Sentence examples for algebraic cycles from inspiring English sources

These additional equations produced new algebraic cycles within those manifolds.

Adding on extra equations would give you smaller shapes, known as algebraic cycles, within that manifold.

It wasn't long before people realised that topologists drawing homology classes onto manifolds and algebraists embedding algebraic cycles into manifolds was actually the same thing.

As mathematicians were now dealing with objects beyond what we can visualise, these "shapes" became known in general as "algebraic cycles".

In the important recent work  [1] of Ballard, Favero and Katzarkov, gaps in the Orlov spectrum were found to depend on the existence of algebraic cycles.

If an algebraic cycle was a nice smooth and generally well-behaved shape, it also earned the title of "manifold".

The Scottish mathematician William Hodge had a great idea about how you could tell which homology classes on any given manifold were equivalent to an algebraic cycle.

The problem is: if you drew any random – possible nasty – shape onto a manifold, how would you know whether it can be stretched into a different shape that can be described as a nice algebraic cycle?

The difficulty was that no one knew for sure when a homology class on a manifold contained at least one shape that was also describable as an algebraic cycle.

A preconditioner defined by an algebraic multigrid cycle for a damped Helmholtz operator is proposed for the Helmholtz equation.

The method is based on a computational homological algebra representation (called homological spanning forest or HSF, for short) that is an useful framework to design fast and efficient algorithms for computing advanced algebraic-topological information (classification of cycles, cohomology algebra, homology A-coalgebra, cohomology operations, homotopy groups, …).