Recent Questions - Mathematics Stack Exchange |
- A Question about limits (sequence)
- Probability of Polynomial of Uniform Random Variables
- For which values of $k$ there exist an $r$ such that $\binom{d}{r}^2\cdot d=\binom{d}{k-1}\binom{d}{k}$?
- The curvature tensor for different connections
- Find the derivative of the following function:
- Has the "partial Sophomore's Dream function" been studied before?
- How to find formula of curve in graph?
- Prove that if $x^*$ is an equilibrium point of $y(n+1)=f^2(y(n))$, then it need not be equilibrium of $x(n+1)=f(x(n))$
- Find sum of primitive roots of $z^{36} − 1 = 0$
- Finding the CDF function of a sum of geometric random variables who are not equally distributed
- Parallelogram and side lengths
- How to solve equation with Matlab
- Let G be a continuous graph. Prove that there is a closed sequence in G containing each edge twice. [closed]
- Size of Cartesian product of two infinite sets
- Given a function $f$ with $f(1\,940)=1\,000\,000$, and its derivative is equal to $kf(200-f)$ and $k$ is a constant. How can I get $f$?
- Maximize product with constraint
- Property of bicentric quadrilateral involving the Miquel point of $OMXN$
- Define a linear map $T:V \to V$ by $T(v)=\left(v,u_{1}\right)u_{2}$. Here $v,u_{1},u_{2}\in V$. Write down a formula for the adjoint map $T^{*}$.
- If $y\in Y$ and $r > 0$, then $B^{Y}_{y}(r) = B^{X}_{y}(r)\cap Y$.
- How to arrive at this equality of tensor products of operators
- Explanation on induction proof of a sequence
- Is $\gcd(a, b) \le |a - b|$ always true? [duplicate]
- Every rank-deficient stochastic matrix has a full rank matrix arbitrarily close to it?
- Semidirect product and direct product
- The approximate probability with zero percent sample
- Patterns in modular multiplication and frequency tables
- Show that $\left(\frac{x_1^{x_2}}{x_2}\right)^p+\left(\frac{x_2^{x_3}}{x_3}\right)^p+\cdots+\left(\frac{x_n^{x_1}}{x_1}\right)^p\ge n$ for any $p\ge1$
- Stiffness matrix for Galerkin method
- Prove $\operatorname{lcm}(a, b) = ab / \gcd(a,b);\ $ $\gcd(a,b) \le |a - b|$ for $a \ne b$
- z alpha by 2 distribution-help!
| A Question about limits (sequence) Posted: 29 Jan 2022 10:23 AM PST I am having a tough time to find the answer of this question. I have no clue about it. Question Below
Thank you. |
| Probability of Polynomial of Uniform Random Variables Posted: 29 Jan 2022 10:17 AM PST Let $U,V,W,X \sim U(0,1)$ be independent. Let $p(U,V,W,X)$ be a polynomial. Is there an easy way to compute $$\mathbb{P}\left(p(U,V,W,X) > 0\right)$$ or find the density of $p(U,V,W,X)$? For example, if my polynomial happens to be $U^2V^2 - WX^3 + UVWX$, could someone show me how to do the computation in this case? I can find the densities of the individual summands, but then am a little stuck since the terms are dependent. (If you happen to be able to factor this polynomial, I definitely can't factor the one I care about, but all the terms are still degree four.) |
| Posted: 29 Jan 2022 10:10 AM PST Let $k<d$ be positive integers. For which values of $1<k<d$ (if any), does there exist an integer $1<r<d$, such that $\binom{d}{r}^2\cdot d=\binom{d}{k-1}\binom{d}{k}$? $r=d$, $k=1$ work. Trying to plug $k=r$ or $k=r+1$ does not provide any other solutions. I guess that there are no other solutions, but I don't know how to prove this. The motivation for this question comes from counting degrees of freedom, in an attempt to find a variational interpretation for an equation. |
| The curvature tensor for different connections Posted: 29 Jan 2022 10:10 AM PST I have gotten stuck at an exercise that deals with the representation of curvature tensors for different connections.
These are the definitions I am working with.
