Recent Questions - Mathematics Stack Exchange |
- Given this partially ordered set, how can I prove all its subsets are chains (totally ordered subsets)?
- Nested Quantifiers Equivalents Proof
- Verify: $a,b$ be positive rational numbers such that for infinitely many integers $n,$ $a^n-b^n$ is an integer, Prove that $a,b$ are also integers.
- An Ideal is Maximal
- getting continuous functions from a function defined over a compact set
- Polynomials in finite fields
- Is there an example of two topologically different knots with the exact same Conway notation?
- Borel Measurability of a function with countable discontinuity points.
- Proving conditions for normal subgroups
- Solve the differential equation xcos(y') +y sin(y') = x³
- Resulting surfaces of cutting Mobius strip at the boundary
- Can you rephrase the question
- Small subset of a compact set is contained in a member of the open cover
- A question from Sheldon Axler (section 2C exercise 2 &3)
- Chain rule for complex-valued functions
- prime counting function $\pi(x)$ is $o(x)$
- Is there some way to find a value $x$, such that $2^{\log_{2}x} = \sum_{k=0}^n 2^k$, without calculating the sum?
- best strategies for 'squid game' episode 6 game 4 "marble game" players ...
- Solve the differential equation ( y - 2xy')² = (y')³
- Question about absolute value semantics and approach
- Is there an intuitive way of visualizing how a function that is flat at some given point can be something other than a constant function?
- Logic Rule of Interference
- Center of $U(\mathfrak{sl}_2)$
- For all $a, b$ in $\mathbb{Z}$, show $a +b$ is a factor of $a^{2n} - b^{2n}$.
- Solving Trigonometric question
- Alternate forms of $α\mathop=\limits^\text{def}\lim\limits_{x\to0} Q^{-1}(x,x)\implies \text{Ei}(-α)=-1\implies Γ(0,α)=1$.
- Do the left and right "parts" of the matrix used when performing Gaussian Elimination have names?
- Balls into bins with left and right bins always empty [closed]
- Integrate $ \int \frac{2\tan x+1}{\sqrt{\tan^2 x+2\tan x+2}} \, \mathrm dx $
- Double integral of $\exp(x^2+y^2)$ over the upper semi circle of radius $r=1$
| Posted: 09 Oct 2021 07:53 PM PDT I'm looking to use Zorn's lemma on the follwoing set: Let $\Sigma$ be the set of all pairs $(A,f)$ where $A$ is a subring of field $K$ and $f$ its homeomorphism of $A$ into $\Omega$. We partially order the set as follows: $$(A,f) \leq (A',f') \leftrightarrow A \subset A' \land f'|_{A} = f$$ Now, given a chain $\{(A_i,f_i)\}_{i \in \{1...n\}} \subset \Sigma$ I want to show it has an upper bound $(\bar{A}, \bar{f})$. I believe $\bar{A} = \cup_{i =1}^n A_i$ now I want to define $\bar{f}$ such that $\bar{f}|_{A_i} = f_i \; \; \forall i \in \{1...n\}$ Since its a chain I'm working with I think there's no harm assuming the functions all satisfy that $f_j|_{A_i} = f_i \; \forall \; i\leq j $ but I still arrive nowhere. Any help would be apreciated, I'm not sure if I'm complicating things more than I should.  |
| Nested Quantifiers Equivalents Proof Posted: 09 Oct 2021 07:45 PM PDT denote L(x): x was late for the meeting Prove or disprove the following two are logically equivalent ∃x,(L(x) ⋀ ∀y, y≠x → ¬L(y))-------- (∃x,L(x)) ⋀ (∀x∀y,L(x) ⋀ L(y) → x=y) I thought that proving this by cases would work but turned out to be a dead-end. Possible use of proof by contradiction?  |
| Posted: 09 Oct 2021 07:38 PM PDT
Let $a=\frac{x}{z}$ and $b=\frac yz .$ Hence $z^n\mid x^n-y^n$ for infinite $n.$ Note that
Now $z$ is fixed and so in $x,y.$ But the LHS is increasing quite fast.
