Transposing instrument - Wikipedia

Transposing instrument - Wikipedia: Instruments whose music is typically notated in this way

The Definition and Purpose of the Zero Article

The Definition and Purpose of the Zero Article: count nouns are sometimes used without an article, especially when they are referred to generically. The same is true when the noun is plural but of indefinite number.

Lecture 22 - Chapter 2: Symmetric Monoidal Preorders - Azimuth Forum

Lecture 22 - Chapter 2: Symmetric Monoidal Preorders - Azimuth Forum: "will be"



↑ They ‘hear’ words rather than read them.

'via Blog this'

Lecture 21 - Chapter 2: Monoidal Preorders - Azimuth Forum

Lecture 21 - Chapter 2: Monoidal Preorders - Azimuth Forum: "use ⊗ so I'll often use that. We pronounce this symbol "tensor"."



'via Blog this'

SpaceWeather.com -- News and information about meteor showers, solar flares, auroras, and near-Earth asteroids

SpaceWeather.com -- News and information about meteor showers, solar flares, auroras, and near-Earth asteroids: "We've already had one Blue Moon in January 2018, a month bookended by full Moons on the 2nd (02:24 UTC) and 31st (13:27 UTC). It's happening again in March, with full Moons on the 2nd (00:51 UTC) and 31st (12:37 UTC). The last time two Blue Moons occurred in such quick succession was January and March of 1999."

'via Blog this'



Slacks of 24 hours at the beginning of the month and  ~12 hours around the end.

Immersion (mathematics) - Wikipedia

Immersion (mathematics) - Wikipedia: "For example, the Möbius strip has non-trivial tangent bundle, so it cannot immerse in codimension 0 (in R2), though it embeds in codimension 1 (in R3).

"



'via Blog this'

Orientability - Wikipedia

Orientability - Wikipedia: " a loop going around one way on the surface can never be continuously deformed (without overlapping itself) to a loop going around the opposite way."



 that is homeomorphic to the Möbius strip. Thus, for surfaces, the Möbius strip may be considered the source of all non-orientability.

'via Blog this'

Topological group - Wikipedia

Topological group - Wikipedia: "The groups mentioned so far are all Lie groups, meaning that they are smooth manifolds in such a way that the group operations are smooth, not just continuous."



'via Blog this'

An Approach to Greek Lettering

by Michael A. Covington



'via Blog this'

Greek alphabet names in greek

barbershop is µπαρµπερικo ´ = barberiko



'via Blog this'

[PDF] Schuller's Geometric Anatomy of Theoretical Physics, Lectures 1-17 - Free Download PDF

[PDF] Schuller's Geometric Anatomy of Theoretical Physics, Lectures 1-17 - Free Download PDF:

 {b} is neither open nor closed

Example 5.7. The interval [0, 1] is compact in (R, Ostd). The one-element set containing (−1, 2) is a cover of [0, 1], but it is also a finite subcover and hence [0, 1] is compact from the definition.

It is clear that removing even one element from C will cause C to fail to be an open cover of R. Therefore, there is no finite subcover of C and hence, R is not compact.

Example

4.6

.

Let

M

=

{

a,b,c

}

and let

O

=

{

∅

,

{

a

}

,

{

a,b

}

,

{

a,b,c

}}

. Then

{

a

}

is open

but not closed,

{

b,c

}

is closed but not open, and

{

b

}

is neither open nor close

Example

4.6

.

Let

M

=

{

a,b,c

}

and let

O

=

{

∅

,

{

a

}

,

{

a,b

}

,

{

a,b,c

}}

. Then

{

a

}

is open

but not closed,

{

b,c

}

is closed but not open, and

{

b

}

is neither open nor close

'via Blog this'

Incidence geometry - Wikipedia

Incidence geometry - Wikipedia: "Every triple of distinct points is incident with precisely one cycle.
For any flag (P, z) and any point Q not incident with z there is a unique cycle z∗ with P I z∗, Q I z∗ and z ∩ z∗ = {P}. (The cycles are said to touch at P.)
Every cycle has at least three points and there exists at least one cycle."



'via Blog this'

Homogeneous space - Wikipedia

Homogeneous space - Wikipedia: " Erlangen program, one may understand that "all points are the same", in the geometry of X. This was true of essentially all geometries proposed before Riemannian geometry, in the middle of the nineteenth century.

