Recent Questions - Mathematics Stack Exchange |
- Prove that this filed is finite
- Does lyapunov function $\frac{dV}{dt}=0$ means $(0,0)$ is stable but not asymptotically stable?
- If $X$ is predictable and $\tau$ a stopping time, is then $X_{\tau+}$ $\mathcal{F}_{\tau}$-measurable?
- Banach space induced by linear operators
- Loss networks and Poisson processes
- A problem about Klingenberg's lemma
- A general way to express gamma function?
- How to find solution for linear congruence?
- Points of f(x)=x^3 except the origin
- What is the meaning of $[x]$ in this notation?
- Choosing $r$ objects from $n$ identical objects, is this interpretation correct?
- Showing that some congruence equation doesn't have a solution.
- Volume of Cylinder and Ellipsoid
- A Problem on cauchy-Schwarz inequality
- Question from Polya and Szego's Problems in Analysis concerning the limit of floor function
- What is the scope of application of Kalman filter?
- 2 continuous functions with the property $f(g(x))=x$.
- How to prove that a subset is open in another one?
- Non-surjective function commutativity with a bijection.
- Find the polynomial of fixed degree that minimizes the maximal error on a set of equidistant points
- Reconstruction of a set from a pair partition
- Recursion for the number of words of length $n$ over the alphabet $\{0, 1, 2\}$ such that there are neither $11$ nor $22$ blocks
- primitive of a function with $e$ and trigonometric functions
- Maximal ideals in rings of rational fractions
- A question about bilinear forms
- Algorithm to find a stable minimum cycle mean in a bipartite digraph under competing strategies of the two colored nodes.
- Rudin's PMA: theorem 10.43
- How to quantify the loss of information in information theory
- Solving non linear equations where equations are equals
- find the tangent to the sphere
| Prove that this filed is finite Posted: 30 Mar 2022 05:20 AM PDT This question was left as an exercise by the prof. in Algebraic Geometry. I tried it at home but not able to make progress. So, I am posting it here for hints.
Attempt: K is finite type over K means that there exists $X_1,...,X_n$ such that $K = \mathbb{Z} [X_1,...,X_n]$. On the contrary, let K is infinite. But, I am not able to think which result should I use. I discussed with a friend and he also couldn't proved it. I am following miles reid undergraduate Algebraic geometry.
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| Does lyapunov function $\frac{dV}{dt}=0$ means $(0,0)$ is stable but not asymptotically stable? Posted: 30 Mar 2022 05:17 AM PDT For an autonomous differential equations with two unknows i.e.: \begin{equation} \frac{dx}{dt}=f(x,y) \qquad \frac{dy}{dt}=g(x,y) \end{equation} And $O=(0,0)$ is the unique solution of $f(x,y)=0$ and $g(x,y)=0$. If I find a lyapunov function $V(x,y)>0$ and $\frac{dV}{dt}=0$. Does this means $O=(0,0)$ is stable but not asymptotically stable? Or I need some more evidences to derive this? |
| Posted: 30 Mar 2022 05:16 AM PDT Setting We work on a filtered probability space. Let $X$ be a predictable process (i.e. it is measurable w.r.t. the predictable sigma-algebra). Let $\tau$ be a stopping time. Motivation Then it is known that $X_{\tau}$ is $\mathcal{F}_{\tau-}$-measurable on $\{\tau<\infty\}$. Question Does it also hold that $X_{\tau+}$ is $\mathcal{F}_{\tau}$-measurable on $\{\tau<\infty\}$? |
| Banach space induced by linear operators Posted: 30 Mar 2022 05:15 AM PDT Let $E, F$ be Banach spaces and $T: E \to F$ is a bounded injective map. If we define on $E$ a norm $\|u\|_{\widetilde{E}}:= \| Tu \|_F, u \in E,$ then one easily see that $(E, \|\cdot\|_{\widetilde{E}})$ is a normed space. In general, this space is not a Banach space. I would like to know if there are any conditions on $E,F$ or $T$, which makes $(E, \|\cdot\|_{\widetilde{E}})$ a Banach space. Thank you very much. |