But again bounding gives that for large $z$ this won't be true. So $z=1.$ Is the method correct? I am not sure of the bounding though.. Any other solution would be helpful too.  |
| Posted: 09 Oct 2021 07:52 PM PDT In $\mathbb Z[\sqrt{-5}]$, define the ideals $=\mathbf p \langle 2, 1+\sqrt{-5}\rangle $, $\mathbf q=\langle3, 1+\sqrt{-5}\rangle$, and $\mathbf r=\langle3, 1-\sqrt{-5}\rangle$. Show that each ideals are maximal. I have solved for p. $Z[\sqrt{-5}]/\langle2\rangle=2^2=4$ Since $\langle2\rangle\subsetneq\langle2, 1+ \sqrt{-5} \rangle$, it has order dividing $4$. Hence $\mathbb Z[\sqrt{-5}]/\mathbf p$ is congruent to $\mathbb Z_2,$ Where $\mathbb Z_2$ is a field. Therefore, p is maximal. I have a good understanding, however I have noticed that qr$=<3>$. Should I show this form of an ideal is maximal, or should I show that q and r are maximal separately?  |
| getting continuous functions from a function defined over a compact set Posted: 09 Oct 2021 07:21 PM PDT $\mathbf{PROBLEM:}$ Let $X$, $Y$ be a spaces and $f : X × Y → \mathbb{R}$ be a continuous function. If $X$ is compact, prove that the function $g$, $h : Y → \mathbb{R}$ defined by $g(y) = sup \{ f(x, y)|x ∈ X\} $ and $h(y) = inf \{f(x, y)|x ∈ X\}$ are continuous. $\mathbf{ATTEMPT:}$ Because $X × \{y\}$ is compact, then $g$ and $h$ have sense since $f|_{X × \{y\}}$ would be bounded. I want to prove this for $g$ and I suppose that the proof will be similar for $h$. I can only think of this: we take an open bounded interval $(c,d)$ and I consider $g^{-1}(c,d)$. Now, because $f^{-1}(c,d)=A × B$ with $A$ open in $X$ and $B$ open in $Y$, we can see easily that $g^{-1}(c,d)\subset B$. However, I dont't achive advance from here...  |
| Posted: 09 Oct 2021 07:27 PM PDT Could someone help get me started on this question, i know how to draw the field for gf(5) but i;m unsure how it links to the polynomial:  |
| Is there an example of two topologically different knots with the exact same Conway notation? Posted: 09 Oct 2021 07:29 PM PDT on whether a Conway notation describes a unique knot, so it'd be two knots which are fundamentally topologically different  |
| Borel Measurability of a function with countable discontinuity points. Posted: 09 Oct 2021 07:27 PM PDT Suppose that $X$ is a Borel subset of $\mathbb{R}$ and $f : X → \mathbb{R}$ such that the set of points at which $f$ is discontinuous is countable. Prove that $f$ is Borel measurable.  |
| Proving conditions for normal subgroups Posted: 09 Oct 2021 07:50 PM PDT I was watching the following online lecture on normal subgroups, and came across the following proof: Normal and quotient groups If we have some group, G, and some subgroup of G, let's say N, we can construct left cosets like so: Each left coset: $gN$ for some $g \in G$. Now we pick two cosets: $xN, yN$. Now since $e \in N$, $x.e = x \in xN$ and $y.e = y \in yN$. Now here is the part that I don't understand: For cosets to act like a group: $x.y \in (xN)(yN)$ Why must this be true? And what is $(xN)(yN)$.Is it the element-wise group operation of each element in $xN$ and $yN$? Then the proof said: $i.e. (xN)(yN) = xyN$. This does not make sense to me, what is so obvious about this statement?  |
| Solve the differential equation xcos(y') +y sin(y') = x³ Posted: 09 Oct 2021 06:58 PM PDT I been stuck in this equation for quite a while and don't know how to solve it.  |
