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Taken from Wikipedia ([here][1]), a scoring rule has the following definition
> Blockquote
Let $\Omega$ be a sample space, and $\mathcal{A}$ is a $\sigma$-algebra of subsets of $\Omega$. Let $\mathcal{P}$ be a **convex class of probability measures** on $(\Omega, \mathcal{A})$. A scoring rule is a function $S: \mat... |
What does "convex class of probability measures" mean in the definition of scoring rules? |
The symmetry of the Markov chain associated with the random walk enables you to calculate the expected return time fairly easily. The chain has $\ 32\ $ states $\ h_1,h_2,\dots,h_{20},p_1,p_2,\dots,p_{12}\ ,$ where $\ h_i\ $ are the hexagonal faces and $\ p_i\ $ the pentagonal faces. If $\ \pi\ $ is the stationary di... |
Informally, a category internal to a category $\mathbb{E}$ is a category where the collection of morphisms and the collection of objects are objects in $\mathbb{E}$, and the operations "assigning the domain", "assigning the codomain", "picking out the identity morphism", and composition are all morphisms in $\mathbb{E}... |
Given your colleague has **two kids**, and **hiddenly** adopting binary biological sex (each kid is either a boy (B) or a girl (G)), the **sample space** is
$$\Omega=\{(G,G), (G,B), (B,G), (B,B) \}$$
with the following probability measure:
$$\mathbb P ((G,G))=p^2, \mathbb P ((G,B))=\mathbb P ((G,B))=p(1-p), \m... |
How to formally define this square matrix? |
As a partial answer to your second question: in 1992 Ales Drápal proved that if two finite groups agree on 89% on their multiplication tables, the groups must be isomorphic! He conjectured that the same holds true if the tables agree on 75% of their entries. I do not think the conjecture has been proved yet. See also [... |
It is required to minimize the quadratic function
$f(\mathbf{x}) = \mathbf{x}^T Q \mathbf{x} + b^T \mathbf{x} + c $
Subject to the quadratic constraint
$ g(\mathbf{x}) = \mathbf{x}^T C \mathbf{x} + d^T \mathbf{x} + e = 0$
where $\mathbf{x} \in \mathbb{R}^n$, and it is assumed that matrix $C$ is positive de... |
Here are two related theorems. I have trouble proving the second one.
>**Monotonicity theorem**: Let $f$ be a real valued function defined on $\mathbb R$. The function is continuous at a point $a$ and $f$ is monotonically increasing on $(a-r,a]$ and monotonically decreasing on $[a,a+r)$. Then, $f$ has a local maximu... |
Given some vector $x = (x_1,...,x_d)$ in $\mathbb{R}^d$ we may embed it in the tensor algebra $T(\mathbb{R}):= \prod_{n=0}^\infty (\mathbb{R}^d)^{\otimes n}$ via the tensor exponential map $\exp_\otimes$. Specifically, we have that
$$\exp_\otimes(x) := \sum_{n=0}^\infty \frac{x^{\otimes n}}{n!}:= \sum_{n=0}^\infty \f... |
In Exercise 10.4 (p.243) in Commutative Algebra with a View Toward Algebraic Geometry by David Eisenbud, it writes
Let $a,b$ be a regular sequence in a domain $R$, and let $S=R[x]$ be the polynomial ring in one variable over $R$.
Show that $ax-b$ is a prime ideal of S (See Exercise 17.2 for a sort of converse. Geo... |
I have the following integral.
$$ G(q) = \int_{0}^{\infty} y^\alpha \exp(-y) \exp(-j b q\exp(-c y)) dy $$
with
$$ \alpha \in \mathbb{R}^+, \quad q, b, c \in \mathbb{R}, \quad j = \sqrt{-1}$$
I have asked this before in different forms, but I have settled down with the following approximation using Gauss-Lag... |
Suppose that we have an arithmetic function $f(x)$ defined as follows:
[![A custom arithmetic function][1]][1]
What are the methods in the literature that will make this function continuous and differentiable? However, it should be noted that this function can be custom. I want a function $g(x)$ such tha... |
Answers:
1. This seems OK.
2. You are relying only on compactness (Bolzano--Weierstrass), which by the way is the property you routinely use in analysis, more than the least upper bound.
