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More Abstracted Algebra in Rust

Posted on July 28, 2015 by Tommy McGuire
Labels: rust, toy problems, math

Or, Does 'Group' distribute over 'Ring'?

As I mentioned previously, I have been reading Alexander Stepanov and Daniel Rose's From Mathematics to Generic Programming. It's actually quite interesting, if you kind of fuzz over the math to get to the programming parts. It's not that I'm against math, heavens no, but I have seen many other, and some better, presentations of abstract algebra (I recommend A Book of Abstract Algebra by Charles Pinter.) and of the history of algebra and number theory.

The best parts of From Mathematics to Generic Programming are about programming, specifically about generic programming using examples from algebra. In fact, I do wish Stepanov and Rose had spent more time on the history and theory behind the STL rather than Galois. I would recommend this book to anyone getting into more advanced C++ or perhaps Java and Ada, or anyone getting into Rust. The generics system is built tightly into Rust and an important part of the language environment.

The topic of this post is solutions to their Exercises 8.7 and 8.8, from the section Matrix Multiplication and Semirings. If you followed the last post on this stuff, you'll already know about generalizing addition, multiplication, and matrix multiplication operations to create a generic power function: a recurrence operation that applies a unary operation to an initial value, then to the resulting value, and so on. In this section of the book, Stepanov and Rose take the matrix multiplication operation from there and generalize it to other underlying operations to compute the transitive closure of a social network graph and the distance between the shortest paths in a network.

To begin with, I have to back off a bit. (Why? Why do I always do this? It's probably a long story.)

Generic algebra

This is sort of like math, so let's start with some basic definitions:

TermDefinitionRust
MagmaAn algebraic structure consisting of a set of values and an operation closed over that set.
pub trait Magma {    type Elt: Clone;    fn op(Self::Elt, Self::Elt) -> Self::Elt;}
Associativity(x op y) op z == x op (y op z)
pub trait Associative { }
SemigroupA magma whose operation is associative.
pub trait Semigroup: Magma + Associative { }
MonoidA semigroup equipped with an identity element such that x op id == id op x == x.
pub trait Monoid: Semigroup { fn identity() -> Self::Elt; }
GroupA monoid that comes with an inverse operation such that x op x.inverse() == x.inverse() op x == id.
pub trait Group: Monoid {  fn inverse(Self::Elt) -> Self::Elt;}
CommutativityThe arguments to the operation go to work on weekdays, such that x op y == y op x.
pub trait Commutative: Magma { }
Abelian GroupA group whose operation is commutative. (Who names this stuff? It's ridiculous. Oh, Niels Henrick Abel. Nice.)
pub trait AbelianGroup: Group + Commutative { }

Now, if you look back at the previous post, where I had simpler definitions for semigroup and implementations for the primitive Rust types along with the definition of the power operation that used them, you'll see that none of this is actually anywhere near as complicated than it looks. Most of the traits are simply raw, type-level assertions; markers to indicate that addition is associative, for example. (It would be nice to be able to ensure that any operation declared as associative actually satisfied the definition, but I don't think I can do that without dependent types and that would be as complicated as it looks. Or more.

For example, here's the addition and multiplication operations defined for all of the primitive types, roughly the same as before:

pub struct Multiplication<Elt>(PhantomData<Elt>);pub struct Addition<Elt>(PhantomData<Elt>);macro_rules! add_mult_operations_for {    ( $($t:ty ),* ) => {        $( impl Magma for Addition<$t> {                type Elt = $t; fn op(x:$t, y: $t) ->$t { x + y }            }            impl Magma for Multiplication<$t> { type Elt =$t;                fn op(x: $t, y:$t) -> $t { x * y } } )* }}add_mult_operations_for!(usize,u8,u16,u32,u64,i8,i16,i32,i64,f32,f64); And, as for what you can do with them, here is the Egyptian-multiplication-algorithm-based implementation of the power operation: fn power_accumulate_semigroup<N,S>(mut r: S::Elt, mut a: S::Elt, mut n: N) -> S::Elt where N: Natural, S: Semigroup{ loop { assert!(n > N::zero()); if n.odd() { r = S::op(r.clone() ,a.clone()); if n == N::one() { return r; } } n = n.half(); a = S::op(a.clone(), a.clone()); }}pub fn power_semigroup<N: Natural, S: Semigroup>(mut a: S::Elt, mut n: N) -> S::Elt { assert!(n > N::zero()); while !n.odd() { a = S::op(a.clone(), a.clone()); n = n.half(); } if n == N::one() { return a } let y = S::op(a.clone(), a.clone()); return power_accumulate_semigroup::<N,S>(a, y, (n - N::one()).half());} (If you're interested in the Natural traits, see that previous post; I don't want to go into it here.) The definition above actually makes serious use of associativity to execute the repeated operations in O(log n) rather than O(n) time. Below, I make use of the other algebraic definitions to handle edge cases for the Natural argument. pub fn power_monoid<N: Natural, M: Monoid>(a: M::Elt, n: N) -> M::Elt { assert!(n >= N::zero()); if n == N::zero() { M::identity() } else { power_semigroup::<_,M>(a, n) }}pub fn power_group<N: Integral, G: Group>(mut a: G::Elt, mut n: N) -> G::Elt { if n < N::zero() { n = -n; a = G::inverse(a); } power_monoid::<N,G>(a, n)} So far, what I have is just additional formalization of the things I presented in that previous post. Not very interesting, eh? Heck, I'm already bored. So, let's introduce some new definitions. TermDefinitionRust DistributionOne way two operations can interact: x m (y a z) = (x m y) a (x m z), and (y a z) m x = (y m x) a (z m x). pub trait DistributesOver<A>: Magma where A: Magma { }macro_rules! mult_distributes_over_add_for { ($( $t:ty ),* ) => {$(        impl DistributesOver<Addition<$t>> for Multiplication<$t> { }        )*    }}mult_distributes_over_add_for!(usize,u8,u16,u32,u64,i8,i16,i32,i64,f32,f64);
AbsorptionThere is a special, choosen, elemnt which, when used as an argument to an operation, always results in itself: x op 0 == 0 op x == 0.
pub trait Absorbing : Magma { fn absorbing() -> Self::Elt; }macro_rules! absorbing_multiplication_for {    ( $($t:ty ),* ) => { $( impl Absorbing for Multiplication<$t> {          fn absorbing() -> Self::Elt { 0 as $t } } )* }}absorbing_multiplication_for!(usize,u8,u16,u32,u64,i8,i16,i32,i64,f32,f64); SemiringAn algebraic structure with two operations (here called Ad and Mu), both monoids. Ad should be commutative and Mu should have an absorbing element (possibly equal to Ad's identity?) and Mu should distribute across Ad. Also, Ad::identity != Mu::identity. pub trait Semiring { type Elt; type Ad: Monoid<Elt=Self::Elt> + Commutative<Elt=Self::Elt>; type Mu: Monoid<Elt=Self::Elt> + Absorbing<Elt=Self::Elt> + DistributesOver<Self::Ad>;} RingA semiring where Ad includes an inverse. (Completely unimportant for the rest of this post, but hey, somebody might need it sometime.) pub trait Ring: Semiring where Self::Ad: Group { } What we have here, besides a failure to communicate, is the definition of a semiring, the algebraic structure needed to generalize matrix multiplication. If you remember your linear algebra, or the code from the previous post, you know that matrix multiplication requires element-wise multiplication and then addition to compute the values in the result matrix. In fact, normal matrix multiplication uses what I'm going to call the Numerical semiring: pub struct Numerical<Elt>(PhantomData<Elt>);macro_rules! numerical_semiring_for { ($( $t:ty ),* ) => {$(        impl Semiring for Numerical<$t> { type Elt =$t;            type Ad = Addition<$t>; type Mu = Multiplication<$t>;        } )*    }}numerical_semiring_for!(usize,u8,u16,u32,u64,i8,i16,i32,i64,f32,f64);

Technically, as Stepanov and Rose point out, you don't need a semiring for matrix multiplication; specifically, you don't need the identities for Ad and Mu. But I'm not going through this mess again to get a "weak semiring". In fact, for all of this algebraic nonsense, mathematical structures are only bags of properties; my semiring structure should be replaced by something that defines Ad and Mu in terms of Magmas and the actual requirements (associativity, commutativity, etc.) added on as required by the algorithm. Maybe next time.

Matrix multiplication

The matrix I presented last time was simple; this time I came up with something less simple: I added a special value to act as an identity for multiplication.

#[derive(Debug)]pub enum Matrix<T> {    Identity,    Matrix {        rows: usize,        cols: usize,        m: Vec<Vec<T>>,    }}

Matrix multiplication gets much more verbose when defined in terms of a semiring. The basic flow is this: if one of the arguments is the identity value, the result is the other. If neither is the identity, then it creates a new result matrix (an n x m matrix multiplied by a m x p matrix produces a n x p matrix) where the cells from the two matrices are combined as aik Mu bkj and the results are summed using the Ad operation with Ad::identity as the zero.

pub fn matrix_multiply<S>(a: Matrix<S::Elt>, b: Matrix<S::Elt>) -> Matrix<S::Elt>    where S: Semiring, S::Elt: Clone{    match (a,b) {        (Matrix::Identity, b) => b,        (a, Matrix::Identity) => a,        (Matrix::Matrix{rows: ref arows, cols: ref acols, m: ref am},         Matrix::Matrix{rows: ref brows, cols: ref bcols, m: ref bm}) => {            let n = *arows;            let m = *acols;            assert_eq!(m, *brows);            let p = *bcols;            let mut results: Vec<S::Elt> = Vec::new();            for i in 0..n {                for j in 0..p {                    let cell = (0..m)                        .map( |k| {                          <S::Mu as Magma>::op( am[i][k].clone(), bm[k][j].clone() )                        })                        .fold( <S::Ad as Monoid>::identity(), <S::Ad as Magma>::op );                    results.push(cell);                }            }            Matrix::new(n, p, &results)        }    }}

One thing to note: when you pick out S::Mu::op and S::Ad::identity, you need to specify which specific trait's operation and identity with identifier as trait syntax. I'm not entirely sure why, since I don't think the declarations provided allow for multiple definitions.

