Copy of Programming with Universal Mapping Properties

Using universal mapping properties to prescribe data types and using resulting theory to build useful algorithms.

Tensor Contractions & Inflations

Tensors are defined by having contractions and cotensors by inflation. What are these and what do they require?

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What makes a tensor?

Copy of Open Education Resources

Open Education Resources

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Characterizing Characteristic

Characteristic structure is information that does not change under isomorphisms. If you have all the isomorphisms you can verify this property. But we often need characteristic structure to find the isomorphisms: Chick-and-egg problem. This presentation reports on a new result that solve this problem.

Applied Combiantors: Intro to Programming Idioms

Many functions we use are built from simpler functions and that explains all the qualities of functions: predictable outputs for example. So where does this process get started? What are primitive functions?

Open Education Resources

Open Education Resources

Higher Inductive Types

Using universal mapping properties to prescribe data types and using resulting theory to build useful algorithms.

Copy of Copy of Tensor Products; Versor Fractions

QuantumFuture

A rough idea of how to get into quantum materials, quantum computing, or quantum whatever

IsometriesChar2

Intersecting classical groups in characteristic 2 is in P.

Programming with Universal Mapping Properties

Using universal mapping properties to prescribe data types and using resulting theory to build useful algorithms.

Copy of Tensor & Their Operators

Definitions and properties of tensors, tensor spaces, and their operators.

Copy of Tensor Products; Versor Fractions

Sub-sub-scripts

Reading mathematics that has subscripts begins rather naturally but can quickly descend into sub-sub-subscripts, relabeling, and lots of errors. A further stress happens when we discover that programming languages don't tolerate subscripts and so we find ourselves rethinking what it is we meant to say. This deck some strategies and the core reason that subscripts can be avoided

Tensor Products; Versor Fractions

deck

Denser Tensor Spaces

Definitions and properties of tensors, tensor spaces, and their operators.

Substitution Rules: Curry-Feys

Substitution needs rules. Curry-Feys is one such system and knowing how to substitute properly is worth the work.

Primitive functions: Combinators

Many functions we use are built from simpler functions and that explains all the qualities of functions: predictable outputs for example. So where does this process get started? What are primitive functions?

Serious Math Asks: What is a function?

When we substitutes values into variables we seem to know intuitively that it makes sense. But with simple constant and identity functions we can soon run into paradoxes. This points at the need to define a theory of substitution known today as lambda-calculus.

Image Meta Data

Basic image convolution

Substitution: lambda-calculus

Variables and functions, the start of lambda calculus

Substitution: Why lambda-calculus?

When we substitutes values into variables we seem to know intuitively that it makes sense. But with simple constant and identity functions we can soon run into paradoxes. This points at the need to define a theory of substitution known today as lambda-calculus.

Affine, Linear

tensor-iso-quant-matter

Geometric Algebra, II

Groups

QuickSylver

Linear time solution to simultaneous Sylvester equation solvers.

Geometric Algebra

Everyday Algebras

Algebra Road Map

Where are we going with algebra?

Brikhoff-Neumann Theorems

A characterization of varieties.

Not true

proving negatives

Homomorphisms and Varieties of Algebra

Closure homomorphic images in varieties.

Varieties of Algebra

Definitions of varieties.

Relations

Relations

Fundamental Homomorphism Theorem

Congruences=>Quotients=>Homomorphisms=>Congruences

Quotients

Definitions and properties of quotients

Homomorphisms

Definitions and properties of homomorphisms

Congruence

Definitions and examples of congruences

Types & Programs

Demonstration of mathematical types and programming comparables.

Dependent types

Dependent types are defined and their implications in propositions-as-types

Natural Numbers

Algebras

Definitions of Algebraic structures

Free Algebra

Operators

Types of data

Group Isomorphism of most orders

Isomorphism testing of solvable groups of most finite orders can be done in nearly linear time.

Signatures

Signatures of operators identify how to use them properly, and types guarantee we use them with the proper inputs.