Recent Questions - Mathematics Stack Exchange |
- Specific conditions for attractors of the dynamics
- The definition of the cardinality of a signature
- Maximum Likelihood Estimator of a Poisson Distribution
- How can I form this second-order conic constraint?
- Quotient group by normal closure of union
- Why is $P(E_4 | \bigcup_{i=1}^3 E_i) = P(E_4)$?
- Finite difference method and differentiation
- Infinite intersection of sets of multiples of naturals
- Analytical solution of the nonlinear 1D Diffusion equation
- Is a graph without vertices but with edges a graph?
- Which book has term by term differentiation theorem for multivariable functions series?
- Number of all possible paths in a complete graph
- Doubt on max tensor product of $C^{\ast}$-algebras
- Coherence Upper Bound of Kronecker Product and Transposed Transposed Khatri-Rao Product
- Questions on '$\emptyset$'
- Double infinite series
- Are all compact subsets of $\mathbb R^{\mathbb R}$ separable?
- maximum range of projectile launched from elevation
- Cardinality of $R^\infty$
- Why does the heat source become so hot?- Heat equation with heat source using finite difference method
- Why using $\tan^{-1}(b/a)$ is not good enough to find the angle of a complex number?
- What is the Kurtosis in Geometric Brownian Motion
- Proving $|\alpha^{<\omega}| < \kappa$ without Axiom of Choice
- How to prove that $\frac{x}{1+x^2}$ is not injective?
- Is the limit in probability of Lévy processes still a Lévy process?
- Bound for the probability of a Lévy process having a jump of a given size
- Prove this algorithm for finding the Eulerian path/cycle in a undirected graph
- Number Of Cycles in a Even Graph
- How is a ring a $\mathbb{Z}$-algebra?
- Idempotent endomorphisms generate a direct sum decomposition
| Specific conditions for attractors of the dynamics Posted: 08 Apr 2022 04:02 AM PDT Lets say we have a set of vectors $\{\bar v_1, \dots, \bar v_N\}$, with $\bar v_i \in \mathbb{R}^d$, and a function $f: \mathbb{R}^d \longrightarrow \mathbb{R}^d$. What we know about $f$, is that it can be factorized in three different functions, i.e., $f = a \circ b \circ c$, such that: The first function $a : \mathbb{R}^d \longrightarrow \mathbb{R}^N$ uses a similarity function $\kappa: \mathbb{R}^d \times \mathbb{R}^d \longrightarrow \mathbb{R}$ to compute the similarity of a given input $\bar x$ with every vector $\bar v_i$. Hence, we have $a(\bar x)_i = \kappa(\bar x,\bar v_i)$ for every $i$. The second function is a continuous function $b: \mathbb{R}^N \longrightarrow \mathbb{R}^N$. The third function is known: $c(\bar x) = M^T \bar x$, where $M$ is the matrix whose columns are the vectors $\bar v_i$. For example, if $\bar x$ is a one-hot vector with $i$-th entry equal to one, we have $c(\bar x) = \bar v_i$. Example: Let $\kappa$ be a dot product, so that we have $a(\bar x)_i = \langle\bar x,\bar v_i\rangle$, and $b$ be the softmax function. With these conditions, we have that the initial function $f$ has the vectors $\{\bar v_1, \dots, \bar v_N\}$ as attractors of the dynamics. Here comes my question, which aims to generalize the above results: What are the minimal conditions that we can have on the two functions $a$ and $b$, such that the initial function $f$ has $\{\bar v_1, \dots, \bar v_N\}$ as attractors? I would assume that $\kappa$ has to be some sort of similarity function (altought I'm not finding any formal definition of it online), and $b$ has to be a function that, given any vector, multiple iterations of it converge to a 1-hot vector. However, i'm struggling into finding the details. What would a formal argument saying that " If $\kappa$ has properties XY and $b$ has property Z, then $\{\bar v_1, \dots, \bar v_N\}$ are always attractors of the dynamics of $f$"? |
