0
votes
4answers
31 views

Find inverse operator

Let $D=\dfrac{d}{dx}.$ Consider the operator $$ D_{h,x}=\frac{e^{hD}-1}{h}. $$ Question. What is explicit form of the operator $D^{-1}_{h,x}?$ I think that $$ D^{-1}_{h,x}=\frac{h}{e^{hD}-1}, $$ ...
0
votes
0answers
23 views

To determine the points of $\Bbb R^2$ at which $(i) f_x$ exists, $(ii) f_y$ exists.

Let $f : \Bbb R^2 → \Bbb R$ be defined by $f(x, y) := x^2 + y^2$ if $x$ and $y$ are both rational, and $f(x, y) := 0$ otherwise. To determine the points of $\Bbb R^2$ at which $(i) f_x$ exists, $(ii) ...
0
votes
1answer
16 views

Product rule in discrete derivative in finite difference scheme.

Suppose we are on real line and I want to discretize the usual derivative operator. Take a smooth function $u$ and step size $h$. Then I could define $$ \Delta_+u(i) = \frac{u(i+1)-u(i)}{h} $$ as the ...
2
votes
3answers
60 views

find the complex number $z^4$

Let $z = a + bi$ be the complex number with $|z| = 5$ and $b > 0$ such that the distance between $(1 + 2i)z^3$ and $z^5$ is maximized, and let $z^4 = c + di$. Find $c+d$. I got that the ...
2
votes
2answers
27 views

Testing whether Argument is valid or not

I am to determine if argument is valid by making truth table ATTEMPT Let W= Warning lights will come on P= Pressure is high R=Relief valve is clogged Then i have premises as W ...
-3
votes
2answers
46 views

100-sided dice was rolled 98 times, how do you choose next numbes to bet, based on current outcomes.

100-sided dice was rolled 98 times, Numbers form 1 to 50 were rolled exactly once, except number 25, which wasn't rolled yet. Number 75 was rolled 49 times You can only bet if the next roll result ...
0
votes
1answer
37 views

How can a Euclidean ball is a convex set

A convex set is one which has line segment between two points from the set and the line is subset of the set, How to prove Euclidean ball and ellipsoids are convex set ?
1
vote
1answer
14 views

Lattices and Hasse diagrams

I would like to know which is the exact difference between lattices ans Hasse diagrams. In some cases, such as when lattices are used to represent maximal and minimal ideals, it seems to me that both ...
4
votes
1answer
40 views

The limit inferior of Borel functions [on hold]

Suppose $X$ is a separable metric space and $F \colon X \times ℝ_+→[0,1]$ is Borel. Let $f(x) = \liminf_{ε→0} F(x,ε)$. Is $f$ Borel?
5
votes
4answers
72 views

Show that $f(x) = \log(x + \sqrt {x^2+1})$ is an odd function

I need to show that $f(x) = \log(x + \sqrt{x^2+1})$ is an odd function and from what I can understand from this question (found while searching): What is an odd function?, I have to show ...
-1
votes
1answer
56 views

What is the limit of this product? (SOLVED)

What does this limit equal? $$\lim\limits_{k\to\infty}\left(\prod_{n=1}^kn^{2^{k-n}}\right)^{\frac{1}{2^{k-1}-1}}$$ All that I have tried so far is computation and it does seem to converge. I ...
2
votes
2answers
53 views

Measures on $\mathbb{R}$ that are not translation invariant

I am looking for examples of measures on $\mathbb{R}$ which are not translation invariant. The only one I could come up so far is the dirac measure. In particular, I am looking for a measure $\mu$ ...
-1
votes
1answer
24 views

Any idea how to approach this problem

A rectangular meadow will have a fence around it. The long side is $130$ m longer than the short side. The sides lengths can be written $x$ and $x+ 130$. Write a simplified expression for 1) ...
3
votes
1answer
392 views

A curious proof of L'Hospital's rule

I quote P. Nahin When Least is Best (2004), pp. 173-174 "Since $g(x)=R(x)h(x)$, then differentiation of both sides gives $$g'(x)=R(x)h'(x)+R'(x)h(x).$$ Since $\lim_{x \rightarrow 0} h(x)=0$, and we ...
1
vote
1answer
26 views

Complex Conjugation problem using the identity $|x|^2=xx^*$

Show that $$|c|^2= \frac{4k^2}{k^2 +\gamma^2}$$ given (1)$$a+b=c$$ and (2)$$ik(a-b)=-\gamma c$$ This was given in a lecture without proof, so there's probably a very simple way of proving the ...
3
votes
3answers
47 views

Finding indefinite integral.

I need hint in finding the integral of $$\int \frac{x^2}{(x \sin x + \cos x)^2} dx $$ I tried dividing the term by $x^2\cos^2x$ and then substituting $\tan x$.
0
votes
1answer
16 views

How to find the point in convex set $C$ that is closest to $y\notin C$?

