Recent Questions - Mathematics Stack Exchange |
- What dictates whether a probability distribution is assigned a name?
- Asymptotics of $\displaystyle\int_{B(0,1)}\frac{dy}{|x-y|^{n+\alpha}}$ as $|x|\to 1^+$
- What does an automorphism of $\mathbb{RP}^1$ preserve that a permutation does not?
- Define 1D Python list as 1d Vector in Math formulation
- Proving concurrence of lines relating tangent line
- Any Built in code for Crossed Prism Graphs on sagemath?
- For every a in N+, prove that there exist a natural number b such that b++=a
- If the interior of a non-empty closed set is empty, the complement of the closed set is dense?
- Rearrange following system of equations for x and y in terms of u and v.
- how to justify expected value of estimated error in LMS estimation?
- Proving this polynomial is irreducible
- Is this matrix positive semidefinite? $M_{ij} = \sqrt{|x_i+x_j|} - \sqrt{|x_i-x_j|}$ where $x_i$'s are reals
- Why RMS value is calculated that way?
- Let $G$ be a group of order $pq$, where $p$, $q$ are distinct primes and $p<q$. Assume that $q$ $\not\equiv$ 1 mod$p$. Prove that $G$ is cyclic.
- Is there a notion of a "preimage sheaf"? That is, would the preimage of a subsheaf be a subsheaf like the preimage of ideals?
- Exponential distribution related questions
- Why the return value of integration takes a opposite sign as we reverse a limits of an integration?
- Why if polynomial $r(n)$ is the lcm of $p(n)$ and $q(n)$, for an specific integer number $a$, then $r(a)$ is not always the lcm of $p(a)$ and $r(a)$?
- $\otimes$-products of global types in $\text{RCF}$
- For which $n \ge 0$ is $n \cdot 2^n + 1$ divisible by 3?
- Prove or disprove: $p$ is the shortest path from $s\in V$ to $t\in V$ with $w'=w_{1}+w_{2}$
- Why $\int \frac{1}{2x+1}\mathrm dx\neq \ln |2x+1|$?
- Green's Theorem Concepts: Circulation in R2
- Insurance statistics book
- Help with $\int _0^{\infty }\frac{\sinh \left(x\right)}{x\cosh ^2\left(x\right)}\:\mathrm{d}x$
- Convergence of the integral $\int_0^\infty \frac{1}{1+x^4\cos^2x}dx$.
- Group Law on Specific Elliptic Curve
- 4 cards are shuffled and placed face down. Hidden faces display 4 elements: earth, wind, fire, water. You turn over cards until win or lose.
- Finite Model Theory
- definition of simple group , why we need normal ???
| What dictates whether a probability distribution is assigned a name? Posted: 24 Jul 2021 08:25 PM PDT I'm curious as to how a probability distribution makes the transformation from an unlabeled function to a named and parameterized distribution (e.g., normal, binomial, Poisson, etc.)? Is there a committee who decides which distributions are inducted into the royal family of named distributions? Or is it a slow process of repeated appearance, and established practical utility, that enables this transformation?  |
| Asymptotics of $\displaystyle\int_{B(0,1)}\frac{dy}{|x-y|^{n+\alpha}}$ as $|x|\to 1^+$ Posted: 24 Jul 2021 08:22 PM PDT In a study of Fractional Laplacian, I encounter the integral $$I(x):=\int_{B(0,1)}\frac{dy}{|x-y|^{n+\alpha}}$$ where
I conjecture that $$I(x)\le C(|x|-1)^{-\alpha}\qquad (*)$$ as $|x|\to 1^+$, i.e. for $|x|>1$ sufficiently close to $1$, there exists a constant $C>0$ indepednent of $x$ such that the inequality $(*)$ holds. Question: Is my conjecture true? If not, what is the asymptotics of $I(x)$ as $|x|\to 1^+$? (NB: This conjecture arises in an attempt to prove Proposition 3.1 (whose proof is omitted) in the paper On the superharmonicity of the first eigenfunction of the fractional Laplacian for certain exponents.) Failed AttemptUsing the reverse triangle inequality, one sees that $$\begin{align} I(x)&=\int_{B(0,1)}\frac{dy}{|x-y|^{n+\alpha}} \\ &\le\int_{B(0,1)}\frac{dy}{||x|-|y||^{n+\alpha}} \\ &=n\cdot \text{vol}(B(0,1))\int_{0}^1\frac{r^{n-1}}{||x|-r|^{n+\alpha}} dr \qquad (1)\\ &=n\cdot \text{vol}(B(0,1))\int_{0}^1\frac{r^{n-1}}{(|x|-r)^{n+\alpha}} dr \\ &=n\cdot \text{vol}(B(0,1))\left(\frac{1}{(|x|-1)^{n+\alpha-1}}-\frac{n-1}{n+\alpha-1}\int_{0}^1\frac{r^{n-2}}{(|x|-r)^{n+\alpha-1}} dr\right) \qquad (2)\\ \end{align}$$ (1): Switching to spherical coordinates. (2): Integrating by parts. The boundary term is $O((|x|-1)^{-n+1-\alpha})$; through integration by parts, the integral term can be shown to be $O((|x|-1)^{-n+2-\alpha})$. Thus, this bound only shows that $I(x)=O((|x|-1)^{-n+1-\alpha})$ which is weaker than the conjecture, unless $n=1$.  |