| Loss networks and Poisson processes Posted: 30 Mar 2022 05:07 AM PDT Consider a shop with capacity C, where customers spend an i.i.d exponential(1) amount of time before leaving through the back door. Customers arrive through the front door as a Poisson($\lambda$) process. If a customer arrives when the shop is already full, then they immediately leave through the side door. Say the equilibrium probability that the shop is full is $p$, and that the system is in equilibrium. I have seen some notes on Erlang alternative routing that refer to essentially the setup above, and claim that the process of exits through the side door is just a Poisson($p \lambda$) process. Is this even true? To me it seems very far from obvious, as the side-door process is not a simple "thinning" of the front-door process. I suspect the PASTA property is useful here, but even then it seems more work is needed since arrivals do not see the state of the shop independently of each other. I understand that some renewal-reward argument can be made to say the long-term rate of flow through the side door is $p \lambda$, but that is all I can manage to say. This setup seems a very general one, so I would really like to get things straight in my head. If indeed the statement is true, please could someone offer a proof? |
| A problem about Klingenberg's lemma Posted: 30 Mar 2022 05:07 AM PDT I see the particular of Klingenberg's lemma:
Since $M$ is compact, if $M$ is smooth, I feel (just feel, not sure) there is a closed geodesic $\gamma$ in $M$ such that $$ l(\gamma) =l(M). $$ Then, for any $p\in \gamma$, I have $$ \exp_p:T_pM\rightarrow M $$ is injective on $B_0(\frac{l(\gamma)}{2})\subset T_pM$. $B_0(\frac{l(\gamma)}{2})$ is the open ball center at origin and radius is $\frac{l(\gamma)}{2}$. Therefore, I have $$ \text{inj}(M) \le \frac{l(M)}{2} $$ Obviously, it is incompatible with Klingenberg's lemma. Where is my mistake? |
| A general way to express gamma function? Posted: 30 Mar 2022 05:06 AM PDT I'm trying to evaluate the giving function: $$\sum_{k=1}^{\infty}\int_{0}^{\infty}{(t-\ln{(s)})}^{n-1}e^{-tk}dt$$ I observed that $e^{-tk}$ present some relation with the zeta function but with this "general" form of the gamma function I'm having a hard time figuring out this relation. If someone can show how to evaluate this |
| How to find solution for linear congruence? Posted: 30 Mar 2022 05:11 AM PDT How to find solutions for linear congruence $x \equiv 4 \pmod{11}$? |
| Points of f(x)=x^3 except the origin Posted: 30 Mar 2022 05:04 AM PDT I want to know what properties does the non-zero points in the domain of $f(x)=x^3$ have? For example, $0$ is an inflection point, since the curvature changes sign at 0 ($f^{"}$ changes sign near $0$). In fact it is an stationary point as well $(f'(0)=0)$. So, it is also a saddle point. What about if $x\neq 0$. Are these points inflection point or something else? I think such points ($x_0\neq 0$) are not inflection points nor saddle points, since then $^{''}(x_0)=6x_0\neq 0$ if $x_0\neq 0$. But from the graph, it seems such points are neither maximum nor minimum points. Can someone please explain in detail what should be called to these non-zero points? Thanks a lot. |
| What is the meaning of $[x]$ in this notation? Posted: 30 Mar 2022 05:06 AM PDT I came across this proposition, but I don't know the meaning of $[x]$. Does it mean the equivalence class of $x$?
