## Second order PDE February 9    1

Kindly help me with this... $$U_{xy} + yU_{yy} + \sin(x+y)=0$$Here $A =0$, so how to calculate the characteristic equations ?as $${dy\over dx} = {B^2 \pm \sqrt D\over2A}$$Let $V=U_y$ ,Then $V_x+yV_y=-\sin(x+y)$Follow the method in http://en.wikiped

I have been trying to find the solutions to the following differential equation that I found in a math book. I am really lost, so anything will do (hints and solutions alike)$$y''(t)-\frac{1}{y(t)^{2}}=0.$$Thank you for your time!If $y'' = \frac{1}{y ## Does every integer appear in the digits of$2\cdot 0.1234567891011… $February 9 Let$C = 0.1234567891011121314-$the Champernowne constant. My question is :Does the real number$2 \cdot C \simeq 0.24691357820222426283032343638404244464850525456586062646668707274...$contain every integer in its digits? For instance,$2022appea ## Nonlinear second order PDE February 9 I need to solve the following PDE (which is a maximized Hamilton-Jacobi-Bellman equation)\begin{align} rV(\theta_1,\theta_2) = \frac{(\theta_1^\rho + \theta_2^\rho)^{\frac{1}{\rho}}}{12(\gamma_1+\gamma_2)}-\frac{1}{6}(\alpha_1 \theta_1^2 + \alpha_2 \ ## Given probability distributionf(x)=2-bx$find b and range for x February 9 Suppose that the distances between houses and the center of a city are distributed with the density function:$f(x)=2-bx$, where x denotes distance. If this is a proper density function, what can we infer about the value of$b$and the range of value ## How is this called February 9 If we have such relation that for$\forall xf(x)\ne x$, how is it called in one word? I can come up with only "graph of this function is not a straight line:)" Thank you ## Is there a close-form solution for the non-linear difference equation February 9 is there a close-form solution for the difference equation below?$$(x_{n+2}-x_{n+1})-(x_{n+1}-x_n)=(\frac{x_{n+1}}{c})(x_{n+2}-x_n)$$Any comments are appreciated. ## The two planes 2x-y+3z=2 and -4x+2y-6z=3 are parallel. Let V= the set of P1P2: P1 is an element of equations one and P2 is an element of equation 2. February 9 1 The two planes 2x-y+3z=2 and -4x+2y-6z=3 are parallel. Let V= the set of P1P2: P1 is an element of equations one and P2 is an element of equation 2. Is this a vector space? P1P2 should have a line above them. my real issue is understanding what makes ## Let$(L_1,L_2,L_3)$be an ordered triple of pairwise distinct plane in$K^3$. There is two possibles types of relative arrangement of such triples. February 9 Let$(L_1,L_2,L_3)$be an ordered triple of pairwise distinct plane in$K^3$. Prove that there is two possibles types of relative arrangement of such triples characterized by the fact that$\dim L_1\cap L_2\cap L_3 = 0$or$1$.In fact I imagine that ## Volume of a 'doughnut without a hole' February 9 I need to find the volume of a torus-shaped object, but it which doesn't have a hole as a real torus? We can find the volume of the ring, but what about the inner part?PS: What is that shape called? ## How does this picture called February 9 Some time ago I saw this in my teacher's room. She called this picture in honor of some scientists (Lagrange,Lie or Liouville, or some other, but I don't remember). Please, name picture. Tha ... ## Exterior derivative of a 2-form February 9 1 I want to prove that the exterior derivative of a 2-form in$\mathbb{R}^n$: $${\alpha = \sum a_{ij} dx_i \wedge dx_j}$$ is: $${d \alpha = \sum (\frac{\partial a_{ij}}{\partial x_k} + \frac{\partial a_{jk}}{\partial x_i} - \frac{\partial a_{ik}}{\part ## How to deduce this formula using differential forms February 9 3 There's a formula from vector calculus that seems terrible to deduce. This formula is:$$\nabla\times (A\times B)=(B\cdot\nabla )A-(A\cdot \nabla)B+A (\nabla\cdot B)-B(\nabla\cdot A)$$Deducing it by computing explicitly the left hand side and getting ## Computing n-th external power of standard simplectic form February 9 1 I need some help:Define a 2-form on R^n by \omega=dx_1\wedge dx_2+dx_3\wedge dx_4+...+dx_{2n-1}\wedge dx_{2n}. How to compute \omega^n:=\omega\wedge\omega\wedge\ldots\wedge\omega?Note that$$ \omega=\sum\limits_{k=1}^n x_{2k-1}\wedge x_{2k} $$## The degree of Gauss map February 9 1 If M is an 2m-dimensional closed orientable hypersurface in \mathbb R^{2m+1}, then we have a Gauss map G:M\rightarrow S^{2m}.I have known from my differential geometry book that degG=\frac{1}{2}\chi(M) where \chi(M) is the Euler character ## basisrepresentation of differential-form f(x_1,x_2):=(x_1+x_2, -x_1, -x_2)^T February 9 I am trying to learn differential forms. I have read some scripts about differental forms and now I am trying to solve some problems.So the problem is:given f: \mathbb{R}^2 \to \mathbb{R}^3, f(x_1,x_2):=(x_1+x_2, -x_1, -x_2)^TNow I have to calculat ## Derivation of SVM algorithm (Lagrangian) February 9 I have a question about the derivation of the SVM algorithm (for example, page 3 here ). The question is about the math, so that's why I'm asking this here.Suppose I have the following optimization problem:$$min_{w, \xi} \frac{1}{2} ||w||^2 +C \sum_i ## Examples of global properties that don't arise from local knowledge February 9 Let$M$be a smooth manifold. As an example of a global property that arises from local data we know that if$(M,g)$is a compact surface without boundary then the Euler characteristic is given by $$2\pi \, \chi(M) = \int_M K_p \; dA$$ where$dA$i ## Find an upper triangular matrix$A$such that$A^3=\begin{pmatrix}8&-57\\0&27\end{pmatrix}$February 9 1 Find an upper triangular matrix$A$such that$A^3=\begin{pmatrix}8&-57\\0&27\end{pmatrix}$. I tried to solve this problem using Cayley–Hamilton Theorem, but I am unable to solve that.We have $$A^3 = \pmatrix{8&-57\\0&27}$$ We calculate ## Need a logistic like function with y=0 at x=0 February 9 1 A logistic curve:$$y=\frac{50}{1+e^{-k(x-10)}}$$fits my exp data very well (having a maximum value 50, having a good trend). However, I hope it can return to zero when$x=0$. Is there an alternative function better than logistic function. How about$$You Might Also Like • Suppose I have a multivariate normal distribution$N$for the continuous multivariate RV$X = (X_1, X_2)^T$. ... • I am trying to calculate some probabilities for a card game. Players have to draw 3 cards each time and the ... • I wrote it as$n^{120}=1\pmod{310}$and thought I'd divide it in simpler congruences with primes (is this ri ... • I'm looking for this behavior to simulate the ... • So the solution says use Ito-s formula, taking$Y_t:= e^{\mu t}X_t$to obtain$dY_t = [\mu e^{\mu t}X_t - e^ ...
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