Let $t\in [-R,R], a\in\mathbb{R}$. I want to know if the following equation holds: $$\int_{-R}^R(t+\frac{ia}{2})^2e^{-t^2}dt=(\frac{1}{2}-\frac{a^2}{4})\int_{-R}^R e^{-t^2}dt-Re^{-R^2}.$$ If I start with left hand side: $\int_{-R}^R(t+\frac{ia}{2})^2e^{-t^2}dt =\int_{-R}^R t^2e^{-t^2}dt+ia\int_{-...

Having trouble finding a recursive defintion for the following: n(0)=.19, n(1)=.1919, n(2)=.191919, n(3)=.19191919

how to find a basis of this sub-space? and what is its dimension? note that every matrix in the basis has to have rank of 1. enter image description here

Let $D \subset R$ and let $f: D\rightarrow R.$ Assume that #D $\geq 4$. Assume that $f$ is strictly 4-monotone, i.e., assume, for all $S\subset D, $ that [#$S=4] \rightarrow [(f|S)$ is strictly monotone]. Prove that f is strictly monotone. I have been able to prove monotone included problems, bu...

I wonder if the following matrix norm inequality holds: Let $A$ and $B$ are both strictly symmetric positive definite matrix $\|(A+B)^{-1}\|_2\leq \|A^{-1}\|_2$ ? Thanks in advance.

I don't really know what counts as a proof and haven't been taught maths since 16yo (29yo PhD now). I've got working knowledge of e.g. basic linear algebra, geometry, and statistics, but this feels like running before I can walk so I'm now going back and filling in the elementary gaps. I'm workin...

If a quadratic equation with integral coefficients is :$$ax^2+bx+c=(rx+s)^2$$ Can we say that the discriminant $$b^2-4ac=0?$$ If so how do you prove it?

Show that for all $n\in\mathbb{N}$, $10^{3^n}\equiv 1\pmod{3^{n+2}}$ but $3^{n+3}\not\mid 10^{3^n}-1$. I think I've proved this problem, but I was unsure if my proof was correct: Proof Let $n=1$. Then, $10^3=1000\equiv1\pmod{3^3}$ since $3^2=9\mid999$ and $3\mid111\implies 3^3\mid999$; but, $3...

I think the correct answer is $7$ because the general quadratic is $$y_i=ax_i^2 + bx_i + c$$ Using the formula $$\color{red}{\fbox{Number of degrees of freedom = Number of data points - Number of Parameters}}$$ The $3$ parameters are $y_i,x_i^2,x_i$ So $10-3=7$ degrees of freedom. The correct...

How would i start off to find a recursive definition for x(1)=.19 x(2)=.1919 x(3)=.191919 ... x(n+1)= x(0) + (something goes here, but what)

How do I show that $x^2-\frac{2}{x}$, without looking at a graph? I am open to algebraic and calculus related answers.

I am wondering how to write up the matrix form of, say, Fresnel or Fourier transform. I know that for the case of Fresnel it would be Toeplitz matrix.

Working with the following ODE: $y'' - \frac{1}{2}y' + (\frac{1}{16} + x - e^x)y=0$ W.T.S. there are solutions of the form: $y_1 = ${1+O$(xe^{-x/2})$}$exp(-2e^{x/2})$ and $y_2 = ${1+o$(1)$}$exp(2e^{x/2})$ as $x \rightarrow \infty$ My Attempt Thm1: $\frac{d^2y}{dx^2} + f(x)\frac{dy}{dx}+g...

Suppose f:[0,1)->R is uniformly continuous on [0,1). Prove that the function is bounded (i.e. that range(f) is a bounded set.

What is the smallest $n$ such that the quaternion group is a subgroup of $S_n$?

I'm stuck on a problem and just need some help. If p=(3/4, 1/4) I'd like to construct a tree for pp and pp*p using the construction in Shannon's theorem. I'm completely stuck since I don't understand his construction. I'd really appreciate some help/

How would I algebraically isolate x in the following exponential decay equation, so that x is most easily derived? $$ y = A e^{-\lambda x} \quad $$

I have a language $L=\{P\}$, $P$ is binary predicate symbol. There is an implementation $N$ of language $L$, where carrier is $\mathbb N=\{0,1,2,\dots\}$ and $P_N=\{(a,b) \in \mathbb N^2: a \leq b\}$ I call a theory $T$ of language $L$ "nice" iff $N |= T$ and $N$ is an inducted subimplementatio...

For example, is {0} considered a closed interval? Why or why not? Doesn't it contain all (it's only) limit point of 0?

We say $A \ge 0$ if all $a_{ij}\ge 0$. Now let $A\ge 0$ and $A^k>0$ for some positive integer $k$. Why does $\rho (A) > 0$?

Define $(p_n)_n$ recursively by $p_0(x) = 0$ and $p_{n+1}(x) = p_n(x) + \frac{1}{2}(x - p_n(x)^2)$. I'm pretty sure this sequence is increasing, but I want to prove it. I've tried using induction, but I get stuck: (Assume $p_k \leq p_{k+1}$) $p_{k+1} \leq p_{k+2}$ $\Rightarrow p_{...

Let $X_t$ be distributed according to $\text{Poisson}(\lambda_1)$ where $\lambda_1$ is the intensity parameter of the distribution and indepdently from this we have a process $Y_t$ distributed according to $\text{Poisson}(\lambda_2).$ Additionally we define a process $Z_t \in \{0,1\}$ such that ...

Prove $\sum_{k=1}^\infty \frac { -2x}{(x^2+k^2)^2}$ converges uniformly using the M test.