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| Choosing $r$ objects from $n$ identical objects, is this interpretation correct? Posted: 30 Mar 2022 05:00 AM PDT CASE $1$:
This case is simple enough as the probability of drawing any ball remains constant, so it becomes a case of tossing a coin $3$ times. The sample space would be {RRR, RRB, RBR, BRR, BBB, BBR, BRB, RBB}, and thus the answer would be $3$ ways. CASE $2$: CASE $3$:
The answer has to be $1$, since there is only one outcome to choose. CASE $4$:
$$P(X) = \frac{n(p).n(q)}{n(r)}$$ where $X$ is the event of choosing two red balls and one blue ball, $n(p)$ is the number of ways of choosing two red balls, $n(q)$ is the number of ways of choosing $1$ blue ball, $n(r)$ is the number of ways of choosing a sample of $3$ balls. So,$$n(p) = n(q) = 1$$$$n(r) = 8$$ from case $3$ and case $2$, since the sample space remains the same. So $$P(X) = \frac{1}{8}$$ Is this result correct? Any external perspective would help! |
| Showing that some congruence equation doesn't have a solution. Posted: 30 Mar 2022 05:00 AM PDT I have the following congruence equation : $x^3\equiv 2 mod 151$, and I want to show that there are no solutions to this equation. So I suppose to the contrary, that there is a solution, then 151 doesn't divide $x$ ( otherwise, $x\equiv_{151}0$).So by Fermat little theorem $x^{150}\equiv_{151}1$, so we should be getting $2^{50}\equiv_{151}1$, which should be a contradiction; but I don't see why? Can someone please enlighten me? Thanks! |
| Volume of Cylinder and Ellipsoid Posted: 30 Mar 2022 04:59 AM PDT
In $\mathbb{R}^3$ I have to find the volume of the domain which lies inside both the cylinder of equation $x^2+y^2=1$ and the ellipsoid with equation $4x^2+4x^2+z^2=64$, where $(x,y,z)$ are Cartesian Coordinates in $\mathbb{R}^3$. The cylinder and the ellipsoid are shown at the graph. The domain in question is a tube domain with the side surface being the surface of the cylinder, and the top and the bottom parts of its boundary being parts of the surface of the ellipsoid. Approach: First, I substitute with cylindrical coordinates as follows: $r^2=x^2+y^2, \tan(θ)=y/x$ and $z=z$ From the equation of the cylinder I get that $r^2=1$ and from the equation of the ellipsoid that $z^2=64-4r^2 \Rightarrow z=\pm \sqrt{64-4r^2}$. Replacing with $r^2=1$ in the equation of the ellipsoid i get the solutions $z=\pm 2\sqrt{15}$. Question: In order to find the volume in question I am considering the triple integral(s) for $-1\leq r \leq 1$, $0 \leq θ \leq 2π$ and for $-2\sqrt{15} \leq z \leq 2\sqrt{15}$. Are these boundaries correct? Since $z=\pm \sqrt{64-4r^2}$, it is broken into two functions, namely $z= \sqrt{64-4r^2}$ for which the boundaries of $r$ and $θ$ will remain the same but the boundaries of $z$ would be $ 0 \leq z \leq 2\sqrt{15}$ and $z= -\sqrt{64-4r^2}$ for which the boundaries of $r$ and $θ$ will remain the same but the boundaries of $z$ would be $ -2\sqrt{15}\leq z \leq 0$. And thus I will calculate the triple integrals. Is this approach correct? Am I missing something? |
| A Problem on cauchy-Schwarz inequality Posted: 30 Mar 2022 04:59 AM PDT I have to prove that for $x,y,z>0$, $$ \frac{2}{x+y}+\frac{2}{y+z}+\frac{2}{z+x}\geq \frac{9}{x+y+z} $$ |
| Question from Polya and Szego's Problems in Analysis concerning the limit of floor function Posted: 30 Mar 2022 05:05 AM PDT For the following question: Let $\theta$ be an irrational number, $0< \theta < 1$ and let $g_{n}=0$ or $1$, according as $\lfloor n\theta \rfloor$ and $\lfloor (n-1)\theta \rfloor$ are equal or different. Show that $\lim_{{n}\rightarrow\infty}\frac{g_{1} + g_{2} + g_{3} + \cdots + g_{n}}{n} =\theta$ I would like to use the result in (1) below. So, I first need to show that, $\lim_{n\rightarrow\infty}g_{n}=\theta,$ where $g_{n}=\lfloor {n\theta} \rfloor$, then use (1) with $a_{n}=g_{n}$ and $0< \theta < \frac{1}{n}<1$, since $\lfloor {n\theta} \rfloor$ has period $\frac{1}{n}.$:
