 # Glossary

 3D printer See: 3D production technologies A,B,C,D representation state space representation of systems: absolute value (44) the value of signals in the real value domain without phase difference; the amplitude of vector quantities adaptive a system property which adjusts the system to the circumstances (such as outside temperature), offsetting their undesired effects adaptivity a system property which adjusts the system to the circumstances (such as outside temperature), offsetting their undesired effects data mining (47) methodology for processing large quantities of data. See also: Big Data. additivity the sum of the values to be added is the same in all grouping Anyone (3) the curious reader of this e-book about systems with a good but general secondary school level of understanding pattern recognition the ability of living beings and machines to recognise and learn identities and similarities. It is mainly applied to visual and auditory patterns. Its computerised form is a branch of computer science. shape (47) an umbrella term for signals and phenomena which are or thought to be interrelated. It is the conceptual framework of similarities or interacting properties of and between such signals and phenomena. A branch of shape construction was applied in Gestalt psychology. The terms pattern and scheme are also used in different conceptual areas to express the same idea. Its technique is pattern recognition. algorithm (96) a mathematical program specifying a given procedure steady state a state of rest where no change occurs. It is generally reached after a change and can be maintained theoretically for an unlimited period of time. state the totality of the measurable and/or calculable variables of the system enabling understanding and control the system. state space the space of measurable and/or calculable signals defining the system amplitude (amplus ‘large, spacious’) the difference (often the maximum difference) of variables from the steady state analytical function infinitely differentiable function that can be approximated with its Taylor series. It is derived from the word analysis. anticommutativity the property of a multiplication where the order of the elements influences the result. AB ≠ BA Aristotle (12, 37, 63) (Aριστοτέλης; 384-322 BC), one of the greatest philosophers and scientific thinkers of all time, the ever-influential figure of European science who summarised the classical scientific knowledge. Arrow , Kenneth J. (12), economist, mathematician, winner of the Nobel Prize in Economics in 1972. His impossibility theorem proves that it is impossible to make a clear decision between multiple alternative values while meeting certain “fair election” criteria. association (12) the act of connecting concepts or thoughts. Association is a key characteristic of cognitive activities and therefore it is a central problem in brain research. transparency clarity transfer function (94) the fractional function describing the transformation between the input and the output of the system. The roots of the numerator are the zeros, and the roots of the denominator are the poles. Banach spaces Banach, Stefan (1892-1945), Polish mathematician input and output The input variable acts on the system. The system reacts producing the output signal which affects its environment. Bernoulli , Johann (1667-1748), member of the Bernoulli family which produced many prominent mathematicians in the 17th and 18th century. Apart from many other achievements, he raised the brachistochrone problem (see there) which marked the origins of variation calculus. Bernoulli brothers the Swiss Bernoulli family produced several generations of excellent mathematicians and scientists. Two brothers are highlighted here: Jacob Bernoulli (1654-1705) inventor of the Bernoulli numbers and author of Ars Conjectandi, one of the oldest works on mathematical probability, and Johann Bernoulli (1667-1748), (see there). Big Data (46, 53) the methods used for processing extremely large data sets considered to be unmanageable before the age of modern computer technology, and for deriving conclusions from such data. bijection the one-to-one correspondence between the elements of two sets. biology (βίος ‘life’, + λογία ‘science’) the science of life and living organisms uncertainty an event, phenomenon or value which is not clearly defined. The most important and theoretically most elaborate way to handle it is the calculus of probabilities, but there are other approaches as well, such as the concept of fuzzy sets, and the methods of evidence and the estimation of certainty. Due to the nature of uncertainty, there is no absolutely valid method to calculate it. Probabilities are approximated by statistics. Blaschke product complex analytical function in the open unit disk with defined zeros. The products created this way give the function the shape where the poles of the transfer function describing the system process are the roots and the absolute values of the distance from the roots indicate the distance from the singular (e.g. unstable) behaviour of the system. Blaschke products are closely related to Hardy spaces and thus to the calculation procedures of modern control theory. By the way, Wilhelm Blaschke (1885-1962) was an Austrian mathematician whose career had inglorious periods. Bode plot A method which provides a clear graphic representation of the behaviour of a single-input-single-output system. The plot shows magnitude (amplitude change) and phase shift separately on a logarithmic scale versus frequency. It was first used by American engineer Hendrik Wade Bode in the 1930s. Bolyai-Lobachevsky geometry hyperbolic geometry named after the inventors of Non-Euclidean geometry brachistochrone (85) (βράχιστος χρόνος ‘the shortest time’) the brachistochrone curve is the curve on which a body slides in the shortest time under gravity. J. Bernoulli published this problem in Acta Eruditorum, the first international scientific journal, inviting the greatest mathematicians of the age to contribute to the solution. The problem and its solution method was the starting point of the calculus of variations and optimization tasks. Cauchy , Augustin-Louis (1789-1857), French mathematician who introduced clear and rigorous methods to discussing mathematical analysis. Of his remarkable scientific achievements systems theory primarily uses the Cauchy sequences (see: Banach spaces). control objective (85) description representing the objective of the system’s operation (see: optimisation) conservation (62) maintaining some system properties during motion (e.g. conservation of energy) damping the reduction of the amplitude (volume, strength, impact) of signals group theory (78) a special concept of mathematical structures which groups algebraic elements with identical characteristics to enable transformations between them. Group theory is primarily associated with the legendary figure of Galois (see there) but was also shaped by many outstanding mathematicians throughout the history of algebra. Major contributions include achievements of Klein in geometry and Lie in algebra which have become part of the basic methodology of systems theory. Transformation procedures make group theory relevant not only to mathematics but also to all other fields of science and practice concerned with formal transformations, such as chemistry. Darwin , Charles (1809-1882) (28), the most influential initiator of modern scientific thought. His main work is the theory of evolution. In addition to providing an explanation to the origin of species, the theory largely contributed to the research of systems and phenomena in the widest sense. Descartes , René (1596-1650) (42, 73, 80), one of the most prominent pioneers of the philosophy of the Age of Enlightenment, a rational thinker who had an enormous impact on science history by connecting geometry and mathematics on a new basis, inventing the coordinate system (named after him) and laying the foundations of analytic geometry. Cartesian coordinate system orthogonal coordinate system in which measures are the distances between the points deterministic (88) a type of process determined by causality student (3) undergraduate student or student with a bachelor’s degree who is interested in the topic differential equation (31, 50, 72) mathematics describing the relations between changes in phenomena. It was first applied to describe the relation between the distance, speed and acceleration in Newtonian mechanics. In essence, it decomposes changes into small linear sections and calculates their relations. differentiability (50) the ability to approximate continuous processes using a linear function, that is, at any point the difference between the tangent and the next points can be