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658080 | Complex numberss
The solutions to the equation [imath]x^2-(3-2i)x+(1-3i)=0[/imath] are [imath]a-i[/imath] and [imath]b-i[/imath] where [imath]a[/imath] and [imath]b[/imath] are integers. What is [imath]a+b[/imath]? [imath]x^2-(3-2i)x+(1-3i) = (x+(a-i))(x-(a-i))[/imath] [imath]= x^2 - 2xi-a^2 - 1[/imath] [imath]1 = x^2 - 2xi - a^2[/imath] I'm not sure where to go from here | 657927 | Linear Algebra Complex Numbers
The solutions to the equation [imath]z^2-2z+2=0[/imath] are [imath](a+i)[/imath] and [imath](b-i)[/imath] where [imath]a[/imath] and [imath]b[/imath] are integers. What is [imath]a+b[/imath]? I simplified and got [imath](z+1)(z+1) = -1[/imath] and now I'm not sure where to go from there. I did this but I'm not sure. [imath](a+i)^2 = a^2 - 1[/imath] [imath](b-i)^2 = b^2 + 1[/imath] [imath]a+b=(a+i)+(b-i)=(a^2-1)+(b^2+1)=a^2+b^2[/imath] |
658507 | Subring of Gaussian integers has no greatest common divisor property
Problem is: Produce elements a and b in the domain [imath]R := \{x+2y\sqrt{-1} \mid x, y \in \mathbb{Z}\}[/imath] having no gcd. How can produce this? Actually I use norm function, and brute force, but what I found is that, except in case of y = 4, so many values of norm function are prime. So I can't handle it. Do you have any idea? | 657058 | Find two elements that don't have a gcd in a subring of Gaussian integers
Find two elements in the domain [imath]R := \{ x + 2y \sqrt {-1} \mid x,y \in \mathbb{Z} \}[/imath] that do not have a gcd. I have no idea how to start. But I know if we consider [imath]R^\prime = \{ x + y \sqrt {-1} \mid x,y \in \mathbb{Z} \}[/imath] then every two elements have a gcd. So there must be something wrong with the 2 here. Any help is appreciated. |
658207 | Let [imath](G,\cdot)[/imath] be a group and [imath]a\in G[/imath]. Define [imath]x\ast y:=x\cdot a \cdot y[/imath]. Show [imath](G,\ast)[/imath] is a group.
Let [imath]G[/imath] be a group with operation [imath]\cdot[/imath] and let [imath]a \in G[/imath]. Define a new operation [imath]*[/imath] on the set [imath]G[/imath] by [imath]x*y[/imath] = [imath]x·a·y[/imath] for all [imath]x,y \in G[/imath]. Show [imath]G[/imath] is a group under the operation [imath]*[/imath]. Does this group under the operation have an inverse? I know you need to use [imath]x*b = a^{-1}[/imath] and [imath]b*x = a^{-1}[/imath]. But I am not getting them to have the same result. So far I have [imath]b*x = bax[/imath] so [imath]bax = a^{-1}[/imath] and [imath]x*b = xab[/imath] so [imath]xab = a^{-1}[/imath]. | 373731 | Showing that [imath]G[/imath] is a group under an alternative operation.
Let [imath]G[/imath] be a group and let [imath]c[/imath] be a fixed elements of [imath]G[/imath]. Now, I'm going to define a new operation "*" on [imath]G[/imath] by [imath]a*b=ac^{-1}b[/imath] How do I prove that the set [imath]G[/imath] is a group under *. Thanks for the help! |
659010 | Given [imath]A_1 = \{\emptyset\}[/imath] and [imath]A_{n+1} = A_n \cup (A_n \times A_n)[/imath] define [imath]A = \bigcup_{i=1}^\infty A_n[/imath], what is [imath]A\times A[/imath]?
Given [imath]A_1 = \{\emptyset\}[/imath] and [imath]A_{n+1} = A_n \cup (A_n \times A_n)[/imath] define [imath]A = \bigcup_{i=1}^\infty A_n[/imath], what is [imath]A\times A[/imath]? I'm kind of lost on this one, any help would be appreciated! | 553403 | Homework - set theory infinite union
A question from my homework I'm having trouble understanding. We are given: [imath]A(1) = \{\varnothing\}[/imath], [imath]A(n+1) = A(n)\cup (A(n)\times A(n))[/imath] [imath]A=A(1)\cup A(2)\cup A(3)\cup \cdots \cup A(n)\cup A(n+1) \cup \cdots[/imath] to infinity The questions are: 1) show that [imath]A\times A \subseteq A[/imath] 2) Is [imath]A \times A = A[/imath]? Thank you for your help. I've tried writing [imath]A(2)[/imath] but it gets really complicated and I'm having trouble understanding what the sets are. Let alone solve the question. |
651155 | Show that [imath]\operatorname{Spec}k[x_1,x_2,...,x_n]/(x_1^2+\cdots+x_m^2)[/imath] is normal for [imath]\operatorname{char}k\neq 2, n\ge m \ge 3[/imath]
I want to show that if [imath]F(T) \in B[T][/imath], where [imath]B:=k[x_1,x_2,...,x_n]/(x_1^2+\cdots+x_m^2)[/imath], is monic and has a root [imath]\alpha \in\mathcal K(B)[/imath] then [imath]\alpha[/imath] actually lives in [imath]B[/imath]. This will imply that [imath]B[/imath] is an integrally closed domain and hence that [imath]\operatorname{Spec} B[/imath] is normal. But I am stuck at this point. On the other hand, I know that the geometric interpration is that a quadric of [imath]\operatorname{rk} \ge 3[/imath] doesn't have "bad" singularities. Well, I know that such quadrics are irreducible, but that only helps me to show that [imath]B[/imath] is at least an integral domain. | 596666 | Proving normality of affine schemes
One of the exercises in Ravi Vakil's algebraic geometry notes, Ex. [imath]5.4.[/imath]I(b), is to show that [imath] \operatorname{Spec}\left(k[x_1, \ldots, x_n]/(x_1^2 + \cdots + x_m^2)\right) [/imath] is normal, where [imath]k[/imath] is any field of [imath]\operatorname{char}(k)\neq 2[/imath], and [imath]n \geq m \geq 3[/imath]. I have absolutely no idea how to get started on this. Is there anyone that could give a hint as to how one would approach this problem? |
660318 | Prove that [imath]gcd((a^{n}-b^{n})/(a-b), a-b) = gcd(n(a,b)^{n-1}, a-b)[/imath] for a,b [imath]\in[/imath] [imath]\mathbb{Z}^+[/imath]
Prove that [imath]gcd(\frac{a^n-b^n}{(a-b)}, a-b) = gcd(n\cdot gcd(a,b)^{n-1}, a-b)[/imath] is true for a,b [imath]\in[/imath] [imath]\mathbb{Z}^+[/imath] I can see how it could be true. I know [imath]\frac{a^n-b^n}{(a-b)} = a^{n-1} + ba^{n-2} + ... + b^{n-1}[/imath] Then [imath]gcd(a,b)^{n-1}[/imath] is going to be of the same order as [imath]a^{n-1} + ba^{n-2} + ... + b^{n-1}[/imath] since this polynomial is homogeneous. I do not know why there is an n being multiplied out in front of [imath]gcd(a,b)^{n-1}[/imath]. Using the Euclidean Algorithm would be kind of difficult with n-cases where n is unknown, even trying to use induction. | 247146 | Show that [imath]\gcd\left(\frac{a^n-b^n}{a-b},a-b\right)=\gcd(n d^{n-1},a-b)[/imath]
How to show that [imath] \gcd\bigg( {a^n-b^n \over a-b} ,a-b\bigg )=\gcd(n d^{n-1},a-b ) [/imath] [imath]a,b\in \mathbb Z[/imath] where [imath]d=\gcd(a,b)[/imath]? Note [imath]\ [/imath] Some of the answers below were merged from this question. The answers (and their comments) may depend on context provided in that question. |
660477 | if [imath]f(x) = \int_{t=1}^{t=x^2} t\sin^2(t)\operatorname d\!t[/imath] then [imath]\frac{\operatorname d\!f(x)}{\operatorname d\!x}=?[/imath]
[imath]f(x) = \int \limits_{t=1}^{t=x^2} t\sin^2(t)\operatorname d\!t[/imath] Do I use U-substitution and have the answer as [imath]f'(x) = 2x*x^2\sin^2(x^2)[/imath] Or does this question require integration by parts? Thanks. | 660431 | Finding derivative of this integral function.
I need help on finding the derivative of this: [imath]g(x) = \int_1^{x^2} (x-t)\sin^2(t)dt[/imath] I thought about taking out x and having it as a constant but how? |
262637 | Criteria for the difference of two matrices to be positive semidefinite when the eigenvectors are known
Let [imath]A[/imath] be a rank 1 positive semidefinite matrix and [imath]B[/imath] a Hermitian matrix. Suppose I know the eigenvectors of both [imath]A[/imath] and [imath]B[/imath] and that [imath]A-B[/imath] is also positive semidefinite. Apart from Weyl's inequality is there anything that can be deduced about the eigenvalues of [imath]B[/imath]? | 253004 | Under what conditions is the difference between a rank [imath]1[/imath], postive semidefinite matrix and a hermitian matrix positive semidefinite?
Let [imath]A[/imath] be a positive semidefinite matrix of rank [imath]1[/imath]. Let [imath]B[/imath] be a general Hermitian matrix. Under what conditions on [imath]B[/imath] (probably in terms of [imath]A[/imath]) is [imath]A-B[/imath] positive semidefinite? I was thinking that it may be along the lines of the generalized eigenvalue problem but can't quite see how. |
661252 | Proving An Equation
I have been revising basic compound angles and I am struggling to understand the following question from the examples I have previously studied on such topic. A first step, or point of direction/suggestion would be brilliant. Thank You. The instantaneous current ([imath]i[/imath]) and the instantaneous voltage ([imath]v[/imath]) in a pure resistance AC circuit is given by; \begin{equation} i=i_{max} \sin(\omega t) \end{equation} and \begin{equation} v=v_{max} \sin(\omega t). \end{equation} Since power is computed \begin{equation} P=iv=i^2 R \end{equation} Show that an equation for instantaneous power is \begin{equation} P=i_{max}^2 R \left(\frac{1−\cos(2\omega t)}{2}\right). \end{equation} | 660952 | Engineering Mathematics
I have been revising basic compound angles and I am struggling to understand the following question from the examples I have previously studied on such topic. I cannot envisage the compound angle formulae associating with this (if I am correct in saying it associates with this topic). A first step, or point of direction/suggestion would be brilliant. Thank You. The instantaneous current [imath]i[/imath] and the instantaneous voltage [imath]v[/imath] in a pure resistance a.c. circuit is given by; [imath]i=i_{\text{max}} \sin \omega t[/imath] and [imath]v= v_{\text{max}} \sin \omega t[/imath] Since power; [imath]P = iv =i^2 R[/imath] Show that an equation for instantaneous power is; [imath]P=i_{\text{max}}^2 R[1-\cos 2(\omega t)]/2[/imath] |
661496 | Harmonic function
Let [imath]B(0;1)=\{x \in \mathbb{R}^N;|x|≤1\}[/imath], the ball of [imath]\mathbb{R}^N[/imath] equipped with the euclidian scalar product [imath]x⋅y=x_1y_1+...+x_Ny_N,\ \ \ x=(x_1,...,x_N),\ \ \ y=(y_1,...,y_N)\ \ \ |x|=\sqrt{x \cdot x}[/imath] Let [imath]\alpha[/imath] and [imath]\beta[/imath] two real numbers, [imath]\alpha \geq0[/imath] , [imath]a \in \partial B(0,1)[/imath] fix. [imath]h(x)=\log\left((1−x\cdot a)^{\alpha}+|x−a|^{\beta}\right)[/imath] Can we find [imath]\alpha[/imath] and [imath]\beta[/imath] such that [imath]h[/imath] is harmonic in [imath]B(0,1)[/imath]? ( harmonic signifies [imath]\Delta_{x}h(x)=\displaystyle \sum_{i=0}^N \frac{\partial^2 h}{\partial x_i^2}[/imath] ). | 661178 | Choose parameters to make a harmonic function
Let [imath]B(0;1)=\{x\in \mathbb{R}^N ;|x|\leq 1\}[/imath], the ball of [imath]\mathbb{R}^N[/imath] equipped with the euclidian scalar product [imath]x \cdot y=x_1y_1+...+x_Ny_N,\ \ \ x=(x_1,...,x_N),\ \ y=(y_1,...,y_N)\ \ \ |x|=\sqrt{x\cdot x}.[/imath] Let [imath]\alpha\mbox{ , } \beta \mbox{ two real numbers, } \alpha\geq 0\mbox{ , } a \in \partial B(0,1)[/imath] fix. [imath]h(x)=\log\left((1-x\cdot a)^{\alpha}+|x-a|^{\beta}\right)[/imath] Can we find [imath]\alpha[/imath] and [imath]\beta[/imath] such that [imath]h[/imath] is harmonic in [imath]B(0,1)[/imath]? |
661608 | x raised to itself infinite number of times
[imath]\Large x^{x^{x^{x^{x^{.^{\,.^{\,.}}}}}}} = 2[/imath] What is [imath]x[/imath]? | 166433 | How to solve infinite repeating exponents
How do you approach a problem like (solve for [imath]x[/imath]): [imath]x^{x^{x^{x^{...}}}}=2[/imath] Also, I have no idea what to tag this as. Thanks for any help. |
661385 | Spivak Chapter 3 Question 3.
