qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
185,549 | <blockquote>
<p>Suppose $F$ is a nondecreasing and right continuous function, and the sequence $\{x_n\}_{n\geq1}$ converges to $x$. Then $\liminf\limits_{n\to\infty}F(x_n)\leq F(x)$. </p>
</blockquote>
<p>How can I prove this?</p>
| Brian M. Scott | 12,042 | <p>HINT: Let $\sigma_R=\langle x_{n_k}:k\in\Bbb N\rangle$ be the subsequence of terms greater than or equal to $x$, and let $\sigma_L=\langle x_{m_k}:k\in\Bbb N\rangle$ be the subsequence of terms less than $x$.</p>
<ol>
<li><p>If either subsequence is finite, it can be ignored.</p></li>
<li><p>If $\sigma_R$ is infini... |
2,502,255 | <p>We say that if $g(x)$ is continuous at $a$ and $f(x)$ is continuous at $g(a)$ then $(f\circ g)(x)$ is continuous at $a$. </p>
<p>The converse: If $(f\circ g)(x)$ is continuous at $a$ then $g(x)$ is continuous at $a$ and $f(x)$ is continuous at $g(a)$ is not necessarily true.</p>
<p>For example consider $g(x) = x+1... | eyeballfrog | 395,748 | <p>If $g$ has a jump discontinuity at $a$, then </p>
<p>$$f(x) = \left[x - \frac{g(a^+) + g(a^-)}{2}\right]^2$$</p>
<p>will have the property that $f(g(x))$ is continuous.</p>
|
3,855,736 | <p>Mathematics is not my primary discipline, but I know enough about both it and academics in general to know that many to most mathematical researchers do what they do because they enjoy doing it. This would seem to make "recreational mathematics" a rhetorical tautology, yet the term is used as if it were a ... | Ross Millikan | 1,827 | <p>I don't think there is a definitive answer. Some characteristics I have noted that set recreational problems apart are</p>
<ul>
<li>Particular instead of general problems</li>
<li>Depend on the digit representation of a number</li>
<li>Involve finding a trick of logic to find the answer</li>
<li>Worded in a way to ... |
1,550,923 | <p>I know of two applications of vector spaces over $\mathbb Q$ to problems posed by people not specifically interested in vector spaces over $\mathbb Q$:</p>
<ul>
<li>Hilbert's third problem; and</li>
<li>The Buckingham pi theorem.</li>
</ul>
<p>What others are there?</p>
| Brian M. Scott | 12,042 | <p>Assuming $\mathsf{CH}$, one can show that $\Bbb R$ is the union of countably many metrically rigid subsets by viewing it as a vector space over $\Bbb Q$. (A set $D$ in a metric space $\langle X,d\rangle$ is <em>metrically rigid</em> if no two distinct two-point subsets of $D$ are isometric.) This is mentioned in Bri... |
4,380,274 | <p>Indefinite integral is pretty easy to solve, I did it by substitution and I'm pretty sure it can be done relatively easy via integration by parts.
The problem are boundaries.</p>
<p>After substitution <span class="math-container">$arcsin x=t$</span> we get</p>
<p><span class="math-container">$$\int_0^\frac{\pi}{2} \... | user170231 | 170,231 | <p><span class="math-container">$$\begin{align*}
I &= \int_0^1 \frac{x - \arcsin(x)}{x^3} \, dx \\[1ex]
&= -\frac12\left(1 - \frac\pi2\right) + \frac12 \int_0^1 \left(1-\frac1{\sqrt{1-x^2}}\right) \, \frac{dx}{x^2} \tag{1} \\[1ex]
&= \frac\pi4-\frac12 + \frac12 \int_0^1 \frac{\sqrt{1-x^2}-1}{x^2 \sqrt{1-x^2... |
656,423 | <p>This is a really simple problem but I am unsure if I have proved it properly.</p>
<p>By contradiction:</p>
<p>Suppose that $x \geq 1$ and $x< \sqrt{x}$. Then $x\cdot x \geq x \cdot 1$ and $x^2 < x$ (squaring both sides), which is a contradiction.</p>
| Asinomás | 33,907 | <p>Here is a direct proof (without contradiction)</p>
<p>$x=1+r$ with $r\geq0$ . Then $x^2=(1+r)^2=1+2r+r^2\geq1+r=x\rightarrow x^2\geq x\rightarrow x\geq\sqrt x$</p>
|
1,784,634 | <p>Here is what I am tasked with..</p>
<p>Find a solution to the recurrence relation: </p>
<p>$F(0) = 2$</p>
<p>$F(n+1) = F(n) + 2n^2 - 1$</p>
<p>as a formula involving the summation operator</p>
<p>$$\sum_{i=1}^n$$</p>
<p>Sorry for the wonky formatting. Anyway, I am mostly looking for pointers/hints as to where ... | Roby5 | 243,045 | <p><strong>Hint:</strong>(Use telescoping)</p>
<p>$F(n+1) -F(n) =(2n^2 -1)$</p>
<p>...</p>
<p>$F(1) -F(0)=(2\cdot 0^2 -1)$</p>
<p>Now add these up.</p>
<p>Can you take it from here?</p>
|
1,784,634 | <p>Here is what I am tasked with..</p>
<p>Find a solution to the recurrence relation: </p>
<p>$F(0) = 2$</p>
<p>$F(n+1) = F(n) + 2n^2 - 1$</p>
<p>as a formula involving the summation operator</p>
<p>$$\sum_{i=1}^n$$</p>
<p>Sorry for the wonky formatting. Anyway, I am mostly looking for pointers/hints as to where ... | sinbadh | 277,566 | <p>Note that $F(n+1)-F(n)=2n^2-1$. Then $\sum_{n=0}^k 2n^2-1=\sum_{n=0}^kF(n+1)-F(n)$. But in the RHS:</p>
<p>$$\sum_{n=0}^kF(n+1)-F(n)=\sum_{n=0}^kF(n+1)-\sum_{n=0}^kF(n)\\
=F(k+1)+\sum_{n=0}^{k-1}F(n+1)-F(0)-\sum_{n=1}^kF(n)\\
=F(k+1)-F(0)+\sum_{n=0}^{k-1}F(n+1)-\sum_{n=0}^{k-1}F(n+1)\\
=F(k+1)-F(0)\\
=F(k+1)-2$$.</... |
816,249 | <p>I just need some references which studies examples of skew adjoint differential operators generating unitary strongly continuous groups of operators, and its applications to partial differential equations.</p>
<p>The example I know is the differential operator defined on the hilbert space $H=L^2(\mathbb{R})$ by
$$A... | Community | -1 | <p>You can take any self-adjoint operator and multiply it by $i$. Example: $i\Delta$ generates the <a href="http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation#Time-dependent_equation" rel="nofollow">Schrödinger equation</a> for a free particle (the potential $V$ is identically zero). The wave equation $u_{tt}=c^2u_... |
4,146,858 | <blockquote>
<p>Q) For every twice differentiable function <span class="math-container">$f:\mathbb{R}\longrightarrow [-2,2] $</span> with <span class="math-container">$[f(0)]^2+[f'(0)]^2=85$</span> , which of the following statement(s) is(are) TRUE?</p>
</blockquote>
<blockquote>
<p>(A) There exists <span class="math-c... | user21820 | 21,820 | <p>Personally, I have never used the quotient rule as I have always found it ugly and inconvenient. For example try computing the derivative of <span class="math-container">$\large\frac{(x+1)^3}{(2x-4)^5}$</span> with respect to <span class="math-container">$x$</span>. Since the quotient rule can be derived from the pr... |
1,585,772 | <p>I am finding this problem confusing :</p>
<blockquote>
<p>If,for all $x$,$f(x)=f(2a)^x$ and $f(x+2)=27f(x)$,then find $a$.</p>
</blockquote>
<p>When $x=1$ I have that $f(1)=f(2a)$ using the first identity.</p>
<p>Then when $x=2a$ I have by the second identity that $f(2a+2)=27f(2a)$,after that I simple stare at ... | J.Gudal | 225,386 | <p>Note that:</p>
<p>$f(2a)=f(2a)^{2a} \Rightarrow 1=f(2a)^{2a-1}$</p>
<p>Hence either $f(2a)=1$ or $2a-1=0$.</p>
<p>But if $f(2a)=1$, then,</p>
<p>$f(x)=1^{x}$ for all $x$, but $f(2)=27$ and so this is false.</p>
<p>Consequently $a=\frac{1}{2}$.</p>
|
3,310,193 | <p>I want to show:
<span class="math-container">$$\mu(E)=0 \rightarrow \int_E f d\mu=0$$</span> </p>
<p><span class="math-container">$f:X \rightarrow [0, \infty] $</span> is measurable and <span class="math-container">$E \in \mathcal{A} $</span></p>
<p>Consider a step function <span class="math-container">$s=\sum_i ... | Aphelli | 556,825 | <p>Consider the smooth surjective exponential map, which is Lipschitz continuous <span class="math-container">$e: T_S S^n = \mathbb{R}^n \rightarrow S^n$</span> (<span class="math-container">$S$</span> being the south pole). </p>
<p>Consider a large <span class="math-container">$M>0$</span> and a small <span class=... |
327,291 | <p>I am struggling to find the English translation of Malcev's paper "On a class of homogenous spaces" providing foundational material for nil-manifolds. To be precise this paper: <i>Malcev, A. I. On a class of homogeneous spaces. Amer. Math. Soc. Translation 1951, (1951). no. 39, 33 pp.</i> (<a href="https://mathscine... | R W | 8,588 | <p>This place is the interlibrary loan of your institution</p>
|
1,548,159 | <p>This is a question asked in India's CAT exam: <a href="http://iimcat.blogspot.in/2013/08/number-theory-questions-and-solutions.html" rel="nofollow noreferrer">http://iimcat.blogspot.in/2013/08/number-theory-questions-and-solutions.html</a> </p>
<blockquote>
<p>How many numbers with distinct digits are possible pr... | Graham Kemp | 135,106 | <blockquote>
<blockquote>
<p>How many numbers with distinct digits are possible product of whose digits is 28?</p>
</blockquote>
</blockquote>
<p>Best interpretation of that is "How many numbers with distinct digits are possible, when the product of those digits must be 28?"</p>
<p>Well, the prime factorisati... |
1,503,958 | <p>In the real number system,the equation $\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1$ has how many solutions?</p>
<p>I tried shifting the second term to the rhs and squaring.Even after that i'm left with square roots.No idea how to proceed.Help!</p>
| Piquito | 219,998 | <p>$$x+3-4\sqrt{x-1}=x-1+4-4\sqrt{x-1}=(x-1)-4\sqrt{x-1}+4=(\sqrt{x-1}-2)^2$$ Similarly $$x+8-6\sqrt{x-1}=(\sqrt{x-1}-3)^2$$ from which the answer.</p>
<p>It is not an equation but an identity in its domain of definition.This is the reason there are infinitely many solutions and not the expected finite number of them ... |
3,847,209 | <p>I'm currently trying to solve the question below.</p>
<h2>Abed is sitting in front of a large screen tv. He thinks he gets the best view when the screen takes up the maximum angle in his field of view.