| The definition of the cardinality of a signature Posted: 08 Apr 2022 04:01 AM PDT Here's the definition of the cardinality of a signature. Can anyone elaborate on what "the least infinite cardinal >= the number of symbols in L" mean? What is the cardinality of a signature in the end? This is from the book Model Theory by Wilfrid Hodge |
| Maximum Likelihood Estimator of a Poisson Distribution Posted: 08 Apr 2022 03:58 AM PDT Suppose that independent observations $X_{1}$ and $X_{2}$ are taken from Poisson $P(aλ)$ and Poisson $P(bλ)$ distributions respectively, where $a$ and $b$ are known and positive. (i) Find the maximum likelihood estimator of $λ$. My Solution: $L(λ) = \prod_{i = 1}^{n} P(X_{i} |λ)$ $L(λ) = P(X_{1} |aλ) . P(X_{2} |bλ) $ $L(λ) = \frac{e^{-aλ}λ^{X_{1}}}{X_{1}!}.\frac{e^{-bλ}λ^{X_{2}}}{X_{2}!}$ $L(λ) = \frac{e^{-λ(a+b)}λ^{X_{1} + X_{2}}}{X_{1}!X_{2}!}$ $\log_{e} (L(λ))$ =$ \log_{e} (e^{-λ(a+b)})$ + $\log_{e} (λ^{X_{1} + X_{2}})$ - $\log_{e} (X_{1}!X_{2}!)$ $\frac{d}{dλ} L(λ) = 0$ at maximum Therefore, $0 = -(a + b) + \frac{X_{1}X_{2}}{λ}$ $λ = \frac{X_{1} + X_{2}}{a + b}$ Please may someone tell me if this is the correct solution or if I have misunderstood the question? |
| How can I form this second-order conic constraint? Posted: 08 Apr 2022 03:57 AM PDT I am trying to show that if we have a real symmetric matrix $Q$ with one negative eigenvalue within a quadratic constraint: $x^\top Qx+a^\top x +b \leq 0$ That this constraint can be formulated as a second-order conic constraint. My intial attempt at doing this was by performing an eigenvalue decomposition and then bringing the negative eigenvalue to the RHS in order to get an expression that is like: $||x||_2\leq t$ Note that this was inspired by section 3.2.8 of https://docs.mosek.com/modeling-cookbook/cqo.html. However I am unsure what to do about the additional $a^\top x +b$ terms here. Would we need to make certain assumptions about these terms? |
| Quotient group by normal closure of union Posted: 08 Apr 2022 03:56 AM PDT Let $G$ be a group and $H, N \subseteq G$ be subsets of the group. Let $\overline{A}$ denote the normal closure of any subset $A\subseteq G'$ in some group $G'$. Let $\pi: G \to G/\overline{N}$ denote the quotient map. I want to know if it is true that the following groups are isomorphic $$G/\overline{N\cup H} \cong (G/\overline{N})/\overline{\pi(H)}$$ I think I have to show that $\overline{\pi(H)}=\pi(\overline{N \cup H})$ so that I can use isomorphism theorems. So, to show the inclusion $\overline{\pi(H)}\subseteq \pi(\overline{N \cup H})$ note that $\pi(H)\subseteq \pi(N \cup H) \subseteq \pi( \overline{N \cup H})$, where the latter is a normal subgroup of $G/\overline{N}$. But then by the definition of normal closure $\overline{\pi(H)}\subseteq \pi(\overline{N\cup H})$. To show the other inclusion, I want to show that $N \cup H \subseteq \pi^{-1}(\overline{\pi(H)})$. Since $\overline{\pi(H)}$ is a subgroup it contains the identity element. But then it's inverse under $\pi$ contains the kernel of $\pi$ which contains $N$. We also have that $H \subseteq \pi^{-1}(\pi(H))\subseteq \pi^{-1}(\overline{\pi(H)})$. Since the preimage of a normal subgroup under a homomorphism is a normal subgroup we have that $N \cup H \subseteq \pi^{-1}(\overline{\pi(H)})$ implies that $\overline{N\cup H} \subseteq \pi^{-1}(\overline{\pi(H)})$. Now since $\pi$ is surjectiv, we conclude $\pi(\overline{N\cup H}) \subseteq \overline{ \pi(H)}$. Now we can use the third isomorphism theorem, since $\pi(\overline{N \cup H})=\overline{N\cup H}/\overline{N}$, because $\overline{N}$ is a normal subgroup of $\overline{N \cup H}$: $$ (G/\overline{N})/\overline{\pi(H)} = (G/\overline{N})/\pi(\overline{N \cup H}) = (G/\overline{N})/(\overline{N \cup H}/\overline{N}) \cong G/\overline{N\cup H}$$. Am I reasoning correctly, is there are simpler proof of this? |