Working through Chapter 3 of Spivak's Calculus. The question is: If [imath]x_1, x_2, \ldots, x_n[/imath] are distinct numbers, find a polynomial function [imath]f_i[/imath] of degree [imath]n-1[/imath] which is 1 at [imath]x,[/imath] and 0 at [imath]x_j[/imath] for [imath]j \ne i[/imath]. Hint: the product of all [imath](x - x_i)[/imath] for [imath]j\ne i[/imath], is 0 at [imath]x_j[/imath] if [imath]j \ne i[/imath]. ( This product is usually denoted by [imath]\prod_{j=1,j\ne1}^n (x - x_j)[/imath] I stared at this for awhile and it never made sense. So I went to the answer key so I could work backwards to understand the problem. The answer is [imath]f_i(x) = \frac{\prod_{j=1}^n (x-x_j)}{\prod_{j=1}^n(x_i - x_j)}[/imath] I stared at this too and the problem was just as perplexing as before. I see that the answer expands to [imath]f_i(x) = \frac{(x-x_1)(x-x_2)\ldots(x-x_n)}{(x_i-x_1)(x_i-x_2)\ldots(x_i-x_n)}[/imath] but so what? How is this a solution to the problem? Assuming [imath]n = 2[/imath], we would have [imath]f_i(x) = \frac{(x-x_1)(x-x_2)}{(x_i-x_1)(x_i-x_2)} = \frac{x^2 - x_1 x - x_2 x + x_1 x_2}{x_i^2 - x_1 x_i - x_2 x_i + x_1 x_2}[/imath] This is some arbitrary rational function. To me, it doesn't feel like an answer. Here are some specifics I don't understand. 1) I see that we have a sequence of numbers. The index of the sequence is j. What is i supposed to be? Sure [imath]j \ne i[/imath], but how is that different from saying that [imath]j \ne blah[/imath] 2) The question says "a polynomial of [imath]f_i[/imath] of degree [imath]n - 1[/imath], which is 1 at [imath]x_i[/imath]". Does that mean [imath]f_i(x_i) = 1[/imath] or [imath]n - 1 = 1[/imath] when [imath]x=x_i[/imath]? 3) Similarly, what is meant by it equals "0 at [imath]x_j[/imath]" ? Can anyone shed some light on this? | 653059 | Help with Spivak Calculus Ch3 Problem 6a
Yet again I find myself stuck on a Spivak question. This time it is simply the question that isn't clear to me. It states: If [imath]x_1, ..., x_n[/imath] are distinct numbers, find a polynomial function [imath]f_i[/imath], of degree [imath]n-1[/imath] which is 1 at [imath]x_i[/imath] and 0 at [imath]x_j[/imath] for [imath]j\neq i[/imath]. Hint: the product of all [imath](x-x_j)[/imath] for [imath]j\neq i[/imath], is 0 at [imath]x_j[/imath] if [imath]j\neq i[/imath]. This product is usually denoted by: [imath]\prod_{\begin{smallmatrix}{j=1}\\ {j\neq i} \end{smallmatrix}}^n(x-x_j)[/imath] To be honest I don't know where to begin. I understand how to create a polynomial equation for a given set of results but this is quite strange. I'm not sure exactly what the author is expecting. The answer book shows: [imath]f_i(x)={\prod_{\begin{smallmatrix}{j=1}\\ {j\neq i} \end{smallmatrix}}^n(x-x_j)}/{\prod_{\begin{smallmatrix}{j=1}\\ {j\neq i} \end{smallmatrix}}^n(x_i-x_j)}[/imath] Which I have expanded out just fine but don't know what to do with. |
661673 | The highest power of [imath](n^r-1)![/imath]
There is a problem which i tried a lot but didn't reach to any conclusions:How can i show that the highest power of n which is contained in [imath](n^r-1)![/imath] is [imath]\frac{n^r-nr+r-1}{n-1}[/imath]?My try:[imath](n^r-1)![/imath]=[imath](n^r-1)(n^r-2)(n^r-3)\cdot\cdot\cdot\cdot3.2.1[/imath].But what next can i do? | 176402 | What is the highest power of [imath]n[/imath] in [imath](n^r-1)![/imath]
What is the highest power of n in [imath](n^r-1)![/imath] where [imath]n[/imath], [imath]r[/imath] are positive integers? The answer supplied is [imath]\frac{n^r-nr+r-1}{n-1}[/imath] The highest power of [imath]n[/imath] in [imath]n^r! = r +[/imath] the highest power of [imath]n[/imath] in [imath](n^r-1)![/imath] as [imath]n^r[/imath] contributes [imath]r[/imath] powers. So, the highest power of [imath]n[/imath] in [imath]n^r![/imath] needs to be [imath]\frac{n^r-nr+r-1}{n-1} + r =\sum_{1≤t<r}n^t [/imath]. This is clearly true if [imath]n[/imath] were prime according to this. Again, if [imath]r=1[/imath], [imath]\frac{n^r-nr+r-1}{n-1}[/imath] becomes [imath]0[/imath]. But according to this, the highest power of composite [imath]n[/imath] in [imath](n-1)! ≥ 1 [/imath]. So, the answer does not seem to hold for [imath]r=1[/imath] if [imath]n[/imath] is composite. If [imath]n=\prod p_i^{a_i}[/imath], where [imath]p_i[/imath] are distinct primes, [imath]n^r=\prod p_i^{ra_i}[/imath] So, the highest power [imath](q_i)[/imath] of [imath]p_i[/imath] in [imath]n^r![/imath] is [imath]\sum_{1≤t<∞}\left\lfloor\frac{n^r}{p_i^t} \right\rfloor =\sum_{1≤t≤ra_i}\frac{n^r}{p_i^t} +\sum_{ra_i+1≤t<∞}\left\lfloor\frac{n^r}{p_i^t} \right\rfloor[/imath] as [imath]p^t|n^r[/imath] for [imath]t≤ra_i[/imath]. So, the highest power of [imath]p_i^{a_i}[/imath] in [imath]n^r![/imath] is [imath]\left\lfloor \frac{q_i}{a_i} \right\rfloor=P_i[/imath](say). So, the highest power of n=[imath]\prod p_i^{a_i}[/imath] in [imath]n^r![/imath] is [imath]min(P_i)[/imath] I could not proceed anymore. |
661896 | Riemann integrals and continuity
Let [imath]d>0[/imath] be a number. Is there a function [imath]f: \mathbb{R} \to \mathbb{R}[/imath] such that [imath]f[/imath] is not Riemann integrable on the interval [imath][a-d, a+d][/imath] if [imath]f[/imath] is continuous at [imath]x = a[/imath]? | 660982 | Integrating real valued functions
If we have a real valued function [imath]f[/imath] continuous at some point [imath]a[/imath], is it necessarily true that [imath]f[/imath] is Riemann integrable on the interval [imath][a - \delta, a + \delta][/imath] if a [imath]\delta > 0[/imath] exists? |
661979 | Real Analysis Integration Proof
Say [imath]f:[a,b]\rightarrow \mathbb{R}[/imath] is continous, [imath]f(x) \geq 0[/imath] for all [imath]x[/imath], and there is a [imath]c \in [a,b][/imath] such that [imath]f(c) > 0[/imath]. Prove [imath]\int_a^b fdx > 0[/imath]. | 644435 | Riemann integral proof [imath]\int^b_a f(x) \, dx>0[/imath]
Prove that if [imath]f[/imath] is a continuous real valued function on the interval [imath][a,b][/imath] such that [imath]f(x)\ge 0[/imath] for all [imath]x\in [a,b][/imath] and [imath]f(x)>0[/imath] for some [imath]x\in[a,b][/imath] then [imath]\int^b_a f(x) \, dx >0[/imath]. The definition I have for Riemann integral is: Let [imath]a,b\in \mathbb{R}[/imath] [imath]a<b[/imath] and let [imath]f[/imath] be a real valued function on [imath][a,b][/imath]. We say that [imath]f[/imath] is Riemann integrable on [imath][a,b][/imath] if there exists a number [imath]A\in \mathbb{R}[/imath] such that for any [imath]\varepsilon>0[/imath] there exists [imath]\delta>0[/imath] such that [imath]|S-A|<\varepsilon[/imath] whenever [imath]S[/imath] is Riemann sum for [imath]f[/imath] corresponding to any partition of [imath][a,b][/imath] of width less that [imath]\delta[/imath]. |
662035 | Finding limit of [imath]\prod_{t=1}^{n}{\left(1-\frac{2}{(n)(n+1)}\right)^2}[/imath]
Let [imath]x_n=\left(1-\frac{1}{3}\right)^2\cdot\left(1-\frac{1}{6}\right)^2\cdot\left(1-\frac{1}{10}\right)^2\cdot\left(1-\frac{1}{15}\right)^2\cdots\left(1-\frac{1}{\frac{n(n+1)}{2}}\right)^2 \quad ,n\geq2 [/imath] then [imath]\lim_ {n\to\infty} x_n[/imath] is (1) [imath]\frac{1}{3}[/imath] (2) [imath]\frac{1}{9}[/imath] (3) [imath]\frac{1}{81}[/imath] (4) 0 | 649789 | Finding limit of a sequence in product form
\begin{equation} \prod_{n=2}^{\infty} \left (1-\frac{2}{n(n+1)} \right )^2 \end{equation} I need to find limit for the following product..answer is [imath]\frac{1}{9}[/imath]. I have tried cancelling out but can't figure out. Its a monotonically decreasing sequence so will converge to its infimum.. how to find the infimum? |
659775 | Proving that the estimate of a mean is a least squares estimator
I think this is a really simple question so please bear with me -- I just had my first class in regression and I'm a little confused about nomenclature/labeling. Does anyone recommend some good weblinks that explain beginning linear regression really well? There's a question I've been looking at for a while and I'm not sure how to do it (although I'm sure the solution is simple): Show that the sample estimate [imath]\hat{\mu} = \bar X = \frac{1}{n} \sum X_i[/imath] is a least square estimator of [imath]\mu[/imath] for a variable [imath]X[/imath] given [imath]X_1, \ldots, X_n[/imath]. My first thought was, [imath]SSE = \sum (\mu - \hat{\mu})^2,[/imath] but I'm not sure if that's right. I'm confused about what the [imath]\beta[/imath] is (is it [imath]n[/imath]?) and I don't know if there are enough parameters to expand it. Thanks so much for your patience and if this doesn't make sense, I can clarify more. Thanks! | 633440 | Proving that the estimate of a mean is a least squares estimator?
I think this is a really simple question so please bear with me - I just had my first class in regression and I'm a little confused about nomenclature/labeling. Does anyone recommend some good weblinks that explain beginning linear regression really well? There's a question I've been looking at for a while and I'm not sure how to do it (although I'm sure the solution is simple): Show that the sample estimate [imath]\hat{\mu}(X) = \frac{1}{n} \sum X_i[/imath] is a least square estimator of [imath]\mu[/imath] for a variable [imath]X[/imath] given [imath]X_1, \ldots, X_n[/imath]. My first thought was, [imath]\mathrm{SSE} = \sum (\mu - \hat{\mu})^2[/imath] But I'm not sure if thats right. I'm confused about what the beta is (is it n?) and I don't know if there are enough parameters to expand it. Thanks so much for your patience and if this doesn't make sense, I can clarify more. Thanks! |
662175 | connected subspace [imath]R^{2}[/imath] which is a union of countable family of pairwise disjoint closed segments.
Give an example of connected subspace [imath]R^{2}[/imath] which is a union of countable family of pairwise disjoint closed segments. I have no idea how to find a connected subspace... Any hint? | 235273 | On the existence of a nontrivial connected subspace of [imath]\mathbb{R}^2[/imath]
In the nontrivial sense, does there exist a connected subspace of [imath]\mathbb{R}^2[/imath] which is a union of a non-empty countable collection of closed and pairwise disjoint line segments each of unit length, i.e. length [imath]1[/imath]? What are some good examples, if any? |
168258 | Is there a way to define the "size" of an infinite set that takes into account "intuitive" differences between sets?
The usual way to define the "size" of an infinite set is through cardinality, so that e.g. the sets [imath]\{1, 2, 3, 4, \ldots\}[/imath] and [imath]\{0, 1, 2, 3, 4, \ldots\}[/imath] have the same cardinality. However, is this the only way to define a useful "size" of an infinite set? Could one conceivably define a "size" where the two example sets have different sizes? | 2446393 | A different way to measure cardinality of infinite sets
One of the counter-intuitive things about measuring infinite sets is that the cardinality of a subset may be equal to the cardinality of the original set, contrasting with the finite world where a subset is always smaller. For example, there are just as many odd numbers as there are integers, which is key to finding room at the Hilbert Hotel, when finite intuition says there should be half as many. My question is: is there a different way to measure the cardinality of infinite sets such that the "intuitive" relationship [imath]S \subsetneq T \implies |S| < |T|[/imath] holds? Does it help if you limit yourself to countable sets? |
663055 | Product formula for [imath]\sum_{k=0}^n\frac{(-1)^k}{2k+1}\binom{n}{k}[/imath]
How to prove the following identity: [imath]\sum_{k=0}^n\frac{(-1)^k}{2k+1}\binom{n}{k}=\prod_{k=1}^n\frac{2k}{2k+1}[/imath] | 4261 | Proving a binomial sum identity [imath]\sum _{k=0}^n \binom nk \frac{(-1)^k}{2k+1} = \frac{(2n)!!}{(2n+1)!!}[/imath]
Mathematica tells me that [imath]\sum _{k=0}^n { n \choose k} \frac{(-1)^k}{2k+1} = \frac{(2n)!!}{(2n+1)!!}.[/imath] Although I have not been able to come up with a proof. Proofs, hints, or references are all welcome. |
663633 | Prove that [imath]\left\lfloor \lfloor x/2\rfloor/2 \right\rfloor=\lfloor x/4\rfloor[/imath] for all [imath]x[/imath].
This I approached the problem. I let [imath]x = n + e[/imath] where [imath]n[/imath] is an integer and [imath]e[/imath] is a decimal less than [imath]1[/imath] but not less than [imath]0[/imath]. I substituted that into the equation to get [imath]\left\lfloor \lfloor (n+e)/2\rfloor/2 \right\rfloor = \lfloor (n+e) / 4 \rfloor [/imath] Then I will try for 4 cases. case 1: [imath]0 \le x < 1/4[/imath] case 2: [imath]1/4 \le x < 1/2[/imath] case 3: [imath]1/2 \le x < 3/4[/imath] case 4: [imath]3/4 \le x < 1[/imath] Am I allowed to pull the n out of [imath]\left\lfloor \lfloor (n+e)/2\rfloor \right\rfloor[/imath]? If I can then i can simply plug in the cases into [imath]e[/imath], which will give me [imath]0=0[/imath] for all [imath]4[/imath] cases. Is that correct? | 433293 | Prove that [imath]\lfloor\lfloor x/2 \rfloor / 2 \rfloor = \lfloor x/4 \rfloor[/imath]
In class, we briefly covered what "floor" and "ceiling" mean. Very simple concepts. They were on one slide, and then we never heard about them again. But now the following homework problem has popped up: [imath]\lfloor\lfloor x/2 \rfloor / 2 \rfloor = \lfloor x/4 \rfloor[/imath] Usually when I post a problem (especially from homework), I like to demonstrate that I'm not just asking for the answer by showing what I've done, what I know so far, and so on... but in this case, I have absolutely no idea where to even begin. I will say that my first approach was to create a chart and try various values for "x" in order to see if there's a pattern and to make sure there was no easy counter-example to prove it false. That was all fine and well, but ultimately I couldn't figure out what to do with the results. Googling for this is quite difficult, as everyone uses slightly different notation and therefore one search doesn't encompass all the actual results. The only clue that I've seen so far that kind of sort of makes sense, was when some guy said that "x" should be replaced with "4n + k", since the right-hand side of the equation is divided by 4, so that k is any remainder 0 through 3. How should one approach this problem? What kind of manipulations can you do to floors? What can you assume? etc. etc. ... |
664021 | Show that [imath]{\boldsymbol v}_1[/imath] and [imath]{\boldsymbol v}_2[/imath] are linearly independent
Show that if [imath]r_1 \neq r_2[/imath], the vectors (functions) [imath]{\boldsymbol v}_1 = \exp(r_1t),\,\,\,\,\,\,{\boldsymbol v}_2 = \exp(r_2t)[/imath] are linearly independent in the space of continuous functions [imath]-\infty<t<\infty[/imath]. I know I need to show that [imath]c_1{\boldsymbol v}_1+c_2{\boldsymbol v}_2 = 0[/imath] is linearly independent if [imath]c_1, c_2 = 0[/imath] but I am not sure how to go about it. | 23139 | Proof of linear independence of [imath]e^{at}[/imath]
Given [imath]\left\{ a_{i}\right\} _{i=0}^{n}\subset\mathbb{R}[/imath] which are distinct, show that [imath]\left\{ e^{a_{i}t}\right\} \subset C^{0}\left(\mathbb{R},\mathbb{R}\right)[/imath], form a linearly independent set of functions. Any tips on how to go about this proof. I tried a working from the definition of an exponential and combining sums but that didn't seem to get me anywhere. I saw a tip on the internet that said write it in the form [imath]\mu_{1}e^{a_{1}t}+\dots+\mu_{n}e^{a_{n}t}=0[/imath] to try to show [imath]\mu_{1}=\dots=\mu_{n}=0[/imath] considering each term of the left hand side must be positive, but I can't get my head around that because while I understand [imath]e^{x}>0\forall x\in\mathbb{R}[/imath] I cannot see why [imath]\mu_{i}[/imath] must be positive in any case. I have thought about differentiating but that doesn't seem to help. The question did originally ask for a "rigourous" proof but I'll take any hints right now and the provided the solution of 'is obvious' is most unhelpful to me. Any input would be fantastic. Thank you. |
664167 | Is [imath]\mathbb Q \times \mathbb Q [/imath] a denumerable set?
How can one show that there is a bijection from [imath]\mathbb N[/imath] to [imath]\mathbb Q \times \mathbb Q [/imath]? | 1074046 | Show that [imath]\mathbb{Q}\times \mathbb{Q}[/imath] is denumerable
I am new to functions and relations, and with some concepts I am not so familiar. I have a question in an homework: Show that [imath]\mathbb{Q} \times\mathbb{Q}[/imath] is denumerable. From what I understood, denumerable means that it is infinitively countable. There are some posts on the web about this topic, but I am still not understanding their explanation, maybe because they are not explained so easily (for me). From what I understood, a set is denumerable if there's a relation with the natural numbers (but I am still not understanding what is this relation). I have heard also about equinumerous sets (contain a bijective function = onto + one-to-one function), but I cannot relate this information with the problem to try to solve it. |
663398 | Irreducibility of [imath]x^{2^n}+x+1[/imath] over [imath]\mathbb{Z}_2[/imath]
I'm trying to solve this problem from Hungerford V.5.9. I have to show [imath]x^{2^n}+x+1[/imath] is irreducible over [imath]\mathbb{Z}_2[/imath] if n>2. I would appreciate some hint cause I don't know how to start with it.Thanks! EDITED: This question is completely different to On irreducible factors of [imath]x^{2^n}+x+1[/imath] in [imath]\mathbb Z_2[x][/imath] The reason is obvious, i m asking if this polynomial is irreducible. In the other question, we assume it is reducible without showing it. | 508920 | On irreducible factors of [imath]x^{2^n}+x+1[/imath] in [imath]\mathbb Z_2[x][/imath]
Prove that each irreducible factor of [imath]f(x)=x^{2^n}+x+1[/imath] in [imath]\mathbb Z_2[x][/imath] has degree [imath]k[/imath], where [imath]k\mid 2n[/imath]. Edit. I know I should somehow relate the question to an extension of [imath]\mathbb Z_2[/imath] of degree [imath]2n[/imath], say [imath]GF(2^{2n})[/imath]. By this way I will be able to correspond each irreducible factor of [imath]f(x)[/imath] to a subfield of [imath]GF(2^{2n})[/imath] that obviously has degree [imath]k[/imath], where [imath]k\mid 2n[/imath]. |
664681 | [imath]\omega[/imath] covers and Lindelöf property
As similarly described in a question represented here before: Let [imath]\langle \mathcal{U}_n: n \in \mathbb{N} \rangle[/imath] be a sequence of [imath]\omega[/imath]-covers of [imath]X[/imath], and suppose that [imath]X[/imath] is Lindelöf. Can we always find a sequence [imath]\langle F_n: n \in \mathbb{N} \rangle[/imath] with each [imath]F_n \in \mathcal{U}_n[/imath] such that [imath]\cup F_n[/imath] is an open cover of [imath]X[/imath]? Thank you! Note: The property that [imath]X[/imath] is Lindelöf was not an assumption in the referred question. | 507599 | Is the union of finitely many open sets in an omega-cover contained within some member of the cover?
Let [imath]\mathcal{U}[/imath] be an open cover of [imath]\mathbb{R}[/imath] (Standard Topology) such that [imath]\mathbb{R} \not \in \mathcal{U}[/imath] and for any finite set [imath]A[/imath] there is a [imath]U \in \mathcal{U}[/imath] such that [imath]A \subseteq U[/imath]. We call such an open cover an [imath]\omega[/imath]-cover. Can we show that for any finite set [imath]B \subset \mathcal{U}[/imath], there is a [imath]V \in \mathcal{U}[/imath] such that [imath]\cup B \subseteq V[/imath]? Ultimately I'm working on showing the following. Let [imath]\langle \mathcal{U}_n: n \in \mathbb{N} \rangle[/imath] be a sequence of [imath]\omega[/imath]-covers. Can we find a sequence [imath]\langle F_n: n \in \mathbb{N} \rangle[/imath] with each [imath]F_n \in \mathcal{U}_n[/imath] such that [imath]\cup F_n[/imath] is an open cover of [imath]\mathbb{R}[/imath]? My approach here was to use each [imath]\mathcal{U}_n[/imath] to cover [imath][-n,n][/imath], thus eventually covering all of [imath]\mathbb{R}[/imath]. Since [imath][-n,n][/imath] is compact and [imath]\mathcal{U}_n[/imath] is a cover, [imath]\mathcal{U}_n[/imath] has a finite subcover. But that's as far as I can get unless what I conjectured above is true. |
664671 | Prove that a number all of whose digits are either [imath]6[/imath] or [imath]0[/imath] cannot be a perfect square
Let [imath]m[/imath] be a number all of whose digits are either [imath]6[/imath] or [imath]0[/imath]. Prove that '[imath]m[/imath]' cannot be a perfect square. | 335673 | Proving that a natural number made entirely of 6's and 0's is not a square.
I proceeded using infinite descent. Let [imath] N =a_na_{n-1}a_{n-2}\ldots\ldots a_2a_1a_0[/imath] be the decimal representation of the number. Then either [imath]N[/imath] ends in an even number of zeroes or [imath]a_0=6[/imath] Now all squares are [imath]\equiv 0 \text{ or } 1 \bmod 4 [/imath]. But if [imath]N[/imath] ends in [imath]06\text{ or }66 [/imath], then [imath]N\equiv 2 \bmod 4 [/imath]. Thus [imath] N [/imath] does not end in [imath]6[/imath]. If [imath]N[/imath] ends in an even number of zeroes then, then [imath] N=10^{2n}\cdot N'[/imath]. Applying the same argument to [imath] N'[/imath] we commence an infinite descent. Is the proof correct ? Any shorter proofs ? |
663588 | Prove that a group where [imath]a^2=e[/imath] for all [imath]a[/imath] is commutative
Defining a group [imath](G,*)[/imath] where [imath]a^2=e[/imath] with [imath]e[/imath] denoting the identity class.... I am to prove that this group is commutative. To begin doing that, I want to understand what exactly the power of 2 means in this context. Is the function in the group a power or something? | 737709 | Let [imath]A[/imath] be a group, where [imath]a^2=1[/imath], a belongs to [imath]A[/imath]. Prove that this group is commutative.
