Recent Questions - Mathematics Stack Exchange |
- Let $a_n$ be the position of the nth $1$in the string $t_n$. Prove that: $a_n = [\frac{1+\sqrt5}{2} . n]$
- Uncertainty principle for non-abelian finite groups
- Comparing Quantities
- Solving non-homogeneous recurrence relations
- Polar coordinates to evaluate double integral
- For 2 orthogonal vectors, is tan(a)tan(b)=-1?
- Normal space on [-1 , 1 ]
- McDermott about Skolemization
- Tetrahedron geometry reference points
- What is the transformation matrix of the linear mapping?
- Differential equation $(xy'-y)^2=(y')^2-((2yy')/x)+1$
- What is differential probability?
- determine the general solution of the given differential equation: $y'' − 4y' + 13y = e^{2x}$
- Notation: intersection of two sets contains all elements
- Is my derivation of the summation formula of the first squares correct?
- Fattened volume of a curve
- Global optimization bibliography
- Solving Laplace's equation $\triangle u = 0$ in the semi-infinite strip $S = \{(x,y): 0 < x < 1, 0 < y\}$
- Understanding the definition of a ternary ring of operators (TRO)
- Seeking references regarding Zero Extension of Sobolev ($W_0^{1,2}$) functions on a compact domain on $C^\infty$ Riemannian manifold
- Solving $xy’’’+(1-m)y’’+2y=0,y=y(x)$ to prove hypergeometric representation of Abramowitz function from NIST.gov
- Implicit function theorem and constrained maximization
- Why do we use the square root as a proportional measure in this optimisation problem?
- Common divisors of binomials
- Parameters in Optimization
- Can we define any metric on $\Bbb{R^\omega}$ so that it represents a norm?
- How can you calculate $\int_{0}^{1} \exp(\sin(x))dx$ exactly?
- Range of correlation proof
- Solving system of equations, (2 equations 3 variables)
- Eigenvalues of a 2x2 matrix A such that $A^2$=I
| Posted: 20 Dec 2021 07:58 AM PST Consider the transformation $f$ acting on a binary string: turn every $(0, 1)$ in it into$ (1, 01).$ The notation$ s_n$ is the string produced after acting$ f $on $1$ all n times. Example :$ 1\Rightarrow 01 \Rightarrow 101 \Rightarrow 01101 \Rightarrow ...$ $1)$ Calculate the number of pairs$ (0,0) $appearing in the string $s_n$ $2)$ Arrange consecutive strings $s_1, s_2,... s_n$ into a string $t_n$ . Let $a_n$ be the position of the nth $1$ and $b_n$ the position of the nth $0$ in the string $t_n$ . Prove that: $a_n = [\frac{1+\sqrt5}{2} . n]$ , $b_n = [\frac{1+3\sqrt5}{2} . n]$ Note: $[z]$ is the floor function. $+)$We will prove inductively that no two zeros are next to each other $.(1)$ Let $(1)$ be true for $n = k .$ Consider the following minor cases: Case $1: ....11.... \Rightarrow ....0101....$ Case $2: 01... \Rightarrow 101.....$ Case $3: 101....\Rightarrow 01101....$ So $(1)$ is also true for $n = k+1 .$ We complete the proof $(1).$ Thus the number of pairs $(0,0)$ in the string $s_n = 0$ for all $n.$ I am very stuck with request $2) . $I hope to get help from everyone. Thanks very much ! |
| Uncertainty principle for non-abelian finite groups Posted: 20 Dec 2021 07:57 AM PST For cyclic groups $C_n$, Donoho and Stark's lemma [1] states that if a function $x(g)$ on the canonical group basis has $r$ non-zero elements, then its Fourier transform \begin{equation} R(\rho) = \sum_{g\in C_n} x(g) e^{-\frac{\mathrm{i}2\pi g\rho}{n} } \end{equation} cannot have $r$ consecutive zero elements, i.e. $R(\rho)$ with $\rho=m+1,...,m+r$ for any integer $m$. It is easy to see that, dually, a function $x(g)$ with $r$ consecutive non-zero elements (and the rest zero) has at least $n-r$ non-zeros in the Fourier domain. This lemma is used to derive the well-known discrete uncertainty principle, i.e. $N_t N_f \geq 1$ where $N_t$ and $N_f$ denote the number of non-zero elements of a function in time and frequency. I am interested in the more general non-commutative group $G$, where $\rho$ are the irreducible representations of $G$, and each $R(\rho)$ is a $d_\rho \times d_\rho$ matrix. I know it is possible to extend the uncertainty principle for non-commutative compact groups (e.g., Alagic and Russell [2]): \begin{equation} N_t \frac{\sum_{\rho\in S} d_\rho^2}{|G|} \geq 1 \end{equation} where $S$ indicates a subset of $N_f$ irreducibles. However, I would like to know if Donoho's 'consecutive zeros' lemma also extends for the non-commutative case. As the meaning of 'consecutive' is not clear in the non-commutative case, I have the following question:
ps. From the 'consecutive in time' condition in the abelian case, I understand that the subset of group elements should not be a subgroup of $G$, but this does not seem to be enough. [1] Donoho, David L., and Philip B. Stark. "Uncertainty principles and signal recovery." SIAM Journal on Applied Mathematics 49.3 (1989): 906-931. [2] Alagic, Gorjan. Uncertainty principles for compact groups. University of Connecticut, 2008. |
| Posted: 20 Dec 2021 07:57 AM PST
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| Solving non-homogeneous recurrence relations Posted: 20 Dec 2021 07:57 AM PST Find $g_{n}$ if $g_{n+2}-6g_{n+1}+9*g_{n}=3*2^n + 7*(3)^n$ given $g_{0}=1,g_{1}=4$. How can I proceed to solve these kind of recurrence relations? I cannot show any work since I haven't made any progress. I was thinking of trying out values and then guessing a formula, and then inducting on it. I also know generating functions, but I can't understand how to use them here. Any help is appreciated! |
| Polar coordinates to evaluate double integral Posted: 20 Dec 2021 07:57 AM PST Let $S$ be the annulus in the $xy-$plane bounded by the circles with equations $x^2+y^2\le 1, x^2+y^2\le 4$. Evaluate the double integral $I=\int\int_Se^{x^2+y^2}dxdy.$ My solution: using polar coordinates (and with a jacobian of $r$) we have the transformed integral $I=\int_{0}^{2\pi}\int_1^4re^{r^2}drd\theta=\pi(e^{16}-e^1)$. Is this correct? |
| For 2 orthogonal vectors, is tan(a)tan(b)=-1? Posted: 20 Dec 2021 07:53 AM PST
This is a part of a physics subproblem I was solving where in the solution they casually mentioned:- Without proving it, and proceeded to solve the complete question. I tried using I am willing to post my physics question here in case someone thinks this is an XY scenario or I got something wrong! :] |
| Posted: 20 Dec 2021 07:45 AM PST
Edit the space is not normal since if A = { -1 } and B = { 1 } then A ∩ B = ∅ where A and B are closed sets because the complement of A is (-1, 1 ] and this is a base element and B is closed because the complement of B is [-1, 1 ), but I don't know how I cannot find two open sets contain A, B but the intersection is not empty set , any help please |
| Posted: 20 Dec 2021 07:44 AM PST I am a beginner in AI. In a paper by McDermott (titled "A critique of pure reason") I found the following passage: "A goal containing a variable is interpreted as a request to find values for the variables. The goal $\texttt{append([a, b], [c, d], X)}$ means "Find an $\texttt{X}$ that is the result of appending $\texttt{[a, b]}$ and $\texttt{[c, d]}$"; in this case, $\texttt{X}$ will get bound to $\texttt{[a, b, c, d]}$". So far so good, but then McDermott says: "From the point of view of logic, $\texttt{append([a, b], [c, d], X)}$ is just a Skolemized version of $\texttt{(not(exists(X), append([a, b], [c, d], X)))}$". I can't understand in what sense $\texttt{append([a, b], [c, d], X)}$ is a Skolemized version of $\texttt{(not(exists(X), append([a, b], [c, d], X)))}$. Should not the Skolemized version of $\neg \exists \texttt{X} (\texttt{append ([a, b], [c, d], X)})$ be rather $\forall \texttt{X} (\neg \texttt{append([a, b], [c, d], X)})$? |
| Tetrahedron geometry reference points Posted: 20 Dec 2021 07:42 AM PST Let us define the tetrahedron ABCD with vertices in A,B,C,D. Any point P inside it can be expressed as (a,b,c,d) where each represent somehow the "composition" of its corresponding vertice (Therefore a+b+c+d=1). If we "cut" the tetrahedron as it is shown in the previous image, it results in a triangle with vertices in AiBj (random point between A and B), C and D. My question is how to transform the expression of P (a,b,c,d) to the new (aibj,c',d') where it is expressed referred to the vertices of the triangle. |