I attempted to solve this by direct calculation. $$ R^{\tilde{\nabla}}(X,Y) \psi =\tilde{\nabla}_X \tilde{\nabla}_Y \psi -\tilde{\nabla}_Y \tilde{\nabla}_X \psi -\tilde{\nabla}_{[X,Y]} \psi \\ =(\nabla+A)_X (\nabla+A)_Y \psi -(\nabla+A)_Y (\nabla+A)_X \psi -(\nabla+A)_{[X,Y]} \psi \\ =\nabla_X \nabla_Y \psi -\nabla_Y \nabla_X \psi -\nabla_{[X,Y]}\psi \\ +\nabla_X A_Y \psi-\nabla_Y A_X \psi +A_X \nabla_Y \psi-A_Y \nabla_X \psi \\ +A_X A_Y \psi-A_Y A_X \psi-A_{[X,Y]} \psi $$ Here the $A_X$ and $A_Y$ are sections into the space of endomorphisms on $TM$, i.e. $A_X, A_Y \in \Gamma(\text{End}(TM))$. We have $R^{\nabla}=\nabla_X \nabla_Y \psi -\nabla_Y \nabla_X \psi-\nabla_{[X,Y]}\psi$. The next step should be to show $$ d(A)=\nabla_X A_Y \psi-\nabla_Y A_X \psi +A_X \nabla_Y \psi-A_Y \nabla_X \psi \\ A \wedge A=A_X A_Y \psi-A_Y A_X \psi-A_{[X,Y]} \psi. $$ But this is where I am getting confused. If I interpret $d$ to mean the exterior derivative and $A$ as a 2-form then $d(A)$ is a 3-form that involves three directional derivatives which is inconsistent with my previous calculations. Similarly, if I attempt to compute $A \wedge A$ using the definition I get a sum over twelve permutations which are too many expressions. Where is my error? |
| Find the derivative of the following function: Posted: 29 Jan 2022 10:10 AM PST Find the derivative of the following function: $F(x)=\sin\Big(\int_0^x \sin\big(\int_0^y \frac{1}{1+t^4}\ dt\big)\ dy\Big)$ |
| Has the "partial Sophomore's Dream function" been studied before? Posted: 29 Jan 2022 10:09 AM PST We can consider the generalized Harmonic numbers $$H_{n,m} := \sum_{k=1}^{n} \frac{1}{k^{m}} $$ as a partial version of the Riemann zeta function, because $$\lim_{n \to \infty} H_{n,m} = \zeta(m). $$ We could also define the "partial Sophomore's Dream function" $$S_{r} := \sum_{q=1}^{r}\frac{1}{q^{q}} ,$$ as we have $$\lim_{r \to \infty} S_{r} := S= \int_{0}^{1}x^{-x}dx ,$$ where the integral on the right is equal to the first Sophomore's Dream constant.
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| How to find formula of curve in graph? Posted: 29 Jan 2022 10:09 AM PST How to find formula of curve of this diagram? And how to know which function is this( log, exponential, ...)? [1]: https://i.stack.imgur.com/6Cahw.jpg |
| Posted: 29 Jan 2022 10:07 AM PST This question is related to Problem 13 in section 1.5 in Elaydi's "An Introduction to Difference Equations": Prove that if $x^*$ is an equilibrium point of $x(n+1)=f(x(n))$, then it is an equilibrium of $y(n+1)=f^2(y(n))$, where $f^2(x)=f(f(x))$. Prove that converse is false in general." This is the solution: I don't understand the part where it says: "since $f(x)$ is not monotonic, then there exists $z^*\neq y^*$ such that $f^{-1}(y^*) = z^* \neq y^*$". If $f$ is not monotonic, how do we know that inverse exists and how do we know that there is preimage of $y^*$ other than $y^*$? I cannot figure this out, so I would be grateful for any help. Thanks a lot in advance. |
| Find sum of primitive roots of $z^{36} − 1 = 0$ Posted: 29 Jan 2022 10:07 AM PST I am trying to understand this concept of sum of primitive roots of unity and here is a typical problem based on it. $z^{36} − 1 = 0$ |
| Finding the CDF function of a sum of geometric random variables who are not equally distributed Posted: 29 Jan 2022 10:03 AM PST Let $p_1,...,p_n \in (0,1)$. $X_1,...,X_n $ are independent random variables so that $X_i \sim Geo(p_i)$. We define $X$ as $ X = X_1 + ... + X_n $. Compute $F_x(t)$ for every $ n \in N $. What I know is that if the random variables were i.i.d then $ X \sim NB (n,1-p) + n $, but that's not the case and it does not get me anywhere. A naive aprroach would be saying that for every $ t \ge n, P(X=t) = p_1(1-p_1)^{t-1} \cdot\cdot\cdot p_n(1-p_n)^{t-1} $, but since the random variables are not equally distributed it can not be true. I dont know how else to aprroach this question, any help is much appreciated. |
