"The role of graphs in probabilistic and statistical modeling is threefold:" (1) to provide convenient means of expressing substantive assumptions;" (2) to facilitate economical representation of joint probability functions; and" (3) to facilitate efficient inferences from observations." (Judea Pearl, "Causality: Models, Reasoning, and Inference", 2000)
"The fuzzy set theory is taking the same logical approach as what people have been doing with the classical set theory: in the classical set theory, as soon as the two-valued characteristic function has been defined and adopted, rigorous mathematics follows; in the fuzzy set case, as soon as a multi-valued characteristic function" (the membership function) has been chosen and fixed, a rigorous mathematical theory can be fully developed." (Guanrong Chen & Trung Tat Pham, "Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems", 2001)
"A game in strategic form is just a function with one input for each player (a strategy) and one output for each player (a payoff). More formally, a game in strategic form is a vector function and its do-main, the strategy space. The strategy space is just the set of all possible combinations of strategies, and therefore incorporates both the player and strategy sets." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"Any game in one of the classes can be converted into any other in the same region by some strictly monotonic transformation. Since a monotonic transform conserves order, all the games in an equiva-lence class are ordinally equivalent. These equivalence classes par-tition the 8-dimensional payoff space for the 2 × 2 games into 144 regions. An ordinal 2×2 game is a 2×2 game with a payoff function that maps from the strategy space to these equivalence classes." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"One important and standardized block of information in the formal descriptions used by game theorists is called a game form. A form specifies the payoffs associated with every possible combination of decisions. There are several widely used forms, including the strate-gic form, typically presented in a matrix, the extensive form, which is usually represented as a tree, and the characteristic function form, expressed as a function on subsets of players." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"Since games are characterized by the payoff function, similar games must have similar payoff functions. To define meaningful neigh-bourhoods, we need to characterize the smallest significant change in the payoff function. Obviously a change affecting the payoffs of one player is smaller than a change affecting two players. The clos-est neighbouring games are therefore those games that differ only by a small change in the ordering of the outcomes for one player." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"[...] complex functions are actually better behaved than real functions, and the subject of complex analysis is known for its regularity and order, while real analysis is known for wildness and pathology. A smooth complex function is predictable, in the sense that the values of the function in an arbitrarily small region determine its values everywhere. A smooth real function can be completely unpredictable [...]" (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)
"Each fuzzy set is uniquely defined by a membership function. […] There are two approaches to determining a membership function. The first approach is to use the knowledge of human experts. Because fuzzy sets are often used to formulate human knowledge, membership functions represent a part of human knowledge. Usually, this approach can only give a rough formula of the membership function and fine-tuning is required. The second approach is to use data collected from various sensors to determine the membership function. Specifically, we first specify the structure of membership function and then fine-tune the parameters of membership function based on the data." (Huaguang Zhang & Derong Liu, "Fuzzy Modeling and Fuzzy Control", 2006)
"Intuitively, two spaces that are homeomorphic have the same general shape in spite of possible deformations of distance and angle. Thus, if two spaces are not homeomorphic, they will tend to look distinctly different. Our job is to specify the difference. To do this rigorously, we need to define some property of topological spaces and show that the property is preserved under transformations by any homeomorphism. Then if one space has the property and the other one does not have the property, there is no way they can be homeomorphic." (Robert Messer & Philip Straffin, "Topology Now!", 2006)
"Moonshine is profoundly connected with physics (namely conformal field theory and string theory). String theory proposes that the elementary particles (electrons, photons, quarks, etc.) are vibrational modes on a string of length about 10^−33 cm. These strings can interact only by splitting apart or joining together – as they evolve through time, these (classical) strings will trace out a surface called the world-sheet. Quantum field theory tells us that the quantum quantities of interest (amplitudes) can be perturbatively computed as weighted averages taken over spaces of these world-sheets. Conformally equivalent world-sheets should be identified, so we are led to interpret amplitudes as certain integrals over moduli spaces of surfaces. This approach to string theory leads to a conformally invariant quantum field theory on two-dimensional space-time, called conformal field theory (CFT). The various modular forms and functions arising in Moonshine appear as integrands in some of these genus-1 (‘1-loop’) amplitudes: hence their modularity is manifest." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)
"The definition of homeomorphism was motivated by the idea of preserving the general shape or configuration of a geometric figure. Since path components are significant characteristics of a space, it is certainly reasonable that a homeomorphism will preserve the decomposition of a space into path components. […] Suppose we are given two geometric figures that we suspect are not topologically equivalent. If both of the figures are path-connected, counting components will not distinguish the spaces. However, we might be able to remove a special subset of one of the figures and count the number of components of the remainder. If no comparable set can be removed from the other space to leave the same number of components, we will then know that the two spaces are not homeomorphic." (Robert Messer & Philip Straffin, "Topology Now!", 2006)
"The easiest way to show two figures are homeomorphic is often to construct an explicit homeomorphism between them. But what if two figures are not homeomorphic? Surely we cannot be expected to check every function between the sets and show that it is not a homeomorphism. One of the goals of the field of topology is to discover easier ways of detecting the differences between spaces that are not homeomorphic." (Robert Messer & Philip Straffin, "Topology Now!", 2006)
"The Simplicial Approximation Theorem is a concise statement of the general result for functions between any two triangulated spaces. It says that on a suitable subdivision of the domain, any continuous function can be homotopically deformed by an arbitrarily small amount so that the modified function sends vertices to vertices and is linear on each edge, face, tetrahedron, and higher-dimensional cell of the triangulation." (Robert Messer & Philip Straffin, "Topology Now!", 2006)
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