If $y=Ax+\epsilon$, where $y$ is a vector of dimension $m$, $A$ is a matrix $m \times n$ and is skinny and full rank, and $x(\mu)$ is a vector of dimension $n$. Regularized least squares estimation of x with the regularization parameter $\mu$ is the vector $\hat x(\mu)$ that minimizes $||Ax-y||^2...

Suppose we have a lattice $L$ with the property that : $\forall x,y \in L$, $x \leq y$ implies $x \leftrightarrow y$ (where $x \leftrightarrow y$ means $x$ and $y$ are compatible - that is, the lattice generated by $x$ and $y$ is distributive). Now suppose $a, b \in L$ and $a < b'$, (where $...

Can someone help me out? What makes the Lagrangian of an optimization problem "convex-concave"? In other words, what can I pick out from the Lagrangian that will tell me whether or not a function is strictly convex-concave and/or globally convex-concave? (I am specifically looking at a question...

let $(X,Y)$ independent random variables, $X>0$ and $W = XY$. how can i compute $E(1_{W\leq t}|X)$ using the distribution function of $Y$ ? some help would be appreciated

Suppose f,g:[a,b]->R are continous functions such that f(a)<=g(a) and f(b)>=g(b). Prove that f(c)=g(c) I know that I need to use IVT but I don't know exactly how

diagram If circle B is rotating along the perimeter of circle A, when will circle B intersect circle C? theline going from hA,hK to perimeter where hA,hK = (0,0) is y = (sqrt(rA^2 - (xA^2))/xA)x (xA,yA) will be on perimeter of circle A and the point (xd,yd) that is rB distance from (xA,yA) o...

I am struggling with this problem. Do I need to look at the cyclic group or subgroups? How do I prove the list is complete? List all f $\epsilon$ {Z}_6 such that f^2 = f. Prove that your list is complete.

I know you can square the eigenvalues, etc.... but how do I prove this for a general case?

would one say that this is the recursive definition for a repeating decimal like .19191919..... Base Case: X(0) := .19 X(1) := .1919 X(2) := .191919 ... X(n) := X(0) + .19*((10^-2)^n) recursive step: X(n+1)= X(0) + .19((10^-2)^(n+1))

I am trying to prove that $T_n=\frac{\bar{X}_n - \mu}{S/\sqrt{n}}\sim t_{n-1}$. How do you show this? Is the T-distrbution a ratio between a standard normal and a Chi?

Prove that if det(A) = 0, then there exists a non-zero vector v such that Av = 0, and that there must exist at least one eigenvector corresponding to each distinct eigenvalue of a square matrix.

I am a software engineer and I am learning combinatorics theory on my own. I recently got stumped by the following problem. Problem Let $a_{0} = 2$ where $a_{n} = 3a_{n-1} - 2$ with $n>0$. Given $G(x) = \sum_1^na_{k}x^k $ be the generating function. Prove $G(x) = \frac{2-4x}{(1-3)(1-x)}$ Att...

I am reading Kassel's book entitled "Quantum Groups". Let H be a Hopf Algebra and let $\epsilon$ and $S$ be the co-unit and antipode, respectively. On page 52 he has the formula $\epsilon\left(\sum_{(x)}\epsilon(x')S(x'')\right)=\epsilon(\eta\epsilon(x))$ where the sum uses Sweedler's conventio...

How do you distribute the inversion in $(A^TA+\lambda I)^{-1}A^Ty$ assuming $A$ is a $n \times n$ square invertible matrix, $y$ is a vector with the dimension of $n$, and $\lambda$ is a constant?

$$L = \lim_{x\to\infty} cln(x) - x$$ Does L diverge to $-\infty$ for all positive c? It seems so for very large c, but taking the limit as c goes to infinity yields $$\lim_{x\to\infty} xln(x) - x = \infty$$ According to wolfram alpha. Is it wrong to set c = x?

$\int_\infty^\infty \{\frac{x^2}{x^4+1}\}dx$ Firstly, the bounds are -infinity to infinity, but I'm not too familiar with MathJax. Secondly, I'm trying to understand trig sub better, because I never could get a good handle on it. All I know is that this integral is supposed to reduce to the in...

The majority of my questions revolve around code (thus my activity on stackoverflow), but I've been going over interview question that assume a good understanding of mathematical notation. I don't have a great mathematical background, so I am having trouble understanding what the question is act...

So... this is the explanation my instructor gives in his PDF, but I can't make heads or tails of it. Use mathematical induction to prove that for all positive integers n: H1 + H2 + . . . + Hn = (n + 1)Hn − n. solution: The base case is easy. For the induction step we assume H1+H2+. . .+Hk = (...

A realtor is trying to determine the relationship between a home's price and its square footage. He runs a linear regression in order to predict the value of the home's list price based on its square footage. Based on the regression output below, identify the following: The slope is 35.03637 a...

Definition of a Mash-up product Prove that a "mash-up" of two summable sequences is summable. (notice the sequence does not necessarily have to be non-negative)

Given the following situation: 99.9 percent of people have read a certain book. 99.7 percent of the people, who have read the book, can answer a certain question about it. Of these, who did not read the book, 0.04% answer the question correctly by chance. How big is the probability, that you have...

I have no idea how to do this. For example, how does one compute 10 raised to 0.90 without a calculator?

I have this recurrence relation I'm supposed to do, and I can't seem to figure out the next step: $T_{n} = T_{n-1} + 2n^2, T_{1} = 1$ What I have so far is this: $T_{n} = T_{n-1} + 2n^2$ $T_{n-2} + 2(n-1) + 2n^2$ <--- Is this right?

What I have so far is using the Cauchy Integral Formula $f'(z)=\frac{n!}{2\pi i}\int_{B(a,R)}\sum\limits_{n=0}^{\infty}a_n(\zeta - a)^n / (\zeta - a)^2 d\zeta=\frac{n!}{2\pi i}\int_{B(a,R)}\sum\limits_{n=0}^{\infty}a_n(\zeta - a)^{n-2}d\zeta$. Not really sure where to go from here?

I was wondering what exactly constitutes a direct proof. For example if I prove an equation for some number > than n, with a formula given to be used such as Euler's Formula, would this still be considered a direct proof as it is not induction or contradiction, etc? Thanks

For the definition of Hausdorff distance, please see here Suppose I have a sequence of interval $I_n=[a_n,b_n]\subset [0,1]$, then I read a in a paper it says that, up to a subsequence, $I_n\to I$ in the sense of Hausdorff distance, where $I=[a,b]$ and $a=\lim_n a_n$ and $b=\lim_n b_n$. The pap...

Question is below: Let $(X_{n})_{n≥1}$ be a sequence independent random variables uniformly distributed on {1, .., n}. Let $t_{n} = inf$ {m≥1 : {$X_{1}, ..., X_{m}$}= {1, ..., n}} be the first time for which all values have been observed. Let $t_{n}^{(k)} = inf${m≥1: |{$X_{1}, ..., ...

I was doing some problems for my quiz earlier today (which is now concluded) and went through some combination problems I'm unsure I answered correctly. If I'm wrong, can someone please explain why to me. Letters = {a, b, c} Create a string of length 25. 1) String must have exactly 9 A's. My a...

Suppose that the function f:[0,1) -> is uniformly continuous on [0,1). Prove that the function f is bounded. (i.e. that the range(f) is a bounded set)

I am trying to prove that given a holomorphic function $f$, $u(x, y)=|f(x+iy)|$, $F=u^2$, we have $$\frac{\partial u}{\partial x}=\frac{\text{Re} (\bar{f}f')}{|f|}$$ $$\frac{\partial u}{\partial y}=-\frac{\text{Im}(\bar{f}f')}{|f|}$$ and $$\frac{\partial^2F}{\partial x^2}+\frac{\partial^2F}{\p...

If $\mu(E_n) < \infty$ for $n\in\mathbb{N}$ and $1_{E_n}\rightarrow f$ in $L^{1}$, then $f$ is (a.e. equal to) the characteristic function of a measurable set. I am not sure how to define $E_n$ in order to prove this, any suggestions is greatly appreciated.

In a ballroom dance class, participants are divided into couples for each drill session. One partner leads and the other follows for three minutes, and then the couple switches roles for the next three minutes. (a) Only four people show up on time. How many ways are there to pair them up? My ans...

Need to connect Borel sets and sigma algebra. Also, need to show that a lebesgue measure contains all of the Borel Sets. I know that borel sets are the smallest open sigma algebra sets but I need to further my understanding and was having trouble with this.

$x^2-4x-cosx=-8$ is the equation and we want the number of solutions . Now i tried taking cos as constant but by formula for root we get a trig equation which cant be solved . Any help!thanks!

In my case, u is 1,-1,0 and v is 1,0,2. I know a vector orthogonal to u will be a+(-b)=0 but I don't know how to complete the problem.

If my data $X_i\sim U(0,\theta)$ is iid. What is the Cramer-Roa lower bound for a variance estimator such as the sample variance? $ S_n= \frac{1}{n} \sum_{i=1}^n (X_i-\bar{X})^2$ I am stuck because the uniform density $f(x)=1/\theta$, which means when I take the log likelihood I don't get any...

I'm tring to figure out how to create "complicated" Bezier curves. E.g. a curve similar to $\frac{sin(x)}{x}$ for $x>0$. Is this achievable with the "typical" Bezier definition by adding control points or by joining multiple Bezier curves?

What is the formula for finding P-value for a 1-tailed Z-test. Is it P = 1 - Φ(Z) Or is it P = Φ(Z)?

Can you please provide detailed explanation? Step by step analysis would be highly appreciated.

Having trouble proving this for Big Oh. just dont know where to start with this one. I learn the basics over big oh notation so for this one, would i have so if s(n) = 2^(n+100) and (a(n)) = 2^n , then would i have to use the limit as n -> infiniti that (s(n) / a(n)) = 0?

Yes, I have been looking through this stackexchange site a lot, and have found a lot of things to assist me. But I still could not solve my problem. So my initial task is to : ( ∫ ∫ ) S ⁢ y z ⁢ dx ⁢ dy ...

How do I find the power series for f(x)=$\frac{x^3}{x+3}$ ? That to in summation notation.

I am trying to understand the following equation for Correlation Coefficient: $r = \frac{\sum_{i=1}^{n}(x_i-\bar x)(y_i-\bar y)}{\sqrt(\sum_{i=1}^{n}(x_i - \bar x)^2\sum_{i=1}^{n}(y_i-y)^2)}$ Can someone dissect this equation and provide reasoning as to why this equation does what it does, ...

I've tried to let $u=y^2$, and got $$u'+2x=2x^2u^2,$$ but I still can't solve it.

I need to show that the solution of $ x'=t^3(x^2+1)+e^t$ cannot have a maxima. $\:$ First I tried to solve the equation, but to no avail.$\:$ Then I tried to show the solution is alway increasing, but I don't think it is. $\:$ I have no idea how to begin this problem. $\:$ If anyone could please ...