3. I think your method is creative, but overtly complicated. More down-to-earth, you could use the [bisection method](https:/... |
I need to solve the following equation to obtain an expression for the probability, an approximation would be accceptable if no exact solution is possible:
$$\text{Prob}\left(y_1\leq y_2\right)=\text{Prob}\left(\frac{a}{x_1}-\frac{a}{x_2}<\frac{1-2\text{Prob}\left(y_1\leq y_2\right)}{\frac{\partial \text{Prob}\left(... |
Let $A\subset \mathbb{P}^2$ be the set defined by the equation $X^3=Y^2Z$. I have the map $f:\mathbb{P}^1\rightarrow V$, $[S,T]\rightarrow [S^2T, S^3, T^3]$. I want to find another map $g:V\rightarrow \mathbb{P}^1$ such that $f\circ g$ and $g\circ f$ are the identity wherever they are defined. My candidate function is ... |
Title: Finding integer solutions for a set of equations involving powers and logarithms
Body:
I am trying to find integer solutions for a set of equations and would appreciate any help or insights on methods to determine if solutions exist for certain cases or generally. The equations are as follows:
1. For th... |
I am working with $ln(1 + e^{b_0 + b_1x})$ centered at $x = a$ such that $a \neq 0$ where $b_0$ and $b_1$ are fixed. My background is in statistics and I have some knowledge about these but not very deep. My questions are:
1. **Convergence:** Is the Taylor series for this function convergent? That is, does it conver... |
I am trying to find integer solutions for a set of equations and would appreciate any help or insights on methods to determine if solutions exist for certain cases or generally. The equations are as follows:
1. For the case where $(3^n - 1)$ is odd:
$$
x = \frac{\left(\frac{3^n - 1}{2}\right) - 2^y}{2 \times \le... |
I am trying to find integer solutions for a set of equations and would appreciate any help or insights on methods to determine if solutions exist for certain cases or generally. The equations are as follows:
1. For the case where $\frac{3^n - 1}{2}$ is odd:
$$
x = \frac{\left(\frac{3^n - 1}{2}\right) - 2^y}{2 \t... |
I find this proof from Munkres Topology lacking rigour and a lot of precision.
**can someone write down the equations that have been mentioned in the proof and probably provide some feeling or intuition for them?**
**does the intersection point of the figure eight of $T\#T$ lie on the dotted circle, because if ... |
Given an object $X$, we can define an equivalence relation on the monomorphisms with range $X$: $u:S\to X,v:T\to X$ are equivalent iff exists an isomorphism $\phi:S\to T$ such that $u=v\circ \phi$. By the definition in wikipedia, the equivalence classes of monomorphisms with range $X$ are the subobjects of $X$.
In t... |
I am working with $\ln \left( 1 + e^{b_0 + b_1x} \right)$ centered at $x = a$ such that $a \neq 0$ where $b_0$ and $b_1$ are fixed. My background is in statistics and I have some knowledge about these but not very deep. My questions are:
1. **Convergence:** Is the Taylor series for this function convergent? That is,... |
I am trying to find integer solutions for a set of equations and would appreciate any help or insights on methods to determine if solutions exist for certain cases or generally. The equations are as follows:
1. For the case where $\frac{3^n - 1}{2}$ is odd:
$$
x = \frac{\left(\frac{3^n - 1}{2}\right) - 2^y}{2 \t... |
I am reading a paper by Berezin, entitled *General Concept of Quantization*. He writes:
[![enter image description here][1]][1]
and then:
[![enter image description here][2]][2]
This ought to be some clever manipulation of matrices and yet I am not able to show how (1.4) follows from (1.1) if $\omega$ is gi... |
Let $X$ be a metric space and $(f_n)$ be a sequence of continuous functions with $f_n: X \rightarrow \mathbb{R}$. I know that, in general, $\sup_{x \in X}\lim_{n\in \mathbb{N}}f_n(x)\ne \lim_{n\in \mathbb{N}} \sup_{x \in X}f_n(x)$, though equality holds when [$(f_n)$ converges uniformly to a function $f$](https://math... |
I need to solve the following equation to obtain an expression for the probability, an approximation would be accceptable if no exact solution is possible:
$$\text{Prob}\left(y_1\leq y_2\right)=\text{Prob}\left(\frac{a}{x_1}-\frac{a}{x_2}<\frac{1-2\text{Prob}\left(y_1\leq y_2\right)}{\frac{\partial \text{Prob}\left(... |
>Doesn't it just show that $g(n)=100n^3$ is bounded by $C∗n^3$?
