I teach mathematics , both pure and applied mathematics , for students who are yet to be undergraduates (Advanced Level) . I find myself very fond of teaching mathematics , but learning. How learning further more mathematics will help to my career , even I have a good practice and knowledge of ...

x is a three-digit natural number, 2x ≡ 3 (mod 5) 3x ≡ 1 (mod 4) what is the smallest value that x can take? What is the method to solve this kind of questions?

If $S$ is an oriented, smooth surface that is bounded by a simple, closed, smooth boundary curve $C$ with positive orientation, then for some vector field $\vec{F}$: $$\oint_C \vec{F} \cdot d\vec{r} = \iint_S {\rm curl} \> \vec{F} \cdot d\vec{S}$$ The latter integral can be written equivalentl...

I am confused. Sometimes i read about terminating and not terminating algorithms. I almost always read these things in the context of turing machines. This means to me: There are algorithms which terminate and others which don't. But for example the german wikipedia page for algorithm says that ...

I have found this question in one of my Universities old pass papers and I'm trying to solve this: You have a fishing rod of length 2 and need to ship it in a box which sides are not longer than 1. In spaces Rn of at least which dimensional n will you be able to fit the rod into the box without...

On the curve C consisting of the x-axis from x=0 to x=4, the parabola y=16−x^2 up to the y-axis, and the y-axis down to the origin. I can't seem to get the right answer. Since the vector field is not conservative, I'll just have to integrate each individual part of the curve. However, the force...

I just finished the first part of PUTNAM competition this year and I would like to have your opinion. Question : We have the recurrence relation $a_0 = 1$, $a_1 = 2$ and $a_n = 4a_{n-1} - a_{n-2}$. Could you find an odd integer factor of $a_{2015}$? I tried to find an explicit formula for thi...

I am asked too calculate the exact length of the intersection curve of the two equations $$ x^2 = 2 y \\ 3z = xy \\ $$ From the point (0,0,0) to the point (6,18,36). I don't know if I am doing the right thing, but I am starting to isolate $x$ from the first equation and insert it in the second...

A polynomial in x has m nonzero terms. Another polynomial in x has n nonzero terms, where m

It's obvious that $x^2>2x+1$ for $x\ge 3$ - we just observe that for $x\ge3$ the LHS grows much faster than the RHS. But how to determine: how faster does the LHS grow? and conclude from it that the inequality indeed holds?

I have a problem with one exercise from differential geometry. I don't even know how to start. Anyone could help with this problem? Let $M$, $N$ be manifolds, $M$ connected. Let $\pi:M\times N \to N $ be a projection of second coordinate. Prove that $k-$form $\omega$ on $M\times N$ is of form ${...

I really do not know if this question is appropriate because maybe some of us will think that question about questions is something silly but I see no reason why it should not be asked. I think that, if this question collects enough answers (although maybe it will collect none), that we will, in...

We have the following relations: $S_1=\{(x,y) \in Z^2:x+y>1 and x>0 \}$ $S_2=\{(x,y) \in P^2:x+y>1 and x>0 \}$ We have to make the graph for each occasion.My question is, what is the difference between these two graphs i am a little confused on how to graph them

Hello I am trying to learn about the symmetries of a regular tetrahedron. I understand the identity and all eight 120 degree rotations that keep one vertex fixed, $(123),(132),(243),(234),(134),(143),(124),(142)$ but I cannot at all understand how to visualize the so called 180 degree rotations ...

Check the statement: For all square matrices $A,B$ of order $n$ is $\det(AB)=-\det(BA)$ Is it correct that the statement is not true for all $A,B$?

Two experiments are to be performed. The first can result in any one of m possible outcomes. If the first experiment results in outcome i, then the second experiment can result in any of ni possible outcomes, i = 1, 2, ... , m. What is the number of possible outcomes of the two experiments? How ...

Here are my steps: $e^{{2\pi i}/100} = (e^{\pi i})^{{2/100}} = ((-1)^2)^{1/100} = 1^{1/100} = 1$. I'm not sure if the normal rules of exponents apply like this if the power is complex.

