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H: rational approximation for $x^x$
Using the standard method with derivatives (and taking logarithm of both sides first) we can prove that $x^x>\frac{1}{2-x}$ for $x\in(0,1)$ - this inequality is an exercise from a problemset.
Is it possible to replace $\frac{1}{2-x}$ with a rational function $R$ satisfying $R(0)=1$ ... |
H: What does the space $C^2([0,T]; H^2(\Omega))$ mean?
Does the space $C^2([0,T]; H^2(\Omega))$ mean: $C^2$ in the time direction and $H^2$ in the space direction?
Thank you very much!
AI: Informally, that's a reasonable way to think about it.
Formally, this is the space of all paths $\gamma : [0,T] \to H^2(\Omega)$ ... |
H: Doubt in Proof by Hippasus - Incommensurability of geometrical lengths leading to irrational number.
I have read about geometrical proofs of irrational numbers based on incommensurability of lengths elsewhere. But, am stuck by the line:
For, if any number of odd numbers are added to one another so that
the number o... |
H: Sample Space for simple Probability Experiment
Imagine three empty boxes. You throw a "fair" coin. If the coin display head, you put a ball inside one of the three boxes whereas the probability for box one is $p\in(0,1)$ and for box two and three it is $\frac{1-p}{2}$.
With which probability (in dependence of p) do... |
H: Evaluating Sum at bounds
I have to find an expression in terms of n using standard results for $$\sum_{r=n+1}^{2n} r(r+1)$$
And have found the general equation
$$\sum_{r=n+1}^{2n} r(r+1) = \frac{2n^3+6n^2+4n}{6}$$
However evaluating it as $$\frac{2(2n)^3+6(2n)^2+4(2n)}{6} - \frac{2(n+1)^3+6(n+1)^2+4(n+1)}{6}$$
do... |
H: Proving the bounds of the cosine sequence without starting with basic trig identities.
If we look at a unit circle we can see that the values of cosine are between -1 and 1. However is there a particular proof for this fact? I have tried using the Euler's identity to arrive at a proof with no luck. I have also trie... |
H: How do I finish solving $f(x)f(2y)=f(x+4y)$?
I'm trying to solve this functional equation:
$$f(x)f(2y)=f(x+4y)$$
The first thing I tried was to set $x=y=0$; then I get:
$$f(0)f(0)=f(0)$$
which means that either $f(0)=0$ or we can divide the equation by $f(0)$ and then $f(0)=1$.
Case 1: If $f(0)=0$ then we can try t... |
H: Riemann integrability criteria
Thinking back about limits and the original definition of the limit I thought that the Reimann integral (for some bound function $f$ in $[a,b]$) could be defined using limit-like definition. I found one definition and proved the equivalence of the two:
for any $\epsilon >0$ exists a p... |
H: Prove these statements about trees are the same
Given a tree $T$ with $|V(T)| = n \geq 2$ prove these statements are the same:
There is an Eulerian path in $T$
There is a Hamiltonian path in $T$
The number of leaves in $T$ is $2$
I don't understand how can a tree have an Hamiltonian path, a tree looks like this:
... |
H: Questions about adjoint functions
Let $V, W$ be euclidean vector spaces and $f \in \mathrm {Hom}_{\mathbb {R}}(V,W)$. The adjoint function of $f$ is $f^{ad} \in \mathrm {Hom}_{\mathbb {R}}(W,V)$. Which of these following statements is true?
a. $f$ is surjective iff $f^{ad}$ is surjective.
b. $f$ is injective iff $f... |
H: Properties of min(x,y) and max(x,y) operators
Is $\min(x^2,y^2)=[\min(x,y)]^2$, and similarly for $\max(x,y)$?
Also, is $\sqrt{\min(x^2,y^2)}=\min(x,y)$? Do other non-linear operations work?
In general, what are the other interesting properties of these operators, and where can I study more about them?
AI: No. $\mi... |
H: A question in Theorem 5 of Lesson 8 of Hoffman Kunze linear algebra
I am self studying Linear Algebra from Hoffman Kunze and I have a question in a theorem of Lesson-8 of text book.
Adding it's image:
How does it follows from Theorem 4 that E(c$\alpha + \beta) = cE\alpha + E\beta$ ?
Image of Statement of Theorem... |
H: implication of the Abel–Ruffini theorem
I am taking a course in abstract algebra, and we proved the following theorem:
I want to prove something more specific. Let's look at polynomials of degree 5 over C.
Someone is claiming he has a magic formula, which receives the coefficients of a polynomial of degree 5, and ... |
H: A question in an example in book Hoffman and Kunze in Lesson- Inner Product Spaces
I am self studying some topics in Linear Algebra from Hoffman Kunze and I have a question in an Example of Lesson- Inner Product Spaces
It's image:
How does orthogonal projection of $R^{3}$ on W is linear transformation defined by ... |
H: Prove that for any set $A$, $A = \bigcup \mathscr P (A)$.
