Recent Questions - Mathematics Stack Exchange |
- If $A$ and $B$ have independent columns, prove that $B^TA$ is invertible
- Show that the limit of the sequence ${(n+a)(n+b)}^{\frac{1}{2}}-n =\frac{a+b}{2}$, where $a>0, b>0$
- When to put calculator in radian and when to put calculator in degree? What is the logic?
- Proving that a system of equations has a unique solution
- How to generate random combinations from large set in GAP
- For primes $p$ and $q$ prove that if $q$ | 2^p − 1, then $p < q$.
- Convergence or Divergence of Improper imtegral $\int_0^{\infty} e^{ax}x^b dx$ for $a,b \in \mathbb R.$
- Well-ordered commutative semirings
- Pulling-back a superset of a subset of the image
- Why is it that if $x$ is an element of the real line, it follows that $x$ is arbitrarily close to the rational numbers?
- Find the Biggest Integer in the List
- Slope field of differential equation
- Bacteria Problem: population starts at 7000, and growth rate in bacteria per hour is given
- Non-trivial subspaces with Set Addition and Intersection
- How to prove the inequality over expectation $E |u^H X v|^{2t}\le E |u^H J v|^{2t}$?
- Are the following statements True for all functions f and for all subsets A, B and C of the domain of f?
- Can we draw the conclusion that this function must be linear?
- what is the minimal polynomial for $P $?
- Maximal normal subgroups and quotient groups
- find the parametric form of the tangent line to the image of $f(t) = ( 4t^2,5/t,t-4)$ at $t=4 ?$
- How can I make an existing time series more volatile?
- show that $Y=\sum_{i=1}^{n} X_{i}$ has chi-squared distribution
- Is there an online graphing calculator that lets you define custom axes?
- Definition of a one to one correspondence via math for elementary teachers
- In the triangle $ABC AN$ is the bisector of the angle $A, N \in BC$. Express the vector $AN$ in terms of the vectors $b = AC$ and $c = AB$
- Does this function live in $H^{\frac{1}{2}}(\partial \Omega)$
- Laurent series proof if f(z)= g(z)+h(z) then g(z) is the expansion with Laurent coefficients
- Integrating inverse of a polynomial using valid method gives wrong result
- Analysis Proof review: Euclidean Algorithm
- Why do we take the dot product with the normal vector when we do Stokes' Theorem?
| If $A$ and $B$ have independent columns, prove that $B^TA$ is invertible Posted: 06 Sep 2021 08:33 PM PDT
If $A,B$ have independent columns then $Ax=0\implies x=0$ and $By=0\implies y=0$. I can also prove that $rank(B^TA)\leq \min(rank(B^T),rank(A))=r$ We have to prove, $B^TAx=0\implies x=0$, it would be helpful to get a hint on how to approach this. Note: This is needed to derive the expression for an oblique projection matrix in the form $P=A(B^TA)^{-1}B^T$.  |
| Show that the limit of the sequence ${(n+a)(n+b)}^{\frac{1}{2}}-n =\frac{a+b}{2}$, where $a>0, b>0$ Posted: 06 Sep 2021 08:32 PM PDT I tried to find the limit of \begin{equation} {(n+a)(n+b)}^{\frac{1}{2}}-n =\frac{a+b}{2}, \text{ where } a>0, b>0 \end{equation} I took the help of rationalization  |
| When to put calculator in radian and when to put calculator in degree? What is the logic? Posted: 06 Sep 2021 08:30 PM PDT How to know when to put calculator in radian or degree mode? Here is already such a question but that question is too trivial for my case. I am signal processing student. And There are places when I get right answer using degree, whereas there are places where I get right answer using radians. Like what is the logic here?  |
| Proving that a system of equations has a unique solution Posted: 06 Sep 2021 08:28 PM PDT I'm trying to prove this follow statement.
