Does there exist a non constant continuous function $f:[0,1] \to \mathbb R$ such that $\int_0^1x^n f(x)dx=0 , \forall n \in \mathbb Z^+$ ? I know that $f$ is identically $0$ if the integral is also equal to $0$ for $n=0$ , but I cannot make a headway what will happen if I omit the $n=0$ case ....

The answer is 3x^2 y^2 ^3√6̅y̅ How does ^3√1̅6̅2̅x̅^̅6̅ y̅^̅7̅̅ equal 3x^2 y^2 ^3√6̅y̅

How many ways can 6 be written the sum of positive integers if numbers can be repeated and order is not important? Looking at this problem i want to say that you can get sum of 6 by adding 1+5, 2+4, 3+3, 4+2, 5+1. so there are only 5 ways this can be rewritten. but i feel like im missing somethi...

How do I solve this without using L'Hospital's rule? $$\lim_{h\rightarrow0} \frac{\cos^{-1}(\frac{1}{2}-h) -\cos^{-1}(\frac{1}{2})}{h}$$ I already tried letting $\theta=\cos^{-1}(\frac{1}{2}-h)$ gets $\cos\theta=\frac{1}{2}-h$ then $h=\frac{1}{2}-\cos\theta$, replaced all $h$ with this and I'm l...

Consider the multiplicative group $S=\{z:|z|=1\}\subset \mathbb C$. Let G and H be subgroups of order 8 and 10 respectively. If n is the order of $G \cap H$ , then what is n ? n=1 n=2 $3\leq n\leq 5$ $n\geq 6$ I am completely stuck on this problem ,please someone help me.Thanks.

Let $C$ be the curve of intersection of the two surfaces $x+y=2 , x^2+y^2+z^2=2(x+y)$ . The curve is to be traversed in clockwise direction as viewed from the origin . The what is the value of $\int_Cydx+zdy+xdz$ ? I am not even able to parametrize the curve of intersection . Please help . Thanks...

$\int\ e^{2\theta}sin (3\theta)\ d\theta$ I am little stuck as to what I can do after this point. Please tell me if my method overall is flawed.

I'm working through an algebraic number theory book, but I can't understand the example shown below: I follow the example up till it assumes that $\frac{p_1}{q_1},...,\frac{p_n}{q_n}$ are the generators for the ideal $R$ of the abelian group $R$. "Then the only primes dividing the denominators...

Suppose that R is a prime ring and charR $\neq2$.$d_{1},d_{2}$ are derivations of R such that for some non-zero ideal of I of R and some $c\in C$ (C is an extenden centroid of R) $d_{1}d_{2}(x)=cx$ for all $x \in I$. I can't show following equality : By computing in different ways ...

I need to write a regular expertion for the language of all the binary words that contains continuum of even number of zeros and after that even number of ones or odd number of zeros and after that odd number of ones $L=\{0^n1^m|n\equiv m(\mod 2)\}$ My try: $(00)^*(11)^*+0(00)^*1(11)^*$ ...

I have n lines, their equations being aix+biy+ci=0. I want to find the number of triangles whose sides lie on the given lines. n (1<=n<=300000), the number of lines. Each of the following n lines contain 3 integers ai, bi and ci (ai,bi,ci < 109), the numbers defining the line i. Note: no three l...

I have been asked to construct a hamiltonian graph and a non-hamiltonian graph using the same degree sequence. I have had no problem constructing the hamiltonian graph however I am finding it extremely difficult to construct the non-hamiltonian one. Is there any steps I can follow when trying to ...

Let $A$ be an $m\times n$ totally unimodular matrix. I want to show that in that case $A^T$ and $\begin{pmatrix} I\\A^T\\-A^T\end{pmatrix}$ are totally unimodular as well. However I have no idea how to prove this. Can you help?

Is there a standard way to find the tangent cone to a general ellipsoid from a point outside of the ellipsoid? Thanks!

I am solving an equation and I have a problem. In the right-hand side of my equation; I have everything in the form of (G+H+T)W except one term WC. (C,G,H,T are constant all Matrices(n*n), and W is an unknown matrix containing my nodes in the domain). In order to solve this equation; I need to fi...

I am attempting to Build a DFA that accepts strings over {0,1,2} that are divided by 3 and doesn't include the substring 012. What I tried doing is taking the original 3 states of a DFA that accepts everything that is divided by 3, remove all the self loops with 0 and add states, but it became s...

Integrate $\int\frac{xe^{2x}}{(1+2x)^2dx}$ u= 1+2x du= 2 dx However that does not help me. Please keep in mind I am only allowed to use Integration by Parts.

how many different rectangles can you draw on a paper $n x n$? rectangles are different if they are different in size and place. I think it is $C(n+1,2)^2$

I want to find an open cover for $(1,5]$ that has no finite sub-cover. my SOLUTION attempt: I thought about a cover to be the set of open intervals of the form ${(1-\frac{1}{n},6)}$. would that work?

Whenever $S$ is a finite set, I define a multiset of $S$ to be a function $\mathbb{N} \leftarrow S$. Obviously, the multisets of $S$ form an abelian monoid with respect to addition. By an unordered pair of elements of $S$, I mean a multiset $A$ of $S$ that returns $0$ at all but two (distinct) el...

