"[…] a pink (or white, or brown) noise is the very paradigm of a statistically self-similar process. Phenomena whose power spectra are homogeneous power functions lack inherent time and frequency scales; they are scale-free. There is no characteristic time or frequency -whatever happens in one time or frequency range happens on all time or frequency scales. If such noises are recorded on magnetic tape and played back at various speeds, they sound the same […]" (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)
"All physical objects that are 'self-similar' have limited self-similarity - just as there are no perfectly periodic functions, in the mathematical sense, in the real world: most oscillations have a beginning and an end (with the possible exception of our universe, if it is closed and begins a new life cycle after every 'big crunch' […]. Nevertheless, self-similarity is a useful abstraction, just as periodicity is one of the most useful concepts in the sciences, any finite extent notwithstanding." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)
"[…] homogeneous functions have an interesting scaling property: they reproduce themselves upon rescaling. This scaling invariance can shed light into some of the darker corners of physics, biology, and other sciences, and even illuminate our appreciation of music." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)
"The zeta function is probably the most challenging and mysterious object of modern mathematics, in spite of its utter simplicity. [...] The main interest comes from trying to improve the Prime Number Theorem, i.e., getting better estimates for the distribution of the prime numbers. The secret to the success is assumed to lie in proving a conjecture which Riemann stated in 1859 without much fare, and whose proof has since then become the single most desirable achievement for a mathematician." (Martin C Gutzwiller, "Chaos in Classical and Quantum Mechanics", 1990)
"I am firmly convinced that the most important unsolved problem in mathematics today is the truth or falsity of a conjecture about the zeros of the zeta function, which was fi rst made by Riemann himself. [...] Even a single exception to Riemann’s conjecture would have enormous ly strange consequences for the distribution of prime numbers [...] If the Riemann hypothesis turns out to be false, there will be huge oscillations in the distribution of primes. In an orchestra, that would be like one loud instrument that drowns out the others - an aesthetically distasteful situation." (Enrico Bombieri, "Prime Territory: Exploring the Infinite Landscape at the Base of the Number System", The Sciences, 1992)
"Regarding stability, the state trajectories of a system tend to equilibrium. In the simplest case they converge to one point" (or different points from different initial states), more commonly to one" (or several, according to initial state) fixed point or limit cycle(s) or even torus(es) of characteristic equilibrial behaviour. All this is, in a rigorous sense, contingent upon describing a potential, as a special summation of the multitude of forces acting upon the state in question, and finding the fixed points, cycles, etc., to be minima of the potential function. It is often more convenient to use the equivalent jargon of 'attractors' so that the state of a system is 'attracted' to an equilibrial behaviour. In any case, once in equilibrial conditions, the system returns to its limit, equilibrial behaviour after small, arbitrary, and random perturbations." (Gordon Pask, "Different Kinds of Cybernetics", 1992)
"A homeomorphism may be thought of as the best possible type of continuous function, and homeomorphic spaces are considered the same in topology. [...] A complex has two structures: that of the topological space underlying the complex, and the subdivision of the complex into cells. The corresponding functions will have to preserve this dual nature. In particular, it would be nice if these functions, yet to be defined, would also induce nice functions on the homology of the complex. Since homology has an algebraic group structure, we want the functions to induce homomorphisms on the homology groups. The best of all functions would be homeomorphisms on the underlying topological space and also isomorphisms on the homology groups." (L Christine Kinsey. "Topology of Surfaces", 1993)
"In any mathematical study, the first thing to do is specify the type or category of objects to be investigated. In topology, the most general possible objects of study are sets of points with just enough structure to be able to define continuous functions." (L Christine Kinsey. "Topology of Surfaces", 1993)
"In mathematics, a function is a sort of converter: It converts a value to some other value. […] the mathematical definition of a continuous function requires that any small change in the input will result in a small change to the output." (Alexander Humez et al, "Zero to Lazy Eight: The romance of numbers", 1993)