arbitrarily small differential operator the symbolic operation and notation of differential calculus differential calculus (85) perhaps the most important method of the system approach, the revolutionary scientific theory of the Modern Age. It decomposes the changes into small proportional steps and transforms them into a generalising symbolic tool, the calculus, which enables the calculation of any type of change. Although many great thinkers contributed to it in the past and many more developed it further later on, differential calculus was created by Newton and Leibniz competing for priority. differentia specifica (12) the property characterising the difference between groups deemed to be similar in certain aspects digital a form expressed with numbers; as opposed to analogue which expresses phenomena by words, pictures or sounds. digitalisation (40, 52) the transformation of a phenomenon measured and recorded in any form into a numerically scaled form. The quality of the digitalised form (i.e. its fidelity to the original) depends on the fineness and density of the scaling measure, such as resolution in the case of digitalised images (the number of pixels, images per surface units, colour range) dimension (71, 73, 77) the least number of factors required to determine phenomena or processes. In the Cartesian coordinate system dimensions are the scale values necessary to determine the position of individual points. In the usual determination of positions two dimensions are used on the plane and three in space. In general, however, the term is used as a definitive key characteristic in any relation or interaction. The word stems from the term (to) measure. dynamics (14) (δύναμαι ‘to be able’) the science of motion and its relations disharmony lack of harmony, deviation from a sequence discrete not continuous, e.g. sampled. Discrete mathematics traditionally include graph theory, combinatorics, game theory, the theory of algorithms, cryptography, mathematical logic, set theory, number theory, algebra and the branch of geometry which deals with defined shapes. dissipation disappearance. In irreversible processes the part (usually heat) of the energy that is lost during transformation. The theoretical framework is included in the second law of thermodynamics. Dissipation is a typical entropy-increasing phenomenon. dissipation disappearance. In completely irreversible processes the part (usually heat) of the energy that is lost during transformation. The theoretical framework of the process is included in the second law of thermodynamics. Dissipation is a typical entropy-increasing phenomenon. EEG (74) electroencephalogram or encephalography; the monitoring and recording of brain activity. balance (25), (50, 59, 61, 63) or equilibrium. A basic characteristic of systems; the completeness of the closely related units of different system elements. The concept is based on the equilibrium of the pans of a balance scale and is extended to the relations of mass and energy, economic values, elementary particles and the entire system of the universe governing thinking and models in these areas. unit circle (80, 94) a special area on the complex plane including its circumference which serves as the basis for function inversion. Certain properties of the infinite complex plane can be easily calculated if transformed into the unit circle (e.g. Hardy spaces). The unit circle can be generalised to the unit sphere and other similar entities in higher dimension problems. Einstein , Albert (1879-1955), physicist who developed the special and general theories of relativity and greatly influenced modern physics and worldview. ECG (74) or EKG, electrocardiogram; the monitoring and recording of heart activity. ellipse (78) see: conic sections energy (60, 63) (εν ‘at’ + έργον ‘work’), the capacity for doing work or triggering change.   The physical property of bodies exposed to four fundamental interactions: gravitational, electromagnetic, strong and weak interactions. Everyday practice is influenced by the first two interactions, while the last two govern the atomic world and its relations with the first two interactions. entropy (18, 40, 68) (en ‘in’ + trope ‘a turning, a transformation’) a system’s ability to transform in itself. It describes transformations in thermodynamic properties and information content and their possible directions. It is the measure of the order and disorder characterising multi-element systems.  The concept was created in thermodynamics to describe the increase or decrease of the order of molecules during the transfer of heat. epigenetics (67, 69) the study of factors completing and modifying heritable processes, and of the influence and heritability of external processes. It is primarily a biological field of science but it can be generalised to e.g. social processes as well. epistemology (ἐπιστήμη ‘knowledge’) the methodology of gaining knowledge and the criticism of this methodology. Erlangen program (71) the main work of Felix Klein in which he systematised the traditional and possible new geometries on the basis of group theory. This laid the foundations of all the ideas and methods related to spaces which are used in modern systems theory and which is the very subject of this e-book. Klein wrote the Erlangen program as his inaugural lecture but ultimately presented a different one. emotions (8) all the emotional effects that influence the decisions and reactions of living beings, particularly of humans, and that can only be integrated into process models with quality markers. Typical examples include deep anger, great happiness, terrible fear, concentrated attention etc. Escher , Maurits Cornelis (1898-1972), Dutch graphic artist whose works first focused on natural forms, then architectural elements, primarily floor patterns. His influential art combined these patterns in complex geometric compositions with significant artistic and conceptual effect. aesthetics (αἰσθητικός ‘perceptive’), The branch of philosophy primarily dealing with the perception of the quality of works, beauty and other quality properties. ether (63) a mythological air-like medium, an ancient element formerly thought to be the medium of electromagnetic phenomena ethics (4) (ἦθος ‘moral character’), the discipline concerned with what is morally right and wrong Euclid (80, 94) (Εὐκλείδης, fl. 300 BC), Alexandrian mathematician. His work Elements is considered to be the foundation of geometry. It provides a summary of the geometrical knowledge of the time and still defines the framework of mathematical argumentation (theorem - proof), so its impact goes beyond geometry. Euclid also contributed to number theory, primarily in the methodology of partitioning. The more than 2000-year-old geometric results and methods constitute the Euclidean geometry. The origins of non-Euclidean geometry in the 19th century came from Euclid’s fifth postulate: the definition of parallels (see: Bolyai-Lobachevsky geometry) Eulerian , Leonhard (1707-1783), mathematician and physicist, influential scientist of the St. Petersburg and Berlin schools. He significantly contributed to the number theory, complex functions and other areas of mathematics, his works are still fundamental in hydrodynamics and other fields of mechanics (e.g. buckling). Euler , Leonhard (1707-1783), mathematician and physicist, influential figure of the St. Petersburg and Berlin schools. He significantly contributed to the number theory, complex functions and other areas of mathematics, his works are still fundamental in hydrodynamics and other fields of mechanics (e.g. buckling) evolution (23,28, 77) (ex ‘out’ + volvere ‘to roll’) the continuous, uninterrupted process of changes, generally from the simpler to the more complex. This e-book also adopts a Darwinian approach. exponential (ex ‘forth’ + ponere ‘to put’), used for processes, functions where the variable is in the exponent. The negative exponent indicates a decreasing process, the positive exponent a rapidly increasing process. The term exponential refers to the latter (exponential explosion). factorisation resolution into factors phase angle (41) the relationship between signals that belong together; the representation of differences of effects in time and space in a coordinate system. black box (48) when the internal operation of a system is unknown, black box is a model about which the only sources of information are the input and the output. Felix Klein , Felix (1849-1925) (71), author of the geometric interpretation of group theory. He summarised traditional and new geometric field theories into a unified system which made him one of the founders of modern mathematical thinking. His main work is the Erlangen program. half plane the half of the infinite complex plane