You are in a room in a maze and want to get out. Whenever you enter a hallway, you mark it with an "X". Whenever you go into a room for the first time, you mark the hallway you came through with an "O". When you leave a room, you never enter a hallway marked "X" and you only enter a hallway marke...

This is what I am trying to prove: Let $ (a_k) $ be a sequence of real numbers. Prove that $ \sum(a_k) $ converges if and only if for any $ \epsilon > 0 $, there is a natural number N so that if $ n \leq N $, then $$ |\sum_{k=n}^{\infty} a_k | <\epsilon $$ I know that a sequence $a_n$ is said ...

If (1+cos A) (1+cos B) (1+cos C)= (1- cos A) (1-cos B) (1-cos C) then prove that each side is +/- sin A sin B sin C

In my Algebra textbook I found a sentence that confused me. It says: "A mapping is a function whose domain is its entire pre-domain." Never before I found a term of pre-domain. There is even a small ilustration, that distinguish a "domain" and "pre-domain", where pre-domain is superset of dom...

This is probably well known. Let $A$ be a natural number and let $B$ be an even number. Does there exist a power of 2 which starts with $A$ and ends with $B$ ? $\bf Fact: $ I know that there is an infinite number of powers of 2 which start with $A$. As another question: The proofs that I...

I don't even know where to start. Can someone please show how to this step by step?

$2x+2-x^2 \ge 3^{x^2-2x+x}$ Is there an exact solution for this inequality in real numbers?

What is the difference between "let" and "for all"? Consider the following example For all natural number n, if n is even, then n squared is even. Let n be a natural number. If n is even, then n squared is even.

The given sequence a(n) = {n^k/z^n} applies for all n element of N, k element of N, z element of R and |z|>1. The task is to prove that this sequence converges to 0. (Sorry for bad notation, ask if snth isnt clear)

$$ \,\,\,\,\,\,\,\,\,\, \min_{X,E} ||X||_* + \lambda ||E||_{\ell 1} \\ s.t. \,\, x_{ij} \ge 0 \text{ for all } x_{ij} \in X \text{ and}\\ \,\,\,\,\,\,\,\,\,\, ||Y-(X+E)A||_F \le \delta, $$ where $X_{m\times n}, E_{m\times n}, Y_{m\times q} \text{ and } A_{n\times q}$ are matrices; $\lambda, \d...

So I was working on one of the exercise question for school and I came across this question and I wanted to know if someone can help me in providing a proof for this question. So the question is: Prove if p1, p2, . . . , pn are distinct prime numbers where p1 = 2 and n > 1 then p1p2p3 · · · pn +...

I try to prove that f(x) = [x]sin(pi*x) continuous on R. Someone help me, please. I need it tomorrow. Thank you.

The UCD Health Centre believe that the number of students who contract conjunctivitis follows a Poisson Process with parameter λ = 1/3 per month The UCD Health Centre hopes to reduce services during the summer months when there are no undergraduate students on campus. The director of the UCD Hea...

I'm trying to use the relevant rules and definitions but making little progress.

As a high school student motivated to do math research with a background in undergraduate mathematics, I have a few questions regarding math research. As a high school student what should I be expected to have mastery of to pursue in this direction? How should I seek guidance and how is it reco...

I have to invert this function: $$f: \mathbb{N} \rightarrow \mathbb{N}, f(n) = \begin{cases} n+1, & \text{if $n$ is odd} \\ n-1, & \text{if $n$ is even} \end{cases}$$ But I am not able to grasp how to invert it while also handling the conditions set on the function. Can any one please give me t...

I have the following Jacobian problem: I'm having trouble working through it because the double integral in terms of u and v is throwing me off. Could someone walk me through it? Thanks!

I am writing a code in matlab that involves working out the square root of an array of numbers $A^2$ at each time step. To the best of my knowledge there is no way of working out the negative squared roots of an array of numbers in maltab. Am I right? Are there any alternatives? Thank you.

Consider the problem $$\min (x-2)^2+y^2$$ $$s.t. \>x^2\leq ky^2+1$$ $$x\geq 0$$ where $k \in \mathbb{R}$ is a parameter of the problem. Determine the status of the point $(1,0)$ for each value of $k$. For what values of $k$ is the point $(1,0)$ is a KKT point, local min,global min. Solution so f...

I am trying to show that $f_n = 1/(n+1)$ converges almost everywhere, where $0 \leq x \leq e^n$ and measure space $ (**R**, **B(R)**, \mu)$. Any comments will be a great help. Thank you. I tried considering $\epsilon > 0$, and a positive integer $N$ such that $1/e^n <\epsilon $, but how can I ...

Random variable X has uniform distribution on $[0,1] \cup [2,3]$. Find cdf of variable X. I mean i do not know how to treat this on such strange interval.

I have little conjecture. Maybe it's stupid i don't know. Let $p>5$ be a prime number. Then $(3^p-1)/2$ is always squarefree? It's true for $p<192$.(I used Mathematica.)

Consider the family of functions fλ(x) = λx − x^3. Describe the dynamics of this family of functions for all λ < −1.