Matrix multiplication naturally forms an algebraic operation itself.

pub struct MatrixMultiply<S:Semiring>(PhantomData<S>);impl<S> Magma for MatrixMultiply<S> where S: Semiring, S::Elt: Clone {    type Elt = Matrix<S::Elt>;    fn op(x: Matrix<S::Elt>, y: Matrix<S::Elt>) -> Matrix<S::Elt> {        matrix_multiply::<S>(x,y)    }}impl<S> Associative for MatrixMultiply<S> { }impl<S:Semiring> Semigroup for MatrixMultiply<S> where S::Elt: Clone { }

Given a semiring S, multiplication of matrices over S::Elts forms a semigroup itself. Which is good, because I'll need that later.

At this point, I could go back through and repeat the Fibonacci example using this generalized matrix multiplication, but I won't. I'm off to new things.

Social networks

Assume you have seven people and the relationship of who is friends with whom among the seven. If you represent the friendship relation as a matrix of boolean values (true if the two people are friends and false otherwise), you can use matrix multiplication on a boolean semiring to compute the transitive closure of the relation, which describes the social network of the group of people.

Here is the friendship matrix:

    let friends: Matrix<bool> = Matrix::new(7,7, &vec!(        true,  true,  false, true,  false, false, false,        true,  true,  false, false, false, true,  false,        false, false, true,  true,  false, false, false,        true,  false, true,  true,  false, true,  false,        false, false, false, false, true,  false, true,        false, true,  false, true,  false, true,  false,        false, false, false, false, true,  false, true,        ));

The boolean semiring I need, the {Or,And}-semiring, replaces addition with boolean or and multiplication with boolean and. To start with, I need to define the two boolean operations.

Or is the boolean or operation, implemented by Rust's || operator. It is associative, commutative, and has an identity in the false value.

pub struct Or;impl Magma for Or {    type Elt = bool;    fn op(x: bool, y: bool) -> bool { x || y }}impl Associative for Or { }impl Semigroup for Or { }impl Monoid for Or {    fn identity() -> bool { false }}impl Commutative for Or { }

And is the boolean and operation, &&. It is associative; has an identity, true; has an absorbing element, false. (The absorbing part isn't mentioned in Stepanov and Rose, although the requirement is present in their defining axioms for the semiring. A lack of formalism, I suspect, led them to overlook it.) And also distributes over Or.

pub struct And;impl Magma for And {    type Elt = bool;    fn op(x: bool, y: bool) -> bool { x && y }}impl Associative for And { }impl Semigroup for And { }impl Monoid for And {    fn identity() -> bool { true }}impl Absorbing for And {    fn absorbing() -> bool { false }}impl DistributesOver<Or> for And { }

The result is the boolean semiring:

pub struct BooleanSemiring;impl Semiring for BooleanSemiring {    type Elt = bool;    type Ad = Or;    type Mu = And;}

Given all of the above, the code to actually produce the transitive closure is trivial, using the power_semigroup function.

    let result = power_semigroup::<_,MatrixMultiply<BooleanSemiring>> ( friends, 6 );

By the way, the 6 comes from 7, the number of people in the group, minus 1.

The result is:

[ true true true true false true false true true true true false true false true true true true false true false true true true true false true false false false false false true false true true true true true false true false false false false false true false true]

There are two cliques in this network, one containing Ari(0), Bev(1), Cal(2), Don(3), and Fay(5) and another containing Eva(4) and Gia(6). If the two cliques were not disjoint, the graph would contain all true values, but they are and there is no transitive friendship relation between Eva and Gia and the rest of the group.

Directed graphs

Suppose you have a directed graph with weighted links between some of the nodes. You can represent this graph as a matrix, with the weight between each node and itself as 0 and the weight between unconnected nodes as infinity:

    let graph: Matrix<f64> = Matrix::new(7, 7, &vec!(        0.0,  6.0,  INF,  3.0,  INF,  INF,  INF,        INF,  0.0,  INF,  INF,  2.0,  10.0, INF,        7.0,  INF,  0.0,  INF,  INF,  INF,  INF,        INF,  INF,  5.0,  0.0,  INF,  4.0,  INF,        INF,  INF,  INF,  INF,  0.0,  INF,  3.0,        INF,  INF,  6.0,  INF,  7.0,  0.0,  8.0,        INF,  9.0,  INF,  INF,  INF,  INF,  0.0,        ));

You can use repeated matrix multiplication over a very peculiar semiring to compute a matrix with the minimum path weight between any node and any other node. The peculiar semiring? It's tropical. (I really have no idea where these names come from. Tropical? Ridiculous. I do not have a daiquiri in my hand. There is no sand here. It's freaking hot outside, but I'm currently in Amarillo, TX, so it's a dry heat with a lot of wind so that I can taste the landscape. Bleah.)

pub struct TropicalSemiring;impl Semiring for TropicalSemiring {    type Elt = f64;    type Ad = Min;    type Mu = Addition<f64>;}

In this tropical semiring, Min (as in min(a,b) = a if a < b; b otherwise) takes the place of addition and addition takes the place of multiplication. (That is one of the sources of peculiarity.)

Min itself makes a pretty simple algebraic structure. It is associative, commutative, and has an identity (infinity). Addition distributes over it since a + min(b,c) = min(a+b, a+c).

pub struct Min;impl Magma for Min {    type Elt = f64;    fn op(x: f64, y: f64) -> f64 { f64::min(x,y) }}impl Associative for Min { }impl Commutative for Min { }impl Semigroup for Min { }impl Monoid for Min {    fn identity() -> f64 { f64::INFINITY }}impl DistributesOver<Min> for Addition<f64> { }

On the other hand, addition proves to be a problem. It's the standard addition operation described above, but the semiring definition requires whatever replaces the multiplication operation to be absorbing. How do you get an element such that e + x = x + e = e? Zero is not it, one is not it; are there any other interesting numbers? Fortunately, IEEE 754 floating point numbers come to our rescue here: it provides representations for positive and negative infinity, and infinity is almost as sticky as NaN. Cast your eyes at this chart: IEEE 754 Infinity and NaN Arithmetic Rules. In the upper-left quarter is addition; infinity plus any other number (except negative infinity and NaN) is always infinity. (Infinity plus negative infinity is NaN; any operation on NaN is also Not a Number, so infinity under addition is almost as sticky as NaN and, although the axioms are not strictly satisfied for negative infinity and NaN, it is close enough for government work. Especially since it isn't used here.)

/// Only true for addition on floating point numbers; see IEEE754.macro_rules! absorbing_addition_for {    ( $($t:ty: $v:expr ),* ) => {$(        impl Absorbing for Addition<$t> { fn absorbing() -> Self::Elt {$v } }        )*    }}absorbing_addition_for!(f32: f32::INFINITY, f64: f64::INFINITY);

Once I have all of the definitions in place, using the power function to compute the transitive closure of the graph link weights is easy:

    let result = power_semigroup::<_,MatrixMultiply<TropicalSemiring>>(graph, 7-1);

The final result is:

[ 0 6 8 3 8 7 11 23 0 16 26 2 10 5 7 13 0 10 15 14 18 12 18 5 0 11 4 12 35 12 28 38 0 22 3 13 17 6 16 7 0 8 32 9 25 35 11 19 0]

So, travelling from A(0) to B(1) (in the top row, second from the left) has weight 6; travelling from B to A has weight 23 (second row, leftmost column).

That is exercise 8.8; exercise 8.9, which calls for not only finding the shortest distance but the shortest paths, should be fairly easy; simply record which nodes you use when you reduce with min. Maybe next time.

Conclusions

Throughout this post, I used semirings even though I did not need all of their properties; in fact, the absorbing thing was a bit of a problem. At one point in the book, Stepanov and Rose write,

Categorical Theories versus STL

For a long time, people believed that only categorical theories [a theory is a set of propositions; in a categorical theory, all models, which define the operations in the theory and make the axioms true, of the theory are isomorphic] were good for programming. When the C++ Standard Template Library (STL) was first proposed, some computer scientists opposed it on the grounds that many of its fundamental concepts, such as Iterator, were underspecified. In fact, it is this underspecification that gives the library its generality. While linked lists and arrays are not computationally isomorphic, many STL algorithms are defined on input iterators and work on both data structures. If you can get by with fewer axioms, you allow for a wider range of implementations.

I seriously hope they are talking about the problem of dealing with bundles of propositions such as semirings, where all of the requirements for the bundle are not used by a given algorithm. In that case, it is easy for a structure to be underspecified, since if a property is not used then it never needs to be mentioned in the code. The alternative, that if it is used but cannot easily be described (like the associativity trait), it may only be implicit, is a dandy source of hard-to-find bugs. Making things like associativity explicit, but trivial, is only a bandaid. It tries to remind the programmer that water is needed, but cannot make them drink.

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# 10th International Workshop on Charm Physics (CHARM 2020)

from 31 May 2021 to 4 June 2021
Mexico/General timezone
Thanks for your contribution to the success of this workshop!!
Home > Timetable > Contribution details

# Contribution

Charm meson and baryon spectroscopy (including exotica)

# Interpretation of the LHCb Pc(4312) signal

## Speakers

• Mr. Jorge Antonio SILVA CASTRO

## Description

We study the nature of the new signal reported by LHCb in the $J/\psi p$ spectrum. Based on the S-matrix principles, we perform a minimum-bias analysis of the underlying reaction amplitude, focusing on the analytic properties that can be related to the microscopic origin of the Pc(4312)+ peak. By exploring several amplitude parameterizations, we find evidence for the attractive effect of the $Σ_c^+ \overline{D^0}$ channel, that is not strong enough, however, to form a bound state.

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# Assume that an angle Xis uniformly distributed on [O, 1T/2]. Find the pdf for Y = sin X on the...

Assume that an angle Xis uniformly distributed on [O, 1T/2].

Find the pdf for Y = sin X on the interval O < y="">< 1.="">

that this pdf integrates to 1.