I am not sure how to approach this problem. if you could help it would be great.

So I tried looking at level surfaces to find such an example, but I wasn't able to generate one. Could someone suggest some possible surfaces with this attribute.

Let $f:R \rightarrow R$. Define $g:R \rightarrow R$ by $g(x)=f(x)(f(x)+f(-x))$ Then which of following is/are correct? A. $g$ is even for all $f$ B.$g$ is odd for all $f$ C.$g$ is even if $f$ is even D.$g$ is even if $f$ is ofdd` Taking f=sinx eliminated option B.What if i take $f$ to be ...

Given $y=\tan(x+y)$, we want to discuss the derivative of $y$ of $x$, says $y'_x$. Assume that $y$ is a function of $x$, then by $y(x)=\tan(x+y(x))$, the derivative of both sides remains equal, $$y'_x=y'_x\sec^2(x+y),\eqno{(1)}$$ then we get $y'_x=-\csc^2(x+y)$. But if we analysis equation (1)...

I am trying to understand bivariate probability distribution functions and i am following All of Statistics: a concise course in statistical inference book In this book the author give one example for joint mass function as mentioned below: Suppose that f (x, y) = x + y if 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 ...

Which interpolation method should I use for complicated "smooth" curves such as $\frac{sin(x)}{x}$ for $x>0$.

Let f be a $C^{1}$ vector field in an open set $E\subset$ $R^{2}$ containing an annular region A with a smooth boundary. Suppose f has no zero in $\bar{A}$ ( the closure of A) and f is transverse to the boundary of A, pointing inward. How can we prove that, if AA contains a finite number of cycl...

I am trying to find the joint distribution of several exponential decay functions all of which have different decay rates.

Of people in their mid-20's who seemed like sure-fire future Fields medalists, who just never quite got to Fields level, and why didn't they succeed?

I'm not sure how to find the congruence class solutions of linear equations or systems of linear equations. For example, how would you solve 3x + 4y = 4 (mod 6)?

Would anyone can explain detail for me this question or provide an example to verify the answer( regardless whether you think it does or it does not) Random d-regular graphs. Explain why the following claim holds: Claim: Any property that holds a.a.s. in the pairing model holds a.a.s. for random...

So i'm revising for my final and i have encountered this problem that uses continuity and cauchy sequences. Let $f:D\rightarrow \mathbb{R}$ be continuous and let $(x_n)$ be a cauchy sequence. a) give an example to show that $(f(x_n))$ isn't Cauchy b) If D is compact, then show that $(f(x_n))$ ...

For example, there is a triangle ABC with angles α = 45 degrees, β = 120 degrees, γ = 15 degrees. Area S = 15. How to find all three heights of the triangle?

Can someone explain help me how to solve this question in step by step so I can understand clear( since I tried but I could not figure out the answer) I appreciate for your time and help Consider a binomial random graph G ∈ G(n, p), where p = p(n) = (ln n + ln ln n + c)/2n for some constant c ∈...

The following is a problem from Weinberger textbook "A first course in partial differential equations". The answer key is $u=\log x$. I have spent three hours on this problem but still cannot reach the same result. The following is my solution. I checked it many times and it looks correct to ...

A and B are initially standing together of a point P. A starts jogging north at a constant speed of $2.4m/s$. One minute later, B starts jogging east at $2.7m/s$. Three minutes after A started moving, at what rate is the distance between A and B increasing? I don't know how to start solving this...

Let S be a set of all complex numbers z such that |z|>1. Show this set is open.

So this is the question. And the answer is $10 \ cm$. I cannot see a way out. Is similarity to be used? Or there will be a construction? Please guide me.

Is there any possible way that we could multiply two qualitative values, without quantifying these values? Please see the following link to see the an example. http://sbassoc.org/wp-content/uploads/2012/05/Gleicher-Change-Formula-Graphic.jpg

Maximum number of people that can be seated in n*m seats where n is number of rows and m is number of columns where some people uses left armrest(l),right armrest(r),both(b) or none(z).

Hello everyone I'm having trouble with this question from my Dynamical Systems course: 4-periodic maps. Let f : [α, β] −→ [α, β] have an orbit of period four {a, b, c, d} where a < b < c < d. a) Enumerate all of the possible ways that f can act on a, b, c, d. For instance, f(a)=b, f(b)=c, f(c)=d...

How to solve the equation $\Delta \log \sqrt{E}=-\lambda^2 E$, where $\lambda$ is a constant, and $E(u,v)$ is more than three times differentiable.

I'm reading through Gödel, Escher, Bach, and I found myself stuck at chapter 9. I've been rereading through several times already, but I must be missing something. To clarify my background, I'm a computer scientist, not a mathematician. On page 273, D. R. Hofstadter states that Could it be, ...

Let S be the unit sphere. I am going to give you three fields, F1,F2,F3. For each field, you are allowed to choose a subset of the sphere, S1, S2, S3. Your goal is to make the flux integral as large as possible in each case. Then calculate each integral. a) F1 = e_r b) F2 = i. c) F3 = . Well,...

Thowing cube twice, Let $X$ be the sum of the two throwing Find the ditribution law My try $$ \begin{array}{c|lcr} &2&3&4&5&6&7&8&9&10&11&12 \\ \hline \text{P}_\text{X}(x) & 1/12 & 1/12 & 1/12&1/12&1/12&1/12&1/12&1/12&1/12&1/12&1/12 \\ \end{array} $$ But the sum shouldn't be $1$0? now th...

