Let $f:[0,1]\to \mathbb{R}$ be continuous, and suppose that $f(0)=f(1).$ Show that there is a value $x\in [0, 1998/1999]$ satisfying $f(x)=f(x+1/1999)$.

I want to solve the equation $$\int^{\infty}_{-\infty}\frac{u(s)}{1+4(t-s)^2} ds= \frac{1}{t^2+6t+10}$$ using $$\mathcal{F}[\frac{1}{1+t^2}] = \pi e^{-|w|} $$ The left hand side is a convolution which I can solve using the hint. However I don't know how I Fourier Transform the right side. When...

Suppose $X$ is an alphabet and $w \in X^n$ is a word over it. Consider a set $P$ of $w$-free words over $X$ (a word is called $w$-free if it does not contain $w$ as a subword). I want to write a generating function for $P$ (by length). I am trying to write some kind of a recurrence relation consi...

Suppose we are given a matrix $M_{n*2n}$ of $n$ linearly independent row vectors. Then I am trying to find an algorithmic way to add $n$ more linearly independent row vectors to this matix resulting in to a matrix $M_{2n*2n}$. Consider this easy example, if the given matrix $M_{2*4}$ is \begin{...

Let's start by quoting the question. Water flows at the rate of 10 m per minute through a cylindrical pipe of diameter 5mm. How much time would it take for it to fill a conical vessel of diameter 40 cm and depth 24 cm? I just don't get what the question means. After doing many sums on volu...

normal distribution, mean=20, sd=2 , p(x^2-x<2) i dont know how to convert it to z-score form. i think if we put 2 on left side we get the p((x-2)(x+1)<0) form. but i dont know how to get any further?

I apologize for a (probably) trivial question but I am looking for a problem name so i can google for its potential solutions. The problem is: Unlike in minimum vertex cover problem where the goal is to compute a set of vertices covering all the edges in a given graph, I am looking for a version ...

Question 1 In a car racing game a Ford mustang car sets off from Edinburgh towards London Kings Cross (800km away) at 6am. It travels at an average speed of 300km per hour. After you have progressed some levels with your game, a Renault car starts at 8.00am in the opposite direction, from London ...

(I asked this on physicsexchange but no reply) I have the metric tensor $g_{\mu\nu}$. I want to make the variation of $\sqrt{-g}$ where $g=detg_{\mu\nu}$. Can I make this work? $\sqrt{-g}=\sqrt{-e^{Tr(log(g_{\mu\nu}))}}=\sqrt{-e^{-Tr(log(g^{\mu\nu}))}}\rightarrow\delta\sqrt{-g}=\delta(\sqrt{-e...

i have my code with the aim of speeding it up, and re-run the experiment on the same computer The following times are observed: 606 567 535 551 536 462 535 508 501 492 523 449 598 576 548 i have to submit report so please help me ...... With what significance is the new code faster than the or...

I'm doing this math project involving the Lotka Volterra Equations. My goal was to be able to graph both the x(t) and y(t) on the same axis against time. So far I was able to find the combined equation of x' and y' using partial differentiation. I called it E(x,y). Substituting my known constant...

I am self-studying probability theory, and I am having quite some problems with the very basic concepts of the theory that are seriously hampering any attempt to proceed further in my study. Here there is one basic problem I am having, with my thoughts (written in italic) about it. Assume the...

Calculate the variation of J defined on C([0,1]) by J(Y)= integral 0 to 1 (Y(x))^2 + (Y'(x))^2 - 2Y(x)sin(x))dx.

mathematically solution for Help with proof of expected value of gamma-gamma distribution gamma-gamma distribution

Let $\mathcal{A}$ be a sub-algebra of $B(H)$ such that $\mathcal{A}$ generated by all its idempotents and $\mathcal{A}$ is close under weak operator topology. Suppose that there exist idempotents $P1$ and $P2$ in $\mathcal{A}$ such that $P1P2≠P2P1$. Can we say there exist idempotent $P3$ in such...

Find a formula in the form z → uz + v or z → uz + v for the following transformation: the rotation through π/2 about i.

Find: $$\lim_{n\to\infty}\sum_{n=0}^\infty x^n$$ Where x is a set of the complex numbers and $|x|<1$ Then prove the limit using $\epsilon$ and $N_0(\epsilon)$ I think the limit is either $0$ or $\frac{1}{1+x}$ but i'm not sure which. Given either I don't know how to do the proof

below is what I attempt, let diam(A) diam(B) => m>n which contradicts our supposition that A⊂B thus , diam(A)≤ diam(B). is my solution is correct?

If U is a open subset in R^n and p is a point in U, then tangent space of U at point p is the whole of R^n. I am having difficulty understanding why this is true.

If A is an invertible matrix then a necessary and sufficient condition for the LU Factorization to exist is : If A is invertible, then it admits an LU (or LDU) factorization if and only if all its leading principal minors are nonsingular Source : https://en.wikipedia.org/wiki/LU_decompositi...

Let $U$ be a set. $Sub(U) := \{A | A \subset U \}$. We can trivial define binary operations $\cup$ and $\cap$ on $Sub(U)$. What van we say about linear equations on $(Sub(U), \cup, \cap)$? For example, about linear equations with matrix $2 \times 2$?