I need to find Σ k MOD m where k is varying from 1 to n and large n and m . How can i find it efficiently?

If I have a graph called $G$, what does $\hat{G}$ mean? Is it a reference to a subgraph of $G$?

I know that all groups of order \leq 5 are Abelian and all groups of prime order are Abelian. Are there any other examples? If so is there something special about the orders of these groups?

n points on circle how many different strings that dont cross each other? My solution is: $C(n,2)$ which makes total sense, for me. and I know it's not enough. I think it is$ C(n,2) + C(n-1,2)$ because from the point of the first string, it has n-1 points left to choose. but there's a problem be...

H is a subgroup of A_5 that has order 30. From this I know that |A_5 : H| = 2. From this I'm supposed to prove that H contains all 3 and 5 cycles and then use that to prove that there cannot be a subgroup of order 30 in A_5. I understand how to do the 3-cycle, but I'm very confused on the 5 cycle...

My attempt on the problem: X X X X = 26^4 number of ways the strings can be made A A X X = 26^2 number of strings with two A's in them B B X X = 26^2 number of strings with two B's in them B B A A = 1 way the string is duplicated A A B B = 1 way the string is duplicated Then if I do 26^4 - 26^2...

Consider an asset that costs $360,800 and is depreciated straight-line to zero over its 7-year tax life. The asset is to be used in a 3-year project; at the end of the project, the asset can be sold for $45,100. If the relevant tax rate is 34 percent, what is the aftertax cash flow from the sale...

Let $f:[0,\infty)$ be continuous.Assume that $f$ is uniformly continuous on $[R,\infty)$ for some $R>0$ .show that $f$ is uniformly continuous on $[0,\infty)$ well my basic idea is since $f$ is continuous ,so it is uniform continuous on $[0,R]$ and given it is uniform continuous on $[R,\infty)$....

I can come up with a dynamic-programming-type program to compute this number, but I am wondering if a nice closed form formula is known.

5 transistors are in a bin, 2 are defective. Test 1 at a time until defective one removed. Let N1 = # of tests made until 1st defective is identified. N2 = # of additional tests made until 2nd defective is identified. N1 = 1, 2, 3, 4 N2 = 2, 3, 4, 5 For P(2,3) I don't see why it's not $ \...

Let $\{N_T:t\geq 0\}$ a homogeneous Poisson process with rate $\lambda\geq0$ and $T\geq0$ independent random variable with mean $\mu$ and variance $\sigma^2$. Find $cov(T,N_T)$. How may I compute $cov(T,N_T)$? May I set T=t and use conditional expectation? I don't know how to proceed.

Is the function f(x)= ∑x^2 is a psuedo concave function? If the answer is yes then how should I proove psuedo concavity in the above example ?

How might I show that the half open intervals (-$\infty$, $k$] and $[k, \infty)$ are open sets in the topology induced by $\mathbb{R}$ on $\mathbb{Q}$?

simplex algorithm Attaching coves $10$ reinforcement wire length $50$ m and $30$ m. The armature must have a size of $8$ m and $6$ m and $8$-m two times more than $6$-m cove cut so as to obtain the largest number of sets of armatures Cn: (I) $1$ $2$ $3$ $4$ $5$ $6$ $7$ (II) $8$ ...

Let $k$ be a field, $A$ a ring (possibly non-commutative) over $k$, suppose $A$ has no two sided ideals. Is there an example for such an $A$ with infinite dimension over $k$?

Find all continuous, infinitely differentiable functions f with domain $\mathbb{R}$ such that $f(x)f(y)=\int_{x-y}^{x+y}f(t)dt$ for all $x,y\epsilon \mathbb{R}$ As some hints provided by the book, the following functions work: *2x *c *$\frac{2}{c}sin(cx)$ *It is also noted that there is a...

I have a problem where I need to find the least squares regression line. I have found $\beta_0$ and $\beta_1$. I know that $\hat{y}$ the vector predictor of $y$ is $x \cdot \beta$ and that the residual vector is $\epsilon = y - \hat{y}$. I know also that the least squares regression line looks s...

May be this question is too short for the "standard of quality of this site", but I would like to know if $f(x)=e^x \cdot cos(e^x)$ is a temprated distribution.

$\lim_{n\to \infty}{\sqrt[n]\frac{(2n)!}{n^2\times{n!}}}$ It is a sequence and n is natural It looks like I should use $\lim_{n\to \infty}{\sqrt[n]{a_n}}=x$ but I don't know how. Does it mean that $a_n=\sqrt[n]\frac{(2n)!}{n^2\times{n!}}$ and then do it from there or is $a_n=\sqrt[n]\frac{(2n)!...

I tried to prove that using the fact that for languages $L_1, L_2$ we get: $(L_1∪L_2)^* =(L_1^* L_2^* )^*$ , but I got stuck here: $(a∪b)^* = (a^* b^* )^*⊆b^* (a^* b^* )^*$

Given $n\times n$ matrix $P(t),A(t)$, $P(t)$ satisfies the matrix differential equation $P'(t)\eq A(t)P(t)$ and the initial condition $P(0)=P_0$. Then Prove $detP(t)\eq detP_0\times exp(\int_0^t tr(A)(s)\,ds)$ If the matrix A is constant matrix then it is easy. But I don't know how to prove it ...