Well that is what it means to be $O(n^3)$ by definition. Recall that $g$ is $O(f)$ if there exists a positive constant $C$ and some $x_0$ such that $|g(x)|\leq C\cdot f(x)$ for all $x\geq x_0$. In other words, $g$ is $O(f)$ if it is *eventually* dominat... |
This is not an answer... Just to give a simple example why it is not always solvable. Imagine a function $\mathbb R^2 \to \mathbb R$.
Let us prescribe any non-closed (importante!) contour of gradient orthogonal to the direction of the contour with a prescribed $\nabla g = \|\nabla g\|\hat n$. Now prescribe every ot... |
This is a classic exercise of functional analysis, but I do not fully understand it after reading many answers in textbooks. So I am trying to reorganize the proof step by step in details. I am hoping that someone may review my proof very carefully and give comments or corrections. Then I will revise the proof and hope... |
Let $\hat{\mathbf a}$ be the unit vector corresponding to the vector $\mathbf a$. We have that $$\mathbf a \cdot \mathbf b = ||\mathbf a || \hat{ \mathbf a }\cdot ||\mathbf b || \hat{\mathbf b} = ||\mathbf a || \ ||\mathbf b || \ \hat{\mathbf a} \cdot \hat{\mathbf b} $$
Now let $A(\mathbf a,\mathbf b) = \hat{\mathb... |
Let $X$ be a metric space and $(f_n)$ be a sequence of continuous functions with $f_n: X \rightarrow \mathbb{R}$. I know that, in general, $\sup_{x \in X}\lim_{n\in \mathbb{N}}f_n(x)\ne \lim_{n\in \mathbb{N}} \sup_{x \in X}f_n(x)$, though equality holds when [$(f_n)$ converges uniformly to a function $f$](https://math... |
Let $ \mathbb{P} = \left \{ \bullet , \blacksquare \right \}$ be a set where elements $ \bullet \neq \blacksquare$. For operation of addition in $\mathbb{P}$ we define:
$$\bullet + \bullet = \blacksquare$$
$$\blacksquare + \blacksquare = \blacksquare$$
$$\bullet + \blacksquare= \bullet$$
$$\blacksquare + \bulle... |
As you asked, I will convert my comments in an answer.
I guess you can prove the theorem showing directly that $\beta$ is bijective but what's the point of that? I mean, you have defined tensor product using an extremely powerful universal property, why you don't want to use it? If you think about the contruction of... |
The **least upper bound** property of the real numbers (lub) can be used to prove the **intermediate value theorem (IVT)** as well as the **Bolzano-Weierstrass** theorem (BWT).
I've been trying to prove the IVT without the lub property, using the BWT as an axiom. The sketch of my proof goes as follows:
**Lemma**:... |
No doubt the literature varies, but often the term *factorization* is used the situation in which $G = HK$ for proper subgroups $H, K \le G$. For the stronger situation you describe in which moreover $H \cap K = \{1\}$, the term *exact factorization* gets used. See for example the following paper.
<cite authors="Li... |
Playing with Desmos calculator and Stirling approximation I define :
>>$$x>0,f(x)=\prod_{n=1}^{M}\left(2-\frac{\Gamma(\frac{x}{n}+1)}{\sqrt{2\pi×\frac{x}{n}}\left(\frac{x}{e×n}\right)^{\frac{x}{n}}}\right)$$
If you plot it properly over $(1/2,5/4)$ say $n=30$ some patterns appears :
The extrema and roots are... |
> What is the coefficient of $\frac{y^3}{x^8}$ in $(x+y)^{-5}$? when $|\frac{y}{x}|<1$ *(EAMCET 2020)*
Using $n=-5$ and $r=3$ in the formula for the generalized binomial theorem gives the correct answer (which agrees with the answer key):
$${5+3-1 \choose 3}(-1)^3=-{7 \choose 3}=-35$$
I did not use the conditio... |
I want to use the $\epsilon-\delta$-criterion to show that for any $c\in\mathbb R$ the function
$$
f: \mathbb R \rightarrow \mathbb R, x\mapsto
\begin{cases}
\frac{1}{x}, &x\neq 0 \\
c, & x=0.
