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H: Topology exercises - closure, frontier
Are my proofs correct? The topology is $\mathbb{R^n}$
Exercises.
Prove that a set $A$ is closed iff $Fr(A)\subseteq A$
A set $A$ is closed iff $A=Cl(A)$
For any set, $A$, $Fr(A)$ is closed
For any set $A\subseteq \mathbb{R^n}$, $Fr(A)=Fr(\mathbb{R^n}-A)$
For any set $A\subset... |
H: Given $\mathbf{\alpha}$, Find the positive-entried vectors $\mathbf{x}$ maximizing $x_1^{\alpha_1}\dotsb x_n^{\alpha_n}$
A problem from an old Advanced Calculus qualifying exam:
"Choose positive real numbers $\alpha_1, \dotsc ,\alpha_n$ such that $\sum_1^n \alpha_i = 1$ and let $f:[0,\infty)^n\to \mathbb{R}:(x_1, \... |
H: Combination of items with and without replacement
How do you determine the number of combinations of items if some items can be replaced and some cannot? For example, I have $2$ lists:
$X = \{A,B,C\}$ - cannot be replaced
$Y = \{1,2,3\}$ - can be replaced
How many combinations are there when combining both lists? ... |
H: A sequence of measurable functions and the sup, lim sup of them
So I have stumbled upon the following theorem:
Let $\left\{f_n\right\}$ be a sequence of measurable functions. For $x \in X$, put
$$
g(x) = \sup \left\{ f_n (x) \mid n \in \mathbb{N} \right\} \\
h(x) = \limsup_{n \to \infty} f_n (x)
$$
Then $g$ an... |
H: Nonhomogeneous Linear ODE Method of Solution Question
So I have the following differential equation:
$$
\frac{dy}{dt}-0.07y=5000
$$
I tried solving it using an integrating factor and ended up getting $y=Ce^{0.07t}-350$.
I plugged the ODE into Wolfram Alpha and it seemed to solve the problem as a separable equation.... |
H: Big $\Omega$ question! Prove $(n-1)(n-2)(n-3)$ is $\Omega(n^3)$
Problem
Prove $(n-1)(n-2)(n-3)$ is $\Omega(n^3)$.
Attempt @ Solution
$f(n) = n^3(1-6/n+11/n^2-6/n^3)$
$g(n) = n^3$
Show that there exists a $C > 0$ and $n_0$ such that $f(n) \ge Cg(n)$ for all $n > n_0$.
I tried plugging in different numbers for $n$ t... |
H: Word problem involving equations
Two bike riders $X$ and $Y$ both start at 2 PM riding towards each other from $40$ km apart. $X$ rides at $30 \frac{\mathrm{km}}{\mathrm{h}}$, and $Y$ at $20 \frac{\mathrm{km}}{\mathrm{h}}$. If they meet after $t$ hours, find when and where they meet.
AI: Let $40 \text{km}= x + y$, ... |
H: How can i solve this equation $|x|\hat x= 1991x$
I found this problem in an old exam and i want to know how to do it, since i couldn't at the time, it's in spanish so i'll leave my translation and the original:
Solve this equation $|x|\hat x= 1991x$. Here $|x|$ is the biggest integer less than or equal than $x$, an... |
H: Figuring out which functions are Big-O of other functions (of a of 9 different functions). Where do I start?
Problem
I need to arrange the following functions in order, so that each function is big-oh of the next function.
Functions
Attempt @ Solution
Understanding: I don't understand what to do here. My best gu... |
H: Theorem 2.13 in Walter Rudin's Principles of Mathematical Analysis
While reading Walter Rudin's Principles of Mathematical Analysis, I ran into the following theorem:
Theorem 2.13. Let $A$ be a countable set, and let $B_n$ be the set of all $n$-tuples $\left(a_1,\dots,a_n\right)$, where $a_k\in A$ ($k=1,\dots,n$),... |
H: Doubling Time for certain bacteria
Say a culture of bacteria doubles in weight every 24 hours. If it originally weighed 10g, what would be its weight after 18 hours?
I know how to calculate half-life but don't know about doubling time. What is the easiest formula to use in order to solve this?
AI: Since the initia... |
H: Question from Introduction to Topology by Mendelson
I'm self studying Intro to Topology by Mendelson(3rd ed.) right now and I'm stuck on a book problem. In case anyone has the book handy, its problem 2 of chapter 3 section 6. The problem is as follows,
Let $O$ be an open subset of a topological space $X$. Prove tha... |
H: If $X$ is a separable Banach space $ \Rightarrow$ $ X=\overline{\cup_{n=1}^{\infty} X_n}$ where $\dim X_n=n$
Let $X$ be a Banach space separable.
