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In mathematics, and in particular in the field of algebra, a Hilbert–Poincaré series (also known under the name Hilbert series), named after David Hilbert and Henri Poincaré, is an adaptation of the notion of dimension to the context of graded algebraic structures (where the dimension of the entire structure is often infinite).
For example, decidability is known for rational sequences of order up to 5 subject to the Skolem conjecture (also known as the exponential local-global principle). Decidability is also known for all simple rational sequences (those with simple characteristic polynomial) subject to the Skolem conjecture and the weak p-adic Schanuel conjecture .
The series is named after the mathematician Carl Neumann, who used it in 1877 in the context of potential theory. The Neumann series is used in functional analysis . It is closely connected to the resolvent formalism for studying the spectrum of bounded operators and, applied from the left to a function, it forms the Liouville-Neumann series ...
A formal power series can be loosely thought of as an object that is like a polynomial, but with infinitely many terms.Alternatively, for those familiar with power series (or Taylor series), one may think of a formal power series as a power series in which we ignore questions of convergence by not assuming that the variable X denotes any numerical value (not even an unknown value).
In statistics, the order of integration, denoted I(d), of a time series is a summary statistic, which reports the minimum number of differences required to obtain a covariance-stationary series. Integration of order d
The most popular TODAY show recipes in 2024 include Jennifer Garner's blackberry crumble, Donna Kelce's marshmallow dinner rolls and more.
Spiked Chips Ahoy "Egg" Cream. This cocktail is the classic Chips Ahoy chocolate chip cookie in a glass. The amaretto and rum dance together, giving the drink a rich nuttiness that represents the ...
The most famous example of a Dirichlet series is = =,whose analytic continuation to (apart from a simple pole at =) is the Riemann zeta function.. Provided that f is real-valued at all natural numbers n, the respective real and imaginary parts of the Dirichlet series F have known formulas where we write +: