Math Research

Math Research Link to heading

A list of topics in mathematics that I have researched to some degree OR wish to research. Mind the aspirational tone

Topics Link to heading

Convolution with $e^{-x^2}$ as a cursor
Convolution $\int{e^{-x^2}f(x)(g(x-t))}dt$ as incremental application of function f(x) onto g(x) substrate
Cartesian plane as a binary switch board on some field
Unit Circle Projection
Developing programming primitives from real analysis operands
Solving collatz conjecture using operator theory
Having a function reference itself
Develop an operator that can iterate a number an indefinite number of times
Integral transform for length of orbits of collatz conjecture
Modified collatz conjecture that adds collatz(1) = 0 and collatz(0) = 0
Deconstructing a function by dot product of its derivative to other function
Scanning line across a function with dot product
Multiplying by x as inbuing of curvature
All single-term polynomials have complete curvature of 2
Curvature as a measure of information
Curvature as a program to steer function
A torus that culminates at a pinched point at one side
Algebra as a crude mechanic process
Algebraic expressions as a sequence of associative function compositions
Integral transforms extract and count properties of functions
Fractional application of a function
Treating [0,1] as a torus
Sinify a function
Curve-filling (space filling) curve
An equation as a manipulatable algebraic object
Compressed executable that is semantically equivalent to uncompressed
An expression as a program
Normalizing a program to a mathematical expression
All math fields should have some way of uniquely mappings their idiosyncratic algebraic objects back to integers
Curves are an infinite series of linear terms
The more curvy, the more linear terms is needed to duplicated it
e is the function number(?)
e is a self-referential number(?)
Inside a sum is a more expressive language
Function that spits out number of rounds needed for a space curve to hit a random (a,b) in R^2
Nested operations maximize the expressiveness of the syntax
Generalized unit measure - allows you to measure anything measurable relatively
Rigorous formalism for how mapping Reals (and reals subsets) to Reals functions - for example that just take a line and stretch it
f[x]+f’[x] I
f[x]+ ArcCurvature[{x,f[x]},x] I
x ^2 == c; algebraic diff between sqrt and div by x
integration as picking random algorithm to add up area
all integrations can be set from a to b to 0 to 1
there is a direct line ( no need for Euler-Lagrange optimization) in a set if the geometry of the boundary is a convex hull on the interior set
does homeomorphism always imply homeomorphism between boundaries?
Does Euler-Lagrange equation map to an easier problem and then solve it?
What is the manual analog of calculus of variation?
Discrete/Continuous invariant calculus
Using the Laplacian operator to incrementally/analytically convert any function to another in a continuous manner
using complex numbers to model curvature
Representing $R^2$ as a matrix centered around the origin with infinite rows and columns
Representing a curve on $R^2$ using a field of boolean values
Representing a curve as a $R^2$ matrix of 1s and 0s where the 1s are the position of points of the curve
Multiplying two $R^2$ matrix curves to find intersection points
Extending the dimensions of a row as a operation by adding or subtracting new rows or columns with 0s
For every class of functions (harmonics, elliptic, polynomials), there is a function involving the exponent of e that can emulate it to some degree
Nested riemann sum: While taking a riemann sum, use another riemann to calculate the error of the raw sum
Using the Dirac Delta function as a zero-detector. Feed it a formula that = 0. Integrate it dirac delta function; then what results is a step function that takes a step function that takes a step at every 0
Creating a relativized version of $R^2$ where the identity curve , $f(x)=x$, is equivalent to another function like $x^2$ or $sin(x)$
Understanding the derivative of a function in reference to another function
Function semantics that comphrensively handles recursive call bombs
Extracting range from integral and creating a dedicated “iterator” object (similar to programming)
Exp[x]== x^(x/Log[x])
Measure using number of open problems as relative health of mathematic formalisms
Analytic formalisms for function application to incrementally apply and compose functions
Tool must contain a commensurate amount of infinity as the problem
Duality between structure and dynamics. Structure can be interpreted as dynamics if you re-interpret one of the dimensions of structure as a time
Destructuring a real curve into holomorphic function. Re[f(x)] is scaling and Im[f(x)] is rotation
Creating a general functional iterator that takes a function and composes itself once. This general functional iterator can be easily composed
Symbol fatigue and math anxiety
Holomorphic functional calculus for iterated real functions of one variable
Formally overloading functions to accept points, lines, surfaces, etc.
Destructuring function application into eucleaean operations
General Integral transform for destructuring constructed algebraic objects
Localizing euclidean operations
Computing analytical and continuous version of function application
Using infinitary methods to surmount cryptographic techniques
Creating an alternative to cartesian coordinate system using dot product and function/vector. Its similar to polar but has two 0s and a 1 and -1
Outline of a proof for collatz that mentions random iteration across evens eventually resolves to 1
Normal form of an expression as the “compressed” symbolic version of un-normalizable variants
When does linear homotopy allow homeomorphism?
Linear homotopy as a fractional function application
Using ambience space to exogenously study the connectedness and holes of a topological space
Distinguishing between homology and connectedness
Treating all topological spaces as compact. Non-connected segments are considered addendum aggregate spaces nominally considered together
Mapping neighborhoods onto a chart then onto a topological space then tracking neighborhoods across homeomorphism
Neighborhoods preserved in turbulent flows
Measuring turbulence using measure of Euclidean operations
Nesting euclidean plane and operations into domain specific spaces
Linear homotopy as a semigroup of bounded linear operators
An infinite set contains itself in the same way a recursive function contains itself
Process oriented function semantics to represent recursive functions
Process theory and the intractability of its formalisms introduce
Dichotomy between intractable but computable formalisms vs non-computable but tractable formalisms
All curves correspond to a straight line in respective space
Using operator theory to study dynamics of genome state and genetic mutation
Using operator theory to study any arbitrary state space and dynamics thereof
Representing a matrix as a infinite matrix with 0s being filled in around the matrix
Zero-matrix (all slots are 0) algebraic closure
Computing intersections by: create a $R^2$ matrix for points for each curve (1 for point, 0 elsewhere), multiply both matrices, somehow zoom in on non-zero points
Addition as disjoint, additive semantics
UI 2D user artifacts as wishlist for mathematical tools
Scrolling on a subset of R^2
Scroll.fx function that can scroll a subset of R^2 like a scrollbar can
Encoding order-related information in object to achieve commutativity
Multiplicative properties as local properties and additive properties as global properties
Taxonomy of useful math functions
Deterministic formalism for random phenomena
Achieving convergence of quantum mechanics and general relativity
DeMorgan laws as percolation of global and local properties