| Why is $P(E_4 | \bigcup_{i=1}^3 E_i) = P(E_4)$? Posted: 08 Apr 2022 03:52 AM PDT When for all $i, E_i$ are independent to one another, why is $P(E_4 | \bigcup_{i=1}^3 E_i) = P(E_4)$? This question is suppose to be a true or false question. But most of my colleague got true but for me I got false as my answer. My attempt was that $P(E_4 | \bigcup_{i=1}^3 E_i) = \frac{P(E_4 \cap (E_1 \cup E_2 \cup E_3)}{P(E_1 \cup E_2 \cup E_3)}$ The numerator has to satisfy $P(E_4 \cap (E_1 \cup E_2 \cup E_3) = P(E_4)P(E_1 \cup E_2 \cup E_3)$ to make the question true. Hence I started off with $P(E_4 \cap (E_1 \cup E_2 \cup E_3) =P((E_4 \cap E_1) \cup (E_4 \cap E_2) \cup (E_4 \cap E_3))$ $= P(E_4 \cap E_1) + P(E_4 \cap E_2) + P(E_4 \cap E_3) - P((E_4 \cap E_1)\cap(E_4 \cap E_2))- P((E_4 \cap E_1)\cap(E_4 \cap E_3)) - P((E_4 \cap E_2)\cap(E_4 \cap E_3)) + P((E_4 \cap E_1)\cap(E_4 \cap E_2)\cap(E_4 \cap E_3))$ $=P(E_4)P(E_1) + P(E_4)P(E_2) + P(E_4)P(E_3) - P(E_4)^2P(E_1)P(E_2) - P(E_4)^2P(E_1)P(E_3) - P(E_4)^2P(E_2)P(E_3) + P(E_4)^3P(E_1)P(E_2)P(E_3)$ $\neq P(E_4)P(E_1 \cup E_2 \cup E_3)$ So shouldn't $P(E_4 | \bigcup_{i=1}^3 E_i) = P(E_4)$ be false? Did I do some calculation wrong?? |
| Finite difference method and differentiation Posted: 08 Apr 2022 03:51 AM PDT What are some examples for applying finite difference method with differentiation? |
| Infinite intersection of sets of multiples of naturals Posted: 08 Apr 2022 03:51 AM PDT There is a example in my Set Theory Book that is giving me trouble: $$\cap_{i=10}^{\infty}D_i=\emptyset, D_i=multiples\;of\;i\;in\;\mathbb{N}.$$ For me, every time I do and intersection, there will be aways a set of numbers that will be there, since I can just multiply tem for the intersected number (i). I have shown something like these: $$D_{10}\cap D_{11}\dots D_x=\cap_{i=10}^x=\{z:z=10.11.12(\dots).x.m\;|\;m\in\mathbb{N}\},$$ and then tryed to limit this to $\infty$. But I don't know how to finish it without bias. |
| Analytical solution of the nonlinear 1D Diffusion equation Posted: 08 Apr 2022 03:50 AM PDT Can someone help me please to find the analytical solution of the following nonlinear Diffusion equation : $\frac{\partial u}{\partial t} = \frac{\partial}{\partial x}[q(x)\frac{\partial u}{\partial x}]+f(x,t),$ with initial condition $u(x,0) = sin(x)$, $0\leq x\leq 1$ and Dirichlet boundary conditions $u(0,t) = 0, 0\leq t\leq 1$, $u(1,t) = sin(1)\exp(-t), 0\leq t\leq 1$, where $f(x,t) = cos(x)exp(-t)$ It would be great if you know some free online solver that can gives an analytical solution. Many Thanks you for your time and for your help ! |
| Is a graph without vertices but with edges a graph? Posted: 08 Apr 2022 03:57 AM PDT I am working in a field where the set of zero points of divergence-free 3D vector fields are investigated. These sets are comprised usually from a set of single (isolated) points or connected lines that may or may not branch. In addition closed curves (loops) are appearing. Traditionally these sets of zero points are called "stagnation graph". I am wondering if such sets are really fitting into "conventional graph theory". For example the loops would be edges without vertices. Thus my question is there a term for an edge without vertex in graph theory? Can it be a "graph" at all? According to (standandard) graph theory, a graph is an ordered pair $G=(E,V)$ of edges and vertices where the edges $E$ are defined by (ordered) pairs of vertices. So according to this definition such a vertex-free loop would not qualify as a graph. But I cannot exclude that a generalized definition or more general structure has been described and defined, somewhere. In any case adjacency matrices for example would not exist, as far as I can see. Can anyone comment on this? (This question can be seen as a kind of counter-part of this question.) |