Let [imath]A[/imath] be a group, where [imath]a^2=1[/imath] and [imath]a[/imath] belongs to [imath]A[/imath]. Prove that this group is commutative. Thank you for help. |
665230 | Must an uncountable subset of R have uncountably many accumulation points?
This question is taken from problem 4.1.8 of "Real Analysis and Foundations" by Krantz The question reads: "Let S be an uncountable subset of [imath]\mathbb{R}[/imath]. Prove that S must have infinitely many accumulation points. Must it have uncountably many?" The first part of the question took some work but ended up coming out pretty smoothly; however, I'm at a complete loss as to how to go about addressing the second problem. Intuitively, I think the answer is yes. My first attempt was to try to show that a countable number of accumulation points allowed one to order the elements of S in such a way that they are countable (ie prove the contrapositive), but I did not manage to get much further than that. My second attempt was to show that [imath]S-{s_{1},s_{2}...}[/imath] where [imath]s_{1},s_{2},...[/imath] are countably many accumulation points of S is uncountable and thus must have an accumulation point, so S has an accumulation point that is not one of the countably infinite set. Hence, the number of accumulation points is uncountable. My question is, is this logic valid? I would have to prove that an uncountable set minus a countable set is uncountable, which shouldn't be too difficult. Any hints/points in the right direction/outright answers are greatly appreciated. | 310113 | Accumulation points of uncountable sets
Given any uncountable subset [imath]S[/imath] of the unit interval. Then [imath]S[/imath] clearly has an accumulation point and indeed uncountably many (which might also be a nice exercise). So my question is: Is there an accumulation point, that again lies in [imath]S[/imath]? |
482944 | Number of different ways of filling [imath]N \times 4[/imath] rectangle with Dominoes
Given a Nx4 (width = N and height = 4) rectangle. How many different ways are there to fill with Dominoes (2x1 or 1x2)? I have found an OEIS sequence http://oeis.org/A005178 for this. The recurrence given in the link is a(n) = a(n-1)+5*a(n-2)+a(n-3)-a(n-4). Any combinatorial proof? or how to get the recurrence? | 664113 | count the ways to fill a [imath]4\times n[/imath] board with dominoes
After solving this problem from SPOJ (count the ways to fill a 4xn board with 2x1 dominoes) I found a different solution while searching on internet. This solution uses the recurrence relation [imath]f(n) = f(n-1)+5f(n-2)+f(n-3)-f(n-4)[/imath] where [imath]f(n)[/imath] denotes the number of ways to fill the [imath]4\times n[/imath] board with [imath]2 \times 1[/imath] dominoes. I don't fully understand how someone gets that kind of relation (I don't know almost anything of combinatorics or recurrence relations) but I think I understand something. The [imath]f(n-1)[/imath] term comes from observing that there is an unique way to fill the last column and the [imath]5f(n-2)[/imath] term comes from observing there are 5 different ways to fill the last two columns. But i don't get where the terms [imath]f(n-3)-f(n-4)[/imath] come from. So I have two questions, the first one is where these terms come from and second I'd like to ask you for a reference to learn combinatorics and recurrence relations. May you help me? Thanks in advance. -edit- For my solution I used a binary number to represent what configurations of the i-th column was valid. For example 1001 represent that rows 1 and 4 are blocked in the i-th row because in the (i-1)th there is an horizontal domino in that positions. I calculated all the valid transitions such binary numbers could go to (I did this by hand for all the binary numbers from 0000 to 1111). Then I got a function which I programmed using dynamic programming and a technique called bitmask to represent the binary numbers as integers. The function is: [imath]f(i,mask) = \begin{cases} 0 &\mbox{if } i = n, mask \neq 0 \\ 1 & \mbox{if } i = n, mask = 0. \\ \sum f(i+1,mask') &\mbox{else} \end{cases}[/imath] Where mask' is taken from the set of all valid masks from where mask could transition to, also i=n means the first column outside the board as I considered 0-indexing. The code for that is here (this is not actually mine, it's from a mate but it's the exactly same idea). |
665605 | How to construct a [imath]2\times2[/imath] matrix [imath]A[/imath] such that [imath]A^3=I[/imath]?
Construct a [imath]2[/imath] [imath]\times[/imath] [imath]2[/imath] matrix [imath]A[/imath] ([imath]\neq[/imath] [imath]I[/imath]) with entries from [imath]\mathbb{R}[/imath] such that [imath]A^3=I[/imath]. First give me some hint. How to construct this kinds of matrices, is there any rule... | 58666 | Is there any matrix [imath]2\times 2[/imath] such that [imath]A\neq I[/imath] but [imath] A^3=I[/imath]
I want to ask you this question: Is there any matrix [imath]2\times 2[/imath] such that [imath]A\neq I[/imath] but [imath]A^3=I[/imath]. In my opinion: No. Thank you very much |
665816 | prove this identity [imath]\sin(x+y)\sin(x-y)=\sin^2 x - \sin^2 y[/imath]
prove this identity : [imath]\sin(x+y)\sin(x-y)=\sin^2 x - \sin^2 y[/imath] I tried solving it with additional formulas but I can't get the right answer. I get [imath]\sin^2 x \cos^2 y-\cos^2 x \sin^2 y[/imath] | 175143 | Prove [imath] \sin(A+B)\sin(A-B)=\sin^2A-\sin^2B [/imath]
How would I verify the following double angle identity. [imath] \sin(A+B)\sin(A-B)=\sin^2A-\sin^2B [/imath] So far I have done this. [imath] (\sin A\cos B+\cos A\sin B)(\sin A\cos B-\cos A\sin B) [/imath]But I am not sure how to proceed. |
666214 | Proving [imath]\left|\frac{w-z}{1-\bar{w}z}\right|[/imath] < 1
How to prove [imath]\left|\frac{w-z}{1-\bar{w}z}\right|[/imath] < 1 if |z|<1 and |w|<1? Please give me a hint. | 343982 | Prove if [imath]|z| < 1[/imath] and [imath] |w| < 1[/imath], then [imath]|1-zw^*| \neq 0[/imath] and [imath]| {{z-w} \over {1-zw^*}}| < 1[/imath]
Prove if [imath]|z| < 1[/imath] and [imath] |w| < 1[/imath], then [imath]|1-zw^*| \neq 0[/imath] and [imath]| {{z-w} \over {1-zw^*}}| < 1[/imath]Given that [imath]|1-zw^*|^2 - |z-w|^2 = (1-|z|^2)(1-|w|^2)[/imath]I think the first part can be proven by saying [imath]|1-zw^*| = 0[/imath] if and only if [imath]zw^*[/imath] = 1. And given the conditions that cannot be true. However I don't know if this part is right. |
666288 | Number of ways to interleave two ordered sequences.
Suppose we have two finite, ordered sequences [imath]x = (x_1,\dots,x_m)[/imath] and [imath]y = (y_1,\dots,y_n)[/imath]. How many ways can we create a new sequence of length [imath]m+n[/imath] from [imath]x[/imath] and [imath]y[/imath] so that the order of elements is preserved? How many ways can we do this using [imath]k[/imath] ordered sequences instead of two? | 382174 | Ways of merging two incomparable sorted lists of elements keeping their relative ordering
Suppose that, for a real application, I have ended up with a sorted list A = {[imath]a_1, a_2, ..., a_{|A|}[/imath]} of elements of a certain kind (say, Type-A), and another sorted list B = {[imath]b_1, b_2, ..., b_{|B|}[/imath]} of elements of a different kind (Type-B), such that Type-A elements are only comparable with Type-A elements, and likewise for Type-B. At this point I seek to count the following: in how many ways can I merge both lists together, in such a way that the relative ordering of Type-A and Type-B elements, respectively, is preserved? (i.e. that if [imath]P_M(x)[/imath] represents the position of an element of A or B in the merged list, then [imath]P_M(a_i)<P_M(a_j)[/imath] and [imath]P_M(b_i)<P_M(b_j)[/imath] for all [imath]i<j[/imath]) I've tried to figure this out constructively by starting with an empty merged list and inserting elements of A or B one at a time, counting in how many ways each insertion can be done, but since this depends on the placement of previous elements of the same type, I've had little luck so far. I also tried explicitly counting all possibilities for different (small) lengths of A and B, but I've been unable to extract any potential general principle in this way. |
666501 | Why [imath]\{\emptyset\} \not \subset\{\{\emptyset\}\}[/imath]?
In my text book it is written that: { } ⊆ { }; { } ⊆ {0/}; { } ⊆ { {0/} }; { } ⊆ C; and {0/} ⊆ C; { {0/} } ⊆ C; but {0/} is not a subset of { {0/} } since the only element of {0/} is 0/ and the only element of { {0/} } is {0/}, so the element of {0/} is not an element of { {0/} }. A set with no elements is an empty set, denoted by {0/}. There are three parts to my question. Firstly, what is the distinction between {0/} and { }? Why is the latter not an empty set like the former? Secondly, is { {0/} } an element and at the same time a set? And finally, howcome is {0/} is not a subset of { {0/} }? How can an empty set not be a subset of an empty set? This is my first question on math.stackexchange: I have no formal Mathematics background so please don't presume too much as much as I want to learn... Thank you, internet! UPDATE: Let C = { 0/, {0/} } | 491465 | Is [imath]\{\emptyset\}[/imath] a subset of [imath]\{\{\emptyset\}\}[/imath]?
[imath]\{\emptyset\}[/imath] is a set containing the empty set. Is [imath]\{\emptyset\}[/imath] a subset of [imath]\{\{\emptyset\}\}[/imath]? My hypothesis is yes by looking at the form of "the superset [imath]\{\{\emptyset\}\}[/imath]" which contains "the subset [imath]\{\emptyset\}[/imath]". |
511458 | Is the function [imath]f(x)= {\sin x \over x}[/imath] uniformly continuous over [imath]\mathbb{R}[/imath]?
Is the function [imath]f(x)= {\sin x \over x}[/imath] Uniformly continuous over [imath]R[/imath] How do i approach this ? I need some hints. | 436825 | Determine whether [imath]f(x)={\sin x \over x}[/imath] is uniformly continuous in [imath]\mathbb R[/imath]
Determine whether the function[imath]f(x)={\sin x \over x}[/imath]is uniformly continuous in [imath]\mathbb{R}[/imath]. I am using the definition that for [imath]\epsilon>0[/imath] there exists [imath]\delta>0[/imath] such that [imath]|f(x)-f(y)|<\epsilon[/imath] whenever [imath]|x-y|<\delta[/imath]. |
666718 | Basic Abstract Algebra - Homomorphism
Given a homomorphism [imath]f:G \rightarrow H[/imath], [imath]G[/imath] finitely generated, what can you say about the order of [imath]g_i[/imath] and [imath]f(g_i)[/imath]? I've thought about this question for a while but haven't come to a conclusion. If we consider the natural homomorphism from [imath]\mathbb{Z}[/imath] to [imath]\mathbb{Z}_3[/imath], the order of 1 in [imath]\mathbb{Z}[/imath] is infinite but the order of 1 in [imath]\mathbb{Z}_3[/imath] is only [imath]3[/imath]. Therefore, the order of [imath]f(g_i)[/imath] can be less than that of [imath]g_i[/imath]. But can't it also be greater? If not, wouldn't it be impossible to construct homomorphisms from a group of order [imath]n[/imath] to a group of order [imath]p[/imath] where [imath]p > n[/imath]? Thanks. | 356597 | Order of [imath]\phi(g)[/imath] divides the order of [imath]g[/imath]
Let [imath]\phi: G \to H[/imath] be a group homomorphism, and let [imath]g[/imath] be an element of [imath]G[/imath]. Show that the order of [imath]\phi(g)[/imath] divides the order of [imath]g[/imath]. |
666936 | Prove Borel's Lemma (Pugh's book #35)
Given any sequence whatsoever of real numbers (a_r), there is a smooth function [imath]f: \mathbb{R} \to \mathbb{R}[/imath] such that [imath]f^{(r)}(0) = a_r[/imath]. Pugh's hint says to try [imath]f=\sum \beta_k(x)a_kx^k/k![/imath], where [imath]\beta_k[/imath] is a well-chosen bump function. I'm working with a few other people and we're trying to define a bump function [imath]\beta_k(x)[/imath] = bump [imath]*b_k*x[/imath] where from 1/2 to 1, [imath](exp(\frac{1}{|x|-1}))(1-exp(\frac{-1}{|x|-1/2}) + C_k[/imath] but have no idea how to define | 63050 | Every power series is the Taylor series of some [imath]C^{\infty}[/imath] function
Do you have some reference to a proof of the so-called Borel theorem, i.e. every power series is the Taylor series of some [imath]C^{\infty}[/imath] function? |
667010 | Proofs with binary trees
Now I have a binary tree which is How would I go about proving binary tree with [imath]n[/imath] leaves has exactly [imath]2 n - 1[/imath] nodes ? | 664608 | Number of nodes in binary tree given number of leaves
How would I prove that any binary tree that has n leaves has precisely [imath]2n-1[/imath] nodes ? Given that a binary tree is either a single node "o" or a node with the left and right subtrees contains a binary tree o / \ binary binary tree tree Any help would be appreciated ! |
667496 | A matrix [imath]A[/imath] is given with a known Jordan decomposition, what is the Jordan decomposition of [imath]A^2+A+I[/imath]?
This is a duplicate of a question I've already asked, but got no answer to. I would appreciated any help. Given a matrix [imath]A[/imath] with a known Jordan decomposition, what is the Jordan decomposition of [imath]A^2+A+I[/imath]? | 666579 | Given a matrix [imath]A[/imath] with a known Jordan decomposition, what is the Jordan decomposition of [imath]A^2+A+I[/imath]?
So far, I understand that I have to look at each Jordan block. How do I prove that for a Jordan block with a value of x, [imath]J_n(x)[/imath], the Jordan decomposition of [imath](J_n(x))^2[/imath] is [imath]J_n(x^2)[/imath] ? I have a feeling this will lead me to the solution. Thanks for your time. |
667847 | Prove that the recurrence is true
I am working on an assignment question, and am having trouble moving ahead. The question is as follows: Let the total number of bit strings with three consecutive zeros be [imath]t_n[/imath]. Prove for [imath]n \ge 4[/imath] that [imath]t_n= t_{n-1} + t_{n-2} + t_{n-3} + 2^{n-3}[/imath] So, I started to check if the base case is true with n=4 [imath]t_4 = t_3 + t_2 + t_1 + 2^1 = 1 + 0 + 0 + 2 = 3[/imath] But isn't the total number of combinations for n =4 only 2? 1000 or 0001 I might just be going about this the wrong way. Any guidance would be appreciated. | 666684 | Number of bitstrings with [imath]000[/imath] as substring
I have [imath]F_n[/imath] number of bitstrings that have [imath]000[/imath], How would I prove that for [imath]n \ge 4[/imath] , [imath]a_n = a_{n-1} +a_{n-2}+a_{n-3}+ 2^{n-3}[/imath]? Now there are many ways to go about this but if I choose starting a [imath]n-3[/imath] bit string and append it with [imath]000[/imath] how would I show that? |
668068 | How would you prove that the graph of a linear equation is a straight line, and vice versa, at a "high school" level?
This is something I've been wondering about. Namely, I've always accepted "on intuition" that the equation [imath]ax + by = c[/imath] is, when graphed, a line. You can plot the points [imath](x, y)[/imath] satisfying the equation and see that yeah, they do indeed form a line. But when I came across this, I realized was that I had accepted this idea without proof: http://www.math.jhu.edu/~wsw/ED/harelfinal.pdf There is enough material in the text to convince the students empirically that a line in the plane is represented by a linear equation, and that the graph of a linear equation is a line. However, these two fundamental theorems on linear functions are not justified mathematically. (pg. 5) and Important theorems on linear functions are not proved. Relevant to the above two standards are two fundamental theorems: A line in the plane is represented by a linear equation and the graph of a linear equation is a line. Neither of these theorems is proved. (pg. 16) Which made me wonder (I haven't seen the texts) -- just how would you not only "justify" this mathematically, but in a way a high-schooler would understand? And not only that, but in a manner which is actually enlightening? In addition, this criticism is leveled against all four high school geometry/algebra books. For example, consider if we were using, say, Hilbert's axioms as our axiom set for Euclidean geometry. Then we could show that the equation is a line by something like this: find three points A, B, and C satisfying it so that [imath]A * B * C[/imath] (the "betweenness" relation), then show that for any point [imath]D[/imath] which is not [imath]A[/imath] or [imath]B[/imath], then one of [imath]D * A * B[/imath], [imath]A * D * B[/imath], or [imath]A * B * D[/imath] must hold. Going the other way (the converse), to show the line is given by the equation, you'd show that for any points A, B, C with [imath]A * B * C[/imath], then every point [imath]D[/imath] with [imath]D * A * B[/imath], [imath]A * D * B[/imath], or [imath]A * B * D[/imath] satisfies some equation of the form [imath]ax + by = c[/imath]. However, it seems this kind of proof is fairly tedious (you have to check three cases in both implications), and relies on quadratic functions and radicals since you have to use the distance formula as that's how you'd define the "betweenness" relation for three points. I suppose the details would vary with regard to the axiom set we use (I don't know if Hilbert's would necessarily be the best for "high school geometry") -- but it seems no matter which one we use, we need some way to determine that three points A, B, and C "lie on the same line" (which is what the "betweenness" relation does, although it does more, since it also orders the points), and a way to express this with regards to coordinatized points as well as points in the axiomatic geometry which the coordinate plane models. It seems that any formula I've seen for that fact using Cartesian coordinates requires a quadratic polynomial expression, for one. Any proof along these lines seems like it would be tedious, or require additional motivation, and so might not be enlightening at this lower level of the person's mathematical development. The "message" seems to easily get lost as one gets bogged down in mechanics. How would you solve this problem? What's a good way to justify this at such a level? | 503601 | Why do we believe the equation [imath]ax+by+c=0[/imath] represents a line?
