>0$$ $$ F_s(t)= \int_0^s e^{-tx}\dfrac{\sin{x}}{x}\,dx, \quad t \geq 0, s>0$$ Show that $F$ is ...
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0answers
9 views

Fourier expansion of sum of an arbitrary function and a trig function

I have this BVP with initial condition being $v(x,0) = -x/\pi - (1/25)sin5x$ and I'm looking for $v(x,t) = \sum b_n sin(nx)e^{-n^2t}$ Expanding $v(x,0)$ gives $v(x,0) = -(1/25)sin5x - ...
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2answers
67 views

Calculating $\int_0^\infty \frac{\ln x}{(x^2+9)^2} dx$

I try to calculate $$ \int_0^\infty \frac{\ln x}{(x^2+9)^2} dx $$ I use a book that tells me to replace $\ln x \ $ by $ \ \ln(|x|) + i\phi_z$ where $\phi_z$ denotes the argument of $z$, chosen between ...
0
votes
2answers
283 views

I want help with $4\times 4$ symmetric matrix

I have a $4\times 4$ matrix $$A=\left(\begin{array}{cccc}8 & 11 & 4 & 3\\11 & 12 & 4 & 7\\4 & 4 & 7 & 12\\3 & 7 & 12 & 17\end{array}\right).$$ I want to ...
3
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1answer
32 views

Show that $\lim_{s \to \infty}F_s(t) = F(t)$ uniformly for $t \in (0,+\infty)$

Given the following functions: $$ F(t)= \int_0^\infty e^{-tx}\dfrac{\sin{x}}{x}\,dx, \quad t>0$$ $$ F_s(t)= \int_0^s e^{-tx}\dfrac{\sin{x}}{x}\,dx, \quad t \geq 0, s>0$$ Show that $\lim_{s \to ...
1
vote
1answer
32 views

I have to pay for the carpet [on hold]

Ok. So I have a problem here that says an area of 4 ft. by 5 ft. costs 70$. How much does and area of 12 ft. by 18 ft. cost? I really have no clue how to do this....
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0answers
159 views

To Find Expected Sum

I Have N objects to paint, ordered in a row and numbered form left to right starting from 1. There are total C colors, numbered from 0 to C-1. At the beginning all objects are colored in color with ...
4
votes
2answers
387 views

Evaluating real integral using residue calculus: why different results?

I have to evaluate the real integral $$ I = \int_0^{\infty} \frac{\log^2 x}{x^2+1}. $$ using residue calculus. Its value is $\frac{\pi^3}{8}$, as you can verify (for example) introducing the function ...
2
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1answer
252 views

Lines in $\mathbb{A}^3$

This seems intuitive, but I'm having trouble coming up with an exact matrix for the problem. Let $\{L_1, \ldots, L_N\}$ be a set of lines through the origin $(0,0,0)$ in the affine space ...
0
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4answers
67 views

Complex Numbers: Im$(\frac{12}{z-7})=1$

Sketch and describe the set of complex numbers satisfying $$Im(\frac{12}{z-7})=1$$ where $z=x+iy$ The answer should be in circle form. Here is what I have so far: $$Im(12)=z-7$$ $$Im(12)=x+iy-7$$ ...
0
votes
2answers
40 views

inflection points of functions question

If a function has no minima or maxima points, does it also mean in has no inflection points? since if no such points exist, it doesnt have to change concavity anywhere, so it sounds as a logical ...
2
votes
2answers
50 views

Finding the length of a spiral

I need to find the length of a spiral. The spiral start at a certain radius $R_1$ and ends at a smaller radius $R_2$. As the spiral spins inwards, the distance between each arm of the spiral decreases ...
1
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1answer
51 views

How do you find the following limit as x approaches infinity?

$\lim_{x\to \infty} \sqrt{x^2+9} - \sqrt{x^2-2}$ I have tried multiplying by the conjugate but the square roots are throwing me off and I'm not sure what to do next. How do you solve this?
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5answers
57 views

Taking the sin of arccos

When solving for the value of x in the equation $$\sin^{-1}{(\sqrt{2x})}=\cos^{-1}(\sqrt{x})$$ one would take the sin of both sides of the equation cancelling out the arcsin leaving ...
1
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1answer
24 views

Simple Trig Question / Introduction to Vectors Question

Sorry this is such a simple question; I'm just struggling a little with my trigonometry homework. An example question: "A ship sails due north (relative to the current) with a speed of 20 knots. The ...
0
votes
1answer
13 views

How would I find the scale factor of a dilated figure on a coordinate plane?

The above question is pretty simple, and I used common sense to figure out that the coordinates (3, -7) is the answer, since it is the only viable spot. I was wondering how I would find the scale ...
2
votes
2answers
48 views

Are these logical predicate translations valid?