| Parallelogram and side lengths Posted: 29 Jan 2022 10:02 AM PST Using the diagram, find $x$ and $y$ if $ABCD$ is a parallelogram. Firstly, we can conclude that $$\measuredangle BAC=\measuredangle ACD=25^\circ \text{ (alternate angles)}$$ Then in triangle $ACD$ we will have $\measuredangle CAD=180^\circ-130^\circ-25^\circ=25^\circ\Rightarrow AD=CD, x=y$. The most obvious thing for me now was to apply the Cosine Rule for $\triangle ACD:$ $$AC^2=AD^2+CD^2-2AD\cdot CD\cos130^\circ\\20^2=2x^2-2x^2\cdot\cos130^\circ$$ Well, $130^\circ$ isn't a "tabular" angle, and we haven't studied how to solve such equations. WolframApha says that $x\approx 11.034$. The next thing I tried: Let $DH\perp AC (H\in AC)$. Then $AH=\dfrac12AC=10$ and in $\triangle AHD$ $$\cos25^\circ=\dfrac{AH}{AD}=\dfrac{10}{x}\Rightarrow x=\dfrac{10}{\cos25^\circ}$$ The given answer in my book is $x=y=10\cos25^\circ$. Am I wrong? Thank you! |
| How to solve equation with Matlab Posted: 29 Jan 2022 10:02 AM PST Matlab math matrix equation equation-solving I want to solve this equation to get T(v) using Matlab but I have no idea about some notation in this equation. The matrix P is a predetermined with random value and dimensions. Please how solve it as Matlab codeThis equation bellow
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| Posted: 29 Jan 2022 10:15 AM PST I only wondered if it would not be possible to use the definition of the Euler graph. definition: Graph G is an Euler graph just when it is continuous and every vertex of the graph G has even degree. |
| Size of Cartesian product of two infinite sets Posted: 29 Jan 2022 09:54 AM PST Let $A$ and $B$ be infinite sets. Is it meaningful to say that $|A\times B|=|A|\cdot |B|$? If yes, in what sense we can understand this product relation? |
| Posted: 29 Jan 2022 10:10 AM PST The problem is this: Given a function $f$ with $f(1\,940)=1\,000\,000$, and its derivative is equal to $kf(200-f)$ and $k$ is a constant. How can I get $f$? I've tried integrating $df/dx$ but I got $f(x) = -\frac{1}a(100f^2-\frac 13 f^3)$ but I don't know how to get this in terms of $x$. |
| Maximize product with constraint Posted: 29 Jan 2022 10:27 AM PST $A_i$ and $x_i$ are positive numbers, I want to solve the following: \begin{align} {\text{maximize}} &\hspace{3mm}& \prod_{i=1}^N (1 + A_{i}x_{i}) \\ \text{subject to} &\hspace{3mm}& \sum_{i=1}^N x_{i} = 1 \end{align} I was trying to guess the pattern of ${x_i}$ by simplifying the problem with only two terms. I get $$x_0 = \frac{A_1 - A_0 + A_0A_1}{2A_0A_1}, \qquad x_1 = \frac{A_0 - A_1 + A_0A_1}{2A_0A_1}$$ but I cannot generalize the solution further. |
| Property of bicentric quadrilateral involving the Miquel point of $OMXN$ Posted: 29 Jan 2022 10:28 AM PST Let $ABCD$ be a bicentric quadrilateral, let $O$ be its circumcenter and $I$, its incenter. Let $M$ and $N$ be the midpoints of $AC$ and $BD$ repectively. Let $X$ be the intersection of $AC$ and $BD$. Let $P$ be the second intersection of circles (IAC) and (IBD). Prove that P is the Miquel point of the quadrilateral $OMXN$. My work: I have managed to prove that $P$,$X$,$I$ and $O$ are colinear using properties of the radical axis $IP$. We also know that $M$,$N$ and $I$ are colinear because of the Newton line of a circumscriptible quardrilateral. So $I$ is actually the intersection point of the diagonals of our quadrialteral $OMXN$. We also know that the quadrilateral $OMXN$ is cyclic and the diameter of its circumcircle is $OX$ so we know that its Miquel point is located on $FG$ where {$F$}=$OM\cap XN$ and {$G$}=$ON\cap XM$. Let $M_q$ be the Miquel point of $OMXN$. Because $FM\perp MX$ and $GN\perp NX$ we know that $X$ is the orthocenter of $\triangle OFG$ so $OX\perp FG$ but $XM_q\perp FG$ (because $XNGM_q$ is cyclic