Not a duplicate of
Prove that $ (\forall A)\bigcup\mathcal P(A) = A$
Prove that for any set A, A = $\cup$ $\mathscr{P}$(A)
This is exercise $3.4.16$ from the book How to Prove it by Velleman $($$2^{nd}$ edition$)$:
Prove that for any set $A$, $A = \bigcup \m... |
H: Difference between finite and infinite for dimension of sum of vector subspaces?
I am reviewing Advanced Linear Algebra by Steve Roman. He says that for subspaces $S$ and $T$ of vector space $V$
$$\dim(S)+\dim(T) = \dim(S+T) + \dim(S \cap T)$$
but we cannot write
$$\dim(S+T) = \dim(S)+\dim(T) -\dim(S \cap T)$$
unl... |
H: How many SVD's does a matrix have?
If $A$ is a $3 \times 3$ matrix with singular values $5$, $4$, and $2$, then there are $9$ distinct singular value decompositions of $A$. True or false?
Is there any method to solve this because I'm not sure how to approach this.
AI: Important rule:
An $n*n$ matrix with n dist... |
H: Solving a third order Euler-Cauchy ODE
I have been given the following ODE:
$$(2x+3)^3 y''' + 3 (2x+3) y' - 6 y=0$$
and I have to solve it using Euler's method, which I am fairly familiar with.
Now, I let $ 2x+3 = e^t$ and $y=e^{λt}$
After differentiating $y$, I get that $$y''' = \frac{y_t'''-3y_t''+2y_t'}{e^{3t}}$... |
H: A where-from and how-to study pure mathematics question
I've tried to find answers to my question on this community as well as several others,but couldn't find a satisfactory answer,so here I am. I thank in advance, to anyone who decides to give time to my question.
Background: I am trying to learn pure mathematics... |
H: Show that: $f(\theta)=\sin\theta\cos(\theta\ -k)$ is max when $\theta = \frac{k+90^{\circ}}{2}$ without using calculus.
Given $$f(\theta)=\sin\theta\cos(\theta\ -k)$$
Show that $f(\theta)$ is maximum when: $\theta = \frac{k+90^{\circ}}{2}$
I can do this easily using calculus, but I'm looking for a way of doing it w... |
H: How to prove that supremum of strictly convex function is infinity?
Suppose there is a strictly convex continuous function $f$: $R^n$ $\rightarrow$ $R$.
Is the supremum of $f$ always infinity? How can we prove it?
I am trying to come up with proof. If $x$ and $y$ are two points in $R^n$, strictly convex implies $f... |
H: If $G$ is a finite group of automorphisms of $L/k$, then $\hom_k(L^G,k_s)=\hom_k(L,k_s)/G$.
Let $L/k$ be a finite separable extension and let $G$ be a finite group of automorphisms. We also fix a separable closure $k_s$ of $k$.
I want to prove that $\hom_k(L^G,k_s)=\hom_k(L,k_s)/G$, where two elements $\varphi,\psi... |
H: Is there a word for a contradictory set of linear system of equations?
So in basic math, we tend to learn that we can solve for the variables if there are n equations and n unknowns. But let's say the equations are contradictory, for example, $x + y = 1$ and $x + y = 5$.
Is there an official math word for a contra... |
H: Limit of sequence of Lebesgue integrable functions is not Lebesgue integrable
Construct a sequence of functions $\{f_n(x)\}_{n=1}^{\infty}\subset L([0,1])$ and measurable function $f(x)$ such that $f(x)=\lim \limits_{n\to \infty} f_n(x)$ for all $x\in (0,1)$ and $$\left|\int_{(0,1)}f_n(x)d\mu\right|\leq 1$$ for all... |
H: If $P(A) = p$ and $P(B|A) = P(B^{c}|A^{c}) = 1 - p$, find the value of $P(A|B)$.
I am trying to solve this question where I have to find P(A|B) given,
$P(A) = p$ and,
$P(B|A) = P(B’|A’) = 1-p$
Since, I don’t have P(B), direct Bayes theorem isn’t applicable but I have this hunch that the equality between the conditi... |
H: Proving that $0 \rightarrow \Bbb Z \rightarrow \Bbb Q \rightarrow \Bbb Q / \Bbb Z \rightarrow 0$ does not split.
Can I prove that the $\Bbb Z$-module exact sequence $0 \rightarrow \Bbb Z \rightarrow \Bbb Q \rightarrow \Bbb Q / \Bbb Z \rightarrow 0$ is non-split exact by proving that $\Bbb{Z} \bigoplus \Bbb{Q} / \Bb... |
H: does a sum related to the prime factorization of the whole numbers.
to define this sum you need a function $f(x)$
$f(x)=$ the sum of the prime powers of $x$
$12=3^1\times2^2$ so
$f(12)=1+2$
$16=2^4$ so
$f(16)=4$
my question does this sum converge and if so what does it converge to?
$$\frac{f(2)}1+\frac{f(3)}2+\frac... |
H: If $X$ is not full rank, is $X(X^TX + \lambda I_p)^{-1}X^T$ invertible if $n < p$?
