Let's take the function f(x) = x^2 / (x-1), with f(x)' = x(x-2) / (x-1)^2 as its derivative. Since x = 1 is not in the domain of f(x) and f(1)' does not exist, do we use x = 1 as a critical number when we divide the line into intervals? Or do we just use x = 0, 2? Similarly, for g(x) = [ (2-x)(...

I want to determine bounds of variables of a system of linear equations which is an underdetermined system. I illustrate with a simple example. For instance, considering a small system -- x+y+z = 10 ----- (1) x+y+k = 20 ----- (2) z+n = 5 ----- (3) Considering, each variable...

Hi guys I am trying to show that id a function is a Lipschitz M continues then it is absolutely continues and $|f'(x)| \leq M$. I think I am on the right track: Proof: (=>) Let f be Lipschitz M continues ie if $|f(x)-f(y)| \leq M|x-y|$ for all $x,y \in E=[a,b]$. now we want to show abs cont: $...

Proof that for set of permutations of set $\lbrace 1,2,..n \rbrace$ $(n\geq2)$ is for fixed number $k\neq1$ is equal with this lemma: $\textbf{lemma: }$The number of permutations where 1 is with $k$ in same cycle and the number of permutations, where are in same cycle different is same. I don't...

Q: Let $A=\{1,2,3\}$ and $B=\mathcal{P}(A)$. Let $B: B \rightarrow B$ be the function defind by the formula $F(X)=A\setminus X$. What is $F(\{1,3\})$ ? A: I have no idea where to start. I assume $X$ is an arbitrary $X$. I originally tried to define the function as $F=\{(x,y) \in B \times B | y=x...

Let $z \in \mathbb{C}$ ,and $\left| z \right| = 1$ , and $A = \left( {\begin{array}{*{20}{c}} 0 & 1 & 0 \\ 0 & 0 & 1 \\ z & 0 & 0 \\ \end{array}} \right)$. What is numerical range of $A$?

Explain why $f^{1}$ is a function if and only if $f$ is a bijective function. My attempt: $f^{1}$ is the inverse relation from B to A $\equiv$ function from B to A By definition of a function from setA to setB, there is a relation from setA to B. (ARB? relation) such that is satisfies two pro...

How do we find the norm of $(1+\sqrt{2})^n$ $ \forall n\geq1$? The norm of $a+b\sqrt{D}$ is defined as $a^2-b^2D$ where $a,b\in\mathbb{Z}$ and $D$ is a square free integer. P.S- This question comes from Ring theory when we try to find the units of the ring $\mathbb{Z}[\sqrt{2}]$ Any hint is welc...

I am deeply struggling with understanding how to apply the Viterbi algorithm. From my course notes, I have the following simple(I'm told) example: If the sequence HH was observed, what is the most likely sequence in which Fair and Biased coins were used ? Following table was generated: and ...

I don't understand why the following two definitions of nilpotence are equivalent: Definition 1. $G$ is $0$-step nilpotent if $G=\{e\}$. G is $k+1$th-step nilpotent if G is not $k$-step nilpotent, but $G/Z(G)$ is. Definition 2. Let $\gamma_{i+1}(G)$ such that $\gamma_{i}(G) \subseteq \gamma_{i+...

I'm learning control theory and I have to solve one example from book (this is really mathematical problem, that's why I post it here :) ). Using Routh criterium, test stability of system which has characteristic equation $$f(s)=s^{3}+(a+6)s^{2}+(5a+10)s+6a+3$$. Any idea? Thanks in advance.

Let p and q be odd prime such that p < q. G is nonabelian group of order pq. Prove if a is in G and isn't the identity then < a > = C(a) So I was able to prove < a > is contained in C(a) but I'm stuck on proving C(a) is in < a > What I started with is C(a) is a subgroup of G so it has order p o...

Suppose $\Omega$ is bounded and that $1 \leq p \leq q \leq \infty $. Prove the following: For $k ∈ N^+$ : $$W^k_q (\Omega) \subset W^k_p (\Omega),$$ where $W$ stands for Sobolev space.

We have a continuous function $f:\Bbb R \rightarrow \Bbb R$ so that $$\lim_{x \to \infty} \frac {f(x)}{x^2} = 0$$ Prove that $\forall m \in \Bbb {R^+}$ $\exists c \in \Bbb R$ so that $$mx^2+f(x)\ge mc^2 + f(c)$$ Can somebody tell me how to interpret (and how to solve) this? I tried constructin...

Any idea where to see the proof for: Twelvefold way for "Injective functions from N to X" https://en.wikipedia.org/wiki/Twelvefold_way#Functions_from_N_to_X

This may be a silly question but I have trouble grasping this very basic concept. In equations, sometimes we have $$ \int_0^\pi \int_x^\pi \frac{sin y}{y} dydx$$ function given as f(x,y). And other times it's just double or triple integral with 1 as f(x,y) followed by dydx. What does this f...

Q: Suppose $R_{1}$ and $R_{2}$ are relations on A. Give a proof or counterexample to justify your answer. If $R_{1}$ and $R_{2}$ are reflexive, must $R_{1} \cup R_{2}$ be reflexive? A: My reasoning is as follows: Let $A$ be a set. Let $\alpha, \beta \in A$ and $\alpha, \beta$ are arbitrary. Let...