Let $L(\mathbb{R}^n, \mathbb{R}^m)$ be the set of linear maps from $\mathbb{R}^n$ to $\mathbb{R}^m$ and $\phi: L(\mathbb{R}^n, \mathbb{R}^m) \to \mathbb{R}$ be defined by $A \mapsto \|A\| := \sup\{\|Ax\| : \|x\| = 1\}$. I'm trying to give a direct proof that $\phi$ is continuous but I'm having ...

I know that the Extended Liouville’s theorem state that: $An$ $entire$ $function$ $f(z)$ $with$ $|f(z)|<C|z|^n$, $for$ $natural$ $number$ $n$, $is$ $a$ $polynomial$ $of$ $degree$ $at$ $most$ $n$. I am wondering whether there is other function, except polynomial, satisfies the condition of the t...

How to show that the following kernel is valid? x, y are [-1,1] K(x,y) = 1/(1-xy) Can someone help and possibly provide reasonable explanation? Also by giving any distict input x(1),...,x(m) how can we show that it is invertible?

I've got the following homework question: Consider 3 urns. Urn A contains 2 white and 4 black balls; urn B contains 8 white and 4 black balls; and urn C contains 1 white and 3 black balls. If 1 ball is selected from each urn, what is the probability that the ball chosen from urn A was white, ...

I had a lesson about operations on funcions. Everything was good until I reach the point of division of function so the lesson was saying that you can divide a function over another function but when it comes to determining the domain i find it weird i.e f(x)=x/x+1 , g(x)=x-3/x+4 the final functi...

This is probably duplicated, please let me know if it is. What is the limit of the sequence $a_n = \sqrt{1 + \sqrt{2 + \sqrt{3 + \cdots + \sqrt n}} }$. Clearly the sequence is increasing and is bounded from above by 2, so it is convergent. But what is the limit?

The numbers $a_1, a_2, a_3, . . .$ form an arithmetic sequence with $a_1 \ne a_2$. The three numbers $a_1, a_2, a_6$ form a geometric sequence in that order. Determine all possible positive integers $k$ for which the three numbers $a_1, a_4, a_k$ also form a geometric sequence in that or...

How to multiply Roman numerals? I need an algorithm of multiplication of numbers written in Roman numbers. Help me please.

I am studying for a midterm and I have no idea about how to prove that the derivate of $\frac{d}{dx}x^2$ is equal to $2x$ Anyone has any idea? Thank you

Let $G$, $H$ be magmas. $G_1 \subset G$ - submagma of $G$, $H_1 \subset H$ - submagma of $H$. Let $G \simeq H_1$, $H \simeq G_1$. Is true that $G \simeq H$?

Find the equations of the two lines through the origin which intersect the line $\frac{x-3}{2}=\frac{y-3}{1}=\frac{z}{1}$ at an angle of $\frac{\pi}{3}$. Let the direction ratios of the two required lines be $(a_1,b_1,c_1)$ and $(a_2,b_2,c_2)$. Therefore the two equations are $\frac{x-0}{a_1}...

The functional A(Y) = 2pi*(integral 0 to 1 (absolute value(Y(x))*sqrt(1+(Y'(x))^2))dx. Find the variation of A(Y). I have no idea how to solve this problem.

I am trying to determine a $y_1$ and $y_2$ for my Green's function. I believe that I have found $y_1$, but I am a bit lost on $y_2$ because of the constant. Here is my initial problem with BC's: $L[y] = y'' + \alpha^2y = 0$, with $\alpha$ = constant; $y(0)=0$; $y(1)=0$ For $y_1$, I used $y=A cos...

Prove or disprove the following statement: There are no $2$ undirected graphs $G=(V,E)$ and $H=(W,U)$ such that the following holds: 1) $G$ and $H$ are connected graphs. 2) The degrees of the vertices of $G$ and $H$ are both $1,1,2,2,2,4$ 3) $G$ and $H$ are not isomorphic. It seems to me lik...

Can you help me to proof that the nth derivative of $9\sqrt(x) $ is $$ (-1)^{(n-1)} *\frac{9(2n-2)!}{(n-1)!} * (4x)^{\frac{1-2n}{2}}$$ I've tried induction but didn't go very far. Many thanks

After some considerations the article I'm reading concludes: "...hence H is a simple split $\mathbb{K}$-algebra". I can't find this definition anywhere: what does "split" mean?

What is the difference between minimum value and lower bound of a function? for me it seems that they are the same.

Consider the system of ODEs $$u_1'=3u_1+4u_2,\\u_2'=5u_1-6u_2$$ For the system of equations above find the Lipschitz constant if we use the $\mathcal{l}_2$ norm.

I am currently looking at the following example (and other similar examples) and I can follow the proof that it is a right Artinian ring and I also follow the example given as to why it is not a left Artinian ring. However, I do not understand what is stopping us from applying the reasoning use...

If (a,p)=1 and p is an odd prime, prove the legendre symbol sum ∑n=1 to p ((an+b)/p)=0. Where b is any integer. I Know the fact that ∑a=1 to p (a/p)=0. But I don't know how to treat with b.

If $a^4 = 1$ and $ab = ab^2$ in a group, show that $a = 1$. I haven't been able to make much progress with this question. Is there some trick that am I missing?

