Monday, March 14, 2022

Recent Questions - Mathematics Stack Exchange

Recent Questions - Mathematics Stack Exchange

Osculating plane characterization

Posted: 14 Mar 2022 04:38 AM PDT

Let $\alpha:I \to \mathbb{R}^3$ a birregular curve parametrized by arc length. Suppose given $s \in I$ there exists an affine plane $\Pi$ such that the tangent line to $\alpha$ in $s$ is contained in $\Pi$ and for any $\epsilon > 0$ there are points of $\alpha(s-\epsilon, s+\epsilon)$ at both sides of $\Pi$ $(*)$. I want to show $\Pi$ is the osculating plane to $\alpha$ at $s$. I would say the only thing I have to do is to prove that $\Pi$ contains the normal line to $\alpha$ in $s$ but I am having problems to exploit the property $(*)$. Could you give me a hint on how to proceed?

There is a very similar post here but it expands on a book proof to which I have no access. Thanks in advance.

In (applied and pure) math study, do we only discuss and need material equivalence, not logical equivalence?

Posted: 14 Mar 2022 04:38 AM PDT

I am a mathematics major student and interested in logic. I have some questions, in math(both pure and applied aspects) study and research, do we clearly distinguish between logical equivalence and material equivalence as what we do in the pure logic study, or do we only discuss and require material equivalence (and roughly think that the two equivalences are the same meaning), because I find that in my daily math study we only discuss and require the material equivalence while ignoring logical equivalence. I want to know the answer and why, I am very grateful!

What is the joint distribution of exponential random variables $T_1$ and $T_2$ when they are related as $T_2 = \max(T_1 - A_2, 0) + S_2$?

Posted: 14 Mar 2022 04:30 AM PDT

Suppose $T_{1}$ and $T_2$ are identically distributed RVs according to an exponential distribution of rate $\mu - \lambda$, $A_{2}$ is an exponential random variable with rate $\lambda$ and $S_2$ is an exponential random variable of rate $\mu$, and $T_1$ and $T_2$ are related as $$ T_2 = \max(T_1 - A_2, 0) + S_2, $$

then, what is the joint probability distribution of $T_1$ and $T_2$, i.e., what would be $f_{T_1, T_2}(t_1,t_2)$ given the following $$ f_{T_1}(t_1) = (\mu-\lambda) e^{-(\mu - \lambda) t_1}, t_1 \geq 0, $$

$$ f_{T_2}(t_2) = (\mu-\lambda) e^{-(\mu - \lambda) t_2}, t_2 \geq 0, $$

$$ f_{A_2}(a_2) = \lambda e^{-\lambda a_2}, a_2 \geq 0, f_{S_2}(s_2) = \mu e^{-\mu s_2}, s_2 \geq 0. $$

Two basic questions

Posted: 14 Mar 2022 04:36 AM PDT

Here are two basic questions to which I currently do not know the answer:

(1) If $a > 0$ and $b$ are integers (where $b$ is negative), then if there exists a (necessarily) negative integer $c$ such that $b = ac$, then is it true that $a \leq |b|$?

(2) Does the inequality $|c - d| \leq |c| + |d|$ hold for all integers $c$ and $d$?

MY ATTEMPT

  • Trying special cases for (1): Let $b = -36$, $a = 4$, and $c = -9$. Then $b=ac$, and the inequality $a \leq |b|$ holds. (How about if $c = -1$? Then we have $b = -a$, and it is then true that $a \leq |b| = |-a| = a$.) However, I currently have trouble articulating a proof for (1).

  • Trying special cases for (2): Let $c = -d$. Suppose that $c > 0$. Then $$2c = |2c| = |c - d| \leq |c| + |d| = |c| + |-c| = c + c = 2c,$$ which does hold. (Again, I currently have trouble articulating a proof for (2), as I have doubts if it is valid for all integers $c$ and $d$.)

Using the definition of a limit, fine explicitly a natural number N which satisfies the below: $\lim_{n \to \infty} (\frac{n^2+1}{n^2+2})=1$

Posted: 14 Mar 2022 04:27 AM PDT

I have attempted this question but I am unsure if I have done it correctly. My current working is as follows:

If $\epsilon>0$ then for $N>\mathbb{N}$,there exists a $n>N$ such that $|{(\frac{n^2+1}{n^2+2})}-1|<\epsilon$.