representing stable or unstable processes Enlightenment (73) intellectual movement of the 17th and 18th centuries in Europe following the Renaissance. A critical approach resulting in the freedom from the social and philosophical ties of former ages. The foundations of our modern system of values and systems approach are rooted in the Enlightenment. philosophy (1) from the perspective of system science: the search for relationships physical model the representation of systems with physical elements of similar behaviour phlogiston (63) (φλόξ ‘flame, blaze’), hypothetical (false) principle of combustion accepted in the 17th century claiming that all combustible substances have a volatile constituent released as flame in combustion. concept (2) term summarising and representing phenomena process the course of change in systems contiguous operating without interruptions source the original, starting point of any phenomenon Fourier , Jean-Baptiste Joseph (1768-1830) (31). While studying thermal conduction he invented the Fourier series (the fact that all changes can be expanded into an infinite series of harmonic components) which initiated the still prolific development of the mathematics and physics of frequency analysis. He is also credited for the discovery of the greenhouse effect. Fourier-Laplace transformations the transformation of processes changing over time into the frequency domain, and the reverse of this operation. The Laplace transformation is the differential operator form of the Fourier transformation and thus a useful tool to do calculations with algebraic equations. Fourier series the decomposition and representation of phenomena in the series of frequency components fractional transformation decomposing a function into product terms frequency (19, 31,74, 92) (frequens ‘numerous, crowded, populous’) the number of times of phenomenon occurs frequencies (19, 31,74, 92) (frequens ‘numerous, crowded, populous’) the number of times of phenomenon occurs course of a frequency function the representation of phenomena as a function of frequencies fuzzy A popular tool for handling underdetermined phenomena and sets with uncertain borders. It describes quantity with quality. It is commonly used for the calculation of uncertainties. Its relation to certain interpretations of probability calculus is disputed. Galilei , Galileo (1564-1642) (125), a legendary figure of renaissance science and research based on observations. He made fundamental contributions to the sciences of motion and astronomy, and to the emergence of the Newtonian approach. Galois , Évariste (1811-1832), eminent mathematician who died at the age of 20 from wounds suffered in a duel. Studying the roots of fifth and higher degree equations he created an important pillar of abstract algebra, the group theory, a unified framework for problems that can be handled with the same operations. Galois group theory , Évariste (1811-1832), eminent mathematician who died at the age of 20 from wounds suffered in a duel. Studying the roots of fifth and higher degree equations he created an important pillar of abstract algebra, the group theory, a unified framework for problems that can be handled with the same operations. Gauss , Johann Carl Friedrich (1777-1855), mathematician, scientist and astronomer, also referred to as Princeps mathematicorum (‘the foremost of mathematicians’). He was a pioneer in several branches of mathematics, including number theory, analysis and differential geometry. He made major discoveries in astronomy, magnetism and optics. As a practical geometer he was a significant contributor to geometry. His role in influencing non-Euclidean geometries is disputed. genetics (69) (γένεσις ‘origin’), the study of biological heredity that can be generalised to deal with social matters. genus proximum (12) when talking about a phenomenon classified into a group, genus proximum means the nearest group. Such classification usually takes place in a hierarchical order (see: similarity). geometry (13) a separate branch of mathematics dealing with the properties and relationships of shapes. The term originally referred to the measurement of land. gnomonic (75) (γνῶσις ‘knowledge’),  a type of map projection in cartography used for showing the length of flight paths. Gram matrix a matrix created from the inner products of vectors in linear spaces. It provides covariance, that is, the properties of interacting behaviours, such as the independence of vectors from each other or the basic relationships of controllability and observability in control theory. Guru (3) a PhD level reader who is dealing with systems at an advanced level in possession of deep mathematical knowledge H_2 in modern systems theory the integral measure of the qualitative behaviour of a system; the norm of a system represented in the Hardy space H_inf norm calculated in the Hardy space based on the square integral of the given phenomenon set as a mathematical concept it is a collection of elements with specific common properties net general term in graph theory for interconnected systems. The logical net is used for considering multiple effects, generally represented also in a graphical form Hamilton , William Rowan (1805-1865) (31, 63, 86), Irish mathematician and physicist who created the uniform general mathematical representation of Newtonian mechanics. His equations serve as the basis for calculating the widest range of dynamic phenomena e.g.  in systems theory, including modern control theory and quantum theory. He made major contributions to optics and astronomy and he invented quaternions (the dynamics of complex elements consisting of four components; see there) which gained their true significance in quantum physics. Hamiltonian equations formulas, explanations separately Hankel , Hermann (1839-1873), German mathematician who worked together with many eminent contemporary mathematicians. He is best known for his contribution to linear algebra. Hankel matrix a square matrix which is symmetric with respect to the main diagonal. It projects processes to the past and thus serves as a hidden Markov process to determine the state matrix Hardy , Godfrey Harold (1877-1947) (80, 82), British mathematician with a major influence on number theory and mathematical analysis. In the field of number theory he worked in legendary collaboration with Indian mathematician Ramajunan who said some of his theorems came to him in dreams. Hardy’s remarks on the beauty of mathematics, the ideal of mathematicians’ interest is frequently quoted. Frigyes Riesz named after him the spaces (see: Hardy spaces) which have practical significance in modern systems theory as they calculate the quality norms of holomorphic (i.e. complexly differentiable in the complex space) systems. Hardy spaces in modern systems theory the finite space of processes represented by analytical functions. It can be isometrically represented in frequency and time space, which makes it particularly suitable for determining system stability, robust stability domains and optimal relations. harmony (24) (harmonia ‘agreement, concord of sounds’) 3D production technologies (33) technology using the idea and technique of inkjet printers. 3D objects are created by putting special liquid but quickly solidifying material (typically plastic or metal) on a surface in several layers. A special area is called 3D bioprinting which experiments with printing biological tissues and cells. Harvey , William (1578-1657) (37), English physician and surgeon credited for the description of systemic circulation. His theory had many precursors; some argue that Arab scholars had already known it in the European Middle Ages. efficiency the relationship between the energy or work put into and coming out of a system. Hilbert space vector space where scalar multiplications are defined; i.e. a space where the norm can be fully determined by scalar multiplications. It was named after Hilbert by Neumann whose work made it a key calculation space in quantum theory. hyperbola (78) a conic section formed by a plane which intersects both cones of a double cone except for the vertex. In such surfaces geometry, more specifically the straight lines passing through a single point show properties that deviate from the Euclidean plane, alter the concept of parallelism and the sum of angles of triangles. Hyperbolic surfaces had an important role in the world view ruled by Einstein's gravity theory but they also opened up new perspectives in the geometric aspects of mathematics. hyperbolic (78) a conic section formed by a plane which intersects both cones of a double cone except for the vertex. In such surfaces geometry, more specifically the straight lines passing through a single point show properties that deviate from the