How can we prove that there is a sequence of subspaces:
$X_1\subset\ X_2 \subset \cdots \subset \ X_n \subset \cdots $ of $X$ such that $\displaystyle X=\overline{\bigc... |
H: Mathematical Induction --- $a_n=2a_{n-1}-1$
Problem
Finish the following mathematical induction showing that $a_0 = 2$ and $a_n = 2a_{n-1}-1$ implies $a_n = 2^n +1$.
Basis: Prove that $a_0 = 2^0 + 1$
Proof:
$a_0$ = $________$ = $________$ = $2^0 + 1$
Induction:
Assume $k \ge 0$ and $a_k = 2^k +1$
Want to show that ... |
H: Combinatorics - How many bit strings of length 8 with exactly two 0's are there for which the 0's are not adjacent
How many bit strings of length 8 with exactly two 0's are there for which the 0's are not adjacent?
I'm having a lot of trouble with this seemingly simple problem.
I'm trying to do this with stars and... |
H: Finding volume of a figure given by relations.
I am stuck on the problem of finding volume of the figure given by $0 \leq z \leq 2, x^{2} + y^{2} \leq 2, x^{2} + y^{2} + z \leq 2x$. I have tried three different coordinates but the problem is $2x$ in the last relation. Thanks ahead for help!
Add. Sorry that I didn't... |
H: Converting a distance matrix into Euclidean vector
I have a distance matrix between different elements. Now I want to calculate the Euclidean vectors that have resulted in that matrix. Is there any efficient method that can do so?
AI: Distance matrices only give the pairwise distances within a finite set of points.... |
H: Differentiability of $xy^{\alpha}$
I was asked to prove that
$|xy|^{\alpha}$ is differentiable at $(0,0)$ if $\alpha > \frac{1}{2}$.
Since both the partial derivatives are zero, I concluded that this function is differentiable if and only if the following holds:
$$ \lim\limits_{(x,y)\to (0,0)} \frac{|xy|^{\alpha}}{... |
H: How can I get the $(x,y)$ of a sub-line, which has 0.45 length of the original line, between two points?
Given two points $(x_1, y_1)$ and $(x_2, y_2)$, they forms a straight line. The target is to find a point $(x_t,y_t)$ between these two points, and the length from $(x_1, y_1)$ to $(x_t,y_t)$ is 0.45 of the leng... |
H: Verifying whether a map is a polynomial ring automorphism
On pg.1, this article talks about an automorphism $f:R[x_{1},x_{2}]\to R[x_{1},x_{2}]$ ($R$ is a ring) defined by
$$f(a)=a, \forall a\in R$$ $$f(x_{1})=x_{1}+x_{2}$$ $$f(x_{2})=x_{2}$$
An automorphism as described here is an isomorphism, which means it has ... |
H: Basic Set Theory: Existence of Three Specific Sets
Do there exist sets $A$, $B$ and $C$ such that $A\cap B \neq \emptyset$, $A\cap C = \emptyset$ and $(A\cap B)\setminus C = \emptyset$?
AI: No. We argue by contradiction. Suppose instead that these sets really did exist. Then since $A \cap B \neq \emptyset$, we kno... |
H: $H(\kappa)$-absoluteness of a formula
Let $\varphi(x,y)$ be an $\in$-formula which is absolute between transitive models of ZF minus powerset axiom. Then $\exists x\, \varphi(x,y)$ is $H(\kappa)$-absolute, where $H(\kappa)$ is the set $\{x\, |\, card(TC(\{x\}))<\kappa\}$. $\kappa$ is uncountable regular.
This is a... |
H: Intro to Topology Mendelson
I'm self studying intro to topology by Mendelson and I'm stuck on a book problem. The problem is,
Let $Y$ be a subspace of $X$ and let $A\subset Y$. Denote the $\operatorname{Int}_X(A)$ as the interior of $A$ in the topological space $X$ and $\operatorname{Int}_Y(A)$ as the topological s... |
H: Solving differential equation $x^2y''-xy'+y=0, x>0$ with non-constant coefficients using characteristic equation?
Whenever you deal with non-constant coefficients you usually use Laplace transform to solve a given differential equation, at least that's how how I learned it.
