"Strange attractors, unlike regular ones, are geometrically very complicated, as revealed by the evolution of a small phase-space volume. For instance, if the attractor is a limit cycle, a small two-dimensional volume does not change too much its shape: in a direction it maintains its size, while in the other it shrinks till becoming a 'very thin strand' with an almost constant length. In chaotic systems, instead, the dynamics continuously stretches and folds an initial small volume transforming it into a thinner and thinner 'ribbon' with an exponentially increasing length." (Massimo Cencini et al, "Chaos: From Simple Models to Complex Systems", 2010)
"In a chaotic system, there must be stretching to cause the exponential separation of initial conditions but also folding to keep the trajectories from moving off to infinity. The folding requires that the equations of motion contain at least one nonlinearity, leading to the important principle that chaos is a property unique to nonlinear dynamical systems. If a system of equations has only linear terms, it cannot exhibit chaos no matter how complicated or high-dimensional it may be." (Julien C Sprott, "Elegant Chaos: Algebraically Simple Chaotic Flows", 2010)
"Strange attractors, unlike regular ones, are geometrically very complicated, as revealed by the evolution of a small phase-space volume. For instance, if the attractor is a limit cycle, a small two-dimensional volume does not change too much its shape: in a direction it maintains its size, while in the other it shrinks till becoming a 'very thin strand' with an almost constant length. In chaotic systems, instead, the dynamics continuously stretches and folds an initial small volume transforming it into a thinner and thinner 'ribbon' with an exponentially increasing length." (Massimo Cencini et al, "Chaos: From Simple Models to Complex Systems", 2010)
"Systems with dimension greater than four begin to lose their elegance unless they possess some kind of symmetry that reduces the number of parameters. One such symmetry has the variables arranged in a ring of many identical elements, each connected to its neighbors in an identical fashion. The symmetry of the equations is often broken in the solutions, giving rise to spatiotemporal chaotic patterns that are elegant in their own right." (Julien C Sprott, "Elegant Chaos: Algebraically Simple Chaotic Flows", 2010)
"At first glance, sets are about as primitive a concept as can be imagined. The concept of a set, which is, after all, a collection of objects, might not appear to be a rich enough idea to support modern mathematics, but just the opposite proved to be true. The more that mathematicians studied sets, the more astonished they were at what they discovered, and astonished is the right word. The results that these mathematicians obtained were often controversial because they violated many common sense notions about equality and dimension." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)
"These days the term 'central limit theorem' is associated with a multitude of statements having to do with the convergence of probability distributions of functions of an increasing number of one- or multi-dimensional random variables or even more general random elements (with values in Banach spaces or more general spaces) to a normal distribution3 (or related distributions). In an effort to reduce ambiguity - and in view of historic developments - the denotation “central limit theorem” in the present examination will usually refer only to the 'classical' case, which deals with the asymptotic equality of distributions of sums of independent or weakly dependent random variables and of a normal distribution." (Hans Fischer, "A History of the Central Limit Theorem: From Classical to Modern Probability Theory", 2011)
"Chaos does provide a framework or a mindsetor point of view, but it is not as directly explanatory as germ theory or plate tectonics. Chaos is a behavior - a phenomenon - not a causal mechanism. [...] The situation with fractals is similar. The study of fractals draws one’s eye toward patterns and structures that repeat across different length or time scales. There is also a set of analytical tools - mainly calculating various fractal dimensions - that can be used to quantify structural properties of fractals. Fractal dimensions and related quantities have become standard tools used across the sciences. As with chaos, there is not a fractal theory. However, the study of fractals has helped to explain why certain types of shapes and patterns occur so frequently." (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012)
"Fractals' simultaneous chaos and order, self-similarity, fractal dimension and tendency to scalability distinguish them from any other mathematically drawable figures previously conceived." (Mehrdad Garousi, "The Postmodern Beauty of Fractals", Leonardo Vol. 45" (1), 2012)
"The more dimensions used in quantitative comparisons, the larger are the disparities that can be accommodated. As irony would have it, however, the ease of comparison generally diminishes in direct proportion to the number of dimensions involved." (Joel Katz, "Designing Information: Human factors and common sense in information design", 2012)