How do I show the limit as x approaches infinity of x^2-xcot(1/x)? Wolfram says it is 1/3 and I know it is supposed to converge to a number other than 0 but I keep getting infinity.

From a population of 50 households, in how many ways can a researcher select a sample with size of 10? A box contains 5 red balls, 7 green balls, and 6 yellow balls. In how many ways can 6 balls be chosen if there should be 2 balls of each color? If 3 marbles are picked randomly from a jar conta...

enter image description here Let (Xn)n≥1 be a sequence of random variables such that Xi ∼ qδ−1 +pδ+1 Define Yn=􏰏􏰏∑n i=1 Xi

Assume I have two Poisson processes with respective parameters$$ \sim\text{Poisson}(\alpha_1),\sim \text{Poisson}(\alpha_2)$$ that I observe over a time interval $[0,t].$ What is the probability then that the latest jump on this interval was done by the second process? So what I am looking for...

Lets say I have found the λ from the characteristic polynomial. Then I substitute it back into (λI - A) and solve for it. Supposed these are the answers that I got. x=2r y=r z=r where r is an element of all real numbers. a basis would be r(2,1,1).. So (2,1,1) is a basis for the eigenspace ass...

A system of Circles pass through $(2,3)$ and have their centers on the line $x+2y-7=0$. Show that the chords in which the circles of the given system intersects the circle $S_1:x^2+y^2-8x+6y-9=0$ are concurrent and also find the point of concurrency. ATTEMPT: The circles of the given system mu...

i got this question in my undergrad what is this integral?

I know it is obviously $f(x)=e^x$, but could you prove this without knowing $\frac d {dx}e^x=e^x$? And does there exist a $g(x)=g'(x)$ but $g(x)\ne f(x)$?

Suppose a function $f:(-a,a)-\{0\}\rightarrow(0,\infty)$ satisfies $\lim\limits_{x\rightarrow 0}\left(f(x)+\frac{1}{f(x)}\right)=2$. Show that $$\lim\limits_{x\rightarrow 0}f(x)=1$$ Let $\epsilon>0$ , then there exists a $\delta>0$ such that $$\left(f(x)+\frac{1}{f(x)}\right)-2<\epsilon\;...

Find a sequence of continuous functions $f_n : \Bbb R \to \Bbb R$ such that $\lim_{x \to 0} (\lim_{n \to \infty} f_n(x))$ and $\lim_{n \to \infty} (\lim_{x \to 0} f_n(x))$ exist and are unequal. My Attempt: The function $f_n(x) = \frac {nx} {nx+1}$ has these properties. We have that, $$\begin{...

Induction shows that an equality holds for all values of $n$. It doesn't show that this is the only equality or formula for the expression that may hold true, correct? For example, say a question asks to find an explicit formula for a functional equation given by $f(n) = f(f(n-1))+f(n-f(n-1))$ fo...

function f_d2(a,b,y0) y=@(a,x) a*x*(1-x); hold on; for j=a:(b-a)/999:b x=y0; for n=100 x=y(j,x); end end hold of...

I am trying to do the Exercise 2.3.7 in Weibel's "An introduction to homological algebra". By definition, need to construct an map $\tau:\text{Hom}_{\mathcal{A}}(k^{th}(F),A)\rightarrow\text{Hom}_{\mathcal{A}^{I}}(F,k_{\ast}(A))$, where $A\in\mathcal{A}$ and $F\in\mathcal{A}^{I}$, and show it is ...

I am using the lecture notes here on page 19 (Algorithm Notes 1) example 1 is the inductive proof of log(n) = O(n) I understand the base case but I don't understand the rest of the example. I need help understanding how log(n + 1) <= log(2n) I don't get where log(2n) came from.

Suppose $G$ is solvable, $N \vartriangleleft G$. Let $f \in Hom(G,H)$. We have a normal series $\{e\}=G_0 \vartriangleleft G_1 \vartriangleleft ... \vartriangleleft G_n = G$ with $G_{i+1}/G_i$ abelian. Let $H_i = f(G_i)$. We denote $f_{i+1}(G)$ as the composition of $f$ and the quotient map $q...

I was working through some of the exercises in Stoll's intro analysis text and came across this problem $Suppose f:[a,b]\rightarrow\Re $ is continuous. Let M=max{|f(x)|:x$\in$[a,b]}. Show that $\lim_{n\to \infty} (\int_a^b|f(x)|^ndx)^\frac{1}{n}=M$ I attempted a proof by contradiction, but had ...

Let $n > 1$ and $m$ and $r$ be positive integers. Prove that $(n^r −1)$ divides $(n^m −1)$ if and only if $r$ divides $m$.

I have understood the two things respectively: 1. Use a set of observations to construct a covariance matrix, and then compute the eigenvectors of the matrix. 2. The diagonalization the Hermitian operator $A=PGP^T$. The columns of $P$ are eigenvectors. The diagonals of $G$ are eigenvalues. How...

If A is the triangular area with vertices (0,0) (1,1) (10,1) show that Integral of( √(xy-y^2) )dA =6

I was wondering if you would be able to please kindly help me with the following questions by solving the equations for X. Answers used can be either the following: A, 180-A, 180+A, 360-A 1) Solve the equation for X (using one of the answers above) a)cosx=cosA b) tanx=tanA c) sinx=-sinA Thanks!