I need to find Σ A[k] MOD m where A[k] is an array element which is containing an integer value and k will be from 1 to 10^5. How can I find it efficiently?

Base case: n=0 $5^0$ = 1 which is true since any tree of depth 0 has 1 vertex. Now, assuming true for $5^k$, how would i show true for $5^{k+1}$ ?

There are many different hash functions, md5, sha, and others. They take a value V and produce a H via transformation Function(V) = H, where Function is md5, sha, etc. My question is: Does every hash value H have a value V? For example, given md5 hash value f2c057ed1807c5e4227737e32cdb8669 (tot...

How do you do this? I have no idea. Thanks.

CAN ANYONE PLEASE SHOW ME HOW TO PROOV THE NEXT STATEMENT? HOW MANY WAYS EXIST SOLVING FOR THIS PROBLEM? Proov: (A ∩ B ⊆ A ∩ C) AND (A′ ∩ B ⊆ A′ ∩ C) → B ⊆ C thanx alot.. :P

I have two events A and B: P(A) = 0.25 P (B) = 0.6 P(AUB) = 0.75 I have calculated: P( not A) = 0.75 P(A int B) = 0.1 I now need P(not A | B). Is it right to use the standard conditional probability formula and assume that: P(not A int B) = P(B) = 0.6 and calculate P (not A int B) / P(B) ...

Problem Description I'm beginning network construction for a problem that I feel could have a far more insightful loss function than a simple MSE regression. My problem deals with multi-category classification (see my question on SO for what I mean by this), where there is a defined distance or...

I'm currently reading notes on a lecture I missed due to not feeling well. These are notes on "Symmetric Groups and Modular Arithmetic Groups". A sentence in the notes says: "For a set $S$, a bijection $S\to S$ is called a permutation of $S$." "Let $G$ be the set of all permutations of $S$ and...

given that $|z+w|=\sqrt{2}$ show that n is a multiple of 4 algebraically. ( z and w are two distinct roots of unity of x^n = 1) In context of the question, i should be using exponential form.

Is $f(x)=|x-1|^3$ differentiable $\forall x$? I can't understand if even if there is no cusp there is still a problem in differentiating it in $x=1$.

Is this the correct expression of the mathematical statement, "Every positive real number has exactly two square roots."; expression: ∀x∃a∃b((x>0) → (a!=b)∧(x=$a^2$)∧(x=$b^2$)).

From what I can dig up, given a vector bundle $E\rightarrow X$, the determinant bundle associated to this is $\Lambda^{n}E\rightarrow X$, where $n$ is the rank of $E\rightarrow X$. Is this the same thing as "the determinant of the vector bundle $E\rightarrow X$"?

The problem is: (The rate of decrease of temperature of an object is continuous and proportional to the difference between the temperature of the object and that of the surrounding medium.) Suppose that the surrounding temperature remains constant. The object is initially at a temperature of $...

The problem: Let $a$ and $b$ be elements of a group $G$. Let ord($a$) $=m$ and ord($b$) $=n$; let lcm($m,n$) denote the least common multiple of $m$ and $n$. Give an example to show that, if $a$ and $b$ do not commute, then ord($ab$) is not always a divisor of lcm($m,n$). My thoughts: We want ...

Suppose, on a smooth projective complex variety $X$ that we are given an effective divisor $D$ and $A\in\mathrm{Pic}(D)$ a globally generated line bundle on $D$ with $r$ independent sections. Then we have a surjection $\mathcal{O}_X^{\oplus r}\to A$ (here we view $A$ as a torsion sheaf on $X$ wit...

I know you are able to find the root of the equation by using newton Raphson method. But is there any other way? coshx+cosx-3=0 I thought maybe you could say that -1<=cosx<= 1 So saying 2< cosh(x) <4 But then I am unsure what to do next or if you could even do it this way?

Say you had the sequence $U_{n+1} = 2bU_n$ where $U_1 = 6$. How would you find the range of values of b for which the sequence converges?

Looking for fundamental solution for the one-dimensional elliptic operator $L = A\frac{\partial^2u}{\partial x^2}(x,t) + B\frac{\partial u}{\partial x}(x,t)+ Cu(x,t)$

I want to learn the basic theory of phase space partition and symbolic dynamics, can you point to any recent thesis and books containing a good exposition ? Thanks!

I have a question regarding the proog of lemma 4.3 in van der Vaart at p.36 (https://books.google.co.uk/books?id=UEuQEM5RjWgC&pg=PA36&lpg=PA36&dq=lemma+4.2+van+der+vaart+one-to-one+differentiable&source=bl&ots=mnRJLEcYHy&sig=ZjAfaVM50LOxJUkphbBT2sJTVcc&hl=it&sa=X&ved=0ahUKEwiglKf31b3JAhWHox4KHS7F...

can you help with this math problem? I dont know how to start? There is any good method? $$\int\frac{x+1}{(x^2-x-5)^3} dx$$ Thank you

I am confused about how to derive the right hand side of the "$=$" $$\int_{E_{\alpha}}\int_0^{\infty}d\alpha dx=\int_{\mathbb{R}^d}\int_0^{\infty}\chi_{[0,|f(x)|]} d\alpha dx$$ $E_{\alpha}=\{x:|f(x)|>\alpha\}$ and $\alpha >0$ $x\in \mathbb{R}^d$ $\chi$ :an indicator function

This one looks simple, but apparently there is something more to it. $$f_{(x)=x^x}$$ I read somewhere that the domain is $\Bbb R_+$, a friend said that $x\lt-1, x\gt0$... I'm really confused, because i don't understand why the domain isn't just all the real numbers. According to any grapher on...