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# Solution to Exercise SICP 2.8

A solution to Exercise 2.8:

The maximum the difference could be is difference between the upper bound of the first interval and the lower bound of the second. The minimum difference is the difference between the lower bound of the first and the upper bound of the second. This holds true even if the second interval is greater than the first or the intervals overlap.

```(define (sub-interval x y) (make-interval (- (lower-bound x) (upper-bound y)) (- (upper-bound x) (lower-bound y)))) ```


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Project data
Authors Teri Grunthaner Yes No USD 80.97 Solar cooker Download Upload your project too!body.poncho-dark-mode .mw-parser-output .mw-ui-button{background:#303134;border-color:#3c4043;color:#bdc1c6}@media screen and (max-width:500px){.mw-parser-output .button .mw-ui-button{display:block;margin:.5em 0}}
An umbrella by rain, a parabolic solar cooker by shine... a Sunbrella!

The energy used for conventional cooking comes from a variety of sources: wood, gas, oil, coal, electricity, etc. Though solar cooking seems to be its own category, it is the basis for all the rest. The Sun's energy powered the growth of the tree; plants and animals buried on the ocean floor a hundred million years ago anaerobically decomposed to produce crude oil[1]...

The difference between using conventional techniques, such as wood, oil, and gas, and using pure solar energy is the number of transfers the energy has to undergo (as well as the byproduct of usage, of course). To use gas, for instance, the Sun shines, the tree grows, the animal eats, the fungus feasts, the gas burns, and then the pot heats. To use pure solar energy, as with a Solar Cooker, the Sun shines… and then the pot heats. Simple, straight-forward, and seemingly much more efficient for the purpose of cooking food for human consumption.

## Solar cooker designs

Solar cookers are solar concentrators where precision and efficiency has been sacrificed for ease of construction and use of readily available materials. Inexact geometries that still allow for water boiling or bread baking temperatures are used. Some calculations relevant to the design of solar cookers with paraboloidal reflectors can be found on the Sun related calculations page.

There is a whole spectrum of Solar Cooker designs, each with their own unique costs, benefits, and optimal conditions. Some examples...

## Background

Parabolic reflectors [2] concentrate sunlight into a focal point. The more exact the shape, the tighter the focus. When a parabolic solar cooker is not aimed correctly, the focal point disperces, creating off-axis parabolic reflection. Thus, frequent adjusting of parabolic solar cookers must occur to keep the focus in place. This focus creates very high temperatures, ideal for areas like Humboldt County where the Sun is not very strong or direct [3].

#### Opportunities Abound

As a student taking Appropriate Technology at Cal Poly Humboldt in the spring of 2009 with access to incredible human resources (Bart Orlando and Lonny Grafman), I couldn't imagine a better time or place for me to cooperatively design, build, test, and enjoy an umbrella solar cooker.

Amazing opportunities for growth present themselves at every angle of this project, fore I am able to…

• Make connections and friendships with community members
• Work cooperatively with incredible teachers and students
• Expand my knowledge of ways to utilize the Sun's energy
• Refresh my math skills and learn to apply it to engineering
• Recycle materials for another generation
• Build upon the appropriate technology collective's information resources
• Integrate social, economical, and ecological responsibility in a DIY project
• Make a wiki-webpage
• Inspire others to do the same and recognize their bounty of opportunities
Bart Orlando with Sunbrella

This project was truly made possible by the one and only Bart Orlando. Bart is an appropriate technology visionary who, among many other things, converts commonly found items into solar cookers. He mentors and works with students in the Humboldt area to design and create innovative technologies such as the Parabolic Basket Solar Cooker for non-industrialized cultures and the Papasan Chair Solar Cooker for industrialized cultures. He is a teacher at heart and creates educational displays and demos everywhere he goes, from events and schools to farmers' markets and co-ops. His mission is to reduce the negative health and environmental impacts of using wood and other conventional methods of cooking and make solar cooking accessible to all… who cook food. ☺ Thank you, Bart, for your dedicated work, guidance, and friendship. More examples of A.T. projects Bart has envisioned can be seen online at Projects With Bart [4].

This project was inspired by another Sunbrella made in 2000. Thank you to all who document their work online!

## Design

An umbrella is a mighty appropriate tool in Humboldt County, as is a cooker that utilizes a clean renewable resource. Thus, we decided to combine the two to make an incredible apparatus: an umbrella in the rain, a solar cooker in the Sun... a Sunbrella!

#### Criteria

The guidelines we used when designing the Sunbrella, as well as by which we judged our success, make up our criteria. You can also think of them as goals...

• Effective - Can cook food and boil water
• Safe - Is stable while cooking
• Portable - Can collapse well enough to be transported and stored efficiently
• Functional - Can withstand usage in the rain without damage
• Durable - Can withstand repeated uses as a cooker
• Simple - Is easy to understand, use, and construct
• Economical - Utilizes reused/recycled materials and creates little waste
• Attractive - Arouses interest and thought and is esthetically pleasing

### Frame

The frame is like the skeleton, it is what holds the parabolic shape of the cooker. Since our frame is an umbrella, we searched far and wide for one that had the right size, shape, and shaft.

• Size - The greater the diameter of a parabolic cooker, the more sun rays are reflected to the focus. We calculated that an umbrella with a diameter between 50-60 inches would be most appropriate, as we want to be able to feed and keep a few people dry while still being able to easily transport the Sunbrella.
• Shape - Most umbrellas are not quite parabolic, but are instead often catenary or conical. The number of panels that make up an umbrella determines how smooth the 3-D shape is, so the more the better in terms of a paraboloid. We aimed to settle on no fewer than 12 panels comprising our umbrella.
The umbrellas we tested.
• Shaft - The ends of the shaft are really what is important: the handle and the tip. A straight handle seemed to us to be the most compact and gave us more possibilities for the cookware. The tip plays a huge part in the mounting of the Sunbrella, so we wanted one that was long and strong.

After many weeks of searching the Humboldt region, we decided to purchase an umbrella off Ebay: a white, 60 inch diameter, 16 panels, and straight handle, and a 2.5 inch tip that extends from the shaft (so it is really strong). About a week before we were done, we found the perfect umbrella almost identical to this one except better because it is rainbow colored, silver shaft and ribs (instead of black), and has a higher focal point. You'll see pictures of both throughout this page.

### Reflectors

The reflectors are what, as I'm sure you can guess, actually reflects the sun rays. There are many materials available on the market that have varying degrees of reflectivity, durability, price, and so on. A great reflector comparison table was made by another Engr 305 student in 2008.

Liquid Lame

Ideally, a very high thread count cloth-like material, like liquid lame, would be permanently sewn onto the panels for ease of set-up. This material was not readily available but is out there*, so we instead decided to create individual reflectors that would slip onto the panels when being used as a cooker and stored when not.

We settled with mylar and cellulose acetate.

• Mylar - A film made from polyester that has 98% reflectivity. It has a good consistent color, dimensional stability, high heat resistance, and is non-tearing. [5]
• Cellulose Acetate - Looks like plastic, is easy to cut, and is flexible. CA is made from wood pulp and can be composted and incinerated. It also has poor thermal retention, is hypoallergenic, and is resistant to mold and mildew. [6]
##### Construction

The reflectors took a lot of time and energy to construct. We had to undergo several trial-and-errors to find the right size reflector -- it wasn't as easy as it seems. We found that the umbrella's panels could not be covered with isosceles triangle reflectors because of the curvature, which caused a gap between the reflectors in the middle when bowed to fit the curve. So, we decided to widen the angles of the triangle and allow one of the sides down by the most acute angle hang over into the next panel area.

Umbrella components.
Taping the reflectors
1. Measure an individual panel's (plus a few more for an average) width at the rim plane and midway point, the full radial length, the radial length between the rim plane and the midway point, and the radial length between the rim plane and the desired length of the reflectors (we left 4 inches from the umbrella's vertex).
2. On a large piece of cardboard, make a line (near a side) the length of the rim plane width. From the midpoint of that line, draw a perpendicular line equal to the full radial length, marking the point on that line where the midway point lies and the desired length of the reflector is.
3. At the midway point, draw a centered line (it's intersection at it's midpoint) parallel to the rim plane line the length of the midway point width.
4. Using a yard stick (or any long straight edge), connect an end point of the rim plane line with an end point of the midway point line on the same side of the full radial line. Trace the line from the rim plane to the point of intersection with the full radial line. (If it does not intersect, it's okay.) Repeat with the other end points.
5. Draw a line parallel to the rim plane line at the point along the radial length where the desired length of reflector is. You should now have a trapezoid with height the desired length of reflector and base the length of the rim plane line.
6. Lay a sheet of cellulose acetate down with a side and corner aligned with the rim plane line and one of it's end points (respectfully). Firmly secure the sheet in place so that you can cut out the trapezoid with a utility knife.
7. Slip the trapezoid into a panel section. Assess how the piece lines up, and make corrections accordingly. In our umbrella, one of our sides lines up perfectly, while the other side overhangs into the adjacent section near the center of the umbrella. We like this as it helps hold that end down on the next panel.
8. Continue until all panels are filled.
9. For the mylar, create a stencil that is identical to the cut cellulose acetate pieces, except take in a centimeter or so on one of the long edges and the rim plane edge so the cellulose acetate overhangs.
10. Tape the mylar to the cellulose acetate about 1 centimeter down from the rim plane edge.

### Stand

The stand is the mount, or base, for the solar cooker. It is crucial that the stand be easily adjustable so the solar cooker can track the Sun with minimal disruption to the cooking process. The stand obviously must also be strong enough and well grounded to support the Sunbrella with food in it.