I'm trying to prove this without matrices, if that's at all possible, as I know this is really saying that the coefficient matrix is not invertible. Is there such a way? It's difficult to navigate the difference cases where on or more of $a,b,c,d,r,sr$ is $0$ but $ad - bc \neq 0$.  |
| How to generate random combinations from large set in GAP Posted: 06 Sep 2021 08:19 PM PDT I need to generate random combinations (unordered tuples without duplication). This code works fine for relatively small $n$ but I need much larger n $n>200$ for example Are there other options for doing this in GAP?  |
| For primes $p$ and $q$ prove that if $q$ | 2^p − 1, then $p < q$. Posted: 06 Sep 2021 08:19 PM PDT For primes $p$ and $q$ prove that if $q$ | $2^p − 1$, then $p < q$.  |
| Posted: 06 Sep 2021 08:15 PM PDT I'm considering whether $\displaystyle\int_0^{\infty} e^{ax}\ x^b dx$ converges or diverges for $a,b \in \mathbb R$. If $a, b > 0,$ then \begin{align} \displaystyle\int_0^{\infty} e^{ax}\ x^b dx =\int_0^1 e^{ax}\ x^b dx+\int_1^{\infty} e^{ax}\ x^b dx.\\ \end{align} \begin{align} \int_1^{\infty} e^{ax}\ x^b dx &\geqq \int_1^{\infty} e^{ax} dx (\because x^b \geqq 1 \ \mathrm{when}\ x\geqq 1.)\\ &=\left[\dfrac{1}{a} e^{ax} \right]_1^{\infty}\\ &=\infty. \end{align} Thus $\displaystyle\int_0^{\infty} e^{ax}\ x^b dx$ diverges. If $a<0$ and $-1<b<0,$ then similarly, \begin{align} \displaystyle\int_0^{\infty} e^{ax}\ x^b dx =\int_0^1 e^{ax}\ x^b dx+\int_1^{\infty} e^{ax}\ x^b dx.\\ \end{align} and \begin{align} \int_0^1 e^{ax}\ x^b dx &\leqq \int_0^1 x^b dx \\ &=\left[\dfrac{x^{b+1}}{b+1} \right]_0^1 \\ &=\dfrac{1}{b+1}<\infty. \end{align} \begin{align} \int_1^{\infty} e^{ax}\ x^b dx &\leqq \int_1^{\infty} e^{ax} dx \\ &=\left[ \dfrac{1}{a} e^{ax} \right]_1^{\infty} \\ &=-\dfrac{e^a}{a}<\infty. \end{align} Thus $\displaystyle\int_0^{\infty} e^{ax}\ x^b dx$ converges. I don't know how I can determine convergence or divergence in other cases. I'd like you to give me any idea.  |
| Well-ordered commutative semirings Posted: 06 Sep 2021 08:31 PM PDT I am interested in the characterization of the most important types of numbers from an axiomatic viewpoint. For example, every complete ordered field is isomorphic to the field of real numbers. In this sense, being a complete ordered field characterizes the real numbers. I am now looking at the much simpler set of positive integers. It is a well-ordered commutative semiring. Does this property characterize them? In other words, are there any well-ordered commutative semirings other than the positive integers?  |
| Pulling-back a superset of a subset of the image Posted: 06 Sep 2021 08:15 PM PDT Is it dangerous to assume that pullback of a superset of the image is properly defined? More precisely, say $f:A\to C$ and $f(A)=B\subset C$ where $C-B\neq\emptyset$. Is it okay to assume that $f^{-1}(C)$ is defined even though $\exists x\in(C-B)$ such that $f(x)$ is not defined? Can I just say that $f^{-1}(C)=A$ or do I have to say $f^{-1}(C)\subset A$?  |
| Posted: 06 Sep 2021 08:13 PM PDT I have to prove this, but I'm not sure where to start.
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| Find the Biggest Integer in the List Posted: 06 Sep 2021 08:25 PM PDT Can you please suggest outcome of the below code mathematically in summation form? For example, like the explanation with sigma.. Thank you.  |
| Slope field of differential equation Posted: 06 Sep 2021 08:02 PM PDT
Hello! This is my homework and I'd be very grateful if someone can help me to solve this question. I cannot figure out the solution of the DE since its RHS is $f(x,y)$ and I do not have any information about it.  |
| Bacteria Problem: population starts at 7000, and growth rate in bacteria per hour is given Posted: 06 Sep 2021 08:02 PM PDT Problem is as follows: A bacterial population is $7000$ at time $t = 0$, and its rate of growth is $1,000 · 4^t$ bacteria per hour at time $t$ hours. What is the population after one hour? (Round your answer to the nearest whole number.) My approach: I'm having a hard time understanding how this involves calculus at all. It seems to me as simple as $1,000*4^(1) + 7,000 = 11,000.$ But it's obviously wrong. Why do we need to integrate anything if we already have the rate of growth?  |