What is the optimal point where he sits(to get the largest angle in his view)?</h2>
<p>I think it has something to ... | vvg | 820,543 | <p>If you consider the screen to be the base of a triangle and the person sitting at the apex of the triangle at a distance of <span class="math-container">$a$</span> measured from the center of the base, what describes the locus of the apex for angle <span class="math-container">$\alpha \in (0^{\circ}, +180^{\circ})$<... |
1,179,195 | <p>Good day everyone. </p>
<p>I need to know automata theory. Can you advice me the best way to study math?
What themes will I need to know to understand automata theory. What a sequence of study? What level will I need to study intermediate themes? Maybe can you say something yet, what can help me quickly learn autom... | Pedro | 23,350 | <p>If $a'$ and $a$ are fixed points, then $d(f(a'),f(a))=d(a',a)<\lambda d(a',a)$ with $0<\lambda <1$, which implies that $d(a,a')=0$ and hence $a=a'$.</p>
|
2,065,639 | <p>$\displaystyle \int_a^b (x-a)(x-b)\,dx=-\frac{1}{6}(b-a)^3$</p>
<p>$\displaystyle \int_a^{(a+b)/2} (x-a)(x-\frac{a+b}{2})(x-b)\, dx=\frac{1}{64}(b-a)^4$ </p>
<p>Instead of expanding the integrand, or doing integration by part, is there any faster way to compute this kind of integral?</p>
| Alex Macedo | 400,433 | <p>If you would like a geometric argument (that can be better visualized for your first integral), note that </p>
<p>$\int_a^b (x - a)(x - b)\, dx = -\int_a^b\int_{x - b}^0\int_0^{x - a} 1 \, dzdydx$</p>
<p>Which is negative the volume of a pyramid with base area $(b - a)^2/2$ and height $b -a$. Therefore the value o... |
2,475,617 | <p>EDIT: I first designated $x$, $y$ as irrational numbers. I mean rational.</p>
<p>I have this, In the question it says: For every $x$, $y$ being rational,there exists $z$ being rational so that: $x<z$ or $z<y$ Now, I have this: $\forall(x,y) \in\Bbb Q^2, \exists z\in\Bbb Q/(x<z)∨(z<y)$ Does this the sig... | amWhy | 9,003 | <p>Your first version says "For all rational ordered pairs (x, y), there exists a rational number z such that $x\lt z \lor z\lt y$."</p>
<p>Here is another way of stating what you seem to want to express, in the form of an implication: </p>
<p>$$\forall x, \forall y \Big((x\in \mathbb Q \land y \in \mathbb Q)\to \exi... |
185,867 | <p>I hear that the axiom of choice (AC) derives from
The generalized continuum hypothesis(GCH).
And also hear that both AC and GCH are independent of
Zermelo–Fraenkel set theory(ZF).</p>
<p>So, I'm just curious why don't expert mathematicians use ZF+GCH
instead of ZF+AC(ZFC).</p>
| Timothy Chow | 3,106 | <p>There are two distinct questions that you might be asking.</p>
<blockquote>
<ol>
<li><p>Why has the mathematical community adopted ZFC as a standard foundation and not ZF+GCH?</p>
</li>
<li><p>What mathematical and philosophical arguments can be advanced for and against adopting AC and/or GCH as a fundamental axiom?... |
55,965 | <p>I'm a games programmer with an interest in the following areas:</p>
<ul>
<li>Calculus</li>
<li>Matrices</li>
<li>Graph theory</li>
<li>Probability theory</li>
<li>Combinatorics</li>
<li>Statistics</li>
<li>More linguistic related fields of logic such as natural language processing, generative grammars</li>
</ul>
<... | Chris Eagle | 5,203 | <p>By Cauchy's theorem, $G$ contains an element of order 2. How does this act in the regular representation?</p>
|
3,019,506 | <p>I am stuck on this problem during my review for my stats test. </p>
<p>I know I have to use the convolution formula, and I understand that:</p>
<p><span class="math-container">$f_{U_1}(U_1) = 1$</span> for <span class="math-container">$0≤U_1≤1$</span> </p>
<p><span class="math-container">$f_{U_2}(U_2) = 1$</span... | Oscar Lanzi | 248,217 | <p>Take a cube and orient it so you're looking down a body diagonal. The edges that appear to be at the boundary of this view actually form a nonplanar object with six congruent edges, right angles wherever two adjacent edges meet, and <span class="math-container">$D_{3d}$</span> point group symmetry.</p>
|
1,998,391 | <p>To define a <strong>real</strong> exponential function
$$f(x)=a^x=e^{x \,\mathrm{lg} a}$$</p>
<p>It is strictly necessary that $a>0$.</p>
<p>But is the same true if the exponent is a <strong>natural</strong> number?</p>
<hr>
<p>In function series and moreover in power series I see things like
$$\sum_{n\geq0} ... | qman | 366,057 | <p>The key is to realize that there are two distinct exponentiation functions, where the one becomes a motivating model for (but does not necessarily embed in) the other. These are unfortunately often conflated, and the identical notation does not help.</p>
<p>The first is the case of natural number exponents ($\Bbb ... |
2,262,661 | <p>The question is in how many ways can we select 20 different items from the empty set?</p>
<h3>ans:</h3>
<p>Obviously in 0 ways since the empty set has no items. I mean, this seem obvious, but maybe there is a trick to this question.</p>
| szw1710 | 130,298 | <p>One second ago there was a very interesting solution here: $$\binom{x}{k}=\frac{x(x-1)\cdot\ldots\cdot(x-k+1)}{k!}$$ for $x\in\Bbb R$ and $k\in\Bbb N$. The number of choices of $k$ distinct elements from $n$-element set is $\binom{n}{k}$ and $\binom{0}{20}=0$. I like this solution, but I am not the author. If it wil... |
478,713 | <p>I have this logic statement:</p>
<pre><code> (A and x) or (B and y) or (not (A and B) and z)
</code></pre>
<p>The problem is that accessing A and B are rather expensive. Therefore I'd like to access them only once each. I can do this with an if-then-else construct:</p>
<pre><code>if A then
if x then
tru... | G Tony Jacobs | 92,129 | <p>The x-intercept is the value that you can plug in for x and have f(x) come out to 0. Try evaluating f(3), and see what happens.</p>
<p>Since f(3)=0, that means the point (3,0) is on the graph, so it is an x-intercept.</p>
<p>Seeing the minus sign inside the parentheses can be confusing, because it makes you think ... |
511,150 | <p>I want to check if I understand proof by induction, so I want to proof the following:</p>
<p>$a^n<b^n$ for $a,b \in \mathbb{R}$, $0<a<b$, $n \in \mathbb{N}$ and $n>0$</p>
<p>Here's my attempt:</p>
<blockquote>
<p><strong>Base Case</strong></p>
<p>If $a=1$ and $b=2$, then $1^n < 2^n$ for any ... | Neal | 20,569 | <p>Since $a < b$, for any number $m>0$, $ma < mb$. In your case, $b^n\cdot a < b^n\cdot b$.</p>
<hr>
<p>I didn't read your question carefully enough. I assumed you were proving this:</p>
<blockquote>
<p>If $a,b\in\mathbb{R}$ and $a<b$, then for every $n\in\mathbb{N}$, $a^n<b^n$.</p>
</blockquote>... |
511,150 | <p>I want to check if I understand proof by induction, so I want to proof the following:</p>
<p>$a^n<b^n$ for $a,b \in \mathbb{R}$, $0<a<b$, $n \in \mathbb{N}$ and $n>0$</p>
<p>Here's my attempt:</p>
<blockquote>
<p><strong>Base Case</strong></p>
<p>If $a=1$ and $b=2$, then $1^n < 2^n$ for any ... | Siddhant Trivedi | 28,392 | <p>Now, suppose the result is true for $n=k$ then</p>
<p>$$a^k<b^k$$</p>
<p>we have to show that the result is true for $k+1$</p>
<p>$$a^k<b^k$$</p>
<p>$$a^k.a<\overbrace{b^k.a}^{3^k.2}$$</p>
<p>$$a^{k+1}<\underbrace{b^k.b}_{3^k.3}$$</p>
<p>because $a<b$ so,"a" is replaced by b on rhs</p>
<p>$$a^{... |
1,528,501 | <p>Let $P(n)$ be a property for all $n \geq 1$. For the phrase "there is some $N \geq 1$ such that $P(n)$ holds for all $n \geq N$" there are some suggestive, convenient abbreviations such as "$P(n)$ holds for large $n$" or "$P(n)$ holds eventually" and so on.</p>
<p>I wonder if there is in literature a like abbreviat... | bof | 111,012 | <p>From <a href="https://en.wikipedia.org/wiki/John_L._Kelley" rel="nofollow">John L. Kelley</a>'s <em>General Topology</em> (available at the <a href="https://archive.org/details/GeneralTopology" rel="nofollow">Internet Archive</a>), p. 65:</p>
<blockquote>
<p>A <strong>directed set</strong> is a pair $(D,\ge)$ suc... |
3,934,974 | <p>I'm wondering how to prove the associativity and identity to prove that the Möbius transformations forms a group.</p>
<p>A Möbius transformation is a complex function of the form <span class="math-container">$M(z)=\dfrac{az+b}{cz+d}$</span>.</p>
<p>Thank you in advance :)</p>
| Mo145 | 857,217 | <p>Since you are explicitly asking for the direct proof:</p>
<ol>
<li>concerning the associativity, this is just the same proof as for the associativity of any group of functions. So consider three moebius transforms <span class="math-container">$M_1,M_2,M_3$</span> then for any complex number <span class="math-contain... |
1,176,098 | <p>Here are some of my ideas:</p>
<p><strong>1. Addition Formula:</strong> <span class="math-container">$\sin{x}$</span> and <span class="math-container">$\cos{x}$</span> are the unique functions satisfying:</p>
<ul>
<li><p><span class="math-container">$\sin(x + y) = \sin x \cos y + \cos x \sin y $</span></p>
</li>
<li... | md2perpe | 168,433 | <p>Here's one way:</p>
<ol>
<li>Define some basic operations like integration and analytical continuation.</li>
<li>Set <span class="math-container">$\ln x=\int_1^x \frac{dt}{t}.$</span></li>
<li>Set <span class="math-container">$\exp=\ln^{-1}$</span> (the inverse function).</li>
<li>Extend <span class="math-container"... |
4,116,134 | <p>Find angle between <span class="math-container">$y=\sin x$</span> and <span class="math-container">$y=\cos x$</span> at their intersection point.</p>
<p>Intersection points are <span class="math-container">$\frac{\pi}{4}+\pi k$</span> and to find angle between them we need to compute derivatives at intersection poin... | Gary Moon | 477,460 | <p>If you think about these as parametrized curves <span class="math-container">$(t,\sin t)$</span> and <span class="math-container">$(t,\cos t)$</span>, then the angle between them at a point <span class="math-container">$t_0$</span> is the angle between <span class="math-container">$(1,\cos t_0)$</span> and <span cla... |
4,116,134 | <p>Find angle between <span class="math-container">$y=\sin x$</span> and <span class="math-container">$y=\cos x$</span> at their intersection point.</p>
<p>Intersection points are <span class="math-container">$\frac{\pi}{4}+\pi k$</span> and to find angle between them we need to compute derivatives at intersection poin... | Bread Zeppelin | 919,679 | <p>f(x): y=sin(x)
g(x): y=cos(x)
let m1 and m2 be the slopes of f(x) and g(x)
m1 = d(sin(x))/dx = cos(x)
m2 = d(cos(x))/dx = -sin(x)</p>
<p>We know that if k is the angle between two lines with slope ma and mb</p>
<p>Tan(k)=|(ma-mb)/(1+ma*mb)|</p>
<p>At the point of intersection, the angle between the curves is the sam... |
1,461,254 | <p>I am about to take an undergraduate course in Mathematical logic any textbooks to recommend.Want it to be rigorous and not missing things. I am an Math undergraduate. Got still 2 years for my degree. I have taken courses mostly in Algebra (ring theory etc.). Also taken Real analysis courses. So my mathematical matur... | Peter Smith | 35,151 | <p>Take a look at my <a href="http://www.logicmatters.net/resources/pdfs/TeachYourselfLogic2015.pdf">Teach Yourself Logic Study Guide</a>, which gives a lot of detailed advice about logic books at different levels, suitable to different backgrounds. </p>
<p>Check out the proposed syllabus of the course you are about t... |
3,155,229 | <p>Stumbled across this weird phenomenon using the equation <span class="math-container">$y = \frac{1}{x} $</span>.</p>
<p><strong>Surface Area:</strong>
When you calculate the surface area under the curve from 1 to <span class="math-container">$\infty$</span></p>
<p><span class="math-container">$$\int_1^\infty \fra... | Cye Waldman | 424,641 | <p>A simple way to visualize this is in terms of Pappus's <span class="math-container">$2^{nd}$</span> Centroid Theorem.</p>
<p>Pappus's <span class="math-container">$2^{nd}$</span> Centroid Theorem says the volume of a planar area of revolution is the product of the area <span class="math-container">$A$</span> and th... |
4,277,924 | <p><strong>1. p ∧ ¬q = T</strong>
<br><br>
<strong>2. (q ∧ p) → r = T</strong>
<br><br>
<strong>3.¬p → ¬r = T</strong>
<br><br>
<strong>4.(¬q ∧ p) → r = T</strong>
<br><br>
From <strong>Eq 1</strong>, we got <strong>p = T</strong> and <strong>q = F</strong>
<br>
Now Apply value of <strong>P</strong> in <strong>Eq 3</st... | James | 917,059 | <p>For a system to be consistent it must have <span class="math-container">$\textbf{one}$</span> outcome that is true, <span class="math-container">$T$</span>. For a system to be inconsistent it would no true values in the outcome, in other words is all false values, <span class="math-container">$F$</span>.</p>
<p>In y... |
1,794,855 | <p>I need to prove that for every three integers $(a,b,c)$, the $\gcd(a-b,b-c) = \gcd(a-b,a-c)$. Assuming that a $a \ne b$.</p>
<p>Having:</p>
<p>$d_1 = \gcd(a-b,b-c)$</p>
<p>$d_2 = \gcd(a-b,a-c)$</p>
<p>How do i prove $d_1 = d_2$?</p>
| Michael Hardy | 11,667 | <p>We have this sum:
$$
\sum_{k=0}^n k\frac{n!}{k!\,(n-k)!}p^k(1-p)^{n-k} \tag 1
$$
First notice that when $k=0$, the term $k\dfrac{n!}{k!\,(n-k)!}p^k(1-p)^{n-k}$ is $0$, and next notice that when $k\ne0$ then
$$
\frac k {k!} = \frac 1 {(k-1)!}
$$
so that
$$
k\frac{n!}{k!\,(n-k)!}p^k(1-p)^{n-k} = \frac{n!}{(k-1)!(n-k)!... |
124,060 | <p>I have a small dark object in the upper left corner of an <code>image</code>.</p>
<p>How can I separate it from the noisy rest and determine its <code>IntensityCentroid</code>?</p>
<p><a href="https://i.stack.imgur.com/D7pHv.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/D7pHv.png" alt="enter ima... | Wjx | 6,084 | <p>There is a way to simplify all these process:</p>
<pre><code>img=--your image here--;
ComponentMeasurements[
DeleteSmallComponents@
ColorNegate@
LocalAdaptiveBinarize[img, 50, {1, -2, -.01}], {"Centroid",
"EquivalentDiskRadius"}]
</code></pre>
<blockquote>
<p>{1 -> {{77.0441, 397.853... |
215,898 | <p>This is a quick follow up to my other <a href="https://mathematica.stackexchange.com/q/215884/44420">question</a> which I thought was different enough to warrant a separate post.</p>
<p><strong>My Question</strong></p>
<blockquote>
<p>How do you plot a region (such as <span class="math-container">$f(x,y)>z$</... | kglr | 125 | <h3>MeshFunctions + MeshShading</h3>
<pre><code>Plot3D[g[x, y], {x, 0, 2}, {y, 0, 2},
MeshFunctions -> {f[#, #2] &},
Mesh -> {{1, 3/2}},
MeshShading -> {White, Orange},
Lighting -> "Neutral"]
</code></pre>
<p><a href="https://i.stack.imgur.com/6JtZI.png" rel="nofollow noreferrer"><img s... |
267,051 | <p>Games appear in pure mathematics, for example, <a href="https://en.wikipedia.org/wiki/Ehrenfeucht%E2%80%93Fra%C3%AFss%C3%A9_game" rel="noreferrer">Ehrenfeucht–Fraïssé game</a> (in mathematical logic) and <a href="https://en.wikipedia.org/wiki/Banach%E2%80%93Mazur_game" rel="noreferrer">Banach–Mazur game</a> (in topo... | Alec Rhea | 92,164 | <p>I would nominate the following paper of Victoria Gitman and Joel David Hamkins for this purpose:</p>