If f is holomorphic function in |z|>1 such that $ \lim\limits_{z \to \infty} \frac{Re(f(z))}{z} =0 $ show that $\lim\limits_{z \to \infty} f(z)=0$. When I first tried this question, I thought about Schwartz's Lemma, in which defines $g(z)=\frac{f(z)}{z}$ in |z|<1. Then when I thought it more car...

Let $Y_1 $and $Y_2$ have joint density function: $$f (y_1, y_2) = e^{-(y_1+y_2)}, \text{for all } y_1 >0,y_2 >0 $$ What is $P(Y_1−Y_2>3)$? My attempt: $$P(Y_1−Y_2>3) = P(Y_1−3> Y_2)$$ $$\int_{0}^{\infty}\int_{0}^{y_1-3} e^{-y_1-y_2} dy_2dy_1$$ Evaluating the integral, the value returns: $1-...

Suppose that $B\subset R^m$, $m\geq 2$, be a unit ball, and $W^{1,2}(B )$ be the Sobolev space. My quesion is does $(W^{1,2}(B))^*\supset L^1(B)$?

A man sat in a round table of 10 seats by himself and put his backpack in a seat next to him. In the end, He got drunk and did not remember which seat did he put his bag. He decides to find it. But he is so drunk that at each step he moves to one of the two adjacent seats with equal probability. ...

In his text Multiplicative Number Theory on page 9, Davenport mentions that another means of expanding the L-function is known and then mentions the fact that, $$ \mathcal{F} \sum_{n=1,n \; odd} \frac{1}{n}e_q(mn) = \begin{cases} \frac{1}{4} \pi & if \; 0<m<q/2 \\ -\frac{1}{4} \pi & if \; q/2<m<...

Help to solve example: $$ \lim_{n\rightarrow\infty}\frac{\sqrt[3]{7^{3n-1}+3^n}}{3^n+4*7^{n+1}} $$

We would like to know how many numbers between 29 and 39. Is it 9 or 10. Thanks.

Find sum of all possible values of the parameter $b$ if the difference between the largest and smallest values of the function $f(x)=x^2-2bx+1$ in the segment $[0,1]$ is $4$. I found that the smallest value of $f(x)=x^2-2bx+1$ is $1-b^2$ But i do not know what will be the largest value of the ...

Let $(f_n)$ be a sequence of increasing functions from $[a,b]$ to $\mathbb{R}$ and $f$ be its pointwise limit. Choose $x$ and $y$ such that $x\ge y$. There are $\epsilon>0$ and $N,M$ such that $|f_n(x)-f(x)|<\epsilon$ and $|f_m(y)-f(y)|<\epsilon$ whenever $n>N$ and $m>M$. If we choose $k>\mi...

So I was wondering if this idea for finding the number of primes below X could work and whether its been tried before: Its slightly optimistic as it suggests that from 7 primes (2,3,5,7,11,13 and 17) and there combinations we should be able to see how many primes there are below 4,594,590 (2*3*5...

Again I am stuck at some proof. I need to proof or deproof that for all linear equivalences: R:(X,X) is R = So far I think it is correct because we get symmetry and linearity, but I have troubles to proof it. Any help is upvoted immediately.

When $t=t_0$, $f(x,t)=f_0(x)\in L^2(U)$. $t\in [0,t_0]$ and $U$ is a open subset of $R^n$.$R(x,t)$ is bounded and smooth about $x$ and $t$. I don't whether suitable the conditions is ,if not, please correct it . How to show the existence of solution of $\frac{\partial f}{\partial t}=-\Delta f...

How to establish the following inequality? $$\sum_{\left|\frac{k}{n}-x\right|>\delta}\binom nk x^k(1-x)^{n-k}\leq \frac{x(1-x)}{n\delta^2}$$ where $\delta >0$.

Mostly, I am interesting how to find the non-trivial $p(x)$ having the coefficients from $\mathcal{F}_{q^n}$ rather from $\mathcal{F}_q$.

Using a pre-defined formula in Desmos android app the following example is given : What is value of theta used within formula for evaluating r ? Is it some implicit value, as it's value is not displayed.

Prove that any sequence {$x_n$} that satisfies $|x_n-x_{n+1}|<=\frac{1}{2^n},n\in\Bbb{N}$ is convergent. Prove that a sequence that satisfies $x_{n+1}-x_n\to0$ is not necessarily convergent.

Actually I'm trying to acces into the Sage Notebook last version (6.9). But the system doesn't recognize my username or password. I tried it using my sagemathcloud user, my system user and my root user but always the same answer appears: "username is not in the system". How can I accesss into Sag...

I have some values imported from excel about the annual sell of a product, with 3 var, month, price, sold items I have seen how to plot a chart using curve fitting tool but I need to define a math expression that can describe the relation between these variables. How can I do this in matlab? Is t...

what is the sum of all three digit natural numbers that are multiples of 14, but not 21? What is a quick way of doing sums like these, as i cannot rely on intuition during timed exams

I am having trouble about modifying a function. We have a function that is believed to define a physical condition truly. However, according to my experiments the given function is insufficient. I want to modify the function in a way that it approximately fits to the experimental data. The origin...

I'm having trouble solving this. Investigate the extreme values of the function f, defined by f(x) = x^3 + 3px + q, for all x ∈ R, p, q being fixed real numbers.

I need help finding all natural numbers n such that ($\phi$)=12. My professor told us to use $\phi$(mn)=$\phi$(m)$\phi$(n). I know that 12 can be factored into 3,4 12,1 6,2.