I need to prove a commutator relation, but I'm getting stuck at the definition of the matrices. $(L_{ab})_{cd} = \delta_{ac} \delta_{bd} - \delta_{ad} \delta_{bc}$ with $a<b$ and $a, b \in 1,2,3,4$. What does this definition of the matrix mean? Can someone explain this to me?

Given $n$ is there an explicit or asymptotic formula for least $m$ such that $$m!\geq n?$$

I'm trying to learn about spline interpolation, and I'm struggling to understand what h_i^3 is in the second and third equation. I don't understand how they derived that equation.

Is there an elementary way to see that there is only one complex manifold structure on $\mathbb{R}^2$? (Up to biholomorphism, naturally.) Elementary in the sense of not appealing to the uniformization theorem.

Good day everyone. Are there any chances to get a compact formula for the following sum of finite number of terms? $\sum\limits_{L=2}^{N} \frac{1}{e^{ \frac{i \varphi}{L}} -1}$ N and $\varphi$ are constants. Thank you.

Can you help me to find the maximum value of $f(x,y) = xy^2$ with the constraint that the x and y coordinates must satisfy $g(x,y) = x^2 + y^2 = 8$ How do I even start? Thanks

Here there is a really central problem I am having self-studying probability theory, that concerns the relation between the definition of expectation in Lebesgue terms and in Riemann terms. I will truly appreciate any feedback because I feel this is really at the very core of the all theory. Sum...

How can I formulate the following system of differential equations in Mathematica]1]1 for to find a general solution for $f[x_1, x_2, x_3, y_1, y_2, y_3]$? By the way I know that the solution is $f[x_1, x_2, x_3, y_1, y_2, y_3] = \frac{y_3 x_1 (y_2 x_3 + y_1 x_2 + y_1 x_3)}{(y_2 x_2 + y_3 x_3 +...

Could you please help me to do this , proof . I'm currently learning set theory and am stuck on this question. Let A= Z , B ={ x ∈ Z : x= 2n + 5 for some n ∈ Z} and C = {x∈ Z , x= − 2m for some m ∈ Z} . Prove that A \ B = C.

I am trying to show this using Leibnitz rule: $$D_2f(x,y) = \frac{\partial {}}{\partial{y}} \left ( \int_0^xg_1 (t,0) \ dt + \int_0^y g_2(x,s) \ ds \right)$$ $$= \int_0^x \frac{\partial{}}{\partial{y}} g_1(t,0) \ dt + \int_0^y \frac{\partial{}}{\partial{y}} g_2(x,s) \ ds$$ $$ = \int_0^x \frac{...

I need to prove directly from the definition that $$(x,y)\in \Bbb R^2, x^2-y^2=1$$ is disconnected. Can someone please give a methodological answer? I cannot handle this type of problems Thanks in advance!

Given G = ({S,A,B,C,D,E}, {0,1}, P, S) where P is S --> 0A0 | 1B1 | BB | DE A --> C B --> S | A C --> S | ε D --> 0 | DE E --> EE Convert to CNF: I came up with the following but i'm stuck.... i'm not sure if I went about it correct: 1) Eliminate e-transition for C S --> 0A0 | 1B1 | BB | DE A ...

Let U denote the set of all nxn matrices A with complex entries such that A is unitary. Then U as a topological subspace of $C^{n^2}$ is a) compact but not connected. b) connected but not compact. c) connected and compact. d) neither connected nor compact. we can say a set is compact if it ...

While calculating a^b mod p, i have a function f(x) in place of a. f(x)=a1*a2*a3*a4; Now i am having a hard time to understand how come a evaluates to ((a1 mod p)*(a1 mod p)*(a1 mod p)*(a1 mod p))mod p I know the basic (a*b)mod c=((a mod c)*(b mod c))mod c, but this isnt leading anywhere...

In an exercise (4.1 Krapinsky, "A kinetic view of statistical physics") I am asked to show that: The probability that a brownian motion on a 1D discrete lattice never reaches the site $n$ scales as $\frac{1}{\sqrt{t}}$ in the long time limit. The brownian motion starts from the origin. My idea ...

Let T be a linear operator on a FDVS V over an algebraically closed field F.Let f be the polynomial over F.Prove that c is a characteristic value of f(T) iff c=f(t), where t is a characteristic value of T. Converse part is trivial. Forward part: let c is a Characteristic value of f(T) ...

Taylor series of: $$f(x)=\int_0^1 \frac{1-e^{-sx}}{s}ds$$ at $x_0 = 0$. I've done: By fundamental theory of calculus: $$f'(x)=1-e^{-1x}$$ Which is clearly differentiable by e.g. $n$ times. What do I need to do to get the expression: $$\sum_{n=0}^{\infty}\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n$$ ...

Let $M\in \mathcal{L}(X) $, WLOG suppose $M$ is injective extend $M$ to $X\oplus Z$ s.t $M$ is surjective. Since $Im(M)$ is finite co-dim we know that $Im(M) \oplus \mathbb{R^{n}}$ is the whole range. Since we now have a bijection we get that it is a homeomophism. Furthermore, $M$ is closed since...

Naturally, one could do this with a calculator pretty easily, but is there a trick, or something you may notice, to calculate this easily by hand? I know $\phi{125} = 100$, so could I perhaps use that somehow? I feel like I can, but I'm not exactly sure how.

f(x,y)={(x^3y-y^3x)\over(x^2+y^2)} if (x,y)\ne(0,0) , 0 if (x,y)=(0,0). I want calculate {(\partial_(xy)f)^2 (0,0) }.

