Recent Questions - Mathematics Stack Exchange |
- How to build a poisson regression model using tscount package?
- How to retrieve closed form of a recursion
- Is $F[G \times H] \simeq F[G] \otimes F[H] $?
- Closed balls as intersection of open balls in metric spaces
- How do we use set builder notation to describe the set Z x Z
- The Sum of ‘Add or Keep Sequence'
- Boundedness of perturbed nonlinear ode
- How do I find an equation for these two sequences?
- A simple question about extreme values of a multivariate function
- Tangent to polar curve
- When $A$ has won $m$ games, probability that $m+ n$ have been played
- The maximum number of 3-cycles in a graph with no cycles of length larger than 3?
- Solution of Klein-Gordon equation in momentum space
- Transposing matrices by multiplication
- $4$ set Venn diagram problem : Find the number of households.
- The simplication of left implication rule in intuitionistic sequent calculus
- How can i approximate this integral using mcmc?
- Solve linear system of equations with constraints
- Meaning of $2^n/\sum_{i=0}^k\binom{n}{i}$
- Show that the difference of two sequences of straight lines converges to $0$
- Does the multi-valued function $\sinh^i(z)$ have an analytic branch on the upper half plane?
- Addition of prime number having prime number of terms.
- Solving cubic first-order ODEs
- Solve the Diophantine equation $ (3^{n}-1)(3^{m}-1)=x^{r} $
- What are the most efficient ways to calculate the sum of positive divisors function, σ, and aliquot sum, s?
- What's the connection between relative entropy and hypothesis testing?
- Is there a pair of two $2 \times 2$ complex matrices that are never simultaneously conjugate to real matrices
- Let $A,B$ be $2\times2$ matrices. Given that we know $\text{Tr}(A)$, $\text{Tr}(B),\text{Tr}(AB)$, how do I find $A$ and $B$ that have those traces?
- Bilinear maps and Bilinear algorithms
- Solve the Lagrangian dual problem associated to $\max_{x_1,x_2\ge0}(3x_1+4x_2)$ such that $x_1^2+x_2^2\le25$
| How to build a poisson regression model using tscount package? Posted: 03 Jul 2022 05:23 AM PDT I am quite the beginner in Poisson regression and am trying to build a model through the tsglm function from the tscount package. The data I have is the following: The results for the following two lines are not good. There is autocorrelation, the PIT plot for the poisson model shows a U-shape and the predictions for both models are constant (12 months ahead). Now, I am sure I am doing something wrong in building my model but honestly, I am just a beginner and have no idea how to improve the model, if it is even possible. I have read the documentation of the package but some things seem a bit blurry and it seems like not many people post their experience with this package. If anyone could please weigh in and give me advice on what could be done to improve this model or what I am doing wrong, it would be great. Or if you could even link me to good websites/papers for me to learn more, I would highly appreciate it. Thank you all. |
| How to retrieve closed form of a recursion Posted: 03 Jul 2022 05:11 AM PDT I am looking for a closed form of this recursion: $$T(0)=1$$ $$T(n) = \begin{cases} T(n-1) & \text{for odd } n \\ 2^n+T(n-2) & \text{otherwise} \end{cases}$$ Obviously this recursion leads to something like this: $$T(6) = 2^0+2^2+2^4+2^6$$ Which would be the same like this: $$T(n)=\sum_{i=0}^{\left \lfloor \frac{n}{2} \right \rfloor} 2^{2i}$$ How do I elemninate the sum now? Is there any other way to retrieve the closed form? Btw, the solution is: $\frac{4^{\left \lfloor \frac{n}{2} \right \rfloor + 1}-1}{3}$ But looking for the way to get there. Thanks for your help. |
