Tensors by volume

A visual tour of how volume leads to tensors.

Function Types

The function type is the natural evolution of logical implication.

Substitution

Substitution seems so obvious that we can be fooled by simple mistakes. Fixing them forces us into a logic and calculus to go with it. This is known as Lambda calculus and the rules we will use are by Curry-Feys. Come along for a playful tour of what goes wrong when we substitute.

Types & Programs

Demonstration of mathematical types and programming comparables.

Copy of Types of data

Copy of Implications with Resources

Some implications involve resources which might not be available. So we cannot always discharge an implication without keeping track of how many times we have the necessary premises. When this happens we simply modify our rules of implication to include counting.

Implications with Resources

Some implications involve resources which might not be available. So we cannot always discharge an implication without keeping track of how many times we have the necessary premises. When this happens we simply modify our rules of implication to include counting.

False does not lead to truth, but context might

Implication is how we take one truth and turn it into a new one. It is enjoys a special place in science as the reason we do hypotheses and the mechanism to compute. However not everything about this operator is immediately obvious.

Logic of OR TRUE AND NOT

We are making our own logic so how hard can it be to make Or, True and Not? Surprises exist because we wont simply be shopping for these concepts already buried somewhere else. We are making them all natural from scratch! While I love these constructs I find them both subtle and worthy of multiple perspectives.

Sequent Calculus

Sequent calculus is a language to write logic. It uses few symbols with little to no meaning so that you can introduce just the meaning you want without the burden of all the implied ideas that normal language might provide. In exchange we get symbols which take practice to master.

Grammar's role in reasoning & programming

To make sure your logic is understood we go through language, and that means grammar. Fortunately most of logical grammar follows very basic rules.

Context for Logic

When you set up an argument you place it in a context and that can make all the difference.

Multiple Logics

A quick introduction to different forms of logic and how to recognize them.

Why are things different?

Why are things different? Explore the limits of structure and the external reasons for difference.

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?

Please Distribute

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.