I'm going for quite a weird question here. As we know, the equation in Cartesian coordinates for a line in 2-dimensional Euclidean geometry is of the form [imath]ax+by+c=0[/imath]. I'm wondering why do we "believe" the plotted graph is the same "line" as in our intuition. It might sound crazy, but think of the time when there were still no coordinates, no axes, no analytic geometry. When Descartes started to grasp the concept that equations represented geometric figures (or more accurately, loci) he would have tried plotting easy forms first, and what else could be easier than [imath]y=x[/imath] or [imath]y=2x+3[/imath] etc. Plotting those revealed something evidently a line to his (and our) naked eyes, but it wouldn't be appropriate for a mathematician to conclude from that alone that the figure is actually a "line", right? So jumping back to our own time, if we forget for once that [imath]ax+by+c=0[/imath] "is" a line, looking at it with fresh eyes, by what criteria are we using to say it is so. I've tried some regular characterizations of the line (especially the geometrical ones) and haven't yet found a satisfactory answer yet. Here are some: The line is the shortest path wrt. the Euclidean distance between two points: sounds OK except that the Euclidean distance is based on our intuition that the "distance" is the length of the line connecting two points (or more accurately, the arc length measure along the path of the straight line.) Of course, we could argue that it's OK by itself to define the distance as [imath]\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}[/imath] without referring to the line beneath, but this sounds to be somewhat an ad hoc claim. Given two points, there is exactly one line passing through both: I interpret this as "Given a family of curves, if for any two given points there is exactly one curve in the family passing through both points, then the family is appropriately understood as the family of 'lines' and we call the individual curves 'lines.'" At first glance this holds up to scrutiny, as many families of circles or parabolas or other "curves" don't satisfy the property (because any family satisfying must have as a corollary the following property: any two curves in the family intersect at 1 point at most.) But then consider the family [imath]\{ax^3+by^3+c=0|a,b,c \in \mathbb{R}\}[/imath] (or more generally, [imath]\{ax^m+by^n+c=0|a,b,c \in \mathbb{R}\}[/imath] with [imath]m,n[/imath] odd positive integers), this family not only satisfies the 2-point-1-line property, but also the parallel postulate: "given a 'line' and a point not on the 'line', there is exactly one 'line' parallel to the given 'line' passing through the given point (where 'parallel' means 'having no intersection')" A line is a curve convex and concave at the same time: This too suffers from the fact that that intuition underlying the definition of "convex" and "concave" contain reference to the straight line. For example we defined a curve to be convex if for any two points on the curve it lies below the straight line connecting the two points, hence [imath]f(tx+(1-t)y) \leq tf(x)+(1-t)f(y), \forall t \in [0,1][/imath] Somehow I'm not at ease with characterizations of the line as endpoints of vectors (for example, the line connecting points [imath]A,B[/imath] as the collection of endpoints of [imath]l(t)=ta+(1-t)b, \forall t \in \mathbb{R}[/imath] where [imath]a,b[/imath] are the position vectors of [imath]A,B[/imath] respectively) either. It seems to rely on representing vectors in 2-D Euclidean space (or 2-D affine space) by rays, which to me is just half a line, making the description circular. One approach I'm thinking of is looking at the real-life construction of the line. In Euclidean geometry there are two basic tools, the straightedge and the compass. The "definition" (or characterization) of a circle in mechanical terms is the collection of points traced out by the pencil tip while the compass radius is hold still. This translates to the geometrical definition of the circle as the loci of points equidistance from a given point. Following this train of thought, the line is the loci of... what? Sorry for the long post and confusing personal criteria of deciding what amounts to a "proper explanation" of a straight line. I'm running out of ideas now, so if you have any, please tell me. Thank you in advance. PS. This is not my homework or school research project. I'm just doing this for fun and wanted to hear other people's views on the topic. |
668167 | I as an ideal of [imath]R[/imath] then [imath]a+I=0+I[/imath] iff [imath]a\in I[/imath]
show that if a,b belong to the ring [imath]R[/imath] and [imath]I[/imath] is an ideal of [imath]R[/imath] then [imath]a+I=0+I[/imath] if and only if [imath]a[/imath] belongs to [imath]I[/imath]. I know that since I is an ideal then it is both a left and a right ideal. | 104750 | Translate of an ideal by [imath]a[/imath] coincides with the ideal iff [imath]a \in I[/imath]
How can I show the following? Show that, if [imath]a[/imath] and [imath]b[/imath] are elements of a ring [imath]R[/imath] and [imath]I[/imath] is an ideal of [imath]R[/imath], then [imath]a+I=0+I \iff a \in I[/imath] I am so interested to know the proof.Thanks and good night all! |
583979 | diagonalizability and invariant subspaces
My question is about linear algebra, especially invariant subspaces and diagonalizability. Here is the question: Let [imath]A[/imath] be a diagonalizable linear operator on the finite dimensional vector space over a field, and let [imath]W[/imath] be a subspace of [imath]V[/imath] which is invariant under [imath]A[/imath]. Let us denote by [imath]A_W[/imath] the restriction of [imath]A[/imath] to [imath]W[/imath]. I need to prove that [imath]A_W[/imath] is diagonalizable. How can I prove that? thanks for your help... | 62338 | Diagonalizable transformation restricted to an invariant subspace is diagonalizable
Suppose [imath]V[/imath] is a vector space over [imath]\mathbb{C}[/imath], and [imath]A[/imath] is a linear transformation on [imath]V[/imath] which is diagonalizable. I.e. there is a basis of [imath]V[/imath] consisting of eigenvectors of [imath]A[/imath]. If [imath]W\subseteq V[/imath] is an invariant subspace of [imath]A[/imath] (so [imath]$A(W)\subseteq W$[/imath]), show that [imath]A|_W[/imath] is also diagonalizable. I tried supposing [imath]A[/imath] has distinct eigenvalues [imath]\lambda_1,\ldots,\lambda_m[/imath], with [imath]V_i=\{v\in V: Av=\lambda_i v\}[/imath]. Then we can write [imath]V=V_1\oplus\cdots\oplus V_m,[/imath] but I'm not sure whether it is true that [imath]$$W=(W\cap V_1)\oplus\cdots\oplus (W\cap V_m),.$$[/imath] If it is true, then we're done, but it may be wrong. |
436045 | how to prove this property of compact operator？
I read about this property of compact operator from wikipedia [imath]K(X, Y)[/imath] is a closed subspace of [imath]B(X, Y)[/imath]: Let [imath]T_{n}, n \in N[/imath], be a sequence of compact operators from one Banach space to the other, and suppose that [imath]T_{n}[/imath], converges to [imath]T[/imath] with respect to the operator norm. Then [imath]T[/imath] is also compact. Can anybody prove it in details or tell me where I can find the proof? Thanks so much! | 416046 | How to show that the limit of compact operators in the operator norm topology is compact
When I read the item of compact operator on Wikipedia, it said that Let [imath]$T_{n}, ~~n\in \mathbb{N}$[/imath], be a sequence of compact operators from one Banach space to the other, and suppose that [imath]$T_n$[/imath] converges to [imath]T[/imath] with respect to the operator norm. Then [imath]T[/imath] is also compact. Can anyone give me a brief proof of this? Thanks in advance. |
669455 | Weak closure in Hilbert space
I am trying to find weak closure of [imath]\{e^{imt}+me^{int}|0\le m<n\}[/imath] in [imath]L^2(-\pi,\pi)[/imath]. I know how to find weak sequential closure and I know that [imath]0[/imath] is in weak closure (I can prove it using weak open sets). Now, how to find or even guess what is weak closure and then prove that really is weak closure? I am aware that weak sequential closure and weak closure are not the same. Is the only way using weakly open sets (which are not so nice) or is there some trick to quickly deduce what is weak closure? Also, I know that in a locally convex space weak closure and original closure of a convex set are the same. Here, we do not have a convex set.... Please help, I do not know how to begin this. | 653151 | Find the weak sequential closure of a set in [imath]L^2(-\pi,\pi)[/imath]
[imath]A=\{f_{m,n}(t)|0\le m<n\}[/imath] where [imath]f_{m,n}(t)=e^{imt}+me^{int}[/imath]. I should find the weak sequential closure of [imath]A\subset L^2(-\pi,\pi)[/imath]. I know what I'm supposed to do. Take a sequence in [imath]A[/imath] and find its weak limit. I also know what is weak convergence and I know that [imath](L^2)^{*}=L^2[/imath], but I don't know what to do with this: [imath]\displaystyle\lim\limits_{k\to\infty}\int\limits_{-\pi}^{\pi}(e^{im_kt}+m_ke^{in_kt})f(t)dt=\int\limits_{-\pi}^{\pi}?f(t)dt[/imath] for every [imath]f\in L^2(-\pi,\pi)[/imath]. Thank you very much for any help. |
669865 | Homeomorphism between punctured plane and cylinder
I am asked to prove that the cylinder and the punctured plane are homeomorphic. I understand that I need to find a function that maps every point in the plane to a point on the cylinder. I can represent every point on the plane in polar coordinates and now I just need to find a mapping from each point in the plane in R^2 to the cylinder in R^3. I have found that a solution is [imath]f(r\cos\theta,r\sin\theta)=(\cos\theta,\sin\theta,\log(r))[/imath] which makes sense except I cannot understand why [imath]f(r\cos\theta,r\sin\theta)=(\cos\theta,\sin\theta,r)[/imath] would not be a solution. Since [imath]r \in (-\infty,\infty)[/imath], wouldn't this still result in a cylinder? | 269712 | Finding a homeomorphism [imath]\mathbb{R} \times S^1 \to \mathbb{R}^2 \setminus \{(0,0)\}[/imath]
Are there any specific 'tricks' or 'techniques' in finding homeomorphisms between topological or metric spaces? I'm trying to construct a homeomorphism between [imath]\mathbb{R} \times S^1 \to \mathbb{R}^2 \setminus \{(0,0)\}[/imath] but I'm having trouble even visualizing how this is going to work. I'm thinking some sort of stereographic projection from the circle and then including the [imath]\mathbb{R}[/imath] somehow but I'm stuck. Any ideas? |
669831 | Proof that if [imath]a^n|b^n[/imath] then [imath]a|b[/imath]
I can't get to get a good proof of this, any help? What I thought was: [imath]b^n = a^nk[/imath] then, by the Fundamental theorem of arithmetic, decompose [imath]b[/imath] such: [imath]b=p_1^{q_1}p_2^{q_2}...p_m^{q_m}[/imath] with [imath]p_1...p_m[/imath] primes and [imath]q_1...q_n[/imath] integers. then [imath]b^n=(p_1^{q_1}p_2^{q_2}...p_m^{q_m})^n= p_1^{q_1n}p_2^{q_2n}...p_m^{q_mn}[/imath] but here i get stucked, and i can't seem to find a good satisfactory way to associate [imath]a[/imath] and [imath]b[/imath]... Any help will be appreciated | 25716 | Show that [imath]a^n \mid b^n[/imath] implies [imath]a \mid b[/imath]
I would like to show that [imath]a^n \mid b^n[/imath] implies [imath]a \mid b[/imath] I thought I could convert it to congruences and work backwards, but as far as I remember, [imath]a \equiv b \pmod{m}[/imath] implies [imath]a^n \equiv b^n \pmod{m}[/imath], not the opposite, unless [imath]m[/imath] is prime. Is that right? So I am not sure how to approach this one. Any ideas? Thanks! |
670023 | [imath]X[/imath] is closed iff [imath]X[/imath] is complete
Let [imath]X \subseteq \mathbb{R}^d[/imath]. We want to show that [imath]X[/imath] is closed iff every Cauchy sequence in [imath]X[/imath] converges in [imath]X[/imath]. MY attempt: Let [imath]X[/imath] be a closed set, and pick a convergent sequence [imath](x_n)[/imath] in [imath]X[/imath]. Since [imath]X[/imath] is closed, then we know that [imath]x_n \to x \in X[/imath]. Since convergence implies Cauchy, then we have found that every Cauchy sequence in [imath]X[/imath] is convergent, as desired. Conversely, suppose every Cauchy sequence converges in [imath]X[/imath], we want to show that [imath]X[/imath] is closed. To this end, pick a sequence [imath](x_n) \subseteq X[/imath] such that [imath]x_n \to L[/imath]. We want to show that [imath]L \in X[/imath]. Since [imath](x_n)[/imath] converges, then it must be Cauchy. And by hypothesis, then we must havethat [imath]L \in X[/imath]. Is this correct? thanks for any feedback in advanced. | 243199 | Proof that a subspace [imath]A[/imath] of a complete metric space [imath]X[/imath] is complete iff [imath]A[/imath] is closed
Here's my proof in my own words, does it stack up? Showing [imath]A[/imath] is complete implies [imath]A[/imath] is closed. Let [imath](x_n)[/imath] be a convergent sequence in [imath]A[/imath]. [imath]A[/imath] is complete [imath]\implies (x_n) \to p \in A[/imath]. Hence [imath]A[/imath] is closed. Showing [imath]A[/imath] is closed [imath]\implies[/imath] [imath]A[/imath] is complete. Let [imath](x_n)[/imath] be a convergent sequence in [imath]A[/imath]. [imath]A[/imath] is closed [imath]\implies (x_n) \to p \in A[/imath]. As every convergent sequence is a Cauchy sequence, [imath](x_n)[/imath] is a Cauchy sequence in [imath]A[/imath] that converges to [imath]p \in A[/imath] and hence [imath]A[/imath] is complete. That sound ok? As an aside, it seems to me that the definition of completeness and closed are basically identical, why are there two definitions for the same thing? Am I missing something here? |
670065 | Functions and Set Theory
Denote by [imath]F(X,Y)[/imath] the set of all functions from [imath]X[/imath] to [imath]Y[/imath]. For sets [imath]A[/imath], [imath]B[/imath], and [imath]C[/imath] prove that a. [imath]F(C,A\times B)[/imath] is in one-to-one correspondence with [imath]F(C,A)\times F(C,B[/imath]). Let's give this bijective function between [imath]F(C,A\times B)[/imath] and [imath]F(C,A)\times F(C,B)[/imath] a name, for example [imath]M[/imath]. If you start out with a function [imath]g[/imath] which maps out every [imath]c[/imath] to some ordered pair [imath](a,b)[/imath], then [imath]M(g) = (x,y)[/imath] where [imath](x,y)[/imath] is an ordered pair of functions. [imath]x(c)[/imath] = [imath]a[/imath] and [imath]y(c) = b[/imath]. In other words, [imath]M[/imath] looks at the behavior of the input function [imath]g[/imath] for every [imath]c[/imath], and splits [imath]g[/imath]'s behavior across two functions. I cannot seem to prove that [imath]M[/imath] is a bijection. Could someone please lay out how I can start with this since we are dealing with a function that maps a function to an ordered pair of functions. Also, how would you do: F(C , F(B , A)) is in one-to-one correspondence with F(B × C , A). Because, C maps to a function. | 669223 | Functions that are sets of all function - proofs
I'm going through the book Proofs and fundamentals, by Bloch, and it doesn't include a solution manual for it's examples. It doesn't have many examples on notation and proof strategy for certain cases, so I needed a little help. Although I understand the idea, and can draw it out, I'm not sure how to write it in decent proof notation. Any help would be appreciated! Edit: apologies, my idea is this: theres some function in [imath]F(C,A\times B)[/imath] such that theres some [imath]x[/imath] within [imath]C[/imath] that means [imath]g(x)[/imath] is within [imath]A\times B[/imath]. Similarly, the second two imply that for some function within [imath]F(A,C)[/imath] there exists a value [imath]x[/imath] such that [imath]g(x)=C[/imath]. and similarly for the third part from [imath]B\rightarrow C[/imath]. It's easy to see with drawing it out that these two are the same because one will have a part within A and the other will lead to a part within [imath]B[/imath], so their cross will be the same. I wanted to go about proving it by setting a function [imath]f[/imath] within [imath]F(C,AxB)[/imath] and then working from there, but I really have no idea where to start or the notation. |
670098 | [imath]T^*T=TT^*[/imath] and [imath]T^2=T[/imath]. Prove [imath]T[/imath] is self adjoint: [imath]T=T^*[/imath]
[imath]V[/imath] is an inner product space of finite dimension over [imath]\mathbb{R}[/imath], and [imath]T:V\to V[/imath] a linear transformation which is normal, that is, [imath]T^*T=TT^*[/imath]. In addition [imath]T^2=T[/imath]. Prove [imath]T[/imath] is self adjoint, that is, [imath]T=T^*[/imath]. I tried to prove it algebraically by using the inner product but it didn't work for me. Then I tried to prove the statement that [imath]T[/imath] is self adjoint iff [imath]\langle v,Tv\rangle[/imath] is real for all [imath]v[/imath]. I know that [imath]T[/imath] is diagonalizable, therefore there is a basis of [imath]V[/imath] consisting of eigenvectors. In addition every eigenvector of [imath]T[/imath] is an eigenvector of [imath]T^*[/imath]. Can I prove that [imath]\langle v,Tv\rangle[/imath] is real only for the vectors in the basis and then it means it is for all [imath]v[/imath]? Any help or further hints are very appreciated. | 319197 | Normal, idempotent operator implies self-adjointness.