<p><a href="https://arxiv.org/abs/1509.01099" rel="nofollow noreferrer">https://arxiv.org/abs/1509.01099</a>.</p>
<blockquote>
<p>The principle of open determinacy for class games---two-player games of perfect in... |
361,171 | <p>If I am about to fabricate a bracelet and I can select $24$ pearls out of a total of $500$ pearls ($300$ white, $150$ red and $50$ green) how many possible bracelets can I create?</p>
<p>The (official) solution to this question is
$$ \frac{500!}{476!} = 3.4\cdot10^{64} $$
But how can it be that the colors of the p... | Douglas S. Stones | 139 | <p>Mathematically, <a href="http://en.wikipedia.org/wiki/Necklace_%28combinatorics%29" rel="nofollow">bracelets</a> are defined inequivalent under rotations and reflections (i.e., the <a href="http://en.wikipedia.org/wiki/Dihedral_group" rel="nofollow">dihedral group</a> action). Moreover, e.g. one red bead should be ... |
3,697,062 | <p>How many solutions are there for this equation <span class="math-container">$$x_1+x_2+x_3+x_4 = 49\,,$$</span> where <span class="math-container">$x_i,\;i= 1,2,3,4$</span> is a non negative integer such that:<span class="math-container">$$ 1\le x_1\le 8,\;3\le x_2\le 9,\;10\le x_3\le 20,\ 0\le x_4\,?$$</span></p>
| Jan Eerland | 226,665 | <p>Well, just a quick search with Mathematica gives:</p>
<pre><code>In[1]:=Length[Solve[{x1 + x2 + x3 + x4 == 49,
1 <= x1 <= 8 && 3 <= x2 <= 9 && 10 <= x3 <= 20 && 0 <= x4}, {x1,
x2, x3, x4}, Integers]]
Out[1]=616
</code></pre>
<p>So, there are <span class="math-c... |
4,416,001 | <p>Given <span class="math-container">$f(xy) = f(x+y)$</span> and <span class="math-container">$f(11) = 11$</span>, what is <span class="math-container">$f(49)$</span>?</p>
| MangoPizza | 956,791 | <p><span class="math-container">$f(xy) = f(x + y)$</span> implies <span class="math-container">$f(x) = f(x \cdot 1) = f(x + 1)$</span>. So, <span class="math-container">$11 = f(11) = f(12) = \cdots = f(49)$</span></p>
|
4,047,784 | <p>Let <span class="math-container">$f_{avg}$</span> be the average value of a continuous function f(x) on the interval [a,b]. Assume that there's only one c <span class="math-container">$\in$</span> [a , b] for which <span class="math-container">$f(c)=f_{avg}$</span>. The question is, if this is the case, then must th... | Kenny Lau | 328,173 | <p>I am not sure what you want. You have essentially specified the answer to be a root of a certain polynomial equation.</p>
<p>If you want to express the solution in terms of radicals (i.e. <span class="math-container">$n$</span><sup>th</sup> roots) like <span class="math-container">$\phi = \frac{1+\sqrt5}2$</span> th... |
4,047,784 | <p>Let <span class="math-container">$f_{avg}$</span> be the average value of a continuous function f(x) on the interval [a,b]. Assume that there's only one c <span class="math-container">$\in$</span> [a , b] for which <span class="math-container">$f(c)=f_{avg}$</span>. The question is, if this is the case, then must th... | Cye Waldman | 424,641 | <p>The sequence that you are looking at is similar to the Fibonacci-Narayana sequence, which is given by <span class="math-container">$f_n=f_{n-1}+f_{n-1-m}, m\ge 1$</span>. The generating function is given by</p>
<p><span class="math-container">$$
\frac{1}{1-x-x^{n+1}}
$$</span></p>
<p>and the limiting ratio is given ... |
3,213,854 | <p>From the definition of a inverse standpoint (<span class="math-container">$f^{-1}(f(x))=f(f^{-1}(x))=x$</span>), why does interchanging variables (<span class="math-container">$x$</span> and <span class="math-container">$y$</span>) work to find the inverse? It seems logical to me but I cannot come up with a coherent... | 雨が好きな人 | 438,600 | <p>When we find the inverse in this way, we are not really ‘interchanging variables’. We’re just rearranging to find the input in terms of the output (when usually we have the output defined explicitly in terms of the input), and then relabelling to make it clearer. It just happens that we usually use <span class="math... |
3,146,161 | <blockquote>
<p>Find particular solution of <span class="math-container">$\dfrac{dx}{dy} +x\cot y =y\cot y$</span> given <span class="math-container">$x=0$</span> when <span class="math-container">$y= π/2$</span>.</p>
</blockquote>
<p>Please help me figure this out
I can't separate them out in order to integrate</p... | Lucas | 444,015 | <p>The function <span class="math-container">$e^{x^{2}}$</span> is continuous increasing on <span class="math-container">$[0, \infty]$</span>. Take
<span class="math-container">$$F(t) = \int_{0}^{t}e^{x^{2}}dx.$$</span>
In each compact interval <span class="math-container">$[0,a]$</span>, <span class="math-container">$... |
1,683,414 | <p>$$\begin{cases}
x^2 = yz + 1 \\
y^2 = xz + 2 \\
z^2 = xy + 4
\end{cases}
$$</p>
<p>How to solve above system of equations in real numbers? I have multiplied all the equations by 2 and added them, then got $(x - y)^2 + (y - z)^2 + (x - z)^2 = 14$, but it leads to nowhere.</p>
| Ross Millikan | 1,827 | <p>Now I would note that $(\pm 3)^2+(\pm 2)^2+(\pm 1)^2=14$. Since $(x-y)+(y-z)=(x-z)$ we can't assign the signs and values arbitrarily. For example, from $x-y=1, y-z=2, x-z=3$ I get $x=1,y=0,z=-2$ and from $x-y=-1, y-z=-2, x-z=-3$ I get $x=-1,y=0,z=2$ The other possibilities do not give solutions.</p>
|
1,575,107 | <p>$$L^{-1}\frac{4s}{(s-6)^{3}}$$
$$4L^{-1}\frac{s}{s^{3}}|s=s-6$$
$$4L^{-1}\frac{1}{s^{2}}|s=s-6$$
$$4L^{-1}\frac{1!}{s^{1+1}}|s=s-6$$
$$4te^{6t}$$</p>
<p>Is this correct? symbolab and Wolfram are giving me different answers...</p>
| Ron Gordon | 53,268 | <p>The ILT may be computed via the residue theorem as follows:</p>
<p>$$\frac{4}{2!} \frac{d^2}{ds^2} \left [s e^{s t} \right ]_{s=6} = 2 \frac{d}{ds} \left [(1+s t) e^{s t} \right ]_{s=6} = 2 \left [(t (1+s t) + t) e^{s t} \right ]_{s=6} = (4 t+12 t^2) e^{6 t}$$</p>
<p>You seem to be missing the $t^2$ term.</p>
|
815,739 | <p>Let $f :\mathbb R\to \mathbb R$ be a continuous function such that $f(x+1)=f(x) , \forall x\in \mathbb R$ i.e. $f$ is of period $1$ , then how to prove that $f$ is uniformly continuous on $\mathbb R$ ?</p>
| Lucian | 93,448 | <p>Since $\mathbb R^a\times\mathbb R^b\cong\mathbb R^{a+b}$, it would follow that $\mathbb R^a\times\mathbb R^0\cong\mathbb R^a$, and therefore $\mathbb R^0\cong\{\varnothing\}$. But I have to admit I never actually saw this notation being used anywhere. As for $\mathbb R^{-n}$, we would have $\mathbb R^n\times$ $\time... |
1,116,435 | <p>How do I get the value of </p>
<p>$$\lim_{x \rightarrow \frac{1}{4} \pi } \frac{\tan x-\cot x}{x-\frac{1}{4} \pi }?$$ </p>
<p>I need the steps without using L'hospital.</p>
| Math-fun | 195,344 | <p>Firs note that $ \displaystyle \tan x - \cot x = - 2 \frac{\cos 2x}{\sin 2x}$. Then consider a change of variable $\displaystyle u=x-\frac{\pi}{4}$ to obtain </p>
<p>$$ \lim_{x \rightarrow \frac{1}{4} \pi } \frac{\tan x-\cot x}{x-\frac{1}{4} \pi }= \lim_{u \rightarrow 0 } [\frac{\sin 2u}{u}\times \frac{2}{\cos... |
1,027,235 | <p>Let N = 12345678910111213141516171819. How can I use modular arithmetic to show that N is (or isn't) divisible by 11? In general, how can I apply modular arithmetic to determine the divisibility of an integer by a smaller integer? I am finding modular arithmetic very confusing and unintuitive. I can understand "simp... | Mike Earnest | 177,399 | <p>First, since $10\equiv -1$ (mod $11$), notice that $10^k\equiv (-1)^k$ (mod $11$). Then, given your number, which can be written more generally as
$$