Suppose $X$ is a $n \times p$ matrix with $\text{rank}(X) = r < p$. Is $X(X^TX + \lambda I_p)^{-1}X^T$ invertible when $n < p$?
We know that
$$\operatorname{rank}(X(X^TX + \lambda I_p)^{-1}X^T) \leq \min(\operatorname{rank}(X), \oper... |
H: Solve $x^5\equiv 4\pmod 7$
We know about calculating $x^2\equiv 2\pmod 7$ using quadratic residue properties in order to find out whether a solution exists or not.
I wonder is there any way to determine that $x^n\equiv k\pmod v$, where $v\ge 2$, $k\in\Bbb Z$, and $n\ge 3$? As I asked in title:
Solve $x^5\equiv 4... |
H: Just a basic question for understanding the definition of topological spaces
slowly I am moving into the $\mathbb{R}^n$ with my analysis studies.
And with starting so, the author introduced some basic topology terms, for better understanding.
After writing down the three criteriums for inducing a topology onto a se... |
H: When the function equation $f(x)f(y)=axy+b$ is solvable
Assume $a,b$ are constants. The question is whether there is a continuous function $f$ defined on $\mathbb R$ or $\mathbb C$ so that
$$
f(x)f(y)=axy+b
$$
Of course, such a function $f$ exists if $b=0$ by taking $$f(x)=\sqrt{a}x\,.$$ Likewise if $a=0$ then $... |
H: Show that $K$ has measure zero
This is a problem from my measure theory book:
Let $K$ be a compact subset of $\mathbb{R}^d$ such that the intersection $H_r(K)\cap H_{r'}(K)$ of two homothetic images ($H_r(x)=rx$ for $x\in \mathbb{R}^d$ and $r\in\mathbb{R}$) of $K$ has Lebesgue-Borel measure zero whenever $0<r<r'<1$... |
H: Prove that $\bigcup\mathcal F$ and $\bigcup\mathcal G$ are disjoint iff for all $A\in\mathcal F$ and $B\in\mathcal G$, $A$ and $B$ are disjoint.
Not a duplicate of
Suppose $F$ and $G$ are families of sets.
Prove $\bigcup\mathcal{F}$ and $\bigcup\mathcal{G}$ are disjoint iff for all $A \in \mathcal{F}$ and $B \in \m... |
H: Integral wrt floor(x)
What is the definite integral of $f(x)=x^2+1$ with respect to the differential of $\lfloor x\rfloor$ i.e ($d\lfloor x\rfloor$) from $0$ to $2$?
I tried to multiply and divide dx by then $d\lfloor x\rfloor/dx = 0$.
How do I approach it?
AI: $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove n... |
H: Let $S=\{a,b\}$. Which binary operation $*$ on $\wp(S)$ makes $(\wp(S),*)$ a cyclic group?
Let $S=\{a,b\}$ be a set, and $\wp(S)$ the power set of $S$. It is well known that $$(\wp(S),\triangle,\emptyset)\cong \mathbb{Z}_2\times \mathbb{Z}_2\,,$$ where $\triangle$ is the symmetric difference of two sets.
Now, there... |
H: Is Gaussian process at random times also a Gaussian process?
I have a question I am not sure whether my answer is correct or not:
I have a gaussian process $X_t$ (for $t\geq0$) and a random function $s(t):[0,\infty)\rightarrow[0,\infty)$.
Does $X_{s(t)}$ (for $t\geq0$) also a Gaussian process?
My answer is that it ... |
H: Floor function equation $⌊x + 1/2⌋ + ⌊x⌋ = \frac12 x^6$
So in this floor equation $⌊x + 1/2⌋ + ⌊x⌋ = \frac12 x^6$, I've tried putting $x = n + e$, where $0 \le e < 1$, but I didn't get anything useful. What should be an approach in these situations?
AI: Suppose $x = n + e$ and consider two cases $e < 0.5$ and $e \g... |
H: Identity involving product of the $\zeta$ function for different values
I would like to prove the identity
$$\sum_{\substack{b,d>0 \\ (b,d)=1}}\frac{1}{b^n}\frac{1}{d^m}=\frac{\zeta(n)\zeta(m)}{\zeta(m+n)},$$
where $\zeta$ is the Riemann zeta function and $n,m\ge 2$. Any help would be welcome.
AI: Every pair $(r,s)... |
H: How many positive divisors are there of the number $2019^{2019}$?
How many positive divisors are there of the number $2019^{2019}$ ?