| Is $F[G \times H] \simeq F[G] \otimes F[H] $? Posted: 03 Jul 2022 05:11 AM PDT I tried building an isomorphism with the linear extension of: $$ (g,h)\mapsto g\otimes h $$ It is obviously unto, and thus because the dimensions are equal it is also one to one. Now, I'm fairly certain that it also preserves multiplication (a simple check on the basis elements shows that). I'm confused though, because using the same reasoning I could have built $$F[G\times H]\to F[G]\times F[H], \ (g\times h)\mapsto (g\times h)$$ Which also should be an isomorphism of rings. Am I missing something? Or are both isomorphims actually correct? Of course, $F[G]$ is the group algebra over some field $F$. |
| Closed balls as intersection of open balls in metric spaces Posted: 03 Jul 2022 05:20 AM PDT Let $(X,d)$ be a metric space. For every point $p\in X$ and real number $\epsilon \geq 0$ we define, respectively, the open and closed ball as the following sets: $B(p, \epsilon )=\left \{ x \in X : d(p,x) < \epsilon \right \}$ $\bar{B}(p, \epsilon )=\left \{ x \in X : d(p,x) \leq \epsilon \right \}$ My question is, given a point $p \in X$ and a real number $\epsilon \geq 0$, does the following relation hold? $\bar{B}(p, \epsilon )=\bigcap_{n=1}^{\infty} B\left ( p, \epsilon +\frac{1}{n} \right )$ One can show that $\bar{B}(p, \epsilon ) \subseteq \bigcap_{n=1}^{\infty} B\left ( p, \epsilon +\frac{1}{n} \right )$. Given $x \in \bar{B}(p, \epsilon)$, it holds true that $x \in B\left ( p, \epsilon +\frac{1}{n} \right )$ for all $n \in \mathbb{N}$ which, by definition, implies $x \in \bigcap_{n=1}^{\infty} B\left ( p, \epsilon +\frac{1}{n} \right )$. Since this holds for any arbitrary $x \in \bar{B}(p, \epsilon)$, it follows that $\bar{B}(p, \epsilon ) \subseteq \bigcap_{n=1}^{\infty} B\left ( p, \epsilon +\frac{1}{n} \right )$. However, I cannot establish equality. How should one proceed? |
| How do we use set builder notation to describe the set Z x Z Posted: 03 Jul 2022 05:11 AM PDT How can I use set builder notation to show all the possible integer ordered pairs, (1,1) (1,2) (1,3) etc. |
| The Sum of ‘Add or Keep Sequence' Posted: 03 Jul 2022 05:17 AM PDT I have a sequence. $a_{n}=a_{\lfloor\frac{n}{2}\rfloor}+\dfrac{1+(-1)^{n+1}}{2}$ where $a_{0}=0$ and $n\in\{0\}\cup\mathbb{N}$ So, it's $a_{2k-1}=a_{k-1}+1$ or $a_{2k}=a_{k}$ for nonnegative integer $k$. If I perform a continued divided by 2, and if the number $n$ lies in $2^{i}\leq n<2^{i+1}$, $n=2(2(\dots(2(2+\_)+\_)+\dots)+\_)+\_$ There would be $i+1$ spaces, and by choosing where I put 0 and 1, the value changes accordingly. So, there would be $P(i+1,j)$ choices if I want to put 1 for $j$ times. Then, it would be $a_{n}=j$. I tried evaluating the sum of $a_{n}$ for an interval $2^{i}\leq n<2^{i+1}$, and I get $\displaystyle \sum^{i+1}_{j=0}\{j\cdot P(i+1,j)\}$. I can't process it further because I'm stuck simplifying the expression. Please suggest a method. Thanks in advance. |
| Boundedness of perturbed nonlinear ode Posted: 03 Jul 2022 04:58 AM PDT Let an ode \begin{equation} \dot{x} = f(x,t), \ \ x_0 = x(0) \end{equation} where $f \in\mathbb{R}^n$ is locally Lipschitz in $x$ and uniformly bounded in $t$. Assume that the solution of that ode is bounded as $\|x(t)\| \leq L(x_0)$ for all $t\geq 0$. Assume now the second ode \begin{equation} \dot{y} = f(y,t) - g(y,t)y, \ \ y_0 = y(0) \end{equation} where $g \in \mathbb{R}^{n\times n}$ is locally Lipschitz in $x$, uniformly bounded in $t$, and positive definite for all $x \in \mathbb{R}^n$, $t \geq 0$. Is there a way to show that the solution $y(t)$ of that second ode is bounded as $\|y(t)\| \leq M(L(x_0),y_0)$? Here $M(L(y_0),y_0)$ is a constant dependent on $L(y_0)$ and $y_0$. Intuitively, since $f()$ produces a bounded solution and the term $-g(y,t)y$ produces an exponentially stable solution, I expect the overall solution $y(t)$ to remain bounded, but have not found a rigorous way to show it (or a counter-example). |