Given 3 red balls of value 3, 2 blue balls of value 5, and 1 yellow ball of value 25 a) What is the expectation for X of drawing two balls? Where X is the value. My issue here is that we have no way to be sure what ball was drawn the first time, I have calculated the X of drawing one ball to be...

Prove by Mathematical Induction 3^2n ≅ 1 (mod 4) for every natural number n.

English is not my mother tongue and I'm studying Algebra using a book in english. This sentence came up to me in an exercise "every row of matrix A adds to zero". What does that mean, in concrete?

I have the next maximization problem and I am wondering if I can transform it to geometric programming which is is easy to solve, because I cannot find other way to find the solution.I also tried solving the Lagrangian but I couldn't find the solution. $$ \max_{P_1,P_2,P_3,P_4} \frac{\frac{P_1h1...

Background I am a software engineer and I have been picking up combinatorics as I go along. I am going through a combinatorics book for self study and this chapter is absolutely destroying me. Sadly, I confess it makes little sense to me. I don't care if I look stupid, I want to understand how t...

If $E$ and $F$ are field, what is $E\vee F$ ? Let take for exemple $E=\mathbb Q(a)$ and $\mathbb Q(b)$. Do we have $$E\vee F=\mathbb Q(a,b) \ \ ?$$

Does anyone know how to prove the following criterion? (Due, 1949) $x \in (0,1)$ is normal for base $10$ $\iff \{10^nx\}_n$ is equidistributed. "$\Leftarrow$" is quite easy using definition of equidistribution, but for the "$\Rightarrow$" part I guess we should use Weyl's criterion. Could you g...

enter image description here What is the principal value of argument (Arg(z-a_{n}))?

How should i solve this integral? i know that it is the same question like here Integrate $\int\frac{x+1}{(x^2+7x-3)^3}dx$ but I've tried solve it for more then 3 hours and i still have no idea ho to solve it. Thank for help. $$\int\frac{x+1}{(x^2+7x-3)^3}dx$$

fn(x) is a series of function that Uniform convergence to f(x). f(x) is the limit function. I know fn(x)< M (M is a bound). Does it true that f(x) < M ?

Was playing around with solids of revolution, the shape given by rotating $y=\sqrt{\sin x}$ about the $x-$axis seems to resemble a blimp. The only thing I can find out about the natural shape of the blimp is that it is formed by the pressure of the lifting gas. Any reason a sinusoidal relat...

I am a tutor at university, and one of my students brought me this question, which I was unable to work out. It is from a past final exam in calculus II, so any response should be very basic in what machinery it uses, although it may be complicated. The series is: $$\sum \limits_{n=1}^{\infty} \f...

I'm having trouble for the derivative of this trig function and got 40 sinx 1/1000pi cosx for the function 20sin^2 (x)/(1000pi)

I am taking a Phd class in econometrics, and the following is used constantly, for $X$ a $n\times k$ matrix, $n \neq k$: $$(X'X)^{-1} = ((X'X)^{-1})'$$ with "$'$" standing for transpose. Having a rather weak background in linear algebra, I cannot understand why this is true. For example: ${\unde...

$f(x)=\sqrt{\frac{x-1}{x+1}}$ $f'(x)={1\over 2}*(\frac{x-1}{x+1})^{\frac{-1}{2}}*{x+1-(x-1)\over (x+1)^2}=\frac{1}{2}*(\frac{x-1}{x+1})^{\frac{-1}{2}}*{2\over (x+1)^2}=(\frac{x-1}{x+1})^{\frac{-1}{2}}*{1\over (x+1)^2}$ Is it valid to write$ (\frac{x-1}{x+1})^{\frac{-1}{2}}= \sqrt{\frac{x+1}{x-1...

If we called the perimeter any triangle b. Prove that if you added the lengths of any two medians (i) It is not larger than (3b)/4 (ii) It is not smaller than (3b)/8 This came up in a math competition some time ago and I think that using the triangle inequality will be necessary. However, I ...

Definition. Two positive (or signed, or complex) measures $\mu$ and $\nu$ defined on a measurable space $(\Omega, \Sigma)$ are called singular if there exist two disjoint sets $A$ and $B$ in $\Sigma$ whose union is $\Omega$ such that $\mu$ is zero on all measurable subsets of $B$ while $\nu$ is z...

Im trying the find the pointwise limit of a sequence on [0,1] to determine if the sequence converges. My sequence is below. fn(x)=(x^n)/n I know that for the sequence to converge lim (fn(x))=f(x) when n goes to infinity. But I don't understand the difference between fn(x) and f(x).

