Berlekamp's algorithm, specifically known as Berlekamp's factorization algorithm, is a method used in computational algebra to factor polynomials over finite fields. It was developed by Elwyn Berlekamp in the 1960s and is particularly effective due to its efficiency in handling polynomials with many roots. ### Key Features of Berlekamp's Algorithm: 1. **Application**: Primarily used for factoring polynomials over finite fields, which are fields with a finite number of elements.

Stopped 2019 apparently. Shame. We need something to be upstreamed!

Persistent homology is a concept from computational topology, a branch of mathematics that studies the shape, structure, and features of data. It provides a way to analyze the topology of data sets, particularly those that vary with a parameter, by examining how the topological features of a shape persist across different scales. ### Key Concepts of Persistent Homology: 1. **Topological Spaces and Chains**: - Topological spaces are sets equipped with a structure that allows for the concept of continuity.

Computer Algebra Systems (CAS) are software programs designed to perform symbolic mathematics. They manipulate mathematical expressions in a way that is similar to how humans do algebra: by applying mathematical rules and properties symbolically rather than numerically. This allows users to perform complex calculations, simplifications, and transformations involving algebraic expressions, calculus, linear algebra, and other areas of mathematics.

Automatic differentiation (AD) is a computational technique used to evaluate the derivative of a function specified by a computer program. AD is particularly useful in various fields including machine learning, optimization, and scientific computing because it allows for efficient and accurate computation of derivatives, which is crucial for gradient-based optimization methods.

Polynomial long division is a method used to divide one polynomial by another polynomial, similar to the long division process used with numbers. It involves a systematic way of dividing polynomials, which results in a quotient and, in some cases, a remainder.

A **Binary Expression Tree** is a specific type of binary tree used to represent expressions in a way that makes it easy to evaluate or manipulate them. Each internal node of the tree represents an operator, while each leaf node represents an operand (such as a number or variable). ### Structure: - **Internal Nodes**: These nodes contain operators (such as +, -, *, /). - **Leaf Nodes**: These nodes contain operands (such as constants or variables).

Symbolic integration, also known as analytical integration, is a mathematical process used to find the integral of a function expressed in closed form, typically involving algebraic expressions, trigonometric functions, exponentials, and logarithms. Unlike numerical integration, which approximates the integral's value over a specific interval using numerical methods, symbolic integration provides an exact solution that is represented in a symbolic form.

Qt front-end for FluidSynth.

The Cantor–Zassenhaus algorithm, also known as the Cantor-Zassenhaus factoring algorithm, is a classical algorithm used for integer factorization, particularly for finding the prime factors of composite numbers. It's especially effective for numbers that are not too large and is known for its ability to factorize numbers using a combination of techniques.

This is actually pretty good! Makes a small first step into The missing link between basic and advanced.

By the Simons Foundation.

Unfortunately does not use a free license for content.

An elementary function is a type of function that is constructed using a finite combination of basic functions and operations. The fundamental types of elementary functions include: 1. **Polynomial Functions**: Functions of the form \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 \), where \( a_i \) are constants and \( n \) is a non-negative integer.

Faugère's F4 and F5 algorithms are important algorithms in computer algebra for solving systems of polynomial equations and performing computations in polynomial rings. They are particularly useful in the context of Gröbner bases, which are a fundamental tool for solving problems in algebraic geometry, coding theory, cryptography, and other areas requiring polynomial manipulation.

2023 Silk Road's Second-in-Command Gets 20 Years in Prison www.wired.com/story/silk-road-variety-jones-sentencing/

2016 "Exclusive: Our Thai prison interview with the alleged top advisor to Silk Road" arstechnica.com/tech-policy/2016/09/exclusive-our-thai-prison-interview-with-an-alleged-top-advisor-to-silk-road/

2015 The Variety Show On the trail of the man believed to be Variety Jones, one of the architects of the defunct drug marketplace Silk Road. www.vice.com/en/article/wnx5qn/the-variety-show

www.justice.gov/usao-sdny/file/797251/download some kind of case file of his trial.

Bibliography:

The curious thing about VJ is that he actually has some culture and says cool things, e.g.:

A Gröbner basis is a particular kind of generating set for an ideal in a polynomial ring, which has desirable algorithmic properties that facilitate solving various computational problems in algebra, geometry, and number theory.

A Gröbner fan is a construction from computational algebraic geometry and commutative algebra that arises from the study of Gröbner bases. Specifically, it is a way of organizing and visualizing the different leading term orders that can be used in the computation of Gröbner bases for a given ideal in a polynomial ring. ### Key Concepts 1. **Gröbner Bases**: These are special sets of generators for ideals in polynomial rings that facilitate solving systems of polynomial equations.

This is the true key question: what are the most important algorithms that would be accelerated by quantum computing?

Some candidates:

Do you have proper optimization or quantum chemistry algorithms that will make trillions?

Maybe there is some room for doubt because some applications might be way better in some implementations, but we should at least have a good general idea.

However, clear information on this really hard to come by, not sure why.

Whenever asked e.g. at: physics.stackexchange.com/questions/3390/can-anybody-provide-a-simple-example-of-a-quantum-computer-algorithm/3407 on Physics Stack Exchange people say the infinite mantra:

Lists:

Polynomial decomposition refers to the process of breaking down a polynomial into simpler, more manageable components. This can take various forms depending on the context and the purpose of the decomposition. Here are a few common applications and methods of polynomial decomposition: 1. **Factorization**: This is perhaps the most common form of polynomial decomposition. A polynomial is factored into products of lower-degree polynomials.