Euclidean plane, alter the concept of parallelism and the sum of angles of triangles. Hyperbolic surfaces had an important role in the world view ruled by Einstein's gravity theory but they also opened up new perspectives in the geometric aspects of mathematics. hyperbolic function functions defined similarly to trigonometric functions but while the latter type of functions defines a circle, hyperbolic functions define hyperbolas. Their signs are similar too: sinh, etc. They play an important role in the description of physical processes. They were first used for describing catenaries (cable or chain curves). hyperbolic geometry Non-Euclidean geometry in which the parallel postulate is invalid. The concept is primarily credited to Bolyai and Lobachevsky but they had many important precursors. hyper matrix a matrix whose elements are also matrices. Hyper matrices are particularly important for the description of processes related to partitioned and connected elements. hyperstable stable in the broad domain of system parameters holistic (21)(holos ‘whole’) an approach which considers the examined system as a whole and in context, similarly to a medical specialist who also considers all other conditions of the patient. holography imaging technology which uses the wave nature and interference phenomena of light to display three-dimensional images (hence the name: holos ‘whole’ + graphia ‘description of’). It became common with the emerging application of laser (coherent light having a constant phase difference). Dénes Gábor (Dennis Gabor) was awarded the Nobel Prize for inventing the holographic method. holomorphic of the same or similar form holomorphism of the same or similar form homogeneity (homos ‘same’ + genos ‘kind’) uniformity in general or in a specific aspect homology (17), (47) (homos ‘same’ + logos ‘relation, reasoning’) the order of a specific field of science; in theology it is the Word. In algebra and particularly in geometry it is a necessary characteristic of identical properties. homolosine (75) equal-area map projection which compensates the distortion of Mercator maps with sinusoidal transformations, resembling to the flattened rind of a peeled orange. It is credited to American cartographer John Paul Goode (1862-1932). homomorphism (17), (47) (homos ‘same’ + morphe ‘form, shape’) see: similarity thermal motion Brownian motion; the random motion of atoms and molecules under the effect of thermal energy thermodynamics (68) the branch of science concerned with the nature of heat and its relationship with other forms of energy strings (42) see: Pythagoras Huyghens , Christiaan (1629-1695), Dutch scientist, astronomer and physicist, one of the most prominent thinkers of the 17th century. Inventor of the pendulum clock and other clockworks, discoverer of several celestial bodies, pioneer in probability theory, mechanics and optics. identification the determination of the nature, states and properties of systems, and the related mathematical and computational methods indentify the determination of the nature, states and properties of systems, and the related mathematical and computational methods time domain the space for describing the behaviour of systems in time infinitesimal (derived from finis ‘end’) separation. to infinitely small parts, referring to the basic idea of differential and integral calculus information theory (68) the theory of transmitting and processing communications and signals. The subject matter of mathematics, philosophy, social science, electrical engineering and other technical fields of science. input-output Signals marking the boundaries of a system intension intention; the consideration of subjective factors with the tool set of non-monotonic logic inverse reciprocal; if multiplied with the original value the product is the unit. control (34, 84) the umbrella term for closed-loop and open-loop control. Term usage varies by language and profession. controllability (89) the requirement that a system should be movable from its current state toward a desired state irreversible (65, 88) not reversible. A typical example of irreversible processes is the making of scrambled eggs. epistemology the branch of philosophy dealing with the methodology of gaining knowledge and the criticism of this methodology isometric (isos ‘the same’) of the same measure; refers to distance-preserving transformations isomorphic of the same form or shape Jacobi equation formula; relationship expressing the requirements of a position of steady state in Lie algebras where the sum of the permutations of algebraic formulas is zero, and the differential changes of elements (the algebraic expressions used) mutually compensate for each other. With binary multiplication and addition operations in an S algebra: (126) game theory the theory of mathematical models of conflict and cooperation between intelligent rational decision-makers. The field of study was created by the publication of John von Neumann (János Neumann) in 1928. signal (41) any form of reality originating from and connecting processes that can be measured in some way. One of the most generic concepts in system theory. Kalman , Rudolf (1930-2016) (89) an outstanding scientist of modern control theory and technology, inventor of the theory of filtering noise affecting the elements to be controlled. Today, practically all controlled devices contain such a filter or a variation of it. His theoretical work is made complete by the mathematical definition of controllability and observability. Kálmán , Rudolf (1930-2016) (89) an outstanding scientist of modern control theory and technology, inventor of the theory of filtering noise affecting the elements to be controlled. Today, practically all controlled devices contain such a filter or a variation of it. His theoretical work is made complete by the mathematical definition of controllability and observability. Kalman filter an algorithm which continuously monitors the state of a system and generates statistics from the collected data to separate information related to system operation and noise. Feeding the results back to the closed-loop control adapts system operation to the changing conditions. The filter was first used for controlling flying objects, but today, due to the development of signal processing equipment, it is used in most control systems. Kálmán filter an algorithm which continuously monitors the state of a system and generates statistics from the collected data to separate information related to system operation and noise. Feeding the results back to the closed-loop control adapts system operation to the changing conditions. The filter was first used for controlling flying objects, but today, due to the development of signal processing equipment, it is used in most control systems. Kalman filters an algorithm which continuously monitors the state of a system and generates statistics from the collected data to separate information related to system operation and noise. Feeding the results back to the closed-loop control adapts system operation to the changing conditions. The filter was first used for controlling flying objects, but today, due to the development of signal processing equipment, it is used in most control systems. characteristic equation equation defining the roots of polynomial differential equations and algebraic equations of similar structure. It is used to solve multiply differentiable analytical process equations (i.e. equations that can be expressed as a Taylor series) which play a primary role in systems theory. Cartesian (80) relating to the works of René Descartes (from the Latinized name: Cartesius). In geometry it refers to the orthogonal coordinate system. Cartesian, orthogonal (80) relating to the works of René Descartes (from the Latinized name: Cartesius). In geometry it refers to the orthogonal coordinate system. cascade control the closed-loop control scheme of systems separated to cascading stages. The term cascade refers to waterfall or fall in general. causality the principle of relationship between cause and effect. In systems theory it is the concept that the present state depends on previous states but it only has a limited influence on future. frames the set structure of a given property in the control of smart systems and other connection properties cognitive sciences the interdisciplinary scientific investigation of cognition, i.e. the mental process of knowing. It synthesizes the results of psychology, sociology, linguistics, computer science and brain research. complex consisting of interconnected parts. In mathematics it refers to systems which express the quantities, values and the magnitude and direction of signals in an integrated way. In complex plane representation they are the coordinates