I need to calculate the Integral $\int_M1$ where $M:=\{(x,y): (x^2+y^2)^2-2*x*y=0\}$ I just got no clue how to approach this problem. I just know that the integral is the area described by M which looks somewhat like an eight. Can you guys give me a hint in the right decision, because I feel lost...

$$\int_a^{g(x)} f(t) \,dt$$ What is the derivative of this integral?

Show that the relation $f\sim g$ if $\int|f-g|=0$ is an equivalence in $L^1(\mathbb{R})$. What constitutes an equivalence in $L^1(\mathbb{R})$? Any help would greatly appreciated.

I am not sure how to change an input parameter 'β' at each time step. My code is below - which gives me an error. Can anybody help please! t = [7 14 21 28 35 42 49 56 63 70 77 84]; for i=1:12; (i) = 0.43e-08 + (4.28e-08 - 0.43e-08)*exp(-0.20*t(i)); end; f = @(t,x) [3494-0.054*x(1)-beta*x...

If I am at a point (0, 0, 3) and a move of 5 units in the positive direction of $t$ on a curve definite by $$ \begin{align*} x &= 3sin(t) \\ y &= 4t \\ z &= 3cos(t)\\ \end{align*} $$ Where am I ? I am not sure how to start this problem. What does the 5 units in the positive direction of t mean...

Let $G$ be a lie group and $\mathfrak{g}$ its lie algebra, that is, the $\mathbb R$-space of left invariant vector fields on $G$. Recall the isomorphism $\mathfrak{g}\simeq T_eG$. The Maurer-cartan form on $G$ is the $1$-form $\omega\in \Omega^1(G; \mathfrak{g})$ defined through the composition ...

Suppose you are given that f is a differentiable function of two variables u and v. Let g(r,s)=f(r^2-s^2, 2rs). Compute ∇g(r,s) in terms of ∂f/∂u, ∂f/∂v and other functions. I expressed f as f(u,v)=f(r^2-s^2, 2rs) and set u=r^2-s^2 and v=2rs. To calculate the gradient I did: ∇g(r,s)=∂g/∂ri + ∂g...

I'm taking a course on Algebraic Topology and I'm struggling to find the solution to this problem: Let $Y$ be the Hawaiian earring in $\mathbb{R}^2$ and $Y'$ the union on infinite $Y$s moved $3z$ units upward (and downward) with $z \in \mathbb{Z}$ and the line $x=0$ so it is connected. Show a...

T,where mTn m<2n V where mVn iff m is odd n is even,or or m,n are even m

I've been reading the book "Algebra a graduate course" by Isaacs, but I met something in Galois Theory that I didn't understand this proof (18.13) THEOREM. Let $F \subset Ε$ with $[E : F] < \infty$. The following are then equivalent: i. $Ε$ is a Galois extension of $F$. ii. $Ε$ is both separ...

I'm trying to calculate the number of "participant*hours" for participation in a program. I have the following variables/data: 1) Total # of sessions given throughout the program. 2) Average duration (in hours) of each session. 3) Average # of participants in each session. So, for example, if ...

Would you exercices on backward stochastic differential equations? and the link with the PDE? Thank you

I know that strong convergence implies weak convergence, but that the converse is not necessarily true. However, apparently in a finite dimensional Hilbert space it does? Proof?

Consider $$ \frac{dx}{dt}=y,~~~~~\frac{dy}{dt}=x+x^2-y. $$ Its Hamiltonian with $H(x,y)=\frac{y^2}{2}-\frac{x^2}{2}-\frac{x^3}{3}$ and the equilibria are $$ O_1=(0,0),~~~O_2=(-1,0). $$ The linearization matrix of the system at $O_1$ is $A_1=\begin{pmatrix}0 & 1\\1 & -1\end{pmatrix}$ and so $O_...

Can someone help a student who is terribly bad at math answer this question? Between 1998 and 2014, in New Mexico, Birth rates for teens 18-19 years of age fell from a rate of 108.8 per 1000 to 69.3 per 1,000 females. What percentage had the teen birth rate dropped? Between 1998 and 2014, in New ...