There are 5 cubes, each cube has a different color and on each cube the numbers 1-6. Someone throws the cubes.In which the set of the numbers that appear on the cubes has exactly 3 objects? I was thinking: we need 3 different numbers and then 2 numbers that appeared already, so - $6\cdot5\cdot4\c...

Prove using Mathematical Induction \frac{d^(n)}{dx} (x) = nx^(x-1)

If $f(x,y)$ is a joint pdf,I understand that, $\int_{-\infty}^\infty\int_{-\infty}^\infty \ \ f(x,y) dxdy=1$ but does this hold for the conditional expectation? $\int_{-\infty}^\infty \ \ f({y|x}) dy=1$

Assume the product $$\Pi_{i=1}^{\infty} \sum_{l \in A} f_i(l)$$ exists, where $f_i(l) \ge 0.$ Does this mean, that I can rearrange this so that $$\Pi_{i=1}^{\infty} \sum_{l \in A} f_i(l) = \sum_{l \in A^\mathbb{N}} \Pi_{i=1}^{\infty} f_i(l_i)?$$

It's just a quick question but I can't find it on Google and i dont know where in a topology book to look. Z_k is integers mod k. So how do I interpret S^3, the unit sphere in 4 dimensions, mod Z_k? Thank you.

Im looking for Tips and Tricks for integration. I know all the basic techniques (substitution, completing the square, parts etc.) I'm mainly interested in dealing with logs in the denominator and dealing with trig terms both above and below (such as lesser known identities etc.)

Which of the following is false? Any abelian group of order 27 is cyclic. Any abelian group of order 14 is cyclic. Any abelian group of order 21 is cyclic. Any abelian group of order 30 is cyclic. If we take $G=\mathbb{Z_3\times Z_3\times Z_3}$,then (1) is false.I know if order(G)=pq,where p,...

Prove that for every four sets A, B, C and D, if $A \subseteq B$ and $C \subseteq D$, then $A - D \subseteq B- C$ Since $A - D \subseteq B- C$ then $x \in A-D$ and $x \in C-B$. Then $x\in A$ and $x\notin D$ and $x\in B$ and $x\notin D$. Then we have $A \subseteq B$ Now I'm stuck. $x \notin C$ ...

Consider the space $l^p=\{(x_i);x_i\in \mathbb C:\sum |x_i|^2<\infty\}$ .Define a norm on $l^2$ by $||x||=\sqrt{\sum |x_i|^2}$. Prove that $l^2$ is closed and bounded but not compact. I know that in a finite dimensional space a set is compact iff it is closed and bounded.But here the space...

Q: A fair coin is tossed independently 5 times. Let X denote the difference between the number of heads and the number of tails obtained. find the probability mass function of X here's my take: P(X=0) = all heads or all tails = 1/32 + 1/32 = 2/32 P(X=4) = one head or one tail = 5C1 x 1/32 + 5...

I really need help with this problem I have no idea how to approach it. Whenever there is an intersection with proofs i always get confused.

Just a disclosure, this may be a rather lengthy post. Thank You for taking the time to read this. I thank you in advance for your contribution. I'm currently working on a brief report on the Telegrapher's Equation for my fractional calculus course. I am still new to Telegrapher's Eq...

Let $f(x) \in \mathbb Z[x]$ be an irreducible monic polynomial such that $|f(0)|$ is not a perfect square . Then is $f(x^2)$ also irreducible in $\mathbb Z[x]$ ?

Why is $\mathbb Z_3[x]$ not isomorphic with $\mathbb Z$ ? (This question arose in trying to determine whether there is a commutative ring $R$ with unity such that $R[x]\cong\mathbb Z$ . It is easy to see that if such a ring exists then $R$ must be a field with two units i.e. $R \cong \mathbb Z...

$$f(n)=cot^2(\pi/n)+cot^2(2\pi/n)+...+cot^2((n-1)\pi/n)$$ then how to find limit of $\frac{3f(n)}{(n+1)(n+2)}$ as n tends to infinity? I dont know any series like that.Riemann sum is not working.What to do?b

Given a square matrix $A$ of order $2n$ such that $a_{ii}=0$ and $a_{ij}\in\{-1,1\},\space i\neq j$, prove that $\det(A)\neq0$.

For the following homogeneous system of constant coefficients initial value problem, I have used a method involving matrices and confirmed that I have found the correct eigenvalues, however, I have not been successful in finding the solution to the overall initial value problem. Consider the sy...

Using the definition in Bartle's Introduction to Real Analysis, I am trying to gain an intuitive understanding of limits that tend to infinity. Given Definition: Let ($x_n$) be a sequence of real numbers. (i) We say that ($x_n$) tends to $\infty$, and write $lim(x_n) = +\infty$ , if for every ...

Let the set S be a collection of all x in R such that |x|>1. Show that the set is open. I was thinking about taking some b within the set. Then constructing an open ball with radius (b-1)/2. Therefore the ball is open. Since the set is contained within S, then the Set s is also open.

What will be the minimum value of $\frac{4}{4-x^2}+\frac{9}{9-y^2}$ ? It is given x and y lies between -2 and 2 and xy=1

f(x,y) = 2xy/(x^2+y^2)^n when (x,y) is not equal to (0,0) = 0 when (x,y) when (x,y) is equal to (0,0) Prove that f is cont. at (0,0) if and only if n=1/2 (n>0)......Thank you very much for answering

Suppose $X$ is a manifold, $Y$ is a Hausdorff space and $p: Y \rightarrow X$ is a local homeomorphism with the path lifting property. Prove that $p$ is a covering map.