d_n10^n+d_{n-1}10^{n-1}+\dots+d_1\cdot10+d_0,
$$
consider what the above expression is (mod $11$). When the above is looked at "with mod 11 goggles on," all of the ins... |
3,305,677 | <p>I am learning Asymptotic complexity of functions from CLRS. I know that exponentiation functions like <span class="math-container">$a^n$</span>,<span class="math-container">$(a>0)$</span> are faster than <span class="math-container">$n!$</span> But what about <span class="math-container">$a^{a^n}$</span> vs <span... | epi163sqrt | 132,007 | <blockquote>
<p>Applying <span class="math-container">$\ln\circ\ln$</span> we obtain
<span class="math-container">\begin{align*}
\ln\left(\ln\left(a^{a^n}\right)\right)
&=\ln\left(a^n\ln a\right)\\
&=\ln a^n+\ln\left(\ln\left(a\right)\right)\\
&=n\ln a+\ln\left(\ln\left(a\right)\right)\\
&\si... |
1,590,817 | <p>I was hoping someone could explain to me how to prove a sequence is Cauchy. I've been given two definitions of a Cauchy sequence:</p>
<p>$\forall \epsilon > 0, \exists N \in \mathbb{N}$ such that $n,m> N$ $\Rightarrow |a_n - a_m| ≤ \epsilon$</p>
<p>and equivalently $\forall \epsilon > 0, \exists N \in \ma... | Ian Cavey | 293,726 | <p>As an easier example of how to apply the definition of a Cauchy sequence, define the sequence $\{\frac{1}{n}\}$. Given any $\epsilon>0$, you would like to find an $N$ such that for any $n,m>N$, $\left|\frac{1}{n}-\frac{1}{m}\right|<\epsilon$. This would certainly be the case if $\frac{1}{n},\frac{1}{m}<\... |
2,094,243 | <p>Given the norm $||(x,y)|| = 2|x| +\frac{1}{3}|y|$.
Sketch the open ball at the on the origin $(0,0)$, and radius $1$.</p>
<p>I understand that the sketch of an open ball withina set looks like the image attached, <a href="https://i.stack.imgur.com/j9HN4.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur... | Mercy King | 23,304 | <p>The open ball centered at the origin, with radius $1$, is the region containing the center $(0,0)$ and bounded by the line-segments
\begin{eqnarray}
2x+\dfrac13y&=&1, 0 \le x\le 0.5\\
-2x+\dfrac13y&=&1, -0.5\le x\le 0\\
-2x-\dfrac13y&=&1, -0.5\le x\le 0\\
2x-\dfrac13y&=&1, 0\le x\le 0... |
743,819 | <p>I'm studying for my exam and I came across the following draw without replacement problem :</p>
<blockquote>
<p><span class="math-container">$N$</span> boxes filled with red and green balls.
The box <span class="math-container">$r$</span> contains <span class="math-container">$r-1$</span> red balls and <span class=... | Jared | 138,018 | <p>All of the boxes contain $N - 1$ balls. This is just a complicated conditional probability problem. Lets look at a single box with $r$ red balls and $g$ green balls. What would the probability be of getting green on the second? Well it depends on whether or not you draw a red or green first. If you draw a red f... |
1,521,649 | <p>I need to show that at most finitely many terms of this sequence are greater than or equal to $c$. </p>
<p>I don't know if it is the wording of the problem but I don't know what this is asking me to do. Help on this would be amazing! And thank you in advance.</p>
| John Douma | 69,810 | <p>$b\ln a=b\int_1^a\frac{1}{x}dx=\int_1^a\frac{b}{x}dx$</p>
<p>Let $u=x^b$</p>
<p>Then $du= bx^{b-1}dx$ and $\frac{du}{u}=\frac{b}{x}dx$</p>
<p>Therefore, $\int_1^a\frac{b}{x}dx = \int_1^{a^b}\frac{du}{u}=\ln {a^b}$</p>
|
1,521,649 | <p>I need to show that at most finitely many terms of this sequence are greater than or equal to $c$. </p>
<p>I don't know if it is the wording of the problem but I don't know what this is asking me to do. Help on this would be amazing! And thank you in advance.</p>
| BLAZE | 144,533 | <p>Assuming knowledge of </p>
<p>$(a^m)^n=a^{mn}\tag{1}$</p>
<p>Now let $a^m=b$ and $a^n = c$</p>
<p>Then from $(1)$</p>
<p>we have $b^n=(a^m)^n=a^{mn}$</p>
<p>so $\log_a(b^n)=mn$ </p>
<p>therefore $n\log_a b=\log_a(b^n)$</p>
<p>Hence </p>
<p>$\color{blue}{\fbox{$\log_a (b^n) = n\log_a b$}}$</p>
<p>I did this ... |
1,710,517 | <blockquote>
<p>Show that 13 is the largest prime which divides two consecutive terms of $n^2 + 3$.</p>
</blockquote>
<p>The integers are $39$ and $52$. First of all, I set the variable for the number as $k$. So, $k|n^2 +3$ and $k|n^2 + 2n+ 4$ which imply that $k|2n+1$. $n=6$ over here. And the fact that 13 is the l... | lulu | 252,071 | <p>Let $a_n=n^2+3$. If $p|a_n$ then $n^2=-3\pmod p$. </p>
<p>A quick calculation shows that $a_{n+1}-a_n=2n+1$ so if $p$ also divides $a_{n+1}$ we must have $2n=-1\pmod p$. Square this to see that it implies $4n^2=1\implies -12=1 \pmod p$. Thus, $p|13$ and we are done.</p>
|
453,212 | <p>Consider a number Q in a made up base system:</p>
<p>The base system is as follows:</p>
<p>It encodes a number as a sum of odd numbers:</p>
<p>1 3 5 7 9 ...</p>
<p>If the number can be expressed as a sum of unique odds. For example, the number 16 is expressed as:</p>
<p>1110 = 7 + 5 + 3 + 1</p>
<p>The system i... | Zev Chonoles | 264 | <p>If $u$ is odd, write $u$ as the following sum of odd numbers:
$$u$$
If $u$ is even (and $\geq 2$), write $u$ as the following sum of odd numbers:
$$(u-1)+1$$
(Converting into a string of ones and zeros is straightforward.)</p>
|
1,479,822 | <p>$f\cdot g$ is Lebesgue integrable, g is Lebesgue integrable, can we deduce that f is Lebesgue integrable? </p>
<p>$f\cdot g$ is integrable, g is integrable, can we deduce that f is finite a.e.? </p>
| Silvia Ghinassi | 258,310 | <p>For a nontrivial example, take $f=\frac{1}{x}\chi_{[1,+\infty)}$ and $g=\frac{1}{x^2}\chi_{[1,+\infty)}$. Then $fg$ is integrable, $g$ is integrable, but $f$ is not.</p>
<p>As for the second question, in general you can't conclude anything, as pointed out already.</p>
|
162,611 | <p>I am working with the square-roots of square symmetric matrices. The answers are to be binary symmetric matrices.</p>
<p>If we take the matrix $$M = \begin{pmatrix}1&1&1&0&0&0&0\\1&0&0&1&1&0&0\\1&0&0&0&0&1&1\\0&1&0&0&1&0&... | Will Jagy | 10,400 | <p>I believe what you have found is what happens when you take the Fano plane, that is a finite projective plane of "order" 2, and write it as a bipartite graph on 7 plus 7 vertices. I think it is an artifact of small dimension that you are able to have your "binary" matrix also be symmetric. There is a projective plan... |
28,456 | <p>I built a <code>Graph</code> based on the permutations of city's connections from :</p>
<pre><code>largUSCities =
Select[CityData[{All, "USA"}], CityData[#, "Population"] > 600000 &];
uScityCoords = CityData[#, "Coordinates"] & /@ largUSCities;
Graph[#[[1]] -> #[[2]] & /@ Permutations[largUSCitie... | cormullion | 61 | <h2>Revised answer</h2>
<p>This uses the connectivity between states to create the graph, and uses the coordinates of the center of each state rather than the cities. I couldn't find a way to get these easily from <em>Mathematica</em> or from <em>WolframAlpha</em> (I'm no Harry Potter, and failed to discover the correc... |
28,456 | <p>I built a <code>Graph</code> based on the permutations of city's connections from :</p>
<pre><code>largUSCities =
Select[CityData[{All, "USA"}], CityData[#, "Population"] > 600000 &];
uScityCoords = CityData[#, "Coordinates"] & /@ largUSCities;
Graph[#[[1]] -> #[[2]] & /@ Permutations[largUSCitie... | PlatoManiac | 240 | <p>Another trivial way to keep the cities connected and at the same time reduce the number of edges in the graph.</p>
<pre><code><< ComputationalGeometry`;
rule = MapThread[#1 -> #2 &, {uScityCoords, largUSCities}];
graph = (Rest@PlanarGraphPlot[uScityCoords][[1, 2]] /.