| Non-trivial subspaces with Set Addition and Intersection Posted: 06 Sep 2021 07:57 PM PDT Exercise Find examples of non-trivial (vector) subspaces $X,Y,Z \subset \mathbb{R}^2$ such that $X+Y = \mathbb{R}^2$ and $X \cap Z = Y \cap Z = \{0 \}$. So, to approach this problem I'm thinking of doing the following: We should let $X = \{\langle 0, 0 \rangle\} \cup \{\langle 0,b^+ \rangle\} \cup \{a^+,0\}$ such that $a^+,b^+ > 0$ and $Y=\{\langle 0, 0 \rangle\} \cup \{\langle 0,b^- \rangle\} \cup \{a^-,0\}$ such that $a^-,b^- < 0$. Then $X+Y=\mathbb{R}^2$ since every $\langle a,b \rangle \in \mathbb{R}^2$ can be written as the sum of positive and negative real numbers (adjusting individual coordinates, if necessary). My biggest issue here is finding a $Z \subset \mathbb{R}^2$ such that $X \cap Z$ and $Y \cap Z = \{\langle 0, 0 \rangle\}$. The only non-trivial subspace of $\mathbb{R}^2$ is a line through the origin, but one can verify quickly that $X \cap Z$ and $Y \cap Z$ collect many more points than just the zero vector. Thus, my assumption is that my constructions of $X$ and $Y$ are insufficient. (P.S. There don't even exist three non-trivial subspaces of $\mathbb{R}^2$?? Can someone explain this as well?)  |
| How to prove the inequality over expectation $E |u^H X v|^{2t}\le E |u^H J v|^{2t}$? Posted: 06 Sep 2021 08:22 PM PDT Assume that $X \in R^{M\times N}$ has positive entries and $\| X \|_F^2=MN$. The random vectors $ u \in C^{M\times 1}$ and $ v \in C^{N\times 1}$ have entries given by \begin{align} u_m = e^{j\theta_m}, v_n =e^{j\phi_n}, \end{align} where $\theta_m$ and $\phi_n$ are i.i.d. uniformly distributed in $[0,2\pi]$. I want to prove the following \begin{align} E |u^H X v|^{2k}\le E |u^H J v|^{2k}, \end{align} where $J \in R^{M\times N}$ is all one matrix and $k\ge 0$ is integer. I have done some numerical simulations, which implies the inequality holds. But how can we prove it? Any idea is appreciated!  |
| Posted: 06 Sep 2021 07:59 PM PDT I came across this question in my assignment: Are the following true for all functions f and for all subsets A, B and C of the domain f? (i) $(\{f(x): x\in A\} \cup \{f(x):x\in B \}) \cap \{f(x):x\in C\}$ is a subset of $(\{f(x):x \in A\}\cap \{f(x):x\in B \}) \cup \{f(x):x\in C\}$ (ii) $(\{f(x):x \in A \} \cap \{f(x):x \in B\}) \cup \{f(x):x \in C\}$ is a subset of $(\{f(x):x \in A\} \cup \{f(x):x \in B\}) \cap \{f(x):x \in C\}$ The way I reasoned this was I assume some random elements for A,B,C and then I calculated the results. And the answer I came up with: True for (i) and False for (ii). Can anyone tell me if the answer is correct, or I am not supposed to do it this way?  |
| Can we draw the conclusion that this function must be linear? Posted: 06 Sep 2021 08:12 PM PDT Consider $f: \mathbb{R}^n \rightarrow \mathbb{R}$. $f\left(\alpha x+\left(1-\alpha\right)x'\right)=\alpha f\left(x\right)+\left(1-\alpha\right)f\left(x'\right)$, $\forall \alpha \in \left[0,1\right]$ and $\forall x, x' \in \mathbb{R}^n$. And if so, how to prove it? Thank you very much! Edit: Why am I get down-voted? Does this question violate some rules or it is too simple or something? I am confused. My attempt is below: I thought "linear function" means $\exists b_1, ..., b_n$ such that $f\left(x\right)=b_1 x_1+...+b_n x_n$. But I have totally no idea what can be the "upstream" step, that is, what do I need to prove first, and then it follows $f$ is linear.  |
| what is the minimal polynomial for $P $? Posted: 06 Sep 2021 08:16 PM PDT let $P$ be the operator on $\mathbb{R}^2 $ which project each vector onto the $x- $axis ,parallel to the $y$ axis :$ P(x,y)=(x,0) $. What is the minimal polynomial for $P $? My attempt : $ P$ doesn't satisfy any constant polynomial because $P \neq 0$ since the operator $P$ defined on $\mathbb{R}^2$ so $P^2(x,y)=P[P(x,y)]=(x,0)=P(x,y)$ $P^2 -P=0 \implies P$ satisfies the polynomial $\lambda^2-\lambda$ Therefore the minimal polynomial for $P $ is $\lambda^2-\lambda$ Is it true ?  |
| Maximal normal subgroups and quotient groups Posted: 06 Sep 2021 08:19 PM PDT I am trying to understand quotient groups and their implications, I came across a theorem: if G is a group, and M is a maximal normal subgroup of G,then G/M is simple. The issue is assuming by contradiction that G/M is not simple, there exists a proper normal subgroup A/M of G/M. I understand that implies A contains M which is a contradiction since M is maximal. The issue I have is: why does A must contain M? given A/M is normal in G/M.  |
| find the parametric form of the tangent line to the image of $f(t) = ( 4t^2,5/t,t-4)$ at $t=4 ?$ Posted: 06 Sep 2021 08:31 PM PDT find the parametric form of the tangent line to the image of $f(t) = ( 4t^2,5/t,t-4)$ at $t=4 ?$ I need some help Pliz don't downvote im a new user  |
| How can I make an existing time series more volatile? Posted: 06 Sep 2021 08:28 PM PDT I have an existing time series that I would like to make more volatile, or more variance. Below is my attempt. This was my attempt.