Thus, for example, Euclidean space, affine space and projective space are all in natural ways homogeneous spaces for their respective symmetry groups. "



'via Blog this'

Erlangen program - Wikipedia

Erlangen program - Wikipedia: "only the incidence structure and the cross-ratio are preserved under the most general projective transformations."



'via Blog this'

Why is group theory important?

Why is group theory important?: "we expect the laws of physics to be unchanging in time. This is an invariance under "translation" in time, and it leads to the conservation of energy. Physical laws also should not depend on where you are in the universe. Such invariance of physical laws under "translation" in space leads to conservation of momentum. Invariance of physical laws under (suitable) rotations leads to conservation of angular momentum. "



'via Blog this'

Definite quadratic form - Wikipedia

Definite quadratic form - Wikipedia: "the quadratic form is called positive definite or negative definite."



'via Blog this'

Riemannian manifold - Wikipedia

Riemannian manifold - Wikipedia: " real, smooth manifold M equipped with an inner product {\displaystyle g_{p}} on the tangent space {\displaystyle T_{p}M} at each point {\displaystyle p} that varies smoothly from point to point "



'via Blog this'

Lie Groups (why do mirrors reverse left and right, but not up and down?)

Lie Groups: "Digression: the well-known puzzle, ``why do mirrors reverse left and right, but not up and down?'' is resolved mathematically by pointing out that a mirror perpendicular to the y-axis performs the reflection:

"



'via Blog this'

Group action - Wikipedia

Group action - Wikipedia: "stabilizer subgroup of G with respect to x (also called the isotropy group "



'via Blog this'

Differential of a function - Wikipedia

Differential of a function - Wikipedia: "The differential dy is defined by

{\displaystyle dy=f'(x)\,dx,}
where {\displaystyle f'(x)} is the derivative of f with respect to x, and dx is an additional real variable (so that dy is a function of x and dx). "



'via Blog this'

What first-order linearity really is

‘to first order everything is linear’. This is tautological, of course, but it really says
that most functions are differentiable and we can approximate by the first derivative.



'via Blog this'

Differential (infinitesimal) - Wikipedia

Differential (infinitesimal) - Wikipedia: "the differential dy of y is related to dx by the formula

{\displaystyle \mathrm {d} y={\frac {\mathrm {d} y}{\mathrm {d} x}}\,\mathrm {d} x,}
where dy/dx denotes the derivative of y with respect to x."



'via Blog this'

One-form - Wikipedia

One-form - Wikipedia: "Many real-world concepts can be described as one-forms:

Indexing into a vector"



'via Blog this'

One-form - Wikipedia

One-form - Wikipedia: "Many real-world concepts can be described as one-forms:

"



'via Blog this'

Linear form - Wikipedia

Linear form - Wikipedia: "Any linear functional L is either trivial (equal to 0 everywhere) or surjective onto the scalar field."



'via Blog this'

Linear form - Wikipedia

Linear form - Wikipedia: "A state of a quantum mechanical system can be identified with a linear functional."



'via Blog this'

Quadrature (mathematics) - Wikipedia

Quadrature (mathematics) - Wikipedia: "In mathematics, quadrature is a historical term which means determining area."



'via Blog this'

One-form - Wikipedia

One-form - Wikipedia: "The usage of one-form in this context usually distinguishes the one-forms from higher-degree multilinear functionals on the space"



'via Blog this'

List of mathematical jargon - Wikipedia

List of mathematical jargon - Wikipedia: "canonical"



'via Blog this'

Function space - Wikipedia

Function space - Wikipedia: " if X is also vector space over F, the set of linear maps X → V form a vector space over F with pointwise operations (often denoted Hom(X,V)). One such space is the dual space of V: the set of linear functionals V → F with addition and scalar multiplication defined pointwise."



'via Blog this'

Function space - Wikipedia

Function space - Wikipedia: "the set of functions from any set X into a vector space have a natural vector space structure given by pointwise addition and scalar multiplication."



'via Blog this'

Pointwise - Wikipedia

Pointwise - Wikipedia: "By taking some algebraic structure {\displaystyle A} in the place of {\displaystyle R} , we can turn the set of all functions {\displaystyle X} to the carrier set of {\displaystyle A} into an algebraic structure of the same type in an analogous way."