| What does an automorphism of $\mathbb{RP}^1$ preserve that a permutation does not? Posted: 24 Jul 2021 08:22 PM PDT If we define an automorphism of $\mathbb{RP}^2$ to be a bijective self-map that preserves collinearity -- that is, a bijection that takes lines to lines -- then this property, all by itself, ensures that every automorphism of $\mathbb{RP}^2$ is represented by a $3 \times 3$ invertible matrix; this matrix is unique up to a scalar, and hence we can conclude that $Aut(\mathbb{RP}^2) \cong PGL(3,\mathbb R)$. If we drop one dimension lower, though, the requirement that an automorphism must take lines to lines becomes trivial: $\mathbb{RP}^1$ is a single (projective) line, so any permutation $\sigma: \mathbb{RP}^1 \to \mathbb{RP}^1$ preserves collinearity. But not every bijective self-map $\mathbb{RP}^1 \to \mathbb{RP}^1$ is an automorphism. What property of $\mathbb{RP}^1$ is preserved by an automorphism but not, in general, preserved by a permutation of the underlying set? To paraphrase the question, I would like to be able to fill in the blanks in the following statement:
Of course one way to fill in the blank is by saying "An automorphism of $\mathbb{RP}^1$ is a bijective self-map that can be represented by a fractional linear transformation of the form $t \mapsto \frac{at+b}{ct+d}$", but that is begging the question. Why should we only consider maps to be automorphisms if they have this specific form? I would prefer not to receive answers that begin with a general definition of an automorphism for projective $n$-space over an arbitrary field $k$, and then show that in the special case $n=1, k = \mathbb R$ it follows that $Aut(\mathbb{RP}^1) \cong PGL(2, \mathbb R)$. I am seeking an elementary perspective, one that need not generalize, that can help explain exactly what we are talking about when we talk about automorphisms of the real projective line.  |
| Define 1D Python list as 1d Vector in Math formulation Posted: 24 Jul 2021 08:21 PM PDT In Python I believe a list is considered as a 1 dimensional vector. how do I define a list A = [1, 2, 100, 2, 300, 1] for example whose elements is of set B provided that the element is more than zero: B ={-1, -2, 1, 2, -300, 100, -100, 2, 300, 1} would this definition below be correct ? : A = [h]_{\forall h\in B: h>0} Thanks !  |
| Proving concurrence of lines relating tangent line Posted: 24 Jul 2021 08:18 PM PDT Given triangle $ABC$ inscribed circle $(O)$. $D$, $E$ are on sides $AC$, $AB$, respectively. $BD$ meets $(O)$ at $M$ and $CE$ meets $(O)$ at $N$. Line $DE$ meets $(O)$ at $P$ and $Q$. Prove that $MN$, $PQ$ and the tangent line at $A$ are concurrent. I have tried using Pappus or Pascal theorem with "$AA$" in the 6 points but give out no results. I've also tried using harmonic division to quadrilateral $MNPQ$, but it didn't work. Please help me with the strategies, thanks!  |
| Any Built in code for Crossed Prism Graphs on sagemath? Posted: 24 Jul 2021 08:13 PM PDT I have been working on a research question regarding a certain property of Crossed Prism Graphs that requires me to use Sage (i.e., Python). I have found on an article entitled: "A new generalization of generalized Petersen graphs" that the family GDGP_2(n; 1, n-3) is also known as Crossed Prism Graph, where GDGP stands for Divisible Generalized Petersen Graphs. Any idea how to code that up in Sage? I would love some guidance!  |