I have been trying to solve this problem for quite a while. I am still unsure of whether any of the avenues I have pursued have been of any use. Any advice will be much appreciated. Question: Let [imath]V[/imath] be a finite-dimensional inner product space, and let [imath]E[/imath] be an idempotent linear operator on [imath]V[/imath]. Prove that if [imath]EE^* = E^*E[/imath] then [imath]E[/imath] is self-adjoint. |
165675 | Constructing a Galois extension field with Galois group [imath]S_n[/imath]
Given a field [imath]F[/imath], can you necessarily construct a field extension [imath]E \supset F[/imath] such that [imath]\operatorname{Gal}(E/F) = S_n\,[/imath]? | 424227 | Galois extension corresponding to [imath]S_n[/imath]
Given a positive integer [imath]n[/imath]. How to find a Galois extension [imath]K/F[/imath] such that [imath]Gal(K/F)=S_n[/imath]? With the restriction [imath]F=\mathbb{Q}[/imath], this is the Inverse Galois Problem. But if we are allowed to choose [imath]F[/imath] and [imath]K[/imath], I suspect there is a simpler example. |
671099 | Induction of [imath]A_i[/imath]
The base case [imath]n=1[/imath]: [imath]B\cup\left(\bigcap_{i=1}^1A_i\right)=B\cup A_1[/imath] and [imath]\bigcap_{i=1}^1(B\cup A_i)=B\cup A_1[/imath]. Now, suppose inductively that [imath]B\cup\left(\bigcap_{i=1}^nA_i\right)=\bigcap_{i=1}^n(B\cup A_i)[/imath]. Then \begin{align*} B\cup\left(\bigcap_{i=1}^{n+1}A_i\right) &= B\cup \left[\left(\bigcap_{i=1}^nA_i\right)\cap A_{n+1}\right] \\ &= \left[B\cup\left(\bigcap_{i=1}^nA_i\right) \right]\cap\left(B\cap A_{n+1}\right) \\ \end{align*} I am stuck at this point of the induction process, could someone please assist me. | 670334 | Proof by induction that [imath]B\cup (\bigcap_{i=1}^n A_i)=\bigcap_{i=1}^n (B\cup A_i)[/imath]
[imath]\displaystyle B\cup (\bigcap_{i=1}^n A_i)=\bigcap_{i=1}^n (B\cup A_i)[/imath] I was able to prove this without using induction, however I am supposed to prove it using induction. How should I go about doing so? |
665038 | Proving strings
We consider strings of n characters, each character being a, b, c, or d, that contain an even number of as. (0 is even.) Let [imath]E_n[/imath] be the number of such strings.Prove that for any integer [imath]n \geq1[/imath],E_(n+1) =2 * [imath]E_n[/imath] + [imath]4^n[/imath] | 665118 | Proving a recurrence relation for strings of characters containing an even number of [imath]a[/imath]'s
We consider strings of [imath]n[/imath] characters, each character being [imath]a[/imath], [imath]b[/imath], [imath]c[/imath], or [imath]d[/imath], that contain an even number of [imath]a[/imath]'s. (Recall that [imath]0[/imath] is even.) Let [imath]E_n[/imath] be the number of such strings. Prove that for any integer [imath]n \ge1[/imath], [imath]E_{n+1} = 2 E_n + 4^n.[/imath] I tried to do induction. I got stuck. [imath]\begin{align}E_{n+2} &= 2 E_{n+1} + 4^{n+1}\\ & = 2 (2E_n + 4^n) + 4^{n+1}\\ & = 4E_n + 2\cdot 4^n + 4^{n+1}\end{align}[/imath] Then I got lost. |
671582 | How to show [imath]15 \mid 2^{4n}-1[/imath] by induction
For all [imath]n \ge 1[/imath] use mathematical induction to establish the divisibility of the statement: [imath] 15 \mid 2^{4n}-1 [/imath] So first i substituted [imath]1[/imath] in and proved the statement to be true by example then for my assumption I put [imath]k[/imath] in for [imath]n[/imath] and for my proof so far I have: \begin{align*} 15 &\mid 2^{4(k+1)}-1 \\ RHS & =2^{4k+4} -1 \\ &=2^4 \cdot 2^{4k}-1 \end{align*} Is the splitting in the last step correct? Also how do I/where do I substitute my induction hypothesis into the equation? step by step explanation please! | 669106 | Prove [imath]5\mid2^{4n}-1[/imath] by induction
For all [imath]n\ge1[/imath], use mathematical induction to establish each other the following divisibility statements: [imath]5\mid2^{4n}-1[/imath] I was wondering if someone could help me with the set up of this proof. I only have done proofs with mathematical induction when I am given a series of numbers and I can incorporate my induction hypothesis in for one of my terms in the series. Step by step explanation please? |
671413 | Let [imath]G[/imath] be a group, [imath]H[/imath] be a subgroup of [imath]G[/imath] and [imath]h[/imath] be a fixed element of [imath]G[/imath].
Let [imath]G[/imath] be a group, [imath]H[/imath] be a subgroup of [imath]G[/imath] and [imath]h[/imath] be a fixed element of [imath]G[/imath]. Show that the subset [imath]gHg^{-1}=\{ghg^{-1}:h\in H\} [/imath] is a subgroup of [imath]G[/imath]. I know of the one and two-step tests for subgroups but I'm at a loss on how to implement either of them as I'm not sure what the inverse is. | 532676 | Prove [imath]x^{-1}Hx[/imath] is a subgroup of [imath]G[/imath]
Let [imath]G[/imath] be a group and [imath]H[/imath] a subgroup of [imath]G[/imath]. For [imath]x \in G[/imath], prove [imath]x^{-1}Hx[/imath] is a subgroup of [imath]G[/imath]. |
672044 | [imath]\mathcal{C}[/imath] is an algebra
I'm trying to show that the following set is an algebra: Let [imath]X[/imath] be some set. We define a system of sets [imath]\mathcal{C}\subseteq \mathcal{P}(X)[/imath] by [imath]\mathcal{C} = \{ S \subseteq X : \text{for all } E \subseteq X \text{ we have } \mu^*(E) = \mu^*(E \cap S) + \mu^*(E \cap S^c),[/imath] where [imath]\mu^*:\mathcal{P}(X) \to \overline{\mathbb{R}}_{+} [/imath] is an outer measure, i.e. [imath]\mu^*(\emptyset) = 0[/imath], For all [imath]A \subseteq B \subseteq X[/imath], [imath]\mu^*(A) \leq \mu^*(B)[/imath], For any sequence [imath]A_i \subseteq X, i =1,2,\ldots[/imath] we have [imath]\mu^*\left( \bigcup_{i=1}^{\infty} A_i \right) \leq \sum_{i=1}^{\infty} \mu^* (A_i).[/imath] Show that [imath]\mathcal{C}[/imath] is an algebra. So I only need to show three things: [imath]X \subset C[/imath] [imath]A \subset C \implies A^c \subset C[/imath] [imath]A,B \subset C \implies A \cup B \subset C[/imath] 1: Letting [imath]X=S[/imath], we see that in order for [imath]X \subset C[/imath] to be true, [imath]\mu^*(E) \geq \mu^*(E \cap X) + \mu^*(E \cap \emptyset) = \mu^*(E \cap X) = \mu^*(E).[/imath] Since [imath]\mu^*(E) \geq \mu^*(E)[/imath] is trivially true, [imath]X \subset C[/imath]. 2: Suppose [imath]A \subset C[/imath]. Then [imath]\mu^*(E) \geq \mu^*(E \cap A) + \mu^*(E \cap A^c)[/imath]. Now consider [imath]S=A^c[/imath], then [imath]\mu^*(E \cap A^c) + \mu^*(E \cap (A^c)^c) = \mu^*(E \cap A^c) + \mu^*(E \cap A) \leq \mu^*(E),[/imath] so [imath]A^c \subset C[/imath]. 3: I'm a little stuck with 3. We have that [imath]\mu^*(E) \geq \mu^*(E \cap A) + \mu^*(E \cap A^c),\\ \mu^*(E) \geq \mu^*(E \cap B) + \mu^*(E \cap B^c).[/imath] How can I use this to show that [imath]\mu^*(E) \geq \mu^*(E \cap (A \cup B)) + \mu^*(E \cap (A \cup B)^c)?[/imath] I have tried doing some adding and subtracting but nothing seems to bring anything. Some other trick maybe? | 257129 | Proving that the set of [imath]\nu[/imath] measurable sets [imath]M\subseteq P(X)[/imath] is an algebra
I am not sure I am using the standard definitions so I will open by defining what I need: Let [imath]X[/imath] be a set, [imath]\nu:\, P(X)\to[0,\infty][/imath] will be called an external measure if [imath]\nu(\emptyset)=0[/imath] and for any [imath]\{A_{i}\}_{i=1}^{\infty}\subseteq P(x)[/imath] (not neccaseraly disjoint) it holds that [imath]\nu(\cup_{i=1}^{\infty}A_{i})\leq\sum_{i=1}^{\infty}\nu(A_{i})[/imath] Let [imath]\nu[/imath] be an external measure on a set [imath]X[/imath] then we say that a set [imath]A[/imath] is [imath]\nu[/imath] measurable if for any [imath]E\subseteq X[/imath]: [imath]\nu(E)=\nu(E\cap A)+\nu(E\cap A^{c})[/imath] The exercise asks to prove that the set of [imath]\nu[/imath] measurable sets [imath]M\subseteq P(X)[/imath] is an algebra. I have proved [imath]\emptyset,X\in M[/imath] and that [imath]A\in M\implies A^{c}\in M[/imath] but I am having problems proving closer under union and intersection. I assume that [imath]A_{1},A_{2}[/imath] are [imath]\nu[/imath] measurable so I get that for any [imath]E[/imath]: [imath]\nu(E)=\nu(E\cap A_{1})+\nu(E\cap A_{1}^{c})[/imath] [imath]\nu(E)=\nu(E\cap A_{2})+\nu(E\cap A_{2}^{c})[/imath] And I need to prove that for any [imath]E'[/imath]: [imath]\nu(E')=\nu(E'\cap A_{1}\cap A_{2})+\nu(E'\cap(A_{1}\cap A_{2})^{c})[/imath] which is the same as [imath]\nu(E')=\nu(E'\cap A_{1}\cap A_{2})+\nu((E'\cap A_{1}^{c})\cup(E'\cap A_{2}^{c}))[/imath] and a similar result to prove closer under union. I guess that it all have to do with choosing the right [imath]E[/imath]'s from knowing that [imath]A_{i}[/imath] are [imath]\nu[/imath] measurable, but I tried different options for an hour now and I don't see this going anywhere. I need some help in showing closer under union and intersection |
672131 | Direct sum of [imath]T^n[/imath]
Question Let [imath]T:V\to V , dimV=n , \Bbb F= \Bbb C[/imath] a linear map Prove that [imath]KerT^n \oplus ImT^n=V[/imath] Thoughts We tried using the rank nullity theorem but got stuck, as we don't know what is [imath]dim(ImT+KerT)[/imath] | 669417 | The kernel and image of [imath]T^n[/imath]
I need help with this question: Let [imath]V[/imath] be a finite vector space where [imath] \dim V = n [/imath], over the complex numbers and let [imath] T: V\to V [/imath] be a linear transformation. Prove that [imath] V = \ker(T^n) \oplus Im(T^n) [/imath] |
672785 | prove f is holomorphic
Let [imath]\Omega[/imath] be an open subset of C not containing [imath]0[/imath]. Let [imath]f[/imath] be a complex valued continuous function on [imath]\Omega[/imath]. Suppose [imath](f(z))^2=z^3[/imath]. Prove that f is holomorphic on [imath]\Omega[/imath]. | 354249 | [imath]f[/imath] continuous, [imath]f^N[/imath] analytic on a domain D implies [imath]f[/imath] analytic on D
Working on a problem in Gamelin's book. "Show that if [imath]f(z)[/imath] is a continuous function on a domain [imath]D[/imath] such that [imath]f(z)^N[/imath] is analytic on [imath]D[/imath] for some integer [imath]N[/imath], then [imath]f(z)[/imath] is analytic on [imath]D[/imath]." He starts a hint with the statement "Show that the zeros of [imath]f(z)[/imath] are isolated." This makes me think. I know that if [imath]f[/imath] is analytic and nonconstant, then the zeros are isolated. Is there some sort of converse to this theorem? I.e., for example, if continuous and zeros are isolated then analytic? His next hint is: "At a zero of [imath]f(z)[/imath], write [imath]f(z)^N=(z-z_0)^mh(z)[/imath] where [imath]h(z_0)\ne 0[/imath] and show that [imath]N[/imath] divides [imath]m[/imath]." However, the book has covered nothing on winding numbers thus far. Also, how will the fact that [imath]N[/imath] divides [imath]m[/imath] help to show that [imath]f(z)[/imath] is analytic? |
571376 | Probability measure for Schwartz function
Let [imath]X[/imath] be a set, [imath]\mathcal F[/imath] a [imath]\sigma[/imath]-field of subsets of [imath]X[/imath], and [imath]\mu[/imath] a probability measure on [imath]X[/imath]. Given random variables [imath]f,g\colon X\rightarrow\mathbb{R}[/imath] such that [imath]\int_\mathbb{R}hd{\mu_f}=\int_{\mathbb{R}}hd\mu_g[/imath] for any Schwartz function [imath]h[/imath]. I want to show that [imath]\mu_f=\mu_g[/imath]. Here there is a proof that the statement is true even if we restrict ourselves to test functions, but the proof looks quite complicated. Since Schwartz functions form a larger class, is there an easy proof for this statement? | 569997 | Integral of Schwartz function over probability measure
Let [imath]X[/imath] be a set, [imath]\mathcal F[/imath] a [imath]\sigma[/imath]-field of subsets of [imath]X[/imath], and [imath]\mu[/imath] a probability measure on [imath]X[/imath]. Given random variables [imath]f,g\colon X\rightarrow\mathbb{R}[/imath] such that [imath]\int_\mathbb{R}hd{\mu_f}=\int_{\mathbb{R}}hd\mu_g[/imath] for any Schwartz function [imath]h[/imath]. Is it necessarily true that [imath]\mu_f=\mu_g[/imath]? |
62908 | How can an ordered pair be expressed as a set?
My book says \begin{equation} (a,b)=\{\{a\},\{a,b\}\} \end{equation} I have been staring at this for a bit and it doesn't make sense to me. I have read several others posts on this, but none made any sense to me. For example, Definition of an Ordered Pair Based on how my ignorant brain is viewing this, I don't see why the definition could not be. \begin{equation} (a,b)=\{\{a\},\{b\},\{a,b\}\} \end{equation} aka the power set. What is the significance of the {a} in that definition? Please keep things simple if possible. Normally definitions have a valid and clear reason for being defined that way. Clarification First, I understand what an ordered pair is. I just don't see how the set notation says that. Second, \begin{equation} (a,b) = \{\{a\},\{a,b\}\}=\{\{a,b\},\{a\}\} \end{equation} Sets don't preserve order, but ordered pairs do. How does the third part of the equality apply to the definition? Third, another issue with the notation that I have starts with the Product Property of Sets \begin{equation} \text{Let [imath]X[/imath] and [imath]Y[/imath] be sets} :\ X=\{a,b,c\}\text{ and }Y=\{a,d,e\}. \end{equation} \begin{equation} \text{Then }X \times Y = \{(a,a),(a,d),(a,e),(b,a),(b,d),(b,e),\dots,(c,e)\} \end{equation} If we look at the first ordered pair and our given definition we have \begin{equation} (a,a)=\{\{a\},\{a,a\}\} \end{equation} How can this be so, you can't have duplicates in sets? I guess what I am looking for in an answer, is not a proof or a definition of ordered pairs, but rather something like, "This notation says what it says because...". Except for the second to last point I get the terminology, I just don't get the connection between the two different uses of notation. | 1647027 | Show that if [imath](x_1,x_2)[/imath] is defined to be [imath]\{\{x_1\},\{x_1,x_2\}\}[/imath] then [imath](x_1,x_2)=(y_1,y_2)[/imath] iff [imath]x_1=y_1[/imath] and [imath]x_2=y_2[/imath]
My Work: If you take the cartesian product of any set with two arbitrary elements [imath]a[/imath] and [imath]b[/imath], and the resulting set is [imath]\{\{x_1\},\{x_1,x_2\}\}[/imath], then the only possible values for [imath]a[/imath] and [imath]b[/imath] are [imath]x_1[/imath] and [imath]x_2[/imath] by definition of the cartesian product. This answer seems overly trivial and I think I'm doing something wrong. Please help! |
673006 | Set Difference Probability
Here is the question: Prove that for every [imath]\epsilon>0[/imath] and every set [imath]A\in\mathcal{B}(\mathbb{R}^{n})[/imath] there is a compact set [imath]K\subset A[/imath] such that [imath]P(A\setminus K)\leq\epsilon[/imath]. --I have previously shown that there is a closed set [imath]F[/imath] and an open set [imath]G[/imath] such that [imath]F\subset A\subset G[/imath] and [imath]P(G\setminus F)\leq\epsilon[/imath]. For the current problem, I think that we can find [imath]K[/imath] as an approximation of the set [imath]F[/imath], such that [imath]K=F_{N}=F\cap[-N,N]^{n}[/imath], which is compact. I think I am just caught up in the set difference algebra, i.e.,[imath]P(A\setminus K)=P(A\setminus (F\cap[-N,N]^{n}))=P(A\cap(F^{c}\cup[-N,N]^{c})^{c})=\cdots[/imath] --Any help on how to finish this is appreciated! | 271848 | Inner regularity of Lebesgue measurable sets
This is an exercise in real analysis: Let [imath]E\subset{\Bbb R}^d[/imath] be Lebesgue measurable. Show that [imath] m(E)=\sup\{m(K):K\subset E, K \text{compact}\}. [/imath] When [imath]E[/imath] is bounded, this can be done by the following proposition: [imath]E\subset{\Bbb R}^d[/imath] is Lebesgue measurable if and only if for every [imath]\varepsilon>0[/imath], one can find a closed set [imath]F[/imath] contained in [imath]E[/imath] with [imath]m^*(E\setminus F)\leq\varepsilon[/imath]. How can I deal with the case that [imath]E[/imath] is unbounded? |
276135 | Calculating [imath]\lim_{n\to\infty}\sqrt{n}\sin(\sin...(\sin(x)..)[/imath]
I was asked today by a friend to calculate a limit and I am having trouble with the question. Denote [imath]\sin_{1}:=\sin[/imath] and for [imath]n>1[/imath] define [imath]\sin_{n}=\sin(\sin_{n-1})[/imath]. Calculate [imath]\lim_{n\to\infty}\sqrt{n}\sin_{n}(x)[/imath] for [imath]x\in\mathbb{R}[/imath] (the answer should be a function of [imath]x[/imath] ). My thoughts: It is sufficient to find the limit for [imath]x\in[0,2\pi][/imath] , and it is easy to find the limit at [imath]0,2\pi[/imath] so we need to find the limit for [imath]x\in(0,2\pi)[/imath]. If [imath][a,b]\subset(0,\pi)[/imath] or [imath][a,b]\subset(\pi,2\pi)[/imath] we have it that then [imath]\max_{x\in[a,b]}|\sin'(x)|=\max_{x\in[a,b]}|\cos(x)|<\lambda\leq1[/imath] hence the map [imath]\sin(x)[/imath] is a contracting map. We know there is a unique fixed-point but since [imath]0[/imath] is such a point I deduce that for any [imath]x\in(0,2\pi)[/imath] s.t [imath]x\neq\pi[/imath] we have it that [imath]\lim_{n\to\infty}\sin_{n}(x)=0[/imath] So I have a limit of the form "[imath]0\cdot\infty[/imath]" and I can't figure out any way on how to tackle it. Can someone please suggest a way to find that limit ? Note: I am unsure about the tags, please change them if you see fit. | 1089733 | How do I solve this limit: [imath] \lim_{x \to \infty } \sqrt{n}\sin(\sin(\sin ... (\sin (1))...)) [/imath]
I have been strugling a lot to solve this question, but couldn't figure out where to start. [imath] \lim_{x \to \infty } \sqrt{n} . \underbrace {\sin(\sin(\sin ... (\sin (1))...))}_{n...times..} [/imath] I think maybe I should assume the part inside brackets as [imath]y[/imath]. But what next?? Any hints/suggestions? |
672926 | Is the scheme for generating [imath]\displaystyle p_n=\left(\frac{1}{3}\right)^n[/imath] stable?
