The integral is [math]\int_{p(x) > 0}p^{-\lambda + 1}(x) \left| \ln p(x) \right|^k dx[/math]. Assumptions: [math]\lambda > 0[/math] [math]\int_{p(x) > 0}p^{-\lambda + 1}(x) dx < \infty[/math] [math]k > 0[/math] [math]p[/math] is a probability distribution This comes from the first few parag...

Is it possible to find all the induced subgraphs in a given graph? If so, how?

Given $A,B\in\Bbb K^{n\times n}$ where $\Bbb K$ is a field when is there a matrix $R$ such that $$B^{-1}A=R\otimes R$$ $$RR'=I$$ where $'$ refers to transpose?

Are there any recent/modern developments in research in classical algebraic topology? Are there still any major open problems?

Is $|A \setminus B| = |A| - |A \cap B|$, where $A$ and $B$ are finite sets, true? I have been unable to prove this or find a good reference on cardinality of set differences. The only reference I found was ProofWiki, and the only case they consider is when $A \subseteq B$, which is not necessaril...

Describe a polynomial-time transformation TRAN that takes an instance of SAT and transforms it into an instance of NAE-SAT (the problem where, given a Boolean expression in CNF form, you are asked, does there exist a solution that makes it true, and also makes at least one literal in each clause ...

Hi guys I have a statement that may be true or false. I believe it is false but I cannot prove or even show by example my claim. If $U$ is open in $R^N$ then $U= \cup _{n=1} ^\infty U_n$ where each $U_n$ is open in R. I think this is false because I know that even for $X \times Y$ not every ope...

If $\{A_n,n\geq 1\}$ are independent events show that $$\frac{1}{n} \sum_{i=1}^{n} 1_{A_i}-\frac{1}{n}\sum_{i=1}^{n}P(A_i) \rightarrow^{P} 0$$. Proof so far $P(|\frac{1}{n} \sum_{i=1}^{n} 1_{A_i}-\frac{1}{n}\sum_{i=1}^{n}P(A_i)|\geq \epsilon)=P(|\frac{1}{n} \big( \sum_{i=1}^{n} 1_{A_i}-\sum_{i=...

Compute the following: a) 2^208 (𝑚𝑜𝑑 53) b) 2^288 (𝑚𝑜𝑑 73) c) 7^19 (𝑚𝑜𝑑 28) Here is what I did so far: a) ∅ (53) = 52 (2^52)^4 => (1)^4 => 1 b) ∅ (73) = 72 (2^72)^4 => (1)^4 => 1 c) ∅ (28) = 12 7^12 * 7^7 mod 28 1 * 7^7 mod 28 => 7 Just wondering if what I've worked out is co...

Give an example, if possible, of two polynomials $f(x)$ and $g(x)$ in the indicated rings such that the degree of $f(x)· g(x)$ is not equal to the sum of the degrees of $f(x)$ and $g(x)$. If not possible, explain why not. a) $Z_8[x]$ b) $Z_7[x]$ I am very confused as to how to begin this prob...

if $10 is 0%, what percentage gain would $30 be. Having trouble coming up with the way to visualize this in cross multiplication

Find the upper and lower sums for $f\left(x\right)=\begin{cases} 1 & 1\leq x \leq 2\\ 2 & 2<x\leq 3\\ 3 & 3<x\leq 4 \end{cases} $ on [1,4] Attempt $$I_{1} = [1,2-a] \ with \ sup(f) = 1 \ and \ inf(f) = 1$$ $$I_{2} = [2-a,3+a] \ with \ sup(f) = 2 \ and \ inf(f) = 2$$ $$I_{3} = [3+a,4] \ with\ su...

Suppose I have: $B = \mathbb{R}^{n \times n}, w, z \in \mathbb{R}^n$ $f(w,z) = w^TBw - 2z^TBw + z^TBz$ Suppose $B$ is nice, invertible, symmetric, whatever helps in doing the completing the square. How can we manipulate this step by step as in the scalar case? to get $j^TBj$, where $j$ is some...

I have two lines in space. Each defined by two equations: line a: x + y -z = -1 x + y + z = 1 line b: 2x + 2y -2z = 3 x + 2y -z = 1 How would I go about finding their relative position preferably using matrices? Thanks a lot!

Does anyone understand how to do this? I'm not great with math at all, and I'm just lost. http://imgur.com/WiiLB5p

Just going through some graph theory concepts and I have two elementary questions which must be pretty trivial (but sometimes what seems trivial to me turns out to be wrong, so I'd be happy if someone could confirm that the following are indeed true): (1) A periodic graph cannot self-loops - ...

Construct the local phase portrait for the system x'=-y+xy y'= x+ (x^2 -y^2 )/2 and show that it is structurally unstable.

Let f, g ∈ C[0,1] with f(x) < g(x) for all x ∈ [0,1]. (i) Prove that there is a polynomial p(x) so that f(x) < p(x) < g(x), x ∈ [0,1]. (ii) Prove that there is an increasing sequence of polynomials {pn(x)} so that f(x) < p_n(x) < g(x), x ∈ [0,1], and p_n → g uniformly on [0,1]. I think in these ...

Is it possible to write the equation $x+y\sin{x}=1$ in terms of $x=f(y$)? I can easily solve for $y$, but am not sure how to approach $x$.