| Which book has term by term differentiation theorem for multivariable functions series? Posted: 08 Apr 2022 03:48 AM PDT I'm looking for a book with a theorem about term by term differentiation for multivariable functions series. |
| Number of all possible paths in a complete graph Posted: 08 Apr 2022 03:58 AM PDT I derive a formula for the total number of all possible paths in a complete graph. As a basis, I took the formula for the number of placements of n elements by m and the formula for the total number of paths between two points in a complete graph. Result: e * n! Is it true? n - number of vertices; e - exponent; Total number paths between two nodes in a complete graph $$A_{2}^{n}=\dfrac{n!}{\left( n-2\right) !}$$ $$N=\left( n-2\right) !e$$ $$Result=A_{2}^{n}\cdot N=n!e$$ |
| Doubt on max tensor product of $C^{\ast}$-algebras Posted: 08 Apr 2022 03:53 AM PDT Im trying to understand proof of corollary $11.34$ from here. The corollary goes as follows:
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| Coherence Upper Bound of Kronecker Product and Transposed Transposed Khatri-Rao Product Posted: 08 Apr 2022 03:59 AM PDT When I am doing some research projects, I am trying to lower the total coherence of a matrix which is defined as $$ \mu^t(\mathbf{Q})=\sum_{m=1}^G\sum_{n=1,n\neq m}^G\left(\mathbf{Q}(m)^\mathsf{H}\mathbf{Q}(n)\right)^2, $$ where $\mathbf{Q}\in\mathbb{C}^{B\times G}$, $\mathbf{Q}(m)$ represents the $G$-th column of matrix $\mathbf{Q}$, and $\mathbf{Q}^\mathsf{H}$ denotes the Hadamard transpose of $\mathbf{Q}$. This definition can also be understood as:
Now let's consider the total coherence of a matrix $\mathbf{A}\otimes\mathbf{B}$ and $\mathbf{A}\bullet\mathbf{B}$, which denote the Kronecker product and the transposed Khatri-Rao product respectively. So does it exist an upper bound for the total coherence? For example maybe is there a conclusion that $$\mu^t(\mathbf{A}\otimes\mathbf{B})\leq\mu^t(\mathbf{A})+\mu^t(\mathbf{B}),$$ or $$\mu^t(\mathbf{A}\bullet\mathbf{B})\leq\max\left\{\mu^t(\mathbf{A}),\mu^t(\mathbf{B})\right\}.$$ If so, please advise me on how to prove it. Thanks in advance. |
| Posted: 08 Apr 2022 03:36 AM PDT $A=${$\emptyset,\Bbb Z,\Bbb Q$}, $B=${{$\emptyset$}$,\Bbb Z,\Bbb Q$}, $C=${$\emptyset$,{$\Bbb Z$},{$\Bbb Q$}}, $D=${$\emptyset$,{$\Bbb Z$,$\Bbb Q$}} I know that if $E=\emptyset$, then $E$ has no element. {$\emptyset$} is non-empty set which has exactly one element, namely object $\emptyset$. But I am a bit confused when to count the '$\emptyset$' as an element. Take the above sets as example, how many elements are there in A? Is it 2 or 3? For set B, are there 3 elements in B? which are {$\emptyset$},$\Bbb Z,\Bbb Q$. Am I right to say that there are exactly 3 elements in $B\cup(C\cap D)$? They are {$\emptyset$}$,\Bbb Z,\Bbb Q$. Does $C\cap D$ contain $\emptyset$ or has no element? Thanks. |
| Posted: 08 Apr 2022 03:33 AM PDT Compute the infinite sum $$\sum_{n=1}^{\infty}\sum_{k=0}^{\infty}\frac{n+k+1}{nk(n+k)^2}$$. I calculated the first part and got $$\sum_{k=1}^{\infty}\frac{n+k+1}{nk(n+k)^2}=\sum_{j=0}^{j=n}\frac{1}{n^2j}+\sum_{j=0}^{j=n}\frac{1}{n^3}{j}+\sum_{j=0}^{j=n}+\frac{1}{n^2j^2}-\frac{1}{n}\sum_{k=0}^{\infty}\frac{1}{k^2}$$. But don't know how to proceed. |