\end{cases}
$$
is not continuous.
Let $x = 0$.
Then we have
$$
\mid f(x) - f(0)\mid
=
\mid\frac{1}{x} - c\mi... |
I am using Kalman filter to estimate the states of a stochastic dynamical system which has very very small noise( consider zero ). The filter is not aware that the noise is zero. Implementation of KF to estimate the states of this dynamical system gives a value, say $\hat{\mathbf{X}}_1\left(t\right)$, at any given time... |
$f$ and $g$ are bounded functions with common domain$D$
$\sup\limits_{x\in D}\big\{f(x)+g(x)\big\}\leqslant\sup\limits_{x\in D}f(x)+\sup\limits_{x\in D}g(x).$
Let both $S $and $T$ be non-empty subsets of $R$
$S+T=\begin{Bmatrix}z & \mid z=s+t,s\in S,t\in T\end{Bmatrix}.$
$\sup(S+T)=\sup S+\sup T$
What I want... |
Consider a sequence of random variables $(p_n)$ such that:
(1) $\Pr\bigg(\limsup_{n \to \infty} p_n\leq U\bigg)=1$
(2) $\Pr\bigg(\liminf_{n \to \infty} p_n\geq L\bigg)=1$
where $L,U$ are real numbers. Does these assumptions imply
$$
\Pr\bigg(\lim_{n \to \infty}d\big(p_n, \big[L,U\big]\big)= 0\bigg)=1,
$$
... |
An easier way would to be argue by sequential definition of continuity. If we can find a sequence ${x_n}\rightarrow 0$ but $f(x_n)\nrightarrow f(0)=c$ then we're done. But if you pick the sequence $x_n=\frac{1}{n}$ , that converges to zero , but $f(x_n)\rightarrow +\infty$ . Therefore, not continuous at x=0, we're done... |
An easier way would to be argue by sequential definition of continuity. If we can find a sequence ${x_n}\rightarrow 0$ but $f(x_n)\nrightarrow f(0)=c$ then we're done. But if you pick the sequence $x_n=\frac{1}{n}$ , that converges to zero , but $f(x_n)\rightarrow +\infty$ . Therefore, not continuous at x=0, we're done... |
To prove that an ideal $I$ of a ring $R = R_1 \oplus R_2 \oplus \dots \oplus R_n$ is equal to the direct sum of the intersections of $I$ with each $R_i$, we'll proceed step by step.
**Step 1: Define the Direct Sum**
The ring $R$ is the direct sum $R_1 \oplus R_2 \oplus \dots \oplus R_n$, meaning each elemen... |
An easier way would to be argue by sequential definition of continuity. If we can find a sequence ${x_n}\rightarrow 0$ but $f(x_n)\nrightarrow f(0)=c$ then we're done. But if you pick the sequence $x_n=\frac{1}{n}$ , that converges to zero , but $f(x_n)\rightarrow +\infty$ . Therefore, not continuous at $x=0$, we're do... |
Denote $[n] := \{1,2,...,n\}$. Denote $U$ as variable for open set
Case 0: $\bigcup U_i = \bigcup \varnothing = \varnothing$, hence open. so for nonempty unions:
Case 1: $\bigcup U_i$ contains $U_j = \mathbb{N}$, then union is $\mathbb{N}$, hence open.
Case 2: $\bigcup U_i$ does not contains $U_j = \mathbb{N}$... |
An easier way would to be argue by sequential definition of continuity. If we can find a sequence ${x_n}\to 0$ but $f(x_n)\not\to f(0)=c$ then we're done. But if you pick the sequence $x_n=\frac{1}{n}$ , that converges to zero , but $f(x_n)\rightarrow +\infty$ . Therefore, not continuous at $x=0$, we're done because we... |
I understand the definition of an [internal category](https://ncatlab.org/nlab/show/internal+category), but so far I haven't found examples of internal categories (other than categories internal to $\mathrm{Set}$, which are just the small categories) that one could encounter "naturally" in mathematics.
I have only fou... |
I'm a novice learning algebraic geometry. Although schemes are quite abstract, it's easy to see their relationship with algebraic varieties. However, when reading through derived functor cohomology, if not for Leray's theorem, it would seem miraculous that the derived functor approach indeed yields cohomology. May I as... |
I want to use the $\epsilon-\delta$-criterion to show that for any $c\in\mathbb R$ the function
$$
f: \mathbb R \rightarrow \mathbb R, x\mapsto
\begin{cases}
\frac{1}{x}, &x\neq 0 \\
c, & x=0.