"Because the differential of a smooth map is supposed to represent the 'best linear approximation' to the map near a given point, we can learn a great deal about a map by studying linear-algebraic properties of its differential. The most essential property of the differential - in fact, just about the only property that can be defined independently of choices of bases - is its rank (the dimension of its image)." (John M Lee, "Introduction to Smooth Manifolds" 2nd Ed., 2013)
"Dimensionality reduction and regression modeling are particularly hard to interpret in terms of original attributes, when the underlying data dimensionality is high. This is because the subspace embedding is defined as a linear combination of attributes with positive or negative coefficients. This cannot easily be intuitively interpreted in terms specific properties of the data attributes." (Charu C Aggarwal,Outlier Analysis", 2013)
"We identify and analyze distorting mental models that constitute experience in a manner that occludes the moral dimension of situations from view, thereby thwarting the first step of ethical decision-making. Examples include an unexamined moral self-image, viewing oneself as merely a bystander, and an exaggerated conception of self-sufficiency. These mental models, we argue, generate blind spots to ethics, in the sense that they limit our ability to see facts that are right before our eyes – sometimes quite literally, as in the many examples of managers and employees who see unethical behavior take place in front of them, but do not recognize it as such." (Patricia H Werhane et al, "Obstacles to Ethical: Decision-Making Mental Models, Milgram and the Problem of Obedience", 2013)
"For group theoretic reasons the most impressive paradoxical decompositions occur in dimension at least three, but there are also interesting decompositions in lower dimensions. For example, one can partition a disc into finitely many subsets and rigidly rearrange these subsets to form a square of the same area as the original disc. Even without the Axiom of Choice it is possible to construct some counterintuitive subsets of the plane, so Choice cannot be held responsible for all that is counterintuitive in geometry. Some more radical alternatives to the Axiom of Choice obstruct such constructions more effectively." (Barnaby Sheppard, "The Logic of Infinity", 2014)
"A signal is a useful message that resides in data. Data that isn’t useful is noise. […] When data is expressed visually, noise can exist not only as data that doesn’t inform but also as meaningless non-data elements of the display" (e.g. irrelevant attributes, such as a third dimension of depth in bars, color variation that has no significance, and artificial light and shadow effects)." (Stephen Few, "Signal: Understanding What Matters in a World of Noise", 2015)
"Curvature is a central concept in differential geometry. There are conceptually different ways to define it, associated with different mathematical objects, the metric tensor, and the affine connection. In our case, however, the affine connection may be derived from the metric. The 'affine curvature' is associated with the notion of parallel transport of vectors as introduced by Levi-Civita. This is most simply illustrated in the case of a two- dimensional surface embedded in three- dimensional space. Let us take a closed curve on that surface and attach to a point on that curve a vector tangent to the surface. Let us now transport that vector along the curve, keeping it parallel to itself. When it comes back to its original position, it will coincide with the original vector if the surface is flat or deviate from it by a certain angle if the surface is curved. If one takes a small curve around a point on the surface, then the ratio of the angle between the original and the final vector and the area enclosed by the curve is the curvature at that point. The curvature at a point on a two-dimensional surface is a pure number." (Hanoch Gutfreund, "The Road to Relativity", 2015
"Dynamics of a linear system are decomposable into multiple independent one-dimensional exponential dynamics, each of which takes place along the direction given by an eigenvector. A general trajectory from an arbitrary initial condition can be obtained by a simple linear superposition of those independent dynamics." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)
"Facts and concepts only acquire real meaning and significance when viewed through the lens of a conceptual system. [...] Facts do not exist independently of knowledge and understanding for without some conceptual basis one would not know what data to even consider. The very act of choosing implies some knowledge. One could say that data, knowledge, and understanding are different ways of describing the same situation depending on the type of human involvement implied - 'data' means a de-emphasis on the human dimension whereas 'understanding' highlights it." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)