But how would you solve the equation usin... |
H: Prove $f : A\rightarrow B, g: B\rightarrow C$ , and $g\circ f: A \overset{1-1}{\rightarrow}C$, then $f:A\overset{1-1}{\rightarrow}B$
Could anyone please explain how to approach this problem, I'm honestly having a hard time figuring out where to start the problem. I know that I have to show that $\forall x,y\in A$ ,... |
H: Finding original amount in half-life problem
Say the half-life of an element is 1590 years. If 10g of the element is left after 1000 years, how much was there originally?
AI: Since
$$\text{Amount remaining} =\text{Original Amount} \times \bigg(\frac{1}{2}\bigg)^{\text{number of half lives}} $$
solve for $X$ in the ... |
H: Calculating the number of times a digit is written when given two numbers
My homework asks me the following:
If a student writes the integers from 5 to 305 inclusive by hand, how many times will she write the digit 5?
I started out by writing every number that contains 5 and I got 31, but 31 is not among the answ... |
H: How can I calculate this exponential growth?
I'm reading the book "Singularity is near", and there is a passage where the author says:
"It takes 100 years to achieve this, with current rate of progress, but because we're doubling the rate of progress every decade, we'll achieve a progress of century in 25 years".
C... |
H: Fibonacci Cubes: $F_n^3 + F_{n+1}^3 - F_{n-1}^3 =F_{3n}$
Prove
$$F_n^3 + F_{n+1}^3 - F_{n-1}^3 =F_{3n}$$
I've tried induction, either it's just very long or a neat trick is required in the inductive step but for some odd reason it's not working out. Ideally I would like any suggestions for the inductive proof.
AI:... |
H: Algebra has me hitting a wall
Math is not my strong suit. As far as I can tell, this is what I'm looking to solve.
A+B=C and B=C*D
If one knows what A and D equal, can one determine the value of B and C?
So far, I've tried:
A+B=C
A+C*D=C
C*D=C-A
C*D-C=-A
A=C-C*D
And then I get stuck. Also:
A+B=C
B=C-A
C*D=C-A
... |
H: How to solve inequalities with one variable by number line
I can solve simple type of inequalities by number line . For example, if I want to solve $x>6$ by number line , then I have to plot this solution into number line which is described in the following image .
How can I solve inequalities with one variable ... |
H: Calculate $\deg(f)$
According to Guillemin and Pollack, Differential Topology Page 109,
$f: X \to Y$ are appropriate for intersection theory ($X,Y$ are boundaryless oriented manifolds, $X$ is compact), when $Y$ is connected and has the same dimension as $X$, we define the degree of an arbitrary smooth map $f: X \t... |
H: For what natural numbers is $n^3 < 2^n$? Prove by induction
Problem
For what natural numbers is $n^3 < 2^n$?
Attempt @ Solution
For $n=1$, $1 < 2$
Suppose $n^3 < 2^n$ for some $n = k \ge 1$
It looks like the inequality is true for $n = 0$, $n = 1$ and $n\ge10$
But, how can I prove this through induction?
AI: You n... |
H: Can a LP optimization problem have exactly two solutions?
For example a linear model defined by equation
Min[5 x + 7 y,8 x + 4 y] <= 7 - 5 x
has a feasible region shown as below.
This model forms a concave feasible region that has two corner points (x_1,y_1), 〖(x〗_2,y_2) and in the second and fourth quadrant respec... |
H: Prove $z \to z^m$ has degree $m$.
I am hoping to prove this obeying author's intention - following his hint. But I am wondering if I shouldn't employ Euler's Formula, and should use a more primitive method? I also granted my proof below is correct?
Prove $z \to z^m$ has degree $m$. Hint: calculate with local param... |
H: The no. of values of k for which $(16x^2+12x+39) + k(9x^2 -2x +11)$ is perfect square is:
I wanted to know, how can i determine the no. of values of k for which $(16x^2+12x+39) + k(9x^2 -2x +11)$ is a perfect square.($x \in R$)
I have tried, since $x$ is real the discriminant must be $\geq 0$.
$D = 4(6-k)^2 -4(16+9... |
H: What is the value of $\log_i i$?
What is the value of $\log_i i$
How to start?
Is it mathematically correct?
AI: If you want to define $\log_ab=\frac{\log b}{\log a}$, then $\log_ii$ exists and has multiple values. Starting from $\exp(i(4n+1)\pi/2)=i$ you may find $\log_ii=\frac{4n+1}{4m+1}$ for $n,m \in \mathbb{Z}... |
H: solve the inequation : $-|y| + x - \sqrt{x^2 + y^2 -1} \geq 1$
I wanted to know, how to the following inequation
$-|y| + x - \sqrt{x^2 + y^2 -1} \geq 1$
I did $x-|y| \geq\sqrt{x^2 + y^2 - 1} +1 \geq 0$ which gives $x \geq |y|$, what to do next...
any help appreciated.