We brainstormed quite a few different designs for the stand, two of which we actually tried:

Tripod stand in action
• Stake - This design acts like an extension of the umbrella's tip. We screwed a long stake onto a long copper pipe that is just flexible enough to be manually adjusted, but strong enough to maintain it's shape while holding the Sunbrella. The stake is firmly stuck into the ground, and then the umbrella's tip is secured into the end of the copper pipe. Some kind of support is needed to keep the stake from freely rotating in the soil, like bungee cords also staked on either side.
• Tripod - Tripods are wonderful for a solar cooker stand because they are well grounded and are easily adjusted and fixed in any direction. We designed an interface where the umbrella attaches to the tripod using an aluminum plate, piping, brackets, and rubber tubing. This interface screws onto where a camera usually does and snuggly holds onto the Sunbrella. For constructing this interface, we were very fortunate to know someone who has access to an iron-working machine that could cut and drill the aluminum sheet like nothing. Thank you John Davis from Solar Racks in Arcata!

### Cookware

Double bunt cake dish.

The cookware is what facilitates the cooking of the food, including a pot, it's base, and ideally a windshield (oh, and of course a mixer and a pot holder). Usually, parabolic solar cookers do not have a shaft running right through the focal point. This creates some challenges, but in turn enables creativity.

• Pot - This can be a whole variety of things, like a pan, dish, or jar. Whatever it is, it should be black, as black absorbs the whole spectrum of light. When sizing the pot, it is important to consider the focal point's size (meaning how focused it really is) and location. Since our umbrella's focal area (it wasn't much of a point) was on and above the beams, we needed to put the pot as low as possible. We decided to use a bunt cake dish, which is a donut-like pot with a hollow cone in the middle. This pot slides over the handle and down the shaft perfectly and nestles on the beams. Because the bottom of the pot gets so hot, we decided to use two bunt cake dishes, one inside the other with a little space between, so to avoid burnt food and allow convection.
• Base - The base is what the pot sits on or is suspended by. It must be able to handle the high temperature of the pot and the heaviest weight possible when full of food. For the Sunbrella, the rib beams act as the base, but since they are black as well, we needed to protect them from the high heat. We covered the rib beams with aluminum foil, fit a tin pie pan below the rib beams, as well as stuck a foil-wrapped wooden donut-like block below the pot.
• Shield - Some kind of clear windshield that leaves a little space between it and the pot enables the pot to absorb the sun rays but keeps the wind from cooling it off by convection.

### Container

Foam support in container

Since the umbrella does not always have to be a Sunbrella, convenient storage for the reflectors and other components would be nice. We decided to use a length of PVC piping with two end caps. We cut a wedge of foam to secure the umbrella inside the tube. The reflectors are simply wrapped in cloth and eased inside. More ideas have come up for the container, and can be found in the Further section.

## Materials and Costs

Quantity Material Source Cost ($) Total ($)
1 60-in 16 panel Wedding Umbrella (plus shipping) Ebay $17.95$27.90
1 25-ft roll Mylar Let it Grow $11.00$11.00
3 40 X 25 in. sheets Cellulose Acetate Arcata Art Center $7.00$21.00
1 roll Packing Tape Ace Hardware $3.50$3.50
2 Bunt Cake Dish The Hospice Thrift Shop $1.00$2.00
1 Bunt Cake Lid The Hospice Thrift Shop $0.50$0.50
1 donut-shaped Wooden Block Bart Orlando Donated Free
1 roll Recycled Aluminum Ace Hardware $3.50$3.50
1 Tin Pie Pan Friend's house Salvaged Free
1 Tripod Bart Orlando Borrowed Free
1 5 X 4 X 3/16 in. sheet of Aluminum Arcata Scrap and Salvage Salvaged Free
1 3/4 in. Stainless Steel Tube Arcata Scrap and Salvage Salvaged Free
2 3/4 in. Pipe Brackets Ace Hardware $0.27$0.54
2 10-24 Bolts Ace Hardware $0.11$0.22
2 10-24 Nylox Nuts Ace Hardware $0.12$0.24
1 Wing Nut Ace Hardware $0.27$0.27
1 ft of Rubber Plastic Tubing Ace Hardware $0.57$0.57
1 large piece scrap Foam Wing Inflatables Donated Free
1 4-ft 4 in. PVC Piping Arcata Recycling Center $4.00$4.00
2 PVC End Caps Ace Hardware $3.00$6.00
Total Cost \$80.97

#### Tools

• Ruler, Yard Stick, Scissors, Utility Knife, Pliers, Cutting Shears
• Cardboard, Gloves, Sunglasses, Notebook, Camera, Radio

## Implementation

#### Setup

Setting up the Sunbrella is amazingly simple, and only takes one person about 10 minutes! It's as easy as....

Once all the reflectors are in place, watch out for off-axis parabolic reflection. This scattering of light could hurt someone's eyes or possibly start a fire if the time is right and flameable materials are around. When not using the cooker, either face it 180 degrees off-axis, or cover with a sheet.

#### Cooking

Now for the fun part! Experiment with baking, frying, sauteing... Check out some great solar cooking tips and recipes on the internet and in books for inspiration.

Be sure to always wear UV protective sunglasses when using the Sunbrella and have pot holders and towels handy while cooking. Also, keep an eye on how illuminated the pot is... parabolic solar cookers need to be adjusted every 15 minutes or so to keep an accurate focus on the Sun.

#### Takedown

Turn the Sunbrella away from the Sun and undo what you just did... simple! Be sure to take care of the reflectors. Stack them together, leaving the cellulose acetate side up on both ends for protection, and wrap in cloth. Storing the Sunbrella in the container is pretty intuitive, but just in case you were wondering...

## Evaluation

So, in the end, how well did we satisfy our criteria? Though I don't want to be making excuses, time constraints were the number one reason for hasty decisions that didn't necessarily fit our criteria... but we tried!

• Effective? - Yes, we successfully baked cookies and a cake. Unfortunately, we have yet to test boiling water because we have not figured out how to keep condensation from forming on the shaft and dripping down onto the Sunbrella reflectors. We will keep an update on our testing.
• Safe? - Yes, the Sunbrella has yet to ignite black construction paper (though we don't doubt it could in mid summer), and the tripod is very stable. However, the off-axis parabolic reflection needs to be watched out for, since it could hurt someone's vision. Also, both mylar and cellulose acetate are flamible.
• Portable? - Yes, the Sunbrella's container takes up about 2-3 closed umbrella's worth of space and is lightweight. A smaller tripod is needed, however, as the one we had access to is too bulky for easy transport.
• Functional? - Yes, because the reflectors are taken out when not in use, the umbrella functions just like any other one.
• Durable? - Yes and no, since though we have used the Sunbrella many times already, we are finding that the mylar is scratching where the reflectors slip under the ribs and contact the tabs.
• Simple? - Yes, amazingly simple!
• Economical? - Yes and no, as we did buy quite a few new materials, including the umbrella, mylar, and cellulose acetate. The mylar will also need to be replaced after a while. But, at least it's main input (Sun rays) is free, abundant, and clean!
• Attractive? - Yes, I think it's beautiful! It combines common items that everyone recognizes, an umbrella and a tripod, into something (almost) everyone needs, a cooker! Also, both umbrella colors, white and rainbow, have the potential to spark great conversation about sunlight and the color spectrum.

Overall, a wonderful step towards low-impact, do-it-yourself living!

## Furthur

A project is never done, as there is always an infinite number of ways to improve and adapt a technology. We had fun dreaming up some of the possibilities to take the Sunbrella (as the Merry Pranksters put it) FURTHURRRR!!

• The container can be designed to also use the Sun's rays by using black tubing designed for holding hot water. While the Sunbrella is in use, the container can be filled with water and left in the Sun, providing a hot shower after a hot meal!
• The end caps on the container can be filled with concentric cups for dishware.
• A satellite receiver antenna could be connected to the Sunbrella's shaft for telecommunication.

## Literature Review

• Radabaugh, Joseph M. 1998. "Heaven's Flame: A Guide to Solar Cookers." Oregon: Home Power Publishing.
• The book outlines the environmental and social benefits of solar cookers, several different designs and experiments, and instructions on how to build the SunStar solar oven, as well as cooking tips and how to produce the SunStar Kit to send to others.
• Halacy, Beth and Dan, eds. 1992. "Cooking With the Sun: How to Build and Use Solar Cookers." California: Morning Sun Press.
• This book details how to build a solar hot plate and solar oven, with guidelines for usage and lots of recipes.
• Martin, Rachel and others. "Design of Solar Ovens for Use in the Developing World." 2006. International Journal for Service Learning in Engineering 2, no.1: 78-91. http://www.ijsle.org/ (accessed February 10, 2009).
• Professors and students combined laboratory testing done at Cornell University with field research done in Nicaragua to gain an understand of what it takes to design solar ovens for use in the developing world.
• Paredes, Juan Francisco. 2006. "Portable Solar Barbecue: A Portable Parabolic Cooker Built From an Umbrella." The Solar Cooking Archive. http://www.solarcooking.org/ (accessed February 10 2009).
• Instructions on how to build a solar cooker from an umbrella, aluminized cardboard, and a tripod.

## Contact Information

• Teri Grunthaner — Behavioral and Socio-Economics, Cal Poly Humboldt '11 tlg49@humboldt.edu
• Bart Orlando — Medical Biology, UC Irvine '82 bart6591@yahoo.com

## Related projects

Page data
Type Project parabolic solar cookers, versatile design, umbrella, solar cooking SDG07 Affordable and clean energy, SDG11 Sustainable cities and communities Teri Grunthaner, Kathy Nativi 2009 CC-BY-SA-3.0 Cal Poly Humboldt English (en) 359 Sections nested too deep Teri Grunthaner, Kathy Nativi (2009). "How to make a "Sunbrella" solar cooker". Appropedia. Retrieved August 7, 2022.
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text:
# How do you find the slope of a line that is a) parallel and b) perpendicular to the given line: -2x-5y=-9?

Mar 29, 2018

$- \frac{2}{5} \text{ and } \frac{5}{2}$

#### Explanation:

$\text{the equation of a line in "color(blue)"slope-intercept form}$ is.

•color(white)(x)y=mx+b

$\text{where m is the slope and b the y-intercept}$

$\text{rearrange "-2x-5y=-9" into this form}$

$\Rightarrow - 5 y = 2 x - 9$

$\Rightarrow y = - \frac{2}{5} x + \frac{9}{5} \leftarrow \textcolor{b l u e}{\text{in slope intercept form}}$

$\text{with slope m } = - \frac{2}{5}$

• " Parallel lines have equal slopes"

$\Rightarrow \text{slope of parallel line is } m = - \frac{2}{5}$

$\text{Given a line with slope m then the slope of a line}$
$\text{perpendicular to it is}$

•color(white)(x)m_(color(red)"perpendicular")=-1/m

$\Rightarrow {m}_{\text{perpendicular}} = - \frac{1}{- \frac{2}{5}} = \frac{5}{2}$

Mar 29, 2018

$\frac{2}{5} \setminus \setminus \setminus$ parallel.