I was looking in my differential equations textbook and I found an interesting problem and I have no idea on how to approach it. I am supposed to let $F(s) = \mathcal{L} \{f(t) \} $ where $f(t)$ is piecewise continuous and of exponential order on $[0,\infty)$. Show that $$\mathcal{L} { \{\int^{...

Is it correct to say that $2^{2^n}$ is exponentially larger comparing to $2^{n}$ ?

Shouldn't it just be the largest previous integer? What is there a remainder term $O(1)$? Thanks,

I am copying it directly from Dummit and Foote. A function $f:\mathbb{R}\rightarrow\mathbb{R}$ is said to have compact support if there are real numbers $a,b$ (depending on $f$) such that $f(x)=0$ for all $x\notin[a,b]$ (i.e., $f$ is zero outside some closed bounded interval). The set of all ...

The reduction formula states as: enter image description here for integration of cos^n x But if n is negative integers like -1, -2,-3,... then can this reduction formula still be applied?

If $a_n = 1$ then why is $$A_n = \sum_1^n a_n = 1+ 1+ ... + 1$$ equal to the floor function $x + O(1)$? Thanks,

Consider a ratio having complex numerator and denominator $(a+bi)$/$(c+di)$. when will this ratio become real ? the obvious answer is when imaginary part is 0. but when will it become real?

I want to make a simple chart, comparing these methods, in terms of convergence, speed, iterations, stability, accuracy, x steps. Can anyone help me please? For example I know that C-N is more accurate than explicit( CN>E). But I'm not so sure about the others. Thanks in advance.

Consider a probability distribution function f(x) where f(x)=3x^2 for 0≤x≤1 f(x)=0 otherwise A)What is the probability that 1/4≤x≤3/4? B)What is the probability that 1/2≤x≤1? C)What is the mean of this distribution? D)Show that the probability distribution function integrated from 0 to 1 equal...

-2/3√(5/12u) What I did: turned denominator and numerator into square roots √5 / √12u simplified denominator to 2√3u and the 2 is multiplied by -2/3 to make -4/3 √5/√12u I then multiplied denominator and numerator by denominator to get 4√15u/9u correct answer: -√15u/9u what did i do wrong tha...

Consider the following summation: $$A = \sum_{i=1}^n \log (n/i)$$ Is this summation of order small oh of $n$? (or $A \in o(n)$?)

(Not sure the tags are appropriate, but can't think of better ones. Please suggest better.) Suppose you have a function $f(x,y,z,...;g(r))$ with the requirement that $r\rightarrow1/r$ leaves $f$ invariant. That works if $g(r)=\frac r{r^2+1}$. Are there other $g$'s such that $g(r)=g(1/r)$? More g...

Can someone show me how the following two expressions are equivalent: $$\Gamma = \frac{i X - R_c}{i X + R_c} = e^{-i 2 \mathrm{tan}^{-1} (\frac{X}{R_c})}$$ I'm working through a calculation and I am not sure how this step is done.

I'm asked to prove that with S being any non singular matrix , if M is a symetric positive definite matrix then S^TMS is also symetric positive definite.

This is part of a proof to show that a homomorphism is one-to-one iff gcd(m,n)=1. I know it is one-to-one iff kernel of theta is e. So,(a^m)=e and (b^n)=e I am at the part where (a^m)(b^n) = e My current line of thinking is to try and get ab^(mn) = e. But I would appreciate any other opinions. ...

I study group von neumann algebras and I want to know the relationship between the cardinal of group and cardinal of its von neumann algebra?

enter image description here Can anybody help me with pats b and c i get an inequality with 2 variables which is sort of unreasonable, we understand x is a length and cannot be negative

How many solutions are there? obviously one is a = b and -b anything else?

Prove that monotonic surjective function $f : U\rightarrow V$ is continuous. ($U$ and $V$ are intervals in $\mathbb{R}$). Thanks for any hints.

Let $A$ be a real symmetric matrix of size $n$ ; is $I_n+A$ always non-singular ? Is $I_n - A$ always singular ?

Does a divergent series have a majorant series? Is there some other name for majorant series? I cannot find much litterature about majorat series.

Let $(X,D)$ be a projective variety and $D$ be a divisor on $X$ with vanishing log Kodaira dimension, then first chern class $c_1(K_X+D)$ vanishes?

I have a matrix |2 -1/2| |1 1 | how to define matrix is rotation or reflection?

I am studying the chapter of my book about transforming polynomials but I don't understand how the reciprocal polynomial is found. This is an excerpt from my book : Let the given polynomial be $f(x)$ and the roots be $r_1,r_2,r_3,$ and $r_4$. One equation whose solutions are the reciprocal...

If 28 objects are arranged in a circle at equal distance from each other, in how many ways can 3 objects be chosen such that no two are adjacent or diametrically opposite.

The question is: If |G|=30 and |Z(G)|=5, what is the structure of G/Z(G)? I don't know what do we mean by 'structure' asked in the question. Please help.