of real and imaginary axes. In dynamic systems the change of direction is given as a function of time or frequency. complex half-plane (41, 96) the plane described by real and imaginary coordinates to represent a process. The term complex and the two concepts of number refer to the representation of all phenomena which are described by dual connections. Such dual connection is the representation of the amplitude, magnitude and phase shift relations, frequency components and vector quantities of the phenomenon. complex plane (41, 96) the plane described by real and imaginary coordinates to represent a process. The term complex and the two concepts of number refer to the representation of all phenomena which are described by dual connections. Such dual connection is the representation of the amplitude, magnitude and phase shift relations, frequency components and vector quantities of the phenomenon. lumped parameter model in general, the model of a spatially distributed system by concentrating it at a characteristic point. The most typical example is the concentration of physical masses at their centre of gravity. assignment of coordinates the assignment of defining measures to signals and phenomena. Its elementary metaphor is the determination of distances with units of length. coordinate system (31,75) system used to determine the position of points (coordinates) on the basis of geometric shapes. The concept evolved from the generalisations of space for the purposes of geographical positioning, so it has many historical precursors. It developed from the mathematical representation of movement, primarily through the work of Fermat and Descartes. The mathematical geometry of the orthogonal coordinate system is referenced as Cartesian system in the relevant literature. use of coordinates (42, 71, 73) operation aimed at the arrangement and combination of quantities and symbols derived from them in a suitable coordinate system. According to prominent algebraists this is what algebra is all about. plotting coordinates (42, 71, 73) operation aimed at the arrangement and combination of quantities and symbols derived from them in a suitable coordinate system. According to prominent algebraists this is what algebra is all about. covariance the coordinated movement of two different variables. It is calculated by summing the products of the deviations of corresponding values of the two variables from their respective means in statistics and calculus of probabilities. Cretan liar (9) paradox attributed to Epimenides: “All Cretans are liars," says a Cretan. The paradox appears in many forms in philosophy; even Gödel refers to it as the conceptual precursor of his incompleteness theorem. conic sections (78) shapes resulting from the intersection of the surface of a cone or double cone with a plane. The types of conic sections are the parabola, the ellipse, the hyperbola and the circle which is a special type of the ellipse. quantum theory (71, 86) a branch of modern physics concerned with the investigation and explanation of phenomena at scales that are beyond the scope of observation by traditional means. It is rooted in the observation that below traditionally observed scales the change of certain phenomena is a series of discrete rather than contiguous units. Quantum theory deals with the similarities of matter and wave like behaviour, phenomena of particles intertwined at elementary level (which show promising future prospects for telecommunications), phenomena taking place under extreme physical conditions such as absolute zero temperature or vacuum, so it deals with the microscopic world which is the deep foundation of modern physics with the uncertainty principle. quantum phenomenon (71, 86) a branch of modern physics concerned with the investigation and explanation of phenomena at scales that are beyond the scope of observation by traditional means. It is rooted in the observation that below traditionally observed scales the change of certain phenomena is a series of discrete rather than contiguous units. Quantum theory deals with the similarities of matter and wave like behaviour, phenomena of particles intertwined at elementary level (which show promising future prospects for telecommunications), phenomena taking place under extreme physical conditions such as absolute zero temperature or vacuum, so it deals with the microscopic world which is the deep foundation of modern physics with the uncertainty principle. quaternions the representation of interrelated sets comprised of four phenomena; the duplication of complex representation. It was primarily used for coordinating rotation, and later for exploiting the opportunities of quantum phenomena in computer science while simultaneously using the quantum properties of photons. The idea occurred to Hamilton while he was walking on the riverside in Dublin, so he carved it into a stone there. At the same time various scientists had similar ideas. unstable not stable; able to lose stability in reaction to even a small input instability not stable; able to lose stability in reaction to even a small input Lagrange , Joseph-Louis (1736-1813) (85, 86) Italian-French mathematician, the most important contributor to variational calculus, inventor of the method of Lagrange multipliers, who redefined Newtonian mechanics as a variational problem and thus promoted the Hamiltonian approach which dominates modern physics. Laguerre polynomials second order linear differential equations which play a role in the quadratic assessments in system modelling and analysis Laplace , Pierre-Simon (1749-1827) (31) mathematician and astronomer whose work greatly reformed the traditional approach to mechanics and the role of differential operators. He developed the Laplace transformation, which is a main tool in the mathematical representation and calculation of systems. He played an important part in statistics, the development of Bayesian interpretation of probability. Among many other results in astronomy, he was the first to suggest the possibility of black holes. Laplace transformation complex variable transformation between the time and frequency domain used for analysing processes described as differential equations with constant coefficients. Due to the linear characteristic, the differential equation form can be replaced by algebraic equations which is the basis of calculating roots (zeros and poles) that are of primary importance in studying system behaviour. The convolution characteristic, on the other hand, simplifies the calculation of the interaction, serial and parallel connection of system elements. Lebesgue , Henri (1875-1941) French mathematician best known for developing the comprehensive theory of integration by extending the measure theory. Lebesgue spaces Banach spaces extended to various distance measures; normed Euclidean spaces. The concepts can be traced back primarily to Frigyes Riesz. Usually referred to as l spaces Lie , Marius Sophus (1842-1899) Norwegian mathematician, author of the transformation groups in group theory, including anti-commutative algebras which express equilibrium relations. Lie group transformation groups characterised by Lie algebras linear (54, 55) proportional change: the addition of inputs triggers addition of effects. Multiple linear effects can be simply added together (superposition). linear differential equation if the system is modelled with linear differential equations, the zeros and poles are the extreme values of the solution of the differential equation: at the input, this is the state where even the biggest effect fails to trigger displacement and the system exhibits infinite inertia; while at the output, this is the state where even the smallest effect will trigger an unstoppable displacement causing instability. linearity (linea ‘line’) a behaviour that can be represented as a straight line. Its main characteristic is additivity (its elements can be simply added together) linearisation (67) an approximation procedure used to make the representation of behaviours proportionate and additive and to simplify calculations this way logic (27) (λóγος ‘reason, idea, word’) logical argumentation logical (27) (λóγος ‘reason, idea, word’) logical argumentation logical functions functions where the relationships between variables are described by logical operations radar equipment used in detecting the position and movement of objects by measuring the radio waves, sound waves and optical waves reflected from the surface of such objects. Depending on the medium, object, distance and other circumstances such equipment include a broad range of devices from microscopes to astronomical telescopes. The reflections are usually processed in the form of three-dimensional images. Originally, it was part