'via Blog this'

Hamiltonian

It turns out that giving the position and momentum of a particle specifies a point
in a space called the ‘phase space’ or ‘state space’ of the particle. Mathematically this is often a
‘cotangent bundle’ — an important concept from the theory of manifolds. A cotangent bundle is
an example of a ‘symplectic manifold’, which in turn is an example of a more general thing called a
‘Poisson manifold’.



'via Blog this'

Classical mechanics observables a commutative and Lie algebra

In fact for all classical mechanics problems, the algebra of observables C∞(X) is always both a
commutative algebra and a Lie algebra, but even better, they fit together to form a Poisson algebra.



'via Blog this'

Observables

In our example of a particle in R
n, we said an observable is a smooth function
F: R
n × R
n → R
by
(q, p) 7→ F(q, p)
from the phase space to R, sending each point of phase space (or state of our system) to the value
of the observables.



'via Blog this'

configuration vs phase spaces

Hamiltonian mechanics focuses not on the configuration space but on the phase space or
state space, where a point specifies the position and momentum of the system



'via Blog this'

Lorentz group

Schuller lecture 13 defines Lorentz group in a general way as a vector space equipped with a pseudo inner product (that is little bit less positive definite)

Configuration space (physics) - Wikipedia

Configuration space (physics) - Wikipedia: "In classical mechanics, the parameters that define the configuration of a system are called generalized coordinates, and the vector space defined by these coordinates is called the configuration space of the physical system. It is often the case that these parameters satisfy mathematical constraints, such that the set of actual configurations of the system is a manifold in the space of generalized coordinates. This manifold is called the configuration manifold of the system.

"



'via Blog this'

Configuration space (mathematics) - Wikipedia

Configuration space (mathematics) - Wikipedia: "these are used to describe the state of a whole system as a single point in a high-dimensional space."



'via Blog this'

schuller | KUPDF

schuller | KUPDF:



'via Blog this'

Picturing the Correspondences

Picturing the Correspondences: "Since the action preserves distances, one can also consider the action of the group just on the set of vectors of norm 1, i.e., on the sphere.)"



'via Blog this'

Center (group theory) - Wikipedia

Center (group theory) - Wikipedia: "The center of an abelian group, G, is all of G.
"



'via Blog this'

Symplectic group - Wikipedia

Symplectic group - Wikipedia: "Since all symplectic matrices have determinant 1, the symplectic group is a subgroup of the special linear group SL(2n, F).

Notational warning: What is here called Sp(2n, F) is often referred to as Sp(n, F).

More abstractly, the symplectic group can be defined as the set of linear transformations of a 2n-dimensional vector space over F that preserve a non-degenerate, skew-symmetric, bilinear form, see classical group for this definition. Such a vector space is called a symplectic vector space. The symplectic group of an abstract symplectic vector space V is also denoted Sp(V)."



'via Blog this'

Unitary group - Wikipedia

Unitary group - Wikipedia: "Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. For the group of unitary matrices with determinant 1, see Special unitary group.

In the simple case n = 1, the group U(1) corresponds to the circle group, consisting of all complex numbers with absolute value 1 under multiplication. All the unitary groups contain copies of this group."



'via Blog this'

Lie Groups

Lie Groups: " and . These are, colloquially, the groups of rotations in 2-space and 3-space. O(n)  is the group of reflections and rotations."



'via Blog this'

Homogeneous space - Wikipedia

Homogeneous space - Wikipedia: "homogeneous space can be thought of as a coset space without a choice of origin."



'via Blog this'

Group action - Wikipedia

Group action - Wikipedia: "Free (or semiregular or fixed point free) if, given g, h in G, the existence of an x in X with g⋅x = h⋅x implies g = h. Equivalently: if g is a group element and there exists an x in X with g⋅x = x (that is, if g has at least one fixed point), then g is the identity. "



"fix point free" that is, the action transforms free of fixed points

'via Blog this'

Group action - Encyclopedia of Mathematics

Group action - Encyclopedia of Mathematics: "If the homomorphism ρρ is injective the action is faithful: GG may be regarded as a subgroup of SXSX"



'via Blog this'

Transitive Group Action -- from Wolfram MathWorld

Transitive Group Action -- from Wolfram MathWorld: "The space , which has a transitive group action, is called a homogeneous space when the group is a Lie group."