In this problem in Problem set(1) of MIT's 6.042: Translate the following sentences from English to predicate logic. The domain that you are working over is X, the set of people. You may use the ...
18
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0answers
803 views

How can I calculate $\int \frac{\sec x\tan x}{3x+5}\,\mathrm dx$

How can I calculate $$ \int{\sec\left(x\right)\tan\left(x\right) \over 3x + 5}\,{\rm d}x $$ My Try:: $\displaystyle \int \frac{1}{3x+5}\left(\sec x\tan x \right)\,\mathrm dx$ Now Using Integration ...
0
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0answers
21 views

Lagrange interpolation polynomial and error estimation

Given is a function $f(x)$ with $f(0)=1$, $f(\frac{1}{2})=2$ and $f(1)=-1$. Additionally is given that $max_{x\in \left [ 0,1 \right ]}f''(x)=1$. Find its Lagrange interpolation polynomial $P$ and ...
4
votes
1answer
36 views

Proving a Subset Identity

Working on part A of this problem: I worked out the first part like this: 1) If $A$ is a subset of $B$ then $\forall~x~[x\in A \implies x\in B]$ 2) Same goes for $C$ being a subset of $D$ (If ...
2
votes
0answers
36 views

perturbation theory solution of forced Duffing's equation

Question: Find the leading order of the asymptotic expansion for large t: $\frac{d^2x}{dt}+\varepsilon\beta\frac{dx}{dt}+x+\varepsilon x^3=Fcos(\frac{1}{3}\big(1+\varepsilon\omega)t\big)$ I have ...
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votes
1answer
56 views

help my stack on differential equations [on hold]

can i have a help for this differential equations i can't success to solve it: (http://i.stack.imgur.com/7V90y.jpg) this this is the equations can you help my plase.
0
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1answer
42 views

Trig Identities [on hold]

write in terms of sine and cosine and simplify the expression $$\frac{\cos(A)-2\sin(A)\cos(A))}{\cos^2(A)}-\sin^2(A)+\sin(A)-1$$ need help solving this.
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2answers
77 views

Prove that a function with a measurable graph is differentiable

Let f:[a,b]→(0,∞) be continuous and let Gf={(x,y):y=f(x)} be the graph of f. prove that Gf is measurable only if f is differentiable in (a,b)?
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3answers
91 views

Pythagorean Trig Identity

I'm about to teach basic Pythgorean trig identities and went through the textbook exercise but I'm stuck on one. Show $$\sec^2 A = \frac{ \mathrm{cosec} A}{ \mathrm{cosec} A - \sin A}$$ Can ...
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2answers
31 views

Finding the power series representation for $\ln(1 -10x)$ via integration.

I'm trying to find the power series representation for $ \ln(1-10x) $ Attempt at solution: $$ \ln(1-10x) = \int {-10\over1-10x} \ dx = -10 \int \sum_{n=0}^\infty (10x)^n dx $$ $$ = -10 ...
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0answers
20 views

For which $a>0$ does this Lebesgee-integral exist (and is finite) [on hold]

Let $\lambda$ be the Lebesgue-measure over $(\mathbb{R},\mathbb{B})$. Determine, for which $a>0$ the Lebesgue-integral: $$\displaystyle\int_\pi^\infty \left(\frac{\sin x}{x}\right)^{a}\text{ ...
11
votes
3answers
189 views

How to show that $ \sum_{n = 0}^{\infty} \dfrac {1}{n!} = e $

How to show that $$ \sum_{n = 0}^{\infty} \dfrac {1}{n!} = e $$ where $e = \lim \left({1 + \dfrac 1 n}\right)^n$ I'm guessing this can be done using the Squeeze Theorem by applying the AM-GM ...
0
votes
1answer
31 views

Surjectivity of composition

I know that this question has been posted few times, but I want to check MY proof, because this is my first time trying to prove anything in mathematics. (I'm afraid if I just copy paste their proofs ...
1
vote
0answers
31 views

RSA aloghorithm - stuck on d

I'm sorry in advance if this sort of question has been posted before. I couldn't find it. I'm clearly an idiot, and I clearly need help, so here I am. I have a homework assignment which overall is ...
0
votes
2answers
21 views

Fine partitions

I am tasked with the following: Give four different partitions $\Pi_1,\Pi_2,\Pi_3,\Pi_4$ of the set $\Bbb N$ with $\Pi_i$ Finer that $\Pi_{i+1}$ for $i =1,2,3$ I think that partition by 8, 4,2 ...
0
votes
1answer
37 views

conjugacy class of a dicyclic group

I have given a group and I have to prepare the character table of this given group. I know that firstly I have to find the conjugacy classes of the given group. The group is below: ...
2
votes
1answer
54 views

Filling of a tank - recurrence relation

Suppose a tank has a maximum limit of 100 units. Each day 2,1 and 0 units are added to the water level with probability p,r and q. Any excess water would overflow and if it reaches the minimum level ...
4
votes
1answer
57 views

Is it true that $\lim_{x\to a}f(x)=0$ if and only if $\lim_{x\to a}|f(x)|=0$?

Is it true that $\lim_{x\to a}f(x)=0$ if and only if $\lim_{x\to a}|f(x)|=0$? I intuitively think this is true, but really no idea to prove it. Can you give me hints?
2
votes
1answer
30 views

Given $A_{m\times n}$ and $B_{n \times m} (m<n)$. prove that AB is not singular and BA is singular

I have the following question which I can't seem to wrap my head around. I don't see how we can determine the desired just from the given info. Given $A_{m\times n}$ and $B_{n \times m}$ ...
3
votes
1answer
213 views

Classification of fundamental groups of non-orientable surfaces

I want to compute the presentation of the fundamental group of the non orientable surfaces $N_h$, thus $\pi_1(N_h)$. I notated with $N_h$ the sphere with $h$ crosscaps. Herefore I first have to ...