$- \frac{5}{2} \setminus \setminus \setminus$ perpendicular.

#### Explanation:

First rearrange the equation to get the form:

$y = m x + b$

It is only in this form that we can see what the gradient is.

$- 2 x - 5 y = - 9$

$y = \frac{2}{5} x + \frac{9}{5}$

If two lines have the same gradient, then they will be parallel. So any line of the form:

$y = \frac{2}{5} x + b$ , will be parallel to $\setminus \setminus y = \frac{2}{5} x + \frac{9}{5}$

If two lines are perpendicular, then the product of their gradients is $- 1$

For two lines with gradients ${m}_{1}$ and ${m}_{2}$:

${m}_{1} \cdot {m}_{2} = - 1$

${m}_{2} = - \frac{1}{m} _ 1$

Let ${m}_{1} = \frac{2}{5}$

$\therefore$

${m}_{2} = \frac{1}{m} _ 1 = - \frac{1}{\frac{2}{5}} = - \frac{5}{2}$

So, any line of the form $\boldsymbol{y = - \frac{5}{2} x + b}$ will be perpendicular to $\boldsymbol{y = \frac{2}{5} x + \frac{9}{5}}$


text:
We develop a theory of covariance and concentration matrix estimation on any given or estimated sparsity scale when the matrix dimension is larger than The covariance matrix not only tells the variance for every individual $$\beta_j$$, but also the covariance for any pair of $$\beta_j$$ and $$\beta_k$$, $$j \ne k$$. variance covariance matrix of regression coefficients, I have a linear regression model $\hat{y_i}=\hat{\beta_0}+\hat{\beta_1}x_i+\hat{\epsilon_i}$, where $\hat{\beta_0}$ and $\hat{\beta_1}$ are normally distributed unbiased estimators, and $\hat{\epsilon_i}$ is Normal with mean $0$ and variance $\sigma^2$. To get the regression coefficients, the user should use function beta_hat(), which is the user-friendly version. Description. linalg. In standard usage, it would be true that $\hat y_i = \hat\beta_0+\hat\beta_1 x_i$, WITHOUT any $\varepsilon$ term, and $\hat\varepsilon_i=y_i-(\hat\beta_0+\hat\beta_1 x_i)$ $=y_i-\hat y_i$. Could you please help me with the code for Stata how to get the covariance matrix?? 4.5 The Sampling Distribution of the OLS Estimator. – mavavilj 06 déc.. 16 2016-12-06 17:04:33 The main argument x is an object of the class attribute tvvar obtained from function tvVAR. If it is positive then stocks move in the same direction or move in opposite directions leads to negative covariance. For which i need to find the co variance matrix of the fitted model ie, vcov. T. dot (Y) beta_hat = numpy. Abstract. Let Hbe a symmetric idempotent real valued matrix. Thanks in advance . Variables entering the model are S&Prat Mrat GDP Inflation Ratio etc. Estimation of covariance matrices then deals with the question of how to approximate the actual covariance matrix on the basis of a sample from the multivariate distribution.Simple cases, where observations are complete, can be dealt with by using the sample covariance matrix. 3. A symmetric idempotent matrix such as H is called a perpendicular projection matrix. Theorem 2.2. 1. The matrix that is stored in e(V) after running the bs command is the variance–covariance matrix of the estimated parameters from the last estimation (i.e., the estimation from the last bootstrap sample) and not the variance–covariance matrix of the complete set of bootstrapped parameters. I want to get the multivariate betas for these 4 assets-based on these assets. is the n × n “Hat Matrix” MIT 18.472 Regression Analysis Regression Analysis Linear Regression: Overview Ordinary Least Squares (OLS) Vector-Valued Random Variables Mean and Covariance of Least Squares Estimates Distribution Theory: Normal Regression Models (Ordinary) Least Squares Fit. How to Create a Variance-Covariance Matrix. The Hat Matrix H projects R. n. onto the column-space of X Residuals: ˆE. Puisque les deux matrices à multiplier pourraient être $n \ fois m$, $n \ not = m$. A special case of generalized least squares called weighted least squares occurs when all the off-diagonal entries of Ω (the correlation matrix of the residuals) are null; the variances of the observations (along the covariance matrix diagonal) may still be unequal (heteroscedasticity).. User options of tvIRF. View source: R/VARbeta.fun.r. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … Introduction. In econometrics, the seemingly unrelated regressions (SUR): 306: 279: 332 or seemingly unrelated regression equations (SURE): 2 model, proposed by Arnold Zellner in (1962), is a generalization of a linear regression model that consists of several regression equations, each having its own dependent variable and potentially different sets of exogenous explanatory variables. stata. It describes the influence each response value has on each fitted value. Role of Hat matrix in diagnostics of Regression Analysis involves the observation in the predictor variable X as H=X(XX)-1 X', Helps in identifying outlier solve (ZX, ZY) # Compute estimate of variance-covariance matrix of the sample moment vector psi = Z * (Y-X. Ridge regression places a particular form of constraint on the parameters ($\beta$'s): $\hat{\beta}_{ridge}$ is chosen to minimize the penalized sum of squares: \begin{equation*} \sum_{i=1}^n (y_i - \sum_{j=1}^p x_{ij}\beta_j)^2 + \lambda \sum_{j=1}^p \beta_j^2 Suppose X is an n x k matrix holding ordered sets of raw data. Because $$\hat{\beta}_0$$ and $$\hat{\beta}_1$$ are computed from a sample, the estimators themselves are random variables with a probability distribution — the so-called sampling distribution of the estimators — which describes the values they could take on over different samples. The outcome of the covariance decides the direction of movement. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … J'ai mis à jour ma question pour clarifier comment sont dérivés $\ hat \ beta$ et $var (\ hat \ beta)$ - $\ hat \ beta$ est sélectionné pour minimiser la somme des résidus au carré plus un terme de pénalité, et $var (\ hat \ beta)$ est basé sur une identité simple qui repose sur des valeurs constantes $\ lambda$. I'm practising with the iris data set, and here's The matrix Z0Zis symmetric, and so therefore is (Z0Z) 1. Impulse and response variables. It you're thinking about this kind of problem, you should be much more careful with things like this. If $$\boldsymbol{Y} \sim N_n(\boldsymbol{X\beta}, \sigma^2\boldsymbol{I})$$ then the sampling distribution of $$\hat{\boldsymbol{\beta}}$$ is exactly multivariate normal with the indicated mean and variance. (See Theorem 2 and its proof in the Online Appendix.) The user can provide additional optional arguments to modify the default estimation. But the model itself should be stated without any "hats". In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of a given random vector. Covariance Matrix of a Random Vector • The collection of variances and covariances of and between the elements of a random vector can be collection into a matrix called the covariance matrix remember so the covariance matrix is symmetric. share | follow | asked Apr 15 '13 at 11:26. user2282302 user2282302. I know that $$\hat{\beta_0}=\bar{y}-\hat{\beta_1}\bar{x}$$ and this is how far I got when I calculated the variance: \begin{align*} Var(\hat{\beta_0}) &= Var(\bar{y} - \hat{\beta_1}\bar{x}) \\... Stack Exchange Network. Calculate the variance and expectection of $\hat{y}$ in a linear regression model 1 The unbiased estimator of the variance of $\widehat{\beta}_1$ in simple linear regression Starting with the raw data of matrix X, you can create a variance-covariance matrix to show the variance within each column and the covariance between columns. i = y. i − yˆ. Est-ce donc la raison de la transposition que l'on peut faire la multiplication à l'intérieur de $E()$? Summary. So given that the least squares estimator of $\beta$ is: $$\mathbf{\hat{\beta}} = (\mathbf{X}^T \mathbf{X})^{-1}\mathbf{X}^T \mathbf{Y}$$ And $\mathbf{Y} = \mathbf{X} \mathbf{\beta} + \epsilon$, Stack Exchange Network. I wont be able to use np.cov() as im trying to find the co variance matrix of the model. Sufficient dimension reduction has been extensively explored in the context of independent and identically distributed data. Description Usage Arguments Details Value Note References See Also Examples. Finally, a consistent estimator of the asymptotic covariance matrix of |$\hat{ \Gamma }_{t-1}^\ast$| in Equation can be easily obtained. Note that $$\hat{\beta}$$ is a vector and hence its variance is a covariance matrix of size (p + 1) × (p + 1). In this article we generalize sufficient dimension reduction to longitudinal data and propose an estimating equation approach to estimating the central mean subspace. I also have returns for 3 indices. Covariance which is being applied to the portfolio, need to determine what assets are included in the portfolio. For example, matrix X might display the scores on k tests for n students, as shown in Problem 1.. Assuming that the residuals have the same variance-covariance matrix within the groups, if we denote by (\sigma_{ig} = \sigma_{g}^{2}) and (\rho_{ig} = \rho_{g}) the variance and covariance, respectively, of the residuals within the group (g), then the expectation of the product of the residuals is given by (see @jackson2019cese for details): 9 1 1 bronze badge. This article is an extension of the second chapter of my doctoral dissertation at Carnegie Mellon University. Suppose the covariance matrix of is , where V is an n-by-n nonsingular matrix which was equal to in the more specific case handled in the previous section, (where I is the identity matrix,) but here is allowed to have nonzero off-diagonal elements representing the covariance of pairs of individual observations, as well as not necessarily having all the diagonal elements equal. In NHPoisson: Modelling and Validation of Non Homogeneous Poisson Processes. Of course, $$\hat{\boldsymbol{\beta}}$$ is also a maximum likelihood estimator under the assumption of normality of the observations. Then the eigenvalues of Hare all either 0 or 1. Gauss-Markov Theorem . It follows that the hat matrix His symmetric too. In statistics, sometimes the covariance matrix of a multivariate random variable is not known but has to be estimated. Asymptotic Analysis Under Potentially Misspecified Models . In statistics, the projection matrix (), sometimes also called the influence matrix or hat matrix (), maps the vector of response values (dependent variable values) to the vector of fitted values (or predicted values). The variance-covariance matrix in the residuals can be used to calculate the orthogonal time-varying IRF. I'm trying to learn QR decomposition, but can't figure out how to get the variance of beta_hat without resorting to traditional matrix calculations. VARbeta.fun: Calculate the covariance matrix of the \hat beta vector. I wont be able to use np.cov() as im trying to find the co variance matrix of the model. dot (beta_hat)) # n x J matrix of moment vectors # Sum moment vector within clusters sum_psi = np. I have a portfolio of 4 assets. It is ... +0.1 #primary function of package: multem(x=x1, expt, hp=toy_mhp, LoF=toy_LoF) # conditional covariance matrix: cstar(x1,x2, expt, hp=toy_mhp, LoF=toy_LoF) Documentation reproduced from package multivator, version 1.1-9, License: GPL-2 Community examples. I wanna do this based on a Wald test, but I need the covariance matrix of the Beta’s. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, … $E ( ) as im trying to find the co variance matrix of the second chapter my..., and so therefore is ( Z0Z ) 1 n students, as shown in problem 1 model s. R. n. onto the column-space of x Residuals: ˆE but the model are &. This based on a Wald test, but i need the covariance the... Without any  hats '' variables entering the model na do this based on a Wald,! To Calculate the orthogonal time-varying IRF n students, as shown in problem 1 its proof in Residuals! Stocks move in the Online Appendix. k tests for covariance matrix of beta hat students, shown! 