Line[{a_, b_}] -> UndirectedEd... |
4,263,631 | <p>so i have question about existence of function <span class="math-container">$f:\mathbb{R} \to \mathbb{R}$</span> such <span class="math-container">$f$</span> is not the pointwise limit of a sequence of continuous functions <span class="math-container">$\mathbb{R} \to \mathbb{R}$</span>.</p>
<p>i'm created a family o... | Lázaro Albuquerque | 85,896 | <p>Consider the cover <span class="math-container">$\cup [k, k+1] \times [0, \frac{1}{k^2}]$</span>. The sum of these rectangle's measures is <span class="math-container">$\sum \frac{1}{k^2} < \infty$</span>.</p>
|
839,431 | <p>Can someone be kind enough to show me the steps to integrate this, I'm sure it's by parts but how do I split up the sin part?
$$x\sin(1+2x)$$</p>
| Anurag A | 68,092 | <p>Let $u=x$ and $dv =\sin (1+2x) \, dx$
Then $v=-\frac{1}{2}\cos(1+2x)$. Now perform integration by parts.</p>
|
253,746 | <p>I am currently aware of the following two versions of the global Cauchy Theorem. Which one is stronger?</p>
<p>1.)If the region $U$ is simply connected, then for every closed curve contained therein, the integral of the holomorphic function $f$ defined on $U$ over the curve is zero.</p>
<p>2.) If the domain $D$ is... | Christian Blatter | 1,303 | <p>If $\Omega$ is simply connected then all closed curves in $\Omega$ have index $0$ with respect to points outside $\Omega$. Therefore the first statement follows from the second.</p>
<p>The second statement actually is the strongest form of Cauchy's theorem in the plane.</p>
|
13,989 | <p>Suppose $E_1$ and $E_2$ are elliptic curves defined over $\mathbb{Q}$.
Now we know that both curves are isomorphic over $\mathbb{C}$ iff
they have the same $j$-invariant.</p>
<p>But $E_1$ and $E_2$ could also be isomorphic over a subfield of $\mathbb{C}$.
As is the case for $E$ and its quadratic twist $E_d$. Now th... | JSE | 431 | <p>Seconding the pointer to Silverman's book. Once you learn the basic definitions of Galois cohomology, you can check as an exercise that the set of isomorphism classes in your second question, when K is a Galois extension of Q, is in natural bijection with</p>
<p>H^1(Gal(K/Q), Aut(E)(K)).</p>
|
3,540,045 | <p>The definite integral, <span class="math-container">$$\int_0^{\pi}3\sin^2t\cos^4t\:dt$$</span></p>
<p><strong>My question</strong>: for the trigonometric integral above the answer is <span class="math-container">$\frac{3\pi}{16}$</span>. What I want to know is how can I compute these integrals easily. Is there more... | Community | -1 | <p>Use walli's formula given here to do it faster <a href="https://math.stackexchange.com/questions/2833731/reduction-formula-for-integral-sinm-x-cosn-x-with-limits-0-to-pi-2">Reduction formula for integral $\sin^m x \cos^n x$ with limits $0$ to $\pi/2$</a></p>
<p><span class="math-container">$I=3\displaystyle\int_{0}... |
2,292,096 | <p>I'm learning inequalities for the first time, and except a paragraph by Paul Zeitz in his book Art and Craft of problem solving, none actually give much motivation of why should I care about inequalities.</p>
<p>The example given by Paul Zeitz was that to prove $b^2-b+1$ is never a perfect square for integer $b$. W... | Somos | 438,089 | <p>The pigeonhole principle depends on an inequality and is nontrivial because it is so useful. Another is $x^2 \ge 0$ for $x$ real which can be used to prove some polynomials have no real roots.</p>
|
1,788,435 | <p>My math book says, a Linear equation has exactly one solution. Because $ax + b = 0$; $x =-\frac{b}{a}$. But I've solved many linear equations with multiple solutions before.
(I'm not very good in math. Need help...)</p>
| Jared | 138,018 | <p>If we assume that <em>all</em> linear equations have the form:</p>
<p>$$
ax + b= 0
$$</p>
<p>(which is completely valid and <em>should</em> be how we view linear equations)</p>
<p>then linear equations have either 1, 0, or infinite solutions. It's quite simple if $a \neq 0$ then they have <em>exactly</em> one so... |
3,878,380 | <p>Do the columns of a matrix always represent different vectors? If so, I don't understand how if I have a <span class="math-container">$3\times3$</span> matrix where the rows represent the dimensions and I multiply it by a <span class="math-container">$3\times1$</span> column vector with the same dimensions, it will ... | Laars Helenius | 112,790 | <p>Suppose <span class="math-container">$\mathbb{R}^m$</span> and <span class="math-container">$\mathbb{R}^n$</span> are vector spaces and there is a linear transformation <span class="math-container">$f: \mathbb{R}^m \to \mathbb{R}^n$</span>. Then <span class="math-container">$f$</span> can be represented by a <span c... |
3,656,978 | <p>I understand to say that a bounded linear operator <span class="math-container">$T$</span> is called "polynomially compact" if there is a nonzero polynomial <span class="math-container">$p$</span> such that <span class="math-container">$p(T)$</span> is compact. </p>
<p>Can anyone help me with examples of polynomial... | Aditya Dev | 746,162 | <p>The error is in the <span class="math-container">$2^{nd}$</span> step of your calculation. Matrix multiplication is <a href="https://en.wikipedia.org/wiki/Matrix_multiplication#Non-commutativity" rel="nofollow noreferrer">Non- Commutative</a> (i.e. <span class="math-container">$ \mathbf {A} \mathbf {B} \not = \mathb... |
1,132,599 | <p>For which prime numbers p does the congruence $x^2+x+1\equiv0$ mod p have solutions? </p>
<p>I am new to the topic of quadratic reciprocity and I know how to answer this question had it been for which prime numbers p does the congruence $x^2\equiv-6$ mod p have solutions? </p>
<p>Can I perhaps split the congruence... | marwalix | 441 | <p>If $p\neq 2$ we have equivalence with $4(x^2+x+1)\equiv 0 \pmod p$. We ca now complete the square $(2x+1)^2\equiv {-3} \pmod p$ this means $-3$ is a quadratic residue.