How do I add and multiply two piecewise functions? f(x)=x+3 when x<2 and (x+13)/3 when x>2 g(x)=x-3 when x<3 and x-5 when x>3

i have a question about the role of Jacobian in the mapping theorem. I know when i want to map the z=x+iy by f(z)=u(x,y) +iv(x,y), i should put z in the function f and find the relation between x and u , y and v. and i know when inverse mapping i should do them by just w=f^-1(z) . but i faced a p...

My sister just submitted an assignment and got a few questions marked incorrect (electronically) but I've just checked over them and don't believe this to be the case. Can someone either point out where I'm going wrong or confirm my suspicions please? Here is the data (sorry for the formatting!)...

I was reading that the ordinal numbers do not form a set because there are too many of them, instead they form a proper class. Is there a maximum cardinality for a set?

Show that a finite set has volume 0 in R^d (Real Analysis section on Jordan Regions, Rectangle in R^d)

Can I interchange the limit ad integral of this function fn(x) := (1 + cosx(sinx)^n)/(n + 1)^2 , integration over [a,b] = [0,pi/2] how is this done I don't understand the theory behind this and looking at example questions has not cleared up how to approach these problems. So how do I solve th...

This a discounting factor method of deriving the famous BS formula. I have done most of the derivation myself but got stuck on the point mentioned below. I would be grateful to get some advice on how to get from that point to the final answer. (The book I am learning this from the called Asset P...

Keep in mind we are working inside $\mathbb{R}^2$ and $\vec{n}$ and $\vec{d}\in\mathbb{R}^2$ Define $\psi^i_{\vec{n}}$ to be the directional angle of $\vec{n}$ Remember $\cos \left( \psi^i_{\vec{n}} \right) =\frac{n_i}{||\vec{n}||}$. Assuming that $\vec{n}$ and $\vec{d}$ are both in the first...

Exercise Description We have been given these exercises on predicate logic, and this is what I have for an answer, I would like to have some feedback to know if I am on the right track, or I have made some mistakes, this is what I've got. MY ANSWERS: 1. ∀x∃y (A(x) -> H(x,y) 2. ∀x∃y∃z ((C(x,...

Show that no matter how nodes and weights of quadrature formula . The formula cannot have precision greater thanShow that no matter how nodes and weights of quadrature formula . The formula cannot have precision greater than 2n+1 2n+1

I want to prove $A\Rightarrow B$ is true. If I prove that $\urcorner(A\Rightarrow B)$ is false with a counter example is it enough?

I have to show that the following inequation is true: $\frac{ln(x) + ln(y)}{2} \leq ln(\frac{x+y}{2})$ I transformed it into $\frac{ln(x \cdot y)}{2} \leq ln(x+y) - ln(2)$ because I thought that I better can show the inequation here, but I don't know how to proceed. How can I proceed or am I...

If I have to find the derivative of a function that requires you to initially use the chain rule and then the rest is a product or quotient, how do I proceed? For example, sin(x^2(x+1)). And also what in case it needs to perform the product/quotient rule and after the chain rule? For example, x^...

I have a function $x(t)$ and a messy integral over $t$, containing $x(t)$, $x'(t)$ and $x''(t)$ which I would like to clean up the integrand by removing the $x''(t)$ by integrating by parts. I have tried defining a PDE by $$ \frac{dz}{dt} = \frac{\partial z}{\partial x}x' + \frac{\partial z}{\par...

I want to get a set of equispaced points in $[0,\pi/2]$ and use piecewise linear interpolation generated by those points to fit the sine function. And I want to determine how many points do I need to put in to generate the linear interpolation fit such that max error in that interval can be guara...

Consider $u_{\lambda}(t,x) = \lambda^{\alpha} u (\lambda^{\beta}t,\lambda^{\gamma} x)$, for some smooth function $u$. I differentiate this wrt $t$ giving me $$ \partial_t u_{\lambda} = \lambda^{\alpha + \beta} \partial_t u.$$ My first question is simply: can I now treat the arguments of the RHS...