I have been struggling to solve this quadratic equation in the variable $x$ with integral coefficients: $$ax^2-4bx+4bc-\dfrac{d^2}{a}=0$$ $a\neq 0$ of course.How do I ensure that $x$ is an integer? What I have done: $$\Delta^2=(-4b)^2-4a(4bc-\dfrac{d^2}{a})$$ $$\Delta^2=16b^2-16abc+4d^2$$ I ...

Consider y''+y=0 and its solution candidate cos(t). Using MATLAB I would like to substitute the candidate in the differential equation and get verification that it is indeed a solution. How can I do it? ps. I already know that the candidate is a solution, however, when I learn the procedure I wil...

How can this problem be solved? $$ \lg_{2} \left( \prod\limits_{a=1}^{2015} \prod\limits_{b=1}^{2015} (1 + e^{\frac{2\pi iab}{2015}}) \right) $$

If $L$ is a regular language then is $L'=\{w|wx \in L$ for some string $x\}$ regular? First step is understand L'. So it is a subset of L that contains strings with a certain prefix?

I am looking for a real symmetric matrix $A$ of size some $n$ such that $I_n+A$ is singular but $I_n-A$ is non-singular . Please help . Thnaks in advance

I having problems understanding one of the parts in Proof 5.2.2 in Kurzweil & Stellmacher group regarding Centralizers of Fitting subgroup. A question about this proof has been asked here before: Question on Proof that $O_p(C/(C\cap F(G)) = 1$ for $C = C_G(F(G))$. Lemma: Let $C := C_G(F(G))$. ...

How to solve this question: integration(0 to infinite)[integration (x to infinite)[(e^(-y/2))/y]dy]dx One idea that i have is to put -y/2=t then subsequently come to the result as the integration of 1/log t but what is the integration of 1/log t Is there any other way for solv...

I want to determine the total change of argument of the imaginary axis under the mapping f given by $f(z)=z^5+5z^3+2z^2+4z+1$. I substitute $z=it$ into $f$ and I get $$f(it)=(−2t^2+1)+i(t^5−5t^3+4t)$$ The parametric plot is given below What is the total change of argument?

it is possible to calculate in closeform $$\sum _{k=1}^{\infty } \left(\frac{2 x}{k}-\frac{\sin \left(\frac{2 \pi n x}{k}\right)}{\pi n}\right)$$ it seem do not convergent for any value of x or n??.. it an interesting question about the dirichlet problem

Does anybody know some book on zeta regularization, and the zeta regularization product? I'm quite interested on the topic but I would need a book with some review...

I am confused when I come across this question, could anyone help? Thanks! Let X1 and X2 have independent gamma distributions with parameters α, θ and β and θ respectively. Now we let W = X1/(X1+X2), could you show W has a beta distribution? Actually is it something about the Jacobian? Or shoul...

Suppose a trickster has three six-sided dice all of which evenly weighted (so each face is equally likely). One has all 6s, one has half 6s and half 1s, and one is a normal die. The trickster randomly chooses one of these dice and rolls it. If it lands 6, what is the probability it is a fair die?...

Find the nth derivative of y=4/(6x+8)^3 y'=4(-3)(6)/(6x+8)^4 y''=4(-3)(-4)(6)^2/(6x+8)^5 y'''=4(-3)(-4)(-5)(6)^3/(6x+8)^6 I recognise the pattern but cant interpret that into a formula to be more specific i am struggling with factorials

How would I go on to prove that $\sqrt{n^2 + 1} < n + 1$?

(Quant Job interviews Questions and Answers Q3.22) Suppose we have an ant travelling on edges of a cube going from one vertex to the other. The ant never stops and it takes it one minute to go along one edge. At every vertex the ant randomly picks one of the three available edges and starts goi...

How do I find the LHS for finding the total number of sets of k items each selected from N items. Order does not matter. For e.g. 1+2+3+...+n = n(n+1)/2 How do I find the LHS for my query? RHS is n(n-1)(n-2)/6 and the question is "Show that total number of triples selected from N items is preci...

(2^(xy)-1)/(abs(x)+abs(y)) I got to the conclusion there is no limit, but don't sure how to prove it.

what does this locus mean: $$Re(z-z1/z-z2)=0$$ I know it is a circle but can anyone tell a geometric approach. just by looking at the equation ?

Is there a faster way than doing a gigantic truth table? I tried some transformation but didn't find a way to simplify the problem.

I need to define a sequence of functions that converge to f so that I can use the monotone convergence theorem to prove. Not sure how to define the sequence of functions. \int_{\mathbb{N}} f\,\text{d}\mu = \sum_{n=1}^\infty f(n)

How to prove that $5^{20} + 2^{14}$ is the composite number?

Let's take an ellipse with the standard equation $$\frac{x^2}{9}+\frac{y^2}{4}=1$$ And I am trying to invert the following ellipse across that ellipse $$\frac{4x^2}{3}+4\left(y-\frac{3}{2}\right)^2=1$$ I obtain a very strange curve with the equation: $$\frac{4\left(\frac{36x}{9y^2+4x^2}\rig...

Consider y'''(t)+y'(t)= sec t. Is ln|sec t + tan t| - t cos t + (sin t) ln|cos t| a solution for this differential equation?

The image of a circle under conformal map 1/z should be a circle, but how to prove it (or how to find the relationship between the two circles)? z = x+iy = d + aexp(itheta), where a is the radius of the circle and d is a real number, so the it is a circle with radius a displaced along the real a...