| Resulting surfaces of cutting Mobius strip at the boundary Posted: 09 Oct 2021 06:57 PM PDT I'm having a real hard time conceptualising the resulting surfaces of cutting a mobius strip along the boundary. I cut a 4cm Mobius strip 1cm along the boundary which resulted in interlinked 2cm Mobius Strip and a 1cm Mobius strip with double length. I realise that 3d space is not an accurate model for what is happening to the Mobius strip, can someone explain the resulting surfaces? The 2cm Mobius strip is still a Mobius strip, but what does the other strip turn into? Additionally, how do I format it into a diagram.  |
| Posted: 09 Oct 2021 06:57 PM PDT Let m be a natural number. Prove that multiplication is well-defined with respect to the relation of congruence modulo m. I'm not looking for the solution, I just don't really understand what I have to prove.  |
| Small subset of a compact set is contained in a member of the open cover Posted: 09 Oct 2021 06:53 PM PDT Let $(X,d)$ be a metric space and $K\subset X$ be compact. Let $\{G_{i}\}_{i\in I}$ be an open cover of $K$. I want to show that $\exists \epsilon>0$ such that every subset $S$ of $K$ of diameter $<\epsilon$ is contained in one of the sets $G_{i}$. A naive idea is to set $\epsilon=diam(\cap_{i} G_{i})$, but it might equal to $0$. My second idea is that since there's a finite subcover $\cup_{i=1}^{n} G_{i}$ of $K$, I will just fix a $G_{1}$ and write the others as $\overline{G}=\cup_{i\neq 1}^{n} G_{i}$. Suppose I will prove it by contradiction, then $\forall \epsilon>0$, every subset $S$ of $K$ is contained in at least two sets $G_{i}$, which means that it's contained in both $G_{1}$ and $\overline{G}$. If this is the case, I guess I need to somehow show that $diam(x,y)$ cannot be arbitrarily small for $x,y\in S$, which gives a contradiction. Thanks for any hints!  |
| A question from Sheldon Axler (section 2C exercise 2 &3) Posted: 09 Oct 2021 06:50 PM PDT Suppose $\mu$ is a measure on $(Z^{+}, P(Z^{+}))$ (the second term is the power set of $Z^{+}$).  |
| Chain rule for complex-valued functions Posted: 09 Oct 2021 06:49 PM PDT The Chain Rule for complex-valued functions is stated as (Complex Analysis (Undergraduate Texts in Mathematics) Corrected Edition by Theodore W. Gamelin): Suppose that $g(z)$ is differentiable at $z_0$, and suppose that $f(w)$ is differentiable at $w_0 = g(z_0)$. Then the composition $(f \circ g)(z)=f(g(z))$" is differentiable at $z_0$ and $$(f \circ g)'(z_0) = f'(g(z_0)) g'(z_0) \, .$$ The proof for $g'(z_0) \neq 0$ is quite straightforward. In proving this for $g(z_0)=0$, the author used the following argument. Since $f(w)$ is differentiable at $w_0$, the difference quotients $f(w)-f(w_0)/(w-w_0)$ are bounded near $w_0$, say $$\left| \frac{f(w)-f(w_0)}{(w - w_0)} \right| \leq C$$ for some constant $C$ and $0 < |w-w_0| < \epsilon $ from which the proof is straightforward. I couldn't grasp the above inequality. Any help in understanding this is much appreciated.  |
| prime counting function $\pi(x)$ is $o(x)$ Posted: 09 Oct 2021 07:04 PM PDT What's the simplest and/or shortest proof that $\lim_{x \to \infty} \frac{\pi(x)}{x} = 0$, where $\pi(x)$ is the prime counting function? I'm curious to see if there's a slick proof that is simpler and shorter than Chebyshev's proof that $\pi(x) \asymp \frac{x}{\log x} \ (x \to \infty)$. Of course, using the prime number theorem to prove it is cheating. In particular, can it be proved using a simple idea like the sieve of Eratosthenes?  |
| Posted: 09 Oct 2021 07:00 PM PDT I was trying to find out how much interest could be made by making reoccurring deposits into a savings account. Let's say you deposit \$500 a month for a year into an account. The account gives you 2% interest a month. You can model the amount of interest you make after $n$ months with the following equation: $$500(n - \sum_{k=0}^n 1.02^k)$$ Let's say there's some miraculous account that offers 100% interest a month, so your earnings can be modeled represented as $500(n - \sum_{k=0}^n 2^k)$. Is there some way to find a $2^{\log_{2}x}$ that is equal to $\sum_{k=0}^n 2^k$, without calculating the sum?  |