I am trying to work out some problems and I am wondering if anyone can help to check over my work and help me to understand some parts. I am using CFF to solve these combinatoric questions; First one being, How many different pizzas we can make of 12 slices, if 6 have peppers and 6 don't. My ...

I hope someone can help me. I have some trouble calculating the bias of two estimators.Unluckily it is really urgent because I hold a presentation next week. The topic is nonparametric local regression. In order to compare kernel estimators I have to calculate the bias of the following two estima...

So I was given a problem at KA today. They offer you a choice among possible simplified versions of this expression: $\frac{1+\frac{x}{y}}{\frac{x}{y}}$ My solution was: $(1+\frac{x}{y})\div\frac{x}{y}=(1+\frac{x}{y})\cdot\frac{y}{x}=\frac{y}{x}+\frac{\not x}{\not y}\cdot\frac{\not y}{\not x}=...

I have to solve the following exercise: The function $f : \mathbb{R} \rightarrow \mathbb{R}$ is a periodic function with $P = 2\pi$ so that $f(x) = f(x + 2\pi)$ is true for all $x \in \mathbb{R}$. Show that there is a $\xi \in \mathbb{R}$ with $f(\xi) = f(\xi + \pi)$. Either I don't understand ...

I know that lim |nAn|^(1/n)=lim |An|^(1/n). But to apply root test, do I need to show lim |nAn|^(1/n-1)=lim |An|^(1/n)?

How to prove Pappus's Centroid theorem about volume for a triangle rotated around an external axe? The theorem says that the volume V of a solid of revolution generated by rotating a plane figure F about an external axis is equal to the product of the area A of F and the distance d traveled by it...

If $f_n\rightarrow f$ almost uniformly, then $f_n\rightarrow f$ a.e. and in measure. Proof: Since $f_n\rightarrow f$ almost uniformly, then for every $\epsilon > 0$ there is a measurable set $F$ with $\mu(F) < \epsilon$ such that the sequence $\{f_n\}$ converges uniformly on $X\setminus E$. I a...

We know that a commutative ring is Noetherian if and only if every prime ideal is finitely generated.Why this property is not true for maximal ideals?

i know the concept of harmonic functions or the conformal mapping and also i know some characteristics of the conformal mapping( i think ). i have already faced a theorem: "the conformal mapping keeps the harmonic ratio of every four points as a constant value." now, i ask you if you know the pro...

I have encountered the following problem involving a homogeneous system of constant coefficients initial value problem, but after determining one of the eigenvalues to be zero, I am not sure how to proceed to find the general solution. $$ \frac {dx}{dt}= \begin{bmatrix} 20 & -20 \...

At the result, if you get 0=-6, does that mean that there is no solution. If you get 5=5 or 6=6, does that mean any value of x (besides those values that are restricted) are solutions?

Looking at the separable differential equation $y' = 3|y|^{2/3}$, I see that it is not as easy to solve because it is not Lipschitz continuous. Does anyone know how this would be solved. I believe that there would be 4 solutions, not one like if you simply solved it as a separable equation. Thanks!

Show that a set $S ⊆ E$ is open if and only if $S ∩ ∂S = ∅$ This is a question from my real analysis class. I started to show the forward direction by letting $S$ be open and using the definition of openness. I know that $S$ is the space inside and that the boundary of $S$ is all the points on t...

Euler showed $1+2+3+4+\dots=-1/12$. What sort of assumptions could I force of $1+2+3+4+\dots$ to make the statement 'is $1+2+3+4+\dots=-1/12$? undecidable?

Consider the formula of Simpson $S(h)$ e Trapezoidal $T(h)$ white $h = (b-a)/2n$. Show that $S(h) = \frac{4T(h) - T(2h)}{3}$

Evaluate; $$\int_{0}^{1} \frac{1}{x+\sqrt{1-x^2}} \space dx$$ My main concern is finding the indefinite integral as once i have that the rest is fairly straight forward. Please give a detailed answer with reference to why you made each substitution (what indicated that said substitution would wo...