Since $2019$ has $4$ positive divisors $1,~3,~673,~2019$, the positive divisors of $2019^{2019}$ are
$1, \\
3,~3^2,~3^3, \cdots, 3^{2019}, \\ 673,~673^2, ~ 673^{3},\cdots, 673^{673},... |
H: Prove for every number $c$ such that $c \geq f(y)$, there is $x \in (a,b)$ such that $f(x) = c$ - Proof Verification
Let $f:(a,b) \to \mathbb{R}$ be continuous on $(a,b)$. Assume that $\lim_{x \to a^{+}}f(x) = \infty$ and $\lim_{x \to b^{-}}f(x) = \infty$. Let $y \in (a,b)$ such that $f$ attains its minimum. Prove ... |
H: Common solution for $f(x) = f'(x) = 0$
I encountered the following problem in Real Analysis:
Let $f:\mathbb{R}\to\mathbb{R}$ be differentiable and assume there is no $x$ in $\mathbb{R}$ such that $f(x)=f'(x)=0$. Show that $S=\{x\mid 0\le x\le 1, f(x)=0\}$ is finite.
I have solved this problem by observing that $S... |
H: How to show ${_2F_1}\left(-\frac{19}{20}, \frac{11}{30}; -\frac{19}{30}; -2\right)$ is zero.
I have seen hypergeometric functions over the years on Wolfram Alpha and am trying to learn more about them. I recently read this question and its associated answers, but understood very little. I wrote a program to arbitra... |
H: Does the embedding of $M_{n}(\mathbb C)$ into $M_{2n}(\mathbb R)$ send $GL_n(\mathbb C)$ into $GL_{2n}(\mathbb R)$?
We can embed $M_{n}(\mathbb C)$ into $M_{2n}(\mathbb R)$ via
$$Z=X+iY \mapsto \pmatrix{X& Y \cr -Y& X}, $$
where $X,Y \in M_{n}(\mathbb R)$. It can be seen that this embedding is an injective ring ho... |
H: Conditional Probability: Two defective monitors
A small showroom has $50$ LED Monitors on a shelf that work perfectly
and another $5$ that are defective in the same shelf. What is the
probability of randomly selecting two defective monitors when
purchasing a pair of LED monitors from that showroom?
$W=$ working m... |
H: Can we equip the power set $P$ of any set $S$ with a binary operation such that $P$ becomes a group (with some restrictions)?
This question is inspired by this one. Please read my answer there to get better context.
Settings. Let $S$ be a (not necessarily finite) set, and $P$ the power set of $S$ (i.e., $P$ is the... |
H: compactness model theory question
Let $\sigma$ be a set of first-order formulas including the axioms of equality. Suppose that for every $n\in\mathbb{N}, \sigma$ has a satisfying model $M_n$ whose domain is finite and has at least $n$ distinct elements. Prove that the set $\sigma$ must have a model with infinite d... |
H: Find the arclength from 0 to 1 on the function $y =\arcsin(e^{-x})$.
I need to find the arclength from 0 to 1 on the function $y = \arcsin(e^{-x})$. I know that
$$y' = \frac{-e^{-x}}{\sqrt{1-e^{-2x}}}$$ By applying arc length formula I get this nasty integral:
$$\int_{a}^{b} \sqrt{1 + (y')^2} \ dx=\int_{0}^{1}\sqrt... |
H: Let $\Omega$ be a finite set. Let $\mathcal{F}\subset\mathcal{P}(\Omega)$ be an algebra. Show that $\mathcal{F}$ is a $\sigma$-algebra.
Let $\Omega$ be a finite set. Let $\mathcal{F}\subset\mathcal{P}(\Omega)$ be an algebra. Show that $\mathcal{F}$ is a $\sigma$-algebra.
MY ATTEMPT
Since $\mathcal{F}$ is an algebra... |
H: Can someone help me understand the question? Its Linear Equations.
Decide whether the given number is a solution to the equation.
$2x + 3x + 2= 10$; $x = \frac{8}{5}$
Is $x = \frac{8}{5}$ a solution?
Help, I don't understand the question and have no idea how to even check if the solution is true or false.
AI: If $x... |
H: Let $A\subset\Omega$ and $\mathcal{B}_{A} = \{B\cap A:B\in\mathcal{B}\}$. Show that $\mathcal{B}_{A}$ is a $\sigma$-algebra on $A$.
Let $\Omega$ be a nonempty set and $\mathcal{B}$ be a $\sigma$-algebra on $\Omega$. Let $A\subset\Omega$ and $\mathcal{B}_{A} = \{B\cap A:B\in\mathcal{B}\}$. Show that $\mathcal{B}_{A}... |
H: Prove that for every $x\in M$ the sequence $\{T^nx\}$ converges to a fixed point of $T$.
