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H: Stuck on a dot product proof
I've been stuck on this for hours and would really appreciate some help!
Question: Suppose $\phi:\mathbb{R}^n\rightarrow\mathbb{R}^n$ is a function that preserves dot products. In other words, for all $u,v \in \mathbb{R}^n$, we have $(u*v)=(\phi(u)*\phi(v))$. Using only basic properties... |
H: Probability distribution binomial
An assembly system is composed of n independent and identical parts. During any given “run” of the system, all parts have a probability of p of working. Suppose the random variable Y represents the number of working parts in any given “run”. You are told that P(Y=3) = 2P(Y=2) and V... |
H: Extend a homeomorphism of an arc of a circle to the entire circle.
Let $X$ and $Y$ be homeomorphic to a circle.
Let $C_X$ and $C_Y$ be closed arcs of $X$ and $Y$, respectively, and there is a homeomorphism $\phi:C_X\to C_Y$.
Is it always possible to extend $\phi$ to $\phi':X\to Y$, such that $\phi'$ is a homeomorph... |
H: Is $\mathbb R$ a normal topological space?
As in the title, in euclidean space is it always possible two find for two disjoint closed sets $A,B$ two open sets $U,V$ disjoint such that $A \subseteq U$ and $B \subseteq V$ (T4-property, normal)?
AI: Every metric space is normal, in particular $\mathbb R^n$. The proof ... |
H: Evaluating $\int_{-2}^{2} 4-x^2 dx$ with a Riemann sum
I'm having problems with a Riemann sum ... I need to find the integral:$$\int_{-2}^2 (4-x^2)\;dx$$Clearly we have $$\int_{-2}^{2}(4-x^2)\;dx=4x-\frac{x^3}{3}\mid_{-2}^{2}=(4\cdot2-\frac{2^3}{3})-(4\cdot(-2)-\frac{(-2)^3}{3})=\frac{32}{3}$$OK.
On the other hand... |
H: Defining a recursive function $f$ on $\{a, b\}$*
I would need some help on how I can define a recursive function $f$ on $\{a, b\}$*
Define a recursive function $f$ on $\{a, b\}$* which replaces any $a$ with $b$ and vice versa, for example, $f(aba) = bab$ and $f(aaabbb) = bbbaaa$
I would appreciate hints and/or ex... |
H: How do I show that $f(x) = x^3+ax^2+c$ has exactly one negative roots?
How do I show that $f(x) = x^3+ax^2+c$ has exactly one negative roots if $a < 0$ and $ c > 0$?
I can use the the bisection technique and choose any number for a and c. But I was wondering is there are any other solutions for this.
AI: $f'(x)=3x^... |
H: Why add one to the number of observations when calculating percentiles?
The CFA Quantitative Methods book uses the following formula for finding the observation in a sorted list that corresponds to a given percentile $y$ in a set of observations of size $n$:
$(n + 1)\frac{y}{100}$
It defines percentile as follows: ... |
H: Generators of Groups
I need to show the following:
Show that $\mathbb{Z}$ is generated by $5$ and $7$.
I think that the solution has to do with relative prime numbers but I don't know where to start.
AI: What you're missing is a quite useful theorem called Bézout's identity. |
H: Normal approximation to the log-normal distribution
Intuitively, it seems that a lognormal distribution with a tiny $\sigma/\mu$ ratio might look quite a bit like a normal distribution. Can this be formalized in any way (e.g., by stating upper bounds on the size of the error margin in the cdf from using this approx... |
H: f(n) from sequence?
I'm more than a little rusty on my algebra and am realizing it could serve a great purpose here. I am currently trying to reverse engineer a relatively simple sequence of numbers but am having a hard time with it. This is the sequence:
500
490
481
472
464
456
448
441
434
427
420
414
408
402
396
... |
H: Understanding the quotient of infinite groups $\mathbb{R}^2/H$ where $H = \{(a, 0): a\in \mathbb{R}\}$
Define $H = \{(a, 0): a\in \mathbb{R}\}$.
