Multidimensional systems

Encyclopedia

By the term

where not only one variable

exists (like time), but several independent variables.

Important problems like factorization

and stability

have recently attracted the interest of many researchers and practitioners.

The reason is that the factorization and stability of m-D systems (

does not exist in the ring

of m-D (

with many applications in biomedicine

, X-ray technology and satellite communications.

There are also some studies combining m-D systems with partial differential equations (PDEs).

**multidimensional systems**or**m-D systems**we mean the branch of (mathematical) systems theorySystems 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 algebraFundamental 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 processingDigital 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).