| For every a in N+, prove that there exist a natural number b such that b++=a Posted: 24 Jul 2021 08:10 PM PDT The question is listed above. The question is that whether$\forall a\in \mathbb{N^+}, \exists b\in \mathbb{N} \,,s.t.a=b++$ is right. My question is, in my proof, I've tried to use the mathematical induction, but as I assume that if the number $n$ satisfies the situation, the element $n++$ could easily make sense, so I'm wondering whether the mathematical induction could make sense. The question is quite trivial but I'm really caught in trouble. Hoping for help.  |
| If the interior of a non-empty closed set is empty, the complement of the closed set is dense? Posted: 24 Jul 2021 08:24 PM PDT Suppose $X$ is a metric space. Take $F \subset X, F \neq \emptyset$ such that $int\;F = \emptyset$. I can clearly see that $F^c$ is dense in $X$ if $X$ is a Euclidean space. But unable to see it for a general metric space. Any help is much appreciated.  |
| Rearrange following system of equations for x and y in terms of u and v. Posted: 24 Jul 2021 08:28 PM PDT We are given the following system of equations(needed to determine the inverse mapping): \begin{align} \sin(u) &= x\sqrt{\frac{1-y^2}{1-x^2y^2}}\\ \sin(v) &= y\sqrt{\frac{1-x^2}{1-x^2y^2}} \end{align} I am trying to solve for x and y, with the condition that both x and y must be in term of u and v only, however I am unsure how to proceed. I tried drawing out a triangle, but that did not seem to help.  |
| how to justify expected value of estimated error in LMS estimation? Posted: 24 Jul 2021 08:01 PM PDT If we have estimator and error like below,
I get $E[\hat{\Theta}] =E[\Theta]$ from  |
| Proving this polynomial is irreducible Posted: 24 Jul 2021 08:28 PM PDT Let $a_1, a_2,..., a_{2n} \geq 1$ be $2n$ distinct positive integers such that at least two of them are even. Show that the polynomial $$(X^2-a_1)(X^2-a_2)...(X^2-a_{2n})-1$$ is irreducible over $\mathbb{Z}(X)$. The approach I've started using to solve this problem: assume $f(X) = g(X)h(X)$ over $\mathbb{Z}$. By Gauss's lemma we may assume that $g,h$ are monic with integral coefficients, and then I'm comparing the degree of both polynomials to try an get a contradiction of the assumption. Can someone give me a few more hints as to how to arrive at such a contradiction?  |
| Posted: 24 Jul 2021 08:03 PM PDT Let finite number of $x_i$'s be reals. Define matrix $M_{ij} = \sqrt{|x_i+x_j|} - \sqrt{|x_i-x_j|}$. Is this matrix positive semidefinite? I am reading this year's IMO problem number 2, which would be trivially true if we prove that $M_{ij}$ is positive semidefinite.
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| Why RMS value is calculated that way? Posted: 24 Jul 2021 07:38 PM PDT The root mean square (RMS or rms) is defined as the square root of the mean square (the arithmetic mean of the squares of a set of numbers). Note that, we only take square of the "set of numbers" but not of the "count of numbers". But when calculating the rms value, we take the square root of the arithmetic mean of the squares of the values. Note, we didn't take the square of the "count of numbers". Then, why take the square root of the arithmetic mean? Shouldn't we take the square root of only the square of the "set of numbers"?  |
| Posted: 24 Jul 2021 08:26 PM PDT This is an exercise in Serge Lang's Algebra in the first chapter. I am wondering why q $\not\equiv$ 1 mod $p$ is assumed considering it is unnecessary. Indeed, if that is excluded from the requirements, then let $H_q$ and $H_p$ be the Sylow subgroups of orders $q$ and $p$, respectively. Then they are cyclic and thus have trivial intersection. Since they have trivial intersection, the product of groups $H_qH_p$(which is a group since $H_q$ is normal) is isomorphic to $H_q$$\times$$H_p$ which has order $pq$ and so it is equal to $G$. Considering $p$ and $q$ are coprime, $G$ is cyclic. Is this solution correct/an acceptable answer to this problem? If so, why is the aforementioned requirement provided? Note that all of the information used in my proof is either in the exercises preceding this one or in the chapter on Sylow subgroups.  |
