qid stringlengths 1 7 | Q stringlengths 87 7.22k | dup_qid stringlengths 1 7 | Q_dup stringlengths 97 10.5k |
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6695 | Ways to evaluate [imath]\int \sec \theta \, \mathrm d \theta[/imath]
The standard approach for showing [imath]\int \sec \theta \, \mathrm d \theta = \ln|\sec \theta + \tan \theta| + C[/imath] is to multiply by [imath]\frac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}[/imath] and then do a substitution with [i... | 717528 | Why is [imath]\int \sec x\,dx[/imath] equal to [imath]\ln | \sec x + \tan x |[/imath]?
Everyone knows that [imath]\int \sec x\,dx = \ln | \sec x + \tan x |[/imath]? But how to reach it through a conscious deduction, through a clear and objective way? |
1057 | Group With an Endomorphism That is "Almost" Abelian is Abelian.
Suppose a finite group has the property that for every [imath]x, y[/imath], it follows that \begin{equation*} (xy)^3 = x^3 y^3. \end{equation*} How do you prove that it is abelian? Edit: I recall that the correct exercise needed in addition that the ord... | 1782335 | Abelian finite group
This is a (maybe be simple) problem from Group Theory, but being a beginner, I am unable to take even a first step forward. Let [imath]G[/imath] be a finite group whose order is not divisible by [imath]3[/imath].Suppose that [imath](ab)^3=a^3b^3\ \ [/imath] [imath]\forall\ \ a,b\in G[/imath]. I a... |
44406 | How do I get the square root of a complex number?
If I'm given a complex number (say [imath]9 + 4i[/imath]), how do I calculate its square root? | 970081 | Find square roots of [imath]8 - 15i[/imath]
Find the square roots of: [imath]8-15i.[/imath] Could I get some working out to solve it? Also what are different methods of doing it? |
40149 | Intuition of the meaning of homology groups
I am studying homology groups and I am looking to try and develop, if possible, a little more intuition about what they actually mean. I've only been studying homology for a short while, so if possible I would prefer it if this could be kept relatively simple, but I imagine ... | 2292374 | Understanding the geometric meaning of homology and cohomology group?
I'm learning some algebraic topology books which introduce me to singular cubial homology and cohomology ( not simplicial as usual so sometimes it's hard for me . I think the difference between these theory is triangulation ) . I will go straight to... |
283 | Is [imath]0[/imath] a natural number?
Is there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number? It seems as though formerly [imath]0[/imath] was considered in the set of natural numbers, but now it seems more common to see defini... | 468587 | Isn't zero natural enough to be included in the set of natural numbers?
I always define [imath]\mathbb{N}[/imath] to include [imath]0[/imath] but some authors don't. Since the elements of [imath]\mathbb{N}[/imath] are used for counting, shouldn't [imath]0\in\mathbb{N}[/imath]? [imath]0[/imath] is the number of cows in... |
99913 | Easy way to show that [imath]\mathbb{Z}[\sqrt[3]{2}][/imath] is the ring of integers of [imath]\mathbb{Q}[\sqrt[3]{2}][/imath]
This seems to be one of those tricky examples. I only know one proof which is quite complicated and follows by localizing [imath]\mathbb{Z}[\sqrt[3]{2}][/imath] at different primes and then sh... | 979906 | Find the ring of algebraic integers.
Find the ring of algebraic integers in [imath]K=\mathbb Q(\sqrt[3]{2})[/imath]. So, I know that [imath]K=\{a+b\sqrt[3]{2}+c\sqrt[3]{2}^2 \mid a,b,c \in \mathbb Q\}[/imath]. My professor has done very little on this topic. I know that an algebraic integer is the root of a monic po... |
75246 | Surjectivity of Composition of Surjective Functions
Suppose we have two functions, [imath]f:X\rightarrow Y[/imath] and [imath]g:Y\rightarrow Z[/imath]. If both of these functions are onto, how can we show that [imath]g\circ f:X\rightarrow Z[/imath] is also onto? | 1141670 | How to prove that the composition of two surjective functions is surjective
I know that the map [imath]f:A\to B[/imath] is a surjective function (onto) if for all [imath]b[/imath] in [imath]B[/imath], there exists an [imath]a[/imath] in [imath]A[/imath] such that [imath]f(a)=b[/imath] But I am having trouble getting ... |
71851 | Prove that any rational can be expressed in the form [imath]\sum\limits_{k=1}^n{\frac{1}{a_k}}[/imath], [imath]a_k\in\mathbb N^*[/imath]
Let [imath]x\in\mathbb{Q}[/imath] with [imath]x>0[/imath]. Prove that we can find [imath]n\in\mathbb{N}^*[/imath] and distinct [imath]a_1,...,a_n \in \mathbb{N}^*[/imath] such that [... | 1768472 | Let [imath]r[/imath] be a rational no. with [imath]0. Then r can be expressed as a sum of reciprocals of finitely many distinct positive integers[/imath]
Let [imath]r[/imath] be a rational no. with [imath]0<r<1[/imath]. Then [imath]r[/imath] can be expressed as a sum of reciprocals of finitely many distinct positive i... |
5119 | Example where union of increasing sigma algebras is not a sigma algebra
If [imath]\mathcal{F}_1 \subset \mathcal{F}_2 \subset \dotsb[/imath] are sigma algebras, what is wrong with claiming that [imath]\cup_i\mathcal{F}_i[/imath] is a sigma algebra? It seems closed under complement since for all [imath]x[/imath] in the... | 1122526 | Showing that if [imath]A_{1},A_{2},...[/imath] are all algebras then the union of all of them is an algebra
I am not sure how to show this. It seems obvious but maybe its not. The help would be greatly appreciated! |
26888 | The union of a strictly increasing sequence of [imath]\sigma[/imath]-algebras is not a [imath]\sigma[/imath]-algebra
The union of a sequence of [imath]\sigma[/imath]-algebras need not be a [imath]\sigma[/imath]-algebra, but how do I prove the stronger statement below? Let [imath]\mathcal{F}_n[/imath] be a sequence of... | 535749 | Union of increasing [imath]\sigma[/imath]-algebras which is no [imath]\sigma[/imath]-Algebra
Give an example for an increasing series of [imath]\sigma[/imath] algebras [imath] \mathcal{A}_1\subset\mathcal{A}_2\subset\ldots [/imath] so that [imath]\bigcup_{i=1}^{\infty}\mathcal{A}_i [/imath] is no [imath]\si... |
51788 | Compactness of Multiplication Operator on [imath]L^2[/imath]
Suppose we have an bounded linear operator A that operates from [imath]L^2([a,b]) \mapsto L^2([a,b])[/imath]. Now suppose that [imath]A(f)(t) = tf(t)[/imath]. Is A compact? Edit: I know [imath]A = A^*[/imath] but I'm not really sure how to start on this.... | 236669 | No Nonzero multiplication operator is compact
Let [imath]f,g \in L^2[0,1][/imath], multiplication operator [imath]M_g:L^2[0,1] \rightarrow L^2[0,1][/imath] is defined by [imath]M_g(f(x))=g(x)f(x)[/imath]. Would you help me to prove that no nonzero multiplication operator on [imath]L^2[0,1][/imath] is compact. Thanks. |
30774 | Prove that [imath]a_{n}=0[/imath] for all [imath]n[/imath], if [imath]\sum a_{kn}=0[/imath] for all [imath]k\geq 1[/imath]
Let [imath]\sum a_{n}[/imath] be an absolutely convergent series such that [imath]\sum a_{kn}=0[/imath] for all [imath]k\geq 1[/imath]. Help me prove that [imath]a_{n}=0[/imath] for all [imath]n... | 1321310 | [imath]\sum_{n=1}^{\infty} a_n[/imath] converges absolutely and [imath]\sum _{n=1}^\infty a_{kn}=0 ,\forall k \ge 1 [/imath] ; then [imath]a_n=0 , \forall n \in \mathbb N [/imath]?
Suppose that the series [imath]\sum_{n=1}^{\infty} a_n[/imath] of real terms converges absolutely and [imath]\sum _{n=1}^\infty a_{kn}=0 ... |
159659 | Which sets are removable for holomorphic functions?
Let [imath]\Omega[/imath] be a domain in [imath]\mathbb C[/imath], and let [imath]\mathscr X[/imath] be some class of functions from [imath]\Omega[/imath] to [imath]\mathbb C[/imath]. A set [imath]E\subset \Omega[/imath] is called removable for holomorphic functions ... | 1073421 | Need help with holomorphic functions on a domain interval removed.