| What is the transformation matrix of the linear mapping? Posted: 20 Dec 2021 07:48 AM PST $ f:M_{2.2}(\mathbb{R}) \rightarrow \mathbb{R^3}, $$ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \mapsto \begin{pmatrix} 2a - 4b \\ -6d \\ 8a-16b+2d \end{pmatrix}\beta =\big (\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} ,\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \big), \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} \big), \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \big)$ $\beta'= ( e_{2}, 2e_{1}, -e_{3})$ of $\mathbb{R^{3}}$ ( these are the standard unit vectors of $\mathbb{R^{3}}$) What is the transformation matrix $M_{\beta'}^{\beta}(f)= ?$ Well my thought is that $f(\left( \begin{array}{c} 0\\ 1\\ 0 \end{array} \right)) =\left( \begin{array}{c} -4\\ 0\\ -16 \end{array} \right) = \square \left( \begin{array}{c} 1\\ 0\\ 8 \end{array} \right) +$$\square \left( \begin{array}{c} -4\\ 0\\ 2 \end{array} \right) +$ $\square \left( \begin{array}{c} 0\\ 0\\ 0 \end{array} \right) +$ $\square \left( \begin{array}{c} 0\\ -6\\ 2 \end{array} \right)$ $f(\left( \begin{array}{c} 2\\ 0\\ 0 \end{array} \right)) =\left( \begin{array}{c} 4\\ 0\\ 16 \end{array} \right) = \square \left( \begin{array}{c} 1\\ 0\\ 8 \end{array} \right) +$$\square \left( \begin{array}{c} -4\\ 0\\ 2 \end{array} \right) +$ $\square \left( \begin{array}{c} 0\\ 0\\ 0 \end{array} \right) +$ $\square \left( \begin{array}{c} 0\\ -6\\ 2 \end{array} \right) $ $f(\left( \begin{array}{c} 0\\ 0\\ -1 \end{array} \right)) =\left( \begin{array}{c} 0\\ 6\\ -2 \end{array} \right) = \square \left( \begin{array}{c} 1\\ 0\\ 8 \end{array} \right) +$$\square \left( \begin{array}{c} -4\\ 0\\ 2 \end{array} \right) +$ $\square \left( \begin{array}{c} 0\\ 0\\ 0 \end{array} \right) +$ $\square \left( \begin{array}{c} 0\\ -6\\ 2 \end{array} \right) $ And from the linear combinations of the $squares$ I can fill the transformation matrix in the columns and in this way I will get the transformation matrix. Is my idea correct? And how does the transformation matrix look like? Thank you in advance |
| Differential equation $(xy'-y)^2=(y')^2-((2yy')/x)+1$ Posted: 20 Dec 2021 07:41 AM PST
Setting $y=xz$, we arrive at the equation $x^4(z')^2=x^2(z')^2-z^2+1$, whence $z'x\sqrt{x^2-1}=\sqrt{1-z^2}$, that is $\dfrac{dz}{\sqrt{1-z^2}}=\dfrac{dx}{x\sqrt{x^2-1}}$, that is, $\arcsin z=-\arcsin\dfrac{1}{x}+c$. It remains to find $y$. Also, $y=\pm x$ also appears due to the fact that $z=\pm 1$ is substituted into the equation. But this is a trivial case. In my textbook, the general answer is somehow different: $(y^2-cx^2+1)^2=4(1-c)y^2$. It is not entirely clear how this was obtained. |
| What is differential probability? Posted: 20 Dec 2021 07:36 AM PST I am trying to understand the following statement (taken from a context related to volumetric image rendering):
What does differential probability mean? (especially that density in this context is defined at any point in 3D space) |
| determine the general solution of the given differential equation: $y'' − 4y' + 13y = e^{2x}$ Posted: 20 Dec 2021 07:46 AM PST determine the general solution of the given differential equation: $y'' − 4y' + 13y = e^{2x}$ I'm solving a list of supposedly simple exercises, one of the subjects is ordinary differential equations, which is where I have the most difficulty, and I've been trying to solve the above question for half an hour, one of my study strategies is to try to do it myself, and then check the result on some online calculator(mathdf) to see if I got it right, in the case of the question above, the results are not matching, and I can't see where I'm going wrong, if someone can help me with this question it would be of great help. |
| Notation: intersection of two sets contains all elements Posted: 20 Dec 2021 07:50 AM PST Let's assume we have to sets $A = \{x_1,\cdots, x_n\}$ and $B = \{x_1,\cdots, x_n\}$. How to write mathematically that their is NO element $x$ which is not in the intersection of $A$ and $B$? |
| Is my derivation of the summation formula of the first squares correct? Posted: 20 Dec 2021 07:39 AM PST I have been thinking about the problem of finding the sum of the first squares for a long time and now I have an idea how to do it. However, the second step of this technique looks suspicious.