\end{cases}
$$
is not continuous.
Let $x = 0$.
Then we have
$$
\left\vert f(x) - f(0)\right\vert
=
\left\ver... |
I want to use the $\epsilon-\delta$-criterion to show that for any $c\in\mathbb R$ the function
$$
f: \mathbb R \rightarrow \mathbb R, x\mapsto
\begin{cases}
\frac{1}{x}, &x\neq 0 \\
c, & x=0.
\end{cases}
$$
is not continuous.
Let $x = 0$.
Then we have
$$
\mid f(x) - f(0)\mid
=
\mid\frac{1}{x} - c\mi... |
I understand the definition of an [internal category](https://ncatlab.org/nlab/show/internal+category), but so far I haven't found examples of internal categories (other than categories internal to $\mathrm{Set}$, which are just the small categories) that one could encounter "naturally" in mathematics.
I have only fou... |
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function on all its domain. The following results hold for any $a, b \in \mathbb{R}^{+}$:
1. $a=b \Leftrightarrow f(a)=f(b)$.
2. $a>b \Leftrightarrow f(a)<f(b)$.
3. $a<b \Leftrightarrow f(a)>f(b)$.
The three results above clearly mean $f$ is strictly... |
I am trying to prove the Lemma(3.4.2)(1) of book "Testing statistical hypotheses" by Lehmann and Romano, its statement is " If $\{p_{\theta}(x)\}$ be a family of densities on the real line with monotone likelihood ratio in $x$ and if $\phi(x)$ is a non-decreasing function of $x$, then $E_{\theta}(\phi(X))$ is a non-dec... |
**long comment**
Is there any relation between $\omega$ and the metric tensor?
I am asking because if $\omega \equiv g$ then starting from $$\frac{\partial \omega_{ij}}{\partial x^k}
+\frac{\partial \omega_{ki}}{\partial x^j}+\frac{\partial \omega_{jk}}{\partial x^i}=0$$
after multiplying with $\omega^{\alpha i}... |
I'm starting R.P Burn "Number and Functions, Steps into Analysis". In Chapter 2, the basic properties of inequalities are derived, and question 8 asks to prove *using the fact that positive numbers are closed under addition* that $a<b \Rightarrow a+c<b+c$.
The solution presented in the book is:
$a<b \Rightarrow 0<... |
I've been calculating the characteristic function of a random variable with distribution $\frac{2}{3}E(\frac{1}{3}) + \frac{1}{3}N(6, 3)$.
\begin{align}
\phi(t) &= \int_{-\infty}^{0} e^{-itx}\frac{1}{3}\frac{1}{\sqrt{3}\sqrt{2\pi}}\exp\bigg({-\frac{1}{2}\frac{(x - 6)^2}{3}}\bigg) \mathrm{d}x +
\int_0^{\infty} e^{... |
On $A$ we have $\phi (x) \le a$ and $p_{\theta'}<p_{\theta}$. So, $p_{\theta'}-p_{\theta}<0$.
$c\le d$ and $t<0$ implies $ct \ge dt$ right? |
Assume that $\mathbb{R}$ is defined axiomatically as a complete ordered field, and let us define the set of natural numbers $\mathbb{N}$ as the smallest inductive set of $\mathbb{R}$. In other words, $$\mathbb{N}=\{1,1+1,1+1+1,\ldots\} .$$
This is the standard quick (top-down) approach in Analysis (like in Baby Rudi... |
A **Mal'cev** algebra is an algebra $\mathbf{A}$ with a ternary term $t$ such that $\mathbf{A} \models t(x,x,y) \equiv y, t(x,y,y) \equiv x$.
For the remaining of the question, refer to the definitions and results contained in Hobby and McKenzie's book *The Structure of Finite Algebras*.
I'm trying to prove tha... |
Assume that $\mathbb{R}$ is defined axiomatically as a complete ordered field, and let us define the set of natural numbers $\mathbb{N}$ as the smallest inductive set of $\mathbb{R}$. In other words, $$\mathbb{N}=\{1,1+1,1+1+1,\ldots\} .$$
This is the standard quick (top-down) approach in Analysis (like in Baby Rudi... |
Why can I not use the 'Law of Total Probability' to solve this differential equation? |
**Let f and g be functions such that for all real numbers x and y,
(( + )) = () + ( + )() . The value of (0) + (1)+. . . (2024) is?**
My attempt:
When x=0 and y=0 then, g(f(0))=f(0)..........(i).