"In most fields of mathematics, we study some class of objects and also the functions appropriate to the objects of study. In linear algebra, the objects studied are vector spaces and the corresponding functions, called linear transformations and usually represented by matrices, are those which preserve the vector space structure. In geometry, we study isometries, such as rotations: functions that do not change the geometric properties of length, angle measure, area, and volume. In abstract algebra, the objects of study are groups and the appropriate functions are homomorphisms, which preserve the algebraic properties of the groups. The best possible homomorphism is an isomorphism; isomorphic groups are essentially identical algebraically." (L Christine Kinsey. "Topology of Surfaces", 1993)
"Determination of transition functions makes it possible to restore the whole manifold if individual charts and coordinate maps are already given. Glueing functions may belong to different functional classes,which makes it possible to specify within a certain class of topological manifolds more narrow classes of smooth, analytic, etc. manifolds." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)
"Geometry and topology most often deal with geometrical figures, objects realized as a set of points in a Euclidean space (maybe of many dimensions). It is useful to view these objects not as rigid (solid) bodies, but as figures that admit continuous deformation preserving some qualitative properties of the object. Recall that the mapping of one object onto another is called continuous if it can be determined by means of continuous functions in a Cartesian coordinate system in space. The mapping of one figure onto another is called homeomorphism if it is continuous and one-to-one, i.e. establishes a one-to-one correspondence between points of both figures." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)
"Such a close connection between trigonometric functions, the mathematical constant 'e', and the square root of -1 is already quite startling. Surely, such an identity cannot be a mere accident; rather, we must be catching a glimpse of a rich, complicated, and highly abstract mathematical pattern that for the most part lies hidden from our view." (Keith Devlin, "Mathematics: the Science of Patterns", 1994)
"The acceptance of complex numbers into the realm of algebra had an impact on analysis as well. The great success of the differential and integral calculus raised the possibility of extending it to functions of complex variables. Formally, we can extend Euler's definition of a function to complex variables without changing a single word; we merely allow the constants and variables to assume complex values. But from a geometric point of view, such a function cannot be plotted as a graph in a two-dimensional coordinate system because each of the variables now requires for its representation a two-dimensional coordinate system, that is, a plane. To interpret such a function geometrically, we must think of it as a mapping, or transformation, from one plane to another." (Eli Maor, "e: The Story of a Number", 1994)
"The concept of homeomorphism appears to be convenient for establishing those important properties of figures which remain unchanged under such deformations. These properties are sometimes referred to as topological, as distinguished from metrical, which are customarily associated with distances between points, angles between lines, edges of a figure, etc." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)
"Catastrophe theory is a local theory, telling us what a function looks like in a small neighborhood of a critical point; it says nothing about what the function may be doing far away from the singularity. Yet most of the applications of the theory [...] involve extrapolating these rock-solid, local results to regions that may well be distant in time and space from the singularity. " (John L Casti, "Five Golden Rules", 1995)
"If you start with a number and form its square root, you get another number. The term for such an 'object' is function. You can think of a function as a mathematical rule that starts with a mathematical object-usually a number-and associates to it another object in a specific manner. Functions are often defined using algebraic formulas, which are just shorthand ways to explain what the rule is, but they can be defined by any convenient method. Another term with the same meaning as 'function' is transformation: the rule transforms the first object into the second. […] Operations and functions are very similar concepts. Indeed, on a suitable level of generality there is not much to distinguish them. Both of them are processes rather than things."