Damping is a way of taming a nonconvergent iteration to get it to converge. Given a splitting matrix $M$, which gives the iteration $$x^{k+1} = x^{k} + M^{-1}r^{k}$$ where $$r^{k} = b-Ax^{k}$$ the corresponding damped iteration with damping factor $\omega < 1$ is defined by $$x^{k+1} = x^{k} +\...

I am reading Murty & Esmonde's Problems in Algebraic Number Theory and was wondering if anyone can offer some clarification on the proof of this theorem: Let $p \in \mathbb{Z}$ be prime and $\mathfrak{p}$ be a prime ideal of $\mathcal{O}_K$ and $\mathcal{D}$ the different of $K$. If $\mathfra...

I would like help just drawing this figure. I understand the differences between the complete quadrilateral and quadrangle but I fear I am not going to draw this correctly in order to complete the proof. Prove: If a complete quadrangle PQRS and complete quadrilateral pqrs are so situated that t...

I'm writing an app in Nim to search for curious integer identities such as those ones listed on:enter link description here where it says "curious identities derived using modular theory". The program doesn't prove the identity but should only give a recommendation of what identities for mathema...

The question states "Integrate $f(x, y, z) = x + \sqrt y -z^2$ over the patch $C = C_1 + C_2$ and they proceed to give $C_1$ and $C_2$ as $r_1(t)$ and $r_1(t)$. I've been dealing with 2 forms of line integrals so far, integrations with respect to $dS$ where $dS = ||r'(t)||dt$ and integrals with $...

Let $V$ be a finite dimensional vectorspace over a field $\mathbb{ F}$. It's easy to show that if $U$ and $V$ are subspaces of $V$ then $U \cap V$ is a subspace. But what if there are an infinite number of them? I have seen sometimes in pure maths that you take an arbitrary finite collection of s...

The problem is: $$\sum_{n=1}^\infty nx (\prod_{k=1}^n\dfrac{sin^2(k\theta)}{1+x^2+cos^2k\theta}) $$ Is that "$\prod$" the same as "$\sum$"? I don't know how to start. What I tried doing was focusing on the $\prod$ equation. Which I concluded is convergent by comparing $$\dfrac{sin^2(k\theta)...

I am wondering if I have interpreted the language correctly in the following question The force of interest at time $t$ is given by $\delta(t) = 0.05-0.005t$ for $\leq t < 5$ and $\delta(t)=0.02+0.001t$ for $5 \leq t \leq 10$. Calculate the present value of an annuity of $\$1000$ payable annu...

It should be a primary question,but only indefinite integral version I can find in wiki.

Let f be a function defined (piecewise) on the interval [1, 2015) by the formula: f(x)= (2/k(k + 2)) , for x ∈ [k,k+1), where k=1,...,2014. Find the area of the region bounded by the graph of f, the x-axis and the lines x = 1 and x = 2015.

Given that f is analytic, under what conditions is $g(z)=\overline{f(z)}$ analytic? Does this explanation make sense? : $g'(z)=lim_{h\rightarrow 0} \dfrac{g(z+h)-g(z)}{h}=lim_{h\rightarrow 0}\dfrac{\overline{f(z+h)}-\overline{f(z)}}{h}=\overline{f'(z)}$ Therefore, g(z) is analytic when $g'(z)=\...

Consider the set A = {a+b ● √5 | a,b ∈ | R, |R - the set of real numbers } Let be defined as follows: (a + b ● √5) ⊕ (c + d ● √5) = (a + c) + (b + d) ● √5 (a + b ● √5) * (c + d ● √5) = (a ● c + 5 ● b ● d) + (a ● d + b ● c) ● √5 c θ (a + b ● √5) = c ● a + c ● b ● √5 i) Is a group? Explain ii)...

I'm having difficulty in approaching this problem and coming up with a solution. I'm not sure where to start in answering this question.

Give an example of a natural number $n > 1$ and a polynomial $f(x) ∈ \Bbb Z_n[x]$ of degree $> 0$ that is a unit in $\Bbb Z_n[x]$. I am trying to understand how units work in polynomial rings. My book doesn't really define it and I need a bit of help with this.

I am currently reviewing some examples of nilpotent groups. Why isn't $GL(2,\mathbb{Z})$ nilpotent?

Define: $$R_k(n) = R_k(n-1) - R_k(n-k-1)/n$$ With $R_k(n) = 1$ for $n \leq k$. Is there a closed-form in terms of combinatorics or harmonic numbers?

On the first paragraph of the Wikipedia page regarding the Leibniz integral rule, we get the expression $$\frac{d}{dx} \int_{y_0}^{y_1} f(x,y)dy = \int_{y_0}^{y_1} f_x(x,y)dy $$ and it says that "provided that both $f$ and $f_x$ are continuous over some region $[x_0,x_1]X[y_0,y_1]$ Can we r...

In Team Fortress $\mathbf{I}\mathbb{I}$ (Team Fortress $2$), there are 9 classes, and soldier is the best. Each of the 9 classes has (for simplicity) 5 types of weapons for each of 3 weapon slots. Additionally, there are up to 12 players on each team, and there are 2 teams. Also, Valve allows you...