To show that $\lim_{n\rightarrow\infty}g_{n}=\theta$, let $\epsilon$ be given, to find $M$, we have $\lvert\lfloor {n\theta} \rfloor-\theta\rvert\leq \lvert n\theta - \theta \rvert < \lvert 1-\frac{1}{n}\rvert<\epsilon.$ We can let $n> \frac{1}{-\epsilon + 1}=M$. To check whether such $M$ works, $\lvert \lfloor {n\theta} \rfloor-\theta\rvert \leq \lvert n\theta - \theta\rvert < \lvert 1-\frac{1}{n}\rvert< 1- (-\epsilon + 1)=\epsilon$ Hence we have shown that $\lim_{n\rightarrow\infty}g_{n}=\theta$ which satisfies the hypothesis in (1) which gives $\lim_{{n}\rightarrow\infty}\frac{g_{1} + g_{2} + g_{3} + \cdots + g_{n}}{n} =\theta$ Can someone can check the correctness of my solution. Thank you in advance. |
| What is the scope of application of Kalman filter? Posted: 30 Mar 2022 05:03 AM PDT Recently I learned some basics about Kalman Filter 1D As I know, Kalman Filter is useful in Telecommunication and GPS positioning. My estimation goal is to measure the reliability of the Circuit using the MC method. Is Kalman Filter also applicable in this field? According to Central Limit Theorem, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution. So I packed my 100 measurements' mean value as a single measurement. Also, I want to know. I can't directly get Estimate Uncertainty($p_n$) and Measurement Uncertainty($r_n$). My idea is to use the standard deviation or the variance of initial measurements as Estimate Uncertainty, use the expected value of error as Measurement Uncertainty. If you could show me any good thesis or method concerned with the approximation of unknown uncertainty, that's the best. Thank you! |
| 2 continuous functions with the property $f(g(x))=x$. Posted: 30 Mar 2022 05:13 AM PDT $f, g\colon \mathbb {R}\to \mathbb {R}$ continuous functions such that $f(g(x))=x$. Prove that $g(f(x))=x$. |
| How to prove that a subset is open in another one? Posted: 30 Mar 2022 04:59 AM PDT Let $B$ be a Banach space. Let $U\subset B$ be an open subset and $K\subset B$ be a compact subset. Define $$\mathscr{L}=\{f:K\to B: f\mbox{ is continuous}\}$$ which is complete metric space endowed with the dustance $$d_{\mathscr{L}} =\max_{x\in K} |f_1(x)-f_2(x)|_{\infty}, \quad f_1,f_2\in \mathscr{L}.$$ I need to prove that $$\mathscr{L}_U=\{f\in \mathscr{L}: f(x)\in U\}$$ is open in $\mathscr{L}$. I tried by contradiction, defining the complementar of $\mathscr{L}_U$ as the set of functions of $\mathscr{L}_U$ such that $f(x)\notin U$ and trying to prove that it is closed but I can't even do this. Could someone please help me? Thank you in advance! |
| Non-surjective function commutativity with a bijection. Posted: 30 Mar 2022 05:13 AM PDT I was just considering how close the product of sets is to being the pushout of a diagram since it is in the pullback which is the dual to the pushout: given $f:Z \rightarrow X$ and $g: Z \rightarrow Y$ then there'd be bijective functions from each set of the product like so: $b:X \rightarrow X \times Y$ and $b: Y \rightarrow X \times Y$. It looks like a commutative diagram at first but if $f$ and $g$ are injective non-surjective functions then I am not sure if the diagram commutes through those functions anymore. |
| Find the polynomial of fixed degree that minimizes the maximal error on a set of equidistant points Posted: 30 Mar 2022 05:12 AM PDT I've got a set of $m \in \mathbb{N}$ 2D points which I want to represent as a polynomial of degree $n$: $\left \{ \left( x_i, y_i \right) \in \mathbb{R}^2 : i \in \mathbb{Z}, 0 \leq i \lt m \right \}$, with $x_i - x_{i - 1} = d$ being constant. These points happen to be values of a univariate real function, like so: $f(x_i) = y_i$; so one could say that I'm approximating that function with a polynomial, but I only care about the values on the equidistant $x_i$, not on the entire interval $[x_0, x_{m - 1}]$. I want to find a polynomial that minimizes the maximal error, so my error function is such: $\max_{0 \leq i \lt m} \lvert y_i - p(x_i) \rvert$. So far I know of two methods of finding "best" polynomials:
What method should I use for finding my polynomials? EDIT: This question seems to be related, however it's specifically about the case $n = 1$, so I'm not sure how much it applies to the case of polynomials in general: Linear regression for minimizing the maximum of the residuals |
| Reconstruction of a set from a pair partition Posted: 30 Mar 2022 05:14 AM PDT
Intuitively, it seems to be true, I haven't found a counter-example. What I realized is that we can neglect the pairs $(x,-x)$, in this way we reduce to the case with $n-1$ pairs. Example: $M=\{-5,-4,-3,-2,-1,1,2,3,4,5\}$, the pairs are $P_1=(-1,3), P_2=(-2,1), P_3=(2,-4), P_4=(-3,-5), P_5=(4,5)$. We can choose $a_1=3, a_2=1, a_3=2, a_4=-5, a_5=4$. I found this question, which is very similar, but still no clue how to link to my problem. Maybe there is something trivial I don't see. Thank you! |
| Posted: 30 Mar 2022 04:56 AM PDT Let $a_n$ be the length of words over an alphabet $\{0,1,2\}$ such that there are neither $11$ nor $22$ "blocks". I. e. $001020$ would be allowed, $001120$ wouldn't because we have a $11$ block. I have to show that $$a_n=2a_{n-1}+a_{n-2}$$ It looks like this recursion dissects the question into three cases (and adds them all up):
It would make sense to me if it'd be $a_n=a_{n-1}+2\cdot a_{n-2}$. |
| primitive of a function with $e$ and trigonometric functions Posted: 30 Mar 2022 05:07 AM PDT Find $$ \int_0^\frac{\pi}{2}\:e^{\sqrt{\cos x}}\sqrt{\sin2x}dx$$
First of all, I noticed that $f$ is elementary therefore it is continuous, thereof it is primitivable. I thought that in order to find $$ \int_0^\frac{\pi}{2}\:e^{\sqrt{\cos x}}\sqrt{\sin2x}dx$$ I have to find one of its primitive that is to say: $$\int \:e^{\sqrt{\cos x}}\sqrt{\sin2x}dx$$ Nevertheless, it was suggested in the answers section that this is quite complicated. Maybe I have to use some tricks for the definite integrals, but I do not know which kind of magical tricks will work in this situation, although I tried some like substitutions or different integral formulas. Any help would be immensely appreciated. I obtained via substitution $$t=\frac{\pi}{2}-x$$ that: $$\int \:e^{\sqrt{\cos x}}\sqrt{\sin2x}dx= -\int \:e^{\sqrt{\sin x}}\sqrt{\sin2x}dx$$ But this is as hard as the first one. Then I tried this: $$\frac{d}{dx}\left(e^{\sqrt{\sin \left(x\right)}}\right)=\frac{e^{\sqrt{\sin \left(x\right)}}\cos \left(x\right)}{2\sqrt{\sin \left(x\right)}}=\sqrt{\cot \left(x\right)}{e^{\sqrt{\sin \left(x\right)}}\sqrt{\cos \left(x\right)}}$$ Because I wanted to try to solve it with the help of integration by parts this similarly was useless, in my opinion. However, the last part seems to be similar to my problem. Maybe this is not the right track What should I do? |
| Maximal ideals in rings of rational fractions Posted: 30 Mar 2022 05:07 AM PDT So far I managed to prove this: let $k$ a field, $n\in\mathbb{N}^*$, $S\subset k[X_1,\dots,X_n]$ the subset of polynomials without zeros on $k^n$, and for $\alpha\in k^n$, $M(\alpha)$ the maximal ideal of $S^{-1} k[X_1,\dots,X_n]$ made of elements that vanish at $\alpha$ ; if $k$ is algebraically closed, or an ordered field, or a finite field, then the set of maximal ideals of $S^{-1} k[X_1,\dots,X_n]$ is the set of $M(\alpha)$, for all $\alpha\in k^n$. When $k$ is algebraically closed, the nullstellensatz shows that a polynomial without zeros on $k^n$ is a non zero scalar of $k^*$. Therefore $S$ is already invertible and $S^{-1} k[X_1,\dots,X_n] = k[X_1,\dots,X_n]$. The description of maximal ideals of $k[X_1,\dots,X_n]$ also follows from the nullstellensatz. Now assume $k$ is an ordered field and let $J$ a maximal ideal of $S^{-1} k[X_1,\dots,X_n]$. The Hilbert basis theorem shows that $k[X_1,\dots,X_n]$ is a noetherian ring. Then the localization $S^{-1} k[X_1,\dots,X_n]$ is also noetherian and there exists a finite set of polynomials $P_1,\dots,P_r \in k[X_1,\dots,X_n]$ that generate $J$. Thus $P_1^2+\dots +P_r^2\in J$. And since $J$ has no invertible elements, there is $\alpha\in k^n$ such as $P_1^2(\alpha)+\dots +P_r^2(\alpha) = 0$. Because $k$ is an ordered field, $\alpha$ is a common zero of all $P_i$'s, and by maximality $J = M(\alpha)$. Now assume $k$ the finite field with $q$ elements and let $J$ a maximal ideal of $S^{-1} k[X_1,\dots,X_n]$. As before there exists a finite set of polynomials $P_1,\dots,P_r \in k[X_1,\dots,X_n]$ that generate $J$. Assume for the sake of contradiction that the $P_i$'s have no common zeros. By the nullstellensatz for finite fields, there exist polynomials $g_i$ and $g$ such as $1 = P_1 g_1+\dots+P_r g_r + g$, with $g$ vanishing on all $F_q^n$. Because $J$ is maximal, its polynomial $P_1 g_1+\dots+P_r g_r$ has a root and we get $1=0$ in $F_q^n$, which is a contradiction. Therefore there is a common zero $\alpha$ of all $P_i$'s, and by maximality $J = M(\alpha)$. Now my question: does this hold when $k$ is any field? |
| A question about bilinear forms Posted: 30 Mar 2022 05:01 AM PDT
Here $U^{\bot,L} = \{ v\in V : \forall u\in U,\ \phi(v, u) = 0 \}$ and $U^{\bot,R} = \{ v\in V : \forall u\in U,\ \phi(u, v) = 0 \}$. I know how to prove this if $V$ is finite dimensional but not sure when $V$ is infinite dimensional... I can also prove that $ U^{\bot,L}\cap (U^{\bot,L})^{\bot,R} = \{0\}$, but for $U^{\bot,L}$ to be non degenerate we also need $U^{\bot,L}\cap (U^{\bot,L})^{\bot,L} = \{0\}$, and that seems far-fetched. May I have some hint on this? Thank you. |
| Posted: 30 Mar 2022 05:15 AM PDT This is a math/graph problem; it has nothing to do with chess! But, I am trying to solve a problem on an infinite large chess board. Assuming the pieces are "close together" (ie, distance less than 9 squares in any direction) OR they are infinitely far away in a given direction, provided they can return in a single move. For example, a rook at infinite distance that still cuts through the local area where the more local pieces all are (kings, knights, pawns). The result is a limited state space that is rather easy to brute force by a computer (say, less than 10,000,000 nodes). Here white tries to make progress by forcing the whole in the north and/or west direction. Lets says that the origin is somewhere in the top-left and we are at coordinates Back to the graph: the finite number of nodes represent all the possible ways the pieces can stand relative to each other (with a maximum allowed distance). It is a bipartite graph because we have alternating black and white nodes (whose turn it is). It is directed because we can seldom return to the previous node (which in general would require the same piece to be put back; but it is the turn of the other player). I can generate the graph, but now I need an algorithm to find the optimal play for both sides: White wants to approach a position that, once it is reached, will repeat every N moves such that Each edge in my graph has a value for the decrease of To summarize in purely mathematical sense: I have a directed bipartite graph with a finite number of nodes. There are black and white nodes. Every edge goes from a white to a black node, or from a black to a white node. Each edge has a weight between -2 and 2. The edges from white to black all have a weight of 0. An arbitrary loop also has a weight: namely, the sum of all the weights of the edges divided by the length of the loop. When on a black node one wants to follow a path that will lead (eventually) to a loop with a weight that is as large as possible, taking into account that when on a white node one wants to follow a path that will lead (eventually) to a loop with a weight that is as small as possible and visa versa. This must lead to a limited (and obviously finite) choice of loops that all have the same weight: no strategy of white can make the value of the weight of followed loops larger and no strategy of black can make the value of followed loops smaller. Can someone figure out an algorithm to find these loops, and explain it to me? |