| best strategies for 'squid game' episode 6 game 4 "marble game" players ... Posted: 09 Oct 2021 07:03 PM PDT Two players each get 10 marbles. Each alternating turn, one player (say, the player whose turn it is) hides some of its own marbles in its fist, and, the other player must guess if the hidden amount is odd or even. And, that other player (i.e., the player whose turn it is not) also must bet an amount of its own marbles. If he/she guesses right, the player whose turn it is gives (as much as it can) the amount of bet marbles to him/her. If he/she guesses wrong, the player whose turn it is takes the amount of bet marbles from him/her. Next, the turn now alternates to the other player. The game stops when one player has all 20 marbles. The losing player gets killed (in the series, that is). Which strategies for both players (perhaps one strategy for the one who gets first turn, and one strategy for the other player) give maximal probabilities to win (and to not get killed). We must assume both players know the starting conditions and are perfect mathematicians and logicians and perfect what not ... :-) Also, n=10 is perhaps just a choice, maybe strategies do not depend on n or on certain n. Also, strategy must not be static and can (as in e.g. chess) change each turn (perhaps dependent on opponent move). On a side note: is this an ancient or newborn game ?  |
| Solve the differential equation ( y - 2xy')² = (y')³ Posted: 09 Oct 2021 07:25 PM PDT I'm solving this differential equation, but I don't even know were to start with this one, could someone help on how do I solve this equation.  |
| Question about absolute value semantics and approach Posted: 09 Oct 2021 07:09 PM PDT A question of approach.. really basic I am afraid. Given $$|a|\leq M$$ is this the correct approach? $$\left| a \right| \leq M = \begin{cases} a \leq M, & a \geq 0 \\ -a \leq M, & a \leq 0 \end{cases}$$ $$ \qquad = \begin{cases} a \leq M, & a \geq 0 \\ a \geq -M, & a \leq 0 \end{cases} $$ thus $$-M \leq a \leq M$$ Thanks.  |
| Posted: 09 Oct 2021 06:59 PM PDT I can't wrap my head around how a function that is flat at a certain point can be anything but a constant function, from an intuitive point of view. Thinking of derivatives as "instantaneous rates of change", if every derivative at that point is equal to 0, none of them are changing, so none of them could ever take on a new value, meaning the function's value could never change. Imagining a particle whose movement with respect to time is defined by a flat function doesn't seem to make any physical sense because of this. In spite of this, non-constant flat functions exist, and I assume a particle could theoretically follow such a trajectory. Or maybe it can't. So does anyone have an intuitive way of visualizing this?  |
| Posted: 09 Oct 2021 07:21 PM PDT If I am given the proposition $(P \lor Q)$ and $(P \land R)$ then can I get $(P \to R)$ I have tried multiple ways but I can't seem to get it, is it even possible?  |