I know how to solve this problem "normally" (using vector methods) but I don't know how to solve it using Lagrange multipliers.

I was going through the introduction to Lebesgue theory in Baby Rudin, where the property was given that: $\mathcal{E}$ is a ring, but not a $\sigma$-ring. $\mathcal{E}$ here represents the family of all elementary subsets of $\mathbb{R}^n$, $n$-dimensional Euclidean space. An elementary s...

Change to uv variables, find the Jacobian and evaluate ∬e^(-(xy)/2)dA, use transformation x=sqrt(v/u) and y=sqrt(uv). R is a region in the first quadrant bounded by y=x/4, y=2x, y=1/x, and y=4/x. I found the Jacobian, which is -1/2u. u ranges from 1/4 to 4, and v from 1 to 2. Doing the actual in...

Let $P(x) = a_nx^n + ..... a_1x + a_0$ Is there a general technique for integrating $$\frac{P(x)}{\sqrt{Ax^2 + Bx + C}}$$ Similarly is there a general method for $$\frac{1}{P(x)\sqrt{Ax^2 + Bx + C}}$$ And finally is there a general method for $$\frac{P(x)}{Q(x)\sqrt{Ax^2 + Bx + C}}$$ Wh...

I've been stuck on this for a while: The number of simple digraphs with |V| = 3 and exactly 3 edges is (a) 92 (b) 88 (c) 80 (d) 84 (e) 76 The answer is (d) though i don't know how to arrive there.

i would like to write if function as mathematics function By if {y(i)=1,0,1}; How i can write it.

So this problem statement says "Use analytical or graphical techniques to find the positive and the negative limit sets of the orbits through the listed initial points." I have a system $$x^{'}=y$$ $$y^{'}=x$$ and the points $(1,1), (1,-1), (1,0)$ I have read through my notes and book and am st...

Prove by contradiction that for a transitive relation ℛ on A, ℛ2 is also transitive. In order for a relation to be transitive it must $aRc$ ^ $bRc -> aRc$

I noticed something about the Collatz Conjecture, (I was literally obsessed with trying to prove it). I of course have NO intention of trying to prove it, clearly it is beyond my reach and I hope not to offend anyone by what may be a nonsensical observation, but I was a bit curious. This is wha...

I'm having trouble understanding how to find Var(X) and Var(Y) for the correlation coefficient between Z and W. Z = 2X + Y W = X - 3Y I have Cov(Z, W) = Cov(2X + Y , X - 3Y ) = Cov(2X, X) + Cov(2X, -3Y ) + Cov(Y , X) + Cov(Y , -3Y ) = 2Cov(X, X) - 6Cov(X, Y ) + Cov(Y , X) - 3Cov(Y , Y ) ...

In a sequence of integers, $t_1, t_2, \ldots; t_{n+3} = t_n+t_{n+1}-t_{n+2}$ for $n \geq 1$. If the first three terms, in order, are $1$, $3$, and $6$, what is the $2006^{\text{th}}$ term? There was an answer to my question which said that $t_{2n+1} = 1+5n$ and $t_{2n} = 8-5n$. I want to prove t...

let $f$ be a step function, $ f:\left[0,2\right]\longrightarrow\mathbb{R} , f\left(x\right)=\begin{cases} 1 & 0\leq x<1\\ 3 & x=1\\ 2 & 1<x\leq2 \end{cases} $ intgerate ${\displaystyle \intop_0^2 f\left(x\right)\,dx}$ using the $U\left(f,P\right),L\left(f,P\right)$ definition. attempt Let...

Given a trigonometric function, can you tell if it is Constructible? If I have $\sin(n)$ for example, how would you tell if it is of the constructible set of numbers?

Find a recurrence relation that counts the number off-diagonal elements of an n + 1 x n + 1 matrix. Solve this recurrence relation for an expression of the number of off diagonal entries as a function of n.

I have an equation $$\sum_{k=2}^7{7\choose k}{0.01^k}(1-0.01)^{7-k} = 1-(0.99)^7 - 0.07(0.01)(0.99)^6 \approx 0.002031$$ I don't know what property the teacher used to quickly transform the summation to two simple equations. Can someone please give me a hint? P.S. This formula is used to calcu...

I am Having a bit of difficulty setting up the bounds for this question. So far I have got: -1 <= x <= 1-y^2 -sqrt(2) <= y <= sqrt(2) -sqrt(2) <= z <= sqrt(2) Can someone please confirm if I am on the right track?

Let $S$ be a complex surface, and let $C \subset S$ be an immersed complex curve with a transverse self-intersection at point $P$. Let $\tilde{C}$ be a curve obtained from $C$ by smoothing the intersection at $P$. Is there a formula for computing the genus $g(\tilde{C})$ in terms of $g(C)$? -- A ...

I'm proving but get many problems $f$ is continuous and open mapping if and only if $\overline{f^{-1}(B)}=f^{-1}(\overline{B})$

Let $f:[a,b]\to\mathbb{R}$ be differentiable.Assume that there exists no $x\in[a,b]$ such that $f(x)=0=f'(x).$prove that the set {$t\in[a,b]:f(t)=0$} of zeros of $f$ is finite.

Prove by induction that 3n +7n - 2 is divisible by 8 for all positive integers n.