Using the induction method, we can prove the correctness of this formula and that the value of the constant $C_{0}$ is really zero. But I created this question because the second step looks very strange, since the left part was multiplied by differential $di$, and the right by $dn$. If we assume that the second step is wrong, then why did we get the correct formula of summation of first squares? Note: The technique shown based on the integrated one is really interesting for me, using the same reasoning we can get the formula of the first cubes and so on EDIT According to @DatBoi's comment, we can calculate constants $C_{0}$ and $C_{1}$ by solving a system of linear equations. The desired system must contain two equations, since we have two unknown values($C_{0}$ and $C_{1}$). To achieve this, we need to use the right part of the statement from step 4 twice, for two different n. For simplicity, let's take $n=1$ for first equation and $n=2$ for second equation, then the sum of the squares for these $n$ is 1 and 5, respectively.
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| Posted: 20 Dec 2021 07:48 AM PST Let $\gamma:[0,1] \rightarrow \mathbb{R}^3$ be a smooth curve with nonvanishing velocity and let $C_\gamma = \gamma([0,1])$ be the image. Denote by $B_r = \{ x : \|x\| < r \}$ the open ball of radius $r$ centered at the origin. Then I expect the following to hold: $$\lim_{r \rightarrow 0} \frac1{r^2} \mathcal{H}^3 (C_\gamma + B_r) = \pi \cdot \mathcal{H}^1(C_\gamma)$$ where $A + B = \{ x+y : x \in A, y \in B \}$ is the Minkowski sum of two sets $A, B$ and $\mathcal{H}^d(A)$ is the $d$-dimensional Hausdorff measure of $d$. The above setup can be easily generalised by replacing a curve with a $d$-dimensional embedded submanifold $M$ with boundary and by replacing $\mathbb R^3$ by $\mathbb R^D$: $$\lim_{r \rightarrow 0} \frac1{r^{D-d}} \mathcal{H}^D (M + B_r) = \omega_{D-d} \cdot \mathcal{H}^d(M)$$ with $\omega_k = \pi^{k/2}/\Gamma(\frac k2 + 1)$ being the volume of the $k$-dimensional unit ball. Does anyone know if this type of result was proven elsewhere before? |
| Global optimization bibliography Posted: 20 Dec 2021 07:49 AM PST Can anyone recommend some books about global optimization? I will have to make a work about that topic despite I don't know exactly the title yet, so any level and approach is welcomed. |
| Posted: 20 Dec 2021 07:35 AM PST
My work: I have solved the problem as in this post. I got $u_n(x,y) = F_n(x)G_n(y)$ where $F_n(x)= c_n \sin n\pi x$ and $G_n(y) = p_n e^{n\pi y} + q_n e^{-n\pi y}$ for $n = \pm 1, \pm 2, \pm 3, \ldots$.