Without using the fundamental homomorphism theorem, how would we know what $\mathbb{R}^2/H$ is? The quotient group is $\{H, (x, y) + H, (x_2, y_2) + H, \dots \}$. Intuitiv... |
H: $F : \mathbb{Z} \to \mathbb{Z}$, $F(n) = 2 -3n$. Is $F$ one-to-one? Onto?
Define $F : \mathbb{Z} \to \mathbb{Z}$ by the rule $F(n) = 2 -3n$, for all $n \in \mathbb{Z}$. Is $F$ one-to-one? Onto?
Now, I understand that one-to-one means that nothing in the co-domain is being pointed to twice. I also understand onto... |
H: Prove $\sum\binom{n}{k}2^k = 3^n$ using the binomial theorem
I'm studying for a midterm and need some help with proving summation
$$\sum\limits_{k=0}^n\binom{n}{k}2^k = 3^n$$
using the binomial theorem.
This is what I've been thinking so far:
In the binomial theorem, we set $x = 0$ and $y = 2$, so:
$3^n = (x+y)^... |
H: infinite summation formula help
How do I find the following sum? $$\sum_{x=3}^\infty 1.536x\left(\frac{5}{8}\right)^x$$ It wouldn't be geometric because of the $x$ in front, right?
AI: Hint: You are correct that it's not a geometric series. What we can do is note that
$$\sum\limits_{n = 0}^{\infty} r^n = \frac{1}{1... |
H: Prove that $\inf(A\cap B)\geq \max [\inf(A),\inf(B)]$
Question:
Let A and B be subsets of real numbers. Prove that $\inf(A\cap B)\geq \max [\inf(A),\inf(B)]$.
Attempt at proof:
Let $x\in A\cap B.$
Then, $\forall (x\in A\cap B):[\inf(A) \leq x]$
and $\forall (x\in A\cap B):[\inf(B) \leq x]$.
Since any value of $x$ i... |
H: Lambda Calculus: Reducing to Normal Form
I'm having trouble understanding how to reduce lambda terms to normal form. We just got assigned a sheet with a few problems to reduce, and the only helpful thing I've found is the following example in the book:
(λf.λx.f(fx))(λy.y+1)2
->(λx.(λy.y+1)((λy.y+1)x))2 //How'd it ... |
H: Contour integration with 2 simple poles on contour
Ok for this one I would appreciate if someone could give me a conceptual answer first. I am supposed to integrate $\int_{-\infty}^{\infty} \frac{e^{-i q t}}{p^2 - q^2} dq$ along a half circle C (whose radius goes to infinity), which comprises a horizontal path alon... |
H: Can one assume that $x$ and $y$ are at a fixed position? GRE
I'm confused because I thought that one cannot assume anything is in a fixed position while on a number line. It says the answer is C but I don't understand how they can deduce that from the given information. Twice as far from $x$ as from $y$? there are... |
H: Sequence of simple functions nonnegative that converge to measurable function $f$
Suppose $f\geq 0$ is measurable. We want to find a sequence of $s_n$ of nonnegative simple functions such that $s_n \to_{pointwise} f$. My book says the we should consided the sequence:
$$ s_n = \sum_{k=0}^{2^{2n}} \frac{k}{2^n} \chi_... |
H: Expression for hyperbola on complex plane
The hyperbola
$$x^2 - y^2 = 1$$
has a simple expression in the complex plane as $\{z^2 + \bar{z}^2 = 2\}$.
Is there a similarly simple expression for a hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$? Or an ellipse?
I know we can express hyperbolae and ellipses as images of... |
H: Find a linear transfornation
Find a linear transfornation T on $\mathbb R^2$ and closed subset C of $\mathbb R^2$ such that T(C) is not closed.
Please....
Thanks.
AI: Consider $T:\mathbb{R}^2 \rightarrow \mathbb{R}$ given by $T(x,y) = x$. It maps the closed set $\{(x,1/x)\ : x \neq 0\}$ to a set which isn't closed. |
H: How do I derive the formula for $\sum\limits_{x = 3}^{\infty} 1.536 (x^2) \left(\frac 5 8\right)^x $?
