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• #### Definitions

action: a literal or a print command in a production system. aleph-null: the cardinality, of the set of natural numbers. AND: the logical operator for conjunction, also written . antecedent: in a conditional proposition (“if then ”) the proposition...

https://astarmathsandphysics.com/university-maths-notes/160-set-theory/2051-definitions.html
• #### Neutrons, Protons and Electrons - Isotopes

An atom is made up of a nucleus, itself made up of protons and neutrons held together by the strong nuclear force, and electrons, which form a 'cloud' and the nucleus. The following points must be made: The number of electrons is equal to the number of...

https://astarmathsandphysics.com/ib-physics-notes/116-atomic-and-nuclear-physics/1222-neutrons-protons-and-electrons-isotopes.html

Russells Paradox is a problem in set theory. Suppose we have a statement about some elements of a set This statement will be true for some values of x and false for others. It is tempting to think that we could form the set of all values of for which...

• #### Sylow's Second Theorem

Sylow's Second Theorem Let be a finite group of order n and let be a prime dividing then the number of distinct Sylow – subgroups (remember that if is the highest power of dividing then the Sylow – subgroup of is that subgroup which has order ). of is...

https://astarmathsandphysics.com/university-maths-notes/146-abstract-algebra-and-group-theory/1712-sylow-s-second-theorem.html
• #### Field Axioms

A field is a set that is a commutative group with respect to two compatible operations, addition and multiplication, with "compatible" being formalized by distributivity, and the caveat that the additive identity (0) has no multiplicative...

https://astarmathsandphysics.com/university-maths-notes/146-abstract-algebra-and-group-theory/1693-field-axioms.html
• #### The Well Ordering Principle

The well-ordering principle states that every non-empty set of positive integers contains a smallest element. It is necessary for this that the set of positive numbers is a well ordered set – that is, they can be arranged in increasing order. Every...

https://astarmathsandphysics.com/university-maths-notes/160-set-theory/2071-the-well-ordering-principle.html
• #### The Life Cycle of Stars

Each star is formed from a massive gas cloud. The cloud condenses over a period of maybe one million years, heating up as it contracts. As it contracts, it starts to spin because of the law of conservation of angular momentum. At some point it will...

https://astarmathsandphysics.com/a-level-physics-notes/166-cosmology/2533-the-life-cycle-of-stars.html
• #### Proof by Induction

The natural ordering on the set of natural numbers (that is, ) has the well ordering property that every nonempty subset of has a least (smallest) element. The principle of induction is the following theorem, which is equivalent to the well ordering...

https://astarmathsandphysics.com/university-maths-notes/148-ananlysis/1751-proof-by-induction.html
• #### Lie Algebras

Every Lie Group has an associated Lie algebra whose underlying vector space is the tangent space of G at the identity element, which completely captures the local structure of the group. We can think of elements of the Lie algebra as elements of the...

https://astarmathsandphysics.com/university-maths-notes/164-vector-calculus/2442-lie-algebras.html
• #### Properties of Permutations

Every permutation can be written as a cycle or a product of disjoint cycles. This follows by considering the effect of a sequence of permutations on each member of the set Each element i will end up after a sequence of permutations in some other...

https://astarmathsandphysics.com/university-maths-notes/146-abstract-algebra-and-group-theory/1705-properties-of-permutations.html
• #### Fission Reactors

The basic elements of a fission reactor are: reactor core for holding fission material or fuel moderator for slowing fast neutrons control rods holding neutron absorbers to control rate offission monitoring system containing devices and indicators...

https://astarmathsandphysics.com/o-level-physics-notes/164-fission-reactors.html
• #### Group Actions

Let be a set and let be a group who elements act on the set A left group action is a function such that: for all where is the identity element in for all and Right actions are similarly defined. From these two axioms, it follows that for every the...

https://astarmathsandphysics.com/university-maths-notes/146-abstract-algebra-and-group-theory/1696-group-actions.html
• #### ZDDs Defined

ZDDs may seem unremarkable, as they resemble well-known data structures such as crit-bit trees or DFAs. Nonetheless, the conditions they must satisfy have far-reaching implications. We define a ZDD \[Z\] to be any directed acyclic graph such that: A...

https://astarmathsandphysics.com/university-maths-notes/160-set-theory/4326-zdds-defined.html
• #### Joule's Experiment

James Joule demonstrated the link between mechanical work done and temperature change. He conducted a series of experiments showing that doing work resulted in an increase in temperature proportional to the amount of work done, allowing for...

https://astarmathsandphysics.com/ib-physics-notes/120-the-history-and-development-of-physics/1314-joule-s-experiment.html
• #### Integral Domains

An integral domain is a commutative ring with no zero-divisors: or Examples The ring is an integral domain. (This explains the name.) The polynomial rings and are integral domains. (Look at the degree of a polynomial to see how to prove this.) The ring...

https://astarmathsandphysics.com/university-maths-notes/146-abstract-algebra-and-group-theory/1697-integral-domains.html
• #### Permutationi Groups

A permutation group is a group of order whose elements are permutations of the integers The set of all permutations is labelled and called the symmetric group. A permutation group labelled is usually a subgroup of the symmetric group. As a subgroup of...

https://astarmathsandphysics.com/university-maths-notes/146-abstract-algebra-and-group-theory/1702-permutationi-groups.html
• #### The Cyclic Notation For Permutation Groups

Consider the set of elements {1,2,3,,...,n}. We may reorder these, choosing the first in n ways, the second in ways and so on. There are n! Possible rearrangements of the set {1,2,3,,...,n}. The set of all possible rearrangements is call the...

https://astarmathsandphysics.com/university-maths-notes/146-abstract-algebra-and-group-theory/1716-the-cyclic-notation-for-permutation-groups.html
• #### Definition of a Topological Space

A topological space is a set together with a collection of subsets of satisfying the following axioms: The empty set and the complete set are in The union of any collection of sets in is also in The intersection of any finite collection of sets in is...

https://astarmathsandphysics.com/university-maths-notes/163-topology/2139-definition-of-a-topological-space.html
• #### The Classification Theorem for Surfaces

The classification theorem of closed surfaces states that any connected closed surface is homeomorphic to some member of one of these three families: 1) the sphere; 2) the connected sum of tori, 3)the connected sum of k real projective planes, for The...

https://astarmathsandphysics.com/university-maths-notes/163-topology/2385-the-classification-theorem-for-surfaces.html
• #### Chaos

Systems highly sensitive to initial conditions display what is called the butterfly effect, which leaves open the possibility that a butterfly flapping it's wings on one side of the world can cause a hurricane on the other side some time latter,...

https://astarmathsandphysics.com/a-level-physics-notes/183-waves-and-oscillations/3085-chaos.html

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