Thanks a lot! |
| Understanding the definition of a ternary ring of operators (TRO) Posted: 20 Dec 2021 07:46 AM PST Let $H$ and $K$ be Hilbert spaces. Recall that a closed subspace $X\subset B(H,K)$ is called a ternary ring of operators(TRO) provided $xy^*z \in X$ for all $x,y,z \in X$. On the other hand, an anti-TRO is same as defined above except $-xy^*z \in X$ for all $x,y,z \in X$
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| Posted: 20 Dec 2021 07:48 AM PST I am looking for a reference of the well-known fact due to P. Jones and A. P. Calderon that a Sobolev function $f \in W^{1,2}_0(\Omega),\ \Omega \subset \mathbb{R}^n$, with the Dirichlet boundary condition on the domain $\Omega$ with Lipschitz boundary, admits zero extension to the entire $\mathbb{R}^n$. Do we have a similar result for any compact domain with Lipschitz boundary on a Riemannian manifold $M$? |
| Posted: 20 Dec 2021 07:40 AM PST Intro: I was looking for a fresh new problem to solve and was inspired by The goal function is the Abramowitz function found in the NIST catalogue of library functions. Here is an interactive graph of the function. Maybe the Abramowitz function appears in Abramowitz and Stegun? The function is also standardized like all of the functions in the NIST DLMF: $$\text f_m(x)\mathop=^\text{def}\int_0^\infty t^m e^{-t^2-\frac xt}dt,m\in\Bbb N$$ Special cases: $$\text f_m(0)=\int_0^\infty t^m e^{-t^2}dt =\frac12 \Gamma\left(\frac{m+1}{2}\right),\text{Re}(m)>-1$$ The great result by Wolfram Alpha using the Meijer-G function: $$\text f_0(x)=\int_0^\infty e^{-t^2-\frac xt}dt=\frac{x\,\text G^{3,0}_{0,3}\left(\frac{x^2}4\bigg|-\frac12,0,0\right)}{4\sqrt \pi}$$ Which can simplify into an expression with $\,_0\text F_2$ hypergeometric functions Here is an attempt at an evaluation. There seems to be no Polynomial function generating function to make it simpler, so let's use a regular series expansion integrated with the Gauss Hypergeometric function: $$\text f_m(x)=\int_0^\infty t^m e^{-t^2-\frac xt}dt=\int_0^\infty t^m\sum_{n=0}^\infty \frac{(-1)^n (t^2+x t^{-1})^n}{n!}dt= \sum_{n=0}^\infty \frac{(-1)^nt^{m-n+1}}{n!(m-n+1)} \,_2\text F_1\left(\frac{m-n+1}3,-n;\frac{m-n+4}3;-\frac{t^3}x\right)\bigg|_0^\infty$$ Where the Gauss Hypergeometric function is just an Incomplete beta function, but the limits make the hypergeometric function diverge. Please correct me and give me feedback! Note that an equivalent problem is solving it's recurrence relation also found on NIST: $$\text f_m(x)\mathop=^{m\ge 2}\frac{m-1}2\text f_{m-2}(x)+\frac x2 \text f_{m-3}(x)$$ also a paper called " Evaluation of Abramowitz functions in the right half of the complex plane" states that a linear partial differential equation for the function is: $$x\frac {d^3}{dx^3}\text f_m(x)+(1-m)\frac{d^2}{dx^2}\text f_m(x)+2\text f_m(x)=0 \implies \text f_m(x)=\frac12 c_2(m)x\,_0\text F_2\left(\frac32,1-\frac m2;-\frac{x^2}4\right)+c_1(m) \,_0\text F_2\left(\frac12,\frac{1-m}2;-\frac{x^2}4\right) +\left(\frac x2\right)^{m+1}c_3(m)\,_0\text F_2\left(\frac m2+1,\frac{m+3}2;-\frac{x^2}4\right)$$ and the very simple delay differential equation: $$x\frac{d\,\text f_m(x)}{dx}=-\text f_{m-1}(x)$$ New Question It has been figured out by @Gary and Claude Leibovici, partly from a CAS, that we have a new decomposition formula for $\,_0\text F_2$: $$\boxed{\text f_m(x>0)=\frac12\Gamma\left(\frac{m+1}2\right)\,_0\text F_2\left(\frac12,\frac{1-m}2;-\frac{x^2}4\right)-\frac x2\Gamma\left(\frac m2\right) \,_0\text F_2\left(\frac32,1-\frac{m}2;-\frac{x^2}4\right)+x^{m+1}\Gamma(-m-1) \,_0\text F_2\left(\frac m2+1,\frac{m+3}2;-\frac{x^2}4\right)} $$ How does one get this solution from $xy'''+(1-m)y''+2y=0,y=y(x)$? One idea is a power series where the coefficients are a hypergeometric series. I know the question has changes, but this is a question, so a new one is here. |