of the military technology but today it is widely accessible and applied in a vast number of fields. Manhattan distance a measure of distance in the grid layout of streets in Manhattan, New York City mathematical model the description of any system with mathematical tools. These descriptive methods provide the essence of the generality of system science. MATHEMATICA (87) a fourth-generation programming language conceived by Stephen Wolfram (1959-) to automate mathematical operations, algorithms and graphical representations. MATLAB (87) product of MathWorks, a company founded by Jack Little and Cleve Moler in the USA in 1984. A fourth-generation programming language designed to describe and display processes and capable of translating verbally defined operations to digital procedures. Another product of the company is Simulink, a special program for process representation. matrix operations matrix is a rectangular array of quantities, functions or mathematical expressions. It can be analysed with the methods of linear algebra with the deviations applied to the given task group. Matrix operations are the main procedures for creating, analysing and operating system models that can be described and approached with linear algebra. Maxwell , James Clerk (1831-1879) (31, 60) the founder of modern electrodynamics. In his equations he summarised the basic relations of electricity and magnetism, and proved the mutuality and wave like characteristics of the two spaces. He laid the foundations of the kinetic theory of gases. Maxwell equation (31) Mayer , Julius Robert (1814-1878) (60) pioneer researcher of thermal energy who discovered the equilibriums in mechanical and thermal energy and enunciated the first law of thermodynamics. observability (89) the requirement that system behaviour can be traced back from the current state to the beginning of the examined effects. This is a precondition for being able to filter out noises and control the system based on its history. conservation of energy (60, 61, 62) the principle stating that energy remains constant regardless of the changes. Mercator , Gerardus (1512-1594) (75), philosopher, mathematician, cartographer. He is most known for his cylindrical map projection which served sea navigation purposes because of its ability to represent routes as straight lines that preserve the angles with the meridians. The projection method, however, has the drawback that it distorts shapes in a way that they appear larger toward the poles and smaller toward the Equator. measure (36, 39) the measurable property of phenomena, such as voltage or temperature artificial intelligence (AI) (27, 103) the loosely defined umbrella term for all systems that tries to represent human thinking through machines. Thus, the term is used for relatively simple series of logical operations as well as for programs performing complex cognitive tasks surpassing human abilities, for the representation of cognitive activities, and even for the fantastic concept of superintelligence. Independent branches include: pattern recognition, robotics, natural language processing, machine translation, brain games and machine learning. metabolism (μεταβολή ‘change’) the chemical transformation processes between living cells; more generally: any transformation affecting the essence or substance. metaphor (6) (μεταφορά ‘transfer’) the figurative explanation of a phenomenon with another, similar one. metamorphosis (61) (μεταμόρφωσις ‘change of form’) a complete change of form or substance quality a general concept of system features, such as accuracy, settling speed or cost.  Stability as a basic requirement is usually treated separately. sampling (53) measurement process where the states of the system are measured at certain intervals or as a function of other factors (not continuously) modality (27) (modus ‘manner, measure’) a characteristic of deduction processes which depend on certain additional circumstances, such as time constraints, personal and group views, random effects, and other restrictive or influencing physical relations or assumptions. Modality is a system of circumstances in non-monotonic logics. model the description of a system by means other than the system itself, such as mathematical representation, physical reduction, the omission of certain parts, or other systems that are better known and behave similarly. A model is only a limited representation of the underlying system, so efficiency needs to be checked in a separate examination. Möbius ,August Ferdinand (1790-1868) (80, 81, 82), a pioneer of projective geometry and topology. He is famous for the Möbius strip and the Möbius transformation. The latter is particularly important for systems theory as it enables highly practical transformations using the group properties of rotation, stretching and shrinking. Möbius transformation the linear transformation of shapes through rotation, stretching and shrinking. negative half-plane see: complex variable negative feedback see: feedback non-monotonic logics a system which modifies simple logical relations by involving in the logical chain tools which characterise the given changes, relations, modalities and intensions and which usually use the techniques of logic or perhaps probability calculus. non-linear (54) different from linear; i.e. effects do not change proportionately and do not simply add up. neural (12) related to the nervous system. It primarily refers to the biological nervous system applied to the models of simplified mechanical systems based on the assumption that they operate along similar principles (neural networks). neural network computer structure modelling the nervous system in a highly simplified way, primarily performing learning tasks. neurology the branch of medical science that deals with the nervous system Newton , Isaac (1642-1726) (86) legendary figure of modern scientific thinking best known for his landmark contributions to mechanics, optics and mathematics. Newtonian reference to Newton’s theories mainly relating to the metaphorical use of motion as a basic phenomenon, the mathematical modelling of dynamic phenomena primarily in the form of differential equations, and to the concept of light as a corpuscular phenomenon. Noether , Amalie Emmy (1882-1935), highly influential German mathematician famous for her outstanding contributions to abstract algebra and theoretical physics. norm (39, 44) the measure of a property of a phenomenon, or a characteristic calculated from measurement results; the basic value of such a property defined from a certain aspect. This way, freezing point is a norm of temperature, daily mean temperature is a norm for heating processes. sink the opposite of source; a stream usually similar or related to energy which becomes lost or absorbed in an area contained by a closed curve. The original mathematical-physical image relates to the phenomenon described by Gauss-Stokes formulas for processes flowing out from and into the closed curve, the sum of which is zero in a position of rest. In control theory, this role is played by the environment of the -1 point representing feedback on the complex plane. Nyquist , Harry (1889-1976), Swedish born American electrical engineer who made landmark contributions to the pioneering activity of the Bell Labs, primarily in relation to the problem of thermal noises and the theory of closed-loop control. For his achievements in the latter area, the amplitude-frequency representation in polar coordinates is called Nyquist plot. cause (69, 77) one of the most dubious problems of philosophical and everyday thinking; the question whether a phenomenon is caused by one or more other (usually successive) phenomena or by the simultaneous interaction of more phenomena. ontology (5) a branch of metaphysics dealing with the origins of and ultimate relationship between phenomena. In information science it is the formal naming, definition and classification of phenomena. operator (48) from a technical point of view, operators are mathematically defined models which represent the subject system. optimal (80) the practice of influencing systems so that they ensure certain desired quality requirements such as costs, speed, acceleration, etc. optimise (80) the practice of influencing systems so that they ensure certain desired quality requirements such as costs, speed, acceleration etc. optimisation (80) the practice of influencing systems so that they ensure certain desired quality requirements such as costs, speed, acceleration etc. orthogonal (77) (ὀρθός ‘straight’ + γωνία ‘angle’) right-angled. In a more general meaning, properties that do not interact with or are opposed to each other. orthogonality (77) (ὀρθός ‘straight’ + γωνία ‘angle’) right-angled. In a more general meaning, properties