| Posted: 24 Jul 2021 08:09 PM PDT More specifically, if $\phi: \mathscr{F} \to \mathscr{G}$ is a morphism of sheaves and $\mathscr{G}' \subset \mathscr{G}$ is a subsheaf (so that $\mathscr{G}'(U) \subseteq \mathscr{G}(U)$ for all $U$ open), then what is the presheaf $U \mapsto \phi_U^{-1}(\mathscr{G}'(U))$? If they are $\mathscr{O}_X$-modules on a ringed space, do they enherit quasi-coherence from $\mathscr{G'}$? If $X$ is a noetherian scheme, can the same be said about coherence? I was able to show it is a presheaf with the restriction maps induced by the ones on $\mathscr{F}$, but I have yet to show it is a sheaf or any of the coherence conditions. More than anything else, I am curious if this construction has a name so that I can learn more about it. It is hard to look up because the 'pullback sheaf' or 'inverse image sheaf' is a very common construction that is substantially different. Thanks! I am going to continue working on this but I will include some progress below. I don't think it's necessary for my question so please feel free to ignore it if you don't need the context. For a little more detail on the presheaf part, if $\rho: \mathscr{F}(U) \to \mathscr{F} (V)$ is a restriction then $\rho(\phi_U^{-1}(\mathscr{G}'(U))) \subseteq \phi_V^{-1}(\mathscr{G}'(V))$ by some diagram chasing so that $\rho|_{\phi_U^{-1}(\mathscr{G}'(U))}: \phi_U^{-1}(\mathscr{G}'(U)) \to \phi_V^{-1}(\mathscr{G}'(V))$ is a well defined map which satisfies the presheaf condition. It is also a sheaf if $\mathscr{G}'$ is. As a sub-presheaf of $\mathscr{F}$, it inherits locality for free. For gluing, if we have $s_i \in \phi_{U_i}^{-1}(\mathscr{G}'(U_i))$ on an open cover $(U_i)$ of $U$ agreeing on intersections, they glue together to a section $s \in \mathscr{F}(U)$ since $\mathscr{F}$ is a sheaf. Then $\phi_{U_i}(s_i)$ also agree on intersections by the presheaf morphism condition so that they lift to a section of $\mathscr{G'}(U)$. By a locality argument on $\mathscr{G}$, it follows that this section is $\phi_U(s)$ so that $s \in \phi_U^{-1}(\mathscr{G}'(U))$ as required. Checking the universal property, it is indeed isomorphic to $\mathscr{F} \times_{\mathscr{G}} \mathscr{G}'$ so this can be viewed as a fibered product.  |
| Exponential distribution related questions Posted: 24 Jul 2021 08:06 PM PDT Suppose that the inter-arrival times of male customers entering the bank are iid exponential random variables with $\frac{1}{\lambda_1}$ and for those females are iid exponential random variables with $\frac{1}{\lambda_2}$. Find the distribution and expected value of $Z_f$, the number of female customers arriving at the bank between two successive male customers. The question bothers me for a while. Please help me out!!!  |
| Why the return value of integration takes a opposite sign as we reverse a limits of an integration? Posted: 24 Jul 2021 07:56 PM PDT I've may been asking the really stupid question. The below general formula. $$ \int_{a }^{b } f \left( x \right) \,dx =-\int_{b }^{a } f \left( x \right) \,dx $$ For instance, $$ \int_{1 }^{3 } x \,dx = \left[ \frac{ 1 }{ 2 } x ^{2} \right]_{1}^{3} $$ $$ = \frac{1}{2} \left( 3 ^{2} -1 ^{2} \right) $$ $$ = \frac{ 1 }{ 2 } \left( 9-1 \right) =4 $$ And as we reverse the limits of the integration, $$ \int_{3 }^{1 } x \,dx =\left[ \frac{ 1 }{ 2 } x ^{2} \right]_{3}^{1} $$ $$ = \frac{1}{2} \left( 1 ^{2} -3 ^{2} \right) $$ $$ = \frac{1}{2} \left( 1-9 \right) $$ $$ = \frac{ -8 }{ 2 } =-4 ~~ \leftarrow~~ \text{Only the sign is opposite} $$ I know that a return value of an integration is given by summing up an each infinitesimal area of a infinitesimal rectangle. In this integration for example, the width of the rectangle is $~ dx ~$ and the height of it is given by $~ x ~$ So even the limits is reversed , the result should be same but of course from the formula, that claim is disvalidated. Why?  |