| Center of $U(\mathfrak{sl}_2)$ Posted: 09 Oct 2021 07:55 PM PDT I've found similar questions, but none with answers. For basis $E, F, H$ of $\mathfrak{sl}_2 (\mathbb{C})$ with $[E,F]=H, [H,F]=-2F, [H,E]=2E$, I have shown that the Casimir element $$C = \frac{1}{4} (EF+FE+\frac{1}{2}H^2) = \frac{1}{4} (EF-H+\frac{1}{2}H^2)$$ must be in the center of the enveloping algebra. Furthermore, I have shown that the centralizer of $H$ is generated by $$EF, H$$ My goal is to show that the center of $U (\mathfrak{sl}_2 (\mathbb{C}))$ is generated by $C$, however I am at a loss for how to proceed (without appealing to machinery I haven't learned about yet such as Harish-Chandra). A priori, a subalgebra of the algebra generated by $EF$ and $H$ need not even be finitely-generated. I have classified the irreducible representations of $\mathfrak{sl}_2 (\mathbb{C})$, however using Schur's Lemma to try to get conditions on the center did not seem to produce anything useful.  |
| For all $a, b$ in $\mathbb{Z}$, show $a +b$ is a factor of $a^{2n} - b^{2n}$. Posted: 09 Oct 2021 07:41 PM PDT Given solution approach for first problem below, unable to approach the second one. #2. For all $a, b$ in $\mathbb{Z}$ , show $a + b$ is a factor of $a^{2n} - b^{2n}$.  |
| Solving Trigonometric question Posted: 09 Oct 2021 07:45 PM PDT Question Without using calculator, show that $\sec \frac{7\pi }{6}= -\frac{2 \sqrt{b}}{b}$ where b is a constant to be found. Solve $\beta -\frac{1}{3} \sec \sec \beta=2 $ for $180^{\circ}\leq \beta \leq 270^{\circ} $. I dont know to continue.. thanks for help me.  |
| Posted: 09 Oct 2021 07:55 PM PDT Based on:
and
Here is the goal limit again using the central Inverse Regularized Gamma function $Q^{-1}(x,x)$: $$α\mathop=^\text{def} \lim_{x\to0} Q^{-1}(x,x)=Q^{-1}(0,0)=\ne 0$$ Note that Wolfram Alpha says that $Q^{-1}(0,0)=0$, but this is just due to the definition of the function. If you take a look at the graph, the "true" limit clearly is not $0$:
However, we can use the main Taylor series definition centered at $x=1$ with the new $C_{n,x}$ coefficient notation. Note that there are other series in the bolded link. $$Q^{-1}(x,x)=((1-x)x!)^\frac1x+\frac {((1-x)x!)^\frac2x}{x+1}+ \frac {(3x+5)((1-x)x!)^\frac3x}{2(x+1)^2(x+2)}+O\left((x-1)^\frac4x \right)\mathop =^\text{def}\sum_{n=0}^\infty C_{n,x}\cdot ((1-x)x!)^\frac nx$$ Now let's apply the limit noting that we can apply the limit term by term. The limit is a long process, but ends up being in terms of $e$ and the Euler-Mascheroni constant γ: $$α= \lim_{x\to0} Q^{-1}(x,x)=Q^{-1}(0,0)= \lim_{x\to 0}\sum_{n=0}^\infty C_{n,x}\cdot ((1-x)x!)^\frac nx=\sum_{n=1}^\infty \frac{C_{n,0}}{e^{(γ+1)n}} = e^{-(γ+1)}+ e^{-2(γ+1)}+ \frac 54 e^{-3(γ+1)}+\frac{31}{18} e^{-4(γ+1)}+ \frac{361}{144}e^{-5(γ+1)}+\frac{4537}{1200} e^{-6(γ+1)} +…=.26473… $$ A few observations on the constants:
An equivalent problem is by solving the following differential equation for the function: $$y(x)y''(x)-y'(x)^2(y(x)+1-a)=0,y(x)=Q^{-1}(a,x)\implies y(x)y''(x)-y'(x)^2(y(x)+1)=0,y(x)=\lim_{a\to 0} Q^{-1}(a,x),y(x)=y(a,x),y(0,0)=α$$ It turns out that the special case can be solved in terms of an Exponential Integral function. Let me "make up" an inverse Exponential Integral function: $$y(x)y''(x)-y'(x)^2(y(x)+1)=0\implies c+x=\frac{\text{Ei}(-y(x))}{c}\implies c^2+cx=\text{Ei}(-y(x))\implies y(x)=-\text{Ei}^{-1}(c^2+cx)$$ So the constant is maybe related to Soldner's Constant Another idea is letting $x=a$ for: $$y(x)y''(x)-y'(x)^2(y(x)+1-a)=0, y(0)$$ but I am not sure if this would produce $y=Q^{-1}(x,x)$. Amazingly, our differential equation works: $\text{Ei}(-α)=-1$ Therefore: $$\text{Ei}(-Q^{-1}(0,0))=\text{Ei}\left(-\sum_\Bbb N C_{n,0} e^{-n(γ+1)}\right)=-1$$ So a "closed form" using a made up Inverse Exponential Integral function would be: $$α=\lim_{x\to0}Q^{-1}(x,x)=-\text{Ei}^{-1}(-1)$$ A better way would be to solve $$\text{Ei}(-x)=-1,x=α$$ for $x$. Let's use this method to