that do not interact with or are opposed to each other. self-regulating (49) an autonomous system that operates without external control, such as any (healthy) biological function. A fundamental principle of the idealised model of a market. parabolas (78) see: conic sections Paracelsus (61) Philippus Aureolus Theophrastus Bombastus von Hohenheim (1493-1541), Renaissance physician, alchemist, astrologer and mystic, one of the pioneer critics of dogmatic scientific thinking in the Middle Ages but affected by contemporary mysticism. parameter in mathematical expressions a constant or variable which determines the uniqueness of the expression from the perspective of a given problem parametrisation the definition, measurement, replacement and calculation of parameters; see: parameter partial differential equations (72) differential equations representing the behaviour of processes which can be described along certain variables. This way, in relation to motion, they represent the elementary steps in changes according to the individual coordinates and time. parallel (56, 95) the relationship between systems or system parts that can replace each other, such as the kidneys. parallel postulate an ancient problem of Euclidean geometry: the possibility of proving that parallels intersect each other only in infinity (see: Non-Euclidean geometries) Parseval , Marc-Antoine (1755-1836), French mathematician primarily known for the theorem named after him. The theorem says that the square of the coefficients of a function’s Fourier series is equal to the square integral of the function in the time domain. It relates to the basic characteristic of the identity of the two representations. partitioning the act of dividing something into parts PID (96, 97) the proportional-integral-derivative (PID) control is the most common control structure used to set the permanent, continuous behaviour of the system and its adaptation speed. PID control (96, 97) the proportional-integral-derivative (PID) control is the most common control structure used to set the permanent, continuous behaviour of the system and its adaptation speed. Pythagoras (42) (Πυθαγόρας, ?570-?495 BC) Greek philosopher, mathematician and mystic clouded by legends. Most importantly, he contributed to science with his worldview deeply rooted in numbers and his achievements related to triangles and string lengths which almost naturally follow from this worldview. His mystical worldview spread in religious-like teachings and organisations. Platonic shapes (25) shapes bounded by congruent regular planes. Figures in space to which ancient lore attributed symbolic, often mystical meanings to refer to the regularities of the world and the exclusiveness of such regularities. Five such three-dimensional shapes were identified and taken as the representation of ancient elements by Platon and the Greeks, and as the structure of the solar system by Kepler. Poincaré torus the hyperbolic representation of Euclidean space into the closed space of a circle or—in higher dimensions—spheres. It has a fundamental role in the transformation interpretations of systems theory. Although it was named after Poincaré, as one of the most important contributors, the idea was elaborated by Italian mathematician Beltrami before him and by Felix Klein in parallel to both of them. Poisson bracket or Lie bracket; a sign used in Lie algebras to refer to the difference of expressions and their non-commutative counterparts which gives zero in a position of rest and a negative value in all other cases indicating difference, such as transformation polar coordinate vector coordinate; definition and representation by amplitude and phase angle polynomials algebraic expressions of two or more terms. They consist of variables, numbers, sum of terms each of which is the product of a constant and variables raised to nonnegative integral powers pole the singularity points of functions representing phenomena. In control engineering they mark the zeros of transfer function denominators. potential the ability to perform work; a type of energy prediction (99) an outcome forecast on the basis of former system behaviour or other preset knowledge projective geometry the geometry and relations of shapes resulting from various projections PROLOG high-level programming language used for solving problems that can be handled by logical methods prosthesis (33) artificial objects used to replace missing body parts such as teeth or bones psychology (26) (ψυχή ‘breath, spirit, soul’ + λογία ‘study of’) the scientific study of mental processes and behaviour registration (74) (from Latin regerere ‘to carry/bring back’) the process of recording system states and phenomena in a traceable way with the tools of the given phenomena or recording technique. This is how registration (recording, logging) devices and modern digital data collection techniques have evolved (the latter in close relation with data mining) hidden Markov process a mass of information in which the Markovian characteristics (i.e. the interdependency or non-interdependency of phenomena, data and their probabilities) can only be revealed with suitable algorithms. It is a typical tool in pattern recognition. coprime integers or coprimes. Numbers with no common divisor other than the unit theory of relativity (63, 71, 77) special and general relativity theory on space, time, the unity of the two, and the relative phenomena of space-time (Einstein) disorder (68) a concept related to entropy referring to the presence or absence of a mathematical or objective structure Renaissance (73) intellectual and artistic movement in Europe between the 14th and 16th centuries, after the Middle Ages, characterised by the revival of classical traditions and values. representation (30, 74) As regards systems, the act of representing a phenomenon with a selected behaviour that is similar in a certain aspect. The most general representation of system phenomena is mathematics residue remainder. The value of an integral along a closed curve. Its meaning becomes clear when interpreting the Gauss-Stokes’ Law (see: Gauss-Stokes, sink). In systems theory residue is interpreted as a confinable singularity on the complex plane. robot partly or fully automated machine that performs work. It derives from the Slavic word for work and was first used in this sense by Karel Čapek in 1921. robust (80, 81, 102) a system which preserves its key properties (primarily its stability but also some other quality properties) in a predefined domain of disturbances to a reasonable and also predefined extent. Shannon , Claude (1916-2001), American mathematician and engineer who laid the foundations of information theory SIMULINK (87) see: MATLAB scaling (39, 71, 74) the classification of a phenomenon by one of its measurable properties, e.g. the classification of temperature by the measurements on the Celsius or Fahrenheit scale. Sophus Lie , Marius Sophus (1842-1899) Norwegian mathematician, author of the transformation groups in group theory, including anti-commutative algebras which express equilibrium relations. serial (56, 95) elements arranged in a series where any element can change the system or render it inoperable if unconnected (e.g. digestive system) Spinoza , Baruch (1632-1677), Dutch-Jewish philosopher, prominent figure of the Enlightenment, pioneer of religious criticism. He identified God with Nature (Deus sive Natura) stable (80, 82, 91, 93) the inclination of systems to find a position of rest after a finite value of disturbance. A basic requirement of system behaviour stability (80, 82, 91, 93) the inclination of systems to find a position of rest after a finite value of disturbance. A basic requirement of system behaviour. stationary characterised by unchanging behaviour static not moving statistics (47) the science of collecting, processing, classifying and analysing data Stokes’ law (Stokes-Gauss) the value of an integral along a closed curve is zero, or corresponds to the source or sink represented within the curve. In physics it refers to various flows, in systems theory it refers to singularities within a given domain in the system. closed-loop control (34, 52, 84) change introduced to a system with a specific aim. As opposed to open-loop control, closed-loop control involves feedback. Term usage varies by language (in Hungarian: szabályozás vs vezérlés vs irányítás). controllability (szabályozhatóság) (89) see: controllability (irányíthatóság) computational linguistics the application of linguistic methods, including in particular grammar, to computers and the use of such methods in computational structures in order to analyse and generate texts. voting the method for selecting the best candidate or the best choice. Its dubious validity