by definition and $XN\perp NG$) so $O$,$X$ and $M_q$ are collinear. Now we can define $M_q$ as the intersection of $OX$ and $FG$ so if we prove that $P$ also lies on $FG$ we are done Edit: From Brocard's theorem we know that $FG$ is the polar line of $I$ with respect to the circumcircle of $OMXN$ so we only need to prove that $P$ lies on this polar line which translates to proving that $(O,X;I,P)$ is a harmonic division which seems way easier to prove than the initial statement but I think that it is still pretty tough. Source: This is actually something I have dicovered while playing with bicentric quadrilaterals on Geogebra. It seems like it is true so I hope it is lol. However I am quite sure that it is true and I am curious why is that so |
| Posted: 29 Jan 2022 10:06 AM PST Problem: Let $V$ denote an inner product vector space. Define a linear map $T:V \rightarrow V$ by $T(v)=\left(v,u_{1}\right)u_{2}$. Here $v,u_{1},u_{2}\in V$. Write down a formula for the adjoint map $T^{*}$. Attempt: Let $ x,y \in V $ be arbitrary. $ ( Tx,y ) = ( (x,u_1)u_2,y ) = (x,u_1)(u_2,y) = \overline{ (u_1,x) }(u_2,y) =(x,T^*y)$ Official solution: Questions:
Thanks in advance for help! |
| If $y\in Y$ and $r > 0$, then $B^{Y}_{y}(r) = B^{X}_{y}(r)\cap Y$. Posted: 29 Jan 2022 10:03 AM PST Problem Let $(X,d)$ be a metric space and $Y\subset X$ a subspace. If $y\in Y$ and $r > 0$, then $B^{Y}_{y}(r) = B^{X}_{y}(r)\cap Y$. My attempt Let us prove the inclusion $\subseteq$ first. If $a\in B^{Y}_{y}(r)$, then $d(a,y) < r$ and $a\in Y$. Since $Y\subset X$, we conclude that $a\in X$ as well. Consequently, we can say that $d(a,y) < r$ and $a\in X$. The last statement is the definition of $a\in B^{X}_{y}(r)$. Gathering all the previous results, we conclude that $a\in B^{X}_{y}(r)\cap Y$. We may now proceed and prove the inclusion $\supseteq$. If $a\in B^{X}_{y}(r)$, then we conclude that $d(a,y) < r$ and $a\in X$. But we also know that $a\in Y$. Hence we may claim that $a\in B^{Y}_{y}(r)$, and we are done. My concerns Is there another way to approach such problem? |
| How to arrive at this equality of tensor products of operators Posted: 29 Jan 2022 10:05 AM PST We have a (self-adjoint) projection operator $Q_m^{AE}$ acting on a tensor product of finite-dimensional complex Hilbert spaces $\mathcal{H}_A \otimes \mathcal{H}_B$. For an element $| v_B \rangle \in \mathcal{H}_B$, we define \begin{equation} Q_m^{A}[v_B] = (I_{A} \otimes \langle v_B |) \, Q_m^{AB} \, ( I_A \otimes | v_B \rangle), \end{equation} where $I_A$ denotes the identity operator on $\mathcal{H}_A$. We have shown that $$Q_m^{A}[v_B] = P_m,$$ for a (self-adjoint) projection operator $P_m$ on $\mathcal{H}_A$, and this holds for any $| v_B \rangle \in \mathcal{H}_B$. Supposedly, we can now easily see from expanding this identity in a basis that $$Q_m^{AB} = P_m \otimes I_B, $$ where $I_B$ is the identity on $\mathcal{H}_B$. However, I don't see how to that. It seems to me that $Q_m^{AB}$ is "trapped" inside the product in the first equation anyway. I am not very familiar with tensor products, so sorry if this is a terribly trivial question. |
| Explanation on induction proof of a sequence Posted: 29 Jan 2022 10:03 AM PST I don't understand two key steps of a proof that the following sequence is bounded and monotone: $$(a_n) \text{ defined recursively} \,, \quad a_1 = 0 \,, \quad a_{n+1} = \frac{a_n +2}{2} \,. $$ Can anyone explain either of these two steps in the proof? Thank you very much for any help. Here is an image of the original exercise 1. First the proof shows that the sequence $(a_n)$ is bounded:
First question: by the induction hypothesis obviously the 2 was inserted as $a_n$. Why is that valid? 2. Then the proof shows that $(a_n)$ is monotone:
Second question: why does the inequality in the last line hold? |
| Is $\gcd(a, b) \le |a - b|$ always true? [duplicate] Posted: 29 Jan 2022 10:14 AM PST