The question is: Find the complete integral of (x+y)(p+q)² + (x-y)(p-q)²=1 I tried by Charpit's method. On solving, I got dp/(2p²+2q²) = dq/4pq Since it is a homogeneous equation, on further simplifying it, ⇒ 1-(p/q)²=c/q , where c is a constant but on putting the above value of p in the given...

Let $A,B,C$, and $D$ be positive constants. What's the most concise way to express $x$ in the equation below? $$ A = B\arctan(x/C)+Dx$$

Please are these graph non isomorphic? and what is the main reason?

Can $\pi$ be represented as $\lim\limits_{n\to\infty}\left(\sqrt{k_n}-\sqrt{m_n}\right)$, where $\{k_n\},\space \{m_n\}$ are sequences of positive integers?

How Solve it to minimum number of literals i can't understand basic properties to simplify this expression (A̅+C)(A̅+C̅)(C+D)(B̅+D)(A+B+C̅D)(A+B̅+C) explain me to understand concepts of simplification!

Let $f: [a,b]\rightarrow [c,d]$ be a surjective and strictly increasing function. Show that $f$ is continuous. I have already proved that any function $f$ in $\mathbb{R}$ is continuous if and only if the preimage of a closed set is again a closed set. Thus it is enough to prove that if $x\le d...

my tuition fees is $700 per month. last month, I gave $1000 extra. this month, I have given $500. how much money should I get back?

A particle of mass $m$ is projected vertically upwards with speed $v_0$. The resistance force is of magnitude $m$$\lambda$$v^2$. Show that the particle comes to rest after rising through a distance: $h$ = $1/2\lambda$ $(ln(1+\lambda v_0^2/g)$

Let $\alpha_1^1x_1+\ldots+\alpha_n^1x_n=1$ $\ldots$ $\alpha_1^mx_1+\ldots+\alpha_n^mx_n=1$ are equations in $\mathbb{F}_2^n$. I am trying to prove that if every solution of $\beta_1^mx_1+\ldots+\beta_n^mx_n=1$ is in one of them then $(\beta_1,\ldots,\beta_n)$ is linear combination of $(\a...

Find all integers a, b ,c such that they satisfy both the following equations (i) a^2=bc+1 (ii) b^2=ac+1 I tried solving came out with following results, (i)-(ii) we will get a+b+c=0 implies, a^3+b^3+c^3=3abc Multiplying equation (i) with "a" and Multiplying equation (ii) with "b and then ...

i am a physicist and i want to attend some extra mathematics courses at my university but i do not know if i have the mathematical background for them. Namely, i want to attend Classical Differential Geometry, Topology and Riemann Geometry. Could you please explain to me the necessary background ...

So the question is "How many different topologies exist on A union, where is |A| = n"?

Can someone please help me? Im not sure what i have to prove. That in the lexicographic ordering on AxA there is the smallest element?If yes how can i do it?

Maybe I seems trivial but, it is always so? How can I prove it?

Suppose 32 objects are placed along a circle at equal distances. In how many ways can 3 objects be chosen from among them so no two of the three chosen objects are adjacent nor diametrically opposite. This is again a problem in a math contest in India and this is how I tried it: Number of ways o...

enter image description here Can anyone please answer this question. I have only little idea regarding this thing

a) f(x) is a fifth degree polynomial. It is given that f(x)+1 is divisible by (x-1)^3 and f(x)-1 is divisible by (x+1)^3. Find f(x) b)a,b,c are real numbers such that a+b+c=0 and a^2+b^2+c^2=1. Prove that a^2b^2c^2 < 1/54. When does the equality hold?

Sorry for my weak foundation. When I read Evans' book (picture below), I don't know what is the standard existence theory. Then I wiki it ,and I can't find. So, what I should read or google?Thanks.

I don't know how to find the sum (not decimal number) or the least upper bound of infinite series $\sum_{{n=1}}^{+\infty} \frac{(k!)^2}{(k^2)!}$

I have just read about two eggs problem. I know that with decreasing amount of jumps we can reach worst case scenario of first jump a = root(2N), n is the number of floors, how about the lower bound of the problem? I am having trouble in coming up with a solution This is my where i read about ...

Let's have equation $$ \\ddot{x} + x^3 - x + a = 0, $$ where $a$ is parameter. How to analyze the solution of following system on stability? The problem is in equilibrium point, since it is determined from qubic equation $x^3 + a - x = 0$. Then in linear approximation corresponding system has th...

Let $\Omega \in {{R}^{n}}$ and $u,v\in {{C}^{2}}\left( \Omega \right)\cap C\left( {\bar{\Omega }} \right),\text{ }f\in {{C}^{1}}\left( R \right)$ such that ${f}'\left( t \right)\ge 0$, for all $t\in R$. Assume $$\left\{ \begin{align} & \Delta u-f\left( u \right)\ge \Delta v-f\left( v \righ...

I have a question in game theory it says :Consider two colonels A and B ruling two opposing armies. Each colonel has 120 soldiers in their armies and there are 6 battlefields. In each battlefield an army wins if and only if it has more soldiers than the opposing army. Now, an army wins the war, i...