| How do I find an equation for these two sequences? Posted: 03 Jul 2022 04:57 AM PDT X - Y 1 - 0 2 - 0 3 - 0 4 - 0 5 - 0 6 - 1 7 - 1 8 - 2 9 - 3 10 - 4 11 - 5 These sequences continue for an infinity. I need an equation/formula to derive Y through X. |
| A simple question about extreme values of a multivariate function Posted: 03 Jul 2022 05:05 AM PDT
I computed the relative partial derivatives of the function as $f_{x}=2x+y$, $f_y=x$ $f_{xx}=2$, $f_{yy}=0$, $f_{xy}=1$. I know that If $f_{xx}f_{yy}-f_{xy}^2 <0$ at the points $(a,b)$ then the point is considered a saddle point, and that $f_x(0,0)=0, f_y(0,0)=0. $ So considering the second partial derivatives are both constants then the point $(0,0)$ is the only interesting point and it's the saddle point which means the function has no maxima or minima. Is this reasoning correct or is there something I'm missing ? |
| Posted: 03 Jul 2022 04:55 AM PDT I'm looking for an equation that describes all tangents to the curve $r=\theta/A$. So far I've been able to find that the slope as a function of $\theta$ is $$m=(tan(\theta)+\theta)/(1-\theta tan(\theta).$$ What I'm missing now us an equation for the y axis intercept $q$ as a function of $\theta$. Other more efficient ways to do it are also very welcome. Thanks in advance for your help. |
| When $A$ has won $m$ games, probability that $m+ n$ have been played Posted: 03 Jul 2022 05:03 AM PDT $A$ and $B$ play a set of games, in which $A$'s chance of winning a single game is $p$, and $B$'s chance $q$. How do I see that the chance that when $A$ has won $m$ games, $m + n$ have been played, is$$\binom{m + n}{n} p^{m+1}q^n?$$ Edit: It's assumed that any number of games is a priori as likely as any other number. |
| The maximum number of 3-cycles in a graph with no cycles of length larger than 3? Posted: 03 Jul 2022 04:48 AM PDT Suppose $G$ is a simple undirected graph with no cycles of length $\ge 4$. Some simple examples indicate that the number of 3-cycles $t$ may be less than $\displaystyle\left\lfloor \frac{|G|-1}{2} \right\rfloor$. Is this true? Why or why not? |
| Solution of Klein-Gordon equation in momentum space Posted: 03 Jul 2022 04:41 AM PDT I'm solving Klein-Gordon equation in order to get scalar field expression. $$(\partial^2 + m^2)\phi=0$$ I expand solution $\phi$ into Fourier integral in momentum space: $$\phi=\int\frac{d^4p}{(2\pi)^4}\varphi(p)e^{-i<p,x>}, where <p,x> = p^0t-(p^1x^1 + p^2x^2 + p^3x^3)$$ And substitute in to equation, getting: $$\int\frac{d^4p}{(2\pi)^4}(-p^2 + m^2)\varphi(p)e^{-i<p,x>}=0 \Rightarrow (-p^2 + m^2)\varphi(p)=0$$ There a book tells that the solution of a gotten algebraic equation is $\varphi(p) = 2\pi\delta(m^2-p^2)\bar\varphi(p)$. I have some difficulties figuring out a couple of points considering this. Since we've gotten a solution in terms of distributions, then the equation is considered in distribution space, and such a result is somehow linked with a known result $xf(x) = 0 \Rightarrow f(x) = C\delta(x)$ So, the points are:
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| Transposing matrices by multiplication Posted: 03 Jul 2022 04:40 AM PDT Are there matrices $A$ and $B$ for which equality $AXB=X^{T}$ holds for any matrix $X\in\mathbb{R^{m\times n}}$? |
| $4$ set Venn diagram problem : Find the number of households. Posted: 03 Jul 2022 04:49 AM PDT Four newspapers - Hindu, ET, TOI and IE - operate in a town. No household that subscribed for ET also subscribed for IE. Further: Question : How many households subscribed for at least two newspapers? I prepared a $4$ set Venn diagram by putting in the data like this but it gave me wrong answers. What have I done wrong with my Venn diagram? Please help !!! Thanks in advance !!! |