Therefore: $|{(\frac{n^2+1}{n^2+2})}-\frac{n^2+2}{n^2+2}|<\epsilon$

$|{(\frac{n^2+1-n^2-2}{n^2+2})}|<\epsilon$

$|{(\frac{-1}{n^2+2})}|<\epsilon$

Then since absolute values

$|{(\frac{1}{n^2+2})}|<\epsilon$

$|{(\frac{1}{\epsilon})}|<n^2+2$

From here I am unsure if I have completed the proof or not. Would it be right to say. If n was greater than some N $(n^2+2)$, then $n>1/\epsilon$. And thus the definition of the limit is satisfied?

Condition number for the sum of two numbers

Posted: 14 Mar 2022 04:27 AM PDT

The condition number for the sum of two numbers $a$ and $b$ is $K=\frac{|a|+|b|}{|a+b|}$. But if I have $a=10$ and $b=-10$, I know that their difference is exactly $0$, so I have in practice no error, but the condition number is infinity. Why is is so? I am surely missing something really important here

How can be mathematically formulated the condition "$x$ is away from each point of singularity"?

Posted: 14 Mar 2022 04:24 AM PDT

During the math class of today, we introduced the function $$f(x)=\frac1x, $$ which has a singularity at $x=0$. Since for certain computations it was needed to avoid the singularity, we said something like: "In order to avoid the point $x=0$, suppose to stay away from it assuming, e.g. that $|x|>>0$ or also $x\to +\infty.$"

This is clear for me, but I have a question: If we deal with a function with an infinite set of singularities (points), how it can be reformulated the above idea to avoid each point of singularity?

I am thinking, for example, to a function like $$f(x)=\frac{1}{\sin x}.$$ I think it maybe would be something related to a choice of a suitable $\varepsilon>0$ enough small such that we can consider $x$ when outside the neighborhood of radius $\varepsilon$ and the neighborhood of each singularity never intersect. But actually I am not sure about that.

Could someone please help?

Thank you in advance!

Why does this identity with the product $\prod_{p\mid n}(p+k)$ and recursive sum of divisors function is true?

Posted: 14 Mar 2022 04:24 AM PDT

Let us define the following recursive function involving the sum of divisors function $\sigma(n)$:

\begin{array}{ l } r(n,1)=\sigma(n) \\ r(n,2)=\sum_{d|n}r(d,1) \\ r(n,3)=\sum_{d|n}r(d,2) \\ \ldots \\ r(n,k)=\sum_{d|n}r(d,k-1) \\ \end{array}

If n is squarefree, then the following identiy seems to hold ($p$ are the prime factors of $n$ smaller than $n$):

$$r(n,k)=\prod_{p\mid n}(p+k)$$

An implementation in Mathematica verifies this empirically:

rn = Compile[{{n, _Integer}, {k, _Integer}},      Total[Nest[Catenate@*Divisors, {n}, k]]];  p[n_, k_] := Product[p + k, {p, Complement[Divisors[n], {1, n}]}];  Print[rn[10, 5]];  Print[p[10, 5]];  (* Output: 70 *)

My question: Does there exist a theorem for this identity or (if not) how can we proof this identity?

Trying to understand $\max_{x} \max_{y} f(x,y) = \max_{x, y} f(x,y)$

Posted: 14 Mar 2022 04:17 AM PDT

I am reading this post where @Yanko gives an answer that proves

$$\max_{x} \max_{y} f(x,y) = \max_{x, y} f(x,y)$$ for a function $f(x,y)$. Please let me quote the answer

Let $(x_0,y_0)$ be any two elements. Then clearly we have that $f(x_0 > ,y_0) \leq\max_y f(x_0 ,y)$

Moreover we notice that $\max_x \max_y f(x,y)$ is the maximum over elements of the form $\max_y f(x',y)$ for some $x'$. In particular this implies that $f(x_0 ,y_0)\leq \max_y f(x_0 ,y)\leq \max _x \max > _y f(x,y)$. As $x_0,y_0$ were chosen arbitrarily we may conclude that $\max_{x,y} f(x,y)\leq \max_x\max_y f(x,y)$ and the inequality in the other direction is obvious.