How to find the point in convex set $C$ that is closest to $y\notin C$? $C=\{ x\in \mathbb{R^2}:(x_1-1)^2+(x_2-1)^2\le1 \}$ and let $y\notin C $ but $y\notin \mathbb{R^2} $.
0
votes
0answers
12 views

Calculating matix derivatives with MATLAB or MATHEMATICA?

I'd like to calculate the following derivative, \begin{equation} \frac{d\|(f(C)\cdot f(C)^{+}-I)\cdot u\|^2)}{dC} \end{equation} Where $C$ is a matrix of dimension $n\times k$ (s.t $k < n$). And ...
2
votes
3answers
274 views

Is the limit of càdlàg functions càdlàg?

Is the pointwise limit of càdlàg functions càdlàg? If not which are the weaker conditions to assure it? I cannot find a counterexample
1
vote
1answer
23 views

Diagonalizing a matrix arising in a simple combinatorial situation

Maybe I'll return to this question a few hours from now and possibly even post an answer then. This concerns a matrix that I described in this answer. Start with a $\dbinom n2\times n$ matrix $B$ ...
1
vote
4answers
69 views

Find the area of a triangle given the radius of its incircle and a tangential point

A friend gave me recently the following interesting problem and I would like to share a couple of solutions. Any additional contributions are welcome. A triangle $\vartriangle ABC$ is given and we ...
-7
votes
0answers
24 views

Olympic sprinter [on hold]

Carol would like to become an Olympic sprinter. Her younger sister Jane would rather play football, but helps Carol by racing against her. When they tried the 100 metre dash, Carol crossed the winning ...
3
votes
0answers
50 views

List of techniques to evaluate limits?

I'd like to make a complete list of techniques to solve a limit. Definition of the limit Continuous functions Algebra of limits Addition, multiplication, division Composition Inverse function ...
-1
votes
2answers
66 views

pointwise almost everywhere convergent subsequence of $\{\sin (nx)\}$ [on hold]

Can you prove or disprove that the sequence $\{\sin (nx)\}$ has a pointwise convergent almost everywhere subsequence with respect to the Lebesgue measure on $\mathbb{R}$ ?
3
votes
1answer
38 views

Singularity type of $\frac{1}{z} e^{-\frac{1}{z^2}} $

I've been asked to compute the singularity type of $f(z) := \frac{1}{z}e^{-\frac{1}{z^2}} $. Here's my reasoning: $$ \frac{e^{-\frac{1}{z^2}}}{z} = z^{-1} \sum_{n=0}^\infty \big( -z^{-2} \big)^n ...
1
vote
0answers
26 views

What finitely generated amenable groups are known to be LERF?

I know that finitely generated nilpotent groups are LERF (LERF means "subgroup separated"). I'm looking for examples (many, if possible) of groups which are: Finitely generated, but infinite ...
0
votes
2answers
23 views

Convergence when the derivative is uniformly continuous

Let $f: \Bbb R \to \Bbb R$ be a derivable function. $f'$ is uniformly continuous in $\Bbb R$ Prove that $[n(f(x+1/n)-f(x))]$ converges uniformly to $f'(x)$ I'm having a hard time seeing why does ...
1
vote
0answers
13 views

Significance of the derivative of a scalar field

I read somewhere that if the temperatures of all points of a huge room were plotted then the derivative at a certain point would give a vector whose direction points in the direction of the hottest ...
4
votes
3answers
152 views

Why is reflection in a plane an automorphism?

I have not studied group theory, but would like to know in simple terms why reflection in a plane is an automorphism. Dr. Hermann Weyl gives the definition of automorphism in his book 'symmetry' as ...
0
votes
1answer
18 views

Combining percentages differences

I would like to know the difference between two approaches of combining percentages. I found an online example that match my situation; the example have been found in The University of Georgia page. ...
-4
votes
2answers
18 views

empty containers [on hold]

two empty containers P and Q have the same volume. Water flows into P at the rate of 4 litres per minute and into Q at the rate of 6 litres per minute. After a certain time, container P can still take ...
1
vote
0answers
21 views

Sampling a Chebyshev polynomial with the discrete cosine transform

I have a Chebyshev polynomial $f$ of degree $n$ in point-value form \begin{align} f&=:S = \left( \left( x_i, y_i \right) \right)_{i=0}^n, \tag{1} \\ x_i &= \cos\left( \frac{i \pi}{n} \right), ...
0
votes
1answer
14 views

How is the distinction of left and right in space related to the orientation of screw?

In Dr. Hermann Weyl's book 'symmetry', he explains the difference between left and right as In space the distinction of left and right concerns the orientation of a screw. If you speak of turning ...
2
votes
3answers
36 views

Proving $|x+y|=|x|+|y| \iff x\cdot y \geq 0$

Prove: $|x+y|=|x|+|y| \iff x\cdot y \geq 0$. $|x+y|=|x|+|y| \iff x+y=x+y$ and $-(x+y)=-x-y \iff \{x,y\}\geq 0$ and $\{x,y\}\leq 0 \iff x\cdot y\geq 0$ in both cases.
1
vote
0answers
26 views

Spinors and forms

In this link http://benasque.org/2009gph/talks_contr/074Herdeiro.pdf page 15, it was said that: "Use spinorial geometry techniques: One takes the space of Dirac spinors to be the space of ...
1
vote
0answers
27 views

Intuitive way to find the minimum surface in $\mathbb R^3$?

Suppose we have two arbitrary closed curves which intersect neither each other nor themselves. By intuition, I guess that the minimum surface ending at boundaries is unique and it is achieved by this ...
2
votes
1answer
26 views

Problem in distribution theory and tempered distributions

I just encountered this question in my real analysis class involving distribution theory it is question 25 chapter 9 from Folland's real analysis second edition, which reads as follows: Suppose ...
-2
votes
0answers
8 views

Local exactness implies potential function [on hold]

Let $D$ be a simply connected domain and let $u(x,y), v(x,y)$ be two smooth functions such that $u_y=v_x$ in $D$. (a) Prove that there exists a potential function $\varphi(x,y)$ such that ...
-1
votes
0answers
34 views

Cohomology groups as sub vector spaces over $ \mathbb{Q} $.

Let $X$ be a non singular algebraic complex projective variety. How to construct a sub-vector space of the vector space $ H^k ( X , \mathbb{Q} ) $ ?. Is it true that $ H^k (Z , \mathbb{Q} ) $ is a ...
4
votes
2answers
52 views

Failure of group definition with weaker axioms

In "A First Course in Abstract Algebra", Edition 7, p.43, Fraleigh writes that It is possible to give axioms for a group $\left<G,*\right>$ that seem at first glance to be weaker, namely: ...
0
votes
1answer
39 views

Show that $(t^m-1)/(t^n-1)$ is a square if and only if $(\exists s \in \mathbb{Z})\ m=np^s$

I want to show the following lemma: Assume that the characteristic of the field $F$ is $p$ and $p>2$. Then $(t^m-1)/(t^n-1)$ is a square in $F[t, t^{-1}]$ ($F[t,t^{-1}]$: the polynomials in ...
2
votes
2answers
54 views

Point-Slope Equation of a line. Why is one answer incorrect and other is correct?

I am reviewing basic algebra. I am using quiz from the link this, and I solved the equation on paper and I get answer which is showing incorrect, I do not understand why is it wrong? It says my ...
1
vote
1answer
27 views

Doubling measure of an annulus

Recall that a doubling measure is a measure with the additional requirement that: $$\mu(B_{2R})\le C_\mu \mu(B_R)$$ for some contstant $C_\mu$. While solving some esercises related to doubling ...
1
vote
1answer
34 views

Partition a square

Compute the smallest positive integer $n$ such that, for any given integer $p\geq n$, we can partition a given square into $p$ number of squares (the small squares are not necessarily congruent) I ...
1
vote
0answers
29 views

Why $(n \times Id )_* O_{A\times A^\vee} = \oplus_{\tau \in A^\vee(S)} (Id\times \tau \circ \pi^\vee)^*P$

Consider an abelian scheme $\pi: A\rightarrow S$, with dual abelian scheme $\pi^\vee: A^\vee\rightarrow S$. The paper I am reading proved a lemma saying that $[n]_* O_A = \oplus_{\mu \in ...
0
votes
1answer
49 views

2x2 matrix multiplication issue

Let $$f_w(z)=z+w=\begin{bmatrix}1 & w \\ 0 & 1\end{bmatrix}z$$ where $z$ is a complex number. Shouldn't this be $w$ when $z=0$? However when I do the multiplication I get ...
2
votes
0answers
18 views

The dual space of weighted compact supported function?

Let $\Omega\subset \mathbb R^N$ be open bounded. It is well know that the dual space of $C_c(\Omega)$, i.e., compacted supported continuous function, can be identified by finite Radon measure ...
1
vote
0answers
15 views

For what $p$ is the condition number of a given matrix $A$, using the $p$ norm on matrices, minimal? [duplicate]

For what p is the condition number of a given matrix $A$, using the p norm on matrices, minimal? The condition number on $A$ is given as: $$K(A,p) = \|A\|_p \times \|A^{-1}\|_p$$ I tried ...
1
vote
0answers
21 views

Itō Integral multiplied by Riemann Integral

I was wondering whats the result of an Itō integral multiplied by a Riemann Integral. For example, what is $$\left(\int_0^T f(u)\ \mathsf dW_u\right)\left(\int_0^T g(v)\ \mathsf dv\right)$$ where $W$ ...
0
votes
1answer
30 views

Bijection from conjugacy class to the factor group by centralizer.

How different is $g^{-1}xg$ from $gxg^{-1}?$ Because proving a bijection from $g^{-1}xg$ type conjugacy class to the set of right cosets of the centralizer of $g$ in $G$ is as easy as proving it from ...

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