Suppose there is an air vent in the center of the front of this classroom (assume it is the xz-plane centered at x=0) near the ceiling. If the velocity of the air blown out of it by the air conditioner can be modeled by the vector field F=, find the air flux exiting an open doorway of the classro...

I have a sparse, square, symmetric matrix with the following structure: (Let's say the size of the matrix is N x N) enter image description here Here, the area under the blue stripes is the non-zero elements. Could someone tell me if there is an algorithm to invert this kind of matrix that is s...

English is not my mother tongue and I'm studying Algebra using a book in english. This sentence came up to me in an exercise "every row of matrix A adds to zero". What does that mean, in concrete? EDIT: Full exercise: If every row of A adds to zero, prove that detA = 0. If every row adds to 1...

I am trying to apply Ito's lemma to compute variance of the following integral $X(t) = \int_{0}^t W(s)dW(s),$ where $W(t)$ is a Wienner process. Could you please check my calculations? $$E(X(t)) = 0 \\ E(X^2(t)) = E\left(\left( \int_{0}^t W(s)dW(s) \right)^2\right) = E\left( \int_{0}^t W(s)^...

I need to find the base of linear space of sequences of form $$a_{n+1} = n(a_n + a_{n-1})$$ I know that n! and !n (number of derangement) satisfies this equation.

I was reading the following paper on Adaptive Bayesian estimation using a Gaussian random field with inverse Gamma bandwidth and wanted to apply the results to a problem that I am working on. However, I am not sure if the results meet the assumptions that I am making about my functions. Basical...

I have found V in terms of x, and then I have found the energy equation for x=1 and x=2. I've then set them equal to one another and solved, finding lambda = 20. I didn't use the values v=4 and v=2; since they are provided, I assume my method is incorrect/incomplete?

Show that a differentiable conjugation preserves the sets $\omega$-limit and $\alpha$-limit. Thanks.

Given sample $N_p(a,Q)$, where $Q>0$ - is known matrix. Divide $a\in\mathrm{R}^p$ so as $a^T = (a_1^T,a_2^T)$. How, assuming $a_2 = 0$, I can find maximum likelihood estimation for $a_1$? Without assumption on $a_2$ it is easily follows from $\left(\frac{1}{\sqrt{2\pi}}\right)^{pn}\left(\frac...

how can he prove the point A(a_1, a_2, a_3); B(b_1, b_2, b_3); C(c_1, c_2, c_3); D(d_1, d_2, d_3); belong to the same plane , as if they belong can then find plane. i now how to prove three given point belong the same plane

I tried integrating by parts to obtain limit as k approaches infinity of (-1/ik)f(x)e^(-ikx) from -infinity to infinity + 1/(ik) integral of f'(x)e^(-ikx) from -infinity to infinity. This second integral can be substituted for the original integral of f(x)e^(-ikx) by properties of Fourier Transfo...

Let $\gamma:[0,2\pi]\to \Bbb C$ be given by $\gamma(t)=e^{it}$. Compute $\int_{\gamma}|z-1||dz|$. $\int_{\gamma}|z-1||dz|=\int_0^{2\pi}|e^{it}-1|d|\gamma|$ where $|\gamma|$ is the variation of $\gamma$ from $0$ to $t$. I am not sure how to deal with the $|d\gamma|$. Does $\int_0^{2\pi}|e^{...

Let $\pi(x)$ be the prime counting function. Knowing that $\pi(x) \sim \dfrac{x}{\ln x}$. How could you prove that $\pi(x) - \pi(N) \gt \dfrac{x - N}{2k}$ for all $x \geq $ some $X_0$?

I'm trying to show that $$(au+bv)^TA(au+bv)>v^TAv,$$ where $a,b\in\mathbb{R},a^2+b^2=1,$ $u$ is an unit vector, $v$ is a vector of norm one, A is a symmetric matrix where $A^T=A.$

By using the definition of kronecker delta $\delta_{KL}$, show that $\delta_{KL}$ is a Cartesian tensor, that is $$\delta ' _{MN}=L_{MK}L_{NL} \delta_{KL}$$ under the rotation $X_K=L_{MK}X' _M$. I don't really know where to start. What is the result differentiated with respect to? Please help.

Let $A$ be an $n\times n$ matrix. Let $c$ be the minimum column sum of $$\sum_{k=0}^\infty \left(\frac{A}{r}\right)^k.$$ In other words we can write this as $$c = \min_{1\le j\le n} \sum_{i=1}^n\sum_{k=0}^\infty \left(\frac{A}{r}\right)^k$$ Is there any way to show that $$c = \sum_{k=0}^\inft...

Can you tell the follow questions: how to find a dictance for the given two line l_1: 13x-y+6z+6=0; l_2: 26x-2y+12z-4=0

In a school election Jack got 2/9 of the votes, Rebecca got 4/9 of the votes. Rebecca had 42 more votes than jack. How many students didn't vote for Jack or Rebecca?

What should I do if I want to develop integral $\int\limits_{0}^{\infty} \frac{t^n}{(x+t)^{2n+4}} dt$ to $\int \frac{x^{p-1}}{1+qx)^{p+r}}=\frac{\Gamma(p)\Gamma(r)}{q^p\Gamma(p+r)}$. I write $\int\limits_{0}^{\infty}\frac{t^n}{x^{2n+4}(1-\frac{1}{x}t)^{2n+4}}dt$. But I don't see this form.