In a basketball game a player is attempting to score. He stands at position (-10, 0). His hands are at position (-10, 15). He throws the ball and the ball follows a parabolic path that is defined as follows: y = -(x)2 + 49 The basket is at position (4, 40) and has a radius of 2. (i) Will he scor...

My problem is to convert the expression $y=\sqrt{2x-x²}$ into $x=f(y)$ format. I get rid of the square root, but then what?

$$ \int t^3 e^{t^2+t} dt $$Solve the given integral by using the substitution techniques or the integretion by parts

I am trying to find a simple example of a group where $g^2 = 1$ for all $g\in G$ but $G$ is not abelian. All the groups I can think about with that property are abelian.

Using two dice what are is the probabilty of not hitting an eleven in 72 rolls?

I have a trouble proving this sentence, as I don't know what assumption should start the proving implication. I know what are the characteristics of a ring, but I m, n are not in the ring. a, b are in ring (F, +, x). m, n are whole numbers. Proof that: m x (a + b) = m x a + m x b.

Suppose a grid starts at position $(0,0)$ and extends up and to the right. A shortest route along streets from $(0,0)$ to $(i,j)$ is $i+j$ units long, going $i$ blocks east and $j$ blocks north. Suppose that the block between $(k,l)$ and $(k+1,l)$ is closed, where $k<i$ and $l \le j$. How many sh...

Let {$S_n$} be the collection of connected sets with the property that $S_n\cap S_{n+1} \neq \emptyset$. Prove that $\cup S_n$ are connected. Help would be appreciated! I do not know how to type $n$ goes from 1 to infinity with the sets. Thanks in advance!

$$ \int ln(2+3cos^2x)dx $$ This is considered to be a difficult integral.Unfortunately, i haven't managed to solve it by any means.

Consider the set $\{1,\dots,10\}$, and suppose we draw $4$ numbers with substitution. I want to calculate the probability of drawing $n$ different numbers, with $n=0,2,3,4$. I omit the value $n=1$ because drawing $1$ different number does not make much sense. Let $\Omega=\{1,\dots,10\}^4$ be the...

Suppose that $(\Omega, F, p) $ is a probability space & $f $ is a measureable function that $\int _{\Omega }|f|^2 =1 , \int _ { \Omega } | f| \ge a>0$ Prove that for every $0\le \lambda \le 1$: $p (\{x\in \Omega : |f (x)| \ge \lambda a\} )\ge (1-\lambda )^2a^2$ Would anyone please give me ...

enter image description here I am not sure how to go with this. Thanks

I can show for any given value of n that the equation $$\sum_{k=1}^n \cos(\frac{2 \pi k}{n}) = 0$$ is true and I can see that geometrically it is true. However, I can not seem to prove it out analytically. I have spent most of my time trying induction and converting the cosine to a sum of compl...

I am solving following problem: Let us have M amount of containers, where each container has it's capacity. Let us have N objects (each of certain size), that are somehow distributed into these containers. Obviously, we may not exceed the capacity of a container. I may start with some round-r...

The velocity v of a fluid flowing in a conduit is inversely proportional to the cross-sectional area of the conduit. (Assume that the volume of the flow per unit of time is held constant.) Determine the change in the velocity of water flowing from a hose when a person places a finger over the end...

I am stuck on the following problem: How can I find the area of the equilateral triangle inscribed in a circle $x^2+y^2+2gx+2fy+c=0$ ? Answer is given to be : $\frac{3\sqrt 3}{4}(g^2+f^2-c)$ sq. unit. Can someone please explain?

If I have two sums $A = \sum_{n=0}^\infty (n+2)(n+1)a_{n+2}x_n$ and $B = \sum_{n=2}^\infty n(n-1)a_nx^n$ To combine the two sums, could I simply change the lower limit of B to 0 because it would make no difference to the value and write $$A+B = \sum_{n=0}^\infty (n+2)(n+1)a_{n+2}x_n+ n(n-1)a_nx^...

Let A be the set of all people who have ever lived. For x, y ∈ A, xRy if and only if x and y were born less than one week apart. Determine: (i) Whether or not the relation R is reflexive; I understand that x is in relation to y if x and y were born less than one week apart, but how would you me...

Prove that the line segment joining the mid-points of two sides of a triangle is parallel to the third side

I need some help with vector spaces. This is what I have: 1/3 + 5*7^1/3 + 4*7^2/3 = x + y*7^1/3 + z*7^2/3 If you rearrange to solve for 1 you get. 1 = (3+ 5x7^1/3 + 4x7^2/3)*(X + Yx7^1/3 + Zx7^2/3) Note: the "x" is multiplication. Basically I think I have to expand, I'm not sure what to d...

True or false? a) Let $(x_{\alpha})_{\alpha\in A}$ a net over a space $X$ and $(x_{h(\beta)})_{\beta\in B}$ a subnet, where $h:B\to A$ is monotone and final. Let $\mathcal{F_1}$ be the filter generated by $(x_{\alpha})$ and $\mathcal{F_2}$ the filter generated by $(x_{h(\beta)})$. Then $\...