This is the original.
What I want is just a time series with more extreme swings. I want everything to scale from the mean trend line.  |
| show that $Y=\sum_{i=1}^{n} X_{i}$ has chi-squared distribution Posted: 06 Sep 2021 08:06 PM PDT Let $X_1,X_2,...,X_n$ independent random variables with chi squared distribution with $k_i$ degrees freedom show that $$Y=\sum_{i=1}^{n} X_{i}$$ has chi-squared distribution with $$g=\Sigma_{i=n}^{n} k_{i}$$ degrees freedom I was trying to do it with the moment generating function of the sum of $n$ random variables with is the product but i dont know, anu hint?  |
| Is there an online graphing calculator that lets you define custom axes? Posted: 06 Sep 2021 08:07 PM PDT Want to transform the x-axis to something else, say x^2 . Need something quick to set up like an online one. Haven't used Matlab in years don't remember it at all.  |
| Definition of a one to one correspondence via math for elementary teachers Posted: 06 Sep 2021 08:18 PM PDT From "Mathematical Reasoning for Elementary Teachers 7th edition" From page 79: Definition: One-to-One correspondence: A one-to-one correspondence between sets A and B is an assigment, for each element of A, of exactly one element of B in such a way that all the elements of B are used. It can also be thought of as a pairing between elements of A and B such that each element of A is matched with one and only one element of B and every element of B has an element of A assigned to it. So, in my mind, the definition above describes a bijective correspondence between sets and a one-to-one corrrespondence would be the standard $r(x)=r(y) \rightarrow x=y$ definition. That being said, I feel like throughout my time of reading maths, sometimes it seems that the phrase "one-to-one" is meant to mean bijective. Can somebody enlighten me to what's going on here? Am I not understanding something?  |
| Posted: 06 Sep 2021 08:28 PM PDT In the triangle $ABC$, $\vec {AN}$ is the bisector of the angle $A, N \in \vec{BC}$. Express the vector $\vec{AN}$ in terms of the vectors $\vec b= \vec{AC}$ and $\vec C=\vec{AB}$. where I made a mistake? Picture 1:
I don't understand how to complete this task correctly. In the second photo, the vectors are folded in the middle. but why do I have an incorrect answer in my task? what should the solution look like? After all, we also add two vectors. Picture 2:
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| Does this function live in $H^{\frac{1}{2}}(\partial \Omega)$ Posted: 06 Sep 2021 08:06 PM PDT In the Poisson equation $$-\Delta u=f$$ $$u=g \in \partial \Omega$$ we usually see in textbooks the requirement $g \in H^{\frac{1}{2}}(\partial \Omega)$ and $u \in H^1(\Omega)$. In my case, $\Omega = [0,1]^2$ In an application, I am dealing with a portion $\Gamma \subset \partial \Omega$ ($\Gamma$ is the union of the lower and left edge) and I have that my data $g$ is \begin{cases} 1 \quad x \in \Gamma \\0 \quad x \in \partial\Omega \setminus\Gamma\end{cases} Is $g \in H^{\frac{1}{2}}(\partial \Omega)$ ? I should check that $\frac{|g(x)-g(y)|}{|x-y|^{3/2}} \in L^2(\partial \Omega \times \partial \Omega)$. But the term $g(x)-g(y)=0$ always, so I'd say that indeed $g$ is in $H^\frac{1}{2}(\partial \Omega)$ Is that correct?  |
| Laurent series proof if f(z)= g(z)+h(z) then g(z) is the expansion with Laurent coefficients Posted: 06 Sep 2021 08:02 PM PDT i. z such that $R<|z|<S$ and $f(z)=g(z)+h(z)$ with g holomorphic on the disk$|z|<S$ and h holomorphic and bounded on the set $\{z\in \mathbb{C}:|z|>R\}$. If $\{c_n\}$ with Laurent coefficients of f in the ring $\{z\in \mathbb{C}: R<|z|<S\}$, consider the expansion of de g y h to prove that \ $$ g(z) = c+\sum_{n=1}^\infty c_nz^n \quad (\text{with}\; |z|<S, c\; \text{constant}) $$  |