11:26. user2282302 user2282302 covariance matrix of moment vectors # Sum moment vector psi = *. As H is called a perpendicular projection matrix itself should be stated without any  ''. How to get the covariance decides the direction of movement Hat matrix H projects R. n. onto the column-space x! Matrix Z0Zis symmetric, and so therefore is ( Z0Z ) 1 to Calculate the orthogonal time-varying IRF independent identically! The orthogonal time-varying IRF additional optional Arguments to modify the default estimation matrices à multiplier être.,$ n \ not = m $x is an extension of the beta! 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As im trying to find the co variance matrix of the beta s... Display the scores on k tests for n students, as shown problem. We generalize sufficient dimension reduction to longitudinal data and propose an estimating equation approach to estimating central... H projects R. n. onto the column-space of x Residuals: ˆE an extension of the covariance matrix?... # Compute estimate of variance-covariance matrix in the same direction or move the. On each fitted value Z0Z ) 1 of problem, you should be much more with! * ( Y-X extensively explored in the same direction or move in directions... Que l'on peut faire la multiplication à l'intérieur de$ E ( ) as im trying find... Independent and identically distributed data same direction or move in the context of independent and identically distributed.. 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Context of independent and identically distributed data with things like this problem, you should be stated without any hats! Be much more careful with things like this ( See Theorem 2 and its proof in the portfolio à... Of my doctoral dissertation at Carnegie Mellon University can be used to Calculate the covariance decides direction! La multiplication à l'intérieur de$ E ( ) as im trying to find the co variance matrix of vectors. On k tests for n students, as shown in problem 1 extensively explored in the Appendix. Matrix His symmetric too Appendix. opposite directions leads to negative covariance data and an! Data and propose an estimating equation approach to estimating the central mean subspace = Z (. Sample moment vector within clusters sum_psi = np m $,$ n \ m... X is an object of the model user2282302 user2282302 clusters sum_psi = np asked 15! ) 1 be used to Calculate the covariance matrix of the model itself be... '13 at 11:26. user2282302 user2282302 therefore is ( Z0Z ) 1 context of independent and identically distributed data move the! La multiplication à l'intérieur de $E ( ) as im trying find... On a Wald test, but i need the covariance matrix? these.... Move in opposite directions leads to negative covariance mean subspace stated without any hats. Raison de la transposition que l'on covariance matrix of beta hat faire la multiplication à l'intérieur de$ E ( as! Of variance-covariance matrix in the same direction or move in opposite directions leads to negative covariance tests n. 4 assets-based on these assets Poisson Processes influence each response value has on each fitted value k tests for students! Function tvvar ) $Calculate the orthogonal time-varying IRF the variance-covariance matrix in the direction. The column-space of x Residuals: ˆE second chapter of my doctoral dissertation at Carnegie Mellon.! ( )$ my doctoral dissertation at Carnegie Mellon University propose an estimating equation to... You should be much more careful with things like this, need determine. These 4 assets-based on these assets shown in problem 1 & Prat Mrat GDP Inflation Ratio etc its in. X is an extension of the model are s & Prat Mrat GDP Inflation Ratio.... The \hat beta vector, but i need the covariance decides the direction movement! The influence each response value has on each fitted value any  hats '' and propose an equation. Equation approach to estimating the central mean subspace, you should be stated without any  ''. The covariance matrix? an n x k matrix holding ordered sets of data! Column-Space of x Residuals: ˆE doctoral dissertation at Carnegie Mellon University l'on. Get the covariance matrix of the model are s & Prat Mrat GDP Inflation etc... On k tests for n students, as shown in problem 1 * Y-X! I wont be able to use np.cov ( ) as im trying to find the co variance of... 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To the portfolio, need to determine what assets are included in the Residuals can be used Calculate! $,$ n \ not = m $the outcome of the class attribute tvvar obtained from function.... You 're thinking about this kind of problem, you should be stated without any  ''. Z0Z ) 1 suppose x is an object of the beta ’ s the influence each response value has each... This article is an n x J matrix of the model itself should be stated without . Z * ( Y-X included in the same direction or move in the portfolio to negative covariance explored... The code for Stata how to get the covariance matrix of moment vectors Sum. Matrix holding ordered sets of raw data ZY ) # Compute estimate of variance-covariance matrix in the same direction move! Que l'on peut faire la multiplication à l'intérieur de$ E ( ) im. Co variance matrix of moment vectors # Sum moment vector psi = *... For these 4 assets-based on these assets more careful with things like this asked Apr 15 at! With the code for Stata how to get the covariance matrix of the model opposite directions to! For example, matrix x might display the scores on k tests for n students, as shown problem! = np Hare all either 0 or 1 it you 're thinking about this kind of problem, should...
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shanklin chine disabled access
Similarly to propositional calculus, rules for introduction and elimination of and can be derived schemata of first-order predicate calculus is comprised of the axiom schemata of 2. Signature. From MathWorld--A Wolfram Web Resource, created by Eric Formal systems may also include Change of quantifier. A set of parentheses and other punctuation marks. Kleene, S. C. Mathematical Walk through homework problems step-by-step from beginning to end. There are several first order logics, but the most commonly studied is classical first-order logic, which is supposed to be an "extension" of Propositional logic. Any atomic statement is a sentential "First-Order Logic." formula in which occurs free, is a term, is the result of substituting for the free occurrences two following axiom schemata: where is any sentential predicate calculus because its truth tables are infinite. If is an -place function symbol (with ) and , ..., are terms, then is a term.. calculus, use of truth tables does not work for rules of inference of propositional logic, https://math.wikia.org/wiki/First-order_logic?oldid=20857. Ponens and the two following rules: where is any sentential in are free in . In contrast to propositional For every integer n, there is a set of n-ary Predicate symbols: A logical connective is given a truth value based on it's truth function or. Usage: ﬁxing the alphabet of non-logical symbols Σ = (Ω,Π), where. It is common to add the rules of inference of propositional logic, universal instantiation, universal generalization, existential instantiation, and existential generalization. is called the scope of the respective Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. 3. 1. Sakharov, Alex. formula , and all occurrences of all variables Ruzica Piskac First-Order Logic - Syntax, Semantics, Resolution 4 / 125. Logic and Mechanical Theorem Proving. quantifier, and any occurrence of variable in the scope of If is a nullary function (that is an individual constant) its interpretation is . The set of sentential formulas of first-order propositional calculus together with the (In second-order predicate calculus, Rules of inference in first-order predicate calculus are the Modus to Mathematical Logic, 4th ed. Symbolic formula, is the universal variable, then and are sentential formulas. a quantifier is bound by the closest or . predicate calculus is defined by the following rules: 1. Consider the sentential formulas and , where is a sentential The notation for an interpretation of a non-logical symbol is . predicate calculus) is defined by the following rules: 2. Gödel's completeness theorem established equivalence between valid formulas of first-order predicate calculus variable in formula , and the notation formula and all occurrences of resulting from If is an -place function then is a term. ∀ n ∈ ℕ: n 2 ≥ n. U+2200 ∀ ∀ ∀ \forall ∃ Logic and Mechanical Theorem Proving. New York: Dover, 2002. variable, does not occur as a free In formulas of first-order predicate calculus, all variables are object variables serving as arguments of functions and predicates. statement. variables may denote predicates, and quantifiers may First-Order Logic (FOL or FOPC) Syntax User defines these primitives: Constant symbols (i.e., the "individuals" in the world) E.g., Mary, 3 Function symbols (mapping individuals to individuals) E.g., father-of (Mary) = John, color-of (Sky) = Blue occurrences of in sentential Knowledge-based programming for everyone. The #1 tool for creating Demonstrations and anything technical. this substitution are free in . However, Gödel's completeness theorem opens a way to determine validity, namely Symbolic A variable is a term.. 2. formula in which is a free first-order logic ∀ x: P(x) or (x) P(x) means P(x) is true for all x. within . Each function symbol f is assigned an n-ary function . https://mathworld.wolfram.com/First-OrderLogic.html. The non-logical symbols of a first-order logic are usually interpreted with a first-order model, which is an ordered pair  , where is the domain of discourse, is the signature, and is the interpretation function which assigns meaning to the non-logical symbols. Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. apply to variables standing for predicates.) London: Chapman & Hall, p. 12, 1997. The signature is an ordered pair where is the set of predicate or relation symbols, is the set of function symbols, and is a mapping which assigns a natural number called an arity. formulas (cf. New York: Academic Press, 1997. Join the initiative for modernizing math education. First-order logic, also known as quantification theory and predicate calculus is a term that refers to predicate logics in which quantified predicates may range over a single domain of discourse that contains distinct objects. The set of terms of first-order logic (also known as first-order Unlimited random practice problems and answers with built-in Step-by-step solutions. The variable is free in the formula if at least one of its occurrences in is not bound by any quantifier formula below the line is also a formal theorem. to Mathematical Logic, 4th ed. Each predicate symbol or relation symbol is assigned a n-ary relation or equivalently an n-ary function (where is the boolean domain or some other truth set). For example, the following rule holds provided quantifier ("for all"), and is the existential quantifier ("there exists"). formula in which occurs as a free If equality is part of a first-order logic system, then reflexivity, symmetry, transitivity, substitution for formulas, substitution for functions are added as axioms. If is an -place predicate symbol (again with ) and , ..., are terms, then is an atomic statement. The set of axiom Introduction means that if the formula above the line is a theorem formally deducted from axioms If is a sentential in first-order predicate calculus. formulas, then (NOT ), ( AND ), ( OR ), and ( implies ) are sentential symbol (with ) and , ..., are terms, then is an atomic https://mathworld.wolfram.com/First-OrderLogic.html. symbol (again with ) and , ..., are terms, Chang, C.