| The simplication of left implication rule in intuitionistic sequent calculus Posted: 03 Jul 2022 04:56 AM PDT The original left implication rule in intuitionistic sequent calculus is $$ \dfrac{\Gamma, A \supset B \vdash A \quad \Gamma,A \supset B, B \vdash C}{\Gamma, A \supset B \vdash C} $$ The right primise can be simplified to $\Gamma, B \vdash C$, since $\vdash B \supset(A \supset B)$. My question is why we can not just remove $A \supset B$ from context of left primise, so there must exist some proofs of $A$ make use of $A \supset B$, but i can't find out some proofs like that. |
| How can i approximate this integral using mcmc? Posted: 03 Jul 2022 05:02 AM PDT How can we approximate this integral using mcmc? $$\int_{\alpha_j} \exp\left(\hspace{0.6mm} - \frac{(\alpha_1...\alpha_J) (\Sigma)^{-1} (\alpha_1...\alpha_J)^T }{2} + \sum_{j=1}^J ( - P\exp(\alpha_{j}) +\alpha_jc_j)\right) d_{\alpha_1} ... d_{\alpha_J},$$ knowing that $ \Sigma $ is a covariance matrix and $P$ is an integer and $c_j$ for $j \in [J]$ are integers. |
| Solve linear system of equations with constraints Posted: 03 Jul 2022 04:57 AM PDT We are working on a game which requires computation of set of active thrusters to maintain desired linear and angular velocities. We know desired velocity.x velocity.y and angular_velocity as a column (a0 b0 c0). Then we have Each thruster could be enabled with arbitrary coefficient in the range of [0, 1], 0 meaning it has no impact on the velocities. The problem is to solve programmatically following linear system respecting the constraints of ki being in the range of [0, 1] each. Simply solving it the usual way yields arbitrary coefficients outside of [0, 1] range. Ideal solution would be a row (k1 k2 .. kn) where ki belongs to [0, 1], anything that would come close to that is highly appreciated as well. The closest we came to solving this is imprecise N^2 algorithm of iterating the array of not-yet-enabled jets N times each time picking the jet with smallest introduced error (thus highest score in terms of sum of both velocities deltas) and enabling it (so only 0 and 1 ki-s). |
| Meaning of $2^n/\sum_{i=0}^k\binom{n}{i}$ Posted: 03 Jul 2022 04:41 AM PDT I was thinking about this problem. Suppose there is a bag of $n$ balls, $k$ of them have unknown colors, and the rest are all red. One starts with \$1 (infinitely divisible) and places $n$ even money bets (for each bet, he can wager any amount between zero and the current stake) on the color of the next randomly picked ball, and he can only bet red. I found the winnings that one can guarantee to win as \begin{equation} \frac{2^n}{\sum_{i=0}^k\binom{n}{i}}=\frac{2^n}{\binom{n}{0}+\binom{n}{1}+\cdots+\binom{n}{k}}. \end{equation} How to interpret this expectation? |
| Show that the difference of two sequences of straight lines converges to $0$ Posted: 03 Jul 2022 05:03 AM PDT Define $d_{\varepsilon,a}(x) = \frac{f(a+\varepsilon)-f(a)}{\varepsilon}(x-a)+f(a)$ $\quad \forall(x,a,\varepsilon) \in \mathbb{R}^2\times \mathbb{R}^*$ where $f: \mathbb{R}\to\mathbb{R}$ is a function. That is $(a_n),(b_n),(c_n) \in (\mathbb{R}_+^*)^\mathbb{N}$ s.t. $\lim\limits_{n\to + \infty} (a_n,b_n,c_n) = (0,0,0)$. Show that if $\forall x \in \mathbb{R}$, $(d_{a_n,x}-d_{b_n,x})_{n \in \mathbb{N}}$ converges to $0$, then $(d_{a_n+c_n,,x}-d_{b_n+c_n,x})_{n \in \mathbb{N}}$ converges to $0$. It may be necessary to assume that $f$ is a continuous function. I'm not sure if that's true, but I can't convince myself that it is or isn't, so if you have a counterexample (even with a non-continuous function), I'd really appreciate it! EDIT : if you have any hint don't hesitate neither to write it ! |
| Does the multi-valued function $\sinh^i(z)$ have an analytic branch on the upper half plane? Posted: 03 Jul 2022 04:52 AM PDT I've tried algebraically manipulating the function but I'm stuck as to how to continue. $$\sinh^i(z) = \left(\frac{e^z -e^{-z}}{2} \right)^i = e^{i \log \left(\frac{e^z -e^{-z}}{2} \right)}$$ I then tried assuming I'm looking at the principal branch of log and work backwards to see which parts of the image correspond to those of the source but I only really did it with Mobius transforms and I'm not good at those. |