Oh hello guys. I am in the middle of challenging myself to putting my computer and math skills together, trying to build a small hobby computational cluster. Being interested in fractals for a long time I have been able to calculate silly amounts of Mandelbrot pixels really fast in my new playgr...

Suppose $Y_{i}$ are i.i.d., and for some $\theta > 5/2$, $E(|Y|^{\theta})<\infty$. How does this condition imply that $sup_{1 \leq i\leq n} |Y_{i}| = o_{P}(n^{2/5})$?

What do with this: $$ \lim_{n\rightarrow\infty}\frac{1}{\sqrt{n}(n-\sqrt[3]{n^3+2})} $$ Wolfram say -inf, but how?

When I read the Ricci Soliton geometric meaning, I get stuck in the plugging in the Ricci flow as picture below.I don't know how to plug in it,in my opinion, Ricci flow is $\partial_tg_{ij}=-2R_{ij}$. Whether it mean $\mathcal L_{X_P}g_{ij}+2R_{ij}+2\lambda g_{ij}=0$ ?If so , how to get Ricci sol...

How would you prove the existence of triangle centers in tetrahedra, for example, the incenter, circumcenter, or centroid?

I have function values $f_1,\ldots,f_n$ that are approximated by data $y_1,\ldots,y_n$. I am looking for a measure that describes the error in the data $y_1,\ldots,y_n$ and I want the measure to take values between $0$ and $1$. I am familiar with the Root Mean Squared Error (RMSE): $$\mathrm{RM...

I have a matrix in following form $$A=\begin{bmatrix} 1&0&0&0&0&0&0&0\\ 0&-1&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0\\ 0&0&0&-1&0&0&0&0\\ 0&0&0&0&1&0&0&0\\ 0&0&0&0&0&-1&0&0\\ 0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&0&-1\\ \end{bmatrix}$$ I need some help to reduce the matrix in following form using unitary transfo...

What are conditions of conditional density function being integrable?

I'm struggling to find the residues of the equation -zln(z)/((z^2+a^2)(2-z)) with poles at z=+/-ai and z=2 I have the residue for z=2 as -2ln2/4+a^2 but I am struggling to find the residues for +/-ai can anyone help me?

The fourier series for $$ \ln (2 \sin x/2)=\sum_{k=0} \frac{\cos kx}{k} $$ Is the fourier series for $$ \ln ( 2 \cos x/2) $$ same ???

I have learned that projecting a vector a onto a vector b is done by multiplying the orthogonal projection of a (say $\mathbf{a_o}$) with the unit vector $\mathbf{\hat{b}}$ in the direction of b = Orthogonal projection of a onto b = $\mathbf{a_o}$ * $\mathbf{\hat{b}}$ However in the context ...

Let $X_n$ be a Markov Chain with a countable state space $S$, and supposed that the state $s\in S$ is recurrent. Denote $\tau_s^(k) = \inf\{n\geq\tau_s^(k+1) \, \vert \, X_n = s\} for $k\geq 1$. Also let $W_k = \tau_s^(k) - \tau_s^(k-1) Show that for $X_0$ = s, the vectors Z_k = (W_k, X_{\tau_s...

An interesting puzzle I came across: For $T>1$, observe a Poisson process of rate $1$ on the time interval $(0,T)$. Every time we observe a point, we may choose to stop. To win the game, we must stop on the last point before time $T$. Else, if we stop at $t$ and there is another point in $(t,...

Let $P$ be an ideal of a ring R .Let $P$ be generated by an element $a$ (i.e. $P=<a>$). Is it true that is $P^n=<a^n>$ such that n is a positive integer? How could I prove that?

Let $i \neq j$. Proof that $i \leftrightarrow j$ if and only if there are $m, n \geq 0$ such that $\mathbb{P}(X_{m + n} = i, X_{m} = j \mid X_{0} = i)$.

Dertemine all c such that the recursive sequence ${a_n}$ defined by setting $a_0=0 ; a_{n+1}=\dfrac{a_n^2+c}{2}$ converges.

Let $G$ a group and its order is $255$. Prove that $G$ is cyclic. I easily demonstrated that the group has only one $Sylow 17-subgroup$ $P$ that is normal in $G$ and it's cyclic since it is of a prime order. Then $G/P$ is also cyclic since a group of order $15$ is cyclic. Then $G$ can be seen as...

On the plane of the $ABE$ $triangle$ we raise $CB$ as perpendicular. $M$ is the middle of $AB$. $AB=8cm$ and $2*CB=AB$. Requirements: a) Calculate the distance between the points $C$ and $B$. b) Show that $EM$ is perpendicular on the plane $(ABC)$ c) Consider $D$ such as $ABCD$ is a rectangle...

$$cos12x/cos4x-sin12x/sin4x=(cos^2 6x-sin^2 6x)/(cos^2 2x-sin^2 2x)-(2sin6xcos6x)/(2sin2xcos2x)$$ What should I do next?

Prove $tgh(\frac{x}{2})=\frac{cosh(x)-1}{sinh(x)}$ I have started with: $tgh(\frac{x}{2})=\sqrt{tgh^2(\frac{x}{2})}=\sqrt{1-cosh^{-2}(\frac{x}{2})}=\sqrt{cosh^2{\frac{x}{2}}-sinh^2{\frac{x}{2}}-cosh^{-2}(\frac{x}{2})}$ I am stuck here

I'm asked to find the value of $$\int_{-1}^{1}\frac{dx}{(x-a)\sqrt{1-x^2}}$$ where $a$ is complex and $a\not\in[-1, 1]$. I think I should use Cauchy's integration formula but don't know how to apply it.