When x=0 and y=1 then, g(f(1))=f(0)+g(1)
or, g(f(1)) =g(f(0))+g(1) [because f(0)=g(f(0)) from equation (i)]
... |
I'm trying to drill down to the core branches of mathematics and it seems to me that *Algebra*, *Analysis* and *Geometry* are the fundamental branches of mathematics––every other 'branch' (topology, number theory, combinatorics, probability, statistics etc.) rest on these three in one shape or form. For example, vector... |
The **least upper bound** property of the real numbers (lub) can be used to prove the **intermediate value theorem (IVT)** as well as the **Bolzano-Weierstrass** theorem (BWT).
I've been trying to prove the IVT without the lub property, using the BWT as an axiom. The sketch of my proof goes as follows:
**Lemma**:... |
I would like to show that $$A = x^T G(x) \tanh(x) \geq 0$$ (with equality only for $x=0$), where $x\in\mathbb{R}^n$ is any vector and $G(x)\in\mathbb{R}^{n\times n}$ is positive definite.
Or come up with a counter-example if it does not hold.
I tried transforming to the associated quadratic forms $$A = \frac{1}{4}... |
More detailed hints than in the comments.
Why there is no Hamiltonian path in this graph.
For convenience, let the vertices of degree 2 be red. Let the other vertices be blue.
First, note that the red vertices in each path cannot be neighbors, since they form an independent set.
It follows that if a Hamilto... |
In the “linear transformation” point of view, matrices represent linear transformations, by denoting where the unit vector in the $x$ direction goes in the first column and where the unit vector in the $y$ direction goes in the second. Of course for $3$ dimensions there are more basis vectors but the same applies.
T... |
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function on all its domain. The following results hold for any $a, b \in \mathbb{R}^{+}$:
1. $a=b \Leftrightarrow f(a)=f(b)$.
2. $a>b \Leftrightarrow f(a)<f(b)$.
3. $a<b \Leftrightarrow f(a)>f(b)$.
The three results above clearly mean $f$ is strictly... |
Let $V$ be a $\mathbb{C}-$ vector space, $J$ an almost complex structure on $V$ and take a real orthonormal basis $\langle v_1,Jv_1,\ldots,v_n,Jv_n\rangle $ with a scalar product $\langle,\rangle = \sum_{i\leq m}v^*_i\otimes v^*_i + (Jv_i)^*\otimes (Jv_i)^*$, where $v^*_i$ are the dual.
We can define the fundamental... |
A beetle is moving across a cartesian coordinate . Initially it is at origin. In each move it chooses one of the four adjacent lattice points and goes to it . After $n$ moves what's the probability that it is at origin again ?
My Attempt : We know the total sample space is $4^{n}$, but can't find the numerator of th... |
In the context of posterior and prior probabilities, one has $P(\alpha|\text{data})\propto P(\text {data} | \alpha)\cdot P(\alpha)$. What confuses me here is that probability is defined for events, and $\alpha$ is not an event, it is a parameter. But then how to interpret $P(\alpha)$ and $P(\alpha|\text{data})$ if $\a... |
Let $V$ be a $\mathbb{C}-$ vector space, $J$ an almost complex structure on $V$ and take a real orthonormal basis $\langle v_1,Jv_1,\ldots,v_n,Jv_n\rangle $ with a scalar product $\langle,\rangle = \sum_{i\leq m}v^*_i\otimes v^*_i + (Jv_i)^*\otimes (Jv_i)^*$, where $v^*_i$ are the dual.
We can define the fundamental... |
Let $f$ and $g$ be functions such that for all real numbers $x$ and $y$, $g(f(x + y)) = f(x) + (x+y)g(y)$.
The value of $(0) + (1)+. . . (2024)$ is?
My attempt:
When x=0 and y=0 then, g(f(0))=f(0)..........(i).