Need some help simplifying $$ie^{it}(1+e^{it})^n$$ where n is an integer, so I can integrate it between $0$ and $2\pi$

Given the series $\sum\limits_{n=1}^\infty a_n^2$ and $\sum\limits_{n=1}^\infty b_n^2$ converge. Show that the series $\sum\limits_{n=1}^\infty a_n*b_n$ converges absolutely. My idea so far: It's quite quite obvious that both given series converge absolutely So the Cauchy-Produc tells me that ...

First of all, why do we take the absolute value of both sides? What is the point/reason? and for (x_n - z)^2 , isnt it always positive? Second of all, do you call this a recurrence relation? for the last part: $e_[n+1]=C e^2_n$ but how can we conclude that the relations is telling us the accu...

can you help me with this math problem? I dont know how to start. $$\int \frac{dx}{2cos(4x)+5} $$

Suppose $a,b\in\mathbb{Z}$ that $a$ and $b$ are both positive and that $gcd(a,b)=d$. Prove that if $n\in\mathbb{Z}$ is positive, the $gcd(an,bn)=dn$. My attempt... $gcd(a,b)=d$ so $d|a$ and $d|b$. Then $a=dk$ and $b=dl$. Also let $d=as+bt$ the smallest positive linear combination. By multiplyi...

Hi can someone explain why Predicate logic cannot model knowledge which maybe retracted with time? is it because that there is a high uncertainty ?

Solve the system of equations in real numbers: $\left\{\begin{matrix}2a^3-3a^2-a^2b+4=0&\\ 2b^3-3b^2-ab^2+4=0 \end{matrix}\right.$

The problem I'm a bit lost at this problem. Tried to solve it by considering the dual or Farkas' lemma, but no luck. Would appreciate any help, Thanks!

Make sure to show all the steps you used to answer the problem and explain each step

Sketch the regions of the complex plane defined by (a) $Re \frac{1 + z}{1 − z}< 0$ and (b) $Re \space log\frac{1 + z}{1 − z}<0$ for part a i defined z as a + bi and got a final answer of $a^2 + b^2 >1$ Is that right and how to i do part b?

Can someone explain how can I understand in a simple way that a Clifford semigroup is a regular semigroup?

I have to prove that for every sentence ϕ, the sentence is valid on all epistimic models. I have no clue how to go about this. ¬Kaϕ ⇒ Ka¬Kaϕ Thank you

Q.) A committee composed of Jesse, Bianca, Ray, and Lily is to select a president and a secretary. How many selections are there in which Tyler is president or not an officer? I cannot understand how to solve such questions. By far I only know that since ordering does matter in this case it will...

Theorem (H16). If: $l$ and $m$ are parallel lines, $j$ is a common perpendicular intersecting $l$ at point A and $m$ at point B, and C and E are points on $l$ so that C is between A and E, Then: $L$(C; m) > $L$(AB); and $L$(E; m) > $L$(C; m). I wish I knew how to even begin this p...

Calculate $\displaystyle \lim_{x \to \infty} (\cos(t x) \sin(x)-t \cos(x) \sin(t x))$ How does one calculate the above limit?

$F = 3x$ for $x ≤ 1/3$, $-3/2(x-1)$ for $x > 1/3$ $F$ is chaotic if 1) Periodic points are dense, 2) $F$ is transitive, 3) $F$ is sensitive to initial conditions. $F$ is transitive if for every $x$,$y$ and any $ε > 0$ there exists $z$ such that $|z-x|<ε$ and $n$ such that $|F^n(z)-y|<ε$. $F$ is ...

How can I prove that $Aut(\mathbb {D}_n) \cong \mathbb {Z}_n \rtimes Aut(\mathbb {Z}_n)$ . Where $\mathbb {D}_n$ - Dihedral group . Can someone help me please? Thank you.

So I have a worksheet for Algebra II and Trig for pre calc that I'm having an issue with. The worksheet is on Logs which I am so awfuk at so excuse the scribbles and such; I'm having an issue on number two, any ideas + a really nice explanation would be awesome. I had the thought process you need...

I have an equation that looks like: $f = \frac{\texttt{E}\left [ x \right ]}{\texttt{E}\left [ y \right ]} - \frac{\texttt{E}\left [ w \right ]}{\texttt{E}\left [ z \right ]} $ where the observations of the variables $x$, $y$, $w$, $z$ have a lot of outliers. I was thinking of replacing $\textt...

please I want u guys to help me I am an adult and I don't know anything about maths as low as changing a number to standard form please I need your help you can refer me to a book, a process, or step by step tutorials

I have this question: Let (M,d) me a metric space. Define λ(x, y) ≡ d(x,y) / (1 + d(x,y)). Temporarily define: Bλ(p; r) ≡ {x ∈ M : λ(p, x) < r}, and B(p;r) ≡ {x ∈ M : d(p, x) < r}. Show that, for all ϵ > 0 there is a δ > 0 such that, for all p ∈ M, B(p; δ) ⊆ Bλ(p; ϵ); and that, similarly, for...