Suppose $S \subset R$ is a set and, for each $n$, $x_n$ is an accumulation point of $S$. Suppose further that $lim_{n}x_{n}=x$. Prove that $x$ is also an accumulation point of S. I was given the hints that: It is not stated that $x_n \in S$ or that $x \in S$. Approach this by the definition,...

Please do not provide answer here, I only needs hints and rough picture. Consider a random variable take values $\pm1$ with probability $1/2$. Prove $\lim_{n\rightarrow\infty}{S_n}$ does not exist almost everywhere.

enter image description here to be honest I could not answer any of these questions, I'm taking this new course, but its not related to my major, can you help please, am really trying

I was hoping whether someone could check if my understanding of this is correct, and also help with my question: Let us suppose that $T: V\rightarrow V$ is a linear transformation with a matrix $A$ that is similar to some diagonal matrix D, and that there is some basis of eigenvectors $e_i$. We...

I have just started reading about the Pumping Lemma, and I have some difficulties understanding the proofs. For example, in the book I am reading there's a proof for the fact that the language $$C =\{w \mid \text{ w has an equal number of 0s and 1s} \}$$ is not regular. The proof starts like ...

I am taking an introduction class on set theory. We have formally constructed the natural numbers, integers, rationals and reals. I am now trying to think of how to define the complex numbers inside of set theory. My idea is to follow the idea of how the reals were constructed from Dedekind cut...

Is f(x)= (ln(x^2))/(1-x^2) uniformly continuous on (-1,1)? I know the definition of uniform continuity. and when the derivative is bounded on (-1,1) f is UC.

I get the substitution notation in lambda calculus for "simple" applications such as: (λx.x+1)(5)=[5/x](x+1)=5+1=6 What I don't get is how that works when I pass a lambda as the "parameter". E.g. to compose functions f(x)=x+1 and f(y)=2*y: (λy.2*y)(λx.x+1)=[λx.x+1/y](2*y)=2*(λx.x+1) (??) Th...

Let $H$ be a hilbert space and $V$ is a banach space. What doese it mean $V$ is densly embedded in $H$ is it necessary that $V\subset H$

I have a Reflexive Banach space V that is compactly embedded into a Hilbert sapce H which is continuously embedded in its turn into a Banach space W:= V' (with W being the dual of V, that is V') and with H being identified with its dual. How can I prove that V is dense in H? is it possible for So...

this time I am asking for help to generalize the code for the QR-factorization, I did the following code for the fist step: %Script for the Householder QR factorization. A=[60,41,-88;42,60,51;0,-28,56;126,82,-71]; disp(A) m=length(A(:,1)); n=length(A(1,:)); I=eye(m); e1=I(:,1)'; %We compute th...

I am an international student currently studying in the US, pursuing a B.A. in mathematics at a relatively unknown Liberal Arts College. My school is small and does not have many course options as far as mathematics courses are concerned, because of its liberal arts nature. However, I was awarded...

Where $d_u$ is the uniform metric and $f=ae^{-x}$ I know the solution involves the step $d_u (f,0)=sup |ae^{-x}|=|a|$ Why does this equal $|a|$?

Use the dot product to show that if $S = \{\vec{v_1}, \vec{v_2}, ..., > \vec{v_n}\}$ is an orthogonal set of nonzero vectors in $\mathbb{R}^n$ then $S$ is a linearly independent subset of $\mathbb{R}^n$. I believe that since the vectors are orthogonal to one another, this implies they are ...

I have had some difficulties understanding proofs that a language is not regular using the Pumping Lemma, and now I need to prove that the following language $$A = \{w \mid \text{ w has even length and the first half of w has more 0s than the second half of w} \}$$ is not regular. I would sta...

Is ln(x^2+y^2)/(1-x^2-y^2) uniformly continuous on (x,y):x^2+y^2<1? Please help.

Let $a$ be a positive integer greater than 1 and define the sequence $u_n$: $$u_n = \dfrac{a^{2n} + a^{n+1} + 3a - 5}{a-1}$$ Find all possible value of $a$ such that $u_n$ is not a perfect square for only finitely many values of $n$ After a few attempt I found out the result is $a \in \{2,10\}$ ...

From the the Wikipedia page Minimum spanning tree: A minimum spanning tree is a spanning tree of a connected, undirected graph. It connects all the vertices together with the minimal total weighting for its edges. Let G be a connected, undirected graph, and let H be a connected subgraph of ...

Can anybody help me out with getting an expression of the values of $\lambda$ for a matrix $A$ for which $det(A-\lambda I)$ equals the determinant of a matrix with on the main diagonal $\lambda$, on the diagonal above the main diagonal $\dfrac{1}{2}$ and on the diagonal under the main diagonal $\...

I am taking a graduate Algebra course, and we were given the following example to see that Maschke's Theorem does not hold if the characteristic of the field F does divide the order of G: Let $F = \mathbb{F}_2$, $G = \mathbb{Z}/\mathbb{Z}_2$ and $V=FG$. Then show that $W = span\{0+1\}$ is not co...

if G acts trivially on A then $H^1(G,A)=Hom(G,A)$.if $A=\mathbb{Z}$ then $H_1(G,\mathbb{Z})=G/G'=G/G'\otimes \mathbb{Z}$ can we say $H_1(G,A)=G/G'\otimes A$?(i think it is natural to define $G\otimes A=G/G'\otimes A$ so we have $H_1(G,A)=G\otimes A$

We define the Dupin indicatrix to be the conic in $T_PM$ defined by equation $II_P(V,V)=1$. If P is a hyperbolic point, show that the principal directions are the symmetry axes of the indicatrix When P is a hyperbolic point, the indicatrix is a hyperbola with equation $k_1*x²+k_2*y²=\pm1$ I do...