I am trying to understand the last sentence, i.e.,

As $x_0,y_0$ were chosen arbitrarily we may conclude that $\max_{x,y} f(x,y)\leq \max_x\max_y f(x,y)$ and the inequality in the other direction is obvious.

Could you please someone provide some more details on how this? Also, does this proof hold for non-separable functions $f(x,y)$? Could you please give an example?

intersection of two lines one with 1 over infinity slope and the other with 3 as slope and 3 as C

Posted: 14 Mar 2022 04:17 AM PDT

If y=mx+c has a slope of 1 over infinity whereas c equals 0, and y=mx+c has a slope of 3 whereas c equals 3. what is the intersection point?

Is indistinguishable probabilities still indistinguishable even if randomness is allowed?

Posted: 14 Mar 2022 04:15 AM PDT

We say that a function $f:\mathbb N\to\mathbb R$ is negligible if for all positive integer $k$, there is $N\in\mathbb N$ such that for all $n\geq N$, \begin{equation} |f(n)| < \frac{1}{n^k}\text{.} \end{equation}

Let $X$ and $Y$ be measurable spaces. For each $n\in\mathbb N$, let $P_n$ and $Q_n$ be probabilities on $X$, and let $R_n$ be a probability on $Y$. Assume that for all measurable set $E\subseteq X$, the function \begin{equation} n\mapsto |P_n(E) - Q_n(E)| \end{equation} is negligible. Then is it true that for all measurable set $F\subseteq X\times Y$, the function \begin{equation} n\mapsto |(P_n\times R_n)(F) - (Q_n\times R_n)(F)| \end{equation} is negligible?

I proved it for the case where $Y$ is a finite set whose $\sigma$-algebra is discrete, using the Fubini's theorem. But I could not prove it for other cases.

Entire function bounded by a polynomial

Posted: 14 Mar 2022 04:17 AM PDT

Let $f(z) $ be an entire function such that for some constant $\alpha$, $|f(z)| \leqslant \alpha |z|^3 $ for all $|z| \geqslant 1$ and $f(z)=f(iz)$ for all $z$.

then comment about existence of such function.
My work $f$ will be polynomial of degree at most $3$ so $f(z) = a + bz + cz^2 + dz^3 \tag{1}$

$f(iz) = a + b(iz) + c(iz^2) + d(iz^3)$
$f(z) = a +ibz - cz^2 -idz^3 \tag{2}$ comparing equation $1$ and $2$ we get $b=c=d=0$
hence $f(z)$ is constant polynomial.

Is this correct?

A question based on Invariance principle

Posted: 14 Mar 2022 04:17 AM PDT

. Suppose the positive integer n is odd. First Al writes the numbers $1, 2,..., 2n$ on the blackboard. Then he picks any two numbers a, b, erases them, and writes, instead, $|a − b|$. Prove that an odd number will remain at the end.

I have proved it in this way, please check if it's correct:-
Clearly, there will be n odds and n even no.s
possibilities of picking a and b int.s:-
$2$ odds -> $|a-b|$= even number Hence, sum of the series ->$(n-2)$odds $+$ evens $=$ odd integer
when a & b are both even numbers -> sum = $n$ odds $+$ $(n-2)$ even $+$ even $=$ an odd integer
Without losing symmetry,
when a is odd and b is even -> sum = $(n-1)$odd $+$ $n$ even $+$ odd $=$ odd integer
Hence, the last integer left shall also be an odd int. Is my proof correct, please check, if not then kindly give me a hint, not a full solution...since I am preparing for math olympiad
Thank you

Limit of $(1+\frac{1}{n^3})^{n^2}$

Posted: 14 Mar 2022 04:33 AM PDT

I have been trying to solve the limit of $y_n = (1+\frac{1}{n^3})^{n^2}$. Through graphical analysis, I have found that $$\lim_{n \to \infty} y_n = 1$$ Which can also be intuitively be understood as $n^3 \geq n^2$. Using Bernoulli's inequality, you can easily find that $$y_n \geq (1+\frac{n^2}{n^3}) \geq 1$$ I have also found that $$y_n - y_{n+1} \geq \left( 1+\frac{1}{(n+1)^3}\right)^{n^2} - \left(1+\frac{1}{(n+1)^3}\right)^{(n+1)^2} = \left( 1+\frac{1}{(n+1)^3}\right)^{n^2} \left(1 - \left( 1+\frac{1}{(n+1)^3} \right) ^{2n+1} \right) = \left( 1+\frac{1}{(n+1)^3}\right)^{n^2} \left( \frac{1}{(n+1)^3} \right) \left( 1 + \left( 1 + \frac{1}{(n+1)^3} \right) + \cdots + \left( 1+\frac{1}{(n+1)^3}\right)^{2n} \right) \geq 1*0*2n\geq 0 $$$$\implies yn \geq y_{n+1}$$ Thus, by using the monotone convergence theorem, we know $y_n$ converges and has a lower bound of $1$. I am however stuck at showing that $\inf{\{y_n | n \geq 1\}} = 1$, which would show that $\lim_{n \to \infty} y_n = 1$. Could I get a hint or a nudge in the right direction ?

PS: I cannot use exponential and logarithmic properties, nor l'hopital's rule, as we have not defined all these things in class

Why average speed is important?

Posted: 14 Mar 2022 04:38 AM PDT

We know that in kinematics we have the concepts about "average speed".By definition the average speed is the total of the distance divided by time , but i still don't get it what is the average speed intuitively and why average speed is so important , and in my opinion average speed it doesnt seems accurate , for example if you say the average 12 km/h , it doesnt mean that you constantly Drive with 12 km/h . I am still confuse what average speed is actually mean and why do we need to learn it?

Find the set of circles with a given radius that pass through a given point

Posted: 14 Mar 2022 04:19 AM PDT

Given a point $(x_0,y_0)$ and a radius $r$, how do you find the set of all circles that have that radius that pass through the point?

Let $(h,k)$ represent the center of the set of circles. Then, it's clear that we have the following, because we want the set of points $(h,k)$ that have a distance $r$ from $(x_0,y_0)$ such that the circle centered at $(h,k)$ has a radius of $r$:

$$ (x-h)^2 + (y-k)^2 = r^2$$

$$ (h-x_0)^2 + (k-y_0)^2 = r^2$$

It's unclear how to proceed from here. While it's true that we have two variables in two equations, and we must solve for $h$ and $k$, the algebra is really messy and I'm not entirely sure this problem admits a closed form solution. Any insights?

For which values of $x$, following series is converging?

Posted: 14 Mar 2022 04:14 AM PDT

I tried to solve following problem.

Let $x \neq \frac{1}{2}$. Find for which values of $x$, following series is converging:

$$\sum_{n=1}^{\infty }\left(\left(\frac{x}{2x-1}\right)^{n} + x^{n}\right).$$

I tied to find when one of the element is dominant, and also when they are smaller than $1$:

$$\left|x \right| \leqslant \left|\frac{x}{2x-1} \right| < 1$$ or $$ \left|\frac{x}{2x-1} \right|\leqslant \left|x \right| < 1.$$

But the solution I'm getting is $0\leq x< \frac{1}{3}$, instead of $-1\leq x< \frac{1}{3}$.

So I know I made a mistake but I can't find it.

Find the maximum volume of a box inscribed in a tetrahedron bounded by coordinate axes and a plane

Posted: 14 Mar 2022 04:23 AM PDT

The full question: Find the maximum volume of a box inscribed in the tetrahedron bounded by the coordinate planes and the plane $x+\frac{1}{2} y + \frac{1}{3} z = 1$.

I tried graphing the plane on geogebra, and I can see the tetrahedron formed by the plane and the axes. However, I'd prefer to find these points analytically rather than graphically, and when I tried doing this I ran into a dead end (I tried just plugging in x=0, etc.). This could be a function of it being late and my brain not working when I'm tired, though.

However, once I get the points of the vertices of the boundaries, I'm not sure what to do from there. I know it'll eventually become an optimization problem.

Why can we only integrate a compactly supported differential form on an open set?

Posted: 14 Mar 2022 04:36 AM PDT

In the text I'm using, the support of $\phi$ (an n-form on an open $U \subset \mathbb{R}^n$) is defined as the closure of the values of $\mathbb{R}^n$ such that $\phi$ is nonzero, and $\phi$ is said to have compact support if its support is compact.

Letting $\phi = gdx_1 \wedge\cdots\wedge dx_n$, we define $\int_{U}\phi$ as $\int_{U}g$, if it exists. I can't really understand why we need compact support to do this. What is the issue if the n-form does not have compact support, but $U$ is bounded? And, if this is an issue, does this mean that we need to have supp$(\phi)\subset U$ in all cases for the integral to be well-defined?

About witt's cancellation theorem

Posted: 14 Mar 2022 04:19 AM PDT

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I am trying to understand the proof of witt's cacellation theorem given in scharlue's book.

  1. $\Sigma^{-1}\sigma$ is an isometry as a composition of two isometries. Its domain is $W_1$ since $W_{1}'\subset V'$ is a subspace?

  2. We can extend $\Sigma^{-1}\sigma$ to an isomtry $\theta$ using 5.3 theroem, since $(V,b)$ is a regular space and $W\subset V$, right? But in the cancellation theorem, they don't mention that $(V,b)$ is a regular symmetric bilinear space.

  3. Why is $\Sigma\theta$ is an isometry "that extends $\sigma$? And therefore it generates an isomerphism between the orthgonal spaces $W_2$ and $W_{2}'$ (here they just choose $W_2$ and $W_{2}'$ this way?)

Thanks very much and sorry for my ignorance

A particular inequality to whole solutions

Posted: 14 Mar 2022 04:37 AM PDT

Let $m$ and $n$ be two positive integers. Assume that $m$ and $n$ are relatively prime. What are the $(i,j)$ ordered pairs of non-negative integers that satisfy $$ (i+1)m+(j+1)n<mn$$

Do you know in the literature articles that deal with this type of inequalities to whole solutions? Thank you, greetings.

Eulerian polynomial problem - Signed permutations

Posted: 14 Mar 2022 04:26 AM PDT

Assume that $B_{n,k}$ is the number of signed permutations with size n which have exactly $k$ descents. $$B_n(t)=\sum_{k=1}^n B_{n,k}t^k,$$

$$\frac{B_n(t)}{(1-t)^{n+1}}=\sum_{k\geq0} (2k+1)^nt^k,$$ and that the recurence relation

$$B_{n+1,k}=(2k+1)B_{n,k} + (2n-2k+3)B_{n,k-1}$$ is valid.

I want to show that $$\frac{B_{n+1}(1)}{B_n(1)}=2n+2.$$

I need this end value so I can show what the conditional value of the number of descents for a signed permutation is.

Determining plane intersection with a ray

Posted: 14 Mar 2022 04:18 AM PDT

Say I know the starting point $P$ and the direction vector $\vec{D}$ of a ray and, and I have a plane specified by a normal vector and a point in the plane $X$. under what conditions can I be certain the ray will intersect the plane? I believe it comes down to the sign of the dot product between $\vec{D}$ and some vector which specifies the plane, but I'm unsure of the specific details, nor how $P$ fits into this. Would someone kindly help me clear this up?

If $f$ is uniformly continuous and bounded with $\int_\mathbb{R}|f(x)x^a|dx<\infty$ then $\int_\mathbb{R}|f^2(x)x^b|dx<\infty$ for some $b>a$

Posted: 14 Mar 2022 04:17 AM PDT

Let $f$ be a uniformly continuous and bounded function on $\mathbb{R}$ with \begin{equation} \int_\mathbb{R}|f(x)x^a|dx<\infty\tag{1} \end{equation} for some $a>0$. Here it is shown that the function $f(x)x^c$ can be unbounded for any $c>0$. However, in the given example the integral $\int_\mathbb{R}|f(x)x^c|dx$ (and, therefore, $\int_\mathbb{R}|f^2(x)x^c|dx$ as well) is finite for all $c>0$. Now the question is whether the condition (1) implies that for some $b>a$ $$\int_\mathbb{R}|f^2(x)x^b|dx<\infty?$$ It seems that Hölder's inequality alone does not lead to the proof of the above inequality.