When x=0 and y=1 then, g(f(1))=f(0)+g(1)
or, g(f(1)) =g(f(0))+g(1) [because f(0)=g(f(0)) from ... |
In order to show something, we need that something clearly defined in our hands. We need to show that an "expression converges". Well, it is a sequence. Then which are its terms? So as the dots are placed in the expression
"$\displaystyle \sqrt[3]{5\sqrt[3]{3\sqrt[3]{5\sqrt[3]{3 \cdots}}}}$"
it looks like we are sta... |
I am trying to understand $\mathbb{C}P^2$. Since I understand the Hopf fibration quite well, I like the following construction:
- Attach a point $\mathbb{D}^0$ (0-cell) to $\mathbb{D}^2$ (2-cell) to get $S^2$.
- Now attach $\mathbb{D}^4$ to $S^2$ by gluing the boundary $\partial \mathbb{D}^4=S^3$ to $S^2$ using th... |
I'm studying modal logic and in Chapter 3 of [this book][1], one of the proposed exercises (3.4) is to show that distribution of necessity operator over implication is valid, that is,
\begin{equation}
\square(\varphi \rightarrow \psi) \; \rightarrow \; (\square \varphi \rightarrow \square \psi) \;\; \; \; \;\text{ax... |
In *Category Theory in Context*, Riehl defines creating limits as follows, for a functor $F: \mathbb{C} \to \mathbb{D}$ and diagram $D: I \to \mathbb{C}$:
> $F$ **creates** limits if whenever $FD: I \to \mathbb{D}$ has a limit in $\mathbb{D}$, there is some limit cone over $FD$ that can be lifted to a limit cone ove... |
I am trying to understand $\mathbb{C}P^2$. Since I understand the Hopf fibration quite well, I like the following construction:
- Attach a point $\mathbb{D}^0$ (0-cell) to $\mathbb{D}^2$ (2-cell) to get $S^2$.
- Now attach $\mathbb{D}^4$ to $S^2$ by gluing the boundary $\partial \mathbb{D}^4=S^3$ to $S^2$ using th... |
The outer product between a bra-ket $|a\rangle\langle a|$ where if $|a\rangle\in\mathcal{H}$ and $\langle a|\in\mathcal{H}_{dual}$ is a vector in the tensor vector space formed by the Hilbert space and its dual. Then it can be related to a linear operator due to the isomorphism
with the space of all endomorphism of t... |
> What is the coefficient of $\frac{y^3}{x^8}$ in $(x+y)^{-5}$? when $|\frac{y}{x}|<1$ *(EAMCET 2020)*
Using $n=-5$ and $r=3$ in the formula for the generalized binomial theorem gives:
$${5+3-1 \choose 3}(-1)^3=-{7 \choose 3}=-35$$
I did not use the condition $|\frac{y}{x}|<1$ anywhere. Was it required as an as... |
A beetle is moving across a cartesian coordinate . Initially it is at origin. In each move it chooses one of the four adjacent lattice points and goes to it . After $n$ moves what's the probability that it is at origin again ?
My Attempt : We know the total sample space is $4^{n}$, but can't find the numerator of th... |
Suppose $\lambda_1, \ldots, \lambda_n \geq 0$ and set $\Lambda$ as a diagonal matrix with these eigenvalues. Consider $f(U) = \|\operatorname{diag}(U\Lambda U^T)\|_2$. Is there an intuition for how to choose an orthogonal $U$ so that $f(U)$ is minimized? To maximize $f(U)$, I think I should just set $U = I$.