Is the scheme for generating [imath]\displaystyle p_n=\left(\frac{1}{3}\right)^n[/imath] stable? [imath]\displaystyle p_{n} = \frac{5}{6} p_{n-1} - \frac{1}{6} p_{n-2}[/imath] | 672727 | Is the following scheme for generating [imath]p_n=(1/3)^n[/imath] stable or not. [imath]p_n=(5/6)p_{n-1}-(1/6)p_{n-2}[/imath].
Is the scheme for generating [imath]p_n=(1/3)^n[/imath] stable or not? [imath]p_n= \frac{5}{6} p_{n-1} - \frac{1}{6}p_{n-2}.[/imath] |
61713 | What's the limit of the sequence [imath]\lim\limits_{n \to\infty} \frac{n!}{n^n}[/imath]?
[imath]\lim_{n \to\infty} \frac{n!}{n^n}[/imath] I have a question: is it valid to use Stirling's Formula to prove convergence of the sequence? | 1389890 | Limit of [imath]n!/n^n[/imath] as [imath]n[/imath] tends to infinity
Using a calculator, I found that [imath]n![/imath] grows substantially slower than [imath]n^n[/imath] as [imath]n[/imath] tends to infinity. I guess the limit should be [imath]0[/imath]. But I don't know how to prove it. In my textbook a hint is given that: Set [imath]a_n=n!/n^n[/imath] Set [imath]m=[n/2][/imath](floor function), then [imath]a_n \le (1/2)^m\le(1/2)^{n/2}[/imath]. Then by comparing to the geometric progression, the sequence [imath]a_n[/imath] tends to [imath]0[/imath]. I have trouble proving the relationship [imath]a_n \le (1/2)^m\le(1/2)^{n/2}[/imath], (I tried to prove by considering separate cases, that is when [imath]m[/imath] is odd and when it is even) using induction gets me nowhere. Or is there other way to prove this limit? I made some search on web and used Stirling's approximation, but to no avail. P/S: Although some said that my question is probably duplicate, the main point in my question is understanding and proving the relationship of the inequalities, which I had trouble understanding and was not addressed in the other suggested question(the sequence [imath](1/n)[/imath] was used as comparison instead of [imath](1/2)^{n/2}[/imath].) |
674241 | Relationship Between Circles Under Inversion
Considering the inversion [imath]w=\frac{1}{z}[/imath] Consider two unshaded circles [imath]C_{r}[/imath] and [imath]C_{s}[/imath] with radii [imath]r>s[/imath] that touch at the origin of the complex plane. The shaded circles [imath]C_{1}[/imath],[imath]C_{2}...C_{7}[/imath] (labeled in counterclockwise direction sequentially) all touch [imath]C_{r}[/imath] internally and [imath]C_{s}[/imath] externally. [imath]C_{1}[/imath] also touches the real axis and [imath]C_{i}[/imath] and [imath]C_{i+1}[/imath] touch for [imath]i=1...6[/imath]. Let [imath]r_{i}[/imath] denote the radius of [imath]C_{i}[/imath]. Then show that for [imath]i=1,2...,[/imath] [imath]r^{−1}_{i}+3r^{−1}_{i+2}=3r^{−1}_{i+1}+r^{−1}_{i+3}[/imath]. I'm struggling to begin with this problem. I know that under inversion the two circles touching at the origin [imath]C_{r}[/imath] and [imath]C_{s}[/imath] will be mapped to straight lines and that the other circles will be mapped to circles, however I'm struggling to see how I would obtain the [imath]r^{-1}_{i}[/imath]s. Any help would be much appreciated. | 674207 | Mapping circles via inversion in the complex plane
Consider two unshaded circles [imath]C_r[/imath] and [imath]C_s[/imath] with radii [imath]r>s[/imath] that touch at the origin of the complex plane. The shaded circles [imath]C_1,C_2...C_7[/imath] (labeled in counterclockwise direction sequentially) all touch [imath]C_r[/imath] internally and [imath]C_s[/imath] externally. [imath]C_1[/imath] also touches the real axis and [imath]C_i[/imath] and [imath]C_{ i+1}[/imath] touch for [imath]i=1...6.[/imath] Let [imath]r_i[/imath] denote the radius of [imath]C_i[/imath]. Then show that for [imath]i=1,2...,[/imath] [imath]r_i^{-1} + 3r_{i+2}^{-1} = 3r_{i+1}^{-1} + r_{i+3}^{-1}[/imath] A picture is attached for clarity. Attempt: Under inversion, [imath]C_s[/imath] and [imath]C_r[/imath] are mapped to lines and [imath]C_i, i \in \left\{1,...,6\right\}[/imath] are mapped to circles. The map [imath]f(z) = 1/z[/imath] is conformal, so the angles are preserved wherever [imath]f'(z) \neq 0[/imath]. Each circle has [imath]4[/imath] tangent points, (except [imath]C_7[/imath]) so under the transformation these [imath]\pi/2[/imath] angles are preserved. The resulting image I have after the transformation is that the [imath]C_i[/imath] are mapped to circles that lie within the two lines and touch the sides. The only way I see just now to preserve the angles is to have all the circles of the same radii. If [imath]s[/imath] is the radius of [imath]C_s[/imath] and [imath]r[/imath] the radius of [imath]C_r[/imath] then the centre between these two lines is [imath]\frac{1}{2}\left(1/s - 1/2r\right) [/imath]. (But I do not think this makes sense.) Many thanks |
674676 | Prove this statement?
I am having trouble with the following proof: Prove that for every three positive real numbers a, b, and c that [imath](a+b+c)*(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) \ge 9[/imath]. I have tried to directly prove this but all I get are dead ends. | 674622 | Discrete Math Proofs Involving Real Numbers
I am stuck on these two problems. [imath]1[/imath]. Prove that for every three positive real numbers a, b, and c that [imath](a+b+c)*(\frac{1}{a}+\frac{1}{b} + \frac{1}{c}) \ge 9[/imath]. [imath]2[/imath]. Prove that for every three positive real numbers a, b, and c that [imath]a^2 + b^2 + c^2 \ge ab + bc + ac[/imath]. I have tried direct proof and have not gotten anywhere significant. I won't put the work on there since it is way too long and I don't think it will help. There must be some sort of trick involved, but for the life of me, I cannot figure it out. |
607684 | Limit of the ratio of consecutive Fibonacci numbers
I have read in a book that the limit of the ratio of consequent Fibonacci numbers is the golden ratio. However, it was just mentioned thus not justified. So, my question is how would you derive the following limit: [imath]\lim_{x\to\infty}{\frac{F_n}{F_{n+1}}}=?[/imath] Where [imath]F_n[/imath] is the nth Fibonacci number? | 2353041 | Consider the recursive relation [imath] a_0=1, \ a_1=1 \ \ and \ \ a_{n+1}=a_n+a_{n-1} [/imath].
Consider the recursive relation [imath] a_0=1, \ a_1=1 \ \ and \ \ a_{n+1}=a_n+a_{n-1} [/imath]. Assuming that [imath] \ \ a=\lim |\frac{a_n}{a_{n+1}}| \ [/imath] exists , find the limit [imath] \ a \ [/imath]. Answer: [imath] \ \ a=\lim |\frac{a_n}{a_{n+1}}|=\lim |\frac{a_n}{a_{n}+a_{n-1}}| [/imath] , (since [imath]a_{n+1}=a_n+a_{n-1}) , [/imath] or, [imath] a=\lim |\frac{\frac{a_n}{a_{n-1}}}{\frac{a_n} {a_{n-1}}+1}| [/imath] or, [imath] a=\frac{\lim|\frac{a_n}{a_{n-1}}|}{\lim|\frac{a_n}{a_{n-1}}+1|}=\frac{a}{\lim|\frac{a_n}{a_{n-1}}+1|}[/imath] , (Since [imath] \lim |a_n /a_{n-1}|=\lim |a_{n+1}/a_n| [/imath] ) . Or, [imath] \lim|\frac{a_n}{a_{n-1}}+1|=1 [/imath] But I can't proceed furher . Any help is really ppreciating . |
674954 | Prove [imath]\omega + \omega_1 = \omega_1[/imath]
I am assuming that [imath]\omega_1[/imath] is the first uncountable ordinal and I'm using ordinal arithmetic. I have so far that if [imath]\alpha[/imath] and [imath]\beta[/imath] are ordinals, then [imath]\alpha + \beta[/imath] = sup{[imath]\alpha + \gamma[/imath] | [imath]\gamma < \beta\}[/imath] where [imath]\beta[/imath] is a limit ordinal. So since [imath]\omega[/imath] and [imath]\omega_1[/imath] are both ordinals, can I say that [imath]\omega + \omega_1[/imath] = sup{[imath]\omega + \gamma[/imath] | [imath]\gamma < \omega_1 \}[/imath]? And then I'm not sure where to go with this. Thanks in advanced. | 650792 | Prove that [imath]\omega + \omega_1 = \omega \cdot \omega_1 = \omega^{\omega_1} = \omega_1[/imath]
I am assuming already that a) the union of countably many countable sets is countable and b) [imath]\omega_1[/imath] is the least uncountable ordinal, so [imath]x < \omega_1[/imath] if and only if [imath]x[/imath] is a countable ordinal. I'm not sure if it is relevant but this question also allows The Axiom of Choice. Thanks |
674968 | Prove [imath]\epsilon_0[/imath] < [imath]\omega_1[/imath]
This is a question in ordinal arithmetic. (If anyone has read 'Classic Set Theory' by Derek Goldrei, this question comes from page 252.) [imath]\epsilon_0[/imath] = sup {[imath]\omega[/imath], [imath]\omega^\omega[/imath], ... } and [imath]\omega[/imath] is the smallest infinite ordinal. [imath]\omega_1[/imath] is the least uncountable ordinal so that [imath]\alpha < \omega_1[/imath] if and only if [imath]\alpha[/imath] is a countable ordinal. I will take the definition of an ordinal to be: a set [imath]\alpha[/imath] is an ordinal if (i) [imath]\alpha[/imath] is well-ordered by [imath]\in[/imath] and (ii) if [imath]\beta \in \alpha[/imath] then [imath]\beta[/imath] is a subset of [imath]\alpha[/imath]. (or [imath]\alpha[/imath] is [imath]\in[/imath]-transitive). I am also not assuming that [imath]\epsilon_0[/imath] is countable at this point. | 109206 | How is [imath]\epsilon_0[/imath] countable?
In Wikipedia, it says that any epsilon number with the index that is countable is countable. How is it? Out of all those numbers, I especially want to know why [imath]\epsilon_0[/imath] is countable. Thanks. |
675129 | Prove : [imath]\frac{x}{y}=\frac{ad+bc}{ab+cd}[/imath]
The quadrilateral [imath]ABCD[/imath] is cyclic [imath](O)[/imath]; [imath]AB=a, BC=b, CD=c, AD=d, AC=x, BD=y[/imath]. Prove : [imath]\frac{x}{y}=\frac{ad+bc}{ab+cd}[/imath] Thanks :) P/s : I have no ideas about this problem ! :( | 649095 | If [imath]ABCD[/imath] is a cyclic quadrilateral, then [imath]AC\cdot(AB\cdot BC+CD\cdot DA)=BD\cdot (DA\cdot AB+BC\cdot CD)[/imath]
If [imath]ABCD[/imath] is a cyclic quadrilateral, then [imath] AC\cdot(AB\cdot BC+CD\cdot DA)=BD\cdot (DA\cdot AB+BC\cdot CD) [/imath] I tried using many approaches, but I could not find a proper solution. Can anyone please help me with this problem? |
541790 | Counting ways to partition a set into fixed number of subsets
Suppose we have a finite set [imath]S[/imath] of cardinality [imath]n[/imath]. In how many ways can we partition it into [imath]k[/imath]-many non empty subsets? Example: There is precisely one way to partition such a set into [imath]n[/imath]-many subsets. and there is one way to partition into a single (sub)set. | 1894095 | How many ways to partition a set?
I have a set [imath]A:=\{1,2,3,4,5,6,7\}[/imath]. How many ways I have to partition [imath]A[/imath] in [imath]5[/imath] non-empty subsets? Which are these ways? |
675879 | If [imath]\gcd (a,b)=1[/imath], and [imath]\gcd(a,c)=1[/imath], then [imath]\gcd (a,bc)=1[/imath]
If [imath]\gcd (a,b)=1[/imath], and [imath]\gcd(a,c)=1[/imath], then [imath]\gcd (a,bc)=1[/imath] Help proving this? I'm really confused how to go about it.. | 673119 | If [imath]\gcd(a,b)=1[/imath] and [imath]\gcd(a,c)=1[/imath], then [imath]\gcd(a,bc)=1[/imath]
How do I go about proving this? If [imath]\gcd(a,b)=1[/imath] and [imath]\gcd(a,c)=1[/imath], then [imath]\gcd(a,bc)=1[/imath]. I'm very confused with gcd proofs. |
671545 | Projective module over a PID is free?
A common result is that finitely generated modules over a PID [imath]R[/imath] are projective iff they are free. Is the same true that an arbitrary projective module over a PID is free? I can't find this fact anywhere, so I suspect it is false, but I can't construct an example. Does anyone have an example of a projective module over a PID which is not free? Thank you. | 1003118 | Non finitely-generated projective [imath]\mathbb{Z}[/imath]-module
Let [imath]M[/imath] be a projective [imath]\mathbb{Z}[/imath]-module. Must [imath]M[/imath] be free? It is easy to see that the answer is yes if [imath]M[/imath] is finitely generated, but I do not know about the general case. If the answer is "yes" (which would surprise me), is the same true for Dedekind domains? |
64857 | Painting the faces of a cube with distinct colours
I don't think this is solved by Burnside's Lemma since there is a condition that each side is painted a different colour. The question is as follows. If I had a cube and six colours, and painted each side a different colour, how many (different) ways could I paint the cube? What about if I had [imath]n[/imath] colours instead of 6? The answer given in an old thread on a different site is [imath]6![/imath] for the first question, and [imath]n(n-1)(n-2)(n-3)(n-4)(n-5)[/imath] for the second question. However, this doesn't actually hold up because a few of the paintings are isomorphic. The original thread assumes we can somehow tell the difference between two paintings which actually look identical if you rotate the cube, which I don't think is what the question intended. The answer I got for the first question is [imath]4! + 4 = 28[/imath]. But this was just through a case-bash, and I'm not sure whether it's correct or whether it generalizes. | 783241 | Paint a cube with 6 colors
I have a unit cube and [imath]6[/imath] colors to paint its sides in. How many different cubes to I get if I use one color per side? I think I know the answer, but just want to be sure regarding my solution. Let colors be numbers [imath]1,2,\dots,6[/imath] and let us fix a side where we put [imath]1[/imath]. We have [imath]5[/imath] options for the opposite side. So we are left with a cyclic 4 sides and 4 numbers to put on them: let us now call them [imath]a,b,c,d[/imath]. We fix a side where we put [imath]a[/imath] and we are left with [imath]3![/imath] options for other sides. As a result the answer is [imath]5\cdot 3! = 30[/imath]. |
546030 | Significance of starting the Fibonacci sequence with 0, 1....
DISCLAIMER: I do not deal with in-depth mathematics on a daily basis as some of you may, so please pardon my ignorance or lack of coherence on this topic. QUESTION: What is the significance of starting the Fibonacci sequence with [imath]0,1[/imath] ? For instance, if I picked any two random integers, say 2 and 7, to start a sequence would I actually be creating some multiple or derivation of the Fibonacci sequence? Is there a general mathematical explanation for the relationship between any sequence represented by [imath]a[0] = x, a[1] = y, a[n] = a[n-1] + a[n-2][/imath] and the Fibonacci sequence? Or, back to my example sequence, is there a general mathematical relationship between: [imath]2,7,9,16,25,41,66,107,173,280...[/imath] and [imath]0,1,1,2,3,5,8,13,21,34...[/imath] Perhaps the Golden Ratio explains it somehow? Any help would be appreciated. | 700943 | how to find nth term of different fibonacci series with golden ratio
what i know : if i want to find [imath]Nth[/imath] term of a fibonacci series like : 1 1 2 3 5 8 13 21 ....... then to find [imath]6th[/imath] term we use golden ratio like: so it becomes like : well and good.now what i want to know : what changes should i make in golden ratio or while applying it for different fibonacci series . For example : 1 6 7 13 20 33 ....... now if i want to find the 10th no. of above series then what changes should i do in golden ratio... now what is the value of X10's R.H.S(right hand side) X10 = ? (for this fibonacci series) |
678211 | entire function with no zeros. Then [imath]f[/imath] must be of the form [imath]f(z)=\exp(g(z))[/imath]
Let [imath]f[/imath] be an entire function with no zeros. Then [imath]f[/imath] must be of the form [imath]f(z)=\exp(g(z))[/imath] where [imath]g[/imath] entire. Is it true or not? How to prove? | 74920 | The existence of analytical branch of the logarithm of a holomorphic function
[imath]\Omega[/imath] is a convex open set in [imath]$\mathbb {C}^n$[/imath] and [imath]f[/imath] is an analytical function Edit: without zero point on [imath]\Omega[/imath], then can we define an analytical branch of [imath]\ln {f}[/imath] on [imath]\Omega[/imath] ? |
79064 | If a function is uniformly continuous on [imath](-\infty,-1][/imath] and [imath][-1,\infty)[/imath] is it uniformly continuous on [imath]\mathbb{R}[/imath]
Is it correct to say that if [imath]f(x)[/imath] is uniformly continuous on [imath](-\infty,-1][/imath] and [imath][-1,\infty)[/imath], then it is uniformly continuous on [imath]\mathbb{R}[/imath]? I don't think this is true but cannot think of a counterexample. Below there is an example of where I want to use this. Thanks for any help Prove that [imath]f(x)=|x|^{\frac{1}{2}}[/imath] is uniformly continuous on [imath]\mathbb{R}[/imath] Proof. As [imath]|x|^{\frac 12}[/imath] is differentiable on [imath](-\infty,-1][/imath] and [imath][1,\infty)[/imath] with the derivatives bounded then it is uniformly continuous on these intervals. It is also continuous on [imath][-1,1][/imath] and so it is uniformly continuous. It is therefore continuous on [imath]\mathbb{R}[/imath]. | 582200 | Uniform continuity on [imath][a,b][/imath] and [imath] [b,c][/imath] [imath]\implies[/imath] uniform continuity on [imath][a,c][/imath].