For $p=2$ we have $\forall x, x^2+x+1\equiv 1\pmod 2$</p>
|
739,516 | <p>Is there a closed form for $$\int_{0}^{\infty}e^{ax^3+bx^2}\,\mathrm{d}x $$?</p>
| Keshav Srinivasan | 71,829 | <p>Here is what WolframAlpha gives me, just for $a=-1$:</p>
<p>$$\begin{align*}
\int^\infty_0 e^{-x^3+bx^2}\,dx=-\frac1{27}b\,\left(2\sqrt{3}\pi e^{\tfrac{2b^3}{27}}\left(I_{-\frac13}\left(-\dfrac{2b^3}{27}\right)+I_{\frac13}\left(-\dfrac{2b^3}{27}\right)\right)-9_2F_2\left(\dfrac{1}{2},1;\dfrac23,\dfrac43;\dfrac{4b^3... |
2,506,182 | <p>The question speaks for itself, since the question comes from a contest environment, one where the use of calculators is obviously not allowed, can anyone perhaps supply me with an easy way to calculate the first of last digit of such situations?</p>
<p>My intuition said that I can look at the cases of $3$ with an ... | D.R. | 405,572 | <p>The last digits of 3 repeat every 4 powers: $3^1=3$, $3^2=9$, $3^3=27$, $3^4=81$, and because the last digit of $3^4$ is one, multiplying by three just gives us three again, and the pattern continues.</p>
<p>So taking $2011 \mod 4$, we get $3 \mod 4$, so the last digit is the same as that of $3^3=27$, which is 7.</... |
2,506,182 | <p>The question speaks for itself, since the question comes from a contest environment, one where the use of calculators is obviously not allowed, can anyone perhaps supply me with an easy way to calculate the first of last digit of such situations?</p>
<p>My intuition said that I can look at the cases of $3$ with an ... | Roman83 | 309,360 | <p>$3^1$: 3 is last digit </p>
<p>$3^2$: 9 is last digit </p>
<p>$3^3$: 7 is last digit </p>
<p>$3^4$: 1 is last digit </p>
<p>$3^5$: 3 is last digit </p>
<p>$3^6$: 9 is last digit </p>
<p>$3^7$: 7 is last digit </p>
<p>$3^8$: 1 is last digit </p>
<p>$(3,9,7,1)$ is period.</p>
<p>$$3^{2011}=3^{502\cdot4+3}$$
T... |
369,589 | <p>I am not a math student, and only kind of picking up something whenever I need it. After emerged in the field of machine learning, probability, measure theory and functional analysis seem to be quite intriguing. I am considering learning stochastic calculus myself, but do not quite know what kind of prerequisites sh... | John Doe | 200,484 | <p>To gain a working knowledge of stochastic calculus, you don't need all that functional analysis/ measure theory. What you need is a good foundation in probability, an understanding of stochastic processes (basic ones [markov chains, queues, renewals], what they are, what they look like, applications, markov properti... |
104,592 | <p>Hello everyone,</p>
<p>Suppose that I am defining a function which embeds a surface (manifold) in $\mathbb{R}^3$.</p>
<p>Is there a standard symbol or letter that is used for this function?</p>
<p>Additionally, is there any other classical or standard notation (such as the hooked arrow for inclusion maps) of whic... | Dox | 25,356 | <p>No, there is none.</p>
|
72,478 | <p>I am trying to find the intervals on which f is increasing or decreasing, local min and max, and concavity and inflextion points for $f(x)=\sin x+\cos x$ on the interval $[0,\pi]$.</p>
<p>I know at $\pi/4$ the derivative will equal zero. So that gives me my critical numbers, positive and negative $\pi/4$ so now I n... | davidlowryduda | 9,754 | <p>In short, $\pi / 4$ is not the only critical value. In general, we know $\sin$ and $\cos$ are periodic, so we expect infinitely many critical values. We also know that if $\sin a = 0$, then $\sin (a + \pi) = 0$.</p>
<p>So when you take the derivative, you get $f'(x) = \cos x - \sin x$. You look for critical poi... |
3,784,484 | <p>Is there a bounded linear operator <span class="math-container">$T \in l^2(\mathbb{N})$</span> have for the essential spectrum the unit disk of <span class="math-container">$\mathbb{C}$</span>; i.e, such that <span class="math-container">$\sigma_{e}(T)=\textbf{D}(0, 1)$</span>; where <span class="math-container">$\... | Martin Argerami | 22,857 | <p>Given any compact <span class="math-container">$K\subset \mathbb C$</span>, there exists <span class="math-container">$T\in B(\ell^2(\mathbb N))$</span> with <span class="math-container">$\sigma_e(T)=K$</span>.</p>
<p>Let <span class="math-container">$\{P_n\}$</span> be a pairwise orthogonal sequence of infinite pro... |
1,725,945 | <p>I'm reading the proof of "Fundamental Theorem of Finite Abelian Groups" in Herstein Abstract Algebra, and I've found this statement in the proof that I don't see very clear.</p>
<p>Let $A$ be a normal subgroup of $G$. And suppose $b\in G$ and the order of $b$ is prime $p$, and $b$ is not in $A$. Then $A \cap (b)=(e... | David C. Ullrich | 248,223 | <p>If $x\in A\cap (b)$ then $x=b^n$ for some $n$. If the order of $b$ is $p$ and $x\ne e$ then we can take $0<n<p$. Since $p$ is prime there exists $m$ so that $b^{nm}=b$; hence $b\in A$.</p>
|
2,075,485 | <p>Let $[a,b]$ be a finite closed interval on $\mathbb{R}$, $f$ be a continuous differentible function on $[a,b]$. Prove that
$$\max_{x\in [a,b]} |f(x)|\le \Bigg|\frac{1}{b-a} \int_a^b f(x)dx\Bigg|+\int_a^b |f'(x)|dx$$</p>
<p>I think this is similar to Sobolev embedding theorem but have no idea about how to use it. I... | kobe | 190,421 | <p>There is a point $c\in (a,b)$ such that </p>
<p>$$f(c) = \frac{1}{b-a}\int_a^b f(x)\, dx$$</p>
<p>Let $x\in [a,b]$. Since </p>
<p>$$f(x) = f(c) + \int_c^x f'(t)\, dt$$</p>
<p>then </p>
<p>$$\lvert f(x)\rvert \le \lvert f(c)\rvert + \int_a^b \lvert f'(t)\rvert\, dt$$</p>
<p>Now finish the argument.</p>
|
736,749 | <p>Let us consider the Fourier transform of <span class="math-container">$\mathrm{sinc}$</span> function. As I know it is equal to a rectangular function in frequency domain and I want to get it myself, I know there is a lot of material about this, but I want to learn it by myself. We have <span class="math-container">... | Alijah Ahmed | 124,032 | <p>In terms of deriving the Fourier Transform, I will make some use of techniques highlighted in <a href="http://www.claysturner.com/dsp/FTofSync.pdf" rel="noreferrer">http://www.claysturner.com/dsp/FTofSync.pdf</a></p>
<p>Let us start with your expression</p>
<p>$$\int_{-\infty}^{\infty}\frac{\sin(\omega_0t)\cos(\om... |
736,749 | <p>Let us consider the Fourier transform of <span class="math-container">$\mathrm{sinc}$</span> function. As I know it is equal to a rectangular function in frequency domain and I want to get it myself, I know there is a lot of material about this, but I want to learn it by myself. We have <span class="math-container">... | Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}
\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\down}{\... |
2,181,540 | <p>What approach we have to solve the following differential equation?</p>
<p>$$y''(x)- \frac{y'(x)^2}{y} + \frac{y'(x)}{x}=0$$</p>
<p>The known solution is $y(x) = c_2 * x^{c_1}$</p>
| Claude Leibovici | 82,404 | <p><strong>Hint</strong></p>
<p>Let $y=e^z$ to make the equation $$\frac{e^{z} \left(x z''+z'\right)}{x}=0$$ This reduces to $$x z''+z'=0$$ Reduce the order $z'=t$ and continue.</p>
<p>I am sure that you can take it from here.</p>
|
2,181,540 | <p>What approach we have to solve the following differential equation?</p>
<p>$$y''(x)- \frac{y'(x)^2}{y} + \frac{y'(x)}{x}=0$$</p>
<p>The known solution is $y(x) = c_2 * x^{c_1}$</p>
| Lutz Lehmann | 115,115 | <p>In principle the same reduction as the other answer, but the other way around:</p>
<p>Substitute $u=\frac{y'}{y}=(\log|y|)'$ to find $u'=\frac1y(y''-\frac{y'^2}y)$ and consequently
$$
u'+\frac{u}x=0\iff (ux)'=0.