| Implicit function theorem and constrained maximization Posted: 20 Dec 2021 07:35 AM PST For simplicity, consider the following constrained maximization problem: \begin{equation*} \begin{aligned} & \underset{x}{\text{maximize}} & & F(x,y) \\ & \text{subject to} & & x + y = 1, \end{aligned} \end{equation*} with $F(x,y)$ concave. We then have the following system of equations from the first order condition: \begin{equation*} \begin{aligned} & F_1(x,y) - F_2(x,y) = 0\\ & x + y = 1, \end{aligned} \end{equation*} where $F_i(x,y)$ is the derivative wrt the $i$ argument. Now suppose I want to find $\frac{dx}{dy}$ around the solution $(x^*,y^*)$. Using the implicit function theorem however, we get two different expressions depending on wich equation we use: \begin{equation*} \begin{aligned} & \frac{dx}{dy} = \frac{F_{22}-F_{12}}{F_{11} - F_{21}}\\ & \frac{dx}{dy} = -1, \end{aligned} \end{equation*} Now my question are:
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| Why do we use the square root as a proportional measure in this optimisation problem? Posted: 20 Dec 2021 07:43 AM PST Cross-posted on Operations Research SE I'm reading a paper where the goal is to determine the weights of a weighted arithmetic mean to estimate a new sample from a random variable. These weights weigh a vector of samples of a known portion of the samples' population to estimate it: $$\hat{y}_k = \frac{w^Ty}{w^T1}\tag{1}$$, where w, y and 1 are all vectors of size $N$. Now every sample of the vector y is described by some numerical features of size $m$. Let's call this collection matrix X. And also the sample we are trying to estimate has this vector of features. What they are trying to do is to estimate the weights by looking how similar each sample of the samples in vector y is compared to the new sample we are trying to estimate. Then a more similar sample will have more weight and will thus have more impact on the value of our estimate of the new sample. To measure the similarity, they use the Euclidean distance on the vector of features that describes each sample. This will give us a matrix A of size $N \times m$. $$A^{(n)} = (X - 1x^n)\space\odot\space(X-1x^n) $$ With the superscript (n) to indicate that the result has been calculated wrt the new sample (the one we are trying to estimate) and $x^n$ the vector of features that describes the new sample. But every numerical feature has its relative importance, so we also want to find a vector v that scales the matrix A so that you eventually get a vector of size $N$. So the more similar, so the lower the scaled distance is for a sample, the more weight we want to give it. So the weights are inversely proportional to the scaled distance which is obvious. And here comes the question: In the paper they use the equation: $$\text{weight}_j = \frac{1}{\sqrt{a^iv}}\tag{2}$$, where $a^i$ is the $i$th row of the matrix A and $j$ represents each of the known samples. But why do they use the square root? Why not just: $$\mathrm{weight}_j= \frac{1}{a^iv}$$ or $$\mathrm{weight}_j= \frac{1}{(a^iv)^2}$$ Why do we use the inversed square root and not just the inverse or the inversed square to define the proportionality ? I can think of some reasons but it's not clear if these are the right ones:
So the answer I'm looking for depends on the reason behind using the square root, or it is derived and thus a necessary use; e.g. it is mathematically derived because they use the Euclidean distance or it is just a model assumption and best practice: then I want to know: why is it best practice to use the square root? Important: To be complete: the paper continues with using Eq. (2) and substituting it to the Eq. (1) to get an optimization problem which is used to find the vector v: $$\hat{v} =\operatorname{argmin}_{v^*}\left[ \left\lVert y_k - \frac{\sum_{i=1}^n \frac{1}{\sqrt{a^iv}}y_i}{\sum_{i=1}^n \frac{1}{\sqrt{a^iv}}}\right\rVert^2 \right]$$ Once v is found, we calculate the weights with Eq. (2): $$\text{weight}_j = \frac{1}{\sqrt{a^iv}}$$ and using this in the Eq. (1): $$\hat{y}_k = \frac{w^Ty}{w^T1}$$ to eventually get our estimate. Reference of the paper: https://oa.upm.es/67130/1/INVE_MEM_2019_334744.pdf |