I want to prove that for a region [imath]\Omega[/imath] with interval [imath]I=[a,b]\subset\Omega[/imath], if [imath]f[/imath] is continuous in [imath]\Omega[/imath] and [imath]f\in H(\Omega-I)[/imath], then actually [imath]f\in H(\Omega)[/imath]. Is t... |
11477 | Fibonacci addition law [imath]F_{n+m} = F_{n-1}F_m + F_n F_{m+1}[/imath]
Question: Let [imath]F_n[/imath] the sequence of Fibonacci numbers, given by [imath]F_0 = 0, F_1 = 1[/imath] and [imath]F_n = F_{n-1} + F_{n-2}[/imath] for [imath]n \geq 2[/imath]. Show for [imath]n, m \in \mathbb{N}[/imath]: [imath]F_{n+m} = F_{... | 1761673 | Prove equality for Fibonacci sequence
I have to show that [imath]F_{n+k} = F_{k}F_{n+1} + F_{k-1}F_{n}[/imath], where [imath]F_{n}[/imath] is nth Fibonacci element. I was trying with mathematical induction applied to n and saying k is constant. step for [imath]n=1[/imath] [imath]F_{k+1} = F_{k}F_{2} + F_{k-1}F_{1}[/... |
48850 | Continuity of the function [imath]x\mapsto d(x,A)[/imath] on a metric space
Let [imath](X,d)[/imath] be a metric space. How to prove that for any closed [imath]A[/imath] a function [imath]d(x,A)[/imath] is continuous - I know that it is even Lipschitz continuous, but I have a problem with the proof: [imath] |d(x,a) ... | 418042 | Show that the function [imath]f: X \to \Bbb R[/imath] given by [imath]f(x) = d(x, A)[/imath] is a continuous function.
I'm studying for my Topology exam and I am trying to brush up on my metric spaces. Suppose [imath](X, d)[/imath] is a metric space and [imath]A[/imath] is a proper subset of [imath]X[/imath]. Show tha... |
206851 | Generalisation of Dominated Convergence Theorem
Wikipedia claims, if [imath]\sigma[/imath]-finite the Dominated convergence theorem is still true when pointwise convergence is replaced by convergence in measure, does anyone know where to find a proof of this? Many thanks! Statement of the theorem: Let [imath]\mu[/imat... | 358161 | Convergence of functions
Assume that [imath](X,M,\mu)[/imath] is a [imath]\sigma[/imath]-finite space. Suppose that [imath]|f_n|\leq g\in L^+[/imath] and [imath]f_n\rightarrow f[/imath] in measure. Show that [imath]\int f=\lim_{n\rightarrow\infty}\int f_n[/imath]. I tried taking a subsequence of [imath]f_n[/imath], ca... |
130471 | Convergence/divergence of [imath]\sum\frac{a_n}{1+na_n}[/imath] when [imath]a_n\geq0[/imath] and [imath]\sum a_n[/imath] diverges
A question from Rudin (Principles) Chapter 3: Let [imath]a_n\geq0[/imath] and [imath]\sum a_n[/imath] diverges. What can be said about convergence/divergence of [imath]\sum\frac{a_n}{1+na_... | 1505389 | Prove that if [imath]\sum a_n[/imath] is divergent then [imath]\sum a_n/(1 + na_n)[/imath] diverges too
I had this question where it was given [imath]a_n > 0[/imath] and [imath]\sum a_n[/imath] is divergent and I was to prove that [imath]\sum\frac{a_n}{1 + na_n}[/imath] diverges too, to be frank I could not make any w... |
220410 | A characterization of functions from [imath]\mathbb R^n[/imath] to [imath]\mathbb R^m[/imath] which are continuous
Greets I came up the other day with the following question: Is it true that [imath]f:\mathbb{R}^n\longrightarrow{\mathbb{R}^m}[/imath] is continuous if and only if [imath]f[/imath] maps compact sets onto... | 1212136 | A map of two Euclidean spaces preserving connectedness and compactness is continuous
Let [imath] f:\mathbb R^n \to \mathbb R^m[/imath]. If [imath]f[/imath] preserves connectedness and compactness then [imath]f[/imath] is continuous. How can this be proven? I don't really know where to start. |
239825 | Can anyone explain why [imath]a^{b^c} = a^{(b^c)} \neq (a^b)^c = a^{(bc)}[/imath]
I'm so puzzled about this: [imath]a^{b^c} = a^{(b^c)} \neq (a^b)^c = a^{(bc)}.[/imath] Why isn't [imath]a^{b^c}[/imath] equal to [imath]a^{(bc)}[/imath]? Why is [imath]a^{b^c}[/imath] instead equal to [imath]a^{(b^c)}[/imath]? And how i... | 2029904 | Notation: Precedence in multiple levels of exponentiation
When people write [imath]a^{b^c}[/imath], does it refer to [imath](a^b)^c=a^{bc}[/imath] or [imath]a^{(b^c)}[/imath]? What about the notation a^b^c (without superscripts)? Is there an agreed-upon convention? |
75130 | How to prove that [imath]\lim\limits_{x\to0}\frac{\sin x}x=1[/imath]?
How can one prove the statement [imath]\lim_{x\to 0}\frac{\sin x}x=1[/imath] without using the Taylor series of [imath]\sin[/imath], [imath]\cos[/imath] and [imath]\tan[/imath]? Best would be a geometrical solution. This is homework. In my math clas... | 400971 | Finding [imath]\lim_{x\to 0}\frac{\sin x}{x} [/imath]
How to find [imath]\lim_{x\to 0}\frac{\sin x}{x} [/imath] |
271138 | convergence of a series involving [imath]x^\sqrt{n}[/imath]
I was trying to prove the convergence of the series [imath]\sum_{n=1}^{\infty}x^{\sqrt{n}}[/imath], for [imath]0<x<1[/imath]. Unfortunately, I could not make one of the standard convergence tests give me an answer. Does anybody of you have a suggestion? any h... | 382109 | Finding the supremum of the following set
I am stuck on the following problem: Let [imath]P=\{x \in \Bbb R: x\ge 0,\sum_{n=1}^{\infty}x^{\sqrt n}< \infty\}[/imath].Then what is the supremum of [imath]P[/imath]? Can someone help me out by providing some explanation? Thanks in advance. |
64432 | What is so special about [imath]\alpha=-1[/imath] in the integral of [imath]x^\alpha[/imath]?
Of course, it is easy to see, that the integral (or the antiderivative) of [imath]f(x) = 1/x[/imath] is [imath]\log(|x|)[/imath] and of course for [imath]\alpha\neq - 1[/imath] the antiderivative of [imath]f(x) = x^\alpha[/im... | 848143 | Why, intuitively, does [imath]log(x)[/imath] come in as the integral of [imath]1/x[/imath], wheras the integral of other powers of [imath]x[/imath] are powers of [imath]x[/imath]?
Question in title really, something I always found strange when I was learning calculus. I can see that [imath]\int \frac{1}{x} dx[/imath] ... |
244409 | If [imath]f^2[/imath] is Riemann Integrable is [imath]f[/imath] always Riemann Integrable?
Problem: Suppose that [imath]f[/imath] is a bounded, real-valued function on [imath][a,b][/imath] such that [imath]f^2\in R[/imath] (i.e. it is Riemann-Integrable). Must it be the case that [imath]f\in R[/imath] ? Thoughts: ... | 2064203 | If [imath]f[/imath] is bounded and if [imath]f^2[/imath] is Riemann integrable, then is f Riemann integrable?
This was on a past exam paper, and it was asking if [imath]f[/imath] is bounded and if [imath]f^2[/imath] is Riemann integrable, then is f Riemann integrable? If I had to guess, I'd say no. I tried creating an... |
56307 | Simultaneous diagonalization of commuting linear transformations
Let [imath]V[/imath] be a vector space of finite dimension and let [imath]T,S[/imath] linear diagonalizable transformations from [imath]V[/imath] to itself. I need to prove that if [imath]TS=ST[/imath] every eigenspace [imath]V_\lambda[/imath] of [imath... | 1841311 | Problem about linear algebra
Suppose we have two [imath]n \times n[/imath] square matrices A and B such that [imath]AB=BA[/imath]. It is known that A, B and AB all have n distinct eigenvectors that is a basis of [imath]\mathbb{C}^n[/imath]. Can we then show that there is a basis of [imath]\mathbb{R}^n[/imath] that com... |
207910 | Prove convergence of the sequence [imath](z_1+z_2+\cdots + z_n)/n[/imath] of Cesaro means
Prove that if [imath]\lim_{n \to \infty}z_{n}=A[/imath] then: [imath]\lim_{n \to \infty}\frac{z_{1}+z_{2}+\cdots + z_{n}}{n}=A[/imath] I was thinking spliting it in: [imath](z_{1}+z_{2}+\cdots+z_{N-1})+(z_{N}+z_{N+1}+\cdots+z_{n... | 443770 | Proving a statmenet about convergence of complex sequence
Let [imath]x_k \in \mathbb C[/imath] for [imath]k \in \mathbb N \cup {0}[/imath] and let [imath]y_k = \frac{(x_0 + x_1 + ... + x_k)}{k+1}[/imath]. We want to prove that if [imath]x_k[/imath] converges to [imath]x[/imath] ([imath]x \in \mathbb C[/imath]) as [ima... |
50316 | [imath]x^x=y[/imath]. How to solve for [imath]x[/imath]?