How would I find the summation of
$$\sum\limits_{x = 3}^{\infty} 1.536 (x^2) \left(\frac 5 8\right)^x $$
Would I have to take the 2nd derivative of $(1/1-x)$?
AI: Yes, but it’s helpful to insert an intermediate s... |
H: projection onto vector spaces
How do you project a vector on to the euclidean ball?
For example, if there is a vector $x ∈ R^n$ how does one project this onto the euclidean ball.
What are the steps for projecting a vector onto a subspace? Is there a formula?
AI: The usual thing would be to divide $x$ by its magnit... |
H: Cauchy sequences and condition $X_{n+1} – X_n\to 0$
Let $\{X_n\}$ be a sequence and suppose that the sequence $\{X_{n+1} – X_n\}$ converges to $0$. Give an example to show that the sequence $\{X_n\}$ may not converge. Hence, the condition that $|X_n-X_m| < \epsilon$ for all $m,n \ge N$ is crucial in the definition ... |
H: Noncommutativity of tensor algebra
My question is simple.
Let $M$ be an $A$-module and let $T(M)$ be its tensor algebra. I saw that it is noncommutative in general... but I can' understand this fact...
I think that by commutativity of tensor product it is commutative... Help me...
AI: Let me change notation: $R$... |
H: a question about continuous differentiable function
Given f is a continuous differentiable funtion,
what is the sufficient condition for
$$\lim_{n\rightarrow \infty } \int_{-n}^{x}f'(s)ds\rightarrow f(x) ?$$
the hint from the instructor is that thinking about the behavior of f, f' when x approaches to negative in... |
H: Advanced urn problem
Imagine there are two urns — urn A and urn B. Urn A contains 3 blue balls and 7 red balls. Urn B contains 7 blue balls and 3 red balls. Balls are now randomly drawn from one of these urns where the drawn ball is always placed back into the same urn. Twelve such random draws yielded 8 red balls ... |
H: a question about a ring (unit, multiplicative inverse)
S is the power set of integers Z.
Define two binary operations:
$\bigoplus $: $A$\bigoplus $B=(A\bigcup B)-(A\bigcap B)$, the symmetric difference set
$\ast : A \ast B = A \bigcap B$,
which forms a ring.
Q: Does $\ast$ have a unit?
I'm even wondering will the ... |
H: differential equation of motion - how to deal with squared differentials
This is probably something pretty elementary. But...
The equation given is $\frac{1}{2}m \dot x^2 - gmx = E$
and the assignment is to solve it to show you get the same thing as Newton's equation of motion. I just want to know if I am approach... |
H: Fourier sine series for function F(t) = t for 0
How to get to the part circled with red? I tried to compute it on Wolfram alpha..
(http://www.wolframalpha.com/input/?i=2%2FL+*Integrate+x+sin%28%28n+pi+x%29%2FL%29+dx+from+0+to+L)
Still confused. Need help.
AI: It's all right, the only problem is that Wolfram Alpha... |
H: Definition verification from two different books?
In Kaplansky's Set Theory And Metric Spaces, he mentions a useful example of a neighborhood of $x$ is a closed ball with center $x$. However, one of the theorems in baby Rudin is "Every neighborhood is an open set". I'm confused?
AI: You’re seeing two different defi... |
H: Coordinate vectors for different size matrices
Hi am having trouble with this question
Let $A = \left[\begin{array}{cc}-1&3&-4&-3\\2&-1&0&-3\end{array}\right]$ Find the coordinate vector for the matrix $A \cdot A^{T}$
with respect to the standard basis for $R^{2 \times 2}$
I figured out the $A \cdot A^{T} = \left[... |
H: Contraction and Fixed Point
How do I show that for $T: X \rightarrow X$ where X is complete and $T^m$ is a contraction that T has a unique fixed point $x_0 \in X$.