| Posted: 24 Jul 2021 07:52 PM PDT Let $n$ be a positive integer and $$p(n) =(n+1)(n-1),$$ $$q(n) =(n+1)(n+1), $$ $$r(n) = (n+1)(n+1)(n-1).$$ I think that $r(n)$ is the least common multiple of $p(n)$ and $q(n)$. The values of these polynomials at $n = 3$ are $p(3) = 8, q(3) = 16$ and $r(3) = 32.$ But the lcm of $8$ and $16$ is $16$, not $32.$ Please explain me why $r(3)$ is not the least common multiple of $p(3)$ and $q(3)$ in spite of $r(n)$ being the lcm of $p(n)$ and $q(n).$ I have been looking for an answer but I couldn´t find one.  |
| $\otimes$-products of global types in $\text{RCF}$ Posted: 24 Jul 2021 08:13 PM PDT I'm struggling to understand a remark in Pierre Simon's book on NIP theories. Let $T$ be a complete theory and $\mathfrak{U}\models T$ a monster model. Suppose $p(x),q(y)\in S(\mathfrak{U})$ are global types with distinct free variables, and that $p$ is $E$-invariant for some small subset $E\subset\mathfrak{U}$. Then the product type $p(x)\otimes q(y)$ in the free variables $xy$ is defined by taking $\phi(x,y,c)\in p\otimes q$ if and only if there exists some $b$ realizing $q|_{Ec}(y)$ and such that $\phi(x,b,c)\in p$; this is well-defined by $E$-invariance of $p$. Simon says that $p$ and $q$ "commute" if $p(x)\otimes q(y)=q(y)\otimes p(x)$. Exercise 2.22 has us show that, if $T=\text{DLO}$ and $p(x)$ and $q(y)$ are $1$-types, then $p$ and $q$ commute provided that $p$ and $q$ are inequivalent. I've done this without trouble, but I'm a bit perplexed by a subsequent remark:
I'm struggling to see why this is the case. By quantifier elimination, a global $1$-type $p(x)\in S(\mathfrak{U})$ is uniquely determined by the $<$-cut of $\mathfrak{U}$ it represents, and the data of whether or not the equation $f=0$ lies in $p$ for each polynomial $f\in\mathfrak{U}[x]$. Since polynomials over a field have finitely many roots, if $p$ contains an equation $f=0$ then it is an algebraic type and hence has all possible realizations already in $\mathfrak{U}$. In particular, if $p(x),q(y)$ are distinct $1$-types that are not realized in $\mathfrak{U}$, then (i) they must represent different cuts in $\mathfrak{U}$, and (ii) we will have that $p(x)\otimes q(y)$ contains $f(x,y)\neq 0$ for each $f\in\mathfrak{U}[x,y]$. Furthermore, if I'm not mistaken, $p(x)\otimes q(y)\supset p(x)\cup q(y)$, so the only possible way for $p(x)\otimes q(y)$ and $q(y)\otimes p(x)$ to differ is if one of them contains $x<y$ and the other contains $x\geqslant y$. But $p(x)$ and $q(y)$ represent different cuts, so without loss of generality there exists $c\in\mathfrak{U}$ such that $x<c\in p(x)$ and $c<y\in q(y)$, and then necessarily $x<y$ lies in both $p(x)\otimes q(y)$ and $q(y)\otimes p(x)$, again since each of these types contains $p(x)\cup q(y)$. So it's unclear to me how the circumstances described by Simon can arise; does anyone have any insight?  |