| Posted: 20 Dec 2021 07:57 AM PST Let $k$ be a number field. Let $M_0,M_1,\ldots,M_r \in k[x_1,\ldots,x_n]$ be monomials, with $r,n \geq 2$. Assume, for some constants $c_1,\ldots,c_r$, binomials $$B_1=M_0+c_1M_1$$ $$\cdots$$ $$B_r=M_0+c_rM_r$$ have a common divisor $D$ which is not a monomial. I would like to know if there is something we can say about $c_1,\ldots,c_r$. A possible relation I can see so far is that $c_1,\ldots,c_r$ might be multiplicatively dependent. For example, $$x^4-16\text{ and }x^4-4x^2 \text{ and } x^4-2x^3,$$ they share a common divisor $x-2$ and $c_i's$ are multiplicatively dependent. But I am not sure how to prove it. Or maybe there is a counterexample? |
| Posted: 20 Dec 2021 07:43 AM PST
I've never come across this type of optimization problem before as it has an interval with parameters. I can't seem to figure out how to approach this question. I want to get the values of x. I first tried separating the interval into two inequalities to use them as constraints: $$2x_1^2+4x_2^2\leq a_2^2$$ $$2x_1^2+4x_2^2\geq a_1^2$$ But I couldn't go far unfortunately. I would appreciate any form of hint or help. Thanks in advance. |
| Can we define any metric on $\Bbb{R^\omega}$ so that it represents a norm? Posted: 20 Dec 2021 07:47 AM PST Me again. Sorry for disturbing. I have read some posts on MSE related to my question. But I still need help to clear my doubt. $\Bbb{R^\omega}=\{(x_n)_{n\in \mathbb{N}}: x_n \in \Bbb{R}\}$ Then, $(\Bbb{R^\omega}, +, \cdot) $ is a linear space. I know , if $(x_n) $ are $p$- summable, then we can define norm , $\ell_p$-norm ($1\le p<\infty $) on $\Bbb{R^\omega}$ . And if $(x_n) 's$ are bounded we can define supremum norm, $\ell_{\infty}$ on $\Bbb{R^\omega}$. The best thing I can do for general $\Bbb{R^\omega}$ ( no special assumption on sequences) is to define a metric on $\Bbb{R^\omega}$ by $$d(x, y) =\sum_{j\in\mathbb{N}}{(a_j)} \frac{|x_j -y_j|}{1+|x_j -y_j|}$$ Where, $(a_j) _{j\in\mathbb{N}}$ is any convergent series of positive reals. And I can show that the metric doesn't induced by a norm on $\Bbb{R^\omega}$. But by checking a particular metric on $\Bbb{R^\omega}$ , doesn't gives us an opportunity to make sure that the linear space $\Bbb{R^\omega}$ is not a normed space. I also know that the existence of Hamel basis of a linear space implies the linear space is a normed space. Again to prove existence of Hamel basis we need Zorn's lemma, an equivalent version of AC. My Question:
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| How can you calculate $\int_{0}^{1} \exp(\sin(x))dx$ exactly? Posted: 20 Dec 2021 07:44 AM PST I'm taking a computational statistics class and I have to calculate $\int_{0}^{1} \exp(\sin(x))dx$ numerically (Monte Carlo integration) and compare the numerical result to the exact result. The problem is, how can one canculate the exact value of the integral? The antiderivative doesn't seem to exist in terms of elementary functions. Mathematica gives me this:
Not only that, Mathematica also cannot calculate the definite integral:
And since Mathematica cannot do it, I don't think other softwares can, though I'm not sure. Of course, Mathematica can do the numerical integration
But that's not what I need! Thanks in advance for any help. |
| Posted: 20 Dec 2021 07:35 AM PST
I was hoping someone could explain this to me. It is to do with proof that the range of correlation is between $1$ and $-1.$ I am not sure what it means to claim that the random variables $X$ and $Y$ are not constant? Does this simply mean not all values are the expected value of $Y$ or $X$ because the variance is given by $$V(N) = E(n - E(n))^2$$ Additionally, why is it possible for $aX + bY$ to be constant? Would very much appreciate anyone's insight. |
| Solving system of equations, (2 equations 3 variables) Posted: 20 Dec 2021 07:48 AM PST I am a bit puzzled. Trying to solve this system of equations: \begin{align*} -x + 2y + z=0\\ x+2y+3z=0\\ \end{align*} The solution should be \begin{align*} x=-z\\ y=-z\\ \end{align*} I just don't get the same solution. Please advice. |
| Eigenvalues of a 2x2 matrix A such that $A^2$=I Posted: 20 Dec 2021 07:35 AM PST I have no idea where to begin. Let A be a $2\times 2$ matrix such that $A^2= I$, where $I$ is the identity matrix. Find all the eigenvalues of $A$. I know there are a few matrices that support this claim, will they all have the same eigenvalues? |
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