How do you find the center of a cake with just a knife?

Posted: 14 Mar 2022 04:13 AM PDT

Consider an undecorated cylindrical cake and a perfect knife. We want to find the center of the cross-sectional circle.

If we can only score the surface of the cake, this reduces to finding the center of a circle with just a straightedge, which is impossible.

But using a knife allows additional constructions. For example, Sarvesh Iyer mentions:

The big difference between straight lines on a circle and knives on a cake is that you can shift any cake pieces you cut, around the shape and match them up with other pieces. The piece that's in a darker shade of orange can be removed and used to replicate that particular angle around the center subtended by it. I don't think one can create duplicate angles with just a straightedge, hence the difference.

Similarly, YNK claims that

Cake cut sevenfold

determines the center in seven cuts, although he has not explained the construction procedure.

Here are some rules modeling our use of the knife.

  1. You can't guarantee any nice properties of the lines like being perpendicular or parallel to another, just chords.
  2. The "canvas" for connections is only the circle (you can't be cutting the table, just the cake).

What is the minimal number of cuts necessary to find the circle? Is it YNK's 7?

Relationship between HOMFLY and Alexander-Conway polynomials

Posted: 14 Mar 2022 04:11 AM PDT

Using $L^*$ to denote the mirror image of a $\mu$-component link $L$, the HOMFLY polynomial satisfies $P_{L^*}(l,m)=P_L(l^{-1},m)$ while the Alexander-Conway polynomial (i.e. a symmetric representative of the Alexander polynomial where the sign is appropriately fixed) satisfies $\Delta_{L^*}(t)=(-1)^{\mu-1}\Delta_L(t)$ (use Seifert matrices).

On the other hand, Cromwell's textbook indicates that $\Delta_L(t)=P_L(1,t^{-1/2}-t^{1/2})$. I understand why this is true (both polynomials satisfy the same skein relation) but I've gotten confused about why this doesn't contradict the above mirror formulas: plugging in $l=1$ in HOMFLY won't allow you to pick up this $(-1)^{\mu-1}$ in Alexander-Conway.

Can anyone help?

Am I doing right? Lie algebra-About finding the ideals of an algebra

Posted: 14 Mar 2022 04:39 AM PDT

I want to find all the ideals and quotient algebras of the algebra:

$L=\{A\in \mathfrak{gl}_2 : a_{21}=0\}$.

And to prove that $L$ is isomoprphic to the direct product of $k$ (one dimensional algebra) and $b_2$.

So, an element in $L$ is of the form:

$\{a_{11}, a_{12}, a_{21}, a_{22}\} = \{a,b,0,c\}$. I know that a basis is:

$X=\{1,0,0,0\}$ , $Y= \{0,1,0,0\}$ , $Z=\{0,0,0,1\}$.

Then: after calculation we get the following muliplication list:

$[X,Y]=XY-YX=Y$,

$[X,Z]=XZ-ZX=0$,

$[Y,Z]=YZ-ZY=Y$

Now, we want to describe all the ideals $I\subset L$:

  1. $I=0$

2.$I\neq 0$:

Let $0 \neq t=aX+bY+cZ\in I$ then after computations we get:

$[X,t]=[X,aX+bY+cZ]=bY$

$[Y,t]=(a+c)Y$

$[Z,t]=bY$

So, if $b\neq 0$ then $Y\in I$.

If $a=c=0, b\neq 0$, then again $Y\in I$.

If $a\neq -c, b\neq 0$, then $Y\in I$.

Thus, if $\dim I=1$ then $I=\langle Y\rangle$.

If $\dim I=2$ then $I=\langle Y,aX+cZ\rangle$ for any choose of b.

If $\dim I=3$ then $I=L$.