Fix a function f and a point $x_0$, and suppose that $f$ is differentiable at $x_0$. For all real numbers $b$, consider the family of lines $L_{b(x)}$ defined by $L_{b(x)} = f(x_0) +b (x-x_0)$. Note that $L_{b(0)} = f(x_0)$ for every $b$ and that $L_{b(x)}$ has slope $b$. Find all values of $b$ s...

In a normed vector space $X$, when can we say: $\lim\|x_n\|=\|\lim x_n\|$ and further, if $f\in X^{*}$, when can we say: $\lim fx_n=f(\lim x_n)$?

I have this problem: Define the following sets: $$a) A=\left\{x\in \mathbb{Z} \mid \frac{6x}{2x+1} \in \mathbb{Z}\right\},$$ $$a) B \cap \mathbb{N}, \text{where } B = \left\{ \frac{105}{2}, \frac{106}{3}, \frac{107}{4}, \cdots, \frac{n+104}{n+1}, \cdots \right\}, n \in \mathbb{N}$$ I don't real...

If you have a variant of the PCP where you only care if the length of the top string equals the length of the bottom string, how would you build a TM that decides that? Like obviously in the beginning you can reject if all of the strings in one list are longer than their counterparts in the other...

f(x)= cosh(x) +cos(x) -3 Let x* be the none negative root of f. Prove that Newton's Method applied to f converges quadratically to x*. Really confused where to start for a proof. I understand that if the second derivative of f(x*) doesn't equal 0 it converges precisely quadratic. But don't n...

can someone explain to me how to solve this system of equations for the parameters a,k and $\phi$? $0=a*sin(k*5-\phi)$ $0=a*sin(k*23-\phi)$ $0.3=a*sin(-\phi)$

"Simplify the radical expression completely. Use absolute value symbols if and only if needed. Rationalize all denominators and eliminate the negative exponents." $\frac\sqrt{3}{-81y^7 z^2}{3yz^5}$ (Please tell me if I did not format the mat correctly.)

I have gamma function Eulera. Can I write that $\Gamma (n+3)=\Gamma((n+1)+2)=\Gamma(n+1)+\Gamma(2)=n!+1$? And I'm not sure that $\Gamma(2)=1$?

I have the following problem: I have to describe up to isomorphism the semidirect product $$ \mathbb{Z}_4 \rtimes \mathbb{Z}_5$$ $$\mathbb{Z}_5 \rtimes \mathbb{Z}_4 $$ Thank in advance!

Let $G$ be an abelian group, and let $a\in G$. For $n≥1$, let $G[n:a] := \{x\in G:x^n =a\}$. (a) Show that $G[n: a]$ is either empty or equal to $αG[n] := \{αg : g \in G[n]\}$, for some $α∈G$. (Recall: $G[n]:=\{x\in G:x^n =1\}$.) (b) If $G$ is cyclic of order $m$, prove that: $|G[n:a]| = (n,m)...

Use Green's Theorem to evaluate this problem? Step by step solution? For a vector field F(x,y)=y^2/(1+x^2)i+2yarctan(x)j find a function f such that F(x,y)=∇f and use this result to evaluate c∫F⋅dr where C: r(t)=t^2i+2(t)j; 0≤t≤1. sorry guys I'm new to the coding.

I've confirmed that they don't intersect, but I'm not sure where to go from there.

I'm trying to use the solve function recursively on my TI-89 calculator. Minimal example to demonstrate the concept: solve( y + solve(x=2, x) = 3) Which should give y=1. Instead, it gives the rather unhelpful (y+x=y+2)=1. I think this is happening because solve returns an equality rather than ...

I am stuck trying to sole the following problem. I think it might have to do with Lebesgue differentiation theorem, but I have no idea where to start. Could someone help me out? Suppose $f \in L^1(\mathbb{R})$. Show for almost all $x \in \mathbb{R} $ we have $$\lim_{h\to 0} \frac{1}{h} \int_0...

How can I prove that $$ G = \oplus<a_i> \text{- cyclic group} \Leftrightarrow \text{End}(G) \text{- noncommutative} $$ Can someone help me please? Thank you.

This question caught my atention recently. Most (if not all) of the answers aproached the problem via a brute force attack. Surely there is a more elegant way to deal with this, given the inherent symmetry. My question is can this be generalised to an $n$ dimensional lattice? Clearly the placeme...

Let $(E,V)$ be a finite-dimensional euclidean affine space \footnote{$V=T(E)$ euclidean vector space and scalarproduct $(\cdot,\cdot)$} and $f\colon E\mapsto E$ a bijective affine map. Show, that there are real constants $0<c\leq C$ exists with $$c\cdot d(P,Q) \leq d(f(P),f(Q)) \leq C\cdot d(P...

How can I prove mathematically that a box with $x$ straight lines drawn through it, can be colored with only 2 colors.

Relating to the problem here: Show that the derivative of a function is not continuous, i.e. we have a function: $$g(x)=\begin{cases} x+2x^2\sin\left(\frac{1}{x}\right)&\text{ if }x\neq0\\\ 0&\text{ if }x=0 \end{cases}$$ Check that the function g is not of Class $C^1$ in any open interval around...