Let P be the plane in $R^3$ defined by $x + 2y + 2z = 0$. Projection of any vector u on this plane is defined by Au = proj $_P$ u. A = $$ \begin{matrix} 8/9 & -2/9 & -2/9 \\ -2/9 & 5/9 & -4/9 \\ -2/0 & -4/9 & 1/9 \\ \end{matrix} $$ Let u be any vector in...

I am trying to simplify the following Meijer-G funtion \begin{equation} G^{2,2}_{2,2}\Bigl({}^{0,\, 1-m}_{0,\,0}\Bigr |x) \end{equation} But the Matlab(MuPAD) and WolframAlpha give me different results, which are not equivalent (The second terms of both answers are equivalent, but not the first t...

Not quite applicable to many situations, but I'm extending this as more of a "I need help/tips" on re-arranging an equation. I something like this: $$ 0.02=\frac{1}{(1+c)}\left(1-e^{-\frac{Z}{y/(1+c))}}\right) $$ I want to re-arranging this equation for $c$. How would this be done? I know that ...

Given that $yu_x+xu_y=xy$ , $x\geqslant0$, $y\geqslant0$ with $u(0,y)=e^{-y^2}$ for $y>0$ , and $u(x,0)=e^{-x^2}$ for $x>0$. After solving this I got $u(x,y)=\ln (x)+f(\frac{ { y }} x ) $ My doubt is that how to use initial values in this case because denominator will become zero? Or is ...

Find product $ xy $ if both x and y are real. After applying basic log identities, I tried equating value of $ 1/(log x)^2+1/(log y)^2 $ but I am not getting any fruitful result.

I am so lost between the measure and distribution! I do know the definition for each but I couldn't apply it for this question : Comment on the existence and uniqueness of invariant measure and of invariant distribution: Does an invariant measure always exist? Is it unique? Does and invariant dis...

Given that $0<a<-b$ what may we deduce about $(a+b)/a$? This implies that $(a+b)/a < 0$ which I can write as $-|a+b|/|a| < 0$ so that $|a+b|/|a|>0$. Is this all I can conclude? My prof says to me that the quantity $|(a+b)/a| < 1$ but I haven't been able to see this yet. Thanks!

Let's presume that we have a PVP fight scene, where 1 or 2 heroes, are fighting 3 monsters. The monsters that the heroes are fighting are the following: Skeleton (2 of this monster) Health: 1 Defense: 0 Attack: 1 Death Knight (1 of this monster) Health:...

What are some good examples of partial combinatory algebras (a.k.a. Schoenfinkel algebras) with surjective pairing? I mean this in the sense that, if $\mathsf{D}$ is the pairing combinator and $\pi_0,\pi_1$ the projection combinators, then $\mathsf{D}(\pi_0x)(\pi_1x)=x$ for all $x$. Especially in...

Let $\lambda >0$. Consider the non-linear system $$ \begin{cases}\dot{x}=x^2\\\dot{y}=-\lambda y\end{cases}. $$ The aim is to get an idea of the phase portrait. There are some things I can see directly: First of all, $(0,0)$ is an equilibrium. On the $x$-axis, we are moving to the right. On t...

If a^2=bc+1 and b^2=ac+1.How do I find all possible values of a,b,c.

Let $(M,g)$ be a manifold with metric $g$ parametrized by the mapping operator $S$ and parametric domain $\Omega$. The sobolev space of order one with respect to the $L_2(M)$-norm $H^1_2(M)$ is defined as follows: Let $C_2^1 = \{ u \in C^\infty(M) s.t. \forall j = 0,1: \int_M |\nabla^j u |^2 dv(g) <

I am not familiar with the formal compute about the invariant under diffeomorphism (isometric),so I want a detail example. For example,$M,N$ are Riemannian manifolds, $\Phi :M\rightarrow N$ is diffeomorphism.How to formally show $$ \int_MR(\Phi^*g_{ij}) dV(\Phi^*g_{ij})=\int_N R(g_{ij})dV(g_{ij}...

I'm studying systems of differential equations an their representations via phase portraits at the moment, I was taught that when sketching the phase plane of a $2\times2$ system of the form: $$ \dot X=AX $$ With solutions $X=c_1\Re (e^{(\alpha+i\beta)t}\xi_1)+c_2\Im(e^{(\alpha+\beta i)t}\xi_1)...

Given $A,B\in\Bbb R^{n\times n}$ when is there a real matrix $X$ such that $$AX=XB$$ holds? Is there such a matrix for any pair of $(A,B)$?

If I reduce a matrix to Smith Normal Form, how do I find the free basis of the smith normal form of the matrix?

Does it not suffice to point out that $$(i, k)(i, j)\in A_n$$ The element at location $i$ is mapped to the element at location $j$ and and the element at location $j$ is mapped to some third location $k$. Does this not prove transitivity if $n>2$?

I am confused on how to multiply by pi using a calculator(ti-84). I am working in radians. Convert 2.51 x 10^4 rad/min : 2.51 x 10^4 * 2pi divided by 60 My answer was: 2628.465854 rad/s The tutorial video I was watching answer was: 2627.1 rad/s Which answer is correct? Thanks.

I'm not sure how to take the derivative of a definite integral.

Let $V$, $V'$ be normed vector spaces and $f:V\rightarrow V'$ be a linear map. I would be very grateful if someone could verify if my proofs of the below statements are 100% correct. Also, I can't prove the first half of b). a) If $f$ is continuous at one point, it is continuous everyw...