| Are all compact subsets of $\mathbb R^{\mathbb R}$ separable? Posted: 08 Apr 2022 03:56 AM PDT The title is the question. I should perhaps add that $\mathbb R$ has its usual topology and the product has the product topology, i.e., $\mathbb R^{\mathbb R}$ is the space of all functions from $\mathbb R$ to $\mathbb R$ with the topology of pointwise convergence. It is easy to see that this space is separable, e.g., the set of polynomials with rational coefficients is dense. However, in general separable topological spaces, subspaces need not be again separable. I guess that the answer to this question is well-known -- however searching, e.g., Engelking's book for such a special question is rather frustrating. |
| maximum range of projectile launched from elevation Posted: 08 Apr 2022 03:47 AM PDT I am trying to find the maximum range of projectile from an elevation. I found the answer in this question, but I have two questions:
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| Posted: 08 Apr 2022 03:39 AM PDT What is the cardinality of $R^\infty$? Is it the same as $R^n$ at finite $n$. Is this a well-defined question? If we take a real analytic function such as an exponential and describe it by the coefficients in its Taylor series expansion what is the cardinality of this coordinate space? (I would expect it to be the same as that of $R^\infty$.) From Does the set of entire functions have the same cardinality as the reals? (link provided by Gerry Myerson) I understand that at least this definition of $R^\infty$ indeed has the same cardinality as $R$. My confusion came from the fact that $R^\infty$ as mentioned in https://math.stackexchange.com/a/2091553/408562 could also refer to some uncountable power of $R$. I think this can be marked as a duplicate of Does the set of entire functions have the same cardinality as the reals?. |
| Posted: 08 Apr 2022 03:52 AM PDT I am trying to model the heat equation with heat source and Robin boundary conditions, i.e. the system: \begin{align} T_t\;=&\;\alpha\Delta T+\frac{1}{c_p\rho \text{Vol}(\Gamma)}1_{\Gamma}(\boldsymbol{x})w(t). && \boldsymbol{x}\in \Omega, t\in I\nonumber \\[1mm] \nabla T \cdot \overset{\to}{n}_{\boldsymbol{x}}=&-K(\boldsymbol{x})(T(\boldsymbol{x},t)-T_{\text{outside}}(t)) &&\boldsymbol{x} \in \partial \Omega, t\in I \label{eq:boundary_condition}\\[1mm] T(\boldsymbol{x},0)\;=&\;T_0(\boldsymbol{x}) && \boldsymbol{x} \in \Omega \end{align} in $3$ dimensions using the finite difference method. My question is the following: The ovens (heat source) becomes unreasonably hot. Any ideas why? Here are some more details. Our domain $\Omega$ is a rectangular prism representing a room, and the heat source $\Gamma$ is two electrical ovens modeled as two (disjoint) rectangular prisms in the interior of $\Omega$. We have discretized the room into nodes $p_{i,j,k}$, and we imagine that each node sits in the centre of a cube of side lenght $h$. In our equation, $w(t)$ is the oven wattage- a given function, and $1_\Gamma$ is the indicator function of the ovens. Our explicit FTCS-scheme is given from \begin{align*} \frac{T_{i,j,k}^{m+1}-T_{i,j,k}^{m}}{\Delta t}\doteq&\; \alpha \frac{T_{i+1,j,k}^m-2T_{i,j,k}^m+T_{i-1,j,k}^m}{h^2}+\alpha \frac{T_{i,j+1,k}^m-2T_{i,j,k}^m+T_{i,j-1,k}^m}{h^2}\\[1mm] +\;& \alpha \frac{T_{i,j,k+1}^m-2T_{i,j,k}^m+T_{i,j,k-1}^m}{h^2}+\frac{1}{c_p\rho \text{Vol}(\Gamma)}1_{\Gamma}(p_{i,j,k})w(t_m). \end{align*} Example: If we run only a few $(1-20)$ time-steps, the temperature in the oven nodes remain reasonable, but not if we run over a longer period. For example, if the parameters are \begin{align} \text{room dimensions}\;=&\;4m\times 6m\times 2m && \text{time length}=120\text{sec}\\ h=\text{spatial step size}\;=&\;1/15 m && \Delta t=1/100\text{sec}\\ w(t)\;\equiv&\; 750\text{watts} && \alpha=1.9\cdot 10^{-5}m^2/s\\ T_{\text{outside}}(t)\;\equiv&\;253.15\text{Kelvin}=-20\text{Celsius} \end{align} Then the temperature in the oven nodes end up at $1408$Kelvin=$1135$celsius after these two minutes. What I have tried