Now, we describe the quotient algebras of L:

$I=\langle Y\rangle$, $Y=(0,1,0,0)$ The quotient algebra is: $L/\langle Y\rangle$ where $L=\langle X,Y,Z\rangle$. The basis of the quotient algebra is: $L/\langle Y\rangle=\langle X',Z'\rangle$ where $X'=X+\langle Y\rangle, Z'=Z+\langle Y\rangle$. Then, $[X',Z']=XZ+\langle Y\rangle= {XZ=0} = \langle Y\rangle=0$.

Therefore, $L/\langle Y\rangle$ is a commutative algebra.

I'm not sure if what I've shown is right and how to continue from here.

For the second part, I need to define an isomorphism $f:b_2×K\to L$, where

$b_2= \{A\in sl_2 : a_{21}=0\}$.

($\dim b_2=2$).

Can I define f by:

$f(A,\lambda)=A+\lambda I$. So f is well defined since both $A, \lambda I$ are in $L$. Now, I need to show that $f$ is an isomorphism of algebras.

To prove that f is a homomorphism of algebras, we have to to check that: $f((A,\lambda)+(A',\lambda'))=f(A,\lambda)+f(A',\lambda')$.

$f((A,\lambda)*(A',\lambda'))=f(A,\lambda)*f(A',\lambda')$.

$f(c(A,\lambda)=cf(A,\lambda)$

f is injective: $Ker(f)={(A,\lambda) \in b_2×k : A+\lambda*I=0}$ A is a matrix of the form (a,b,0,-a) so $A+\lambda*I=0$ iff $a+\lambda=0$ /\ $b=0$ /\ $-a+\lambda=0$ iff $a=b=\lambda=0$ so $Ker(f)=(0_{2\times 2},0)$.

f is surjective:I have to show that for every matrix $A \in L$ there is a pair $(B,\lambda)$ such that $f(B,\lambda)=A$. So if A=(a,b,0,c) then I can take $B=(a-\lambda, b,0, c-\lambda)$ where $c=2\lambda-a$. I am not sure about this.

Many thanks

What is a practicized (or suitable) label for morphisms that satisfy the condition mentioned in the body of this question?

Posted: 14 Mar 2022 04:27 AM PDT

Let it be that $\mathfrak A$ and $\mathfrak B$ denote $\mathcal L$-models for some language $\mathcal L$ and that $A$ and $B$ denote their domains/universes.

A function $h:A\to B$ can be recognized as a morphism $\mathfrak A\to\mathfrak B$ if it satisfies the conditions:

  • $h(c^{\mathfrak A})=c^{\mathfrak B}$ for constants symbols.
  • $h\circ f^{\mathfrak A}=f^{\mathfrak B}\circ h^n$ for function symbols $f$.
  • $(a_1,\dots,a_n)\in r^{\mathfrak A}\implies (h(a_1),\dots,h(a_n))\in r^{\mathfrak B}$ for relation symbols $r$.

In both cases $n$ denotes the arity of the symbol.

If moreover $h$ is injective and satisfies the stronger condition:

$$(a_1,\dots,a_n)\in r^{\mathfrak A}\iff (h(a_1),\dots,h(a_n))\in r^{\mathfrak B}\text{ for relation symbols }r\tag1$$then $h$ is an embedding.

My question:

Is there some suitable terminology for a morphism $h$ that satisfies $(1)$?

I would like to say things as: "a morphism $h$ is an embedding if it is injective and ...."

Edit:

The answer of Alex is fine and educating. I have found a more direct answer to my question later in some script. Homomorphisms that satisfy $(1)$ were called strong homomorphisms there. So a homomorphism is an embedding iff it is strong and injective. Also I was told there that strong homomorphisms correspond one-to-one with congruences (hence quotients).

