Search results

...nue our discussion on equivalence relations and classes and also introduce mappings. Problem set exercises will be posted at the end of the lecture. ==Equivalence relations and classes (cont.)== ...

.../TrivialGroupoid/|homomorphisms]], [[../TrivialGroupoid/|homeomorphisms]], and so on) of <math>\mathcal{C}</math>. and <math>h\circ k</math>. The [[../PreciseIdea/|concept]] of commutative diagr ...

...nnections to the theory of [[../Cod/|categories]], especially the concrete and abstract categories of sets, ...ular set theory]], and the classical logic of [[../Predicate/|predicates]] and logical propositions. ...

...that would have made life possible in a such a minimal system-- a grandad and granma of all living organisms on Earth. ...ce2.html} (or [[../TrivialGroupoid/|diagram]]) of sets and set-theoretical mappings}: ...

...d3.png|thumb|360px|Visualization of a convex combination of degree 1., 2. and 3. in Geogebra ]] ...ficients <math>\lambda_i \in [0,1]</math> are from the unit interval [0,1] and the sum of all <math>\lambda_i</math> for the vectors <math>v_i\in V</math> ...

==Lorentz transformation via trigonometric functions== ...ermannian function]] <math>{\rm gd}(\eta)=2\arctan(e^{\eta})-\pi/2</math>, and <math>\vartheta</math> is equivalent to the Lobachevskian [[w:angle of para ...

...sets, the categorical sequence consists of sets joined by set-theoretical mappings in linear fashion, such as: where <math>Hom_{Set}(A,B)</math> is the set of [[../Bijective/|functions]] from set <math>A</math> to set <math>B</math>. ...

...g|thumb|180px|right|The two planes are subsets of the 3-dimensional space, and their intersection is a line.]] ...Vector_space#Definition vector spaces], linear maps between vector spaces, and matrices.<ref>[[Wikipedia: Linear algebra]]</ref> ...

...bles in a fuzzy control system are in general mapped by sets of membership functions similar to this, known as "fuzzy sets". The process of converting a crisp i ...al to either 1 or 0, but the scheme can deal with them as simplified fuzzy functions that happen to be either one value or another (see [[Wikipedia:fuzzy set]]) ...

...ectors/|scalars]]. Both [[../SR/|special relativity]] (Lorentz covariance) and general relativity (general covariance) use covariant basis vectors. ...inverse matrix, that is, an orthogonal matrix, is one in which covariance and contravariance can typically be treated on the same footing. This is of bas ...

...uintuple of sets, <math>I,O</math> and <math>S</math>, and set-theoretical mappings, ...ta</math> being called the ''[[../StableAutomaton/|transition function]]'' and <math>\lambda</math> being called the ''[[../StableAutomaton/|output functi ...

...ets, <math>I</math>,<math>O</math> and <math>S</math>, and set-theoretical mappings, ...}</math> in response to a specific s-automaton input <math>i \in I</math>, and <math>\lambda</math> is the ''output function'' that maps couples of conse ...

...th> in the complex number plane. The point <math>z</math> has a blue color and <math>f(z)= \frac{1}{z}</math> is marked in red color. <math>z</math> is m '''Complex analysis''' is a study of functions of a complex variable. This is a one quarter course in complex analysis at ...

...theorie Wiki2Reveal slides]'''. Individual sections are treated as slides, and changes to the slides are immediately reflected in their content. The follo (1) Paths as continuous mappings from an interval <math>[a,b]</math> into the complex numbers <math>\mathbb{ ...

Individual sections are considered as slides, and changes to the slides immediately affect the content of the slides. (1) Paths as continuous mappings from an interval <math>[a,b]</math> into the complex numbers <math>\mathbb{ ...

...hip between complex differentiability and partial derivatives of component functions of a map from <math>\mathbb{R}^2</math> to <math>\mathbb{R}^2</math>. After === Real and Imaginary Part Functions === ...

===Categorical dynamics and mathematical representations in functional biology=== ...topology]] [[../PreciseIdea/|concepts]], concrete [[../Cod/|categories]], and/or [[../Cod/|graphs]]; thus, they are describing or modeling theost importa ...

...ex ord3.png|thumb|360px|Visualization of convex combinations of order 1, 2 and 3 in Geogebra ]] ...[[w:en:coefficients|coefficients]] are [[w:en:non-negative|non-negative]] and sum to 1. ...

==Lorentz transformation via squeeze mappings== ...ector squeeze mapping.svg|250px|right|thumb|A squeeze mapping relates blue and green parallelograms.]] ...

==Lorentz transformation via hyperbolic functions== [[File:Hyperbolic functions-2.svg|thumb|upright=1.4|A ray through the unit hyperbola {{math|1=''x''<sup ...