| Addition of prime number having prime number of terms. Posted: 03 Jul 2022 05:04 AM PDT introducing s(k): Here we first defined a symbol s(k) so that we could explain our observations in better manner. When we wrote s(3) then it's means summation of 3 prime number and Similarly s(5) means summation of 5 prime numbers and so on. List of prime number from 2 to 401:Link for more prime number 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401 Observation: s(2) = 2+3=5(prime number) s(3) = 5+7+11 = 23(prime number) s(5) = 13+17+19+23+29 = 101(prime number) s(7) = 31+37+41+43+47+53+59 = 311(prime number) s(11) = 61+67+71+73+79+83+89+ 97+101+103+107 = 931(composite numbers) s(13) = 109+113+127+131+137+ 139+149+151+157+163+167+ 173+179 = 1895(composite numbers) s(17) = 181+191+193+197+199+ 211+223+227+229+233+239+241+251+257+263+269+271 = 3875(composite numbers) s(Prime number) = ......... and so on. note: We don't know any computer programming so that we could calculate such things for large number, we have checked this pattern upto s(37). The value of s(19)=6349(composite numbers) Claim: My claim : 2,3,5,7 are only consecutive prime numbers for which s(2);s(3);s(5);s(7) will be prime number? My friend Sankalp Savaran claim: There may be infinately pairs or more than one pairs of Four consecutive Prime number $P_a; P_b ; P_c; P_d$ such that s($P_a$); s($P_b$); s($P_c$); s($P_d$) will be prime number? Question : Who's claim is correct here? And you can give counterexample to incorrect any of the two claim |
| Solving cubic first-order ODEs Posted: 03 Jul 2022 05:22 AM PDT First, I should clarify that for invertible $\xi(\eta)$ and $\dot\eta\neq0$: $$\dot\xi=\dfrac{1}{\dot\eta}\\ \dfrac{d\dot\xi}{d\xi}=-\dfrac{\ddot\eta}{\dot{\eta}^2}\\ \ddot\xi=\dfrac{d\xi}{d\eta}\dfrac{d\dot\xi}{d\xi}=-\dfrac{\ddot\eta}{\dot{\eta}^2}\dot{\xi}=-\dfrac{\ddot\eta}{\dot\eta^3}\\ $$ I was recently trying to answer an ODE question and it's basically solving: $$\ddot\xi+a_1(\xi)\dot\xi+a_2(\xi)\xi=a_3(\xi)\\ $$ I can transform it into an ODE in $\eta$: $$-\ddot\eta+a_1\dot\eta^2+a_2\xi\dot\eta^3=a_3\dot\eta^3\\ $$ Which can be transformed to a non-linear first-order ODE. $$-\dot\kappa+a_1\kappa^2+(a_2\xi-a_3)\kappa^3=0\\ $$ The question is can these nonlinear ODEs be solved to solve the - I would expect - much harder second-order ODEs? Substituting $\kappa=e^{\chi(\xi)}$ and finding $\rho$ such that $-\dot\kappa+(\kappa^3\rho)'$ is equal to the LHS don't work. My next idea was completing the cube so I can have $(\kappa+\nu(\xi))^3$ to make a substitution. The linear terms are missing so I guess that's how I am going to complete it. I will also have two extra $\kappa$-linear terms which will be duable via the substitution. However, I have an extra $\nu(\xi)$ which makes the ODE non-Bernoulli. $$\dfrac{-1}{a_2\xi-a_3}\dot\kappa+\dfrac{a_1}{a_2\xi-a_3}\kappa^2+\kappa^3=0\\ (\kappa+b)^3=\kappa^3+3\kappa^2b+3\kappa b^2+b^3\\ \dfrac{-1}{a_2\xi-a_3}\dot\kappa+\left(\kappa+\dfrac{a_1}{3(a_2\xi-a_3)}\right)^3=3\kappa\left(\dfrac{a_1}{3(a_2\xi-a_3)}\right)^2+\left(\dfrac{a_1}{3(a_2\xi-a_3)}\right)^3\\ \chi=\kappa+\dfrac{a_1}{3(a_2\xi-a_3)}\\ -\dot\kappa=\left(\dfrac{a_1}{3(a_2\xi-a_3)}\right)'-\dot\chi\\ \dfrac{1}{a_2\xi-a_3}\left(\left(\dfrac{a_1}{3(a_2\xi-a_3)}\right)'-\dot\chi\right)+\chi^3=3\left(\chi-\dfrac{a_1}{3(a_2\xi-a_3)}\right)\left(\dfrac{a_1}{3(a_2\xi-a_3)}\right)^2+\left(\dfrac{a_1}{3(a_2\xi-a_3)}\right)^3\\ \dot\chi+\dfrac{a_1^2}{3(a_2\xi-a_3)}\chi-\dfrac{2a_1^3}{27(a_2\xi-a_3)^2}-\left(\dfrac{a_1}{3(a_2\xi-a_3)}\right)'=(a_2\xi-a_3)\chi^3 $$ Also, dividing by $\nu$ isn't helpful because then you get $\frac{\dot\chi}{\nu}$. The $\nu$ also renders solving for the homogeneous case (if there is) and the substitution $\chi=e^{\rho(\xi)}$ (which I reconsidered) useless. |