I am trying to figure out if this sequence will either diverge or converge. My initial thought was that any value of n to sin will be less than 1. enter image description here

Write a context free grammar for the following: L={a^{m} b^{n} c^{k}| n,m,k >= 0, n>= m+k} I am having some trouble writing the above cfg from the language. So far, I have S-> S1 S2 S3 S1 -> a S1 S2 S1 -> \Lambda S2 -> b S2 S2 -> \Lambda S3 -> S2 c S3 S3 -> \Lambda

This is an indeterminate form and I think I should use the fact that x-2/sin(x-2) = 1 but idk how to do that.

\lim_{a \rightarrow \infty}\frac{1}{a}\int_{0}^{\infty}\frac{x^2+ax+1}{1+x^4}.tan^{-1}(\frac{1}{x})dx=?

I'm trying to design the algorithm and answer these two questions but can't produce a solution. I've tried drawing the pictures for the graph as well.

I mean to say does the function F(x), the antiderivative of f (x) have any sort of meaning if you're not taking the difference of it between two limits a and b.

Define: $$R_k(n)=R_k(n−1)−R_k(n−k−1)/n$$ With $R_k(n)=1$ for $n≤k$. I am trying to simplify this to a closed form (binomial coefficients, harmonic numbers, etc). Is it possible?

The 24th problem in the first chapter of Spivak's Calculus has to do with proving that the placement of parentheses in a sum is irrelevant. Let $s(a_1, a_2,... a_{n})$ denote some sum formed from $a_1, a_2,...a_n$. For example, if $n=5$, $s(a_1, a_2,... a_{5})$ may represent $(a_1 + a_2) + (a_3 ...

Let $A,B,C\in M_n$. Is this true that $Tr(ABC)=Tr(ACB)$?

I am trying to prove Lagrange's Group Theorem that given the group G with the subgroup H then $|H|\ |\ |G|$ and if $\{g_i|i\in I\}$ is a complete list of representatives of the cosets of H, then$\ |G|=|I||H|$ where I is the index of H in G. I was wondering if this proof is correct. $$\begin{align}&

I have a problem. I am trying to find sequence $x\in l^2$, witch satisfies: $(\forall p<2) (x\not\in l^p)$. Is it possible? Why?

I'm still new to MATLAB I'm trying to calculate the gamma function in MATLAB. My code is as follows: int(power(z,2.0*b)*exp(-z),z,0,inf) My output ends up being (4851*gamma(49/50))/2500. I have to input my output in order to get the approximate solution: 1.9636. Is there a way to output the ...

I would separate this limit but 1/x is an indeterminate form, which is where I'm stuck.

So I am doing a project and was given the group Z2 x Z9 and was told it was a noncyclic group. However, when finding the subgroups (1,1), (1,2), (1,4), (1,7), and (1,8) all generate the group. Am I doing it wrong or is the group actually cyclic?

In this signal processing paper(Schörkhuber, Klapuri, Sontacchi) its referred to "rotating (shifting) the rows of M^{CQT} up- or downwards.". Any idea what this means?

Can someone please hint me as how to solve this category of questions as I am novice in it.

Given these 2 regression equations how do I compute mean and find r. X=-0.4Y+6.4 Y=-0.6X+4.6 when I rearranged the equations, I solved for X and Y hence X=6, Y=1 How do I get mean and r using the the two values.

I know in general that AUB = A + B - (A∩B) but then how to prove it in general using cardinality and sets?

This is easy for normal infinity, but being negative makes taking the root impossible.

If we have 6 balls, 2 yellow, 3 red and 1 green. What are the probabilities of drawing 2 yellow balls in 2 goes? My answer is 1st go is 1/6. Because of no replacement, 2nd go is 1/5. Now do I add the two or multiply the two. Add gives 11/30, and multiply gives 1/30

Each term of $x^m$ is also a term of $x^{ln(m)}$, so interval of convergence must be smaller than $(-1,1)$. So option (a) is correct since $e>1$, but how to explicitly arrive at interval $(0,1/e)$. please suggest.

Evaluate the following sum $\space\space\sum\limits_{n=1}^{\infty}\left(\frac{1}{n}\sum\limits_{m=1}^{n}\frac{1}{m}\right)^2$.

Let $S:= \{1-\cfrac{1}{2}n: n\geq0\}⊂[0,1],$ and let $T:= \{m+a_n:m∈ℕ, an∈S\}⊂[0,\infty)$. Show that $T$ is countable. Please someone help me with this

Let $f$ be analytic function defined on the open unit disc in $\mathbb{C}$. Then $f$ is constant if $f\left(\frac{1}{n}\right)=0$ for all $n\geq1.$ $f(z)=0$ for all $|z|=1/2$ $f\left(\frac{1}{n^2}\right)=0$ for all $n\geq1.$ $f(z)=0$ for all $z\in (-1,1)$ I used Identity theorem and conclude ...

I'm a bit confused. I think that it's uniformly convergent on [0,1]. My proof: Let epsilon be greater than 0. Then, N exists in the natural numbers such that N*epsilon > 1. Then, for all x in [0,1], |fn(x) - f(x)| = x^n/n < 1/n < epsilon. But, for x in (1, inf), I'm not sure what to do. It doesn...