"The best way to think about mathematics is to include not only the content dimension of algorithms, procedures, theorems, and proofs but also the cognitive dimensions of learning, understanding, and creating mathematics." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)
"The correlational technique known as multiple regression is used frequently in medical and social science research. This technique essentially correlates many independent" (or predictor) variables simultaneously with a given dependent variable" (outcome or output). It asks, 'Net of the effects of all the other variables, what is the effect of variable A on the dependent variable?' Despite its popularity, the technique is inherently weak and often yields misleading results. The problem is due to self-selection. If we don’t assign cases to a particular treatment, the cases may differ in any number of ways that could be causing them to differ along some dimension related to the dependent variable. We can know that the answer given by a multiple regression analysis is wrong because randomized control experiments, frequently referred to as the gold standard of research techniques, may give answers that are quite different from those obtained by multiple regression analysis." (Richard E Nisbett, "Mindware: Tools for Smart Thinking", 2015)
"The strength of an average is that it takes all the values in your data set and simplifies them down to a single number. This strength, however, is also the great danger of an average. If every data point is exactly the same (picture a row of identical bricks) then an average may, in fact, accurately reflect something about each one. But if your population isn’t similar along many key dimensions - and many data sets aren’t - then the average will likely obscure data points that are above or below the average, or parts of the data set that look different from the average. […] Another way that averages can mislead is that they typically only capture one aspect of the data." (John H Johnson & Mike Gluck, "Everydata: The misinformation hidden in the little data you consume every day", 2016)
"In category theory there is always a tension between the idealism and the logistics. There are many structures that naturally want to have infinite dimensions, but that is too impractical, so we try and think about them in the context of just a finite number of dimensions and struggle with the consequences of making these logistics workable."(Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)
"Mathematics is particularly good at making things out of itself, like how higher-dimensional spaces are built up from lower-dimensional spaces. This is because mathematics deals with abstract ideas like space and dimensions and infinity, and is itself an abstract idea. […] Mathematics is abstract enough that we can always make more mathematics out of mathematics." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)
"The presence of structure distinguishes maps from diagrams. They are similar in kind but not degree. Diagrams share some of the structure but don’t quantify or qualify spatio-temporal relationships in the same way as maps. They are ‘simplified figures to convey essential meaning,’ whereas maps tend toward robust meaning relative to the subject. Symbols in diagrams have multiple possible significations until we specify or point to their meaning through context using an index. Diagrams are indexical, i.e. they point to something, but they aren’t indexed: they don’t order or organize within a larger context nor do they have a spatio-temporal dimension like maps." (Winifred E Newman, "Data Visualization for Design Thinking: Applied Mapping", 2017)
"The trig functions’ input consists of the sizes of angles inside right triangles. Their output consists of the ratios of the lengths of the triangles’ sides. Thus, they act as if they contained phone-directory-like groups of paired entries, one of which is an angle, and the other is a ratio of triangle-side lengths associated with the angle. That makes them very useful for figuring out the dimensions of triangles based on limited information." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)
"Quaternions are not actual extensions of imaginary numbers, and they are not taking complex numbers into a multi-dimensional space on their own. Quaternion units are instances of some number-like object type, identified collectively, but they are not numbers" (be it real or imaginary). In other words, they form a closed, internally consistent set of object instances; they can of course be plotted visually on a multi-dimensional space but this only is a visualization within their own definition." (Huseyin Ozel, "Redefining Imaginary and Complex Numbers, Defining Imaginary and Complex Objects", 2018)
"The higher the dimension, in other words, the higher the number of possible interactions, and the more disproportionally difficult it is to understand the macro from the micro, the general from the simple units. This disproportionate increase of computational demands is called the curse of dimensionality." (Nassim N Taleb, "Skin in the Game: Hidden Asymmetries in Daily Life", 2018)
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