Assume random variables X$_1$, ... , X$_n$ and Y$_1$, ..., Y$_n$ are U(0,a)-distributed. Show that Z$_n$ = n*$log\frac{max(Y_{(n)},X_{(n)})}{min(Y_{(n)},X_{(n)})}$ has an Exp(1) Distribution. I've started this problem by setting {X$_1$,...,X$_n$,Y$_1$,...Y$_n$} = {Z$_1$,...,Z$_n$} Then the max(Y...

My math professor recently told us that she wants us to be familiar with summation notation. She says we have to have it mastered because we are starting integration next week. She gave us a bunch of formulas to memorize. I know I can simply memorize the list, but I am wondering if there is a qui...

How can I find the last two digits of $2^1000$ mathematically?

Does cross product have an identity i.e. $\vec{i} \times \vec{somevector}$ = $\vec{somevector}\times \vec{i}$ = $\vec{somevector}$ where $\vec{somevector}\in \mathbb{R^3}$ and is the identity we are looking for.

Now, I've obviously come across the following during my physics degree: $$ \int{\cos\theta\,d\theta} $$ But I'm starting to see this being introduced to this: $$ \int{d(\cos\theta)} $$ Could anyone just give a foolproof dummies guide to this second expression. For me, it has to equal $\cos\th...

${a_n}$ is a sequence with $limsup(a_n)=1$ as $n$ approaches infinity. I have trouble with these, so I want to verify these (a) $a_n \leq1$ eventually; False (b) $a_n \leq2$ eventually; True (c) $a_n \geq1$ eventually; False If any of these are wrong, could you please provide me with a count...

Assume that matrix $A \in M_n$ is similar to a diagonal matrix $D$ with $0$ or $1$ diagonal entries. I read the textbook and it says that there are $n+1$ such diagonal matrices. However, as I expect, if each entries can select values $0$ or $1$, we should have $(2!)^n$ such kind of matrices. Co...

Prove by induction that $3^n +7^n −2$ is divisible by $8$ for all positive integers $n$. So far I have the base case completed, and believe I am close to completing the proof itself. Base case:$(n=1)$ $3^1 + 7^1 - 2 = 8/8 = 1 $ Inductive Hypothesis: Assume that $3^n +7^n −2$ is divisible by 8...

How do I find the probability of being dealt a 5-cad hand with 2 pairs in poker? The pairs can be any cards, and the fifth card must be another rank than the 2 pairs. I have tried (52/52) x (3/51) x (48/50) x (3/49) x (44/48) = 0.003169 This answer does not seem right. Help?

For splitting $A = M-P$ a damped iteration with damping factor $\gamma >1$ is $$x^{k+1} = x^{k} +\gamma M^{-1}r^{k}$$ where $$r^{k} = b-Ax^{k}$$ Show that if $M^{-1}A$ has real, postive eigenvalues, then the damped iteration associated with $M$ converges for sufficiently small $\gamma$. Use this...

enter image description here f(x,y)= 5-$Sqrt{x^2 + y^2}$ Need to know the step-by-step setup using integration. Don't need the final solution. Sorry new to the coding.

I find that when I come across questions to do with generating sets of groups, I'm never quite sure how to go about the problem. It's difficult to deal with them purely set-theoretically, as you can't make definitive statements about what they contain (or can you?). I'm interested in how other pe...

There is a claim not directly related to separating hyperplane theorem but uses it in some way (sorry but that is all the context I have). As a condition for applying separating hyperplane theorem it says: Claim: "Suppose that $C$ is a convex set, and $0 \notin C$, then the condition for u...

There are two assumptions in Rudin's proof. The first one: We may assume, without loss of generality, that [a,b]=[0,1]. I searched and was given that a linear transformation t = (x-a)/(b-a) is a continuous mapping of [a,b] to [0,1] and thus we make such assumption. Can anyone expain more? The...

Let $(X,d_X)$ and $(Y,d_X)$ be two metric spaces, $E \subset X$ a dense set and $f:X \to Y$ be a continuous function that is uniformly continuous on E. Is $f$ then uniformly continuous on $X$? If yes, is the following proof right? If not, where is the error in my proof? Proof: Let $\epsilon > ...