| Posted: 30 Mar 2022 05:15 AM PDT This is the definition which we need for the proof of the theorem: This is the definition of the $\mathscr C''$ - equivalent :
There is the theorem: Suppose E is an open set in $R^3$, $u$ $\in$ $\mathscr C''(E)$, and $G$ is a vector field in $E$, of class $C''$ . $(a)$ if $F$ $=$ $\nabla$$u$, then $\nabla$ $\times$ $F$ $=$ $0$. $(b)$ if $F$ $=$ $\nabla$ $\times$ $G$, then $\nabla$ $\cdot$ $F$ $=$ $0$. Furthermore, if $E$ is $\mathscr C''$-equivalent to a convex set, then $(a)$ and $(b)$ have converses, in which we assume that $F$ is a vector field in $E$, of class $\mathscr C'$: $(a')$ if $\nabla$ $\times$ $F$ $=$ $0$, then $F$ = $\nabla$$u$ for some $u$ $\in$ $\mathscr C''(E)$. $(b')$ if $\nabla$ $\cdot$ $F$ $=$ $0$, then $F$ $=$ $\nabla$ $\times$ $G$ for some vector field $G$ in $E$, of class $\mathscr C''$ There is the proof : If we compare the definitions of $\nabla$$u$, $\nabla$ $\times$ $F$, and $\nabla$ $\cdot$ $F$ with the differential forms $\lambda_F$ and $\omega_F$ given by ($124$) and ($125$), we obtain the following four statements: $F$ $=$ $\nabla$$u$ if and only if $\lambda_F$ $=$ $du$. ($\star$). $\nabla$ $\times$ $F$ $=$ $0$ if and only if $d\lambda_F$ $=$ $0$. ($\ast$) $F$ $=$ $\nabla$ $\times$ $G$ if and only if $\omega_F$ $=$ $d\lambda_G$. ($\oplus$) $\nabla$ $\cdot$ $F$ $=$ $0$ if and only if $d\omega_F$ $=$ $0$. ($\circ$) I could n't understand these four statements ($\star$),($\ast$),($\oplus$),($\circ$). Any help would be appreciated. |
| How to quantify the loss of information in information theory Posted: 30 Mar 2022 05:05 AM PDT Assume there are 64 matrices, which are all equally important, and then one of them is lost. We know all these matrices before we lost one. How do you quantify the increase of information entropy from that matrix becoming unknown? Before losing this matrix, the result is certain. Similarly, suppose your job is counting how many dots are in a graph, but some parts of this graph are lost. How much has the entropy increased due to that missing part? I'm a rookie in information theory, so any discussion is welcome. Thank you! |
| Solving non linear equations where equations are equals Posted: 30 Mar 2022 05:00 AM PDT I wonder and tried to google it, but I am not sure what to google it, how to solve non linear equations where equations are equal between each other. I am able to write a specific algorithm for two equations but not dynamically for N equations. I will show the example of three (how my equations approximately looks like):
C1, C2, C3, X - are unknows, but in the end I do not need to know result of X. It can be interpreted like this (last equation C1 + C2 + C3 = 1 is not included here):
Please, don't try to solve this, I am not sure if these equations have results. I just randomly typed coefficients. But this is how my equations can looks like. Only with different coefficients. I tried to calculated it with only two unknows and I have got quadratic equation in the end so with three unknowns there will be cubic in the end. With N unknows there will be polynomial equation with degree N. Also, I have to say, result do not have to be with 100% accuracy. I am not sure if its help somehow or not. I found on google that maybe using iterative method could help. I look at few iterative methods but I am still not sure how to use it on this kind of problem. I also found, that non linear equation can by linearize. Maybe that would be a option but I am not sure how to do it here. |
| find the tangent to the sphere Posted: 30 Mar 2022 05:04 AM PDT obtain the equations of tangent to sphere $$x^2+ y^2+z^2+6x-2z+1 = 0$$ which pass through the line $$3 (16-x) = 3z=2y+30$$ Now I know if the plane is $$lx +my+n z=p$$ then $$-I/3 +m/2+n/3=0$$ also $(16,-15,0)$ is a point on the plane I know that there is $2$ answers , but how to proceed |
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