Let $(M, d)$ be a complete metric space, let $T:M\to M$ be a continuous map and let $\varphi:M\to\mathbb{R}$ be a function which is bounded below. Assume that together they satisfy $$d(x,Tx)\leq\varphi(x)-\varphi(Tx)$$ Prove th... |
H: I don't understand this question
You have a craving for Mrs. Fields gourmet cookies. You have a choice of oatmeal raisin, macadamia nut, triple chocolate, cinnamon sugar, chocolate chip, and peanut butter. If you must choose at least one cookie. How many ways is this possible?
AI: The question is perhaps a little a... |
H: Does $y=9$ solve $2y+9(y-4)=52$?
What are the steps to properly solve this?
Determine whether $y=9$ is the solution to the equation
$2y+9(y-4)=52$.
AI: "Is a solution to" means the same thing as "Makes the given things true". So to check whether $y = 9$ is a solution to $2y+9(y-4)=52$, we want to see whether choosi... |
H: Number of ways of choosing n, m elements from non-disjoint sets A and B?
How many ways are there to choose $m$ and $n$ elements from (potentially non-disjoint) sets $A$ and $B$, respectively?
If the sets were disjoint, this would be $\binom{|A|}{m} \binom{|B|}{n}$.
If they aren't necessarily disjoint however, I ca... |
H: If $f$ is a continuous function and $f(a + b) = f(a) + f(b)$, how do I prove that $f(x) = mx$ where $m=f(1)$?
If $ f $ is a continuous function and $ f ( a + b ) = f ( a ) + f ( b ) $, how do I prove that $ f ( x ) = m x $ for any $ x $ in real numbers, where $ m = f ( 1 ) $?
I know that I have to start by showing ... |
H: if $\frac{p}{q}, \frac{r}{s}$ are positive simplified fractions such that $qr - ps=1$, prove that $\frac{p+r}{q+s}$ is also a simplified fraction
if $\frac{p}{q}, \frac{r}{s}$ are positive simplified fractions such that $qr - ps=1$, prove that $\frac{p+r}{q+s}$ is also a simplified fraction
It's not hard to prove ... |
H: generating functions for the sequence $\{\frac{k(k-1)}{2}\}$ and ${(k+1)(k+2)}$
For the following two sequences: (1) $\{(k+1)(k+2)\}$, (2) $\{\frac{k(k-1)}{2}\}$, I am trying to obtain the generating functions for both of them. I am going through a text finite difference equations and method of generating function... |
H: Solve the system of equations for $x$ and $y$?
I'm trying to solve this system of linear equations:
$3x^2 - 12y = 0$
$24y^2 -12x = 0$
for $x$ and $y$, but I'm a little confused. I get $x = 0, 2$ and when I plug those into my first equation I get $y = 0, 1$ but when I plug it into my second equation I get $y = 0, 1,... |
H: Typographical error? Integral curve of $y^2dx-x^2dy=xy$ passing through point (1,1)
The following first-order differential equation and boundary condition appears in Section 2.7, Problem 14 of An introduction to the theory of Differential Equations by Walter Leighton.
$y^2dx-x^2dy=xy$, passing through (1,1)
The sol... |
H: Let $f:[a,b]\to \mathbb{R}$ be Riemann integrable. Let $g:[-b,-a]\to \mathbb{R}$ be defined by $g(x):=f(-x)$. Show that $g$ is Riemann integrable
Let $a<b$ $f:[a,b]\to \mathbb{R}$ be Riemann integrable. Let $g:[-b,-a]\to \mathbb{R}$ be defined by $g(x):=f(-x)$. Show that $g$ is Riemann integrable with $\int_{[-b,-... |
H: For a set A is it always possible to find a measurable superset A* such that $\mu^*(A^*-A)=0$
Given a measure set $\langle X,\mu,\mathcal{A}\rangle$ let $\mu^*$ be the outer measure induced by $\mu$ and $A \subseteq X$ with finite outer measure (Not necessarily measurable). Can a measurable superset $A^*$ with the... |
H: "Show that the limit $ \lim\limits_{(x,y) \to (0,0)} \frac{2x^2y^3}{x^4+y^6} $ does not exist."
$$ \lim\limits_{(x,y) \to (0,0)} \frac{2x^2y^3}{x^4+y^6} $$
My reasoning after reading the textbook: Direct substitution wouldn't work since it would lead to the indeterminate form $0/0$. We can examine the values of $f$... |
H: Why is the well-ordering theorem so important in the set theory?
Why is the well-ordering theorem so important in the set theory?
Every set can be well-ordered.
Mathematicians think the above theorem is very important but the below theorem is not so important.
Of course I know that the above theorem is stronger t... |
H: Every continuous open mapping $\mathbb{R} \to \mathbb{R}$ is monotonic
Prove that every continuous open mapping from $\mathbb{R} \to \mathbb{R}$ is monotonic
I want to prove it only (or mostly) using arguments and concepts from topology, and not from analysis.