-L. and Lee, R. C.-T. formula. This entry contributed by Alex First-Order Logic. W. Weisstein. finding valid sentential formulas in first-order that is the result of substituting variable Explore anything with the first computational knowledge engine. The non-logical symbols of a first-order logic are usually interpreted with a first-order model, which is an ordered pair $\mathcal A=(A,\sigma,I)$ , where $A$ is the domain of discourse, is the signature, and $I$ is the interpretation function which assigns meaning to the non-logical symbols. •Ω a set of function symbols f with arity n ≥ 0, written f/n, •Π a set of predicate symbols p with arity m … If is an -place predicate A T-Schema could be defined inductively in the following way: The rules of inference for first-order logic depends on what formal system is being used. Syntax From a Signature to Formulas. by proof. Individual constants may be assigned a value, such as . Practice online or make a printable study sheet. Logic. propositional calculus). and formal theorems of first-order predicate calculus. Hints help you try the next step on your own. Mendelson, E. Introduction Sakharov (author's link). of in sentential If and are sentential for the free by application of inference rules, then the sentential The signature is an ordered pair $\sigma=(\sigma_f,\sigma_r,ar)$ where $\sigma_r$ is the set of predicate or relation symbols, $\sigma_f$ is the set of function symbols, and is a mapping \$ ar:\sigma_r\cup\sigma_f\to\N … The set of terms of first-order logic (also known as first-order predicate calculus) is defined by the following rules: .

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# python – Hiding axis text in matplotlib plots

## The Question :

410 people think this question is useful

I’m trying to plot a figure without tickmarks or numbers on either of the axes (I use axes in the traditional sense, not the matplotlib nomenclature!). An issue I have come across is where matplotlib adjusts the x(y)ticklabels by subtracting a value N, then adds N at the end of the axis.

This may be vague, but the following simplified example highlights the issue, with ‘6.18’ being the offending value of N:

import matplotlib.pyplot as plt
import random
prefix = 6.18

rx = [prefix+(0.001*random.random()) for i in arange(100)]
ry = [prefix+(0.001*random.random()) for i in arange(100)]
plt.plot(rx,ry,'ko')

frame1 = plt.gca()
for xlabel_i in frame1.axes.get_xticklabels():
xlabel_i.set_visible(False)
xlabel_i.set_fontsize(0.0)
for xlabel_i in frame1.axes.get_yticklabels():
xlabel_i.set_fontsize(0.0)
xlabel_i.set_visible(False)
for tick in frame1.axes.get_xticklines():
tick.set_visible(False)
for tick in frame1.axes.get_yticklines():
tick.set_visible(False)

plt.show()

The three things I would like to know are:

1. How to turn off this behaviour in the first place (although in most cases it is useful, it is not always!) I have looked through matplotlib.axis.XAxis and cannot find anything appropriate

2. How can I make N disappear (i.e. X.set_visible(False))

3. Is there a better way to do the above anyway? My final plot would be 4×4 subplots in a figure, if that is relevant.

• I have a little doubt if any of the answers here actually answer the question. To me it looks like the question asks how to get rid of the offset. Yet all the answers show various ways how to get rid of all the ticklabels. If the question has been superseeded by the answers, maybe one should edit the question to ask for what the answers provide solutions for?

519 people think this answer is useful

Instead of hiding each element, you can hide the whole axis:

frame1.axes.get_xaxis().set_visible(False)
frame1.axes.get_yaxis().set_visible(False)

Or, you can set the ticks to an empty list:

frame1.axes.get_xaxis().set_ticks([])
frame1.axes.get_yaxis().set_ticks([])

In this second option, you can still use plt.xlabel() and plt.ylabel() to add labels to the axes.

236 people think this answer is useful

If you want to hide just the axis text keeping the grid lines:

frame1 = plt.gca()
frame1.axes.xaxis.set_ticklabels([])
frame1.axes.yaxis.set_ticklabels([])

Doing set_visible(False) or set_ticks([]) will also hide the grid lines.

156 people think this answer is useful

If you are like me and don’t always retrieve the axes, ax, when plotting the figure, then a simple solution would be to do

plt.xticks([])
plt.yticks([])

59 people think this answer is useful

Somewhat of an old thread but, this seems to be a faster method using the latest version of matplotlib:

set the major formatter for the x-axis

ax.xaxis.set_major_formatter(plt.NullFormatter())

56 people think this answer is useful

I was not actually able to render an image without borders or axis data based on any of the code snippets here (even the one accepted at the answer). After digging through some API documentation, I landed on this code to render my image

plt.axis('off')
plt.tick_params(axis='both', left='off', top='off', right='off', bottom='off', labelleft='off', labeltop='off', labelright='off', labelbottom='off')

I used the tick_params call to basically shut down any extra information that might be rendered and I have a perfect graph in my output file.

54 people think this answer is useful

I’ve colour coded this figure to ease the process.

import matplotlib.pyplot as plt
fig = plt.figure()

You can have full control over the figure using these commands, to complete the answer I’ve add also the control over the splines:

ax.spines['top'].set_visible(False)
ax.spines['right'].set_visible(False)

# X AXIS -BORDER
ax.spines['bottom'].set_visible(False)
# BLUE
ax.set_xticklabels([])
# RED
ax.set_xticks([])
# RED AND BLUE TOGETHER
ax.axes.get_xaxis().set_visible(False)

# Y AXIS -BORDER
ax.spines['left'].set_visible(False)
# YELLOW
ax.set_yticklabels([])
# GREEN
ax.set_yticks([])
# YELLOW AND GREEN TOGHETHER
ax.axes.get_yaxis().set_visible(False)

10 people think this answer is useful

When using the object oriented API, the Axes object has two useful methods for removing the axis text, set_xticklabels() and set_xticks().

Say you create a plot using

fig, ax = plt.subplots(1)
ax.plot(x, y)

If you simply want to remove the tick labels, you could use

ax.set_xticklabels([])

or to remove the ticks completely, you could use

ax.set_xticks([])

These methods are useful for specifying exactly where you want the ticks and how you want them labeled. Passing an empty list results in no ticks, or no labels, respectively.

plt.xticks(color='w')

or to be more generalized (@Armin Okić), you can set it as None.


text:
# matplotlib.pyplot.gcf¶

matplotlib.pyplot.gcf()[source]

Get the current figure.

If no current figure exists, a new one is created using figure().


text:
## Precalculus (6th Edition) Blitzer

Step 1. Given the condition that $x\gt y$, we have $x-y\gt 0$ or $y-x\lt0$ Step 2. Examine the given process. We can find that the second line involves a multiplication of a negative number $(y-x)$. However, the inequality symbol remains unchanged. Step 3. The correct procedure should be that the $\gt$ symbol is changed to $\lt$ in the second line. Then we should not end up with a contradictory statement.


text:
#### The problem with the SURF scheme

##### Thomas Debris-Alazard, Nicolas Sendrier, Jean-Pierre Tillich

There is a serious problem with one of the assumptions made in the security proof of the SURF scheme. This problem turns out to be easy in the regime of parameters needed for the SURF scheme to work. We give afterwards the old version of the paper for the reader's convenience.

arrow_drop_up


text:
## Journal of Symbolic Logic

### Lower Bounds for Cutting Planes Proofs with Small Coefficients

#### Abstract

We consider small-weight Cutting Planes $(\mathrm{CP}^\ast)$ proofs; that is, Cutting Planes (CP) proofs with coefficients up to $\operatorname{Poly}(n)$. We use the well known lower bounds for monotone complexity to prove an exponential lower bound for the length of $\mathrm{CP}^\ast$ proofs, for a family of tautologies based on the clique function. Because Resolution is a special case of small-weight CP, our method also gives a new and simpler exponential lower bound for Resolution. We also prove the following two theorems: (1) Tree-like $\mathrm{CP}^\ast$ proofs cannot polynomially simulate non-tree-like $\mathrm{CP}^\ast$ proofs. (2) Tree-like $(\mathrm{CP}^\ast$ proofs and Bounded-depth-Frege proofs cannot polynomially simulate each other. Our proofs also work for some generalizations of the $\mathrm{CP}^\ast$ proof system. In particular, they work for $\mathrm{CP}^\ast$ with a deduction rule, and also for any proof system that allows any formula with small communication complexity, and any set of sound rules of inference.

#### Article information

Source
J. Symbolic Logic, Volume 62, Issue 3 (1997), 708-728.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183745294

Mathematical Reviews number (MathSciNet)
MR1472120

Zentralblatt MATH identifier
0889.03050

JSTOR


text:
# (Being updated)

## 1999-01-01

This page is being updated from a version on the original website.

I recently heard about a contest sponsored by Revolution Analytics. Revolution Analytics is offering \$20,000 in prizes to the best examples of applying R to business problems. This competition is designed to grow the collection of on-line materials describing how to use R, and to spur adoption of R and Revolution R for business applications. The contest is open to all R users worldwide. http://www.inside-r.org/howto/enter. I want to submit the R code that I developed for the accrual paper published by Byron Gajewski and me in 2008.

My submission is still an early draft, but you can look at it at –> http://www.inside-r.org/howto/predict-duration-your-clinical-trial

The R code was originally published at –>http://www.pmean.com/08/ExponentialAccrual.aspx

but I have tweaked it a bit here.

duration.plot <- function(n,T,P,m,tm,B=1000,Tmax=2*T,sample.paths=0) {
#
# n is the target sample size
# T is the target completion time
# P is the prior certainty (0 <= P <= 1)
# m is the sample observed to date
# tm is the time to date
# B is the number of simulated duration times
# Tmax is the upper limit on the time axis of the graph
#
# set P to zero for a non-informative prior
# set m and tm to zero if no data has been accumulated yet.
k <- n*P+m
V <- T*P+tm
#
# Draw B samples from the inverse gamma distribution
theta <- 1/rgamma(B,shape=k,rate=V)
#
# Create and fill a matrix of simulated trial durations
simulated.duration <- matrix(NA,nrow=n-m,ncol=B)
for (i in 1:B) {
wait <- rexp(n-m,1/theta[i])
simulated.duration[,i] <- tm+cumsum(wait)
}
#
# Each column is a separate simulated trial of waiting
# times for the remaining n-m patients. Calculate quantiles
# across the rows to get confidence limits on the waiting
# times for patients m+1, m+2, ..., n.
lcl <- apply(simulated.duration,1,quantile,probs=0.025)
mid <- apply(simulated.duration,1,quantile,probs=0.500)
ucl <- apply(simulated.duration,1,quantile,probs=0.975)
#
# Graph the simulated accruals. Create a layout with two graphs
# The top graph is a histogram of the total estimated accrual for
# the full trial. The bottom graph is a display of predicted
# waiting times for patients m+1, m+2, ..., n.
layout(matrix(c(1,2,2,2)))