I wondered because I read that the $\gcd(a, a + 1)$ is always $1$ because of the Euclidean Algorithm. I would really appreciate your answers, thanks in advance. |
| Every rank-deficient stochastic matrix has a full rank matrix arbitrarily close to it? Posted: 29 Jan 2022 09:54 AM PST Define the following subspace of matrices. $$ S = \{A\in \mathbb{R}_{\geq 0}^{n\times m}\mid \sum_{j=1}^{m}a_{i,j}=1,n<m\}. $$ Is it true that for every $A\in S$, and every $\epsilon>0$ there exists $A_{\epsilon}\in S$ such that $\|A-A_{\epsilon}\|_2<\epsilon$ and $A_{\epsilon}$ is of full-rank? |
| Semidirect product and direct product Posted: 29 Jan 2022 09:58 AM PST Let $A$ be an abelian group and $G$ a group and let $\alpha:G\rightarrow{\rm Aut}(A)$. I want to show that the semidirect product $A\rtimes _{\alpha }G$ is isomorphic to the direct product $A\times G$ iff $\alpha(g)=id$ for all $g\in G$. This is true if $G$ is abelian. If $G$ is nonabelian, I think that the only if direction is not true in general. So I will be thankful if someone provides us a counterexample. Thank you in advance |
| The approximate probability with zero percent sample Posted: 29 Jan 2022 10:23 AM PST I just got a statistic question which make me confuse because too many same variable(in this case 10%), so I don't know what is that value and how to compute the approximate probability. Here is the question(I changed the variable values and name, so I could be able to answer the real question by myself):
I don't know why the sample of 30 children can give a 0% rate. I tried to research again using a search engine and never found a similar question to this question. Can anyone help me with a formula for that question? |
| Patterns in modular multiplication and frequency tables Posted: 29 Jan 2022 10:08 AM PST I moved this question from MathOverflow since it seemed not appropriate there. Consider the multiplication table $f(n,m) = (n\cdot m) \text{ mod } N$ for $n,m \leq N/2$. For $N=30$ it looks like this:
Shades of gray depict the distance $|k - N/2|$, from white for $k=0$ to black for $k = N/2$. For $N = 300$ this gives the following picture:
Now consider a completely different table: the table of frequencies $n : N$ in which you draw in the $n$-th row $n$ dots at exact positions $i/n \cdot N$ for $i = 0,\dots n-1$. For $n,i \leq N/2$ this table looks like this:
When we set the positions to rounded values $[i/n \cdot N]$ the picture changes slightly
and it becomes more similar to the multiplication table. Overlaying the two tables reveal that the yellow dots perfectly lie on blurred "lines" in the multiplication table:
The two pictures next to each other: In more detail the top left and the bottom left part of the tables:
Here it is the voids between the yellow dots that lie on blurred "lines" visible in the multiplication table:
The voids disappear when the yellow dots are at their exact positions $i/n \cdot N$:
I wonder if someone is around who says: All of this comes as no surprise, it could easily have been expected. But a little bit more thorough "explanation" would be very much appreciated. Especially of the fact that rounding systematically creates similarities that are not present when the yellow dots are at their exact positions. Another specific question is: Why is the center of the "whirl" in the lower right quadrant almost exactly at the position $[N/3,N/3]$, independent of $N$?
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| Posted: 29 Jan 2022 10:14 AM PST The inequality $\sqrt{\frac{a^b}{b}}+\sqrt{\frac{b^a}{a}}\ge 2$ for all $a,b>0$ was shown here using first-order Padé approximants on each exponent, where the minimum is attained at $a=b=1$. By empirical evidence, it appears that inequalities of this type hold for an arbitrary number of variables. We can phrase the generalised problem as follows.