I know there exists $\lambda_1 \in (0,1)$ for $x, y \in X$ such that $d(T^mx, T^my) \leq \lambda_1 d(x, y)$ and I need to show that T is a contraction ... |
H: how to calculate integral $\lim_{h \to -\frac{1}{2}} \int_0^{1+2h}\frac{\sin(x^2)}{(1+2h)^3}dx$
how to calculate integral,
$\displaystyle\lim_{h \to -\frac{1}{2}} \int_0^{1+2h}\frac{\sin(x^2)}{(1+2h)^3}dx$
Not sure if the limit exists or not.
AI: We may change your limit by $$\displaystyle\lim_{\varepsilon \to 0}\f... |
H: Can every topological space be considered as a subspace of a separable space?
We know that subset of a separable space may not be separable.
Now is it true that any topological space can be considered as a subspace of a separable space?
Please give a hint.
AI: Yes. Let $\langle X,\tau\rangle$ be a topological space... |
H: For symmetric positive definite $A,\ B$ does $\sqrt{AB}=\sqrt{A}\sqrt{B}$?
Let $A, B ∈ F$ n×n be Hermitian and positive definite and assume
that $S = AB$ is also positive definite.
Show that for the unique positive definite square roots of A, B, S, we have
$\sqrt{S} =\sqrt{AB} =\sqrt{A}\sqrt{B}$
This result is requir... |
H: good textbook to self-learn systems of ODEs
I've taken regular Ordinary Differential Equations. Right now I'm taking Systems of ODEs and the textbook is less than stellar. I was wondering if anyone could point me to a decent self-study book for the subject.
Systems of ODEs: matrices composed of regular ODEs
Example... |
H: Prove the least upper bound property using $\mathbb{Q}$-Cauchy sequences.
Hi everyone I'd like to know if the next proof is correct. I'd appreciate any suggestion mainly in the points marks with (1) and (2).
Theorem: Let $E$ be a nonempty subset of real numbers which has an upper bound. Then it must have exactly o... |
H: simple probability question, two aces drawn
There is a pile of 6 cards that contains two aces. The cards are then sorted out into two separate piles. What is the chance that there is an ace in each pile?
My approach is that there are ${3 \choose 1} {3 \choose 1}$ or $9$ ways such that the aces are in different pi... |
H: Show that every polynomial is the sum of an even and odd function.
I have been given an optional challenge problem at the end of one of my earlier tutorials that I am unsure how to solve. It is a question with three parts, but I would like to tackle them separately with a little help on part a.
a function $f$ is ev... |
H: Linearly independent columns of a matrix product
Given $\mathbf{A} = \mathbf{B}\mathbf{C}$, with $\mathbf{B} \in \mathbb{R}^{m\times n}$ and $\mathbf{C} \in \mathbb{R}^{n\times p}$. Say we know that the columns of $\mathbf{A}$ are linearly independent. Does this also imply that the columns of $\mathbf{B}$ and $\mat... |
H: Prove a $T_0$ topological group is $T_1$
How to prove that
a $T_0$ topological group is $T_1$.
I am a beginner in topological group. Also I want some good reference.
AI: In a topological group, the group operations are continuous. So if you have two points $x \neq y$, and a neighbourhood $U$ of $x$ that does not co... |
H: Are functions of this sort bijections from a subset of the reals to the reals?
I'm teaching about infinite cardinalities tomorrow and will be showing that $\tan x$ is a bijection from $(-\pi/2, \pi/2)$ to $\mathbb{R}$. As I was putting the slides together, it occurred to me that there are probably lots of bijectio... |
H: counting on $4$ pairs of gloves
There are $4$ different pairs of gloves. $4$ right handed gloves are given randomly to four persons
then $4$ left handed gloves are given.how many ways are possible such that nobody gets the
right (correct)pair of gloves.
$\underline{\bf{My\;\;Try}}::$ Let the $4$ pairs of gloves as... |
H: Why does $\sum_{j,k\geq 0}\frac{(j+k)a^{j+k}}{j!k!}=\sum_{l=0}^\infty \frac{l(2a)^l}{l!}$?
I notice that $$\sum_{j,k\geq 0}\frac{(j+k)a^{j+k}}{j!k!}=2ae^{2a} = \sum_{l=0}^\infty \frac{l(2a)^l}{l!}.$$
Is there a simple intuitive explanation why these two should have the same sum, or is it more or less a coincidence?... |
H: Find by integrating the area of the triangle vertices $(5,1), (1,3)\;\text{and}\;(-1,-2)$
Find by integrating the area of the triangle vertices $$(5,1), (1,3)\;\text{and}\;(-1,-2)$$
I tried to make straight and integrate, but it is very complicated, there is some better way?
AI: It is just tedious.
Let $f_1(x... |
H: Connectedness of Disjoint Union of Connected Sets
The definition of connected sets is:
A topological space $X$ is connected iff there do not exist sets $U, V \subset X$ such that: $U, V \neq \varnothing$, $U \cap V = \varnothing$ and $U \cup V = X$, with both $U$ and $V$ both open and closed.
I am having trouble ... |
H: Asymptotics of a real sequence
Let $(a_n)_{n\in\mathbb{N}}$ be a real sequence with $a_n\in O(n^d)$ $(d\in (-1,0))$.
Now we consider the expression
$$ b_n:=(1-\sqrt{1-a_n}).$$
Is $b_n\in O(\sqrt{n^d})$?
Thanks!
AI: Yes. Even better, it is in $O(n^d)$. We have
$$\sqrt{1-x} = 1 - \frac{x}{2} - \frac{x^2}{8} + O(x^3)$... |
H: $\int_{\frac{1}{3}\pi}^{\frac{2}{3}\pi} {\sin(x)\;dx}$ using Riemann sums?
How to find the integral $$\int_{\frac{1}{3}\pi}^{\frac{2}{3}\pi} {\sin(x)\;dx}=1$$ using Riemann sums?
AI: We will use the Lagrange trigonometric identities:
$$
\sum_{k=1}^n\cos(kx)=\frac12\left(\frac{\sin\left(\frac{2n+1}{2}x\right)}{\sin\... |
H: An element of order $n$ generates a normal subgroup of $D_n$
Let $a$ be an element of order $n$ of $D_n$. Show that $\langle a\rangle \lhd D_n$ and $D_n/\langle a\rangle \cong \mathbb Z_2$.
Proof: Let $K = <a>$ for some a ∈ G. Let H ≤ K be an arbitrary subgroup. Since
$H ≤ K = <a>$ it follows that $H = <a^d>$ for... |
H: Distributing distinct apples among 5 people
How many ways are there to distribute 6 distinct apples among 5 people? How would I do this? I know for identical it would b C(6 + 5 - 1, 6).
AI: Each apple has $5$ possibilities for the person it goes to. There are $6$ apples.
So you might think of $$5 \times 5 \times... |
H: Changing index of summation. $\sum_{y=1}^\infty (1-\theta)^{y-1}\theta$
I am always confused how to change the index of summation.
$$\sum_{y=1}^\infty (1-\theta)^{y-1}\theta$$
The above is supposed to be a geometric sum and sum up to $1-(1-\theta)^x$? But how?
AI: Is your upper bound of summation $x$? If yes, set ... |
H: if T is normal, then $\sigma(T)=\sigma_{ap}(T)$
I want to show that if the operator T is normal, then $\sigma(T)=\sigma_{ap}(T)$
Its proof is obvious from one hand.But i cant prove that $\sigma(T)\subseteq\sigma_{ap}(T)$
Recall that $\sigma_{ap}(T)=\{\lambda \in... |
H: lower sum and upper sum of x^2 ...
I hope you're able to understand what I'm writing now:
I have to calculate the lim of lower sum and upper sum for the Integral $\int_0^1 x^2 dx, $ by decomposition the interval into n pieces of the same length.
So, I know the following things:
1.) the width of $ x_{k+1} - x_k = 1/... |
H: Two basic questions about uniformizers in algebraic curves
I'm recently trying to study basics about algebraic curves. However, apparently I'm still quite unfamiliar with the subject as there have occured 2 questions that seem quite basic to me, yet I don't directly see a way to make them clear to myself:
1.) For a... |
H: Chance of getting six in three dice
I am having a hard time wrapping my head around this and am sure that my answers are wrong.
There are three dice.
A. Chance of getting exactly one six on the three dice.
$$(1/6) * 3 = 1/3$$
B. Chance of getting exactly two sixes.
$$(1/6 * 1/6) * 1.5 = 1/24$$
C. Chance of getting ... |
H: The pumping theorem and regular language
I have this problem:
$L_0 = L(a^*bba^*)$, the language of the regular expression $a^*bba^*$
$L_1 = \{uu \mid u \in L_0 \}$
Is $L_1$ a regular language?
I know that I should use the pumping theorem for this, but I don't know how to use it.
I assume that $L_1$ is the same as... |
H: find the equation of the circle passing through the extremities of the diameter of the circle
find the equation of the circle passing through the extremities of the diameter of the circle
$x^2 +y^2 +2x-4y-2=0$
$x^2 +y^2 =0$
$x^2 +y^2 -6x-8y-2=0$
I cant understand what the question asks us to do.
AI: I am not sure... |
H: Probability of rolling 6's on 3 dice - adjusted gambling odds?
I'm struggling to work out odds on a game that were working on. It's probably best if I write an example as I'm really not a mathematician!
I'm working on a dice game where the player bets 1 coin and rolls 3 dice. If any of the dice are a 6 we payout ba... |
H: On nilpotent factor group
Let $G$ be a finite group and let $N$ be a normal subgroup of $G$ with the property that $G/N$ is nilpotent. Prove that there exists a nilpotent subgroup $H$ of $G$ satisfying $G = HN$.
This is problem 223 on page 24 of http://www.math.kent.edu/~white/qual/list/all.pdf. I think that Frat... |
H: Find the probability of $a>b+c$, where $a$, $b$, $c$ are $U(0,1)$
What is the probability that $a > b + c$?
$a, b, c$ are picked independently and uniformly at random from bounded interval [0,1] of $\mathbb{R}$.
AI: Probability as the volume of a pyramid $V = \frac{1}{3}Sh = \frac{1}{3}\cdot\frac{1}{2}\cdot1 = \fra... |
H: Characteristic polynomial of a matrix is monic?
Given a $n \times n$ matrix A, I need to show that its characteristic polynomial, defined as $P_A (x) = det (xI-A)$ is monic. I am trying induction. But no clue after induction hypothesis.
AI: $$A=\begin{pmatrix}a_{11}&a_{12}&\ldots&a_{1n}\\a_{21}&a_{22}&\ldots&a_{2n}... |
H: Nilpotent elements in power series over a Noetherian ring
Suppose $A$ is a Noetherian ring. Let $$f(x)=\sum_{i=0}^{\infty} a_i x^i\in A[[x]].$$ If $a_i$ are all nilpotent, then $f$ is nilpotent. How to prove it?
AI: Hint: the sequence of ideals $(a_0), (a_0) + (a_1), (a_0) + (a_1) + (a_2), \ldots$ stabilizes after... |
H: Open subgroup of $SO(3)$
Does $SO(3)$ have an open nontrivial subgroup?(Group $SO(3)$ with usual matrices product, is all $3\times 3$ matrices whose determinant is 1 and for every element $A\in SO(3)$ we have $A^tA=AA^t=I_3$ and also let it's norm be a operator norm, that's, norm of linear mapping $A:R^3\rightarrow... |
H: Showing that Bezier curve length is less than its control polygon
This is a homework and pardon me for the huge gap of my Mathematics knowledge. After thinking and referencing for a few days I came up with something like following, appreciate help to comment whether this is already correct:
By observing the followi... |
H: On equalizers in Top
Wikipedia says "The equalizer of a pair of morphisms is given by placing the subspace topology on the set-theoretic equalizer." for the category $\mathbf{Top}$.