I tried looking for ways to solve this equation and came across something like Lambert's W function, which, by the way, I did not understand a bit, because I've never learned it nor do I have a decent mathematical background. I also came across one more method c... | 1281246 | How to solve for [imath]y[/imath] the equation [imath]x= y^y[/imath]?
I need an equation where I receive a number that when raised to itself equals the input. Formally: in [imath]x=y^y[/imath] solve for [imath]y[/imath]. Intro to Calculus level knowledge. If the Lambert function is necessary, please explain it to me. ... |
198787 | Prove that if c is a common divisor of a and b then c divides the gcd of a and b..
If [imath]c[/imath] is a common divisor of [imath]a[/imath] and [imath]b[/imath] then [imath]c[/imath] divides the greatest common divisor of [imath]a[/imath] and [imath]b[/imath]. What can we use to prove this? | 1753209 | Let [imath]a^n, a^m \in (a^k)[/imath] for some positive integer [imath]k[/imath]. Then [imath]k \mid n, m.[/imath] Hence [imath]k \mid \operatorname{gcd(n, m)}?[/imath]
Let [imath]a^n, a^m \in (a^k)[/imath] for some positive integer [imath]k[/imath]. Then [imath]k \mid n, m.[/imath] Hence [imath]k \mid \operatorname{... |
36364 | What does [imath]\ll[/imath] mean?
I saw two less than signs on this Wikipedia article and I was wonder what they meant mathematically. http://en.wikipedia.org/wiki/German_tank_problem EDIT: It looks like this can use TeX commands. So I think this is the symbol: [imath]\ll[/imath] | 625763 | What is the difference between [imath]\gg[/imath] and [imath]\gt[/imath]?
What is the difference between [imath]\gg[/imath] and [imath]\gt[/imath] ? Thank you [imath]\infty[/imath] times. |
305954 | Can you someone help me to find the indefinite integral, step by step. I did my self, and getting wrong answer.
can you please someone tell me how to do this indefinite integral in steps [imath]\int[/imath][imath]cos(\sqrt{6x})\over\sqrt{6x}[/imath] dx | 305351 | what are the possible answers we can get for the below intergral?
Could you please tell me what are the possible answers (if there is more than one) for the following indefinite integral? [imath]\int \dfrac{\cos(\sqrt{6x})}{\sqrt{6x}}dx[/imath] |
184609 | Why is the last digit of [imath]n^5[/imath] equal to the last digit of [imath]n[/imath]?
I was wondering why the last digit of [imath]n^5[/imath] is that of [imath]n[/imath]? What's the proof and logic behind the statement? I have no idea where to start. Can someone please provide a simple proof or some general ideas ... | 827467 | How do you prove that [imath] n^5[/imath] is congruent to [imath] n[/imath] mod 10?
How do you prove that [imath]n^5 \equiv n\pmod {10}[/imath] Hint given was- Fermat little theorem. Kindly help me out. This is applicable to all positive integers [imath]n[/imath] |
37327 | [imath]\infty = -1 [/imath] paradox
I puzzled two high school Pre-calc math teachers today with a little proof (maybe not) I found a couple years ago that infinity is equal to -1: Let x equal the geometric series: [imath]1 + 2 + 4 + 8 + 16 \ldots[/imath] [imath]x = 1 + 2 + 4 + 8 + 16 \ldots[/imath] Multiply each side... | 352650 | Paradox of Infinity?
If a series such as '[imath]a[/imath]' below adds to infinity: [imath]a = 1 + 2 + 4 + 8 + 16 + \cdots\to \infty[/imath] Multiplying '[imath]a[/imath]' by [imath]2[/imath] yields: [imath]2a = 2 + 4 + 8 + 16 + \cdots\to \infty[/imath] However when I subtract these two series, I find a paradoxical an... |
304480 | A Combinatorial Question to Solve a System of Equations
Suppose we have [imath]N[/imath] integer-valued variables [imath]i_1[/imath], [imath]i_2[/imath], [imath]\cdot\cdot\cdot[/imath], [imath]i_N[/imath], such that each variable can take integer values from 0 to [imath]k[/imath], and the sum of these [imath]N[/imath]... | 686 | Combinations of selecting [imath]n[/imath] objects with [imath]k[/imath] different types
Suppose that I am buying cakes for a party. There are [imath]k[/imath] different types and I intend to buy a total of [imath]n[/imath] cakes. How many different combinations of cakes could I possibly bring to the party? |
78533 | prove that [imath]\frac{(2n)!}{(n!)^2}[/imath] is even if [imath]n[/imath] is a positive integer
Prove that [imath]\frac{(2n)!}{(n!)^2}[/imath] is even if [imath]n[/imath] is a positive integer. For clarity: the denominator is the only part being squared. My thought process: The numerator is the product of the fir... | 667404 | show that [imath]2n\choose n[/imath] is divisible by 2
I tried using induction, but in the inductive step, I get: If [imath]2n\choose n[/imath] is divisible then I want to see that [imath]2n +2\choose n +1[/imath] [imath]{2n +2\choose n +1} = (2n + 2)!/(n+1)!(n +1)! = {2n\choose n} (2n+1)(2n+2)/(n+1)^2[/imath] [imath]... |
295466 | Algebraic sets in [imath]\mathbb{A}^2[/imath]
Deduce that if [imath]Z[/imath] is an algebraic set in [imath]\mathbb{A}^2[/imath] and [imath]c\in\mathbb{C}[/imath] then [imath]Y = \{ a\in\mathbb{C} : (a,c) \in Z \}[/imath] is either finite or all of [imath]\mathbb{A}^1[/imath]. Deduce that [imath]\{ ( z,w) \in \mathbb{... | 295445 | Proving algebraic sets
i) Let [imath]Z[/imath] be an algebraic set in [imath]\mathbb{A}^n[/imath]. Fix [imath]c\in \mathbb{C}[/imath]. Show that [imath]Y=\{b=(b_1,\dots,b_{n-1})\in \mathbb{A}^{n-1}|(b_1,\dots,b_{n-1},c)\in Z\}[/imath] is an algebraic set in [imath]\mathbb{A}^{n-1}[/imath]. ii) Deduce that if [imath]Z[... |
28568 | Bijection between an open and a closed interval
Recently, I answered to this problem: Given [imath]a<b\in \mathbb{R}[/imath], find explicitly a bijection [imath]f(x)[/imath] from [imath]]a,b[[/imath] to [imath][a,b][/imath]. using an "iterative construction" (see below the rule). My question is: is it possible to ... | 1006445 | Proving [imath](0,1)[/imath] and [imath][0,1][/imath] have the same cardinality
Prove [imath](0,1)[/imath] and [imath][0,1][/imath] have the same cardinality. I've seen questions similar to this but I'm still having trouble. I know that for [imath]2[/imath] sets to have the same cardinality there must exist a bijec... |
160738 | How to define a bijection between [imath](0,1)[/imath] and [imath](0,1][/imath]?
How to define a bijection between [imath](0,1)[/imath] and [imath](0,1][/imath]? Or any other open and closed intervals? If the intervals are both open like [imath](-1,2)\text{ and }(-5,4)[/imath] I do a cheap trick (don't know if tha... | 299006 | What is the most effective way to implement Hilbert's hotel?
Assuming I need to find an onto and 1-to-1 function from [imath](a,b)[/imath] to [imath](0,1)[/imath], well that's not a hard job. But things are getting bit more complicated when I'm asked to do the exact same but from [imath][a,b)[/imath] to [imath](0,1)[/... |
29441 | Prove that [imath]n[/imath] is a sum of two squares?
Problem Let [imath]n = p_1.p_2.p_3 \cdots p_k.m^2[/imath], where [imath]p_1, p_2, p_3 \cdots p_k[/imath] are distinct primes. Prove that n is sum of two squares if and only if [imath]p_i[/imath] is either 2 or [imath]p_i \equiv 1 \pmod{4}[/imath] For [imath]p_i ... | 1678756 | solutions for the diophantine equation [imath]x^2+y^2=n[/imath]
Are there any solutions for the diophantine equation [imath]x^2+y^2=n[/imath] ? For [imath]n \in \mathbb{P} \wedge n \equiv1\pmod4[/imath] solutions are widely known. Can we generalize a bit? |
7938 | [imath]n!+1[/imath] being a perfect square
One observes that \begin{equation*} 4!+1 =25=5^{2},~5!+1=121=11^{2} \end{equation*} is a perfect square. Similarly for [imath]n=7[/imath] also we see that [imath]n!+1[/imath] is a perfect square. So one can ask the truth of this question: Is [imath]n!+1[/imath] a perfect s... | 767510 | [imath]1+n!=m^{2}[/imath] for some n,m[imath]\in\mathbb{N}[/imath]
I have no idea whether this is known or not and I couldn't find anything related on Google. While I was studying , I come up with this idea [imath]1+n!=m^{2} [/imath] for some [imath]n,m\in\mathbb{N}[/imath] [imath]1+4!=5^{2}[/imath] [imath]1+5!=11^{2}... |
93409 | Does every Abelian group admit a ring structure?