Can someone explain where the map in this isometry came from? First the question is: By thinking about how a circular cone can be 'unwrapped' onto the plane, write down an isometry from $$\sigma(u,v)=(u\cos v, u \sin v, u), \ \ \ \ \ u>0, \ \ 0<v<2\pi$$ (a circular half-cone with a straight lin...

Countable set problem. Detailed explanation would be highly appreciated.

$f$ is a rational function $S^1\to S^1$, where $S^1$ is the one point compactification of $\mathbb R$. Is $\infty$ always a regular value of $f$?

How can I find an isometry from the cone $$\sigma(u,v)=(u\cos v, u \sin v, u), \ \ \ \ \ u>0, \ \ 0<v<2\pi$$ to the $xy$-plane? How should I begin?

Show that any bounded harmonic function on the whole plane must be a constant function.

Identify the type of indeterminate situation that justifies using l'Hopital's rule, then use l'Hopital's rule to find the limit. lim x->􀀏0 (e^(8/x) - 1x)^(x/2)

Suppose Newton's Method is used with an initial guess Xo that lies at a critical point (a, b), where b does not equal 0. What happens to X1 and later approximations? Give reasons for your answer.

integral of $(e^x-e^-x)/(e^x+e^-x)^2$

$L:\mathbb{R}^3\to\mathbb{R}^3$ is a linear map and $S\subset\mathbb{R}^3$ is a regular surface invariant under $L$, that is $L(S)\subset S$. My first doubt is what does $L(S)\subset S$ actually mean? Does it mean the set of points of $L(S)$ is a subset of the set of points of $S$? My next que...

Let $B_1 \to X_1=\mathbb P^n$ be the blow-up of $\mathbb P^n$ along a linear subspace $\mathbb P^k$. Let $B_2 \to X_2$ be the blow up of a quadratic cone $X_2=\{x^2=yz\} \subset \mathbb A^3$ in its vertex $p$. I need to calculate $K_{B_i}$. As any blow up is an isomophism ouside of the excepti...

The title says it all, please help me.In graph theory, are undirected graphs assumed to be reflexive? What are the assumptions about symmetry and transitivity?

If I want to find P(X 1)can I write it as P(X 2) if I know X

I am trying to evaluate the following using Stokes' Theorem, but I am running into issues setting it up... $$\iint_Scurl\vec(F)\cdot d\vec(S)$$ where $$\vec(F)(x,y,z)=\langle sin(x^2y),xy,xyz^2 \rangle$$ S is a cone $$z=\sqrt(x^2+y^2) , 0\leq z\leq 3 $$ If I parametrize it, I end up with $$9=x...

I am provided with a value m , such that I have numbers from 1 to m , and another number n ( n<=m) such that I can choose any n numbers from given m numbers. Now I need to calculate total possible combinations such that the maximum of the selected n numbers is strictly less than the sum of remai...

How do I solve $\frac{dx}{dt} = a + (b - 1)x^2$, where $a$, $b$ are constant? I've tried pushing symbols around, but to no avail.

Let $\cal{U}$ be the unit group of group ring $\Bbb{Z}G$ then the Normalizer Problem (NP) states that $N_{\cal{U}}(G)=G\frak{z}$ where $\frak{z}=\cal{Z(U)}$. Now why (NP) is equivalent to saying that $Aut_Z(G)=Inn(G)$ where $Aut_Z(G)$ denotes the automorphisms of $G$ induced by conjugation with ...

So the question asked to find the extreme values of the function f(x,y,z)=e^(xyz) `under the constraint 2x^2+y^2+z^2=24. enter image description here I'm a bit lost on what to do now because I tried isolating for one of the variables and sub it back into another one of those 4 equations. But, i...

I found it somewhere and copied in my notes, but couldn't find the source and it seems to me a pretty strong statement... does anybody knows if it's true and can sketch a roughly proof?

Verify the identity: P(AUB)=P(A)+P(BA') I could verify using the Venn diagram. Is there another way?

Let $\mu_n=\{\zeta \mid \zeta^n-1=0\}=\{1,\zeta _n,\zeta _n^2,...,\zeta _n^{n-1}\}$. We call generator an element $\zeta_n^k$ when $\gcd(k,n)=1$. Why those number are such important ? I think that $\left<\zeta _n^k\right>$ is a subgroup of $\left<\zeta _n\right>$ if and only if $(k,n)=1$, and tha...

I am currently stuck on the following question: My current thinking is to assume g $\in$ $\Omega$(f), use the definition to somehow show that ccc is c and therefore the definition of $\Omega$ holds true for $g^3$ as well. But I can't seem to put together a chain of equalities to show this. Am...

I use sliding window average in my calculations: average for the last N elements and average for the N elements before that. What would be the correct mathematical notation to describe this?

I have been raking my brain for the solution to this problem for a while, and have searched as best I could for an answer. I have to solve for $x$ in the equation: $$y= \frac{x-1}{2x+3}$$ Since $(x - 1)$ is not a factor of $(2x+3)$ and I can't figure out how to seperate out x once I reach the st...

I have been taught that "inverting a square matrix with small determinant is numerically unstable because it is close to singular"? Is this right opinion?