Prove that in any simple, undirected graph $G=(V,E)$, the chromatic number $χ(G)≤ \sqrt{2m} +1$, where $m=|E|$

I am trying to find number of positive solutions for linear diophantine equation ax + by = c. I followed this answer http://math.stackexchange.com/a/80886/290384 ...there is written that the solution is: floor(c/(ab)) or floor (c/(ab)) + 1 I can't decide when to add the one to the result. For e...

I try get it why relation divisibility is not relation partially ordered set. $A=\{−2, 2, 4, 6, 8, 10\}$ with relation divisibility "|" $R$ is relation divisibility | when $a,b,c \in Z : a = b \cdot c$ For relation partially ordered set must be relation: reflexive (fulfil) - everery number ...

Let $y \in R$, the goal is to find the dual problem to: $$\min y\\ s.t. |y| \leq 0$$ The lagrangian of the problem is: $$L(y, \lambda) = y + \lambda|y|$$ The dual function is: $$g(\lambda) = \inf_y y + \lambda |y|, \lambda \geq 0$$ But the constraint $|y| \leq 0$ is only satisfies at $y = ...

If A and B are connected, then does it imply that their union is connected? For example, if $A=[0,2], B=[4,5]$. Is the union connected? I believe the intersection is not necessarily connected, but how about this example? Thanks in advance!

I am currently working on a problem and I at a loss as to where to begin. Consider a recursive sequence x_n, where: x_0 = 0 x_n = z^(x_n-1) I need to find the values of z for which the sequence is convergent. I have currently plugged in some sample values of z into this sequence and this is w...

When in general is true that $\nabla_{\mu}\delta=\delta\nabla_{\mu}$ where $\nabla_{\mu}$ is covariant derivative?

$(1+x^2)y''(x) -xy'(x) + y(x) = 0$ My attempt so far: I know we are looking for an ordinary point because $a(x) =\frac{x}{(1+x^2)}\ $ and $b(x) = \frac{1}{(1+x^2)} $ are both analytic. Also I know y = $\sum_0^{\infty}a_nx^n$ The way I have been taught to compute this sum is to differentiate the...

I would like to know if there is any proof without using the fact that: $$\lambda_1\cdot\lambda_2\cdot\ldots\cdot\lambda_{n-1}\cdot\lambda_n = det(A)$$ I managed to prove that if $\lambda = 0$ then, A is singular, by: $$ det(A-\lambda\cdot I)=0 \Rightarrow det(A-0\cdot I)=0 \Leftrightarrow det...

Assume the sequence of random variables $X_1, X_2, \cdots$ are IID with finite mean and finite variance. Define a random variable: \begin{align} Y_n = \frac{X_n}{n} \end{align} Show that $Y_n \to 0$ almost surely. To converge to some value almost surely implies: \begin{align} \mathbb{P}\left( \...

How do I show that the function $g=3$ is contained in the unit ball of $X=C[0,1]$ in the uniform metric? I know the uniform metric $$d_u(g,0)=sup |3|$$ where do I go after this?

$\int_0^te^{A(t-\tau)}Bu(\tau)d\tau$ is rewritten as $(e^{\cdot A}*Bu)(t)$ by using convolution product, where $e$ is the natural logarithm base, $A\in\mathbb R^{n\times n}$ and $B\in\mathbb R^{n\times m}$ are constants, $u_1(t)$ and $u_2(t)\in L^\infty([0,T],\mathbb R^m)$. Then for the convoluti...

Let $$(x,y,z)\in \Bbb R^3, x^2+y^2+z^2=1$$ I need to show that this set is connected. Help would be appreciated! Thanks in advance!

What are the steps in order to transform the cosine function to the exponential function: $$ [\cos(k \pi/N)]^2 \Rightarrow e^{-n/2 (k \pi /N)^2} $$ if i know that $$ 0 < k/N << 1 $$ I just want to understand this step. I am awere of the Euler's formula, yet i can't figure out, how to use it. T...

Can someone help me solve this limits? I know how to solve basic trigonometric limits, but this is too advanced for me. I know that i should start by writing the tgx=sinx/cosx. Some instructions after that?

Let X and Y be nonnegative random variables with $EX^4 + EY^4 < \infty$, Show that for any $r,s \in (0, \infty)$, with $r+s=4$, $EX^rY^s <\infty$

I've found a similar question here but what I'm looking for is a little bit different... Besides trying to prove that both these conditions are satisfied and a,b,c and d are four different positive integers: a² + b² = c² + d² a + b = c + d I would like to prove that this holds for n t...

I have been reading Mac Lane's Categories for the Working Mathematician, and the prospect of developing category theory without any use of set theory is mentioned more than once in the book, but never actually realised. I was wondering whether there are any good references (books or online notes)...