Let $f_1(x),...,f_n(x) \in \mathbb Z[x]$ be $n$ polynomials ,then is it true that there exists a reducible polynomial $g(x)\in \mathbb Z[x]$ such that $g(x)+f_i(x)$ is irreducible in $\mathbb Z[x]$ (,or $\mathbb Q[x]$) , $\forall i=1(1)n$ ?

lim(2 + 2*x*sin(4/x)) as x approaches infinity

I need to prove that the following sequence (1+(1/n^2))^n converges towards 1. I tried to use the Bernoulli inqeuality, however, that is not very useful since in the original sequence is a plus sign. I then tried to use the Sandwich Theorem by finding two sequences which would make bounds for the...

Good day! I have this coirdinate equation: $$\frac{gt^2}{2}+{v_y}t-\frac{5}{3}R=0$$. How i can express variable $t$ from this equation? I calculated this as quadratic equation, and caught this: $$t=\sqrt{\frac{10R}{9g}}$$. But on site where i checked result, placed this expression: $$t=\sqrt{\f...

I'm doing a past exam paper and the question asks: Which of the following apply? a. The sequence were the n'th term is given by nlog(n) has growth rate that can be bounded from above by a polynomial b. 2^n is larger than n^100 when n > 1000 c. The sequence 2,4,8,16,32 where the n'th term is g...

How is: 7^7 mod 55 the same as 28 mod 55? This is taken from this youtube video (5:24): https://www.youtube.com/watch?v=O-4_oS3G7MI

Can someone explain how it is possible to have the number of elements in a spanning set be greater than the dimension of a vector space?

I am trying to understand Milnor's proof of the Hairy Ball Theorem (here's the proof I am trying to read: http://people.ucsc.edu/~lewis/Math208/hairyball.pdf). In lemma 1, he first considers the case that the compact set $A\subset \mathbb{R^n}$ is a cube and proves that given $x,y \in A$, there e...

How to show that $\mathbb Q(i,a)=\mathbb Q(a)(i)$ ? Is it by definition ?

I am studying algebraic graph theory with a shaky basic. As I am new to the topic, I would be thankful if anyone help me to understand the following proposition. $A, B$ are matrices of size $m \times n$ (not symmetric matrices). Given, rows of $A, B$ are fixed, then each of them can have $n...

In any triangle, what is minimum possible value of: $ frac{r_{1} r_{2} r_{3}}{r} $ I reduced its value to $ frac{s^{4}}{Area^{2} $ , But I don't know how to proceed now.

Could someone show how $nCr+(n+1)Cr+(n+2)Cr+(n+3)Cr=(n+3)C(r+1)$ ? I tried expanding but in the end nothing really got cancelled to prove the identity

Give a differential equation in R2 that has a counterclockwise limit cycle inside a clockwise limit cycle.``

A friend of mine told me that the cohomology of $\pi_1(M)$ was isomorphic to the cohomology of the manifold $M$. Is that true (maybe there are some hypothesis) ? Does someone know a reference for this result, and maybe an explanation why this should be intuitively true ? Thanks in advance.

Can anyone help me here? Question: "X is a normed space and A is a subset dense in the dual of X. x belongs to X and the sequence (x_n) of X is bounded of E such that f(x_n) converges to f(x) for all f in A. Show that x_n converges to x weakly" My try: I think that if I show that A=cl(A) so I ...

I have the following problem.The jacobian matrix is given in the image below.I just cannot seem to figure out how they arrived at the determinant.Can anyone show the steps or elaborate the procedure?

Eric has got 3 sum wrong each time he exactly pressed one wrong key 5+3+2=317 25+36=900 8+8+2=3 can you work out which key he actually pressed

Given some undirected graph $G=(V,E)$, $|V| \geq 2$ I want to prove that there are at least 2 vertices $x,y \in V$ s.t $G-x,G-y$ are connected. I'd like someone more experienced to review my proof. Firstly, a lemma: Let $G=(V,E)$ be an undirected connected graph. Let $v$ be some vertex in the ...

I ran across this Euler sum while trying to evaluate an integral. I mentioned it in another thread, but though perhaps asking about it separate may be a good idea. $$\sum_{n=1}^{\infty}\frac{H_{2n}}{n(6n+1)}$$ Numerically, it converges to around $0.502788$ I found this while trying to evaluate...

If the answer is yes, what is the intuition when to use ordinary $GF$, and when to use exponential $GF$?

How must an equation look like, for it to not be possible to put every variable one side. For example in $2x=4y$, I can highlight $x$ by saying $x= \cfrac {4y}{2}$, and $y$ by saying $y= \cfrac {2x}{4}$. So what makes it impossible to do this in some equations?

Prove the following mutual information in equality $$ I(X;Z|Y) \geq I(Z;Y|X) - I(Z;Y) + I(X;Z)$$

Question Consider $n$ people who are attending a party. We assume that every person has an equal probability of being born on every day of the year, independent of everyone else. Assuming that nobody is born on the $29^\mathrm{th}$ February and that $n\leq365$, find the probability that each pers...

If A and B are normal linear transformation on the finite-dimensional complex inner product space X such that AB=0, does it follows that BA=0?

Calculate Double integral $$\iint_D (x^2 + y^2 ) dxdy$$ where: $$D=\{(x,y)\in\mathbb{R}^2 : x\le x^2+y^2\le2x, -x\le y \le x \}$$ What i did? I tried to use polar coordinates and i got this ==> $\sqrt x\le r \le \sqrt(2x)$ Can you please help me with the limit of integration if i change this...

what is the limit of f(x) = (sinx * sin(1/x)) as x approaches 0 ?? how many ways there are to solve this?