Thanks
AI: EDIT The answer refers to the inequ... |
H: Is it neccessarily the case that $f$ is measurable, if it's measurable in each variable?
A well-known example of a function that is continuous in each variable but fails to be jointly continuous is:
$$
f(x,y) = \begin{cases}
\frac{xy}{x^2 + y^2} & (x,y) \neq (0,0) \\
0 & (x,y) = (0,0) \\
\end{cases}... |
H: representations and modules
I am reading representations of Lie Algebra in Humphreys.He is defining representation as $L$-modules. In case of group representation we have the correspondence between representation and modules over the group ring. But in both the places we have different definition for modules like i... |
H: Number of trees with a fixed edge
Consider a vertex set $[n]$. By Cayley's theorem there there are $n^{(n-2)}$ trees on $[n]$, but how can one count the following slightly modified version:
What is the number of trees on $[n]$ vertices where the edge $\{1,2\}$ is definitely contained in the trees?
AI: We can actua... |
H: Find a formula for this sequence (and prove it).
This is a 2 part problem.
Part I
I need help finding a formula for this sequence of numbers:
$$\frac{1} {1\times 2} + \frac {1} {2\times3} + \cdots + \frac {1} {n(n+1)}$$
Part II
I need to prove the formula conjectured in Part I.
AI: HINT:
$$\frac1{k(k+1)}=\frac1k-\f... |
H: Let $f(x) = \int \frac{x}{1-x^{8}}dx\,$
Let $f(x) = \int \frac{x}{1-x^{8}}dx\,$.
Represent $I(x)$ by a power series $\sum^{\infty}a_{n}x^{n}$.(Find $a_{n}$)
What is the radius of convergence of $I(x)$ ?
Two curves are generated by polar equations $r=1+\sin\theta$ and $r=-\sin\theta$.
Find the area of the region... |
H: Finding a basis for vector space $U$
Let $U$ denote the subspace of $M_{2\times 2}(\mathbb{C})$ defined by
$$U=\left\lbrace\left(\begin{matrix}a&b\\ c&0\end{matrix}\right):a + b + c=0\right\rbrace.$$
How would one find a basis for that vector space? Any clues please.
AI: Well, the only requirement is that $a+b+c... |
H: How to solve $i^z = \ln z$
How to solve $i^z = \ln z$?
Putting $z = re^{i\theta}$ and $i = e^{i\pi/2}$ gives :
$$
e^{i\pi/2re^{i\theta}}= \ln r +i\pi
$$
How to continue ?
Thanks
AI: Use Euler's formula $e^{i\theta}=\cos\theta+i\sin\theta$ and equate the real and imaginary parts of both sides of the equation you'v... |
H: Find the value of this logarithmic expression involving fifth root of unity.
Let $\alpha$ be the fifth root of unity. We then want to evaluate the
expression $$\log |1 + \alpha + \alpha^2 + \alpha^3 - 1/\alpha |$$
Thanks in anticipation for your help in solving this!
AI: HINT:
$$1+\alpha+\alpha^2+\alpha^3=\frac... |
H: How can I check the nature of critical point on three variable function
I have study on multivariate calculus. What is the best way
to finding the nature of critical point on a real-valued three a variable function?
In two variable function
we can use $$D = f_{xx}(x,y)f_{yy}(x,y) - (f_{xy}(x,y))^2$$ to check what ... |
H: How to prove inequality on quadratic form and orthogonal projection
This is from a paper I'm reading. I don't know how to prove it.
Assume that $\mathbf A$ is an $n\times n$ positive semi-definite matrix which has $k$ non-zero eigenvalues. We can assume that all positive eigenvalues are larger than some positive ... |
H: How does one show a topological space is metrizable? Using text Intro. to Topo. by Mendelson
I'm self studying Intro. to Topology by Mendelson.
The problem I'm looking at is,
Prove that for each set $X$, the topological space $(X,2^X)$ is metrizable.
I'm not having so much trouble with this problem per se, but with... |
H: Let $R, P , Q$ be relations, prove that the following statement is tautology
Let $P, R, Q$ be relations. Prove that:
$\exists x(R(x) \vee P(x)) \to (\forall y \neg R(y) \to (\exists xQ(x)
\to \forall x \neg P(x)))$ is a tautology.
How do I do so? please help.