find many more digits of the constant $$α=0.264737010451543159461927…$$ while the other real root, for fun, of $\text{Ei}(-x)+1$ is: $$x= -0.1724867417161…$$ Here are $3$ other identities of this constant using this Incomplete Gamma function identity and this Generalized Exponential Integral function. The third one uses the Hyperbolic Cosine and Hyperbolic Sine Integral functions $$Γ(0,α)=\text E_1(α)=1\implies \int_α^\infty \frac{dx}{xe^x}=1$$ $$\text{Shi}(α)-\text{Chi}(α)=1$$ Note that other constants may also fit the formulas for the constant. Many other identities can be similarly derived, so please go ahead; I have written enough for now. How can I find an alternate form of the constant? Please correct me and give me feedback!  |
| Do the left and right "parts" of the matrix used when performing Gaussian Elimination have names? Posted: 09 Oct 2021 07:09 PM PDT Given a matrix used to perform Gaussian Elimination like this: $$ \begin{bmatrix} 1 & -3 & | & 1 \\ 2 & -7 & | & 3 \end{bmatrix} $$ Which would be derived from a system of equations like this: $$ \begin{cases} x - 3y = 1 \\ 2x - 7y = 3 \end{cases} $$ Do the different "sides", i.e. the parts that come from the right hand side of the equation and the part that comes from the left hand side of the equation, respectively, have names?  |
| Balls into bins with left and right bins always empty [closed] Posted: 09 Oct 2021 06:58 PM PDT N balls are sequentially and randomly allocated into M bins arranged in a circle. If a bin receives a ball, its left and right bins cannot receive any ball and are always set to be empty (new coming balls are discarded). What is the expected number of occupied bins at last? From my simulation of N=M=100, the expected number is 36, but I want to know if it can be formulated mathematically. When N=M=4, simulation gives a value around 1.576. Background: This is a quantitative interview question I am trying to solve. This question is an extension of the original balls-into-bins problem, which adds the dependency to the neighbour boxes by introducing a special constraint. Additional Explanation: The original balls-into-bins question only says N balls are randomly allocated into M bins. Please calculate the expected number of non-empty bins. The answer is $$ M\left (1-\left(\frac{M-1}{M}\right)^N \right ) \approx M\left(1-\frac{1}{e}\right) $$  |
| Integrate $ \int \frac{2\tan x+1}{\sqrt{\tan^2 x+2\tan x+2}} \, \mathrm dx $ Posted: 09 Oct 2021 07:11 PM PDT I tried writing $\tan^2 x+2\tan x+2$ as $(\tan x+1)^2+1$ and letting $\tan\theta = \tan x+1$ yields $$ \int \frac{2\tan\theta-1}{\tan^2\theta-2\tan\theta+2}\,\sec\theta\,\mathrm d\theta $$ Then letting $ t = \tan(\frac{\theta}{2})$ yields $$ \int\frac{t^2+4t-1}{t^4+2t^3-2t+1}\,\mathrm dt $$ Obviously what's next is factoring out the denominator and doing partial fraction decomposition, but this seems very hard. I wanted to know if there's an easier method.  |
| Double integral of $\exp(x^2+y^2)$ over the upper semi circle of radius $r=1$ Posted: 09 Oct 2021 07:01 PM PDT I'm really not sure on formatting so I'll give it my best shot, but my working out at the moment leaves me at the point of
So between $\pi$ and $0$ for the left integral (because it's a semi circle not a circle) and between $1$ and $0$ for the right integral. Would this be correct?  |
| You are subscribed to email updates from Recent Questions - Mathematics Stack Exchange. To stop receiving these emails, you may unsubscribe now. | Email delivery powered by Google |
| Google, 1600 Amphitheatre Parkway, Mountain View, CA 94043, United States | |
No comments:
Post a Comment