$$</p>
|
2,953,757 | <p>My try to solve this question goes as follows:</p>
<p><span class="math-container">$g=gcd(n^2+1, (n+1^2)+1) = gcd(n^2+1, 2n+1) = gcd(n^2-2n, 2n+1)$</span>.</p>
<p>By long division: </p>
<p><span class="math-container">$$n^2-2n = -2n(2n+1) + 5n^2$$</span></p>
<p>Since <span class="math-container">$g$</span> divid... | Hagen von Eitzen | 39,174 | <p>Clearly <span class="math-container">$$\gcd(n^ 2+1,(n+1)^2+1)\mid \bigl((n+1)^2+1\bigr)-\bigl(n^ 2+1\bigr)=2n+1,$$</span>
hence also
<span class="math-container">$$\gcd(n^ 2+1,(n+1)^2+1)\mid (2n+1)n-2(n^2+1)=n-2 $$</span>
as well as
<span class="math-container">$$\gcd(n^ 2+1,(n+1)^2+1)\mid (2n+1)n-2(n^2+1)=2n+1-2(... |
2,953,757 | <p>My try to solve this question goes as follows:</p>
<p><span class="math-container">$g=gcd(n^2+1, (n+1^2)+1) = gcd(n^2+1, 2n+1) = gcd(n^2-2n, 2n+1)$</span>.</p>
<p>By long division: </p>
<p><span class="math-container">$$n^2-2n = -2n(2n+1) + 5n^2$$</span></p>
<p>Since <span class="math-container">$g$</span> divid... | Barry Cipra | 86,747 | <p>Since <span class="math-container">$\gcd(n,2n+1)=\gcd(n,1)=1$</span>, you can keep going:</p>
<p><span class="math-container">$$\gcd(n^2-2n,2n+1)=\gcd(n(n-2),2n+1)=\gcd(n-2,2n+1)=\gcd(n-2,5)\in\{1,5\}$$</span></p>
<p>The key reduction here is the gcd property <span class="math-container">$\gcd(a,b)=1\implies\gcd(a... |
1,440,106 | <p>I am currently studying how to prove the Fibonacci Identity by Simple Induction, shown <a href="http://mathforum.org/library/drmath/view/52718.html">here</a>, however I do not understand how $-(-1)^n$ becomes $(-1)^{n+1}$. Can anybody explain to me the logic behind this?</p>
| Robert Soupe | 149,436 | <p>The powers of $-1$ are $$-1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, \ldots$$ If $n$ is odd, then $(-1)^n = -1$, but if $n$ is even then $(-1)^n = 1$. So if you multiply $(-1)^n$ once by $-1$, it is the same as incrementing $n$. (Technically you can also decrement, but you can't generalize that to other negativ... |
3,098,245 | <p><span class="math-container">$$ \frac{d}{dx}e^x =\frac{d}{dx} \sum_{n=0}^{ \infty} \frac{x^n}{n!}$$</span>
<span class="math-container">$$ \sum_{n=0}^{ \infty} \frac{nx^{n-1}}{n!}$$</span>
<span class="math-container">$$ \sum_{n=0}^{ \infty} \frac{x^{n-1}}{(n-1)!}$$</span>
This isn't as straightforward as I thought ... | Hyperion | 636,641 | <p>Rewrite the infinite sum as
<span class="math-container">$$\sum_{n=0}^{\infty}\frac{x^n}{n!} = 1 + \sum_{n=1}^{\infty}\frac{x^n}{n!}$$</span>
Then
<span class="math-container">$$\frac{d}{dx} (1 + \sum_{n=1}^{\infty}\frac{x^n}{n!}) = \sum_{n=1}^{\infty}\frac{nx^{n-1}}{n!} = \sum_{n=1}^{\infty}\frac{x^{n-1}}{(n-1)!}=... |
2,230,897 | <blockquote>
<p>A line passing through $P=(\sqrt3,0)$ and making an angle of $\pi/3$ with the positive direction of x axis cuts the parabola $y^2=x+2$ at A and B, then:<br>
(a)$PA+PB=2/3$<br>
(b)$|PA-PB|=2/3$<br>
(c)$(PA)(PB)=\frac{4(2+\sqrt3)}{3}$<br>
(d)$\frac{1}{PA}+\frac{1}{PB}=\frac{2-\sqrt3}{2}$ </p>... | mathlove | 78,967 | <p>Let $C,D$ be the point on $x$-axis such that $AC\perp CP,BD\perp PD$ respectively.</p>
<p>Then, $\triangle{PAC},\triangle{PBD}$ are triangles with $30^\circ,60^\circ,90^\circ$ from which we have
$$PA=\frac{2}{\sqrt 3}AC=\frac{2}{\sqrt{3}}|A_y|,\qquad PB=\frac{2}{\sqrt 3}|B_y|$$</p>
<p>Eliminating $x$ from the syst... |
3,596,514 | <p>Let <span class="math-container">$0\le x \le 1$</span>
find the maximum value of <span class="math-container">$x(9\sqrt{1+x^2}+13\sqrt{1-x^2})$</span></p>
<p>I try to use am-gm inequality to solve this because it's similar to CMIMC2020 team prob.12 but i don't know how to do next. </p>
| Dr. Mathva | 588,272 | <p>Even though I still believe that the calculus approach is the most suited for the problem, consider the somehow <em>magical</em> ansatz I found: <span class="math-container">\begin{align*}x\cdot\left(9\sqrt{1+x^2}+13\sqrt{1-x^2}\right)&=2\cdot \frac34\cdot 3x\cdot 2\cdot \sqrt{1+x^2}+2\cdot\frac{13}4\cdot x\c... |
259,119 | <p>I have a string like this:</p>
<p><code>string="there is a humble-bee in Hanna's garden";</code></p>
<p>Now I want to exclude those words that contain "-" and "'". My own solution would be:</p>
<p><code>StringDelete[string,Cases[StringSplit[string," "], _?(StringContainsQ[#, {... | kglr | 125 | <pre><code>StringDelete[string, WordCharacter .. ~~ "'" | "-" ~~ WordCharacter ..]
</code></pre>
<blockquote>
<pre><code>"there is a in garden"
</code></pre>
</blockquote>
<pre><code>StringRiffle @ Select[StringFreeQ["'" | "-"]] @ StringSplit[string]
</code></pre>
<bl... |
182,101 | <p>With respect to assignments/definitions, when is it appropriate to use $\equiv$ as in </p>
<blockquote>
<p>$$M \equiv \max\{b_1, b_2, \dots, b_n\}$$</p>
</blockquote>
<p>which I encountered in my analysis textbook as opposed to the "colon equals" sign, where this example is taken from Terence Tao's <a href="http... | paul garrett | 12,291 | <p>Upvoting the other answers and comments... and: conventions vary. The only way to know with reasonable confidence is from <em>context</em>. However, one can't know whether an author "believes in" setting context. The most important criterion may be whether or not one is tracking things well enough to reasonably infe... |
3,247,176 | <p>I have this statement:</p>
<blockquote>
<p>It can be assured that | p | ≤ 2.4, if it is known that:</p>
<p>(1) -2.7 ≤ p <2.3</p>
<p>(2) -2.2 < p ≤ 2.6</p>
</blockquote>
<p>My development was:</p>
<p>First, <span class="math-container">$ -2.4 \leq p \leq 2.4$</span></p>
<p>With <span class="math... | Ross Millikan | 1,827 | <p>The point is that if you use <span class="math-container">$1$</span> and <span class="math-container">$2$</span> together you know <span class="math-container">$-2.2 \lt p \lt 2.3$</span>. This does allow you to ensure <span class="math-container">$|p| \le 2.4$</span> because all numbers in the interval are less th... |
317,980 | <p>Suppose that $(P,\leq)$ is a countably infinite linear ordering, and $U$ is a p-point ultrafilter on $P$. Show that there is an $X\in U$ such that $X$ has order type $\omega$ or $\omega^*$. (This is <a href="http://books.google.com/books?id=WTAl997XDb4C&pg=PA86" rel="nofollow">Exercise 7.8</a> in Jech)</p>
<p>I... | Camilo Arosemena-Serrato | 33,495 | <p>We shall make use of the following facts:</p>
<ol>
<li>If $U$ is an ultrafilter and $X\cup Y\in U$, then $X\in U$ or $Y\in U$.</li>
<li>(Exercise $7.7$ of Jech's <em>Set Theory</em>)If $D$ is a $p$-point on $\omega$, then for any decreasing sequence $A_0\supset A_1\supset \cdots \supset A_n\supset \cdots$ of elemen... |
317,980 | <p>Suppose that $(P,\leq)$ is a countably infinite linear ordering, and $U$ is a p-point ultrafilter on $P$. Show that there is an $X\in U$ such that $X$ has order type $\omega$ or $\omega^*$. (This is <a href="http://books.google.com/books?id=WTAl997XDb4C&pg=PA86" rel="nofollow">Exercise 7.8</a> in Jech)</p>
<p>I... | Brian M. Scott | 12,042 | <p>I’m going to call the ultrafilter $\mathscr{U}$. Let $P_0=P\in\mathscr{U}$. Given $P_\xi\in\mathscr{U}$ for some $n\in\omega$, choose a non-endpoint $p_\xi\in P_\xi$. Exactly one of the sets $\{p\in P_\xi:p<p_\xi\}$ and $\{p\in P_\xi:p_\xi\le p\}$ belongs to $\mathscr{U}$; let it be $P_{\xi+1}$. If $\eta$ is a li... |
223,008 | <p>Ok so my teacher said we can use this sentence:
<strong>If $a$ is not a multiple of $5$, then $a^2$ is not a multiple of $5$ neither.</strong></p>
<p>to prove this sentence:
<strong>If $a^2$ is a multiple of $5$, then $a$ itself is a multiple of $5$</strong></p>
<p>I don't understand the logic behind it, I mean wh... | Cameron Buie | 28,900 | <p>Given statements <span class="math-container">$p,q$</span>, the following are equivalent:</p>
<blockquote>
<p><span class="math-container">$p\Rightarrow q$</span></p>
<p><span class="math-container">$\neg q\Rightarrow\neg p$</span></p>
</blockquote>
<p>To check that, you can use a truth table. These two are called <... |
223,008 | <p>Ok so my teacher said we can use this sentence:
<strong>If $a$ is not a multiple of $5$, then $a^2$ is not a multiple of $5$ neither.</strong></p>
<p>to prove this sentence:
<strong>If $a^2$ is a multiple of $5$, then $a$ itself is a multiple of $5$</strong></p>
<p>I don't understand the logic behind it, I mean wh... | sperners lemma | 44,154 | <p>For $p$ prime, $p|ab$ implies $p|a$ or $p|b$.</p>
<p>Since 5 is prime we have $5|a^2$ implies $5|a$ or $5|a$.</p>
|
3,535,316 | <p><span class="math-container">$$\int_{0}^{\pi}e^{x}\cos^{3}(x)dx$$</span></p>
<p>I tried to solve it by parts.I took <span class="math-container">$f(x)=\cos^{3}(x)$</span> so <span class="math-container">$f'(x)=-3\cos^{2}x\sin x$</span> and <span class="math-container">$g'(x)=e^{x}$</span> and I got"</p>
<p><spa... | aud098 | 744,646 | <p>Break each term as <span class="math-container">$\frac{n}{(n-1)!}=\frac{1}{(n-2)!}+\frac{1}{(n-1)!}$</span> now just add these two series of the form <span class="math-container">$\sum \frac {1}{(n!)}=e$</span></p>
|
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