I don't have anything that I think is useful or corre... |
H: Does a linear function $\mathrm {End}_K(V) \otimes V \to V$ which maps $A \otimes v$ to $\det (A) \cdot v$ exist?
$K$ is a field and $V$ a finite dimensional $K$-vector space.
Does a linear function $\mathrm {End}_K(V) \otimes V \to V$ which maps $A \otimes v$ to $\det (A) \cdot v$, for all $A \in \mathrm {End}_K(V... |
H: Is the Cartesian product of two bounded sets bounded? What about for compact sets?
The title pretty much speaks for itself. I just got this curiosity while self-studying Real Analysis using Baby Rudin and Charles Pugh's "Real Mathematical Analysis" out of boredom. I am still relatively new to abstract math since th... |
H: Does "y (2 + x) = 3" represents a straight line equation?
I'm new to straight line equation and trying to find out whether $y (2 + x) = 3$ represents a straight line equation or not. Could anybody please help me to figure out how to reach a conclusion here.
AI: No it does not, It represents a hyperbola.
General equ... |
H: To find a smooth planar curve starting at $\vec{r_0}$ stopping at $\vec{r_1}$ with some additional constraints.
To find a smooth planar curve starting at $\vec{r_0}$ stopping at $\vec{r_1}$ whose unit tangents at start and stop are $\hat{v_0}$ and $\hat{v_1}$ and has the minimum length. Let us assume that the curv... |
H: Finding the maximum value given two system of equations
I was given $(x,y)$ that satisfies both of this equation:
$4|xy| - y^2 - 2 = 0\\(2x+y)^2 + 4x^2 = 2$
And was asked to find the maximum value of $4x + y$.
Solving for $y^2$, I get this equation:
$8x^2 + 4|xy| + 4xy - 4 = 0$
I assumed that if I were to find the ... |
H: Probability of having 3 tails consecutively out of 5 tosses
Suppose we have a coin tossing game. What is the probability that out of 5 coin tosses, you will get 3 tails consecutively?
For this problem, I was thinking of using the Binomial distribution PMF since it seems to describe the number of success out of n tr... |
H: $\frac{1}{z^2-5z+4}=\sum_{n=-\infty}^{n=\infty}a_nz^n$. Find $a_{-10}$ and $a_{10}$
Here is the question: By Laurent expansions, there exist constants $a_n$ such that for $1<|z|<4$, $\frac{1}{z^2-5z+4}=\sum_{n=-\infty}^{n=\infty}a_nz^n$. Find $a_{-10}$ and $a_{10}$.
My idea:
Using partial fraction decomposition, ... |
H: Why is the vector dot product scaled?
I have scoured all the answers on this website but I still cannot understand why $a\cdot b = |a||b|\cos\theta$, if the dot product is interpreted as the amount of one vector, say $a$, in the same direction as the other, say $b$, then why do we scale it by multiplying it with th... |
H: How would I find the order of a factor group and determine what it's isomorphic to
Using this operations table i'm trying to figure out how to find the order of the factor group $D_{6} / H$ with H = $ \{ \rho_{0} , \rho_{3} \}$ and then figure out which well-known group it's isomorphic to. I know the order of $D_{... |
H: Find the remainder $1690^{2608} + 2608^{1690}$ when divided by 7?
Find the remainder $1690^{2608} + 2608^{1690}$ when divided by 7?
My approach:-
$1690 \equiv 3(\bmod 7)$
$1690^{2} \equiv 2(\bmod 7)$
$1690^{3} \equiv-1 \quad(\mathrm{mod} 7)$[ quite easy to determine , $\frac{2*1690}{7}$..so on]
$\left(1690^{3}\rig... |
H: Problem with proving inequalities
Question:
Prove that if $x,y,z$ are positive real numbers such that $x+y+z=a$ then $(a-x)(a-y)(a-z)>\frac8{27}a^3$ is not true.
My Approach:
$$\frac{a-x}{2}=\frac{y+z}2$$
$$\frac{a-y}{2}=\frac{x+z}2$$
$$\frac{a-z}{2}=\frac{x+y}2$$
Using $AM>GM$ we get $$\frac{x+y+z}{3}>\root 3 \o... |
H: How to solve $\int ^{1}_{-1}\frac {x^{2n}}{\sqrt {1-x^{2}}}dx?$
I couldn't solve this.
$$\int ^{1}_{-1}\dfrac {x^{2n}}{\sqrt {1-x^{2}}}dx$$
I thought that like the following.
$$\int ^{1}_{-1}\dfrac {x^{2n}}{\sqrt {1-x^{2}}}dx=\int ^{1}_{-1}\dfrac {1-\left( 1-x^{2n}\right) }{\sqrt {1-x^{2}}}dx\\=\int ^{1}_{-1}\dfrac... |
H: Why don't we need to count the two pre-assigned people in this committee forming probability question?