Let X have PDF: 1/4 0<x<1 3/8 3<x<5 0 otherwise Let Y=1/X, Find PDF for Y.

How many surjective functions that exists from the set A=(1,2,3,4,5,6) (Domain) towards the set B=(w,x,y,z) (Image) I have no idea on how you do this tbh. Any hints would be helpful.

I experience some difficulty with converting to polar coordinates in integrals. So the question I'm struggling with is 'Evaluate the double integral ∬D x^(6)*y dA where D is the top half of the disc with center the origin and radius 4, by changing to polar coordinates' I'm not sure about solvi...

Give an example of a square matrix A $\in$ $M_n$ with the following properties. (a) entii(A) $\neq$ 0 for each i $\leq$ n but A is not invertible. (b) entii(A) = 0 for each i $\leq$ n but A is invertible. I know that for an upper triangular matrix the diagonal entries must not equal zero in or...

Polygonal numbers are of the form $\cfrac {n^2(s-2)-n(s-4)}{2}$, where $s$ is the number of sides e.g. when $s=5$ we get pentagonal numbers, and $n$ is which one in order it is i.e. the $n^{th}$ $s$-gonal number. Mersenne numbers are of the form $2^p-1$. We usually speak of Mersenne primes,...

$\int_0^{1}f(x)dx$ is approximated by Af(1/3)+Bf(2/3) I would like to derive above formula using Lagrange interpolation polynomial. How should I start?

I have a function fn(x)=x^n where fn:(0,1/2)→R I am trying to find wether the differentiable function can interchange the differentiation and the limit and also determine the pointwise limit of {fn} and {f'n} I don't understand how to approach these kinds of problems, I have answered que...

I am stuck on Exercise 3.26 in Folland's Real Analysis: Let $\mu$ and $\nu$ be positive Borel measures on $\mathbf{R}^n$ such that $\mu + \nu$ is regular. Prove that $\mu$ and $\nu$ are regular. It is immediate that $\mu$ and $\nu$ are finite on compact sets. So it remains to show for, ...

This is in the context of product of probability functions, but the summation done here should be general. I have trouble seeing the rule happening when $$\sum_{w_1\in\Omega_1}...\sum_{w_n\in\Omega_n} P(w_1)...P(w_n)$$ $\implies$ $$\sum_{w_1\in\Omega_1}P(w_1)\sum_{w_2\in\Omega_2}...\sum_{w_n\...

Let $f$ be a relation from $\mathbb{Z}_{4}$ to $\mathbb{Z}_{8}$ given by $f([x])=[x^2+6x]$. Prove or disprove that this is a well-defined function. I've attempted to both prove it and find a counterexample, but I've reached a dead-end in either case. For the proof, this is what I have so far...

If $f:D \to \mathbb{R}$ be continuous and let {${x_n}$} be a Cauchy sequence in $D$. Assuming $D$ is closed and bounded, show that {$f(x_n)$} is a Cauchy sequence.

How could I start my proof? I drawn $G$ for $r=2$ but still have no clue how to kick off.

Could someone elaborate how the following implication is seen: $$A_k=\{(w_1,w_2,...,w_n)\in \Omega^n\}: w_i=1 \text{ for exactly k indices}\}$$ $$\implies|A_k|={n \choose k}$$

∂u^2/∂t^2=∂u^2/∂x^2 u(0,t)=0 ∂u/∂x(1,t)=1 u(x,0)=0 ∂u/∂t(x,0)=0 I solve the heat equation by using Laplace transformations and found that L^(-1){sinh(sx)/s cosh(s)} Can anyone help me to find u(x,t) using the residues ?

For a projective scheme $V$ over an algebraically-closed field $k$ it is a well-known fact fact that a base-point-free linear system of divisors is very ample iff it separated $k$-points and tangent vectors. I have not seen this proven for non-algebraically closed fields, and suspect that it actu...

Prove that a simple graph $G$ with $2n$ vertices and $n^2 +1$ edges contains a triangle for $n \ge 2$. I see it for $n = 2$ or $n = 3$ ... , but I fail to generalize it.