| Solve the Diophantine equation $ (3^{n}-1)(3^{m}-1)=x^{r} $ Posted: 03 Jul 2022 05:11 AM PDT Is there a solution other than solution \begin{equation*} (3^{2}-1)(3^{2}-1)=2^{6}=4^{3}=8^{2} \end{equation*} of the Diophantine equation \begin{equation*} (3^{n}-1)(3^{m}-1)=x^{r} \end{equation*} for positive integers $ n, m, x, r $ such as $ \min \lbrace n; m; x; r\rbrace \geq 2 $ ? By keeping the same conditions with $ r \geq 3 $ ! |
| Posted: 03 Jul 2022 05:23 AM PDT Given a positive integer $n$, what are the most efficient algorithms for calculating the sum of positive divisors function $\sigma_{1}(n)$ and the aliquot sum $s(n)$ ? |
| What's the connection between relative entropy and hypothesis testing? Posted: 03 Jul 2022 05:03 AM PDT I've seen stated in these online notes (Link to pdf) that the relative entropy, $\mathrm D(p\| q)\equiv\sum_x p_x \log\frac{p_x}{q_x}$, can be understood as quantifying how easy it is to discriminate between two possible probability distributions using samples from them. That is, loosely quoting from the first paragraph in the above notes: suppose we are given $X_1,...,X_n$ all sampled IID from either $p$, or IID from $q$. The optimal test to decide based on the samples which one was the correct distribution, has a failure rate of $$e^{-n(\mathrm D(p\|q)+o(1))}.$$ The notes mention a "Stein's lemma", but googling I haven't found a basic source discussing these results. What is a primary source (possibly a textbook or online notes) discussing this result and how it's derived? Or, even better, what is a direct way to arrive to this result? |
| Posted: 03 Jul 2022 05:06 AM PDT The problem is as in the title. Let $A$ and $B$ be $2 \times 2$ complex matrices. Are there $A,B$ such that for any $2 \times 2$ matrix $C$, either $CAC^{-1}$ or $CBC^{-1}$ is never real. A numerical example I think I found would be: $A=\left( \begin{array}{cc} -1 & -4-i \sqrt{29} \\ \frac{8}{9}+\frac{1}{9} \left(-4-i \sqrt{29}\right) & -1 \\ \end{array} \right)$ and $B=\left( \begin{array}{cc} -1 & -9 \\ 1 & 3 \\ \end{array} \right)$ However, I do not know how to show that it actually works. It is a candidate as Mathematica could not find the conjugation. |
| Posted: 03 Jul 2022 04:59 AM PDT Let $A,B$ be $2\times 2$ matrices. Given that we know $\operatorname{Tr}(A)$, $\operatorname{Tr}(B),\operatorname{Tr}(AB)$ how do I find $A$ and $B$ that have those traces? Naively, I would let $A$ be the diagonal matrix with entries $\operatorname{Tr}(A)/2$ and the diagonal entries of $B$ be $\operatorname{Tr}(B)/2$ but this just doesn't work. (I know those traces do not uniquely determine $A,B$ but I just need one instance of $A,B$) |
| Bilinear maps and Bilinear algorithms Posted: 03 Jul 2022 05:00 AM PDT
Let $\mathit{A}$,$\mathit{B}$,$\mathit{C}$ be vector spaces. A map $f:\mathit{A}\times \mathit{B}\to C$ is said to be bilinear if for each fixed element $b\in \mathit{B}$, $f(.,b):\mathit{A}\to\mathit{C}$ is a linear map. Similarly, for each fixed element of $\mathit{A}$. Matrix multiplication is an example of a bilinear map.Following my definition, I can prove that it is a bilinear map, but I don't understand the intuitive idea behind it. In my opinion, it is simply a linear map with one element fixed.
Kindly help me with these questions. Thanks! |
| Posted: 03 Jul 2022 05:07 AM PDT Consider the (non-linear) optimization problem ($P$)
Solve the Lagrangian dual problem. I don't have a clue how the dual problem is even obtained since the constraint is nonlinear. Could anyone please help me? |
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