How to analytically solve an ODE based on the thermal radiation equation

Posted: 14 Mar 2022 04:27 AM PDT

I have to find parameter for $C\text{, }R_0\text{, }\alpha\text{, }T_0\text{, }T_A\text{, }T_L\text{ and }\epsilon \sigma A$ based on pairs of $I,t$ with $T\left(t\right)=T_L$. Also I know the pair $I, T$, for which $ \frac{\partial T}{\partial t} = 0 $. All unknown are real and strictly positive. $$ \frac{\partial T}{\partial t} = \frac{\left( R_0 \left( 1 + \alpha \left( T - T_0 \right) \right) \right) I^2 - \epsilon \sigma A \left( T^4 - T_A^4 \right)}{C} \\ T\left(0\right) = T_A $$ Further I can assume that: $$ T_A \le T_0 \\ T_L > T_A \\ R_0 = 200e-6 \\ T_0 \text{ is probably } 23+273.15 \text{ or in the range of } T_A\\ T_A \text{ is probably } 85+273.15 \text{ or } 105+273.15 \\ T_L \ll 1734+273.15 $$

For simplification, I have already substituted complex sub-expressions and factored out $T$: $$ \frac{\partial T}{\partial t} = D T^4 + E T + F \\ \text{ with: } \\ D = \frac{- \epsilon \sigma A}{C} \\ E = \frac{R_0 \alpha I^2}{C} \\ F = \frac{R_0 I^2 - R_0 \alpha T_0 I^2 + \epsilon \sigma A T_A^4}{C} \\ $$ This can easily be decomposed into: $$ \frac{1}{D T^4 + E T + F} \partial T = \partial t $$

Thanks to @mattos I know that this is a Chini equation, for which the Chini invariant is independent of $t$: $$ \frac{\partial y\left(t\right)}{\partial t} = f\left(t\right) y^n\left(t\right) + g\left(t\right) y\left(t\right) + h\left(t\right) \\ f\left(t\right)=D \\ n=4 \\ y\left(t\right)=T \\ g\left(t\right)=E \\ h\left(t\right)=F \\ C = f^{−n−1}\left(t\right) h^{−2n+1}\left(t\right) \left(f\left(t\right)\frac{\partial h\left(t\right)}{\partial t}−h\left(t\right)\frac{\partial f\left(t\right)}{\partial t}+n f\left(t\right) g\left(t\right) h\left(t\right)\right)^n n^{−n} \\ C = D^{−4−1} F^{−2*4+1} \left(D\frac{\partial F}{\partial t}−F\frac{\partial D}{\partial t}+4 D E F\right)^4 4^{−4} \\ C = D^{−5} F^{−7} \left(0 D−0 F+4 D E F\right)^4 4^{−4} \\ C = D^{−5} F^{−7} 4^4 D^4 E^4 F^4 4^{−4} \\ C = D^{−1} F^{−3} E^4 \\ C = \frac{E^4}{D F^3} $$ Obviously $C$ is independent of $t$ and $n=4$.

And from there I have no idea how to continue. Referenced, which do not really fit I found are:

Side information:

  • Finally I need the function $ t\left(I, T_A\right)\text{ for which } T\left(t\right)=T_L $.
  • The solution should be analytical to allow error calculations.
  • The final formula has to be calculated in a low-power embedded system.
  • In a first step, we could assume that $ \alpha = 0 $
  • The parameters $R_0, \alpha, T_0, T_A, T_L, C \text{ and } \epsilon{}\sigma{}A$ have (later, independent of this question) to be calculated out of pairs of $t\left(I\right)=t_L$ and (one) $\lim_{I \to I_C+}t\left(I\right)=\infty$ with $T_A$ possibly pre-defined and possibly different between both.

P.S.: My math lectures are roughly 20 years ago, please feel free to improve my notation.

If $|A_{ij}|\leq 1$ for all entries of symmetric $A$, then the largest possible spectral norm of $A$ is smaller than $n$?

Posted: 14 Mar 2022 04:15 AM PDT

In other words, for any symmetric or antisymmetric matrix $A\in\mathbb{R}^{n\times n}$ that has $|A_{ij}|\leq 1$, can we conclude that \begin{align*} \|A\| \leq \|J\|, \end{align*} where $\|\cdot\|$ denotes spectral norm and $J$ represents all one matrix?

Since we already know that all one matrix $J$ has the smallest eigenvalue $\sigma_n = 0$ and all other eigenvalues $\sigma_1 = \sigma_2 = \ldots = \sigma_{n-1} = n$, it is to say whether such $A$ can have singular value greater than $n$.
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