Unfortunately, I still get the same problem in all cases (also if i change the step sizes, time step size and the other parameters.) Some thoughts Some tests have shown that: The ovens do indeed output the correct amount of energy into the room. The boundary conditions on the room also work as expected. If the initial temperature equals the outside temperature (both constant), and the ovens are turned off, then the temperature in the room remains constant as expected. Our Courant/ Mesh Fourier number $\frac{\alpha \Delta t}{h^2}$ is quite small, so I don't believe that the step-sizes is the problem. (Although I suspected this number was in fact too small. Making it larger did not improve the situation). Two other possible causes may be that the heat is not able to diffuse away from the oven fast enough, or that the problem is somehow caused by the heat-term. As mentioned, I tried making the diffusivity temperature-dependent, but it did not work. Also, it seems unreasonable that this adjustment should be necessary, since the heat is evenly distributed over the oven, since many panel ovens are mostly air anyway, and since some papers argue that realistic modelling can be obtained even with diffusivity assumed constant (e.g. this paper.) It should be mentioned though that the result may be close to what one would expect if no heat leaves the ovens at all. |
| Why using $\tan^{-1}(b/a)$ is not good enough to find the angle of a complex number? Posted: 08 Apr 2022 03:57 AM PDT Let $z\in\mathbb{C}$ then it can be written as $z=a+bi$ where $a,b \in \mathbb{R}$. We know that to find the angle of $z$ we use $\tan^{-1}(b/a)$, but however some people say that this method is not good enough. Why? Can someone explain? |
| What is the Kurtosis in Geometric Brownian Motion Posted: 08 Apr 2022 03:37 AM PDT Suppose that $dS_t=S_t(\mu\mathop{dt}+\sigma\mathop{dW_t})$ which has solution $$S_t=S_0\exp\left(t\left(\mu+\frac{\sigma^2}{2}\right)+\sigma W_t\right),$$ such that $W_t$ is a Wiener process, $\mu$ is drift, and $\sigma$ is volatility. So can \begin{split} K(\delta)&=\text{kurt}\left(\frac{S_{t+\delta}-S_t}{S_t}\right),\\ &=\text{kurt}\left(e^{\frac{\delta \sigma^{2}-2\sigma W_t}{2}+\delta\mu+\sigma W_{t+\delta}}-1\right), \end{split} be simplified for all $\delta$? In simulations, $K$ bounces around $3$ (for small $\delta$ – see comment). Note that $W_{t+s}-W_s$ is independent of $s$. Any help would be much appreciated. |
| Proving $|\alpha^{<\omega}| < \kappa$ without Axiom of Choice Posted: 08 Apr 2022 03:50 AM PDT Let $\kappa$ be a cardinal. Fix an ordinal $\alpha < \kappa$. Let $\alpha^{<\omega}$ denote the set of all finite sequences of ordinals less than $\alpha$. Can we show that $|\alpha^{<\omega}| < \kappa$ without invoking the Axiom of Choice? I know it is, for instance, independent without $\mathsf{AC}$ that $\omega_1$ is a countable union of countable ordinals, but I'm unsure of variants of this statement (such as the one I posed). |
| How to prove that $\frac{x}{1+x^2}$ is not injective? Posted: 08 Apr 2022 03:33 AM PDT By looking at graph of $f(x) = \frac{x}{1+x^2}$ I can clearly see that there are at least 2 points $x_1$, $x_2$ where: $$f(x_1) = f(x_2), \quad x_1 \neq x_2$$ How can I prove this function is not injective? Problem here is that in contrast with function for example $f(x) = x^2$ you can check with negative $x$ to prove non injectivity, but in this case both $x_1$, $x_2$ are either positive or negative.