I have the nfa problem below which I dont quite understand. I understand the basics of dfa's/nfa's but have not seen the syntax used below before, and dont quite understand how to create the state transition diagram from information. If anyone could help me understand the problem below it would b...

I would like to find a source book or article which discuss the Diophantine equation $$ x^2+y^2=z^2+t^2 $$ for which $x,z$ are odd positive and $y,t$ are even positive integers. Any brief explanation is also welcome and appreciated. Thanks!

Find all functions $f$ such that $f\left(\frac{1}{x}\right)+(x+1)f(x)=1,\space x\neq0$.

Let f be a C^1 vector field in an open set E Ϲ R^2 containing an annular region A with a smooth boundary. Suppose that f has no zeros in A', the closure of A, and that f is transverse to the boundary of A, pointing inward. (a) Prove that A contains a periodic orbit. (b) Prove that if A contains...

Could you tell me the way to solve this , but not the answer ? lim x^ (x^х - 1) , x-> 0 Thank you.

is $|ln|x||$ differentiable for all $x$ is is defined and continuous? I can see that on the graph that it is not differentiable at $-1$ and $1$, but how can I prove it?

From Wikipedia, a GCD domain is an integral domain in which every pair of elements has a GCD. Let us consider some polynomial quotient ring $R=K[X]/(pq)$ where $K$ is a field and $p$, $q$ are (irreducible) polynomials. Then $R$ is not an integral domain (since $pq=0$ in $R$), so cannot be a GCD ...

How to solve the following double infinite series diagonally? I tried it but could not get the answer.

Let $\mathfrak{g}$ denote the Lie algebra of a Lie group $G$. The adjoint representation of $G$ is defined as the function $Ad_g:\mathfrak{g}\rightarrow\mathfrak{g}$ that maps each $x\in\mathfrak{g}$ to $Ad_g(x)=gxg^{-1}$, for some $g\in G$. why does $gxg^{-1}$ fall into $\mathfrak{g}$ ?

I have to solve the following equation for x:$$ \sum_{k} (x'A_kx - 1)(A_kx)=0$$ $A_k$ is a matrix and $x$ is a vector. Can anyone give me some suggestion on how to solve this problem?

Let B = {0,1,2,3,4} and let {0},{1,3,4},{2} be a partition of B that induces a relation Q. Find the distinct equivalence classes of Q.

Got an interesting problem from a friend. How many zeroes does $n!$ end in when written in base $n$? For every factor of $n$ in $n!$, I know that there will be $1$ $0$ added. However, I'm not really sure how to proceed from here.

keeping in mind the similar question as above I try to show that $$\sum _{n=1}^{\infty } \sum _{k=1}^{\infty } \left(\frac{4 \sin ^2\left(\frac{\pi n}{k x}\right)}{x}-\frac{2}{k x}\right)=\frac{2-x \coth \left(\frac{x}{2}\right)}{2 x}$$ but it seem a hard work...any clue thanks

How do we get Wallis Formula $$\frac{\pi}{2}=\lim_{l\to\infty} \prod_{j=1}^{l+1}\frac{(2j)(2j)}{(2j-1)(2j-1)} $$ from $$\lim_{n\to\infty}\frac{(n+1)^2}{n+\frac{3}{2}} \bigg[\frac{\Gamma(n+1)}{\Gamma(n+\frac{3}{2})}\bigg]^2=1$$

I have exponential form $$ je^{-j\pi/2} $$ I want to convert this to polar form $$j(cos\pi/2 + j sin \pi/2)$$ is it correct?

I have read the answer for graduate-level Algebra background and all answers in stackexchange and mathoverflow discussing Homological Algebra textbooks. But none of them directly answers my question, and not all of the lauded textbooks indicates the prerequisites clearly. My background: undergr...

Show that $x^4+x^3+x+1$ is irreducible in $k[x]$ for any field $k$. I have no idea how to prove it other than writing $p(x)q(x)=x^4+x^3+x+1$ and trying to figure out what happens with the coefficients of $p$ and $q$ (supposing $p,q$ both degree 2, or $p$ degree 3 and $q$ degree 1). But I was...

I am trying to show, that Let $\Omega \subset \mathbb{R}^n$ be measurable and $f\colon \Omega \rightarrow [0,\infty)$ Lebesgue-integrable. Show that: $$\int_\Omega f(x)dx=\int_0^\infty \lambda(\{f>s\})\; ds.$$ Can somebody explain to me, why for $(x,s) \in \Omega \times \mathbb{R}$ is $\chi...

I'm studying abstract algebra, and I'm confused in quotient and polynomials. The exercise that I feel confused was this; Let $E$ be an extension field of a field $F$ and let $\alpha\in E$ be transcendental over $F$. Show that every element of $F(\alpha)$ that is not in $F$ is also transcendent...

I'm wondering about an endomorphism. If we have an endomorphism $f$ in $\mathbb{R^3}$ such as its matrix in canonical basis is \begin{pmatrix} 1 & 1 & 0 \\ -1 & 2 & 1 \\ 1 & 0 & 1 \\ \end{pmatrix} and $V=Vect{(1,1,1)}$, $Z=Vect((1,0,0),(0,1,-1))$ We see $V$ is invariant by $f$ and $...