$\lim_{x\to \infty}\frac{5x^2-4}{x-x^2} = \frac{\infty}{\infty}$ I know I will have to use L'Hôpital's rule to solve. However, I confused as to how the infinity sign is calculated. At first I thought the infinity sign to be just a large number. The only difference being whether it was a large...

Suppose we have nonlinear system of differential equations $$ \frac{d\mathbf x}{dt} = \hat{A}(\mathbf x, \mu)\mathbf x $$ How to show that it has periodic solutions?

$y=(a^x)^y$ deriving according to $x$: $y={e^{lna}}^{x+y}$ $\frac{dy}{dx}={e^{lna}}^{x+y}\cdot(\frac{x+y}{a}+(1+\frac{dy}{dx})lna)$ $\frac{dy}{dx}=\frac{x+y}{a}\cdot a^{x+y}+a^{x+y}lna+a^{x+y}lna\frac{dy}{dx}$ $\frac{dy}{dx}(1-a^{x+y}lna)=\frac{x+y}{a}a^{x+y}+a^{x+y}lna$ $\frac{dy}{dx}...

There is the one of my tries: 5=2^{log^_25}5 So then i sholud prove that {log^_25}5 is bigger than sqrt(7).

Let $X_{N_1},\cdots,X_{N_n}$ is a sample without replacement from the set $\{1,2,\cdots,N\}$, and let $\bar X_n=\sum_{i=1}^n X_{N_{i}}/n$. Then, how one can find $E(\bar X_n)$, $Var(\bar X_n)$, $\max_{1\le i\le N} (X_i-\bar X_n)^2$, and $\sum_{i=1}^N(X_i-\bar X_n)^2$?

Let $K$ be a compact topological group, and let $(V,\pi)$ be a continuous representation of $K$ over the complex field $\mathbb{C}$. Denote by $\mathrm{d}$ the Haar measure on $K$. If $v\in V$ satisfies $\int_K\pi(k)v\mathrm{d}k=0$, does $v$ always lie in the $\mathbb{C}$-vector space spanned by ...

Around ten years ago I had read somewhere that there was a question in an exam for application for software engineer position in a big company which states: "What is the one billionth digit of $\pi$?" Can we predict the $n$th digit of $\pi$ without knowing any preceding digits at all?

function question2c(a,b,x) T=a:(b-a)/999:b; y=zeros(length(T),1); y(1)=x; hold on; for j=2:length(T) y(j) = T(j-1)*y(j-1)*exp(-y(j-1)); plot(T(j), y(j)); end hold off; end It sure looks like one of my pics of a different equation in my textbook, however could someone double chec...

Let $(X_t)_{t\ge0}$ be a (homogeneous) stochastic process that satisfies the stochastic differential equation $dX_t=b(x)dt+\sqrt{a(x)}dW_t$, $t\ge0$, where $X_0=x_0$ is given, and $W_t$ is Brownian motion. Suppose also that $X_t$ is such that its state space is the interval $[0,H]$ with the left ...

This is the one of my tries: $5=2^{log_25}$. Then i should prove that: ${\sqrt 7}>{log_25}$. So can you help me end this proof or suggest another?

Suppose we have given constants $A_i, x_i (i=1..N)$ Is it possible to approximately calculate the sum of N gaussians in less than N iterations for any x? (may be with some preprocessing) $$\sum_{i=1}^{N}A_i e^{-(x-x_i)^2}$$ The same question for 2D case $$\sum_{i=1}^{N}A_i e^{-(x-x_i)^2-(y-y...

I worked out with symplectic matrices that the transpose is also simplectic for the 2x2 case since the algebra was easy and the determinant of the matrix just needed to equal 1. [The expression for det is easy for 2x2] and transpose has equal det. But what about the nxn case. Is it also true?

Today, I encountered a problem in "Problem-Solving Strategies" by Arthur Engel (Chapter $9$. Sequences, page-$225$). it is as follows: There does not exist a monotonically increasing sequence of nonnegative integers $a_1,a_2,a_3,...$ so that $a_{nm}=a_n+a_m$ for all $n,m \in \mathbb{N}$. I coul...

I have to solve $(z+1)^3+i(z-1)^3=0$ I tried it in many ways.but I couldn't come up with an answer.can anyone give me any hint?

I read the following article: On some exact distribution-free (...). It is a statistical paper but my question is a bout some notion of graph theory they use. They have a complete graph $\mathcal{K}_{2n}$ whose vertices are $\{z_{1},\dots,z_{n},z^{\ast}_{N},\dots,z^{\ast}_{N}\}$. There is some c...

n=240 trials with a 6sided dice X = #5's Y = #6's how do I go about showing that Cov(X,Y) = -20/3 ? I think I need to find V(X+Y) but im not sure how. V(X)=V(Y)=240* 1/6 * 5/6

Find the harmonic conjugate of u=Im(e^z^2) (use a as your constant of integration.) I have tried the solution as (iez^2) but the answer doesn't seem to be correct. Please help

I am trying to solve this problem: use $ x^2+1$ (polynomial interpolation) to reduce $$ \frac{(x^2+1)}{x(x-1)(x-2)(x-3)}$$ I don't know how can i reduce a fraction by interpolation method.