Let [imath]f:\mathbb R \to \mathbb R[/imath]. Prove that if [imath]f[/imath] is uniformly continuous on [imath][a,b][/imath] and [imath][b,c][/imath], then [imath]f[/imath] is uniformly continuous on [imath][a,c][/imath]. My attempt at a solution: I've came up with a solution but I am having doubts if it is correct, so I would like to check that: Let [imath]\epsilon>0[/imath], we know that there exist [imath]\delta_1, \delta_2[/imath] such that if [imath]x,y \in [a,b] |x-y|<\delta_1 \implies |f(x)-f(y)|<\dfrac{\epsilon}{2}[/imath] and if [imath]x,y \in [b,c], |x-y|<\delta_2 \implies |f(x)-f(y)|<\dfrac{\epsilon}{2}[/imath] Let [imath]\delta=\min\{2\delta_1,2\delta_2\}[/imath], if [imath]|x-y|=|x-b+b-y|\leq |x-b|+|b-y|<\delta \implies |f(x)-f(y)|\leq |f(x)-f(b)|+|f(b)-f(y)|\leq \dfrac{\epsilon}{2}+\dfrac{\epsilon}{2}=\epsilon[/imath]. This proves that [imath]f[/imath] is uniformly continuous. I would appreciate if anyone could tell me if my proof is ok or if I've made any mistakes. (S.A. Understanding Analysis. pp 119 question 4.4.7) |
678497 | Prove that [imath]{n^5 - n}[/imath] is divisible by 5
I need to prove by induction if [imath]{n^5 - n}[/imath] is divisible by 5 and I have no idea how I would do it. I am trying to prove it for several hours now, I started with [imath]{n^5 - n} \mod 5 = 0[/imath] but then I realized that I have no idea how to use modulo for transformations. The next thing I tried was [imath](5 + n)\frac{n^5 - n}5 = \frac{(n+1)^5 - n+1}5[/imath] but it didn't get me anywhere. How would you do it? | 350675 | Prove [imath](n^5-n)[/imath] is divisible by 5 by induction.
So I started with a base case [imath]n = 1[/imath]. This yields [imath]5|0[/imath], which is true since zero is divisible by any non zero number. I let [imath]n = k >= 1[/imath] and let [imath]5|A = (k^5-k)[/imath]. Now I want to show [imath]5|B = [(k+1)^5-(k+1)][/imath] is true.... After that I get lost. I was given a supplement that provides a similar example, but that confuses me as well. Here it is if anyone wants to take a look at it: Prove that for all n elements of N, [imath]27|(10n + 18n - 1)[/imath]. Proof: We use the method of mathematical induction. For [imath]n = 1[/imath], [imath]10^1+18*1-1 = 27[/imath]. Since [imath]27|27[/imath], the statement is correct in this case. Let [imath]n = k = 1[/imath] and let [imath]27|A = 10k + 18k - 1[/imath]. We wish to show that [imath]27|B = 10k+1 + 18(k + 1) - 1 = 10k+1 + 18k + 17[/imath]. Consider [imath]C = B - 10A[/imath] ***I don't understand why A is multiplied by 10. [imath]= (10k+1 + 18k + 17) - (10k+1 + 180k - 10)[/imath] [imath]= -162k + 27 = 27(-6k + 1)[/imath]. Then [imath]27|C[/imath], and [imath]B = 10A+C[/imath]. Since [imath]27|A[/imath] (inductive hypothesis) and [imath]27|C[/imath], then [imath]B[/imath] is the sum of two addends each divisible by [imath]27[/imath]. By Theorem 1 (iii), [imath]27|B[/imath], and the proof is complete. |
514802 | Convergence of series implies convergence of Cesaro Mean.
Proof. Let [imath]\sum_{k = 0}^N c_k \rightarrow s[/imath], let [imath]\sigma_N = (S_0 + \dots + S_{N-1})/N[/imath] be the [imath]Nth[/imath] Cesaro sum where [imath]S_K[/imath] is the [imath]Kth[/imath] partial sum of the series. Then [imath]s - \sigma_N \\= s - c_0 - c_1(N-1)/N + c_2(N-2)N +\dots+c_{N-1}/N \\ =c_1/N + c_2 2/N + \dots + c_{N-1}(N-1)/N + c_N + \dots[/imath] Where do I go from here? | 565288 | Can you please check my Cesaro means proof
I wanted to prove the following: if [imath]x_n \to x[/imath] then [imath]y_n \to x[/imath] where [imath] y_n = {x_1 + \dots + x_n \over n}[/imath] Please can you tell me if my proof is correct? My proof is this: Let [imath]\varepsilon > 0[/imath]. Fix [imath]N[/imath] such that [imath]n > N[/imath] implies [imath]|x_n - x| < {\varepsilon \over 2}[/imath]. Then [imath]\left |{1 \over n} \sum_{k=N+1}^{N + n} x_k - x \right | < {\varepsilon \over 2}[/imath]. Now let [imath]M[/imath] be such that [imath]{|x_1 + \dots + x_N | \over M} < {\varepsilon \over 2}[/imath] and [imath]M > N[/imath]. Then [imath] \left | \sum_{k=1}^M {x_k \over M} - x \right | \le \left | \sum_{k=1}^N {x_k \over M} \right | + \left | \sum_{k=N+1}^M {x_k \over |M-N|} - x \right | < \varepsilon [/imath] Here the proof is finished. But one can observe: It is possible that [imath]y_n[/imath] converges even if [imath]x_n[/imath] doesn't: If [imath]x_{2n} = 0[/imath] and [imath]x_{2n + 1} = 1[/imath] then [imath]x_n[/imath] does not converge but [imath]y_n \to {1 \over 2}[/imath]. |
678604 | What is the order of this group?
Let [imath]H[/imath] be the subgroup of the group [imath]G[/imath] of all [imath]2 \times 2[/imath] non-singular matrices whose entries are integers modulo a given prime [imath]p[/imath] consisting of those and only those matrices in [imath]G[/imath] whose determinant is [imath]1[/imath]. What is the order of [imath]H[/imath]? And how to find it? I've already managed to find the order of [imath]G[/imath]. It is [imath]p^4 - p^3 - p^2 + p[/imath]. Last but not least, I would also like to be able to compute the order of [imath]G[/imath] and that of [imath]H[/imath] in the general case corresponding to the [imath]n \times n[/imath] matrices for an arbitrary integer [imath]n > 2[/imath]. | 616286 | The number of [imath]2 × 2[/imath] invertible matrices with entries from field [imath]\mathbb{Z}_p[/imath]
Let [imath]p[/imath] be a prime number and let [imath]\mathbb{Z}_p[/imath] denote the field of integers modulo [imath]p[/imath]. Find the number of [imath]2 × 2[/imath] invertible matrices with entries from this field. an I get some help how to solve the problem. thanks. |
678682 | Prove that [imath](n!)^2[/imath] is greater than [imath]n^n[/imath] for all values of n greater than 2.
This problem , I assume can be proved using induction, however I am trying to find another way. Is there a simple combinatorial approach? One notices that [imath](n!)^2[/imath] is equal to the number of permutations of size n squared, and that [imath]n^n[/imath] is the number of redundant combinations where there are n spaces and n choices. Any help would be much appreciated. Thanks | 640602 | Show that if [imath]n>2[/imath], then [imath](n!)^2>n^n[/imath].
Show that if [imath]n>2[/imath], then [imath](n!)^2>n^n[/imath]. My work: I tried to apply induction. So, at the induction step, I need to prove, [imath]n^n>(n+1)^{n-1}[/imath] Here, I tried to use induction again without any luck. I also took log of both sides, but I did not get anything. Please help! |
679225 | Problem about points on an equilateral triangle
Suppose that [imath]A[/imath], [imath]B[/imath], and [imath]C[/imath] are three points in a plane, such that [imath]AB = AC = BC = 1[/imath]. At each point in time, [imath]A[/imath] is moving toward [imath]B[/imath], [imath]B[/imath] is moving toward [imath]C[/imath], and [imath]C[/imath] is moving towards [imath]A[/imath], all with speed [imath]v = 50[/imath]. At what time [imath]T[/imath] will all the points reach the center of the triangle? | 44896 | The vertices of an equilateral triangle are shrinking towards each other
For an equilateral triangle ABC of side [imath]a[/imath] vertex A is always moving in the direction of vertex B, which is always moving the direction of vertex C, which is always moving in the direction of vertex A. The modulus of their "velocity" is a constant. When and where do they converge. Attempt. Found the "when" using a physics style approach by "fixing the frame" on one of the vertices. (From this frame, other two vertex are moving towards origin in a straight line and components of their speed along this line can be used to find when the three meet at origin) For the "where" it is difficult using above approach as this is some kind of rotating and shrinking triangle which is difficult to translate. @all Apologies for bumping this question. I wished to give an answer the bounty but it wont let me until the next 23 hours. For the record: I am not seeking new answers. Update: A cool example of PSTricks package of [imath]\LaTeX[/imath], for anyone who finds this question later. Link to code (a .tex file) And using Pgf/TikZ Source Page |
427635 | Grandi's series contradiction
This is the Grandi's series: [imath]1-1+1-1+1-1+\dots[/imath] The series can be equal to [imath]0[/imath] [imath](1-1)+(1-1)+(1-1)+\dots=0+0+0+\dots=0,[/imath] or to [imath]1[/imath] [imath]1-(1-1)-(1-1)-(1-1)-\dots=1-0-0-0\dots=1,[/imath] or to [imath]1/2[/imath] [imath]S=1-1+1-1+1-\dots,\quad\quad S=1-(1-1+1-1+1-1+1-...)[/imath] [imath]\Rightarrow S=1-S\Rightarrow 2S=1\Rightarrow S=1/2[/imath] Isn't this a contradiction? The integers are closed under addition and subtraction, but we get a fraction. Why? | 425226 | Finding the fallacy in this broken proof
Today, a friend gave me a "proof" of [imath]1=2[/imath] and challenged me to find the fallacy. [imath]1 = 1[/imath] [imath]1 = 1 + 0 + 0 + 0 ...[/imath] [imath]1 = 1 + 1 - 1 + 1 - 1 + 1 - 1 ...[/imath] [imath]1 = 2 - 1 + 1 - 1 + 1 - 1 ...[/imath] [imath]1 = 2 + 0 + 0 ...[/imath] [imath]1 = 2[/imath] My answer was that once you turn the initial [imath]1 + 1[/imath] into a 2, everything is offset so a [imath]-1[/imath] is always left at the end no matter how many times it is repeated. This negative one balances out the [imath]2[/imath] at the beginning so [imath]1=1[/imath] still holds true. I.e. [imath]1 = 1 + 0 + 0 + 0 ... = 1 + (1 - 1) + (1 - 1) + (1 - 1) = 2 + (-1 + 1) + (-1 + 1) - 1[/imath] However, my friend claimed that my answer only applies if the [imath]+ 1 - 1[/imath] repeats for a finite number of times. He argues that because the sequence repeats infinitely and things work differently when working with infinity, my answer is not valid. Can anyone enlighten me to the true fallacy in this proof? |
678963 | Linearly independent linear functionals
Let [imath] f_1,\ldots,f_n[/imath] be linearly independent linear functionals on a vector space [imath]X[/imath]. Show that there are [imath]n[/imath] elements [imath]x_1,\ldots,x_n[/imath] in [imath]X[/imath] such that the [imath]n\times n[/imath] matrix [imath][f_i(x_j) ][/imath] is non-singular. | 679057 | Linearly independent functionals
Let [imath] f_1,\ldots,f_n[/imath] be linearly independent linear functionals on a vector space [imath]X[/imath]. Show that there are [imath]n[/imath] elements [imath]x_1,\ldots,x_n[/imath] in [imath]X[/imath] such that the [imath]n\times n[/imath] matrix [imath][f_i(x_j) ][/imath] is non-singular. |
680725 | if [imath]U\cup W=V[/imath] then [imath]U=V[/imath] or [imath]W=V[/imath]
Let [imath]U,W[/imath] two subspaces in [imath]V[/imath]. Assuming that [imath]U\cup W = V[/imath], show that: [imath]U=V[/imath] or [imath]W=V[/imath]. So, by definition of union: [imath]U \cup W = \left\{ {u + w|u \in U,w \in W} \right\}[/imath] How to proceed? And a technical issue: Is [imath]U \cup W[/imath] equivalent to [imath]U + W[/imath]? EDIT: Consider the following scenario (Following @Learner approach): [imath]U = V \backslash \{x\}[/imath] and [imath]W = V \backslash \{y\}[/imath]. [imath]U\cup W = V[/imath], but none of the subspaces is equal to [imath]V[/imath]. Where is the error? | 677481 | A union of two subspaces not equal to the vector space.
Let [imath]L,M[/imath] two subspaces of the vector space, [imath]V[/imath] such that both [imath]L,M \ne V[/imath]. Prove: [imath]L\cup M \ne V[/imath]. I think this is a case of a proof by contradiction. Lets assume [imath]L \cup M = V[/imath]. Hence, [imath]\dim(V) = \dim(L) + \dim(M) - \dim(L\cap M)[/imath] How to proceed? |
679397 | What is bigger, [imath]p(\mathbb{N})[/imath] or [imath]\mathbb{R}[/imath]?
I know that [imath]|p(\mathbb{N})|>|\mathbb{N}|[/imath], and that [imath]|\mathbb{R}|>|\mathbb{N}|[/imath], and I wonder whether [imath]|p(\mathbb{N})|>|\mathbb{R}|[/imath] or not. What I tried so far: I found the function from [imath]\mathbb{R}[/imath] to [imath]p(\mathbb{Q})[/imath] defined by [imath]f(x)=\{q\in \mathbb{Q}|q<x\}[/imath], which I am quite sure to be injective function, but not onto. As [imath]|\mathbb{Q}|=|\mathbb{N}|[/imath], also [imath]|p(\mathbb{Q})|=|p(\mathbb{N})|[/imath], so I inferred that [imath]|p(\mathbb{N})|\ge |\mathbb{R}|[/imath]. But are they equal? In this, [imath]p(A)[/imath] is the power set of A, denoted also by [imath]2^A[/imath] and defined as is the set of all subsets of A. | 209396 | Is [imath]2^{|\mathbb{N}|} = |\mathbb{R}|[/imath]?
