Recent Questions - Mathematics Stack Exchange |
- Is there $b$ depending only on $a$ such that $\prod_{i=1}^n \Gamma(\lambda_i a) \le b$ for all $\lambda_i >0$ s.t. $\sum \lambda_i = 1$?
- Flux of vector field. Stokes' Theorem
- Standard term for "area between a tangent to a hyperbola and its two asymptotes"?
- Find $f\left(x\right)$, given that $f''\left(x\right)=3x^2+1$, $f\left(0\right)=1$ and $f\left(2\right)=-1$ and that it is a 2ndorder derivable funct.
- Generalization on double expectation
- Almost sure covergence of exponentially distributed random variables
- More information on 'constructing' integrals
- Pointwise convergence of specific sequence of functions
- Which f satisfies E[f(X) X] = E[f(X)] E[X]?
- Cost Optimization Problem: Which of 2 Manufacturing Processes Yields More Efficient Production
- Simplify fit equation of experimental data
- Given that $a_n < \frac{1}{2} (a_{n-1}+a_{n-2})$ and $a_n>0$. Show that $(a_n )$ is convergent
- Flux of vector field. Gauss' Theorem
- Importance of Fixed-point theorems
- Explicit description of maps $\phi_\alpha\in End(E)$ for a CM elliptic curve
- equality between sigma algebras
- Star-autonomous categories are categorifications of Boolean algebras?
- Is a volume defined in manifold with spacetime-signature?
- Bound and Operator Norm of this linear mapping example
- If $d(x,y) = \int_a^b |x(t)-y(t)| dt$ prove that $(X,d)$ is a metric space for $x,y \in X$ & $[a,b]$ into $\mathbb{R}$ [closed]
- Evaluating $\int_{0}^{\frac{\pi}{2}} x^{2} \ln (\cos x) d x$
- What is the "derivative" of a family of submanifolds?
- Is every finite set of reals Freiman isomorphic to a set of integers
- Design a DFA for the language $L = \{a^n b \mid n \geq 0 \}$ [closed]
- Find $\lim_{(x,y)\to (1,0)} \frac{(x-1)^2ln(x)}{(x-1)^2 + y^2}$ if it exists.
- Prove that $(x+1)^p\equiv x^p+1 \mod p^2$.
- Given ten letters : K,K,K,S,S,S,S,S,S,S. Find number of ways to arrange given ten letters such that no K should be there between two S?
- Prove that if W is a subspace of a vector space V
| Posted: 28 May 2022 03:44 PM PDT Consider the Gamma function $$ \Gamma(x)=\int_{0}^{\infty} t^{x-1} e^{-t} \mathrm d t \quad \forall x>0. $$ From this Wikipedia page, I get for all $x_{1}, x_{2}>0$ and $t \in[0,1]$, $$ \Gamma\left(t x_{1}+(1-t) x_{2}\right) \leq \Gamma\left(x_{1}\right)^{t} \Gamma\left(x_{2}\right)^{1-t} . $$
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| Flux of vector field. Stokes' Theorem Posted: 28 May 2022 03:41 PM PDT Let´s consider the vector field f $(x,y,z) = (y-z,z-x,x-y)$ and $ b > a >0 $ . How could we calculate the flux of f through the surface $S = \{x^2+y^2+z^2 = 2ax, x^2+y^2 \leq 2bx, z \geq 0\} $ oriented upward using Stokes´ Theorem? Any help would be greatly appreciated. |
| Standard term for "area between a tangent to a hyperbola and its two asymptotes"? Posted: 28 May 2022 03:34 PM PDT Is there a standard concise term for:
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| Posted: 28 May 2022 03:31 PM PDT $\left(x\right)$ is a 2ndorder derivable function. Given that $f''\left(x\right)=3x^2+1$, $f\left(0\right)=1$ and $f\left(2\right)=-1$, find $f\left(x\right)$. Using $f\left(x\right)\:=\:x^2\:+\:c$ as the general equation for a second-order derivable function I've arrived at this: $f\left(0\right)\:=\:1$ $1\:=\:\left(0\right)^2\:+\:c$ $1\:=\:c$ and $f\left(2\right)\:=\:-1$ $-1\:=\:\left(2\right)^2\:+\:c$ $-1\:=\:4\:+\:c$ $-5\:=\:c$ So $f\left(x\right)\:$ ends up like this: $f\left(x\right)\:=\:x^2\:-\:5$ Is that correct? |
| Generalization on double expectation Posted: 28 May 2022 03:24 PM PDT First, For $E[Z|X]$ denotes the expectation of $Z$ given the joint distribution of $X$, if I wanted to obtain $E[Z]$, I can simply use $E[E[Z|X]]$, but what if I have $E[Z|X,Y]$, how could I obtain $E[Z]$. Second, how to interpret $E_{X,Y|A}[\cdot]$? I understood it as the expectation of $[\cdot]$ given the distribution of $X,Y|A$? And how do I obtain $E_{X|A}[\cdot]$ or simply $E_X[\cdot]$ from $E_{X,Y|A}[\cdot]$ Following is the context of which I was trying to understand: given that $r_a(h)=E_{X,Y|A=a}[l((h(X),Y)]$, to decompose $\sum_{a\in A} \mu_a r_a(h)$ $$ \begin{align} \sum_{a\in A} \mu_a r_a(h)&=\sum_{a\in A}\mu_aE_{X|a}[E_{Y|X,a}[l((h(X),Y)]] \\&=\sum_{a\in A}\mu_aE_X[\frac{p(X|a)}{p(X)}E_{Y|X,a}[l(h(X),Y)]] \\&=\sum_{a\in A}\mu_aE_X[\frac{p(X|a)}{p(X)}E_{Y|X}[\frac{p(Y|X,a)}{p(Y|X)}l(h(X),Y)]] \\&=E_x[\frac{1}{p(X)}E_{Y|X}[\frac{\sum_{a\in A}\mu_ap(X|a)p(y|X,a)}{p(Y|X)}l(h(X),Y)]] \\&=E_X[\frac{\sum_{a\in A}\mu_ap(X|a)}{p(X)}E_{Y\sim P^{\mu}(Y|X)}[l(h(X),Y)]] \end{align}$$ with $P^u(Y|X)=\frac{\sum_{a\in A}\mu_ap(X|a)p(Y|X,a)}{\sum_{a\in A}\mu_ap(X|a)}$ and denoting $p(y|X,a)=\{p(Y=y_i|X,a\}_{i=1}^{|\mathcal{Y}|}$ the conditional probability mass vector of $Y|X,a$ If someone could please help me with the questions I had above and extend the concept to the decomposition of $\sum_{a\in A}\mu_ar_a(h)$, that would be really helpful! Thank you so much for your time and efforts in advance |
| Almost sure covergence of exponentially distributed random variables Posted: 28 May 2022 03:44 PM PDT Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, $(\lambda_n)_{n\ge1}$ be a sequence of positive real numbers such that $\lim\limits_{n\to\infty}\lambda_n=\infty$ and $(X_n)_{n\ge1}$ be a sequence of exponentially distributed random variables with parameter $\lambda_n$. Then show that $X_n\xrightarrow{a.s.}0$ is not necessarily true. Here I can observe from intuition that it is enough to show the statement is false for $\lambda_n=c+\log n,\quad\forall n\ge1$ where $c>0$ but I want to prove it mathematically using the following definition of almost sure covergence of random variables: A sequence of random variables $(X_n)_{n\ge1}\xrightarrow{d}X$, converges almost surely to the random variable $X$ if $\mathbb{P}\{\omega\in\Omega\mid\limsup\limits_{n\to\infty} |X_n(\omega)-X(\omega)|\ge\varepsilon\}=0,\quad\forall\varepsilon\ge0$ but I am struggling with properly formulating the mathematical statements here as a beginner in measure-theoretic probability theory. If someone could show me how to proceed I will really appreciate it. |
| More information on 'constructing' integrals Posted: 28 May 2022 03:17 PM PDT A pretty simple question: Derive the integral formula for the area of the surface obtained by rotating a smooth curve $y = y(x) > 0$ around the $x$-axis, between the values $x = −R$ and $x = R$. Solution:
Honestly I'm concerned with how little I understand this construction. I'm quite far through an undergrad maths course but nothing like has ever been explicitly taught. I was wondering if anyone knows any online resources with some examples of constructing (simple) integrals like this in a similar way to what's written above so that I can learn on this. Also it would be great to see $(*)$ written out more explicitly so I can see exactly what is going on here. Thank you! |
| Pointwise convergence of specific sequence of functions Posted: 28 May 2022 03:16 PM PDT I am trying to prove that the following sequence of functions converges pointwise using $\delta-\epsilon$ notation: $$ f_n(x) = \begin{cases} 1 & \mathrm{if}\; \frac{1}{2} - \frac{1}{n} \leq x \leq\frac{1}{2} + \frac{1}{n} \\ 0 & \mathrm{otherwise}\end{cases} $$ I know that if I take: $$ \lim_{n \to \infty} f(x) = \begin{cases} 1 & \mathrm{if}\, x = \frac{1}{2} \\ 0 & \mathrm{otherwise}\end{cases} $$ Using $\delta-\epsilon$ notation, if I let: $$ N = \frac{1}{x-\frac{1}{2}} + 1 $$ Will this be enough to prove pointwise convergence via: $$ n\geq N \implies \left|f_n(x) - \begin{cases} 1 & \mathrm{if}\, x = \frac{1}{2} \\ 0 & \mathrm{otherwise}\end{cases}\right| = 0 \,\,\, \forall x $$ |
| Which f satisfies E[f(X) X] = E[f(X)] E[X]? Posted: 28 May 2022 03:14 PM PDT Consider a non-negative discrete random variable $X$, with known probability mass function $p_X(x;\theta)$. I am looking for a function $f$ that satisfies the following equation: $$ \mathbb{E}_\theta[f(X) X] = \mathbb{E}_\theta[f(X)]\mathbb{E}_\theta[X] \tag{1}$$ It is immediate that a constant function (with probability $1$) satisfies the previous condition. That is, $f(X) = c \quad \forall X \in \mathcal{X}$, with $\mathbb{P}[X \in \mathcal{X}] = 1$, satisfies (1). What restrictions does (1) impose on $f$? Can I safely assume that $f(X)$ cannot depend on $X$? Note 1: (1) must be valid for any possible value of $\theta$. Note 2: $p_X(x;\theta )$ is a member of the one-parameter exponential family. Note 3: $X$ is a complete minimal sufficient statistics of $\theta$. |
| Cost Optimization Problem: Which of 2 Manufacturing Processes Yields More Efficient Production Posted: 28 May 2022 03:12 PM PDT Armstrong Metals is a new metal manufacturer that has recently discovered a new alloy, Z, of which, one batch of Z can be made by combining one batch of metal X and one batch of metal Y. Furthermore, the company has three different manufacturing processes of note, processes A, B, and C, each producing X and Y metals in differing amounts and also costing different amounts to perform. The average costs and amount of metals produced in each process is already known and is given as follows: Process C:
Process B:
Process A:
The company is trying to figure out, between A and B, which manufacturing process will yield the most alloy Z (factoring in both by combining X + Y as well as from direct byproducts) for the least amount of money spent. Is the problem solveable? If so, solve it and give the more efficient manufacturing method between A and B as well as the cost value of alloy Z using this method. If not, explain why not. I'm not sure how to even start this problem. Any help is appreciated. |
| Simplify fit equation of experimental data Posted: 28 May 2022 03:12 PM PDT Is there any way to simplify the following expression: $$ y = 1.9365x^{0.5} - 4.3418x^{0.555} $$ I guess by simplify I mean somehow combine the terms with $x$ in them or something similar. There might be ways to do something I'm not familar with as well. |
| Given that $a_n < \frac{1}{2} (a_{n-1}+a_{n-2})$ and $a_n>0$. Show that $(a_n )$ is convergent Posted: 28 May 2022 03:45 PM PDT I have the following exercise:
My idea is to prove that $a_n$ is a bounded sequence and that is (I hope) quite easy to show because it can be possible to find an upper bound for $a_n$ writing a series of strictly inequalities and I showed that$$a_n <\frac{2^{n-N_0}a_{N_0}+\text{something}}{2^{n-N_0}}$$where $N_0$ is the natural number for that my sequence has that relation between $a_n$ and the two terms before. How can I prove now that $a_n$ is convergent? |
| Flux of vector field. Gauss' Theorem Posted: 28 May 2022 03:09 PM PDT Let´s try to solve the following exercise : Calculate the flux of the vector field: f$(x,y,z) = (x^2,y^2,z^2)$ through the surface $S = \{x^2+y^2=z^2,0\leq\ z \leq\ 1\} $ (orientation fixed). My attempt I tried to solve it using Gauss' Theorem. The closed surface of the volume would be $ S $ $ \cup$ $S_1 $ where $S_1 $ is the surface that describes the circle $ x^2 +y^2 =1 $ in the plane $ z=1$. We are interested in calculating the integral: $ \iint_S $ f $\cdot\ $n $dS = \iiint_V div $ f $dxdydz - \iint_{S_1} $ f $\cdot\ $n $dS_1 $ In my case: $div$ f $ =2x+2y+2z $ To calculate the first integral, I decided to use cylindrical coordinates: $x=\rho \cos\varphi$ $y = \rho \sin \varphi$ $z = z$ Using that $x^2+y^2=z^2 $ on the cone, we have that $\rho = z$. The limits of integration could be: $ 0\leq\ \rho \leq\ 1\ $ $ 0\leq\ \varphi \leq\ 2\pi $ $ \rho\leq\ z \leq\ 1 $ Putting it all together: $ \iiint_V div$ f $dxdydz = 2 \cdot \int _{0}^{1} \int _{0}^{2\pi} \int _{\rho}^{1} (\rho (\cos \varphi + \sin \varphi) +z) \cdot \rho \cdot d \rho d\varphi dz $ For the second integral, we parametrize the surface: $r_1(x,y) = (x,y,1)$ $ \iint_{S_1} $ f $\cdot\ $n $dS_1 = \iint_Q (x^2,y^2,1) \cdot (0,0,1) dxdy = \mu (Q) = \pi $ where $Q = \{(x,y) \in R^2|x^2+y^2 \leq\ 1\} $ My questions are: Is it correct? What can we say about the orientation? Is there a simpler way? Any suggestions are welcome. |
| Importance of Fixed-point theorems Posted: 28 May 2022 03:45 PM PDT I have a more general question on the importance of fixed-point theorems. In mathematics youre being introduced to so many fixed-point theorems but i still could not figure out why they are so important. Why would be a simply looking statement as $f(x)=x$ be so important. I would appreciate any answer. Thanks in advance. On wikipedia it says nothing about the importance, contextualisation of theorems in mathematics is sometimes not given. |