AI: This isn't a tautology. Consider a universe with ... |
H: Solution of $a x+\sin x -L =0$
How to find $x$ such that $a x+\sin x -L =0$ where $L,a$ are constant and $a>0$?
Thank you .
AI: This type of equations are called Transcendental equations and are generally not solvable in a closed form. However, you can use numerical techniques like Newton-Raphson method, Bisection ... |
H: The limit of $f(x)=\sin \frac{1}{x}$ at $x=0$
On page 96 of Spivak's Calculus, 4th Edition, he writes:
... For this function it is false that $f$ approaches $0$ near $0$. This amounts to saying that it is not true for every number $\epsilon > 0$ that we can get $|f(x)-0| \lt \epsilon$ by choosing $x$ sufficiently ... |
H: Immediate consequence of Riemann-Roch
Let $X$ be an algebraic curve, $D$ a divisor and $\mathscr{O}(D)$ the line bundle associated to $D$ in the canonical way. The following implication should follow immediately from the Riemann Roch formula
$$deg(D)<0 \implies h^0(X, \mathscr{O}(D)) =0.$$
Could you help me to see ... |
H: Proving uniform convergence of a sequence
I have to prove the uniform convergence of this sequence $f_n(x)=\tan^{-1}nx$ in $[a,b],a>0$
What I have reached so far:
$$|f_n(x)-f(x)|=\left|\tan^{-1}nx-\frac\pi 2\right|=\tan^{-1}nx-\frac\pi 2<\epsilon$$
How do I proceed further ?
AI: Recalling the Taylor series of $\arc... |
H: Transform quadratic ternary form to normal form
Does anyone know of an integral transform which transforms the normal form $Ax^2 + By^2 + Cz^2 + Dxy + Eyz + Fzx = 0$ to the form $ax^2 + by^2 + cz^2 = 0$ ?
Thanks in advance.
AI: It is a two step iterative algorithm:
Take the first variable $x$. If $x^2$ appears the... |
H: Show that the boundary of $A$ is empty iff $A$ is closed and open.
I'm reading Intro to Topology by Mendelson.
The problem at hand is,
Show that $\text{Bdry}(A)=\emptyset$ if and only if $A$ is closed and open.
This was all the problem statement had, but I'm in the chapter covering closure, interior and boundary wi... |
H: About the convergence of a sequence in $L^1$
Suppose that $f_n$ is a sequence of nonnegative functions such that $\int f_n d\mu=1$ for all $n$, and $f_n\to f$ in $L^1$. Let $p>1$. Is it then true that $f_n^p\to f^p$ in $L^1$?
AI: This is false. Because as Chris remarks above, $f_n^p$ needn't be in $L^1$.
For exampl... |
H: Evaluating $\lim\limits_{x \to 0} \frac1{1-\cos (x^2)}\sum\limits_{n=4}^{\infty} n^5x^n$
I'm trying to solve this limit but I'm not sure how to do it.
$$\lim_{x \to 0} \frac1{1-\cos(x^2)}\sum_{n=4}^{\infty} n^5x^n$$
I thought of finding the function that represents the sum but I had a hard time finding it.
I'd appr... |
H: Root of an exponential equation
Let $0 \le a \le 1$ and $-\infty < b < \infty$. I am looking for a solution of the exponential equation.
$$
a^x + abx = 0.
$$
I guess closed form expression of the root in terms of $a$ and $b$ may not be there. In that case, an asymptotic expansion of the root in terms of $a$ and $b$... |
H: Intuition for Multiple Summation becoming One Summation - Nothing too formal/rigorous please
Source. I grok addition is associative and commutative, and a term can be moved into other summations iff these other summations aren't summing this term. Hence I grok
$$\sum_{i,j} g_{ij} \sum_r a_{ir} x_r \sum_s a_{js} x_... |
H: Number of times $g(p_1)$ occurs in $\sum_{d\mid n}g(d)$
$$
g(n)=\begin{cases}
1 & \text{if }n=1 \\[10pt]
\sum_{d\mid n,\ d\ne n} g(d) & \text{else}
\end{cases}
$$
How can I calculate $g(n)$ efficiently?
I was trying to collect all the $g(p)$ terms after complete decomposition of $n$.
After some googling I found ... |
H: K-theory - dependence of algebraic structure: What is the K-theory of a direct product?
I want to figure out the dependence of K-class of an finitely generate projective ring and its algebraic struture.
For example consider $K_0(\mathbb{C})\cong \mathbb{Z}$ and $K_0(\mathbb{R}\times\mathbb{R})$,
where the algebraic... |
H: number of days needed : 48-hour project with 4 employees working 6 hrs/day?