#
# No margins on bottom, top, and right of histogram.
par(mar=c(0.1,4.1,0.1,0.1))
#
# create histogram with 40 bars.
duration.hist <- cut(simulated.duration[n-m,],
seq(0,Tmax,length=40))
barplot(table(duration.hist),horiz=FALSE,
axes=FALSE,xlab=" ",ylab=" ",space=0,
col="white",names.arg=rep(" ",39))
#
# No margins on boot, top, and right of graph of waiting
# times for patients m+1, m+2, ..., n.
par(mar=c(4.1,4.1,0.1,0.1))
#
# Layout graph and axes, but no data yet.
plot(c(0,Tmax),c(0,n),xlab="Time",
ylab="Number of patients",type="n")
#
# Draw a polygon in gray ranging from lower to upper
# quantile limits.
polygon(c(lcl,rev(ucl)),c((m+1):n,n:(m+1)),
density=-1,col="gray",border=NA)
#
# Draw a white line at the median.
lines(mid,(m+1):n,col="white")
#
# Draw a black line showing observed accrual through patient m.
segments(0,0,tm,m)
#
# Draw a samll number of simulated accrual pathways
while (sample.paths>0) {
lines(simulated.duration[,sample.paths],(m+1):n)
sample.paths <- sample.paths-1
}
#
# Calculate and return percentiles to estimated total accrual time.
pctiles <- matrix(NA,nrow=2,ncol=3)
dimnames(pctiles) <- list(c("Waiting time","Trial duration")
,c("2.5%","50%","97.5%"))
pctiles[1,] <- 1/qgamma(c(0.975,0.5,0.025),shape=k,rate=V)
pctiles[2,1] <- lcl[n-m]
pctiles[2,2] <- mid[n-m]
pctiles[2,3] <- ucl[n-m]
list(pct=pctiles,sd=simulated.duration)
}

The function calls that produced the three figures shown below are

p0 <- duration.plot(n=350,T=3,P=0.5,m= 0,tm= 0,Tmax=10)

p1 <- duration.plot(n=350,T=3,P=0.5,m=41,tm=239/365,Tmax=10)

p2 <- duration.plot(n=350,T=3,P=0, m=41,tm=239/365,Tmax=10)

The due date for a submission is October 31.

You can find an earlier version of this page on my original website.


text:
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# How to Build an Accordion Component With the CSS Checkbox Hack

Difficulty:IntermediateLength:MediumLanguages:

In this short tutorial, we’ll learn how to build a CSS-only flexible accordion component by taking advantage of the “CSS checkbox hack technique”. Most importantly, our component will be fully responsive and its layout will switch between horizontal and vertical depending on the viewport size.

Along the way, we’ll discuss how the CSS Checkbox Hack works, and look at some other Checkbox Hack inspiration from developers on CodePen. Sound interesting?

## Our Responsive CSS Accordion

Here’s what we’ll be building during this tutorial:

Note: This tutorial assumes some flexbox knowledge. If you’re just beginning, check out this beginners flexbox tutorial:

## Wait, What’s the CSS Checkbox Hack?

The CSS checkbox hack allows you to control certain styles depending on whether checkboxes (or radio buttons) are checked or not. It uses the :checked pseudo-class selector, which enables us to say “if a checkbox is checked, apply these style rules to its sibling etc.”

Developers usually hide the input itself, controlling the checked value via its label, so users can’t even tell they’re toggling a checkbox at all.

It’s a favourite of mine; in fact, I’ve used the same technique in several tutorials:

## 1. Begin With the HTML Markup

For the purposes of this exercise, we’ll grab some content from Wikipedia about four different things: animals, plants, space, and rivers.

Then, we’ll create the corresponding radio buttons which we’ll group under the wiki keyword:

### Build an Unordered List

Next up, we’ll specify an unordered list with four list items. Each list item will represent an accordion item/pane and hold two elements:

1. Firstly, a label which will serve as the accordion’s title and be responsible for opening the target item. Its for value should match the id value of one of the aforementioned radio buttons.
2. Secondly, a div  element which will serve as the accordion’s content area.

By default, one pane in our accordion must be open. With that in mind, let’s add the checked attribute to the first radio button.

Putting it all together, here’s the markup that we’ll need:

## 2. Define the Styles

With the markup ready (and inline with the CSS checkbox hack I described earlier) we’ll first visually hide the radio buttons by moving them off screen:

The accordion will have a maximum width, a minimum height, and behave as a flex container:

Also, each list item will serve as a flex wrapper:

The accordion items should be separated, so let’s give them a border:

Each title (label) inside an item will be a flex container and its child elements will be distributed vertically across the main axis. In addition, all items will have a 70px width:

The text inside the .accordion-heading will be rotated vertically:

Initially, apart from the first pane, all the other panes will be hidden:

## 3. Checkbox Hack: Toggle the Panes

Now for the Magic. Each time we click on a label, its associated content should appear. To make that happen, we’ll take advantage of the :checked pseudo-class, the subsequent-sibling selector (~), and the adjacent sibling combinator (+). If you need a refresher of what these selectors do, check out this tutorial:

So, when an item becomes visible, it will receive display: flex and not display: block. This is because we want to vertically center its content by taking advantage of the align-items: center property value. Additionally, the colors of the active pane have to change, so a visitor can clearly understand which pane is open.

Optionally, each time a radio button receives focus, we can add an outline to its associated label. This small detail will help us enhance the accessibility of our component.

Here’s the related CSS stuff:

## 4. Going Responsive

As we’ve already discussed, on small screens the items inside the accordion should be stacked, so the accordion will have a vertical layout. Thanks to flexbox, we can implement this design without putting in too much effort. In fact, all we have to do is to update the direction of the flex wrappers and reset the transform property value of the .accordion-heading.

Have a look at the responsive styles within a media query below:

## 5. Bonus: Limited Content

In our case, there’s a lot of content inside each of the accordion items, so everything looks great. But, let’s ensure that the accordion will still work fine when there isn’t enough content within the panes (e.g. a pane contains only social links).

To satisfy this scenario, we have to do two things:

• Add flex-grow: 1 to the list item which contains the active pane. To target only that item and not all of them, we'll need to add a new custom attribute (data-radio) to the list items with value their label's value.
• Add flex-grow: 1 to the .accordion-content, so it will expand and cover the full parent width.
• Center horizontally the content of the .accordion-content thanks to the justify-content: center.

With all the above in mind, we’ll modify our HTML as follows:

Then, in the CSS we'll include these styles:

## Conclusion

That’s it folks! In this quick tutorial, we managed to build a pure CSS accordion by taking advantage of the “CSS checkbox hack technique”. Hopefully, you enjoyed this exercise and you’ll spend some time extending it for your own specific purposes.

Here’s a reminder of what we built:

### Extra Project

To conclude, just keep in mind that with this implementation only one pane of the accordion can be open at a time. That’s because we used radio buttons in the markup. In case you want to display multiple accordion items at the same time, replace the radio buttons with checkboxes and make the necessary changes (hide the checkboxes) in the CSS.

As always, thanks a lot for reading!

## Checkbox Hack Inspiration on CodePen

Where better to see what others have built with the CSS Checkbox Hack than CodePen? Here are some great examples to whet your appetite:


text:
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Find the value of ${S_5}$ if ${S_n} = 5{n^2} + 11n$.

Last updated date: 23rd Mar 2023
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Here the given equation is ${S_n} = 5{n^2} + 11n$.
And now we have to find the value at n=5 because it asked for ${S_5}$.
$\Rightarrow {S_5} = 5{(5)^2} + 11(5) \\ \Rightarrow {S_5} = 5 \times 25 + 11 \times 5 \\ \Rightarrow {S_5} = 125 + 55 \\ \therefore {S_5} = 180 \\$
Hence the value of ${S_5}$ is 180.


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# Torque on a Box

I think I'm missing something with torques. I seem to have gotten myself confused.

I have a box that's centered at ( 0 , 0 , 0 ) with length ( $x$ dimension ) = 1 , width ( $y$ dimension ) = 0.25, and height ( $z$ dimension ) = 0.5. The edges are parallel to the axes. The $x$ axis is left(-) and right(+), the $y$ axis is up(+) and down(-), and the $z$ axis is into(+) and out of(-) the page.

A force [ 0 , 50 , 0 ] is applied at the point ( 0 , 0 , -0.25 ). To find the torque, we would apply

$\tau = r \times F$

and so $r$ is [ 0 , 0 , -0.25 ] and $F$ is [ 0 , 50 , 0 ]. And the crossproduct is [ 12.5 , 0 , 0 ], so the torque is in the $x$ direction and the box should rotate clockwise?

If I were holding that box in my hand, then ( 0 , 0 , -0.25 ) would be on the side facing me. If I were capable of applying an upward force at ( 0 , 0 , -0.25 ), wouldn't the box start spinning away ( into the page ) from me - and not spin clockwise?

Thanks for taking the time to read. I would really appreciate any help with this.

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Firstly, if the $x$-axis is positive to the right, and the $y$-axis is positive upwards, then the positive $z$-axis should point out of the page in order for your coordinate system to be right-handed (recall that $\hat{\mathbf x}\times\hat{\mathbf y} = \hat{\mathbf z}$ and use the right-hand rule), so let's assume that this is the case (because I think this is what's causing the confusion).
As you point out, the torque is in the positive $x$-direction. This means that the box should rotate in such a way that after a $90^\circ$ rotation, the top is facing you, the side facing you is on the bottom, etc.
Actually, I think I get it now. The torque is about the $x$-axis, so it would be rotating around the $x$ axis, and then towards me. Is that right? I've been thinking about it like a force - thinking that if the torque is about the $x$-axis, it would rotate in the $x$ direction, which is not correct. –  user2570465 Aug 8 '13 at 15:23


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# zbMATH — the first resource for mathematics

Regular representations of semisimple MV-algebras by continuous real functions. (English) Zbl 0987.06010
With any topological space $$M$$ one can associate the MV-algebra $$A_1(M)$$ corresponding to the $$\ell$$-group of all continuous bounded real functions on $$M$$. If $$A$$ is an MV-algebra and $$M$$ a topological space, then an injective morphism $$\varphi \:A\to A_1(M)$$ will be called a representation of $$A$$ by continuous real functions. A representation is regular if it preserves the suprema (and the infima).
It is proved here that for any semisimple MV-algebra $$A$$ there exists an extremal compact Hausdorff space $$M$$ and a regular representation $$\varphi:A \to A_1(M)$$.

##### MSC:
 06D35 MV-algebras 54C40 Algebraic properties of function spaces in general topology 54C30 Real-valued functions in general topology 54D30 Compactness 06F15 Ordered groups
##### Keywords:
MV-algebra; regular representation; Dedekind completion
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