(w) Regular (or simply transitive or sharply transitive) if it is both transitive and free; this is equivalent to saying that for every two x, y in X there exists precisely one g in Gsuch that g⋅x = y. In this case, X is called a principal homogeneous space for G or a G-torsor. 

'via Blog this'

Group action - Wikipedia

Group action - Wikipedia: "The group action is transitive if and only if it has exactly one orbit, i.e., if there exists x in X with G⋅x = X. This is the case if and only if G⋅x = X for all x in X.

"



'via Blog this'

Orthogonal Group -- from Wolfram MathWorld

Orthogonal Group -- from Wolfram MathWorld: "In fact, the orthogonal group is a smooth -dimensional submanifold"
Because the orthogonal group is a group and a manifold, it is a Lie group. 

... preserve the quadratic form , and so they also preserve circles , which are the group orbits.

'via Blog this'

Group Orbit -- from Wolfram MathWorld

Group Orbit -- from Wolfram MathWorld: "For example, consider the action by the circle group on the sphere by rotations along its axis. Then the north pole is an orbit, as is the south pole. The equator is a one-dimensional orbit, as is a general orbit, corresponding to a line of latitude.

"



'via Blog this'

torsors

torsors: " This combination of translations and dilations arises because R is not just a group, but a ring. "



'via Blog this'

torsors

torsors: "You can always pretend a torsor is a group. But, it involves an arbitrary choice!"

:

Any group G is a G-torsor, and every other G-torsor is isomorphic to G - but not canonically!



'via Blog this'

Group representation - Wikipedia

Group representation - Wikipedia: "formally, a "representation" means a homomorphism from the group to the automorphism group of an object"



'via Blog this'

Real coordinate space - Wikipedia

Real coordinate space - Wikipedia: "it is the prototypical real vector space and is a frequently used representation of Euclidean n-space."



'via Blog this'

Euclidean space - Wikipedia

Euclidean space - Wikipedia: "A Euclidean space is not technically a vector space but rather an affine space,"



'via Blog this'

Unit sphere - Wikipedia

Unit sphere - Wikipedia: " forms[edit source]
If V is a linear space with a real quadratic form F:V → R, then { p ∈ V : F(p) = 1 } may be called the unit sphere[1][2] or unit quasi-sphere of V."



'via Blog this'

Orthogonal group - Wikipedia

Orthogonal group - Wikipedia: "The term "orthogonal group" may also refer to a generalization of the above case: the group of invertible linear operators that preserve a non-degenerate symmetric bilinear form or quadratic form[1] on a vector space over a field. In particular, when the bilinear form is the scalar product on the vector space F n of dimension n over a field F, with quadratic form the sum of squares, then the corresponding orthogonal group, denoted O(n, F ), "



'via Blog this'

Symmetry group - Wikipedia

Symmetry group - Wikipedia: "The proper symmetry group is then a subgroup of the special orthogonal group SO(n), and is therefore also called rotation group of the figure."



'via Blog this'

Symmetry group - Wikipedia

Symmetry group - Wikipedia: "The proper symmetry group of an object is equal to its full symmetry group if and only if the object is chiral (and thus there are no orientation-reversing isometries under which it is invariant)."



'via Blog this'

Symmetry group - Wikipedia

Symmetry group - Wikipedia: "The proper symmetry group of an object is equal to its full symmetry group if and only if the object is chiral (and thus there are no orientation-reversing isometries under which it is invariant)."



'via Blog this'

Permutation group - Wikipedia

Permutation group - Wikipedia: "The group of all permutations of a set M is the symmetric group of M, often written as Sym(M)."



'via Blog this'

Group action - Wikipedia

Group action - Wikipedia: " Actions of groups on vector spaces are called representations of the group."



'via Blog this'

Introduction

Introduction: "The Lie algebra is a vector space, but it has additional structure:"



'via Blog this'

Categories

Categories: "Group representations are really only a special case of category representations."



'via Blog this'

Symmetry (physics) - Wikipedia

Symmetry (physics) - Wikipedia: "Continuous symmetries can be described by Lie groups while discrete symmetries are described by finite groups (see Symmetry group)."



'via Blog this'

Introduction to Category Theory/Categories - Wikiversity

Introduction to Category Theory/Categories - Wikiversity: "Monoids in general have one feature that free categories can lack, in that different compositions of arrows can result in the same monoid element. This can happen with categories too, but not with free ones."