Given some Abelian group [imath](G, +)[/imath], does there always exist a binary operation [imath]*[/imath] such that [imath](G, +, *)[/imath] is a ring? That is, [imath]*[/imath] is associative and distributive: \begin{align*} &a * (b * c) = (a*b) * c \\ &a * (b + c) =... | 1006716 | Ring structures on abelian groups
My question is: given an abelian group [imath]G[/imath] with addition [imath]+[/imath], is there some natural multiplicative structure that arises so that we can define a ring [imath](G, +, \cdot)[/imath]. For instance, multiplication on [imath]\mathbb{Z}[/imath] and [imath]\mathbb{Z}... |
80453 | How to prove that [imath]\lim\limits_{n \to \infty} \frac{k^n}{n!} = 0[/imath]
It recently came to my mind, how to prove that the factorial grows faster than the exponential, or that the linear grows faster than the logarithmic, etc... I thought about writing: [imath] a(n) = \frac{k^n}{n!} = \frac{ k \times k \times ... | 1042129 | A limit of a sequence
I'm trying to prove the following limit [imath](\frac{2^n}{n!}) \to 0[/imath] But it seems difficault to me. How can I prove it? Thanks. |
24456 | Matrix multiplication: interpreting and understanding the process
I have just watched the first half of the 3rd lecture of Gilbert Strang on the open course ware with link: http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/ It seems that with a matrix multiplication [imath]AB=C[/im... | 1698783 | Understanding matrix multiplication
I have a hard time understanding, on an intuitive level, what matrix multiplication actually does. I have used it a lot, but I do not really know what it does. I know that [imath]Ax = y[/imath], where [imath]A[/imath] is a matrix and [imath]x[/imath] is an [imath]n[/imath]-tuple, is... |
117194 | Fastest way to try all passwords
Suppose you have a computer with a password of length [imath]k[/imath] in an alphabet of [imath]n[/imath] letters. You can write an arbitrarly long word and the computer will try all the subwords of [imath]k[/imath] consecutive letters. What is the smallest word that contains all comb... | 1320278 | Puzzle: Cracking the safe
A safe is protected by a four-digit [imath](0-9)[/imath] combination. The safe only considers the last four digits entered when deciding whether an input matches the passcode. For instance, if I enter the stream [imath]012345[/imath], I am trying each of the combinations [imath]0123[/imath], ... |
166553 | How is the Integral of [imath]\int_a^bf(x)~dx=\int_a^bf(a+b-x)~dx[/imath]
Can Some one tell me what this method is called and how it works With a detailed proof [imath]\int_a^bf(x)~dx=\int_a^bf(a+b-x)~dx[/imath] I've been using this a lot in definite integration but haven't seemed to have realized why it is true. Bu... | 1064896 | Show that [imath]\int_0^a f(x)dx=\int_0^a f(a-x)dx[/imath]
I don't really know where to start with this one. Can you just ignore the [imath]f(..)[/imath] and deal exclusively with what's inside the brackets? |
45008 | Equivalent conditions for a preabelian category to be abelian
Let's fix some terminology first. A category [imath]\mathcal{C}[/imath] is preabelian if: 1) [imath]Hom_{\mathcal{C}}(A,B)[/imath] is an abelian group for every [imath]A,B[/imath] such that composition is biadditive, 2) [imath]\mathcal{C}[/imath] has a zero... | 220051 | Image in abelian categories
[imath]\def\im{\operatorname{im}}\def\coker{\operatorname{coker}}[/imath]For a morphism [imath] f: A\to B[/imath] in an abelian category, we let [imath]\im f:=\ker(\coker f)[/imath]. Then the morphism [imath]A\to \im f[/imath] is an epimorphism and [imath]\coker(\ker f\to A).[/imath] May I ... |
20170 | Is there a measurable set [imath]A[/imath] such that [imath]m(A \cap B) = \frac12 m(B)[/imath] for every open set [imath]B[/imath]?
Is there a measurable set [imath]A[/imath] such that [imath]m(A \cap B)= \frac12 m(B)[/imath] for every open set [imath]B[/imath]? Edit: (t.b.) See also A Lebesgue measure question for fu... | 1647113 | Existence of a subset [imath]S\subset\mathbb R[/imath] s.t. [imath]\forall a, S\cap [a,b] has Lebesgue measure (b-a)/2?[/imath]
I am trying to either find an example of such a set, or prove that no such set exists. I know of examples of dense sets with measure [imath]1/2[/imath] on specific intervals, such as [imath][... |
625 | Why is the derivative of a circle's area its perimeter (and similarly for spheres)?
When differentiated with respect to [imath]r[/imath], the derivative of [imath]\pi r^2[/imath] is [imath]2 \pi r[/imath], which is the circumference of a circle. Similarly, when the formula for a sphere's volume [imath]\frac{4}{3} \pi ... | 766642 | Validity of proof for surface area of a sphere
On a geometry test I forgot the formula for the surface area of a sphere so I derived it and ended up being right. But it seems like my derivation is wrong. I got the surface area formula by taking the derivative of the volume formula, [imath]\frac{4}{3}\pi r^3[/imath]. M... |
16244 | If [imath]\int_0^x f \ dm[/imath] is zero everywhere then [imath]f[/imath] is zero almost everywhere
I have been thinking on and off about a problem for some time now. It is inspired by an exam problem which I solved but I wanted to find an alternative solution. The object was to prove that some sequence of functions ... | 1837775 | Zero integral implies zero function almost everywhere
Assume [imath]f[/imath] is Riemann integrable and further assume that [imath]\int_a^x f=0[/imath] for all [imath]x[/imath]. How would I go about showing that [imath]f[/imath] itself is [imath]0[/imath] almost everywhere? I am new to Lebesgue's measure theory so I a... |
114462 | A map is continuous if and only if for every set, the image of closure is contained in the closure of image
As a part of self study, I am trying to prove the following statement: Suppose [imath]X[/imath] and [imath]Y[/imath] are topological spaces and [imath]f: X \rightarrow Y[/imath] is a map. Then [imath]f[/imath] i... | 740588 | Proving that [imath]f(\bar Z)\subset\overline {f(Z)}[/imath] when [imath]f[/imath] is a continuous map
I'm trying to solve this question from my textbook: Let [imath]f:X\rightarrow Y[/imath] be a continuous map and let [imath]Z \subset X[/imath]. Prove the inclusion [imath]f(\bar Z)\subset\overline {f(Z)}[/imath]. ... |
93816 | Proving that an additive function [imath]f[/imath] is continuous if it is continuous at a single point
Suppose that [imath]f[/imath] is continuous at [imath]x_0[/imath] and [imath]f[/imath] satisfies [imath]f(x)+f(y)=f(x+y)[/imath]. Then how can we prove that [imath]f[/imath] is continuous at [imath]x[/imath] for all ... | 2445643 | Conditions about continuous functions
Say we have [imath]f(x+y) = f(x) + f(y) \quad \forall x,y \in \mathbb R[/imath] and [imath]f[/imath] is continuous at one point at least. I wish to show there must be some [imath]c[/imath] such that [imath]f(x)=cx[/imath] for all [imath]x[/imath]. Think I can do so by first showin... |
10400 | Comaximal ideals in a commutative ring
Let [imath]R[/imath] be a commutative ring and [imath]I_1, \dots, I_n[/imath] pairwise comaximal ideals in [imath]R[/imath], i.e., [imath]I_i + I_j = R[/imath] for [imath]i \neq j[/imath]. Why are the ideals [imath]I_1^{n_1}, ... , I_r^{n_r}[/imath] (for any [imath]n_1,...,n_r \i... | 328890 | If a finite set of ideals generates a ring, then so does any set of arbitrary powers of those ideals.
In Lang's Algebra, pg 95 (3rd Revised Ed.), he concludes a proof on the Chinese Remainder Theorem with: In the same vein as above, we observe that if [imath]\mathfrak{a_1},\dots,\mathfrak{a_n}[/imath] are ideals of ... |
125065 | Partitioning a natural number [imath]n[/imath] in order to get the maximum product sequence of its addends
Suppose we have a natural number [imath]n \ge 0[/imath]. Given natural numbers [imath]\alpha_1,\ldots,\alpha_k[/imath] such that [imath]k\le n[/imath] [imath]\sum_i \alpha_i = n[/imath] what is the maximum valu... | 1860096 | Maximize the product of the partitions of an integer
Let [imath]n>0[/imath] be an integer. Consider all partitions of [imath]n[/imath], i.e. all possible ways of writing [imath]n[/imath] as a finite sum of positive integers, [imath]n=n_1+n_2+\cdots+n_k.[/imath] What partition maximizes the product [imath]n_1n_2\c... |
52373 | Proof that [imath]\gcd(ax+by,cx+dy)=\gcd(x,y)[/imath] if [imath]ad-bc= \pm 1[/imath]
I'm having problems with an exercise from Apostol's Introduction to Analytic Number Theory. Given [imath]x[/imath] and [imath]y[/imath], let [imath]m=ax+by[/imath], [imath]n=cx+dy[/imath], where [imath]ad-bc= \pm 1[/imath]. Prove tha... | 2947523 | [imath]h\mid (3a + 5b)[/imath], prove [imath]h\mid a[/imath] and [imath]h\mid b[/imath]
I have this homework question. "For any integer [imath]a[/imath] and [imath]b[/imath], prove that [imath]\gcd(a,b) = \gcd(3a+5b,11a+18b)[/imath]." I know that if [imath] g = \gcd(a,b)[/imath] and [imath]h = \gcd(3a+5b,11a+18b)[/i... |
115228 | Solution(s) to [imath]f(x + y) = f(x) + f(y)[/imath] (and miscellaneous questions...)