| Explicit description of maps $\phi_\alpha\in End(E)$ for a CM elliptic curve Posted: 28 May 2022 03:44 PM PDT Let $E$ be an elliptic curve with complex multiplication. When $n\in \mathbb{Z}\hookrightarrow End(E)$, we can define its associated map as $\phi_n: P\mapsto P+..._{n \text{ times}}+P$. However when $\alpha\not\in \mathbb{Z}$ (but is an element of an imaginary quadratic field and of $End(E)$), I am unsure what the explicit description of $\phi_\alpha$, the associated endomorphism of $E$, is. Question: What is the explicit description of the map $\phi_\alpha$? A possible construction I have in mind is utilizing the isomorphism $\phi: \mathbb{C}/L_E\simeq E$ to write $\phi_\alpha(P)=(\phi\circ \alpha\circ \phi^{-1})(P)$ where $\alpha$ denotes multiplication by $\alpha$. I'm unsure if this is explicit, though, and would not know how to compute this map for specific examples. |
| equality between sigma algebras Posted: 28 May 2022 03:12 PM PDT
For a function $f$ and a set of subsets of the range of $f$, say $\{A_1,A_2,\cdots\}$, define $f^{-1}(\cup_n \{A_n\})$ to be $\cup_n \{f^{-1}(A_n)\}$ I think one can generalize and prove the following:
$\subseteq $ is straightforward as $f^{-1}(\sigma(F))$ is a sigma-algebra containing $f^{-1}(F)$. But I'm not sure how to prove the other direction. If the claim can be used, then taking $F$ to be the set of open subsets of $\mathbb{R}$, $f^{-1}(\sigma(F)) = \sigma(f^{-1}(F))$, which is a subset of the set of measurable sets because by assumption $f^{-1}(F)$ is a subset of measurable sets. |
| Star-autonomous categories are categorifications of Boolean algebras? Posted: 28 May 2022 03:26 PM PDT 1. Question
How do star-autonomous categories behave as categorified versions of Boolean algebras or Boolean rigs? 2. Wikipedia says
I suppose the internal hom of two objects $a,b$ in this category is the join $\neg a \lor b $, correct? The dual functor is the complement? |
| Is a volume defined in manifold with spacetime-signature? Posted: 28 May 2022 03:36 PM PDT A volume in $R^n$ is easily calculated by multiplying the lengths of the $n$ dimensions. I'm wondering how the different sign of time acts on the volume in a manifold with spacetime signature (1,3). Is a volume defined at all in such a manifold? |
| Bound and Operator Norm of this linear mapping example Posted: 28 May 2022 03:37 PM PDT For an exercise, I need to investigate whether some linear mappings are bounded and determine the operator norm. I seem to be stuck on this particular one : $T_3 : l^p(\mathbb{N}) \rightarrow l^p(\mathbb{N})$ , $\{x_n\}_{n \in \mathbb{N}} \rightarrow \{c_n x_n\}_{n \in \mathbb{N}}$ , $p \in [1, \infty]$ , $c = \{c_n\}_{n \in \mathbb{N}} \in l^{\infty}(\mathbb{N})$ My approach towards boundedness would be : $\left\| T_3(x) \right\|_p = \left\| c_1 \cdot x_1 + c_2 \cdot x_2 + ... \right\|_p = \left\| \sum_{i = 1}^{\infty} c_i \cdot x_i \right\|_p \leq \left\| c \right\|_p \cdot \left\| x \right\|_p$ Now, boundedness can be shown if $\left\| c \right\|_p \leq C$. However, my confusion is to exactly show/investigate this since we also have that $c = \{c_n\}_{n \in \mathbb{N}} \in l^{\infty}(\mathbb{N})$, meaning I am not sure how to apply the $\left\| . \right\|_p$ to $c$ as defined in this case. Help is much appreciated! :-) |