"The goal of catastrophe theory is to classify smooth functions with degenerate critical points, just as Morse's Theorem gives us a complete classification for Morse functions. The difficulty, of course, is that there are a lot more ways for critical points to 'go bad' than there are for them to stay 'nice'. Thus, the classification problem is much harder for functions having degenerate critical points, and has not yet been fully carried out for all possible types of degeneracies. Fortunately, though, we can obtain a partial classification for those functions having critical points that are not too bad. And this classification turns out to be sufficient to apply the results to a wide range of phenomena like the predator-prey situation sketched above, in which 'jumps' in the system's biomass can occur when parameters describing the process change only slightly." (John L Casti, "Five Golden Rules", 1995)
"Topology studies the properties of geometrical objects that remain unchanged under transformations called homeomorphisms and deformations." (Victor V Prasolov, "Intuitive Topology", 1995)
"[...] a manifold is a set M on which 'nearness' is introduced" (a topological space), and this nearness can be described at each point in M by using coordinates. It also requires that in an overlapping region, where two coordinate systems intersect, the coordinate transformation is given by differentiable transition functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)"
"An essential difference between continuity and differentiability is whether numbers are involved or not. The concept of continuity is characterized by the qualitative property that nearby objects are mapped to nearby objects. However, the concept of differentiation is obtained by using the ratio of infinitesimal increments. Therefore, we see that differentiability essentially involves numbers." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)
"Analyticity is the property of a differentiable function y = f(x) that can be represented by the infinite series for all x near each point x0." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)
"And of course the space the wave function live in, and (therefore) the space we live in, the space in which any realistic understanding of quantum mechanics is necessarily going to depict the history of the world as playing itself out […] is configuration-space. And whatever impression we have to the contrary (whatever impression we have, say, of living in a three-dimensional space, or in a four dimensional spacetime) is somehow flatly illusory." (David Albert, "Elementary Quantum Metaphysics", 1996)
"By studying analytic functions using power series, the algebra of the Middle Ages was connected to infinite operations (various algebraic operations with infinite series). The relation of algebra with infinite operations was later merged with the newly developed differential and integral calculus. These developments gave impetus to early stages of the development of analysis. In a way, we can say that analyticity is the notion that first crossed the boundary from finite to infinite by passing from polynomials to infinite series. However, algebraic properties of polynomial functions still are strongly present in analytic functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)
"Continuous functions can change freely, while analytic functions are rigid. In this sense, we can say that continuous and analytic functions are antipodal." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)
"Continuous functions can move freely. Graphs of continuous functions can freely branch off at any place, whereas analytic functions coinciding in some neighborhood of a point P cannot branch outside of this neighborhood. Because of this property, continuous functions can mathematically represent wildly changing wind inside a typhoon or a gentle breeze." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)
"Differentiability of a function can be established by examining the behavior of the function in the immediate neighborhood of a single point a in its domain. Thus, all we need is coordinates in the vicinity of the point a. From this point of view, one might say that local coordinates have more essential qualities. However, if are not looking at individual surfaces, we cannot find a more general and universal notion than smoothness." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)
"Even though there are no methods to represent each differentiable function by an 'equation', we can still investigate differentiable functions by various analytic methods. Because of this, we can say that differentiable functions have more mathematical reality than continuous functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)
"Fuzzy systems are excellent tools for representing heuristic, commonsense rules. Fuzzy inference methods apply these rules to data and infer a solution. Neural networks are very efficient at learning heuristics from data. They are 'good problem solvers' when past data are available. Both fuzzy systems and neural networks are universal approximators in a sense, that is, for a given continuous objective function there will be a fuzzy system and a neural network which approximate it to any degree of accuracy." (Nikola K Kasabov, "Foundations of Neural Networks, Fuzzy Systems, and Knowledge Engineering", 1996)
"Graphic scales are not only the most helpful means of communicating map scale but also the safest. An alternative to blind trust in the user's sense of distance and skill in mental arithmetic, the simple bar scale typically portrays a series of conveniently rounded distances appropriate to the map's function and the area covered." (Mark S Monmonier, "How to Lie with Maps" 2nd Ed., 1996)
"If each change of a certain quantity results in a corresponding change of another quantity, we can say that there exists a functional relationship between those two quantities. Viewed in this manner, the idea of functions expands endlessly. The concept of functions is truly comprehensive, but while it is all encompassing, it is not fathomless; at least, not with respect to our current subject of manifolds. You might feel that linear functions or quadratic functions are far too specific and that you are sinking into the depths of the ocean called functions. However, you will be rescued from the ocean depths by understanding of the functions that are needed to describe manifolds. These functions are continuous, analytic, and differentiable functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)