The exponent $e$ of a bivariate normal density is given as follows : $$ -\dfrac{1}{102} [ (x+2)^{2} - 2.8(x+2)(y-1) + 4 (y-1)^{2} ]$$ We need to find : E(X) , E(Y) , Var(X) , Var(y) and $\rho$ (correlation coefficient). I tried to compare that $102$ with $ 2 ( \sqrt{1 - \rho^{2}})^{2}$ , but t...

If $P(x)=p_nx^n+p_{n-1}x^{n-1}+...+p_o$ is divided by $(x-a)$, show that the remainder is $P(a)$. Need help verifying my proof. Proof: If $P(x)$ is divided by $(x-a)$, it gives a quotient of $Q(x)$ and remainder $R$. $P(x)\equiv(x-a)Q(x)+R$ Substituting $a$ for $x$ in this identity gives $R$. ...

This is a part of the question from CLRS (Introduction to Algorithms) Chapter on FFT and Polynomials. I am self reading and am stuck at this part. Let $n$ be a power of 2. Suppose that we search for the smallest $k$ such that $p = kn + 1$ is prime. Give a simple heuristic argument why we might ...

Could anyone help me with this question? Thanks! Suppose that two balls are randomly chosen without replacement from an urn containing $5$ white balls and $8$ red balls. Let the random variable X equals $1$ if the first ball chosen is white and equals $0$ the first ball chosen is red. Let the ra...

Find P2(x) for f(x) = e^cosx expanded about x0 = 0 and then I have to find M > 0 so that |f(x) - p2(x)| <= M|x|^3 for all x ∈ R. I think p2(x) = ex but I'm not really sure. And I really need help with the second part, where I have to find M.

I'm aware of some theorems like the Banach-Tarski's which yield very counter-intuitive results, however, it's proof is far beyond my knowledge, so I'm looking for some result that is easy to prove and gives a counter-intuitive result, using Choice. Does anyone know any examples?

A lot of questions about this have unsatisfying answers that either argues how unsafe RSA is when $p=q$ or points out that $\phi(n)$ is not $(p-1)(q-1)$ when $p=q$ and that $\phi(n)$ must be used to make RSA secure. However, I'd like to know why the RSA fails, i.e why the Decipher $R$ is not equ...

Find local max local min for f(x) = x^2 + 1/x, I tried to take first order derivative $$ f'(x) = 2x -x^{-2}$$ but how do I go from here.

The figure below shows a broken piece of a circular plate made of glass.

So the subject of translations in a very simple case came up the other day, where for $x,y\in R$ $f(x-a)$ is just $f(x)$ translated to the right. This is trivial to show in a very hand waving sort of way: the function $f$ "looks back", and the fact that $f(x-a)$ goes to the right and $f(x+a)$ goe...

I found a similar exercise but I couldn't solve it also let $a\in R$ which is PID if $a=\prod_{i=1}^{k} p_{i}^{n^{i}}$ then $R/(a) \cong \bigoplus R/(p_{i}^{n^{i}})$

I am trying to solve this recurrence relation $f(x)$ = $f(x - 3)$ + $\frac{(2(x - 3)^3 + 3(x - 3)^2 - 2(x - 3))}{24}$ but I don't know how to proceed further. So can some one please help me in solving this recurrence relation. And also please give me some hints on how to solve these kind of pro...

Let $D \subset R$ and let $a \in D$. Show that acc(D \ {a}) = acc(D). I'm not sure I really understand what this question is asking. How can I show that these accumulation points are equivalent?

Let $P(x)=(m^2+4m+5)x^2-4x+7,m\in R$.If $3\leq x\leq 5$,then find the minimum of the minimum value of $P(x).$ The minimum value of $P(x)=(m^2+4m+5)x^2-4x+7$ occurs at $x=\frac{2}{m^2+4m+5}$ So the minimum value of the $P(x)$ is $\frac{-4}{m^2+4m+5}+7$ But i dont know how to find the minimum v...

A matrix $M\in\mathcal{M}_n(\mathbb{R})$ is said to be primitive if it exists an integer $k$ such as $M^k$ has all its coefficients strictly positive. My question is : If $M$ is primitive and if $N$ is similar to $M,$ say $N=PMP^{-1}$ with $P\in\mathrm{GL}_n(\mathbb{R}),$ do we have that $N$...

The series in question is $$\sum_{n=2}^{\infty}\frac{2}{n^2-1}.$$

I'm giving a lecture to senior year high school students. I want to talk about the importance of mathematics and illustrate my point with some practical examples. I've decided to concentrate on financial mathematics. I've got about 20min to speak about math. I thought about explaining some basic...

I was thinking to my self one day, whats bigger, 3.3 or 3.3333333 (1/3)? I decided to write a program in swift to test it var threeDotThreeThree:Double = 3.3 var threeRequring:Double = 1/3 threeDotThreeThree > threeRequring The result of the line threeDotThreeThree > threeRequring was true. ...

Need some help with this problems: *1. Conjecture the value of the limit $$ \lim_{n \to \infty}\,\int_0^{\infty}\,\left( 1+\dfrac{x}{n} \right)^ne^{-2x}\,dx $$ Prove your conjecture. Justify.* 2. Suppose the $f_n \in L^1(\mathbb{T})$, $n = 1,2,...$ and $\| f_n - f \|_{L^1(\mathbb{T})} \xright...

Suppose we have an expression and want to define it using some symbol. One can write something like "$X:= \text{Expression}$" to mean the symbol $X$ is defined to be the expression after the symbol "$:=$". From my understand, before the definition $X$ is supposed to be just a symbol, after the d...