The following question comes from MITx 6.431x.
Out of five men and five women, we form a committee consisting of four
different people. Assume that 1) each committee of size four is
equally likely, 2) Alice and ... |
H: How to integrate $ \int\frac{x-2}{(7x^2-36x+48)\sqrt{x^2-2x-1}}dx$?
How to integrate $$ \int\frac{x-2}{\left(7x^2-36x+48\right)\sqrt{x^2-2x-1}}dx\,\,?$$
The given answer is $$ \color{brown}I=-\dfrac{1}{\sqrt{33}}\cdot \tan^{-1}\left(\frac{\sqrt{3x^2-6x-3}}{\sqrt{11}\cdot (x-3)}\right)+\mathcal{C}.$$
I tried by diff... |
H: How to calculate $ \lim_{x\to\infty} (\frac{x}{x+1})^x$ using L'Hopitals rule?
I am trying to calculate $ \lim_{x\to\infty} (\frac{x}{x+1})^x$ using L'Hopital.
Apparently without L'Hopital the limit is
$$ \lim_{x\to\infty} (\frac{x}{x+1})^x = \lim_{x\to\infty} (1 + \frac{-1}{x+1})^x = \lim_{x\to\infty} (1 - \frac{... |
H: Probability conditional or normal?
I am struggling with this problem.
My work:
I did a) part and I think it should be
$$\{(0,0),(0,1),(1,0),(1,1),(2,0),(2,1)\}$$
for part b) i am not sure will it be $$=1-0.4-0.5=0.1$$
Confused with , and |
And I am stumped in part c and onwards!
AI: From the definition of conditio... |
H: Prove that a polynomial ring is integrally closed
Let $V \subseteq {\mathbb{A}}^2_{\mathbb{C}}$ be the curve defined by $x^2-y^2+x^3=0$, and let $\mathbb{C}\left [ V \right ]$ the coordinate ring of $V$. Let $\Theta :=\bar{y}/\bar{x} \in \mathbb{C}\left ( V \right )$. I must show that the ring $B:=\mathbb{C}\left ... |
H: Evaluating $\lim_{x\to 0} x^{x^{x}}-x^x$ using a graph
I came across a question
$$\lim_{x\to 0} x^{x^{x}}-x^x$$
I tried plotting the graph, but graph of $x^x$ doesn't exist but $x^{x^x}$ which is quite indigestible.
When I plotted the graph for $x^{x^x} - x^x$ then this graph exists which gives $-1$ at $x=0$, so I'... |
H: How to show that $\arctan(|x-y|)\le\arctan(|x-z|)+\arctan(|y-z|)$
I have to show that $\delta$ is a metric with:
$$\delta(x,y):=\arctan(|x-y|)$$
The first two axioms are really straight forward, but I kinda struggle with showing
$$\arctan(|x-y|)\le\arctan(|x-z|)+\arctan(|y-z|)$$
My first try was (since the arctan i... |
H: How to calculate $\lim_{x \to \infty} \left( \frac{1}{\sin^2(x)} - \frac{1}{x^2} \right)$?
I am trying to calculate the limit, using L'Hospital's Rule.
$$\lim_{x \to \infty} \left( \frac{1}{\sin^2(x)} - \frac{1}{x^2} \right)$$
My attempt
$$\lim_{x \to \infty} \left( \frac{1}{\sin^2(x)} - \frac{1}{x^2} \right) = \l... |
H: If $ \lim a_{n}b_{n}=\gamma $ and $ \lim b_{n}=1 $ then $ \lim a_{n}=\gamma $
let $a_n,b_n $ be sequences such that
$ \lim a_{n}b_{n}=\gamma $ (maybe infinity or minus infinity)
and $ \lim b_{n}=1 $
Prove or disprove that $ \lim a_{n}=\gamma $
I tried to prove it using epsilon-delta definition, but I couldnt finish... |
H: Bolzano Weierstrass Theorem for General Metric Spaces
Though $\mathbb{R}$ is not compact, because of LUB axiom one can conclude BW theorem i.e (every bounded sequence will have a convergent subsequence.)
My questions are:-
In what kind of Metric Spaces this result will hold?
(example: In Compact Metric Spaces this... |
H: If $A \leq B$ for a positive-definite operator $A$ in finite dimensions, then $B^{-1} \leq A^{-1}$
Exercise 13 from SEC. 82 of Finite-Dimensional Vector Spaces - 2nd Edition by Paul R. Halmos.
If a linear transformation $A$ on a finite-dimensional inner product space is strictly positive (positive-definite), and if... |
H: Prove: $\sum_{n=0}^\infty \frac{a_n}{n!}x^n$ converges for every $x$ if $\sum_{n=0}^\infty a_nx^n$ has radius of convergence $R>0$
$\sum_{n=0}^\infty a_nx^n$ has radius of convergence $R>0$
Prove: $\sum_{n=0}^\infty \frac{a_n}{n!}x^n$ converges for every $x$.
Also, I don't understand why $R < 1$ implies $$\lim_{n \... |
H: Cholesky decomposition of a Kronecker product
Assume that the $n\times n$ matrix $\mathbf{A}$ has the Cholesky decomposition of the form $\mathbf{A}=\mathbf{L}\mathbf{L}^H$. Now, suppose the matrix $\mathbf{B}$ is the result of a Kronecker product as $\mathbf{B}=\mathbf{I}\otimes\mathbf{I}\otimes\mathbf{A}$ where $... |
H: Unclear problem with $n$-th power matrix and limit
Find $$\lim\limits_{n \to \infty} \frac{A_n}{D_n}$$ where
$$\begin{pmatrix}
19 & -48 \\
8 & -21 \\
\end{pmatrix} ^{\! n} =
\begin{pmatrix}
A_n & B_n \\
C_n & D_n \\
\end{pmatrix}$$
$n$ - is the power of a matrix, but what is $A_... |
H: Show that if $T,T'$ are edge-distinct minimum spanning trees of $G$, then $T$ has two edges of the same weight
Let $G=(V,E)$ be an undirected graph. Let $w:E\mapsto \mathbb{R}$ be a weight function over the edges. Let $T,T'$ be two minimum spanning trees with distinct edges (namely, $T\cap T' = \emptyset$). Show th... |
H: Distribute $n$ distinguishable balls into $k$ distinguishable baskets
Given a number $n$ and $k$ numbers $n_1,n_2,n_3\ldots, n_k \in \mathbb{N}$ such that
$n_1+n_2+n_3+\ldots+ n_k=n$
How many ways are there to distribute distinguishable balls into $k$
distinguishable baskets so that exactly $n_i$ balls are placed ... |
H: Find the maximum $x$ coordinate of a point so that the area of a quadrilateral is $48$
In the $Oxy$ rectangular coordinate system we're given points $O(0,0), A(0,6)$ and $B(8,0)$. The point P is chosen so that $OAPB$ is a convex quadrilateral with area of $48$. Find such P with maximum $x \in \mathbb{Z}$ value.
H... |
H: Does $x^n$ belong to $O(e^x)$ for all $n\geq 1$?
My question is essentially two-fold. I've been asked to prove that $x^5 \in$ $O(e^x)$ as $x\to \infty$, and trying to do that I decided to plot some functions of the form $x^n$ next to $e^x$, and noted that after some (possibly very large) point $e^x$ tends to outgro... |
H: Law of Cosines: Proof Without Words
I am trying to prove the Law of Cosines using the following diagram taken from Thomas' Calculus 11th edition.
I have an answer, but I think there must be a simpler or better way to do it. Here is my answer:
Construct a coordinate system such that $(0,0)$ is located at the botto... |
H: Why cant $a_n$ be zero in a polynomial function?
I was looking at the definition of the polynomial function which is pretty much always stated like this:
$$P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+ a_{1}x+a_{0}\\ a_{i}\in \mathbb{R}\, , i=0,1,2,\cdots,n\\a_{n}\neq 0$$
I was always wondering why is it that $a_{n}\neq 0... |
H: Does a linear function $\mathrm {End}_K(V) \otimes \mathrm {End}_K(V) \to \mathrm {End}_K(V)$ which maps $A \otimes B$ to $A \circ B$ exist?
$K$ is a field and $V$ a finite dimensional $K$-vector space.
Does a linear function $\mathrm {End}_K(V) \otimes \mathrm {End}_K(V) \to \mathrm {End}_K(V)$ which maps $A \otim... |
H: Spivak Calculus Chapter 3 Problem 10-(d)
I am currently working through Michael Spivak's „Calculus“ 3rd edition all by myself and came across this Problem which might not be that important at all, but I am still curious to find out more about it. English is not my first language, so I apologize in advance for my mi... |
H: Chebychev's inequality over a discrete random variable: $<$ vs $\le$
By Chebychev's inequality it holds that
$$
\begin{split}
Pr(|X-\mu |<\epsilon )>1-{\frac {\sigma ^{2}}{\epsilon ^{2}}} \\
\end{split}
$$
For a discrete random variable $X$, does the following hold?
$$
\begin{split}
Pr(|X-\mu | \le \epsilon )>1-{\f... |
H: How many number of ordered pair $(m, n)$ can be formed if $m+n=190$ and $m$ and $n$ are positive integers and coprime?
The question involves the concept of number theory
Kindly provide the hints to solve the question not the entire solution
I don't know how to approach these kind of problems
AI: If $d=\gcd(m,190-m)... |
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