My lecturer was talking today (in the context of probability, more specifically Kolmogorov's axioms) about the additive property of functions, namely that: [imath]f(x+y) = f(x) + f(y)[/imath] I've been trying to find what functions s... | 536735 | Find all [imath]f: \mathbb{Q} \rightarrow \mathbb{R}[/imath] such that [imath]f(x+y) = f(x)+f(y)[/imath]
i have to find all functions [imath]f: \mathbb{Q} \rightarrow \mathbb{R}[/imath], such that [imath]f(x+y)=f(x)+f(y)[/imath]. So functions of the form [imath]f(x) := ax, a \in \mathbb{R}[/imath] satisfy the above co... |
59738 | Probability for the length of the longest run in [imath]n[/imath] Bernoulli trials
Suppose a biased coin (probability of head being [imath]p[/imath]) was flipped [imath]n[/imath] times. I would like to find the probability that the length of the longest run of heads, say [imath]\ell_n[/imath], exceeds a given number [... | 513808 | Probability of Runs of Heads of Length N
For example: [imath]“THHTHTTHHHTHTHTTHHTHT”[/imath] contains 1 run of heads of length 3, 2 runs of length 2, and 4 runs of length 1. Assuming [imath]P(H) = p[/imath] and [imath]P(T) = (1-p)[/imath], calculate (using properties such as conditional probability and Bayes’ Rule)... |
244644 | Question about set notation: what does [imath]]a,b[[/imath] mean?
In the following question here the notation [imath]c\in ]a,b[[/imath] is used. What does this mean? I have never seen it before. | 1077112 | What is the meaning of the notation [imath]]a,b[[/imath]?
I've seen the notation [imath]]a,b[[/imath] in several questions on this site, but I am not familiar with it. Can someone clue me in? |
43032 | How to obtain tail bounds for a sum of dependent and bounded random variables?
Note: I divide this question to two separated question not to be duplicate version. I am looking for tail bounds (preferably exponential) for the sum of dependent and bounded random variables. Consider [imath]K_{ij}=\sum_{r=1}^N\sum_{c=1}^... | 42997 | How to obtain tail bounds for a linear combination of dependent and bounded random variables?
I am looking for tail bounds (preferably exponential) for a linear combination of dependent and bounded random variables. consider [imath]K_{ij}=\sum_{r=1}^N\sum_{c=1}^N W_{ir}C_{rc}W_{jc}[/imath] where [imath]i \neq j[/imat... |
33970 | Finding the limit of [imath]\frac{Q(n)}{P(n)}[/imath] where [imath]Q,P[/imath] are polynomials
Suppose that [imath]Q(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0} [/imath]and [imath]P(x)=b_{m}x^{m}+b_{m-1}x^{m-1}+\cdots+b_{1}x+b_{0}.[/imath] How do I find [imath]\lim_{x\rightarrow\infty}\frac{Q(x)}{P(x)}[/imath] an... | 1477946 | How to solve this limit: [imath]\lim_{n \to \infty} \frac{(2n+2) (2n+1) }{ (n+1)^2}[/imath]
[imath] \lim_{n\to\infty}\frac{(2n+2)(2n+1)}{(n+1)^{2}} [/imath] When I expand it gives: [imath] \lim_{n\to\infty} \dfrac{4n^{2} + 6n + 2}{n^{2} + 2n + 1} [/imath] How can this equal [imath]4[/imath]? Because if I replace [ima... |
66052 | Prove that the center of a group is a normal subgroup
Let [imath]G[/imath] be a group. We define [imath]H=\{h\in G\mid \forall g\in G: hg=gh\},[/imath] the center of [imath]G[/imath]. Prove that [imath]H[/imath] is a (normal) subgroup of [imath]G[/imath]. | 330109 | Elements which commute with a given element form a subgroup
Let [imath](G,∗)[/imath] be a group and [imath]a\in G[/imath] then so that set of elements [imath]x[/imath] of [imath]G[/imath] such that [imath]a∗x = x∗a[/imath] is a subgroup of [imath]G[/imath]. I have tried by using theorem that [imath]H[/imath] is as sub... |
301005 | Is there an example of a function [imath]f: \mathbb{Z} \to \{\text{finite subsets of }\mathbb{Z}\}[/imath]?
In my last question, I asked for a proof of "Are the set of all finite subsets in [imath]\mathbb{Z}[/imath] countable?" . I had a good answer that showed me that it is an [imath]f: \mathbb{N} \to \{\text{finite ... | 200389 | Show that the set of all finite subsets of [imath]\mathbb{N}[/imath] is countable.
Show that the set of all finite subsets of [imath]\mathbb{N}[/imath] is countable. I'm not sure how to do this problem. I keep trying to think of an explicit formula for 1-1 correspondence like adding all the elements in each subset an... |
96826 | The Monty Hall problem
I was watching the movie 21 yesterday, and in the first 15 minutes or so the main character is in a classroom, being asked a "trick" question (in the sense that the teacher believes that he'll get the wrong answer) which revolves around theoretical probability. The question goes a little somethi... | 1141915 | Conditional probability problem of three choices
I have the following problem where I have difficulties grasping the intuition: Lets say we have three boxes, with two of them empty and one containing a gold price. Lets say we randomly select one of the boxes. After our selection, we are given which one of the re... |
257392 | If [imath]\gcd(a,b)=1[/imath], then [imath]\gcd(a+b,a^2 -ab+b^2)=1[/imath] or [imath]3[/imath].
Hint: [imath]a^2 -ab +b^2 = (a+b)^2 -3ab.[/imath] I know we can say that there exists an [imath]x,y[/imath] such that [imath]ax + by = 1[/imath]. So in this case, [imath](a+b)x + ((a+b)^2 -3ab)y =1.[/imath] I thought sett... | 522289 | Given that [imath]gcd(a,b)=1[/imath], prove that [imath]gcd(a+b,a^2-ab+b^2)=1[/imath] or [imath]3[/imath], also when will it equal [imath]1[/imath]?
It is an exercise on the lecture that i am unable to prove. Given that [imath]gcd(a,b)=1[/imath], prove that [imath]gcd(a+b,a^2-ab+b^2)=1[/imath] or [imath]3[/imath], als... |
18179 | Finding Value of the Infinite Product [imath]\prod \Bigl(1-\frac{1}{n^{2}}\Bigr)[/imath]
While trying some problems along with my friends we had difficulty in this question. True or False: The value of the infinite product [imath]\prod\limits_{n=2}^{\infty} \biggl(1-\frac{1}{n^{2}}\biggr)[/imath] is [imath]1[/imath].... | 513053 | Is [imath]\prod \limits_{n=2}^{\infty}(1-\frac{1}{n^2})=1[/imath]
Question is to check if [imath]\prod \limits_{n=2}^{\infty}(1-\frac{1}{n^2})=1[/imath] we have [imath]\prod \limits_{n=2}^{\infty}(1-\frac{1}{n^2})=\prod \limits_{n=2}^{\infty}(\frac{n^2-1}{n^2})=\prod \limits_{n=2}^{\infty}\frac{n+1}{n}\frac{n-1}{n}=(... |
31097 | A lady and a monster
A famous problem: a lady is in the center of the circular lake and a monster is on the boundary of the lake. The speed of the monster is [imath]v_m[/imath], and the speed of the swimming lady is [imath]v_l[/imath]. The goal of the lady is to come to the ground without meeting the monster, and the ... | 2646032 | What it the best constant in the riddle of the man and the tiger?