'via Blog this'

Introduction to Category Theory/Sets and Functions - Wikiversity

Introduction to Category Theory/Sets and Functions - Wikiversity: "For every set X and every two functions {\displaystyle f_{1},f_{2}:X\to Y} the following holds: {\displaystyle g\circ f_{1}=g\circ f_{2}} if and only if {\displaystyle f_{1}=f_{2}} . In this case, the function g is called monic."



'via Blog this'

Category (mathematics) - Wikipedia

Category (mathematics) - Wikipedia: "The category Cat consists of all small categories, with functors between them as morphisms.

"



'via Blog this'

Category

Read $centerdot as “after” and you'll be able to avoid treating a categorical arrow as a function.
ob (C) and hom (C) are complementary and conjugate.

Monoid as a single object category - Mathematics Stack Exchange

Monoid as a single object category - Mathematics Stack Exchange: "But now if we just observe that the operations on MM are exactlyexactly the same as the operations on Morph(C)Morph(C), we may regard the category CC as the monoid MM. "



'via Blog this'

Category (mathematics) - Wikipedia

Category (mathematics) - Wikipedia: "a small category with a single object x. (Here, x is any fixed set."



'via Blog this'

word choice - "As following" vs "as follows" - English Language Learners Stack Exchange

word choice - "As following" vs "as follows" - English Language Learners Stack Exchange: "as follows is always singular"



'via Blog this'

Category (mathematics) - Wikipedia

Category (mathematics) - Wikipedia: "Any monoid can be understood as a special sort of category (with a single object whose self-morphisms are represented by the elements of the monoid), and so can any preorder.

"



'via Blog this'

SpaceWeather.com -- News and information about meteor showers, solar flares, auroras, and near-Earth asteroids

SpaceWeather.com -- News and information about meteor showers, solar flares, auroras, and near-Earth asteroids: "no other spacecraft could match--before or since,"



'via Blog this'

Categories

Categories: "spinors (fancy for 4-tuples of complex numbers),"



'via Blog this'

Categories

Categories: "the fundamental group is really a functor from the category Top to the category Group."



'via Blog this'

Topological group - Wikipedia

Topological group - Wikipedia:



'via Blog this'

Group action - Wikipedia

Group action - Wikipedia: "When there is a natural correspondence between the set of group elements and the set of space transformations, a group can be interpreted as acting on the space in a canonical way."



'via Blog this'

Topological group - Wikipedia

Topological group - Wikipedia: "The real numbers form a topological group under addition"



'via Blog this'

Categories

Categories: "Diff - smooth manifolds as objects, smooth maps as morphisms
"



'via Blog this'

space-time-wave

For any conjugate pair, there's some constant (could they be a smooth function) which relates one to the other. Waves show up representing some number which relates some space and time aspects of that media.

Mathematical structure - Wikipedia

Mathematical structure - Wikipedia: "Examples are homomorphisms, which preserve algebraic structures; homeomorphisms, which preserve topological structures; and diffeomorphisms, which preserve differential structures."



'via Blog this'

Join and meet - Wikipedia

Join and meet - Wikipedia: "A partially ordered set that is both a join-semilattice and a meet-semilattice is a lattice"



'via Blog this'

Ubiquitous octonions | plus.maths.org

Ubiquitous octonions | plus.maths.org: "bounce back and forth between visualising things and scribbling formulas on paper, and neither one by itself is good enough"



'via Blog this'

Vector space - Wikipedia

Vector space - Wikipedia: "multiplied ("scaled") by numbers, called scalars."



'via Blog this'

Form - Wikipedia

Form - Wikipedia: "Differential form, a concept from differential topology that combines multilinear forms and smooth functions"



'via Blog this'

Homogeneous polynomial - Wikipedia

Homogeneous polynomial - Wikipedia: "A form is also a function defined on a vector space,"



'via Blog this'

Ring (mathematics) - Wikipedia

Ring (mathematics) - Wikipedia: "A commutative ring such that every nonzero element has a multiplicative inverse is called a field."



'via Blog this'

Division algebra - Wikipedia

Division algebra - Wikipedia: "constructing a division algebra of three dimensions"



'via Blog this'

Group action - Wikipedia

Group action - Wikipedia: "Actions of groups on vector spaces are called representations of the group."



'via Blog this'