and clarity is long known. Condorcet proved that no voting method can be free from inconsistencies or contradictions, illustrating his claim through the conflict of decision by majority vote and decision by comparing pairs. Similar reservations apply to decision-making processes used in computer programming. extreme (value) the first or last, largest or smallest, most favourable or most unfavourable values in a series personal freedom (7) a fundamental human right especially emphasised in information society (cf. privacy, transparency) syllogism (27) (συλλογισμός ‘conclusion, inference’) a form of deductive reasoning. They were originally defined by Aristotle and still serve as the basic structure of deductive logical procedures and computer programs, such as the programming language PROLOG. symbols (6) (σύμβολον ‘contrasting, comparing’) something representing another thing by association or resemblance symmetry (25) (συμ ‘together’ + μετρία ‘measure’) similarity or balance among systems or parts of a system singular unique value or state: optimal, pessimal, unstable. See: poles singularity unique value or state: optimal, pessimal, unstable. See: poles stochastic (88) (στόχος ‘assumed, alleged’) effects or processes that are or assumed to be random, uncertain and of unspecified origin and nature superposition (54) the simple linear cumulation of phenomena (see: linear) supremal upper limit. A term generally used to describe spaces, separately from the term used for the maximum of a function. supremum upper limit. A term generally used to describe spaces, separately from the term used for the maximum of a function. filtering in systems theory, the elimination of disturbances affecting a process usually by separating the frequency or amplitude domain of the useful signals and of the disturbances or by employing probability calculus or statistical methods. See: Kalman filter filter (46) in systems theory, calculation methods which separate disturbances from the signals to be evaluated and controlled usually employing probability calculus methods, mainly in frequency domains. learning the statistical recording of the behaviour, states and influence outcomes of systems and the analysis of such data with the methods of probability calculus and logic reserve the stand-by energy or energy-like state of a system which is ready to use. See: potential energy Taylor , Brook (1685-1731), English mathematician, author of the series named after him. He also participated in the adjudication of the Leibniz-Newton calculus controversy Taylor series the power series of functions which can be differentiated any times in the surrounding or domain of the examined point. Most functions used in practice can be expressed or approximated as a Taylor series which makes the method one of the most common practical tool for calculations. inertia (15) the tendency of bodies to change their state only when acted on by an external force (from Latin iners ‘inactive’). The concept of inertia is often used figuratively in many other fields, such as in social sciences. The relations between inertia, mass and space-time gained new interpretations in the general theory of relativity. power (15, 63) In physics, power (symbol: P) is the speed of performing work or transferring energy; the work done per unit time. In dynamics, this work is the distance covered against a force, in all other fields it is an analogous concept, e.g. temperature change space (71, 72, 75, 79) the totality of phenomena and processes connected by some regularity of a geometrical nature. It is conceptually rooted in the experience of sight which is determined by mathematics and physics in the Cartesian coordinate system using the Euclidean rules. The generalisation of the concept is essential in thinking about and doing calculations in systems, ensuring the most efficient approach to modelling their operation. Formerly space was a central concept in making maps for various purposes and designing buildings, now it determines medical imaging and 3D production technologies. One of the most important achievements in the development of scientific thought is our ability to place phenomena in space. spaces (71, 72, 75, 79) the totality of phenomena and processes connected by some regularity of a geometrical nature. It is conceptually rooted in the experience of sight which is determined by mathematics and physics in the Cartesian coordinate system using the Euclidean rules. The generalisation of the concept is essential in thinking about and doing calculations in systems, ensuring the most efficient approach to modelling their operation. Formerly space was a central concept in making maps for various purposes and designing buildings, now it determines medical imaging and 3D production technologies. One of the most important achievements in the development of scientific thought is our ability to place phenomena in space. natural language spoken or written communication in any language that has evolved naturally. Its translation into a form that can be processed by computers is a central task of computational linguistics  (see there). thermodynamics (59) (θέρμη ‘heat’ + δύναμις ‘energetic force’) the term was coined in the 19th century which shows that the actual relationship between heat as a type of energy and heat as its scalable perceived effect was discovered relatively late in the history of physics. The discovery has also contributed to our understanding of the relationship between heat and motion. Toeplitz matrix (see: XI) a diagonal-constant matrix in which the entries along any diagonal parallel to the main diagonal are identical. The form resembles the continuous playing of video recordings in different times and places, this is why it is used to represent series of processes that take place there and back in time. Toeplitz , Otto (1881-1940) German-Jewish mathematician mainly working in functional analysis, and the history and education of mathematics trajectory path and its graphical representation; the representation of processes.  (from Latin traicere ‘throw across’) trajectories path and its graphical representation; the representation of processes.  (from Latin traicere ‘throw across’) transformations (43) change of shape, especially from one representation mode to another. In systems theory the most common type of transformation is from the time domain to the frequency domain. decision by vote see: voting real numbers the infinite set of natural integers and fractions. Its representation is the number line. probability the expected chance of an event to occur. Probability is the most important case of uncertainty definitions (e.g. Fuzzy). Its most elaborate methodology is probability theory. See: stochastic, fuzzy probability theory (47) the calculation of the expected occurrence of random events calculus of variations (85) the selection of functions whose definite integral takes an extreme value (minimum or maximum) in a set of functions (functionals). Practically, this is the basic question in all optimisation tasks. vector spaces the space of vectors, that is, quantities that have magnitude and direction (phase angle) open-loop control (34,52, 84) operation usually by logical means and without feedback. Term usage varies by language and profession. Pope Alexander VI (1431-1503) (75), member of the Borgia family during the Italian Renaissance who issued the bull granting Spain all lands to the west and Portugal all lands to the east of the meridian lying halfway between the Cape Verde islands and Haiti. feedback (56, 90, 95) the most important element of closed-loop controls; the return of information about impacts affecting a process and their result to the process Youla , Dante C. (1925-) (98) American electrical engineer, best known for the parametrisation named after him and Kučera. If a stable control system is complemented with an element which aims to meet a given quality property, the whole system’s stability needs to be revised and recalibrated. This is done by Youla parametrisation which relies on the coprime property of system elements (i.e. that their greatest common divisor is 1) Youla parametrisation the design of the parameters of the closed-loop control and the stable system which satisfies its quality requirements Youla-Kucera parametrisation See: Youla parametrisation noise disadvantageous signal that disturbs the operation of the system See: filter noises, disturbances disadvantageous signal that disturbs the operation of the system See: filter closed-loop control see: feedback disturbance (40, 95) any signal influencing the process but not serving its purpose. Disturbances are usually stochastic feedforward See: filtering feedforward, cascade control See: filtering zero (94) see: transfer function