[imath]\newcommand{\man}{\text{man}}[/imath] [imath]\newcommand{\tiger}{\text{tiger}}[/imath] Consider the following situation: A man is standing in the center of a circle of radius [imath]r[/imath]. On the circle there is a tiger. The man move in arbit... |
28476 | Finding the limit of [imath]\frac {n}{\sqrt[n]{n!}}[/imath]
I'm trying to find [imath]\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}} .[/imath] I tried couple of methods: Stolz, Squeeze, D'Alambert Thanks! Edit: I can't use Stirling. | 516578 | What is the limit [imath] \underset{n\to\infty}{\lim} \frac {{n!}^{1/n}}{n} [/imath]
What is the following limit equal to and how do I prove it? [imath] \underset{n\to\infty}{\lim} \frac {{n!}^{1/n}}{n}. [/imath] I've been trying for a while and I can't seem to get it. |
22472 | Nested sequences of balls in a Banach space
This seems to be a fairly easy question but I'm looking for new points of view on it and was wondering if anyone might be able to help. (By the way- this question does come from home-work, but I've already solved and handed it, and I'm posting this out of interest, so no HW ... | 1478376 | Let [imath]X[/imath] be a Banach space, and [imath]B_1\supseteq B_2 \supseteq\cdots [/imath] . Show that [imath]\bigcap\limits_{i=1}^\infty B_i\neq\emptyset[/imath]
Let [imath]X[/imath] be a Banach space, and [imath]B_1\supseteq B_2 \supseteq \cdots [/imath] a sequence of closed balls with radius [imath]r_i[/imath] an... |
37971 | Identity for convolution of central binomial coefficients: [imath]\sum\limits_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}=2^{2n}[/imath]
It's not difficult to show that [imath](1-z^2)^{-1/2}=\sum_{n=0}^\infty \binom{2n}{n}2^{-2n}z^{2n}[/imath] On the other hand, we have [imath](1-z^2)^{-1}=\sum z^{2n}[/imath]. Squaring ... | 360053 | Finding a closed form expression for this sum
For non-negative [imath]n[/imath], find [imath] \sum_{k=0}^n \binom{2k}{k}\binom{2n-2k}{n-k}. [/imath] I can't figure this out. Any ideas? |
387 | Sum of reciprocals of numbers with certain terms omitted
I know that the harmonic series [imath]1 + \frac12 + \frac13 + \frac14 + \cdots[/imath] diverges. I also know that the sum of the inverse of prime numbers [imath]\frac12 + \frac13 + \frac15 + \frac17 + \frac1{11} + \cdots[/imath] diverges too, even if really slo... | 329807 | Convergence of [imath]\sum^n_{k=1}\frac1k[/imath] after removing terms containing the digit [imath]p[/imath]
We know that [imath]\sum^n_{k=1}\frac1k[/imath] diverges. But if I were to pick a digit [imath]p[/imath] so that [imath]p[/imath] is an integer between [imath]0[/imath] and [imath]9[/imath] inclusive, and then ... |
82489 | How to compute [imath]\int_0^{\pi/2}\frac{\sin^3 t}{\sin^3 t+\cos^3 t}dt[/imath]?
Calculating with Mathematica, one can have [imath]\int_0^{\pi/2}\frac{\sin^3 t}{\sin^3 t+\cos^3 t}\,\mathrm dt=\frac{\pi}{4}.[/imath] How can I get this formula by hand? Is there any simpler idea than using [imath]u = \sin t[/imath]? ... | 439851 | Evaluate the integral [imath]\int^{\frac{\pi}{2}}_0 \frac{\sin^3x}{\sin^3x+\cos^3x}\,\mathrm dx[/imath].
Evaluate the integral [imath]\int^{\frac{\pi}{2}}_0 \frac{\sin^3x}{\sin^3x+\cos^3x}\, \mathrm dx.[/imath] How can i evaluate this one? Didn't find any clever substitute and integration by parts doesn't lead anywher... |
128657 | How to prove that a simple graph having 11 or more vertices or its complement is not planar?
It is an exercise on a book again. If a simple graph [imath]$G$[/imath] has 11 or more vertices,then either G or its complement [imath]$\overline { G } $[/imath] is not planar. How to begin with this? Induction? Thanks for yo... | 3026767 | Show that [imath]K_n[/imath] is not the union of two planar graphs
Show that [imath]K_n[/imath] is not the union of two planar graphs for [imath]n\ge 11[/imath] I know that a graph [imath]G[/imath] is planar iff it does not have [imath]K_5[/imath] or [imath]K_{3,3}[/imath] as its induced subgraphs But how to use it ... |
105633 | Prove that [imath]a=b[/imath], where [imath]a[/imath] and [imath]b[/imath] are elements of the integral domain [imath]D[/imath]
Let [imath]D[/imath] be an integral domain and [imath]a,~b \in D[/imath]. Suppose that [imath]a^n=b^n[/imath] and [imath]a^m=b^m[/imath] for any two some [imath]m,~n[/imath] such that [imath... | 82678 | [imath]a^m=b^m[/imath] and [imath]a^n=b^n[/imath] imply [imath]a=b[/imath]
Let [imath]D[/imath] be an integral domain and let [imath]a^m=b^m[/imath] and [imath]a^n=b^n[/imath] where [imath]m[/imath] and [imath]n[/imath] are relatively prime integers, [imath]a,b \in D[/imath]. How do I show [imath]a=b[/imath]? |
34271 | Order of general- and special linear groups over finite fields.
Let [imath]\mathbb{F}_3[/imath] be the field with three elements. Let [imath]n\geq 1[/imath]. How many elements do the following groups have? [imath]\text{GL}_n(\mathbb{F}_3)[/imath] [imath]\text{SL}_n(\mathbb{F}_3)[/imath] Here GL is the general linear... | 1069116 | Order of [imath]\mathrm{GL}_n(\mathbb F_p)[/imath] for [imath]p[/imath] prime
While proving some facts about matrix group operations on finite fields, I stumbled across the following question: What is the order of the group of invertible [imath]n\times n[/imath] matrices over a finite field of prime order [imath]p[/... |
67148 | If [imath]a^3 =a[/imath] for all [imath]a[/imath] in a ring [imath]R[/imath], then [imath]R[/imath] is commutative.
Let [imath]R[/imath] be a ring, where [imath]a^{3} = a[/imath] for all [imath]a\in R[/imath]. Prove that [imath]R[/imath] must be a commutative ring. | 360958 | Prove that [imath]R[/imath] is a commutative ring if [imath]x^3=x[/imath]
Let [imath]R[/imath] be a ring satisfying: [imath]\forall x\in R, \; x^3=x[/imath]. Prove that [imath]R[/imath] is a commutative ring. |
48080 | Sum of First [imath]n[/imath] Squares Equals [imath]\frac{n(n+1)(2n+1)}{6}[/imath]
I am just starting into calculus and I have a question about the following statement I encountered while learning about definite integrals: [imath]\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}[/imath] I really have no idea why this statemen... | 387664 | Proving [imath]\sum_{k=1}^n{k^2}=\frac{n(n+1)(2n+1)}{6}[/imath] without induction
I was looking at: [imath]\sum_{k=1}^n{k^2}=\frac{n(n+1)(2n+1)}{6}[/imath] It's pretty easy proving the above using induction, but I was wondering what is the actual way of getting this equation? |
65861 | Summation of natural number set with power of [imath]m[/imath]
Who knows about the summation of this series: [imath]\sum\limits_{i=1}^{n}i^m [/imath] where [imath]m[/imath] is constant and [imath]m\in \mathbb{N}[/imath]? thanks | 811693 | Calculate [imath]1*1+2*2+3*3+4*4+....+r*r[/imath]
If [imath]1+2+3+...+r[/imath] is in otherwords [imath]\frac{r(r+1)}{2}[/imath], then what is the answer with squares? Thank you for all the help. |
11464 | How to compute the formula [imath]\sum \limits_{r=1}^d r \cdot 2^r[/imath]?
Given [imath]1\cdot 2^1 + 2\cdot 2^2 + 3\cdot 2^3 + 4\cdot 2^4 + \cdots + d \cdot 2^d = \sum_{r=1}^d r \cdot 2^r,[/imath] how can we infer to the following solution? [imath]2 (d-1) \cdot 2^d + 2. [/imath] Thank you | 667191 | Difficulty with understanding summations
I am in advance sorry if this question is too easy for this site, but I am having real problem understanding how to solve this summation: [imath]\sum_{i=1}^n{i*2^i}[/imath] I understand basics of summations but i don't know where to start, please help. |
122274 | Why [imath]x^{p^n}-x+1[/imath] is irreducible in [imath]{\mathbb{F}_p}[/imath] only when [imath]n=1[/imath] or [imath]n=p=2[/imath]
I have a question, I think it concerns with field theory. Why the polynomial [imath]x^{p^n}-x+1[/imath] is irreducible in [imath]{\mathbb{F}_p}[/imath] only when [imath]n=1[/imath] or [i... | 509277 | Dummit exercise 14.3.11: extension degree of finite fields
Dummit and Foote's exercise 14.3.11 asks to prove that [imath]f(x) = x^{p^{n}}-x+1[/imath] is irreducible over [imath]\mathbb{F}_{p}[/imath] iff [imath]n=1[/imath] or [imath]n=p=2[/imath]. To prove the 'only if' part, the exercise suggest to prove that if [ima... |
76946 | Prove the inequality [imath]n! \geq 2^n[/imath] by induction
I'm having difficulty solving an exercise in my course. The question is: Prove that [imath]n!\geq 2^n[/imath]. We have to do this by induction. I started like this: The lowest natural number where the assumption is correct is [imath]4[/imath] as: [imath]... | 296790 | Prove [imath]n!>2^n[/imath] for [imath]n\geq4[/imath] using induction.