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| Is the limit in probability of Lévy processes still a Lévy process? Posted: 08 Apr 2022 03:42 AM PDT Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(\mathcal F_t)_{t\ge0}$ bea filtration on $(\Omega,\mathcal A)$ and $(X_t)_{t\ge0}$ be a process on $(\Omega,\mathcal A,\operatorname P)$. Remember that $X$ is called $\mathcal F$-Lévy in law if $X$ is $\mathcal F$-adapted, continuous in probability, $X_0=0$ and \begin{align}X_{s+t}-X_s&\text{ is independent of }\mathcal F_s\tag1\\X_{s+t}-X_s&\text{ has the same distribution as }X_t\tag2\end{align} for all $s,t\ge0$. Now let $(X^n_t)_{t\ge0}$ be a process on $(\Omega,\mathcal A,\operatorname P)$ for $n\in\mathbb N$ with $$X^n_t\xrightarrow{n\to\infty}X_t\;\;\;\text{in probability for all }t\ge0\tag3.$$
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| Bound for the probability of a Lévy process having a jump of a given size Posted: 08 Apr 2022 03:43 AM PDT Let
Let $B\in\mathcal B(E)$ with $0\not\in\overline B$ and $$f(x):=x1_B(x)\;\;\;\text{for }x\in E.$$ Note that $$\pi_tf:=\int f\:{\rm d}\pi_t=\sum_{\substack{s\in[0,\:t]\\\Delta X_s\:\in\:B}}\Delta X_s\;\;\;\text{for all }t\ge0.$$
What if $E=\mathbb R$ and $B=\{-1,1\}$. Couldn't $X$ have exactly two jumps in $[0,1]$, one of size $-1$ and the other of size $1$, so that $\pi_tf=0$? |
| Prove this algorithm for finding the Eulerian path/cycle in a undirected graph Posted: 08 Apr 2022 03:33 AM PDT Take a look at the procedure (source: https://cp-algorithms.com/graph/euler_path.html) procedure FindEulerPath(V):
Note that before running this algorithm, we first check if either all vertices have an even degree or all except two have an even degree (in the latter case we start in any of them). I understand the Hierholzer's algorithm (https://en.wikipedia.org/wiki/Eulerian_path#Hierholzer's_algorithm) but I am not sure about proving this one, though I can sense some connection with the Hierholzer's algorithm, perhaps a similar proof with decomposition into circles could be thought of? |
| Number Of Cycles in a Even Graph Posted: 08 Apr 2022 03:48 AM PDT I was trying to prove following problem for Bondy Murthy Graph Theory Book The problem is :
as we knew :
assuming the above definition, after doing my main question is how can i prove that the number of such lines(=converted cycles by splitting-off method) is odd ? |
| How is a ring a $\mathbb{Z}$-algebra? Posted: 08 Apr 2022 03:48 AM PDT If all of the below is true, does it follow directly from the definitions and the canonical way every abelian group can be considered a $\mathbb{Z}$-module? EDIT: In the "definitions" below, it is necessary to include left and right distributivity; the two distributive axioms do not follow from the other structures. (For commutative rings technically one of the two will always follow from the other so it is only necessary to assume one -- for non-commutative rings one needs to assume both.)
In particular, no ring is a non-associative $\mathbb{Z}$-algebra? |
| Idempotent endomorphisms generate a direct sum decomposition Posted: 08 Apr 2022 04:00 AM PDT Show that there is a one-to-one correspondence between indempotents $e\in\operatorname{End}_R(V)$ and direct sum decompositions $V=X\dotplus Y$. Attempt: We have $e^2 = e$. Therefore, we can write an element $x\in V$ as $x = e(x) + (1-e)(x)$. Write $X = Re$ and $Y = R(1-e)$. Then $V = X + Y$. Note that if $a$ belongs to $X \cap Y$, then $a = e(x) = (1-e)(y)$. So, $e(a) = e^2(x) = e(x)= a$ and $e(a) = e(1-e)(y) = -a$. Therefore $a= 0$. Hence $V$ decomposes into a direct sum. |
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