For the... |
Taken from Wikipedia ([here][1]), a scoring rule has the following definition
> Let $\Omega$ be a sample space, and $\mathcal{A}$ is a $\sigma$-algebra of subsets of $\Omega$. Let $\mathcal{P}$ be a **convex class of probability measures** on $(\Omega, \mathcal{A})$. A scoring rule is a function $S: \mathcal{P} \tim... |
Mathematica gives the following asymptotic expansion:
$$
\int_0^\Lambda \frac{1-\cos(qx)}{q} \mathrm{d}q \overset{\Lambda\to\infty}{\sim}\gamma+\log\left(\frac{x}{\Lambda}\right) +\mathcal{O}\left(\frac{1}{x\Lambda}\right)
$$
where $\gamma\approx 0.577$ is the Euler-Mascheroni constant. How can this expansion be o... |
> What is the coefficient of $\frac{y^3}{x^8}$ in $(x+y)^{-5}$? when $|\frac{y}{x}|<1$ *(EAMCET 2020)*
I am using the [generalized binomial theorem][1]
$$(x+y)^r=\sum_{k=0}^{\infty}{r \choose k} x^{r-k}y^k$$
Substituting $n=-5$ and $r=3$ in the above gives:
$${5+3-1 \choose 3}(-1)^3=-{7 \choose 3}=-35$$
I d... |
My question is motivated by this question:
https://math.stackexchange.com/questions/78445/does-law-of-large-numbers-converge-in-l1
that asks about the the convergence in $L^1$-norm of the sample mean $\bar{X_n}$ to the population mean $\mu.$
I looked at the answers and comments and I understood that the answer... |
Bound on expected norm of the difference between the sample mean $\bar{X_n}$ and population mean $\mu$ as a function of the sample size $n$ for LLN? |
> What is the coefficient of $\frac{y^3}{x^8}$ in $(x+y)^{-5}$? when $|\frac{y}{x}|<1$ *(EAMCET 2020)*
I am using the [generalized binomial theorem][1]
$$(x+y)^r=\sum_{k=0}^{\infty}{r \choose k} x^{r-k}y^k$$
Substituting $n=-5$ and $r=3$ in the above gives:
$${5+3-1 \choose 3}(-1)^3=-{7 \choose 3}=-35$$
I d... |
im asked to find the number of combinations for distributing 20 grades (between 0 and 100) with a difference of at least 4 between every 2 grades. \\\
the approach i tried was that if i choose the first number from 0 to 5 (they can increase by 4 or 5 for 19 times) theres: $6\cdot\mathrm{2}^{19}$ combinations. \\\
and... |
### Question:
When $f \in C^2[a;b]$ we define $ ||f||_2=\sqrt{\int_{[a;b]}|f|^{2}}$ and when $f$ is bounded on $[a;b]$ we define $||f||_{\infty} = Sup_{x \in [a;b]} |f(x)| $.
Is it possible that it exists a $C>0$ that verify for all continuous function on $[a;b]$; $||f||_{\infty} \leq C ||f||_2 $ ?
###... |
Is it possible that it exists a $C>0$ that verify for all continuous function on $[a;b]$; $||f||_{\infty} \leq C ||f||_2 $? |
In the single laser distance measurement, assuming I have a perfect sensor, I have a single point of object hit and multiple points of miss. Let's assume I can only hit some circles of radius r=5 in the 2d space. [So I've hit some circle shooting from point (0,0) along the x-axis.][1]
After a shot, I have 1 point of... |
How to combine evidence of existence vs non-existence in distance measurement? |
I am trying to understand $\mathbb{C}P^2$. Since I understand the Hopf fibration quite well, I like the following construction:
- Attach a point $\mathbb{D}^0$ (0-cell) to $\mathbb{D}^2$ (2-cell) to get $S^2$.
- Now attach $\mathbb{D}^4$ to $S^2$ by gluing the boundary $\partial \mathbb{D}^4=S^3$ to $S^2$ using th... |
Playing with Desmos calculator and Stirling approximation I define :
>>$$x>0,f(x)=\prod_{n=1}^{M}\left(2-\frac{\Gamma(\frac{x}{n}+1)}{\sqrt{2\pi×\frac{x}{n}}\left(\frac{x}{e×n}\right)^{\frac{x}{n}}}\right)$$
If you plot it properly over $(1/2,5/4)$ say $M=30$ some patterns appears :
The extrema and roots are... |
My question is motivated by this question:
https://math.stackexchange.com/questions/78445/does-law-of-large-numbers-converge-in-l1
that asks about the the convergence in $L^1$-norm of the sample mean $\bar{X_n}$ to the population mean $\mu.$
I looked at the answers and comments and I understood that the answer... |
For quite some time I, as a beginner in Category Theory, have been pondering about the intuition behind defining adjoint functors. I've gone through the other answers addressing this, but they seemed to be too technical for me. Is it right to think of adjoint functors to kind of 'translate a difficult problem to a simp... |
I am trying to find integer solutions for a set of equations and would appreciate any help or insights on methods to determine if solutions exist for certain cases or generally. The equations are as follows:
1. For the case where $\frac{3^n - 1}{2}$ is odd:
$$
x = \frac{\left(\frac{3^n - 1}{2}\right) - 2^y}{2 \t... |
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