| For which $n \ge 0$ is $n \cdot 2^n + 1$ divisible by 3? Posted: 24 Jul 2021 08:20 PM PDT Same question: Found here. Hi, I'm a bit stuck with my question. I've read through the answers in the above link, but I cannot quite understand the answers fully. I don't want to hijack the other thread either, so I thought I'd start a separate thread. $\textbf{The question is:}$ For which ${n \ge 0}$ is the number ${n \cdot 2^n + 1}$ divisible by 3? I'v gotten this far in my understanding: For even ${n(n=2k)}$: \begin{equation} \begin{aligned} 2^n&=2^{2k} \equiv_3 4^k \equiv_3 1^k \equiv_3 1\\ n\cdot 2^n+1 &\equiv_3 n\cdot 1 + 1 \equiv_3 n +1 \end{aligned} \end{equation} For odd ${n(n=2k+1)}$: \begin{equation} \begin{aligned} 2^{2k+1}&=2^{2k}\cdot 2 \equiv_3 1^k \cdot -1 \equiv_3 1 \cdot - 1 \equiv_3 -1\\ n\cdot 2^n+1 &\equiv_3 n\cdot -1 + 1 \equiv_3 -n +1 \end{aligned} \end{equation} One of the answers in the link above mentioned \begin{equation} n\equiv 0 (\text{mod 2})\qquad n\equiv 2(\text{mod 3}) \end{equation} which could be "consolidated as ${n\equiv 2 (\text{mod 6})}$". I don't understand what the "consolidate"-part means. So, would someone please like to share some knowledge and insights with me? It feels like I'm almost there, but still a far way to go. Thanks  |
| Prove or disprove: $p$ is the shortest path from $s\in V$ to $t\in V$ with $w'=w_{1}+w_{2}$ Posted: 24 Jul 2021 08:17 PM PDT I saw the following statement:
I could not disprove it and I believe this statement it true. But I could not prove it formally. How do I translate "$p$ be the shortest path from $s\in V$ to $t\in V$ with $w_1$ and $w_2$" into math?  |
| Why $\int \frac{1}{2x+1}\mathrm dx\neq \ln |2x+1|$? Posted: 24 Jul 2021 08:12 PM PDT Why $$\int \frac{1}{2x+1}\mathrm dx\neq \ln |2x+1|$$? While doing integration by partial fraction. I noticed they wrote that $$\int \frac{1}{2x+1}\mathrm dx=\frac{1}{2} \int \frac{2}{2x+1}=\frac{1}{2}\ln |2x+1|$$ Why $2$ is needed in numerator for this type of integration? There's only single $2$ in denominator.  |
| Green's Theorem Concepts: Circulation in R2 Posted: 24 Jul 2021 08:12 PM PDT I am trying understand how circulation density arises in Green's theorem and I'd like to know if my line of thinking is on the right track. Here it goes :). Idea We know that if we have a vector field $\vec F=P\hat x+Q\hat y\in\mathbb{C}^1$ on an open region $D$ containing the simple region $R$ whose boundary is $\Gamma$, then we can "cut" $\Gamma$ into $K$ positively-oriented rectangular paths such that $\Gamma=\bigcup_{k=1}^K\Gamma_k$. Consequently, $$\oint_\Gamma \vec F\cdot d\vec r=\sum_{k=1}^K \oint_{\Gamma_k}\vec F\cdot d\vec r $$ which says the the macroscopic circulation of $\vec F$ along the curve $\Gamma$ is equal to the sum of microscopic circulations over the region $R$ enclosed by $\Gamma$. Moreover, we know that the circulation along one of these rectangles is $$\oint_{\Gamma_k}\vec F\cdot d\vec r=\bigg(\frac{Q(x+\Delta x,\;y)-Q(x,y)}{\Delta x}-\frac{P(x,\;y+\Delta y)-P(x,y)}{\Delta y}\bigg)\;\Delta x\Delta y$$ (for proof, see Green's Theorem in the Plane: Circulation Density). Hence, the circulation density, or curl, of $\vec F$ at any point $(x,y)$ in $R$ is given by $$\text{curl}\vec F(x,y)=\lim_{\Delta A\rightarrow 0}\frac{1}{\Delta A}\oint_\Gamma \vec F \cdot d\vec r=Q_x-P_y,$$ which is a scalar function. Now, if we let $\sigma (x,y) = Q_x-P_y$ and integrate over the $R$, we obtain $$\sum_{i=1}^n\sum_{j=1}^m\sigma(x^*_{ij},y^*_{ij})\Delta x\Delta y $$ hence, $$\lim_{n,m\rightarrow\infty}\sum_{i=1}^n\sum_{j=1}^m\sigma(x^*_{ij},y^*_{ij})\Delta x\Delta y =\iint_R \sigma(x,y) dxdy=\iint_R (Q_x-P_y )dA$$ whose RHS can be related back to the line integral $\oint_\Gamma\vec F\cdot d\vec r$ giving us what we know as Green's Theorem.  |
| Posted: 24 Jul 2021 07:45 PM PDT Am am looking for a book on insurance statistics. The problem is that there are a lot of googleable variants that it is difficult to decide which one is good and which ones are better in good ones. This is why I am asking here for sugestion. To be short I am looking for a good reference which, if it is possible, describes mathematical tools for borh life insurance and non-life insurance. It would be great if the reference was about statistics and about the liability side of the balance sheet as well as about finance and the asset side of the balance sheet. My math background is intermediate but if the level needed for a prospective reference is indicated, references for various math background will be appreciated.  |