| Integrating inverse of a polynomial using valid method gives wrong result Posted: 06 Sep 2021 08:19 PM PDT The integral is: $$\int \frac{1}{x^2+(x-1)^2}dx$$ One way to integrate inverses of polynomials is to find their roots and find the fractions which when added together, give the original fraction (with the roots being the denominators). For example: $$\int \frac{1}{x^2-x-2}dx = \int \frac{1}{(x-2)(x+1)}dx\rightarrow \frac{a}{x-2}+\frac{b}{x+1}=\frac{a(x+1)}{(x-2)(x+1)}+\ \frac{b(x-2)}{(x-2)(x+1)}$$ $$\rightarrow ax+a+bx-2b=1 \rightarrow \left\{ \begin{array}{c} ax+bx=0x \\ a-2b=1 \\ \end{array} \right. \rightarrow a=\frac{1}{3}, b=-\frac{1}{3}$$ $$\rightarrow \int \frac{1}{x^2-x-2}dx=\int (\frac{1/3}{x-2}-\frac{1/3}{x+1})dx$$ And then solve. I know this is known, I just wanted to show what I've been taught exactly, no ambiguation. So about the question. We can open up the squares to get: $$\int \frac{1}{x^2+(x-1)^2}dx = \int \frac{1}{2x^2-2x+1}dx=\int \frac{1}{2(x-0.5-0.5i)(x-0.5+0.5i)}dx$$ $$\rightarrow \frac{a}{x-0.5-0.5i}+\frac{b}{x-0.5+0.5i}=\frac{1/2}{(x-0.5-0.5i)(x-0.5+0.5i)}$$ I'll spare the details: $$a=-\frac{i}{2},b=\frac{i}{2}$$ $$\rightarrow \int \frac{1}{x^2+(x-1)^2}dx=\int (\frac{i/2}{x-0.5+0.5i}-\frac{i/2}{x-0.5-0.5i})dx=i(\int \frac{1}{2x-1+i}dx-\int \frac{1}{2x-1-i}dx)$$ Once again sparing the details of some simple u substitution: $$=\frac{i}{2}(ln\lvert 2x-1+i\rvert-ln\lvert 2x-1-i\rvert)$$ Absolute value (which's added from integrating something to the (-1)) always gives out a real number, and the entire "ln" expression is multiplied by an imaginary number, so the output is always imaginary. The actual video I took this from contained a completely different solution. It's not even close to mine, which's always imaginary. Why? How did this happen? Thanks.  |
| Analysis Proof review: Euclidean Algorithm Posted: 06 Sep 2021 08:16 PM PDT I'm self-studying Real Analysis using Terence Tao's Analysis-I book. I need advise or correction if something is incorrect with my proof. Thank you beforehand. Claim: (Proposition - 2.3.9) Proof: The base case: Induction step: We can write, $n+1 = (mq + r) + 1$ Hence this closes the induction.  |
| Why do we take the dot product with the normal vector when we do Stokes' Theorem? Posted: 06 Sep 2021 08:04 PM PDT So this part I'm struggling with on Stokes' Theorem: $$\iint_S ~(\text{curl}~\vec{F} \cdot \hat{n})~ dS$$ I don't really understand why we would want to dot it with the unit normal vector at that point. This is going to tell us how much of the curl is in the normal direction but why would we want this surely we only care about how much the curl is actually on the surface as opposed to normal to the surface it seems to me like this is actually the opposite of what we want. Very counter intuitive to me. I'm guessing I am misinterpreting something here so if someone would explain that would be fantastic. Also in the next line it says $$\iint_S ~(\text{curl}~\vec{F} \cdot \hat{n})~ dS=\iint_S ~\text{curl}~\vec{F} \cdot d\vec{S}$$ so is this just a notation to say that $$\hat{n}\cdot~ dS= d\vec{S}$$ why is this so it it just purely for convenience or is there some reason to write it like this, I struggle to see why some differential of the surface would be a vector? To me it's just a little chunk of the surface area. I really need this clarified and cleared up thanks.  |
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