As before, AM-GM is far too weak since the inequality $\displaystyle\|\boldsymbol a\|_p^p\ge 2\left(\prod_{\text{cyc}}\frac{x_1^{x_2}}{x_2}\right)^{1/{2p}}$ does not guarantee the result when at least one $x_i$ is smaller than $1$. We can eliminate the exponent on the denominator by taking $x_i=X_i^{1/p}$ so that $\displaystyle\|\boldsymbol a\|_p^p=\sum_{\text{cyc}}\frac{X_1^{X_2^{1/p}}}{X_2}$ but the approximant approach no longer becomes feasible; even in the case where $p$ is an integer the problem reduces to a posynomial inequality of rational degrees. Perhaps there are some obscure $L^p$-norm/Hölder-type identities of use but I'm at a loss in terms of finding references. Empirical results: In the interval $p\in[1,\infty)$, Wolfram suggests that the minimum is $n$ (Notebook result) which is obtained when $\boldsymbol a$ is the vector of ones. However, we note that in the interval $p\in(0,1)$, the empirical minimum no longer displays this consistent behaviour as can be seen in this Notebook result. The sequence $\approx(1.00,2.00,2.01,3.36,3.00,4.00)$ appears to increase almost linearly every two values, but I cannot verify it for a larger number of variables due to instability in the working precision. |
| Stiffness matrix for Galerkin method Posted: 29 Jan 2022 10:10 AM PST I am working on a finite element approximation to solve the following ODE type: $$-\dfrac{d}{dx}\bigg(a(x)\dfrac{du}{dx}\bigg)=f(x)$$ with $u(0) = u(1) = a(0)$, using a Galerkin method. We start by defining a mesh $x_j=jh$ over $N$ points. We take the piecewise function $\phi$ as a basis: $$\phi_j(x) =\begin{cases} 0&\text{if}\, ~x\not\in(x_{j-1},x_{j+1})\\\\ \dfrac{x-x_{j-1}}{x_{j}-x_{j-1}}&\text{if}\, x\in (x_{j-1},x_{j})\\\\ \dfrac{x_{j+1}-x}{x_{j+1}-x_j}&\text{if}~x\in (x_j,x_{j+1}) \end{cases}$$ Specifically, for the problem: We take the test method $v(x)=\phi_i(x)$ and the Galerkin method we are using is: Find $u$ such that $\displaystyle\int_0^1 a(x)\dfrac{du}{dx}\dfrac{dv}{dx}~dx=\displaystyle\int_0^1 f(x)v(x) dx$, for approximation: $u=\displaystyle\sum_{j=1}^M\xi_j~\phi_j(x)$ Substituting for u and v into the method, the left hand side gives the stiffness matrix with entries $(a_{ij})=\displaystyle\int_0^1 a(x)\dfrac{d\phi_j}{dx}\dfrac{d\phi_i}{dx}~dx$ In the specific example I'm doing, $a(x)=1$ and we should find: $(a_{ij})=\begin{cases} \frac{2}{h}&\text{if}\, ~i=j\\ \frac{-1}{h}&\text{if}\, i=j+1,~\text{or}~i=j-1 \end{cases}$ I don't understand how to get those values for $(a_{ij})$? My attempt: $$\phi_j'(x) =\begin{cases} 0&\text{if}\, ~x\not\in(x_{j-1},x_{j+1})\\ \dfrac{1}{h}&\text{if}\, x\in (x_{j-1},x_{j})\\ \dfrac{-1}{h}&\text{if}~x\in (x_j,x_{j+1}) \end{cases}$$ But computing $\phi_i'$ seems to be throwing me. |
| Prove $\operatorname{lcm}(a, b) = ab / \gcd(a,b);\ $ $\gcd(a,b) \le |a - b|$ for $a \ne b$ Posted: 29 Jan 2022 10:13 AM PST What's the simplest proof that the least common divisor of $a$ and $b$ is equal to the product of $a$ and $b$ divided by the greatest common divisor, i.e.: $$\operatorname{LCM}(a,b)=\frac{ab}{gcd(a,b)}$$ and that $$\gcd(a,b)\le |a-b|,\ \ {\rm for}\ \ a\neq b$$ Note: $ $ the original version was: $\ \,\gcd(a,b) < |a-b|,\,$ but that has trivial counterexamples, e.g. see the comments below. |
| z alpha by 2 distribution-help! Posted: 29 Jan 2022 10:06 AM PST How do we use z aplha by 2 distribution? I don't get it one bit. For the table it says z on lhs and second decimal place of z on the top in the rhs. Firstly if e.g apha is 0.05 then apha by 2 is 0.025 so we know that we have z subscript 0.025 so how on earth do we use the table if we know only the value of alpha and not the value of z, what on earth do we look for in the tables? s |
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