Could you help me with this one? Thanks. The answer should be 1, somehow. I tried everything I know, but I couldn't solve it. http://i.stack.imgur.com/LuyX0.png

Could I expect that the Pollard rho method 1. x ← 2; y ← 2; d ← 1; 2. While d = 1: 1. x ← g(x) 2. y ← g(g(y)) 3. d ← gcd(|x - y|, n) 3. If d = n, return failure. 4. Else, return d. should works for Gaussian integers? With the minor changes due to gcd?

I'm struggling with the concept of recursive convolutional codes. Say we have the generator matrix $$\begin{pmatrix} \frac{D}{1+D^3} & 1 & 0 \\ \frac{D^2}{1+D^3} & 0 & 1 \end{pmatrix}$$ which means the first column is $[D+D^4+D^7+ \dots; D^2 + D^5 + D^8 + \dots]$. Why doesn't this give the sys...

Hint for solving the following integral: $$\int \frac{2r}{|1-r^{2}|^{\frac{3}{2}}}dr$$? My attempt: I tried putting $u= \frac{1}{\sqrt{|1-r^{2}|}}$ then $du = \frac{r}{(1-r^{2})\sqrt{|1-r^{2}|}}$, but not working. Can someone help me in solving this question? I would be happy to get some hints.

How can I prove that $2^{32}$+1 is composite number using Miller-Rabin primality test.I can't find a solution which verify the hypothesis of theorem.Thank you!

I can't work out the way the solution was found. Any idea?

Suppose that we have two data points which tell us about the output of some function f(x): (0, 50) (10, 150) We know that the function is quadratic (so it's something like ax2 + bx + c). The question is: how do we estimate the parameters a, b and c in such a way that the resulting function ma...

If I was to use the Green's function method to solve the ordinary differential equation $$Ly=f(x)$$ Would $L$ have to be Hermitian, either way please can you explain why?

I was given a sum s, then I expressed it using sigma notation as you can see below. Now, I am supposed to find the closed form values for the said sum as shown in eq.2, but I am stuck at eq.1. I don't know how to deal with the highlighted part, am I forgetting something or did I make some mistake...

I would like to fin the residue of the function $$\frac{g(z)}{cos^{2}z}$$ at $z_{n}=(n+\frac{1}{2})\pi$, where $g$ is analytic. I tried the Limit formula for higher order poles and it fails. What other method can I use?

Consider the equation $yu_x − uu_y = x$. (a) Write a parametric representation of the characteristic curves.

. Suppose we have x1,··· , xn a random sample from a Geometric distribution, say Geom(πx). We also have a second, independent, random sample y1,··· , ym from Geom(πy). Further we have the following set of hypotheses, H0 : πx = πy against H1 : πx 6= πy. (a) Write down the joint likelihood under ...

Question: Find a (simple) $f(n)$ so that $\lim\limits_{n \rightarrow \infty} \frac{n \sum\limits_{k=1}^{n} \frac{1}{k}}{f(n)} = 1$ My Attempt: I know the answer, by using Mathematica, is $f(n) = n \cdot ln(n)$. I, however, can't find a way to prove this. I've only tried $$\lim\limits_{n \righta...

So when looking at functional sequences in terms of uniformally convergent, I am struggling to translate the definitions into examples. I understand that $f_n=\frac{x^2}{n}$ is the sequence $f_n$={$x^2$, $\frac{x^2}{2},$ $\frac{x^2}{3}...$} However how does $f$ relate to $f_n$. If i have a defi...

How can I calculate $$\int_{\lvert z-i \rvert = 2} \frac{1}{\cos^2(z)}$$ My first attempt was to use residuals: I have figured out that $z=\pm \pi/2$ is a pole of order $2$, so I tried to apply this formula, but the limit diverges. I have also tried to write $f$ as Laurent-series to get $a_{-1...

I don't get this, I am asked if the following $X_t$ is a brownian motion or not. $Z$ is a standard normal variate. $X_t=\sqrt{t}Z$. I s$X_t$ a Brownian motion? Answer is apparently no and one of the reasons is that the variance of $X_t-X_s=$ is $(\sqrt{t}-\sqrt{s})^2$. But I get $t-s$ as th...

It seems obvious, but somehow I find it difficult to prove it in an arbitrary metric space. How can I proceed in this case? Thank you for any help.

Let $M$ be a compact manifold of dimension of $m \geq 2$. Show that there exists an injective immersion of $\mathbb{R}$ in $M$, whose image is not the trajectory of any flow on $M$.

How do I prove that $$\sum_{n=1}^{\infty}\frac{1}{n^s}$$ diverges fos $s<1$, by estimating its partial sums?

How do I integrate: $\int_{\mathbb{R}} (S_t - K)^+ \phi(t) dt$ where $\phi$ is a normal density and $S_t$ is a geometric brownian motion? I know my answer should be $\Phi(d_1)$, where $\Phi$ is the normal CDF and $d_1$ is the $d_1$ appearing in the Black-Scholes price of a European option.

For Example, Can we assume that the incentre of a pedal triangle is the orthocentre of the triangle on which the pedal triangle is constructed in all cases?