| Help with $\int _0^{\infty }\frac{\sinh \left(x\right)}{x\cosh ^2\left(x\right)}\:\mathrm{d}x$ Posted: 24 Jul 2021 08:12 PM PDT I want to know how to prove that $$\int _0^{+\infty }\frac{\sinh \left(x\right)}{x\cosh ^2\left(x\right)}\:\mathrm{d}x=\frac{4G}{\pi }$$ Here $G$ denotes Catalan's constant, I obtained such result with the help of mathematica. I also found that the integral equals a certain infinite series $$\int _0^{+\infty }\frac{\sinh \left(x\right)}{x\cosh ^2\left(x\right)}\:\mathrm{d}x=\sum _{n=0}^{+\infty }\frac{\binom{2n}{n}^2}{16^n\left(2n+1\right)}=\frac{4G}{\pi }$$ which can also be found in this link. So I've $2$ questions $1)$$¿$How can we transform the integral into the mentioned series? $2)$$¿$Is there a simple way to evaluate the main integral without resorting to series expansion? What I did for question $\#2$ is to employ the substitution $x=\ln\left(t\right)$ $$\int _0^{+\infty }\frac{\sinh \left(x\right)}{x\cosh ^2\left(x\right)}\:\mathrm{d}x=-2\int _1^{\infty }\frac{1-t^2}{\ln \left(t\right)\left(1+t^2\right)^2}\:\mathrm{d}t$$ But I'm not sure how to proceed.  |
| Convergence of the integral $\int_0^\infty \frac{1}{1+x^4\cos^2x}dx$. Posted: 24 Jul 2021 07:41 PM PDT I want to study the convergence of the integral $\int_0^\infty \frac{1}{1+x^4\cos^2x}dx$.I am not sure whether it converges or diverges.I am able to show only that $\frac{1}{1+x^4\cos^2x}\geq \frac{1}{1+x^4}$ for all $x\in (0,\infty)$.But that does not give an bound for comparison test.So,can someone give me some hint for solving this problem?  |
| Group Law on Specific Elliptic Curve Posted: 24 Jul 2021 08:05 PM PDT Let $n$ be a fixed positive integer. Consider the equation $$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=n.$$ If $n$ is odd then there are no positive integer solutions to this equation, and if $n$ is even then the positive integer solutions can be quite large. A nice reference is this mathoverflow question. Clearing denominators gives the equivalent equation $$a^3+b^3+c^3+abc=(n-1)(a+b)(a+c)(b+c).$$ This is an elliptic curve (provided that you pick an identity point).
Certainly it's possible to write down the group law by transforming to Weierstrass form, using the (ugly) group law formula, and transforming. The question is whether there is a nice way to write down the group law.  |
| Posted: 24 Jul 2021 08:02 PM PDT Question: 4 cards are shuffled and placed face down in front of you. Their hidden faces display 4 elements: water, earth, wind, fire. You turn over cards until win or lose. You win if you turn over water and earth. You lose if you turn over fire. What is the probability that you win? I understand that wind is effectively absent from the sample space. Does not affect your chances of winning or losing. I also know that 1/3 (because we removed wind), you can pick fire where you lose the game.  |
| Posted: 24 Jul 2021 08:22 PM PDT It seems that finite model theory is regarded (in a sense) as a computer theoretic subject. Is this the case or are there questions of interest that are of interest to mathematical logicians or more properly model theorist? And if so what would be a good place to learn about them?  |
| definition of simple group , why we need normal ??? Posted: 24 Jul 2021 08:19 PM PDT Every prime ordered group is simple, its because it doesn't admit any subgroups. But where comes the normal subgroup, why cant the people use just subgroups instead of normal subgroups in the definition of simple group. Does the normal subgroup has any influence when we are dealing with the structure of groups ??? In addition help me with an example of a group that admits subgroup but it doesn't have normal subgroups ???  |
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