In an exercise question, I am asked to show that: Given $q>p>3$, where p and q are distinct odd prime numbers. Show that $p > 3 + \frac{6}{q-3}$ and conclude that $\phi(n) < n < \frac{3\phi(n)}{2}$. To prove the first part of this I did the following: Proof by Contradiction: Suppose tha...

Dan's DVD discount dungeon is going out of business and selling al of its DVDs. The first customer bought 1/6 of all the dvds plus 1; the second buyer again took 1/6 of the remaining DVDs plus 2; the third buyer bought 1/6 of the remaining DVDs plus 3, and so on. When Dan had sold all of the DVDs...

I have recently started to develop my mathematical intuition. In the past I saw math as a mere game of symbol manipulation, whosoever was able to see patterns and cram formulas and apply them upon those symbols won. However,I now feel that the truth is contrary to the aforementioned,this is becau...

$ (e^{2x})=(e^x)^2$ Can Some one explain how these two are the same? I know it is trick, but I cannot see how it fits into the exponent rules.

enter image description here Guys I am so lost on this question. Can someone please help me with this.

I am studying for an exam and this was one of the hard problems in my textbook. Let $f_n(x) = 1 − cos(2πnx)$ for n $\in$ N and x $\in$ [0,1]. Verify that $f_n$ is the density of a probability measure $μ_n$ on [0, 1]. Is there a weak limit $μ_n \rightarrow^w$ μ? Either show convergence and ide...

Basicly the title. I know it's easy but I can't figure it out

I know that they are not the same since $E(X|A) = \frac{E(X 1_A)}{Pr(A)} $ (*) But without using formula (*) above, for me, both of them means "Average of X on an event A". Could you please explain for me intuitively what is the subtle difference between $E(X 1_A)$ and $E(X|A)$? Thank you.

Player 2 A B C Player 1 x 6,10 0,2 3,0 y 0,0 1,4 4,6 I tried removing B from the equation as it is supposed to have been strictly dominated but it doesn't give me the right answer, i dont know how to go about answering this.

I am reading the book Lectures on the geometry of Poisson manifolds, by Izu Vaisman. To a Poisson structure $\{\cdot,\cdot\}$ on a manifold $M$ we associate the Poisson bivector field $w\in\Gamma(\Lambda^2TM)$. We have Lemma: $[w,w]=0$ This follows directly from proposition 1.4 in the book...

I got to the point where I showed: $$[\pi(G):A_G \cap \pi(G)]=2$$ I have a trouble convincing myself of the final step. So if $\pi:G \rightarrow S_G$ is faithful (i.e. $\pi$ is injective), then $G$ can be identified with $\pi(G)$ and thus the preimage of $A_G \cap \pi(G)$ gives the subgroup of i...

I have 81 episodes of Midsomer Murders. I have watched them all, so I like to pick one "at random" to watch now. Can I randomly select one by dividing them in half over and over, flipping a coin each time to pick which half to keep until I get down to one? If I get an odd number at any stage, bec...

What is the solution of $$ \frac{d}{dy} \int_{0}^{y} exp(-y(x+1)) dx $$ If there was no y inside exponential function the answer was the exponential function.

Is there a case where a function $f$ that is not differentiable at $0$ and a function $g$ that is differentiable at $0$ where $f+g$ is differentiable at $0$.

Let R be a relation defined on the set N by a R b if either a|2b or b|2a. Prove or disprove: R is an equivalence relation.

If $f$ is such a homomorphism, then $f(x+y)=f(x)f(y)$. I know of examples of $f$ which satisfy this property, such as $f(x)=e^x$, but how do I find all of them?

I am trying and not understanding how to factor. I have tried long division and synthetic division but am not able to factor -x^3+7x^2-10

I apologize if this is not the correct place to ask such a question. This is my first time using Math StackExchange. Please feel free to point me to another site if this is not the correct place to ask this :) I just completed my undergraduate degree in Mathematics and Applied Mathematics. The r...

Attached is an image which i don't quite understand: Why can he pull the value $f(x)$ inside the integral when $f$ is continuous? Then i don't see where he uses the ball $B$ being of radius less than $\delta/2$

How to factor following polynomial into its lowest terms? $$x_i^n+(n^n-1)\prod _{i=1}^nx_i.$$ That question comes from inequality $$(\sum_{i=1}^n x_i)^n \ge x_i^n+(n^n-1)\prod _{i=1}^nx_i,$$ where $x_1,x_2,\ldots,x_n>0$.

Does ∑ln(n)/(n*n^(1/3)) converges or diverges? Which test should i use? i tried the ratio test and root test but both of them are inconclusive so tried it by comparison test but i don't know which function i have to compare with. Thanks!! ∑ln(n)/n^(4/3)

So, matrix A * its inverse gives you the identity matrix correct? Also, if you have AB=BA, what does that tell you about the matrices?

I know that: $$\int_0^1 1 - x^2 dx = \frac{2}{3}$$ And that represents the area below the curve, delimited by the lines $x= 0$ and $x = 1$ But $$\int_0^1 1 - x^2 dx = \lim_{n \to \infty} \left( 1 - \frac{2n^3 + 3n^2 +n}{6n^2} \right) $$ And this limit is obtained by calculating an approximation ...

Is there a case where $f$ is differentiable at $0$ but not any other point?