| Posted: 28 May 2022 03:11 PM PDT I know I have to prove the four properties of metric spaces, but I am unsure of one specific step: I have shown that $|x(t)-y(t)| \geq 0.$ But I need to show that also $\therefore \int_a^b|x(t)-y(t)|dt \geq 0$. to satisfy first property (nonnegativity). Do I need to prove this step? Is this the right way to go? |
| Evaluating $\int_{0}^{\frac{\pi}{2}} x^{2} \ln (\cos x) d x$ Posted: 28 May 2022 03:10 PM PDT By the Fourier Series of ln(cos x) ), $ \displaystyle \ln (\cos x)=-\ln 2+\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \cos (2 n x) \tag*{(*)} $ Multiplying (*) by $ x^2$ and then Integrating both sides from $0 $ to $\frac{\pi}{2}$ yields $\displaystyle I=-\underbrace{\int_{0}^{\frac{\pi}{2}} x^{2} \ln 2 d x}_{\frac{\pi^{3} \ln 2}{24}}+\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \underbrace{ \int_{0}^{\frac{\pi}{2}} x^{2} \cos (2 n x) d x}_{J_n}\tag*{} $ Integrating by parts twice yields $\displaystyle \begin{aligned}J_{n} &=\frac{1}{2 n} \int_{0}^{\frac{\pi}{2}} x^{2} d(\sin 2 n x) \\&=\frac{1}{2 n}\left[x^{2} \sin 2 n x\right]_{0}^{\frac{\pi}{2}}-\frac{1}{n} \int_{0}^{\frac{\pi}{2}} x \sin 2 n x d x \\&=\frac{1}{2 n^{2}} \int_{0}^{\frac{\pi}{2}} x d(\cos 2 n x) \\&=\left[\frac{1}{2 n^{2}} x \cos 2 n x\right]_{0}^{\frac{\pi}{2}}-\frac{1}{2 n^{2}} \int_{0}^{\frac{\pi}{2}} \cos 2 n x d x \\&=\frac{\pi}{4 n^{2}} \cos n \pi-\frac{1}{2 n^{2}}\left[\frac{\sin 2 n x}{2 n}\right]_{0}^{\frac{\pi}{2}} \\&=\frac{\pi}{4 n^{2}} \cos n \pi\end{aligned}\tag*{} $ We can now conclude that $\displaystyle \begin{aligned}I &=-\frac{\pi^{3} \ln 2}{24}+\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \cdot \frac{\pi}{4 n^{2}} \cos n \pi \\&=-\frac{\pi^{3} \ln 2}{24}+\frac{\pi}{4} \sum_{n=1}^{\infty} \frac{(-1)^{n+1}(-1)^{n}}{n^{3}} \\&=-\frac{\pi^{3} \ln 2}{24}-\frac{\pi}{4} \sum_{n=1}^{\infty} \frac{1}{n^{3}} \\&=\boxed{-\frac{\pi^{3} \ln 2}{24}-\frac{\pi}{4} \zeta(3)}\end{aligned}\tag*{} $ Is there any other method to evaluate $I$? |
| What is the "derivative" of a family of submanifolds? Posted: 28 May 2022 03:43 PM PDT Suppose that $N(t)$ is a family of k-dimensional submanifolds in an n-dimensional manifold $M$, with $n > k$. Then $N(t)$ is a path in the space $\mathcal{M}$ of all k-dimensional submanifolds. I want to know, what is "$N'(t)$"? If $r(t)$ is a curve in a manifold $M$, which is finite-dimensional, I can easily calculate it's derivative $r'(t)$. How would I get the analogous thing for $N(t)$ as in the situation above? |
| Is every finite set of reals Freiman isomorphic to a set of integers Posted: 28 May 2022 03:41 PM PDT I believe I remember reading the following claim in a paper (the relevant definitions are recalled below):
While the claim makes intuitive sense to me, I'd like to see a formal proof. Definitions An abelian group is torsion-free if every non-zero element has infinite order (i.e. if adding $a$ to itself $k$ times is equal to the identity (for some finite $k>0$), then $a$ must be the identity). Given two groups $(A,*)$, $(B,\circ)$, and a subset $A'\subset A$, we say a homomorphism $\phi:A \to B$ is a Freiman $r$-isomorphism (wrt $A'$), if for $a_1,\dots,a_r,a_1',\dots,a_r' \in A'$, we have $$\phi(a_1)\circ \ldots \circ \phi(a_r) = \phi(a_1')\circ \ldots \circ\phi(a_r')$$ if and only if $$a_1 * \ldots * a_r = a_1' * \ldots * a_r'.$$ |
| Design a DFA for the language $L = \{a^n b \mid n \geq 0 \}$ [closed] Posted: 28 May 2022 03:23 PM PDT Problem : Design a deterministic finite automaton for the language $L= \{a^nb \mid n \geq 0 \}$ |
| Find $\lim_{(x,y)\to (1,0)} \frac{(x-1)^2ln(x)}{(x-1)^2 + y^2}$ if it exists. Posted: 28 May 2022 03:36 PM PDT Show that $\lim_{(x,y)\to (1,0)} \frac{(x-1)^2log(x)}{(x-1)^2 + y^2}$ exists. Also find the limit. Method 1 $\frac{(x-1)^2}{(x-1)^2 + y^2}$ is less than $1$. Therefore, $\frac{(x-1)^2log(x)}{(x-1)^2 + y^2}$ is less than log$x$. Also, $\frac{(x-1)^2log(x)}{(x-1)^2 + y^2} \geq 0$ for all $(x,y)$ in the domain. Therefore, we have, $0\leq\frac{(x-1)^2log(x)}{(x-1)^2 + y^2}\leq$ log$x$. But $\lim_{(x,y)\to (1,0)}$ log$x$ $=0= \lim_{(x,y)\to (1,0)}0$. Hence, by squeeze theorem, $\lim_{(x,y)\to (1,0)} \frac{(x-1)^2log(x)}{(x-1)^2 + y^2} = 0$ Method 2 Let $\varepsilon>0$. Choose $\delta<e^{\varepsilon}-1 \implies$ log$(\delta+1)<\varepsilon$. If, $(x-1)^2+y^2<\delta^2 \implies |x-1|<\delta \implies |x|-1<\delta$ $\implies x<\delta +1$ $ (\because$ log$x$ is not defined for $x<0)\implies $log$x$ $<$ log$(\delta+1)$, then, $|\frac{(x-1)^2log(x)}{(x-1)^2 + y^2}-0|\leq$|log$x$|$< |$log$(\delta+1)|<|\varepsilon| = \varepsilon$ Hence, by $\varepsilon-\delta$ definition of limit, $\lim_{(x,y)\to (1,0)} \frac{(x-1)^2log(x)}{(x-1)^2 + y^2}$ exists and is equal to $0$. Are my both methods correct? |
| Prove that $(x+1)^p\equiv x^p+1 \mod p^2$. Posted: 28 May 2022 03:09 PM PDT Let $p$ be a prime number and $x$ and integer such that $x^{2p}+x^p-1\equiv 0\mod p^2$. Prove that $(x+1)^p\equiv x^p+1 \mod p^2$. It seems to me there is no such $x$ in the first place. Any thoughts? |
| Posted: 28 May 2022 03:23 PM PDT Given ten letters : K,K,K,S,S,S,S,S,S,S. How many ways to arrange given ten letters such that no letter "K" between two letters "S", example : "KKSSSSSSSK", "KKKSSSSSSS", "SKKSSSSSSK", etc. I am confused. I have calculate ways to arrange 10 given letters, $$\dfrac{10!}{3!\cdot 7!}=120 \text{ ways.}$$ Now I want to calculate the complement of "to arrange given ten letters such that no "K" between two "S" ", that is "there is letter "K" between two letters "S"". If I calculate how many ways : $$8\cdot \dfrac{7!}{2!\cdot 5!}=168\text{ ways.}$$ That is impossible, negative number. $$120-168=-48\text{ ways.}$$ How to solve that combinatorics problem? Please help me :( |
| Prove that if W is a subspace of a vector space V Posted: 28 May 2022 03:38 PM PDT Prove that if $W$ is a subspace of a vector space $V$ and $w_1, w_2, ..., w_n$ are in $W$, then $a_1w_1 + a_2w_2 + ... + a_nw_n \in W$ for any scalars $a_1, a_2, ..., a_n$. My solution is we have $a_iw_i \in W$ for all $i$. And we can get the conclusion that $a_1w_1, a_1w_1 + a_2w_2, a_1w_1 + a_2w_2 + a_3w_3$ are in $W$ inductively. Any ideas on how to improve this because I feel it is not enough. Thank you in advance. |
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