Multidimensional systems
Encyclopedia
By the term multidimensional systems or m-D systems we mean the branch of (mathematical) systems theory
Systems theory
Systems theory is the transdisciplinary study of systems in general, with the goal of elucidating principles that can be applied to all types of systems at all nesting levels in all fields of research...

where not only one variable
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...

exists (like time), but several independent variables.
Important problems like factorization
Factorization
In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original...

and stability
Stability theory
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions...

have recently attracted the interest of many researchers and practitioners.

The reason is that the factorization and stability of m-D systems (m > 1) is not a straightforward extension of the factorization and stability of 1-D systems because for example the fundamental theorem of algebra
Fundamental theorem of algebra
The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root...

does not exist in the ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

of m-D (m > 1) polynomials.

## Applications

Multidimensional systems or m-D systems are the necessary mathematical background for modern digital image processing
Digital image processing
Digital image processing is the use of computer algorithms to perform image processing on digital images. As a subcategory or field of digital signal processing, digital image processing has many advantages over analog image processing...

with many applications in biomedicine
Biomedicine
Biomedicine is a branch of medical science that applies biological and other natural-science principles to clinical practice,. Biomedicine, i.e. medical research, involves the study of physiological processes with methods from biology, chemistry and physics. Approaches range from understanding...

, X-ray technology and satellite communications.
There are also some studies combining m-D systems with partial differential equations (PDEs).