I just want to know if my proof to this question is correct. First, I proved it was true for [imath]n = 4[/imath]. [imath]4!>2^4[/imath] [imath]24>16[/imath] Then, I assumed that it was true for [imath]n=k[/imath]. [imath]k!>2^k[/imath] Afterwards,... |
88300 | If [imath]f(x)[/imath] is continuous on [imath][a,b][/imath] and [imath]M=\max \; |f(x)|[/imath], is [imath]M=\lim \limits_{n\to\infty} \left(\int_a^b|f(x)|^n\,\mathrm dx\right)^{1/n}[/imath]?
Let [imath]f(x)[/imath] be a continuous real-valued function on [imath][a,b][/imath] and [imath]M=\max\{|f(x)| \; :\; x \in [a... | 318634 | Show that [imath]\lim((\int_{a}^{b}f^{n})^\frac{1}{n})=\sup\{f(x):x\in[a,b]\}[/imath]
Another exercise (this one is 7.2.18) from "Introduction to Real Analysis" by Bartle and Sherbert that I'm struggling with: Let [imath]f[/imath] be continuous on [imath][a,b][/imath], let [imath]f(x)>=0[/imath] for [imath]x\in[a,b][/... |
49383 | How does [imath] \sum_{p grow asymptotically for \text{Re}(s) < 1 ?[/imath]
Note the [imath] p < x [/imath] in the sum stands for all primes less than [imath] x [/imath]. I know that for [imath] s=1 [/imath], [imath] \sum_{p<x} \frac{1}{p} \sim \ln \ln x , [/imath] and for [imath] \mathrm{Re}(s) > 1 [/imath], the par... | 679950 | prime zeta function when [imath]0[/imath]
I would like to know if there is a good estimate for the sum which concerns all primes not exceeding [imath]x[/imath]: [imath]\sum\limits_{p\leq x}\frac{1}{p^s}[/imath][imath]0<s<1[/imath]. Only this. Thanks in advance! |
58943 | The simply connected coverings of two homotopy equivalent spaces are homotopy equivalent
This is exercise 1.3.8 in Hatcher: Let [imath]\tilde{X}[/imath] and [imath]\tilde{Y}[/imath] be simply-connected covering spaces of path connected, locally path-connected spaces [imath]X[/imath] and [imath]Y[/imath]. Show that if ... | 136405 | Homotopy equivalence of universal cover
As part of am exam question (Q21F here), I'm trying to prove that if [imath]X[/imath] and [imath]Y[/imath] are path-connected, locally path-connected spaces with universal covers [imath]\widetilde{X}[/imath] and [imath]\widetilde{Y}[/imath], respectively, then if [imath]X \simeq... |
4764 | Sine function dense in [imath][-1,1][/imath]
We know that the sine function takes it values between [imath][-1,1][/imath]. So is the set [imath]A = \{ \sin{n} \ : \ n \in \mathbb{N}\}[/imath] dense in [imath][-1,1][/imath]. Generally, for showing the set is dense, one proceeds, by finding out what is [imath]\overline{... | 313943 | Closure of the set [imath]\{\sin(n): n\in\mathbb{N}, n > 0\}[/imath] is the interval of reals [imath][-1,1][/imath]?
Please prove or disprove. Any help is appreciated Thanks in advance |
3408 | Characterizing non-constant entire functions with modulus $1$ on the unit circle
Is there a characterization of the nonconstant entire functions [imath]f[/imath] that satisfy [imath]|f(z)|=1[/imath] for all [imath]|z|=1[/imath]? Clearly, [imath]f(z)=z^n[/imath] works for all [imath]n[/imath]. Also, it's not difficul... | 1387347 | Characterize all entire functions [imath]f[/imath] such that [imath]|f(z)|=1[/imath], whenever [imath]|z|=1[/imath]
Characterize all entire functions [imath]f[/imath] such that [imath]|f(z)|=1[/imath], whenever [imath]|z|=1[/imath]. I think either [imath]f(z)=c[/imath] or [imath]f(z)=cz[/imath] where [imath]|c|=1[/ima... |
14429 | What's the cardinality of all sequences with coefficients in an infinite set?
My motivation for asking this question is that a classmate of mine asked me some kind of question that made me think of this one. I can't recall his exact question because he is kind of messy (both when talking about math and when thinking a... | 591465 | Cardinality of the set of all natural sequences is [imath]2^{\aleph_0}[/imath]
I was wondering how can you prove that [imath]\mathbb{N}^\mathbb{N} \sim 2^\mathbb{N}[/imath] (where [imath]\mathbb{N}^\mathbb{N}[/imath] is the set of all functon [imath]f:\mathbb{N}\rightarrow \mathbb{N}[/imath]). I think I can show that... |
39802 | Why does [imath]1+2+3+\cdots = -\frac{1}{12}[/imath]?
[imath]\displaystyle\sum_{n=1}^\infty \frac{1}{n^s}[/imath] only converges to [imath]\zeta(s)[/imath] if [imath]\text{Re}(s) > 1[/imath]. Why should analytically continuing to [imath]\zeta(-1)[/imath] give the right answer? | 354265 | 1+2+3+4+... = -1/12
Consider the zeta function [imath]\zeta(s)= \sum \limits_{n=1}^{\infty} \frac{1}{n^s}[/imath]. It is established that [imath] \zeta(-1) = -\frac{1}{12}[/imath]. Reference (Equation 90) Then we have [imath] \zeta(-1) = \sum \limits_{n=1}^{\infty} \frac{1}{n^{-1}}= 1+2+3+4 + ... = -\frac{1}{12}[/im... |
305708 | Cyclotomic polynomials, irreducibility
I need to decide if certain cyclotomic polynomials are irreducibles over the [imath]\mathbb{F}_q[/imath]. For example, if [imath]\Phi_{12}(x)[/imath] is irreducible over [imath]\mathbb{F}_9[/imath]. Anyone can help me? Ok, i think i should aclare something: this question is not a... | 305111 | Irreducible cyclotomic polynomial
I want to know if there is a way to decide if a cyclotomic polynomial is irreducible over a field [imath]\mathbb{F}_q[/imath]? |
13344 | Proof of [imath]\int_0^\infty \left(\frac{\sin x}{x}\right)^2 \mathrm dx=\frac{\pi}{2}.[/imath]
I am looking for a short proof that [imath]\int_0^\infty \left(\frac{\sin x}{x}\right)^2 \mathrm dx=\frac{\pi}{2}.[/imath] What do you think? It is kind of amazing that [imath]\int_0^\infty \frac{\sin x}{x} \mathrm dx[/imat... | 484907 | Integration of [imath]\int_0^\infty\frac{\sin^2x}{x^2}dx[/imath]
I have been trying very hard to find the answer to the following integral[imath]\int_0^\infty\frac{\sin^2x}{x^2}dx[/imath] given that [imath]\int_0^\infty\frac{\sin x\cos x}{x}dx = \frac{\pi}{4}[/imath] |
79287 | Conditions Equivalent to Injectivity
Let [imath]A[/imath] and [imath]B[/imath] be sets, where [imath]f : A \rightarrow B[/imath] is a function. Show that the following properties are valid equivalent*: [imath]f[/imath] is injective. For all [imath]X, Y \subset A[/imath] is valid: [imath]f(X \cap Y)=f(X)\cap f(Y)[/im... | 1105696 | If [imath]f[/imath] is an injection, [imath]f(S_1 \cap S_2) = f(S_1) \cap f(S_2)[/imath]
I need to prove that if [imath]S_1[/imath] and [imath]S_2[/imath] are subsets of a set [imath]X[/imath], and if [imath]f: X \to Y[/imath] is an injection, prove that [imath]f(S_1 \cap S_2) = f(S_1) \cap f(S_2)[/imath]. I know I n... |
9321 | Understanding proof of completeness of [imath]L^{\infty}[/imath]
I'm reading page number 4 here. In particular the section where it deals with the case [imath]p=\infty[/imath], that is , showing that [imath]L^{\infty}[/imath] is complete. http://www.core.org.cn/NR/rdonlyres/Mathematics/18-125Fall2003/5E3917E2-C212-463... | 742046 | Proving the Rietz-Fischer Theorem for [imath]p = \infty[/imath]
Rietz-Fischer Theorem: Let [imath]E[/imath] be a measurable set and [imath]1 \le p \le \infty[/imath]. Then every rapidly Cauchy sequence in [imath]L^p(E)[/imath] converges both with respect to the [imath]p[/imath]-norm and pointwise almost everyone on [... |
7757 | How to prove this binomial identity [imath]\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}[/imath]?
I am trying to prove this binomial identity [imath]\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}[/imath] but am not able to think something except induction,which is of-course not necessary (I think) here, so I am inquisitive to ... | 478859 | Prove that [imath]\sum_{k=1}^{n}k\binom{n}{k}=n\cdot2^{n-1}[/imath]
I want to prove the following: [imath]\sum_{k=1}^{n}k\binom{n}{k}=n\cdot2^{n-1}[/imath] what I did is(use binominal): [imath]\binom{n}{k}X^k\cdot 1^{n-k} = (X+1)^n[/imath] [imath]k\binom{n}{k}X^k\cdot 1^{n-k} = k(X+1)^k-1[/imath] now I replace [imath]... |
82309 | Slowing down divergence 2
Let [imath]f(x)[/imath] and [imath]g(x)[/imath] be positive nondecreasing functions such that [imath] \sum_{n>1} \frac1{f(n)} \text{ and } \sum_{n>1} \frac1{g(n)} [/imath] diverges. (Why) must the series [imath]\sum_{n>1} \frac1{g(n)+f(n)}[/imath] diverge? | 858339 | For [imath]a_n,b_n\uparrow[/imath] and [imath]\sum \frac{1}{a_n}[/imath], [imath]\sum \frac{1}{b_n}[/imath] divergent is the series [imath]\sum \frac{1}{a_n+b_n}[/imath] also divergent?
Let [imath]a_n[/imath] and [imath]b_n[/imath] are strictly increasing to [imath]+\infty[/imath] sequences such that the series [imath... |
255614 | Prove that [imath]\sum_{n=1}^{\infty}\ a_n^2[/imath] is convergent if [imath]\sum_{n=1}^{\infty}\ a_n[/imath] is absolutely convergent
Suppose that [imath]\displaystyle\sum_{n=1}^{\infty}\ a_n[/imath] is absolutely convergent. How can we prove that [imath]\displaystyle\sum_{n=1}^{\infty}\ a_n^2[/imath] is convergent? | 36429 | If [imath]\sum_{n=1}^{\infty} a_n[/imath] is absolutely convergent, then [imath]\sum_{n=1}^{\infty} (a_n)^2[/imath] is convergent
Let [imath]\sum_{n=1}^{\infty} a_n[/imath] be a series in R. Prove that if [imath]\sum_{n=1}^{\infty} a_n[/imath] is absolutely convergent, then [imath]\sum_{n=1}^{\infty} (a_n)^2[/imath] i... |
40998 | The sum of [imath](-1)^n \frac{\ln n}{n}[/imath]
I'm stuck trying to show that [imath]\sum_{n=2}^{\infty} (-1)^n \frac{\ln n}{n}=\gamma \ln 2- \frac{1}{2}(\ln 2)^2[/imath] This is a problem in Calculus by Simmons. It's in the end of chapter review and it's associated with the section about the alternating series test.... | 1627615 | How to prove this series: [imath]\displaystyle \sum_{n=1}^{\infty }\frac{\left ( -1 \right )^{n}\ln n}{n}=\gamma \ln 2-\frac{1}{2}\ln^22[/imath]
How to prove this series [imath]\sum_{n=1}^{\infty }\frac{\left ( -1 \right )^{n}\ln n}{n}=\gamma \ln 2-\frac{1}{2}\ln^22[/imath] and \begin{align*} \sum_{n=1}^{\infty }\frac... |
20661 | The sum of an uncountable number of positive numbers
Claim:If [imath](x_\alpha)_{\alpha\in A}[/imath] is a collection of real numbers [imath]x_\alpha\in [0,\infty][/imath] such that [imath]\sum_{\alpha\in A}x_\alpha<\infty[/imath], then [imath]x_\alpha=0[/imath] for all but at most countably many [imath]\alpha\in A[/i... | 419739 | All finite sums bounded implies countable
Let [imath]X[/imath] be a set of positive real numbers, and let [imath]S[/imath] be the set of all finite sums of members of [imath]X[/imath]. Suppose that [imath]S[/imath] is bounded. Prove that [imath]X[/imath] is countable. Not much idea on how to start here. A contraposi... |
26363 | Square roots -- positive and negative
It is perhaps a bit embarrassing that while doing higher-level math, I have forgot some more fundamental concepts. I would like to ask whether the square root of a number includes both the positive and the negative square roots. I know that for an equation [imath]x^2 = 9[/imath], ... | 380359 | Why is [imath]\sqrt{4} = 2[/imath] and Not [imath]\pm 2[/imath]?
I've always been told that if [imath]\ x^2 = 4,[/imath] [imath] =>x = \pm2[/imath] But recently, Prof. mentioned that if [imath]x = \sqrt{4}[/imath], Then [imath]x = +2(only)[/imath] I am very skeptical about this because they both mean the same thing a... |
93463 | Is [imath]\mathbf{Q}(\sqrt{2}, \sqrt{3}) = \mathbf{Q}(\sqrt{2}+\sqrt{3})[/imath]?
Is [imath]\mathbf{Q}(\sqrt{2}, \sqrt{3}) = \mathbf{Q}(\sqrt{2}+\sqrt{3})[/imath] ? [imath]\mathbf{Q}(\sqrt{2},\sqrt{3})=\{a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6} \mid a,b,c,d\in\mathbf{Q}\}[/imath] [imath]\mathbf{Q}(\sqrt{2}+\sqrt{3}) = \lbr... | 396276 | How can I show this field extension equality?
How can I show this field extension equality [imath]\mathbb{Q}(\sqrt{2}, \sqrt{3}) = \mathbb{Q}(\sqrt{2} + \sqrt{3})[/imath]? |
304487 | Unit Tangent and Unit Normal Vectors -- Calculus III Question
Consider the following vector function. [imath]r(t) = \left\langle 2t \cdot \sqrt{2}, e^{2t}, e^{-2t}\right\rangle[/imath] (a) Find the unit tangent and unit normal vectors [imath]T(t)[/imath] and [imath]N(t)[/imath]. [imath]T(t) =[/imath] [imath]N(t) =[/im... | 305007 | Normal tangent vector and normal vector problem
Consider the following vector function. [imath]r(t) = \left\langle 2t \cdot \sqrt{2}, e^{2t}, e^{-2t}\right\rangle[/imath] (a) Find the unit tangent and unit normal vectors [imath]T(t)[/imath] and [imath]N(t)[/imath]. [imath]T(t) =[/imath] [imath]N(t) =[/imath] (b) Use ... |
296686 | Union of Countable Sets is Countable
Let [imath]\{E_n\}[/imath], [imath]n = 1, 2, 3, \ldots[/imath], be a sequence of countable sets, and put [imath]S = \displaystyle \bigcup_{n=1}^{\infty} E_n[/imath]. Prove that [imath]S[/imath] is countable. | 55181 | countably infinite union of countably infinite sets is countable
How do you prove that any collection of sets [imath]\{X_n : n \in \mathbb{N}\}[/imath] such that for every [imath]n \in \mathbb{N}[/imath] the set [imath]X_n[/imath] is equinumerous to the set of natural numbers, then the union of all these sets, [imath]... |
180073 | Approximating [imath]\pi[/imath] with least digits
Do you a digit efficient way to approximate [imath]\pi[/imath]? I mean representing many digits of [imath]\pi[/imath] using only a few numeric digits and some sort of equation. Maybe mathematical operations also count as penalty. For example the well known [imath]\fra... | 1990401 | Find two rational numbers [imath]\frac ab[/imath] satisfying [imath]\mid \pi - \frac ab\mid < \frac{1}{\sqrt 5 b^2}.[/imath]
Find two rational numbers [imath]\frac ab[/imath] satisfying [imath]\mid \pi - \frac ab\mid < \frac{1}{\sqrt 5 b^2}.[/imath] I dont know how to find such rationals. IS there a method? or trial? |
66145 | [imath]\mathbb{Q}/\mathbb{Z}[/imath] has a unique subgroup of order [imath]n[/imath] for any positive integer [imath]n[/imath]?
Viewing [imath]\mathbb{Z}[/imath] and [imath]\mathbb{Q}[/imath] as additive groups, I have an idea to show that [imath]\mathbb{Q}/\mathbb{Z}[/imath] has a unique subgroup of order [imath]n[/i... | 391327 | [imath]\mathbb{Q}/\mathbb{Z}[/imath] has cyclic subgroup of every positive integer [imath]n[/imath]?
I would like to know whether [imath](\mathbb{Q}/\mathbb{Z},+)[/imath] has [imath]1[/imath]. Cyclic subgroup of every positive integer [imath]n[/imath]? [imath]2[/imath]. Yes, unique one. [imath]3[/imath]. Yes, but not ... |
143173 | Showing the inequality [imath]|\alpha + \beta|^p \leq 2^{p-1}(|\alpha|^p + |\beta|^p)[/imath]
I have a small question that I think is very basic but I am unsure how to tackle since my background in computing inequalities is embarrassingly weak - I would like to show that, for a real number [imath]p \geq 1[/imath] and... | 2174788 | Common inequality
I was trying to prove that for arbitrary [imath]A,B\in\mathbb{R}[/imath] and any [imath]p\in\mathbb{N}[/imath] it holds [imath](A+B)^p \leq 2^{p-1}(A^p+B^p)[/imath] I'd appreciate some advice. Thanks a lot. |
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