The paving of a road takes 48 hours if done by an employee. As a Project Manager, calculate the number of days required if you have a workforce of 4 people who can work for 6 hours a day?
AI: $$4\;\text{persons}\times \frac{6\;\text{hours}}... |
H: Quartiles of a exponentially distributed function
I am doing an exercise where I'm supposed to calculate the quartiles of the exponentially distributed function $f_\mathbb{X}(x)=\lambda e^{-\lambda x}$. So, first I calculate the distribution function, $F_\mathbb{X}(x)$, to be
$F_\mathbb{X}(x) = 1 - e^{-\lambda x}$
... |
H: compute the discrete(sampled) time process noise matrix
Given continuous time state transition equation as follows,
$$
\frac{dx}{dt}=Ax+\nu
$$
where $\nu \sim N(\mathbf{0},Q)$, $Q$ being the process noise matrix, one can compute the discrete(sampled) time process noise matrix as follow,
$$
Q(\delta t)=\int\limits_... |
H: Solve a recursion using generating functions: $F_n+F_{n-1}+⋯+F_0=3^n , n\geq0$?
Given the recursive equation :
$$F_n+F_{n-1}+⋯+F_0=3^n , n\geq0$$
A fast solution that I can think of is placing $n-1$ instead of $n$ , and then we'll get :
$$F_{n-1}+F_{n-2}+⋯+F_0=3^{n-1} $$
Now subtracting both equations :
$$F_n+... |
H: What is the smallest amount of the provision?
Provisions for three companies totaling $ 48 million allocated in the ratio of 8:3:1. What is the smallest amount of the provision?
This is my calculation:
= 8 +3 +1
= 12
= 8x12: 3x12: 1x12
= 96: 36: 12
Provision of the smallest amount is 12 million.
=> Refer to my exer... |
H: If a vector $v$ is an eigenvector of both matrices $A$ and $B$, is $v$ necessarily an eigenvector of $AB$?
I'm preparing for my final and this question came up in one of the practices. I am tempted to say no, but I've been having trouble proving this.
If a vector $v$ is an eigenvector of both matrices $A$ and $B$, ... |
H: $\forall A\exists L(\mathcal P(L)=A)$
To be honest this idea is not mine but i saw this axiom somewhere and I dont remember where
Is the axiom that say that exist the "logarithm set" $L$ for every set $A$.
"$\forall A\exists L(\mathcal P(L)=A)$"
Using the intuition in tha naive set theory we can see that such $L$... |
H: How to show that a polynomial has real root between two given values?
Let $C_0,C_1,\ldots,C_n$ are real constants. It is given that $$C_0 + \frac{C_1}{2} + \ldots + \frac{C_{n-1}}{n} + \frac{C_n}{n+1}= 0$$ We need to prove that the equation $C_0 + C_1 x + \ldots + C_{n-1} x^{n-1} + C_{n}x^{n} = 0$ has at least one ... |
H: Homogenous ordinary equation - Homogeneous
The question is:
$(x-y)dx + xdy = 0$
Trying to solve:
$
\\M(x,y) = (x-y)
\\N(x,y) = x
$
$
\\Kx - Ky = K(x-y) \Rightarrow \text{ Homogeneous}
\\Kx = K(x) \Rightarrow \text{Homogeneous}$
$
\\y = vx
\\dy = vdx+xdv$
$
\\(x-vx)dx+x(vdx+xdv)=0
\\xdx + x^2dv = 0
$
I'm stucked her... |
H: $AB=BA$ if there is an orthonormal basis of $\mathbb{R}^n$ of eigenvectors
Show that if there is an orthonormal basis of $\mathbb{R}^n$ that consists of
eigenvectors of both of the $n \times n$ matrices $A$ and $B$, then $AB = BA$.
I'm not sure if what I have done suffices to solve the problem, but let $(v_1,v_... |
H: Question on a functional analysis exercise.
These days I am doing some independent study of functional analysis. While solving problems, I could not handle the following part of an exercise (exercise 13, chapter 1 of Rudin's Functional Analysis).
Let $C$ be the vector space of all complex continuous functions on $... |
H: How to prove that $\lim\limits_{x\to0}\frac{\tan x}x=1$?
How to prove that
$$\lim\limits_{x\to0}\frac{\tan x}x=1?$$
I'm looking for a method besides L'Hospital's rule.