"[...] moving from the concept of continuity to differentiability, and then to analyticity, we are moving from weaker properties to stronger ones. Therefore, the relations between the corresponding properties of functions can be expressed as follows: {continuous functions} > {differentiate functions} > {analytic functions}." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)
"Similarly to the graphs of continuous functions, graphs of differentiable" (smooth) functions which coincide in a neighborhood of a point P can branch off outside of the neighborhood. Because of this property, differentiable functions can represent smoothly changing natural phenomena." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)
"[...] the only characteristic property that continuous functions have is that near objects are sent to corresponding near objects, that is, a convergent sequence is mapped to the corresponding convergent sequence. It is reasonable to say that we cannot expect to extract from that property neither numerical consequences, nor a method to extensively study continuity. On the contrary, analytic functions can be represented by equations" (precisely speaking, by infinite series). Compared to analytic functions, continuous functions, in general, are difficult to represent explicitly, although they exist as a concept." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996
"To describe the property of smoothness, differentiable functions should be specified first. To do so, coordinates need to be introduced on the topological space. Those coordinates can be local coordinates such as the ones used by Gauss. Once coordinates are introduced around a point a in a topological space, differentiable functions near the point a are distinguished from the continuous functions in the region near a. If different coordinates are chosen, then a different set of differentiable functions is distinguished. In other words, the choice of local coordinates determines the notion of smoothness in a topological space." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)
"Mathematical programming (or optimization theory) is that branch of mathematics dealing with techniques for maximizing or minimizing an objective function subject to linear, nonlinear, and integer constraints on the variables." (George B Dantzig & Mukund N Thapa, "Linear Programming" Vol I, 1997)
"In fact the complex numbers form a field. [...] So however strange you may feel the very notion of a complex number to be, it does turn out to provide a 'normal' type of arithmetic. In fact it gives you a tremendous bonus not available with any of the other number systems. [...] The fundamental theorem of algebra is just one of several reasons why the complex-number system is such a 'nice' one. Another important reason is that the field of complex numbers supports the development of a powerful differential calculus, leading to the rich theory of functions of a complex variable." (Keith Devlin, "Mathematics: The New Golden Age", 1998)
"It is a major statistical sin to show a graph displaying a variable as a function of time with the vertical (left-hand) scale cut short so that it does not go down to zero, without drawing attention to this fact. This sin can create a seriously misleading impression, and, as they do with most sins, sinners commit it again and again." (Herbert F Spirer et al, "Misused Statistics" 2nd Ed, 1998)
"Algebraic topology studies properties of a narrower class of spaces, - basically the classical objects of mathematics: spaces given by systems of algebraic and functional equations, surfaces lying in Euclidean space, and other sets which in mathematics are called manifolds. Examining the narrower class of spaces permits deeper penetration into their structure." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)
"[...] fuzzy logic [FL] has many distinct facets - facets which overlap and have unsharp boundaries. Among these facets there are four that stand out in importance. They are" (i) the logical facet;" (ii) the set-theoretic facet:" (iii) the relational facet, and" (iv) the epistemic facet. […] The logical facet of FL, FL/L, is a logical system or, more accurately, a collection of logical systems which includes as a special case both two-valued and multiple-valued systems. […] The set-theoretic facet of FL, FL/S, is concerned with classes or sets whose boundaries are not sharply defined. […] The relational facet of FL, FL/R, is concerned in the main with representation and manipulation of imprecisely defined functions and relations. […] The epistemic facet of FL, FL/E, is linked to its logical facet and is centered on applications of FL to knowledge representation, information systems, fuzzy databases and the theories of possibility imprecise probabilities." (Lotfi A Zadeh, "The Birth and Evolution of Fuzzy Logic: A personal perspective", 1999)
"One of the basic tasks of topology is to learn to distinguish nonhomeomorphic figures. To this end one introduces the class of invariant quantities that do not change under homeomorphic transformations of a given figure. The study of the invariance of topological spaces is connected with the solution of a whole series of complex questions: Can one describe a class of invariants of a given manifold? Is there a set of integral invariants that fully characterizes the topological type of a manifold? and so forth." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)
"These so-called stochastic processes show up everywhere randomness is applied to the output of another random function. They provide, for instance, a method for describing the chance component of financial markets: not every value of the Dow is possible every day; the range of chance fluctuation centers on the opening price. Similarly, shuffling takes the output of the previous shuffle as its input. So, if you’re handed a deck in a given order, how much shuffling does it need to be truly random?" (John Haigh," Taking Chances: Winning With Probability", 1999)
"Two figures which can be transformed into one other by continuous deformations without cutting and pasting are called homeomorphic. […] The definition of a homeomorphism includes two conditions: continuous and one- to-one correspondence between the points of two figures. The relation between the two properties has fundamental significance for defining such a paramount concept as the dimension of space." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)
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