What is the simplest way to prove this? It seems to be an instance of a more general (not only about $\mathbf{Top}$) theorem. Isn't it... |
H: Meaning of symbol $L^1(\mathbb{P})$
In Furstenberg-Kesten theorem, a theory relating to products of random matrices, one of the assumptions is that:
$$\log^{+}||A||\in L^1(\mathbb{P}),$$
where $A$ (a random matrix) is the generator of the cocycle. My question is, can anyone explain what is meant by this assumption... |
H: Simplifying Multiple Summations for worst case analysis
I'm figuring out a worst case analysis on a function. After converting it to a set of summations, and changing the sigma notations into summation formuale I ended up with:
N(N+1)(2N+1) / 6 + N - N(N+1) / 2
Using LCD i was able to combine the firs... |
H: Is there a difference between connected subset and connected subspace?
Let $(X,\tau)$ be a topological space.
Let $(Y,\tau_Y)$ be a subspace of $X$.
Let's say "$Y$ is a connected subset of $X$ iff 'there does not exist nonempty subsets $A,B$ of $X$ such that $\overline{A} \cap B = \emptyset, \overline{B} \cap A= \e... |
H: Problem about arithmetic and general harmonic progressions
An arithmetic progression and a general harmonic progression have the same first term, the same last term and the same number of terms. Prove that the product of the $r$th term from the beginning in one series and the $r$th term from the end in the other i... |
H: Proximal Operator of a Quadratic Function
What is a proximal operator and how would one derive it in general for a function?
In particular, if I had a function: $ f(x) = x^TQx + b^Tx + c $ How would I get the proximal operator for this if Q was a m dimensional square symmetric positive semidefinite matrix?
AI: The ... |
H: Can a continuous function have a non-continuous derivative?
If $f(x)$ is continuous throughout its domain, can its derivative, g(x) be non-continuous for any point?
If so, what effect does a non-continuous derivative have on the function?
AI: Yes, consider the function $$f(x) = \begin{cases} 0 \qquad x\leq0 \\ x \q... |
H: Diophantine Equation: solving $x^2-y^2=45$ in integers
How should I solve $x^2-y^2=45$ in integers? I know $$(x+y)(x-y)=3^2\cdot 5,$$ which means $3\mid (x+y)$ or $3\mid (x-y)$, and analogously for $5$.
AI: $$ (x-y)(x+y) = 45 \Rightarrow x + y = \frac{45}{x-y} $$
but since $$ x +y \in \Bbb Z \Rightarrow x - y \mid ... |
H: How do I show that $u_x$, $u_y$, $u_{xx}$, $u_{yy}$ and $u_{xy}$ are also solution for the pde $u_{xx}+u_{yy}=0$?
Title says it all. Let $u$ satisfy the patial differntial equation $u_{xx}+u_{yy}=0$(elliptic in linear 2nd order pde). How do I show that $u_x$, $u_y$, $u_{xx}$, $u_{yy}$ and $u_{xy}$ are also solution... |
H: Help with Inequality involving absolute values of trig
I am trying to wrap my ahead around the following problem:
Prove that for all $x,y$ in $\Bbb R$
$ |\sin(x) - \sin(y)| \leq |x-y|$
And prove that for $x,y$ in $R$
$|\cos(x) - \cos(y)| \leq |x - y|$
My first idea was to use partial derivatives to find relative op... |
H: Geometrical meaning - Derivatives
The area of a trapezoid with basis $a$ and $b$, and height $h$ equals to $S = \frac{1}{2} (a+b) h$. Find $ \frac{\partial S}{\partial a}, \frac{\partial S}{\partial b}, \frac{\partial S}{\partial h} $ and, using a drawing , determine their geometrial meaning.
I have no idea of the ... |
H: If $f(3x-1)=9x^2+6x-7$, determine $f(x)$
if $f(3x-1)=9x^2+6x-7$ determine all the $f(x)$ functions.
I tried in this way :
$t=3x-1 \Rightarrow x=(t+1)/3$
$f(t)=9(t+1)^2/9-6((t+1)/3)-7((t+1)/3)\ldots$
but unfortunately I get the original function.
Thanks in advance.
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