Sketch $\begin{cases}\dot x =x-y \newline\dot y= x+y\end{cases}$ on a phase portrait. The original question requires me to change the variables to polar ones, but then asks me to draw a phase portrait of the problem using the prior form and then the new form, and I am currently stuck with drawing...

I have a very short proof here for the following lemma, and there's one small bit I am not sure why it is true. Lemma: Let $G$ be a Groebner basis for the polynomial ideal $I$. Let $p\in G$ be a polynoial such that $LT(p) \in <LT(G-\{p\})>$. Then $G-\{p\}$ is also a Groebner basis. The foll...

I need some assistance with the proof for part (b) of the following problem statement: Problem Statement: Decompose the set $\mathbb{C}^{2\times2}$ of $2\times2$ complex matrices into orbits for the following operations of $GL_{2}(\mathbb{C})$: (a) left multiplication (b) conjugatio...

$x_t$ is a super-matingale. And f is an increasing function. Can does $f(x_t)$ has any martingale property? What is f is decreasing?

Is there a formula for calculating the cardinalities of orbits of $A\in \text{GL}_2(F)$ under conjugation? I know that $|G|=|G_x||Gx|$.

1) Are there examples of homeomorphic compact metric spaces of different Hausdorff dimension? 2) If yes, are there sufficient conditions on the spaces which would imply the equality of Hausdorff dimensions? In my case the spaces are the so called multiple conic singularities (MCS-) spaces. The...

$ I=\int _{ -\infty}^{ \infty} \frac{log(\lvert t\rvert)}{x^4+t^2}dt $ ATTEMPT:- Since the function is even, the following property may be used: $\int _{-a}^a f(x)dx=2\int_0^a f(x)dx$, if $f(x)=f(-x)$$ \qquad$$-(1).$ $\implies I=2\int _{ 0}^{ \infty} \frac{log(\lvert t\rvert)}{x^4+t^2}dt $ H...

So I got this question in my exam and I couldn't solve it. Later my professor gave me the solution but I'm not getting it properly. I guess my concepts on projection are not that strong. Can you please help me understand it? Question: Give an example to show that if R and S are both n-ary relatio...

If not, give a counterexample.

How can I relate any two functions to each other.

Define a sequence of RVs $(X_n)$ that only take on integer values. We want to show it does not converge to some integer "$l$" almost surely. We prove by contradiction: Assume $(X_n) \to l$ almost surely. Then, $\exists N$ such that $n > N \implies X_n = l$. This would force $P(X_n = l) =1$ $\fo...

I first started by integrating both sides with respect to t (dt). It says that B is along the z-axis but how do I account for that.

The Temple of Horus at Edfu, Egypt, has a formula for finding the area of an arbitrary quadrilateral with sides a,b,c,d as (a+c)/2 * (b+d)/2. First, show that this formula is correct if the quadrilateral is a rectangle, then show that the formula gives too large an answer for any other quadrilat...

The curve whose equation is y = ax^3 + bx^2 + cx +d has a point of inflexion at (-1,4), has a turning point wen x=2 and passes through the point (3,-7). Find the values of a,b,c,d and the position of the other turning point

What is the solution of $$ \frac{d}{dy} log( \int_{0}^{y} \exp(-y(x+1))\, dx ) $$ The solution without log was explained here.

The exercise: Let $\omega = \sum a_i(\mathbf x) dx_i$ be a 1-form of class $\mathscr{C}''$ in a convex open set $E \subset \mathbb{R}^n$. Assume $d \omega = 0$ and show that $\omega$ is exact in $E$, by completing the following outline: Fix $p \in E$, Define $$f(\mathbf{x}) = \int_{\bf [p,x]} ...

As an exercise, we want to prove that the following function is strictly increasing, without using derivatives. $f(x)=x^3-\frac{1}{2} x$ Using derivatives, this is a very simple proof, but since we are not able to, my method approach begins with looking at the definition of strictly increasi...

Given a divergent series a sub n, is the sine of that series also divergent? If not, give a counterexample.

I'm a little confused on what is being asked here: Show that the following sets are countably infinite, by defining a bijection between N (or Z+) and that set. -The set of positive integers divisible by 5. -{1,2,3} X Z.

I am trying to formulate in a recursive manner the backwards probabilities $\beta$ of a Hidden Markov Model where $w_i$ are the observed symbols and $s_i$ are the latent states. Is the following derivation correct? Do you know if there is any way without reformulating everything from scratch, to...

I have an integer variable $q_i$ and a binary variable $x_{ ij }$. How can I write the following constraint? if $x_{ ij } = 0$ then $q_i = 0$ else if $x_{ ij } = 1$ then $q_i > 0$

$b_n \ge 0$ and the series of $b_n$ converges. $a_n \ge 0$ and $lim \frac{a_n}{b_n} exists$. Does the series of $a_n$ converge?

Suppose we are given a function $f(x)$ with two continuous derivatives which satisfies the differential equation $xf′′(x)+3x(f′(x))^2=1−e^{−x}$ for all real x. (Do not attempt to solve this differential equation.) Part 1 If f has an extremum at a point c, show that this extremum is a (local) ...

Can someone help me start is problem off. Find the volume bounded by the parabolaid x^2 + y^2 + z =7 and the plane z=1. Thank you