AI: Strong hint: $$\displaystyle \lim \limits_{x\to 0}\left(\frac{\tan (x)}{x}\right)=\lim \limits_{x\to 0}\left(\frac{\tan (x)-0}{x-0}\right)=\lim... |
H: How To Count Shuffle permutations
Let $n\in \mathbb{N}$ and $S(n)$ the permutation group on $\{1,\ldots,n\}$.
For any $p,q\in \mathbb{N}$ with $p+q=n$, the set $Sh(p,q)\subset S(n)$ is
the set of all permutations $\tau$ such that $\tau(1)< \cdots \tau(p)$ and
$\tau(p+1)<\cdots <\tau(n)$, usually called the set of... |
H: Some theorems in euclidean geometry have incomplete proofs
I have seen that, in euclidean geometry, proofs of some theorems use one instance of the 'geometric shape'(on which the theorem is based) to proof the theorem.
Like, the proof of 'A straight line that divides any two sides of a triangle proportionally, is... |
H: Completion of a metric space in categorical terms
Is it possible to define the completion of a metric space using categorical terms?
AI: The completion of a metric space $X$ is an initial object in the category $\mathcal{U}_X$ whose objects are uniformly continuous maps $\iota_Y \colon X \to Y$, where $Y$ is a comp... |
H: Predicate logic example
While studying predicate logic, i see some example as exercise there.But can't figure it out.Can anyone help me?
(i)If a brick is on another brick, it is not on the table.
(ii)Every brick is on the table or on another brick.
(iii) No brick is on a brick which is also on a brick.
EDIT:
My a... |
H: What does "prove by induction" mean?
What does "Prove by induction" mean? Would you mind giving me an example?
AI: Proof by induction means that you proof something for all natural numbers by first proving that it is true for $0$, and that if it is true for $n$ (or sometimes, for all numbers up to $n$), then it is... |
H: 1st order linear DE with step function input
the 1st order linear equation is:
$y'(t) + \frac D M y(t) = f(t)$
with constants:
$D = 100kg/s$
$M = 1000kg$
$f(t) = Fu(t)$ <-- that's Force x the unit step function
an initial condition:
$y(0) = 20.8m/s$
the input is a step function scaled by the Force $F$ ($Fu(t)$)
we... |
H: Is proving that a mapping maps every element of the domain and is surjective sufficient to prove that it is a ring automorphism?
The ring under consideration is $R[x_1,x_2,\dots, x_n]$. Shouldn't proving that a mapping $f:R[x_1,x_2,\dots, x_n]\to R[x_1,x_2,\dots, x_n]$ maps every element of the domain and is surjec... |
H: Why would a statistician or mathematician want to find the ratio between two maximum likelihood in a likelihood-ratio test?
Why would a statistician or mathematician want to find the ratio between two maximum likelihood function in a likelihood-ratio test?
I know maximum likelihood is the maximum of the probability... |
H: Liapunov functions
I would really like to see some very simple worked out or with some well pointed hints on these guys. i have two textbooks that outline the idea behind them, but both give only one example that are very contribbed imho. ( Non-Linear dynamics and intro to chaos By Steven H Strogatz) and (different... |
H: What does $R_{0}^{+}$ mean?
I'm reading this paper: http://www.aaai.org/Papers/KDD/1996/KDD96-027.pdf, and the authors used the symbol $R_{0}^{+}$ in the definition of Exact Exception Problem, such as $D: P(I) \rightarrow R_{0}^{+}$. Could anyone please help me understand what the symbol $R_{0}^{+}$ means? It seems... |
H: Double integral
Calculate the iterated integral $$\int_{1} ^4\int_{1} ^2 \left(\frac xy+\frac yx\right)\,dy\,dx$$
This is the work that I've done, but it'd lead me to the wrong answer, so either I did it completely wrong or I made in error in my calculation.
$$\int_{1} ^4\int_{1} ^2 \left(\frac xy+\frac yx\right)\,... |
H: convert ceil to floor
Mathematically, why is this true?
$$\left\lceil\frac{a}{b}\right\rceil= \left\lfloor\frac{a+b-1}{b}\right\rfloor$$
Assume $a$ and $b$ are positive integers.
Is this also true if $a$ and $b$ are real numbers?
AI: This is not true in general, e.g. take $b = 1/2$ and $a = 2,$ so the LHS is $4$ wh... |
H: Zariski tangent space of a point viewed as a point of a subvariety
Let $X \subset \mathbb{C}^n$ be an affine variety (not irreducible). Let $Y$ be a subvariety of $X$ (again not irreducible). How can we relate the Zariski tangent space at $P \in Y$ and at $P \in X$?