Is [imath]2^{|\mathbb{N}|} = |\mathbb{R}|[/imath]? If so, how? I was reading the Wiki page on the , and it says "Moreover, [imath]\mathbb{R}[/imath] has the same number of elements as the power set of [imath]\mathbb{N}[/imath]", but I don't see how this is true? I feel like it has something to do with binary, but I'm not too sure how it works? Do I have to show a map of all reals can be done in binary? I'm just very confused, and any advice would be appreciated! |
673213 | question about distance of sets
LEt [imath]A,B \subseteq \mathbb{R}^d[/imath] be non-empty sets. Define their distance to be [imath] d(A,B) = \inf \{ ||x-y|| : x \in A, \; \; y \in B \} [/imath] For any [imath]A,B[/imath], do we have that [imath]d(A,B) = d( \overline{A}, \overline{B} ) [/imath] ?? | 677240 | The distance between two sets inside euclidean space
Let [imath]A,B \subseteq \mathbb{R}^d[/imath] be non-empty sets. Define their distance to be [imath] d(A,B) = \inf \{ \|x-y\| : x \in A, \; \; y \in B \} [/imath] For any [imath]A,B[/imath], I want to prove that [imath]d(A,B) = d( \overline{A}, \overline{B} )[/imath]. My Attempt Put [imath]\alpha = d(A,B)[/imath]. Therefore, [imath]\alpha = \|x_0 - y_0 \|[/imath] for some [imath]x_0 \in A[/imath] and some [imath]y_0 \in B[/imath]. But, notice [imath]x_0 \in \overline{A}[/imath] and [imath]y_0 \in \overline{B}[/imath] by definition. hence [imath] \| x_0 - y_0\| \geq \inf\{ \|x' - y'\| : x' \in \overline{A}, \; \; y' \in \overline{B} \} = d( \overline{A}, \overline{B} )[/imath] [imath]\therefore d(A,B) \geq d( \overline{A}, \overline{B} )[/imath] I am stuck trying to show the other direction: [imath]d(A,B) \leq d( \overline{A}, \overline{B} )[/imath] Can someone help me? thanks a lot |
682258 | Prove that [imath]f \ast g[/imath] is continuous and bounded if [imath]f\in L^1(R^n)[/imath] and [imath]g\in L^\propto (R^n)[/imath]
My Engliah is no so good and it is my first time to use this website, so I apologize for it if I didnot make myself clearly:) | 644753 | [imath]f[/imath] is bounded and continious [imath]\Rightarrow[/imath] the convolution integral [imath]\int f(\tau)g(x-\tau)\text{ d}\tau[/imath] is bounded and continuous
Let [imath]g\in L^1(\mathbb{R}^n)[/imath] and [imath]f:\mathbb{R}^n\to\mathbb{R}[/imath] be bounded and continuous. Why is the convolution integral [imath]f*g:\mathbb{R}^n\to\mathbb{R}\;,\;\;\;\int f(\tau)g(x-\tau)\text{ d}\tau[/imath] also bounded and continuous? *Proof*[imath]\;\;\;[/imath] It holds [imath]|(f*g)(x)|\leq\int |f(\tau)g(x-\tau)|\text{ d}\tau =\int |f(x-t)g(t)|\text{ dt}\le \sup_{x\in \mathbb{R}^n}|f(x)|\int |g(t)|\text{ dt}[/imath] Since [imath]f[/imath] is bounded and [imath]g[/imath] is integrable the supremum and the integral exists. This should be enough for at least the boundedness of [imath]f*g[/imath]. While I would like to be sure that this is correct, I would also like to know how we show the continuity. |
682541 | Legendre symbol - Find all primes such that [imath](7/p)=1[/imath]
I've been thinking a lot about this problem, but couldn't come up with an answer. Any help would be appreciated! | 371942 | Find the values of [imath]p[/imath] such that [imath]\left( \frac{7}{p} \right )= 1[/imath] (Legendre Symbol)
Show that if [imath]p[/imath] is an odd prime coprime to [imath]7[/imath], then [imath]\left( \frac{7}{p} \right) = 1[/imath] if and only if [imath]p \equiv \pm 1, \pm 3,[/imath] or [imath]\pm 9 \pmod{28}[/imath]. HINT: If [imath]p[/imath] is an odd prime, determine which values can [imath]p[/imath] take [imath]\mod28[/imath], and consider each of these values in turn. Note that if we know [imath]p \mod 28[/imath] then we know [imath]p \mod 4[/imath], and hence we know whether [imath]\frac{p-1}{2}[/imath] is odd or even. Here, [imath]\left( \frac{a}{b} \right)[/imath] is the Legendre symbol. The bit I don't understand in the hint is, what do they mean by consider the values that [imath]p[/imath] can take [imath]\mod 28[/imath]. Do they mean the values that would make [imath]p[/imath] a quadratic residue [imath]\mod 28[/imath], i.e all the [imath]x[/imath] values satisfying [imath]x^2 \equiv \mod 28[/imath], because then isn't this just [imath]1,4,9,16,25[/imath]? What do they mean the to "consider each of these values in turn"? |
682956 | Prove that [imath](a,bc)=1[/imath] if and only if [imath](a,b)=1[/imath] and [imath](a,c)=1[/imath]
I have proven the forward direction. By BeZout's identity: [imath]1=ar+(bc)s[/imath] for some integers [imath]r,s[/imath]. A corollary states that [imath](a,b) =1[/imath] if and only if [imath]1=ar+bs[/imath] for some integers [imath]r, s[/imath]. So we can conclude that [imath](a,b)=1[/imath] since [imath]1 = ar +b(cs)[/imath] and also that [imath](a,c)=1[/imath] since [imath]1=ar + c(bs)[/imath]. But I do not know how to prove the reverse direction. | 485013 | Show that [imath]\gcd(a,bc)=1[/imath] if and only if [imath]\gcd(a,b)=1[/imath] and [imath]\gcd(a,c)=1[/imath]
Show that [imath]\gcd(a,bc)=1[/imath] if and only if [imath]\gcd(a,b)=1[/imath] and [imath]\gcd(a,c)=1[/imath]. I am new at proofs and I think I should use Euclid's Lemma which states "If [imath]p[/imath] is a prime that divides [imath]ab[/imath], then [imath]p[/imath] divides [imath]a[/imath] or [imath]p[/imath] divides [imath]b[/imath]. However, I am not sure how to create a concrete proof or argument. Any hints or help would be greatly appreciated. Thanks! |
683153 | How to show that [imath]e^{x+y} = e^x e^y[/imath] by series expansion
I know that [imath]e^xe^y=e^{x+y}[/imath] but I want to show it by expanding the exponentials in MacLaurin Series. [imath] \left(\sum_{n=0}^{\infty} \frac{x^n}{n!}\right) \left(\sum_{m=0}^{\infty} \frac{y^m}{m!}\right) =^? \sum_{n=0}^{\infty} \frac{(x+y)^n}{n!} [/imath] This is what I have. I'm not sure what my next step would be since I can't think of a way to combine the two summations on the RHS. | 414061 | Prove [imath]e^{x+y}=e^{x}e^{y}[/imath] by using Exponential Series
In order to show [imath]e^{x+y}=e^{x}e^{y}[/imath] by using Exponential Series, I got the following: [imath]e^{x}e^{y}=\Big(\sum_{n=0}^{\infty}{x^n \over n!}\Big)\cdot \Big(\sum_{n=0}^{\infty}{y^n \over n!}\Big)=\sum_{n=0}^{\infty}\sum_{k=0}^n{x^ky^n \over {k!n!}}[/imath] But, where should I go next to get [imath]e^{x+y}=\sum_{n=0}^{\infty}{(x+y)^n \over n!}[/imath]. Thanks in advance. |
446011 | Independence of disjoint events
I'm taking a class in Probability Theory, and I was asked this question in class today: Given disjoint events [imath]A[/imath] and [imath]B[/imath] for which [imath] P(A)>0\\ P(B)>0 [/imath] Can [imath]A[/imath] and [imath]B[/imath] be independent? My answer was: [imath]A[/imath] and [imath]B[/imath] are disjoint, so [imath]P(A\cap B)=0[/imath]. [imath]P(A)>0[/imath] and [imath]P(B)>0[/imath], so [imath]P(A)P(B)>0[/imath]. [imath]P(A\cap B)\not =P(A)P(B)[/imath], so [imath]A[/imath] and [imath]B[/imath] are not independent. However, I was told that I am wrong and we cannot know whether or not [imath]A[/imath] and [imath]B[/imath] are independent from the given information, but I did not receive a satisfactory explanation. Is my argument valid? If not, where do I go wrong? | 1832686 | Probability: Are disjoint events independent?
I just read that disjoint events, A, B, if, [imath]\mathbb{P}(AB) = 0[/imath] are independent. This really frustrates me. My teacher stated otherwise - [imath]\mathbb{P}(AB) = 0 \iff A \cap B = \emptyset \implies \mathbb{P}(AB) = 0 \ne \mathbb{P}(A)\mathbb{P}(B)[/imath] because [imath]\mathbb{P}(A)[/imath] and [imath]\mathbb{P}(B)[/imath] are not empty (does the latter come from the definition of independent events?). Could somebody clear this for me? |
683863 | How to find the sum of series [imath]\sum_{i=1}^{\infty}\frac{i}{2^i}[/imath]?
I am learning about series of numbers at the moment. In the book there is an exercise in which I need to find the sum of : [imath]\sum_{i=1}^{\infty}\frac{i}{2^i}[/imath] I know it is equal to [imath]2[/imath]. But how do I get to that result? Are there any general ways of finding the sums of series? In the books I am using, there is a lot about series, their convergence etc. but almost no examples. | 674220 | A simple series [imath]\sum_{i=1}^\infty \frac{i}{2^i} = 2[/imath]
I don't do math a long time, so I completely don't remember how to prove that: [imath] \sum_{i=1}^\infty \frac{i}{2^i} = 2 [/imath] Can anybody help me? |
683968 | [imath]a|b \land b|a \iff a = \pm b[/imath]
Prove: [imath]a|b \land b|a \iff a= \pm b[/imath] So far I have [imath]a|b \iff b=ka[/imath] ([imath]k[/imath] is an integer), [imath]b|a \iff a=mb[/imath] ([imath]m[/imath] is an integer). Where do I go from here? Step by step explanation please! | 662258 | Showing that [imath]a \mid b[/imath] and [imath]b \mid a[/imath] if and only if [imath]a= \pm b[/imath].
[imath]a \mid b[/imath] and [imath]b \mid a[/imath] if and only if [imath]a= \pm b[/imath]. How do I go about this proof? Step by step explanation please! |
446647 | irrationality of [imath]\sqrt{2}^{\sqrt{2}}[/imath].
The fact that there exists irrational number [imath]a,b[/imath] such that [imath]a^b[/imath] is rational is proved by the law of excluded middle, but I read somewhere that irrationality of [imath]\sqrt{2}^{\sqrt{2}}[/imath] is proved constructively. Do you know the proof? | 173804 | Deciding whether [imath]2^{\sqrt2}[/imath] is irrational/transcendental
Is [imath]2^\sqrt{2}[/imath] irrational? Is it transcendental? |
683043 | Conditional expectation of the sum of three dice rolls given the sum of their maximum and product
Consider the random experiment in which three fair dice are rolled simultaneously (and independently). Let [imath]X[/imath] be the random variable defined as the sum of the values of these three dice. Let [imath]Y_1[/imath] be the maximum of the three values, let [imath]Y_2[/imath] be their product, and let [imath]Y=Y_1+Y_2[/imath]. Finally, define [imath]Z=E[X∣Y][/imath] (the conditional expectation of [imath]X[/imath] given [imath]Y[/imath]). Find the expected value of the random variable [imath]Z[/imath]. Enter your answer as a fraction, such as [imath]\frac{3}{2}[/imath]. I would like some help on where to start this problem, please. I know [imath]X[/imath] can be any value between [imath]3[/imath] and [imath]18[/imath] and [imath]Y_2[/imath] could be (most of the values) between [imath]1[/imath] and [imath]216[/imath]; i.e., [imath]Y_2[/imath] will never be [imath]7[/imath], [imath]11[/imath], [imath]13[/imath], etc. since those numbers are prime and are greater than [imath]6[/imath]. Thanks! | 684537 | Conditional expectation of X given Y
Consider the random experiment in which three fair dice are rolled simultaneously (and independently). Let X be the random variable defined as the sum of the values of these three dice. Let [imath]Y_1[/imath] be the maximum of the three values, let [imath]Y_2[/imath] be their product, and let [imath]Y=Y_1+Y_2[/imath]. Finally, define Z=[X∣∣Y] (the conditional expectation of X given Y). Find the expected value of the random variable Z. Enter your answer as a fraction, such as [imath]\frac{3}{2}[/imath]. |
666870 | Generators of Translation - Lie Algebra
I have just started learning Lie Groups and Algebra. Considering a flat 2-d plane if we want to translate a point from [imath](x,y)[/imath] to [imath](x+a,y+b)[/imath] then can we write it as : [imath] \left( \begin{array}{ccc} x+a \\ y+b \end{array} \right) = \left( \begin{array}{ccc} x \\ y \end{array} \right) + \left( \begin{array}{ccc} a \\ b \end{array} \right)[/imath] Now the set of all translations [imath] T = \left( \begin{array}{ccc} a \\ b \end{array} \right) [/imath] form a two paramater lie group (I presume) with addition of column as the composition rule. If that is so, how do I go about finding the generators of this transformation. PS: In my course I have been taught that the generators are found by calculating the taylor expansion of the group element about the Identity of the group. For instance, [imath]SO(2)[/imath] group [imath] M = \left( \begin{array}{cc} cos \:\phi & -sin \:\phi \\ sin \:\phi & cos \:\phi \end{array} \right) [/imath] I obtain the generator by taking [imath] \frac{\partial M}{\partial \phi}\Bigg|_{\phi=0} = \left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right) [/imath] Now if I exponentiate this, I can obtain back the group element. My question how do I do this for Translation group. | 667502 | Translations in two dimensions - Group theory
I have just started learning Lie Groups and Algebra. Considering a flat 2-d plane if we want to translate a point from [imath](x,y)[/imath] to [imath](x+a,y+b)[/imath] then can we write it as : [imath] \left( \begin{array}{ccc} x+a \\ y+a \end{array} \right) = \left( \begin{array}{ccc} x \\ y \end{array} \right) + \left( \begin{array}{ccc} a \\ b \end{array} \right)[/imath] Now the set of all translations [imath] T = \left( \begin{array}{ccc} a \\ b \end{array} \right) [/imath] form a two parameter lie group (I presume) with addition of column as the composition rule. If that is so, how do I go about finding the generators of this transformation. I know the generators of translation are linear momenta in the corresponding directions. But I am not able to see this here. PS: In my course I have been taught that the generators are found by calculating the Taylor expansion of the group element about the Identity of the group. For instance, [imath]\operatorname{SO}(2)[/imath] group [imath] M = \left( \begin{array}{cc} \cos \:\phi & -\sin \:\phi \\ \sin \:\phi & \cos \:\phi \end{array} \right) [/imath] I obtain the generator by taking [imath] \frac{\partial M}{\partial \phi}\Bigg|_{\phi=0} = \left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right) [/imath] Now if I exponentiate this, I can obtain back the group element. My question how do I do this for Translation group. EDIT :This edit is to summarise and get a view of the answers obtained. Firstly, the vector representation of the translation group (for 2D) would in general have the form : [imath] \begin{pmatrix} 1 & 0 & a_x\\ 0 & 1 & a_y \\ 0 & 0 & 1 \end{pmatrix}\ [/imath] with generators (elements of Lie algebra) [imath] T_x =\begin{pmatrix} 0 & 0 & i\\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}\ , \;\; T_y = \begin{pmatrix} 0 & 0 & 0\\ 0 & 0 & i \\ 0 & 0 & 0 \end{pmatrix}\ [/imath] Secondly, the scalar-field representation of the same is given by the differential operators [imath] exp^{ i(a_x\frac{\partial}{\partial x}+ a_y\frac{\partial}{\partial y} )} [/imath] with generators [imath] T_x^s = i\frac{\partial}{\partial x},\;\;T_y^s = i\frac{\partial}{\partial y} [/imath] The Lie algebra is two-dimensional and abelian : [imath] [T_x,T_y] = 0[/imath] |
183859 | How to prove Lagrange trigonometric identity
I would to prove that [imath]1+\cos \theta+\cos 2\theta+\ldots+\cos n\theta =\displaystyle\frac{1}{2}+ \frac{\sin\left[(2n+1)\frac{\theta}{2}\right]}{2\sin\left(\frac{\theta}{2}\right)}[/imath] given that [imath]1+z+z^2+z^3+\ldots+z^n=\displaystyle\frac {1-z^{n+1}}{1-z}[/imath] where [imath]z\neq 1[/imath]. I put [imath]z=e^{i\theta}[/imath]. I already got in left hand side cos exp in real part, but there is a problem in the right hand side, I can't split imaginary part and real part. Please help me. Thanks in advance. | 225941 | Proving [imath]\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}[/imath]
I am being asked to prove that [imath]\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}[/imath] I have some progress made, but I am stuck and could use some help. What I did: It holds that [imath]\sum\limits_{k=0}^{n}\cos(kx)=\sum\limits_{k=0}^{n}Re(\cos(kx))=\sum\limits_{k=0}^{n}Re(\cos(x)^{k})=Re(\sum\limits_{k=0}^{n}\cos(x)^{k})=Re\left(\cos(0)\cdot\frac{\cos(x)^{n}-1}{\cos(x)-1}\right)=Re\left(\frac{\cos(x)^{n}-1}{\cos(x)-1}\right) [/imath] For any [imath]z_{1},z_{2}\in\mathbb{C}[/imath] we have it that if [imath]z_{1}=a+bi,z_{2}=c+di[/imath] then [imath]\frac{z_{1}}{z_{2}}=\frac{z_{1}\overline{z2}}{|z_{2}|^{2}}=\frac{(a+bi)(c-di)}{|z_{2}|^{2}}=\frac{ac-bd+i(bc-ad)}{|z_{2}|^{2}}[/imath] hence [imath]Re\left(\frac{z_{1}}{z_{2}}\right)=\frac{Re(z_{1})Re(z_{2})-Im(z_{1})Im(z_{2})}{|z_{2}|^{2}}[/imath] Thus, [imath]Re\left(\frac{\cos(x)^{n}-1}{\cos(x)-1}\right)=\frac{(\cos(nx)-1)(\cos(x)-1)-\sin(nx)\sin(x)}{(\cos(x)-1)^{2}+\sin^{2}(x)}=\frac{\cos(nx)\cos(x)-\cos(nx)-\cos(x)+1-\sin(nx)\sin(x)}{\cos^{2}(x)-2\cos(x)+1+\sin^{2}(x)}=\frac{\cos(nx)\cos(x)-\cos(nx)-\cos(x)+1-\sin(nx)\sin(x)}{-2\cos(x)+2}=\frac{\cos(nx)\cos(x)-\cos(nx)-\cos(x)+1-\sin(nx)\sin(x)}{-2(\cos(x)-1)}= \frac{=\cos(nx)\cos(x)-\cos(nx)-\cos(x)+1-\sin(nx)\sin(x)}{-2(-2\cdot\sin^{2}(x/2))}=\frac{\cos(nx)\cos(x)-\cos(nx)-\cos(x)+1-\sin(nx)\sin(x)}{4\sin^{2}(x/2)}=\frac{\cos(nx)\cos(x)-\cos(nx)-\cos(x)+1-\sin(nx)\sin(x)}{4\sin^{2}(x/2)}=\frac{\cos(x(n+1))-\cos(nx)-\cos(x)+1}{4\sin^{2}(x/2)} [/imath] This is the part where I am stuck, I would appriciate any help or hint on how to continue. Edit: Given the corrections by André I get: [imath](\cos(nx+x)-1)(\cos(x)-1)+\sin(nx+x)\sin(x)=\cos(nx+x)\cos(x)-\cos(nx)-\cos(x)+1+\sin(nx+x)\sin(x)[/imath] so [imath]\cos(nx+x)\cos(x)+\sin(nx+x)\sin(x)=\cos(xn+x-x)-\cos(nx)=0[/imath] Edit 2: I found anoter mistake in the above, I will try to correct Edit 3: When multiplying correctly the above it works out :-) |
151979 | Sum of [imath]\cos(k x)[/imath]
I'm trying to calculate the trigonometric sum : [imath]\sum\limits_{k=1}^{n}\cos(k x)[/imath] This is what I've tried so far : [imath]\renewcommand\Re{\operatorname{Re}} \begin{align*} \sum\limits_{k=1}^{n}\cos(k x) &= \Re\left(\sum\limits_{k=1}^{n}e^{i k x}\right)\\ &= \Re\left(e^{i x}\frac{1 - e^{inx}}{1 - e^{ix}}\right) \end{align*}[/imath] How can I go on ? | 1867020 | How to prove that a sum of [imath]\cosh(kx)[/imath] is equal to a formula?
I need to prove that [imath]\sum_{k=0}^{n}\cosh(kx) = \frac{\sinh((n+1/2)x) + \sinh(x/2)}{2\sinh(x/2)}[/imath] Can you help me out? How do I even start? |
685109 | A polynomial with solvable Galois group and solution by radicals
Suppose [imath]f(x)\in \mathbb{Q}[x][/imath] has a solvable Galois group, then we know that it can be solved in terms of radicals. But do we know how to explicitly write the solutions of [imath]f(x)[/imath] in terms of radicals? | 176013 | A formula for the roots of a solvable polynomial
Let [imath]F[/imath] be a field and [imath]p(x)\in F[x][/imath] a separable polynomial, denote [imath]K[/imath] as the splitting field of [imath]p[/imath] and assume that [imath]K/F[/imath] is Galois with a solvable Galois group. I don't understand if this imply of any formula (in radicals) for the roots of [imath]p[/imath] (however, I do understand how a formula would imply that [imath]p[/imath] is solvable by roots). Is there some kind of a way to obtain the roots of a solvable polynomial ? |
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