Consider the following recurrence relation $T(1)=1$ $T(n+1)=T(n)+⌊\sqrt{n+1}⌋$ for all $n≥1$ The value of $T(m^2)$ for $m≥1$ is $(m/6) (21m – 39) + 4$ $(m/6) (4m^2 – 3m + 5)$ $(m/2) (m^{2.5} – 11m + 20) – 5$ $(m/6) (5m^3 – 34m^2 + 137m – 104) + (5/6)$ My attempt : I've used counter exam...

How can i proof that $x^5 + 7x^4 + 2x^3 + 6x^2 - x + 8$ is irrudicible in $\mathbb{Q}$? I can't use the Eisenstein's criterion and I tryed to put this polynomial in $\mathbb{Z}_3$ and $\mathbb{Z}_5$. Can you give me some advice please?

Suppose $A$ is a 4 by 4 matrix with the characteristic polynomial $P_A(\lambda) = (\lambda-2)^4$ with the minimal polynomial $m_A(\lambda) = (\lambda -2)^2$. This tells me that that the Jordan form with respect to the eigenvalue 2 (the only eigenvalue) is $4 \times 4$ size matrix and the largest ...

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

Let $f \in \mathbb Z[x]$ be irreducible and suppose $f(x)$ has two roots in $\mathbb C$ with product $1$ . Then is it true that degree of $f$ is even ?

I need to show whether or not the maximum of two martingales is also a martingale. Originally, I thought yes. But supposedly the answer is no. So as a counter-example, let $U_i$ be $iid$ $unif(0,1)$, $X_0 = 1$, and $$X_n = 2^n\prod_{k=1}^n U_k.\tag{1}$$ I already know that $(X_n)$ is a martingal...

How can i prove for any tree G=(V,E) |E|=|V|-1 I have tried the induction on the number of vertices but nothing happened

The joint probability density function of X and Y is given by f(x,y)=c*(y^2 - 36*^2)*e^{-y} y/6 < x < y/6 0 < y < infty Find c and the expected value of X: That's my question that I have an issue. I know how can I find c and expected value of x or y for joint pdf. When...

Here is the question: What is the simplest expression for the inverse of $(I+xy^*)$ where $x$ an $y$ are $n*1$ vectors, also state the required condition for the invertibility. I tried to solve it by using determinant, however it did not lead to anywhere. Could you please give me a hand?

Find the extremal values of $f(t,x,y,z)=t^2+3x^2+0.3y^2+12z^2 on the unit sphere in $R^4. Justify your answer.

Does this series converge uniformly or not ? sum from n=1 to infinity of (nx-1)/n^2 in the interval of [2,5] I know i can use the the Weierstrass M-test can be use to show if a series converges uniformly but how would i show that my series does not converge uniformly ?

How to prove $f(x)=\frac{x}{(1-x)^2}$ is continuous in (-1,1) using either $\epsilon-\delta $ definition or other methods. thank u

$\sum_{k=0}^{n} ((4n-3)*2^n)+4$ = $(2^{n+3}+4)n-7*2^{n+1}+15$ How? I've tried everything but i don't see it. Any equivalent solutions are also welcome, thanks.

Do that question have a definity and ıf that question have a definity set, what is the definity set for that question. $0^i$

Let f : R-->R be continuous at p in R: If f(p) > 0; then show that there is an open interval I that contains p such that f(x) > 0 for all x in I:

Given the matrix representing a relation on a finite set, write a program in C to determine whether the relation is reflexive and/or irreflexive. Given the matrix representing a relation on a finite set, write a program in C to determine whether the relation is symmetric and/or antisymmetric. Gi...

Show that for the rotation matrix $begin{amatrix}cos(a) & -sin(a)\\sin(a) & cos(a)\end{amatrix}$ = With x and y being any vectors contained in R^2

Is it possible to draw more then one Hasse diagram for one/same poset ? $ R = \{(1,1); (2,2); (3,3); (4,4); (1,3); (1,4); (2,4); (3,4)\} $ There is no the smallest element, the greatest element is 4. Minimal elements are 1, 2. Maximal element is 4. I drew two Hasse diagrams for this relation...

I've got to prove for any vector given, $\vec{x} = [x_{1}, ..., x_{n}]^{T} \in \mathbb{C}^{n}$ that it's true that: $\left | x_{1} + ... + x_{n} \right | \leqslant \sqrt{n} \cdot \left \| x \right \| $ Please give me some hints for that. I know that in this case: $\left \| x \right \| = \sqrt{\le...

I'm trying to prove that the Cantor set is equal to a certain set of 'escape points' for a mathematical feedback system. In this proof I'm going to need the fact that every element of the Cantor set has a base-3 representation in which only 0's and 2's occur. However, I'm having a hard time with...

I'm trying to figure out a problem on inner space products: Les S be the space of $C^1[-1,1]$ functions, and: <;> $\mapsto \int_{-1}^{1} f'(x)g'(x)dx$ Decide if <;> is an inner product over S. To decide if this is an inner product I'm going through the axioms, and I can't show this one...

Draw a Cantor set $C$ on the circle and consider the set $A$ of all chords between points of $C$. Is $A$ a convex set?

Let $G$ be a connected bipartite graph, so $G×G$ is bipartite and has exactly two components. Show that at least one component of $G×G$ admits an involution (i.e., an automorphism of order $2$) that interchanges its partite sets. An automorphism of order $2$ is simply an automorphism $ϕ≠idϕ$ s...