(Corrected after Mariano's comments)
Based on my... |
H: Evaluate the following Riemann Stieltjes integral
Let $\alpha(x) = 3[x]$ where $[x]$ is the greatest integer function. Evaluate $$\int_{0}^{2}{\alpha\left(\dfrac{x}{\sqrt{2}}+1\right) \mathrm{d}\alpha(x)}$$
Attempt: If I apply the same idea of evaluating R-S integrals w.r.t jump functions, I get that $$\int_{0}^{2... |
H: A counter-example of the second isomorphism theorem for topological groups
Let $G$ be a topological group and $H$ and $N$ subgroups. Suppose $H$ is contained in the normalizer of $N$, then by using arguments of the second isomorphism theorem we can show that there is a canonical continuous isomorphism
$$\phi:H/H\ca... |
H: $f(z) = z^k + kz$ is injective in the unit disc
How would I go around proving that $f(z) = z^k + kz$ in injective in the open unit disc in $\mathbb{C}$, for each natural $k$?
I don't really know where to start. I observed that its derivative is nonzero, but that was all that I came up with.
Any help is appreciated.... |
H: How to Tell If Matrices Are Linearly Independent
If I have two matrices, for example: $\begin{pmatrix}1&0\\2&1 \end{pmatrix}$ and $\begin{pmatrix} 1&2\\4&3\end{pmatrix},$ how do I determine if they are linearly independent or not in $\mathbb{R}^4$?
I am familiar with checking for independence with vectors, such as ... |
H: isomorphic polynomial rings
I'm certain that this is a dumb question, but I'll ask anyway.
I know that if $\theta : F \to K$ is a field isomorphism then we get an induced isomorphism $\varphi:F[x] \to K[x]$ such that $\varphi|F = \theta$. We construct such a $\varphi$ by setting $\varphi|F = \theta$ and $\varphi(x)... |
H: Combinatorial - Ways to create subcommittees of a certain size out of a committee?
Each member of a 10 member committee must be assigned to exactly one of 3 subcommittees (management, supervisor, employee). If these subcommittees are to contain 1,3, and 6 members respectively, how many different subcommittees can b... |
H: Hausdorff Space that is a non Normal Hausdorff Space
Can someone give me an example of a Hausdorff space (i.e $T_2$),
that is not a normal Hausdorff space (i.e $T_4$)?
AI: There are many examples. One simple one is the $K$-topology on $\Bbb R$. Let $K=\left\{\frac1n:n\in\Bbb Z^+\right\}$. Let
$$\mathscr{B}=\{(a,b)... |
H: Finding the lowest common value in repeating sequences
Assume I have N sequences of ones and zeros. Each sequence repeats every p terms. I want to find the minimum position where all sequences evaluate to "1"
Here is an example set of sequences for $N = 3$ and $p = \{2, 3, 5\}$:
$$
\begin{align}
a_1 & = \{1, 0, 1, ... |
H: Sampling 100 widgets to test for defective ones
Given a 100 widgets. The probability of a widget being defective is $\frac{1}{2}$. Let
$A$ be the event that $k$ sampled widgets are all functioning properly, for $0\leq k\leq 100$.
$B$ be the event that $6$ or more of the $100$ widgets are defective.
What is the mi... |
H: What is the correct order when multiplying both sides of an equation by matrix inverses?
So my questions is let's say you were asked to solve for $A$, and you have something like this:
$$BAC=D$$ where B, C , and D are matrices. So the way I would solve this would be to multiply both sides by $B^{-1}$ and $C^{-1}$ (... |
H: Using the definition of big-oh notation, show that for any $k,\gamma>1$, $n^k=O(\gamma^n)$.
This question had been on my midterm in a course I took last year:
Prove that for any $k,\gamma>1$, $n^k=O(\gamma^n)$.
Intuitively, this makes sense. Even the fastest exponential algorithm (for example, $1.001^n$) will eve... |
H: Notation in propositional logic
If in propositional logic one is trying to simplify a formula by evaluating its subformula, would it be considered an abuse of notation to actually substitute the bits $\{0,1\}$ in for the formula, to say something like "$0\wedge 1\equiv 0$" or "$0\wedge 1=0$".
AI: Judging by the com... |
H: If $f_i(X)$ is connected for all $i=1,2,...,n$ then $X$ is connected.
For each $i\in\{1,...,n\}$, consider the map $f_i:\mathbb{R}^n\to \mathbb{R}$ defined by $f_i(x)=x_i$ for all $x=(x_1,...,x_n)\in\mathbb{R}^n$. I would like to know if the following statement is true or false: if $f_i(X)$ is connected for all $i=... |
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