Mathematics

"The chapter "Design and analysis of Sukhov's towers"... is devoted to consideration and analysis of Sukhov's structural calculations... Sukhov's design process is reconstructed based on... calculations of five different water towers. From... tables stored in the Moscow city archives, a summary... is produced to chart the development of the towers over more than three decades..."

"An analysis of the design and construction of the NiGRES tower on the Oka... The initial ensemble of four electricity transmission masts represents the consummation of Shukhov's tower construction method. Only one... remains... today."

"The theory of minimal surfaces and surfaces of constant mean curvature is a vigorously developing branch of mathematics which has a broad range of applications in physics, chemistry and biology where it investigates, for example, soap films and bubbles, bimaterial interfaces or capillaries. Over the past decades, it was also boosted by the anticipated nanotechnology applications. Applications of minimal surfaces have been extensively explored in the sculpture (see e.g. the works by ) and in architecture (s, the Shukhov radio tower ... works by etc.) The first mention of minimal surfaces goes back to Lagrange (1798), who considered the... variational problem..."

"Construction history teaches us that excellent, huge structures like this one are invariably the result of a specific and singular solution, in which all aspects were reflected upon before or after early experiences in the building sequence. The author wants to compare the achievement of Segovia Aqueduct with more recent singular structures: the Oka electricity pylons (1929) by... Vladimir Gregoryevich Shukhov... and the roof of the 1972 Olympic Games in Munich, by a team directed by Gunter Behnish, , /, and Jürgen Linkwitz..."

"The structures of the great Russian engineer Vladimir Grigor'evič Šuchov (Shukhov) are among the world's most sophisticated and distinctive in the history of steel construction. These extremely stable structures, such as cable-stabilised arches, doubly curved grid-shells and above all lattice towers hold a great fascination... They result from the desire to achieve an engineering objective using as little material as possible. ...[T]hey are a testament to the extraordinary creativity and inventiveness of an extensively educated engineer ..."

"Due to... complexity... doubly-curved anticlastic surfaces such as hyperboloid and hyperbolic paraboloid (HP) have been rarely used for deployable structures. In fact, anticlastic structures can be easily constructed using simple straight bars rather than SLEs since their geometric forms can be generated by s..."

"According to Elizabeth C. English, there was a direct connection between the mathematical discoveries of Lobachevsky and the structural explorations of"

"The "idea of the " and the first... diagrid structure have been credited to... Shukhov. The design evolved as an efficient and easily constructed tower for carrying a large gravity load... a water tower. The "Shukhov Tower"... 1896, relies on the use of a diagonal lattice of steel angles, constrained laterally... by steel rings. ...The tower is hollow, requiring little resistance to wind loads ..."

"Although the slender steel sections, when formed into the hyperbolic paraboloid shape, did undergo some bending, a diagrid shape emerges, using a combination of straight segments that are joined at their nodal points of intersection."

"The Shukhov towers tended to use much longer steel sections and overlap them at their crosspoints, rather than using the crosspoints as "nodes" in the fashion of later or spaceframe structures."

"The of a one-sheeted hyperboloid is resolved into three different mesh variants to create open lattices and their structural behaviour investigated."

"[A] series of kinetic structures which are capable of geometric transformations have been developed. ...[T]he most impressive ones are deployable bar structures with [a] single degree-of-freedom ...These structures ...may become stable and carry loads ...Therefore ...may offer viable solutions for architectural applications, especially for temporary buildings, emergency shelters, exhibition halls, outdoor recreation facilities or sporting fields ...[M]ost of them are composed of scissor like elements (SLEs) ...which is a complex structural system ...[P]resent solutions are insufficient to constitute real form flexibility, because they are limited to... forms such as singly-curved vaults and doubly-curved synclastic s."

"The creative principles of innovative trends such as Rationalism and Constructivism were well understood by engineers and constructors bent on applying the latest technological achievements. Great engineers such as Shukhov, [Artur] Loleit and [Hermann] Krasin, were happy to work with innovative architects. They jointly explored new ways of developing architecture, and engineers became active members of Asnova and Osna. ...In the twenties... Soviet architecture set tasks for the building industry which prompted the employment of new materials and structural elements, and raised building standards. Engineering technology in the building industry achieved great successes during this period, such as the metal structures by Shukhov and Krasin, Loleit's work in the field of ferro-concrete, the elaboration of modern timber work by Karlson and the production of new hyperbolic paraboloid roofing by Markarova. The very latest structures and building materials, as well as modern production methods, were applied in the construction of engineering and industrial buildings such as Shukhov's radio tower in Moscow in 1922 and Krasin's viaduct at the Shatura power station in 1925."

"[H]yperbolic s are associated with nuclear and thermal power plants... they are also used... in some large chemical and other industrial plants. [T]hey are high rise structures in the form of doubly curved thin walled shells of complex geometry..."

"A groundbreaking advance on the way to efficient structures in steel construction was the principle of using tetrahedra as a basic module instead of rectangular geometries. The inventor is considered to be the all-round genius Alexander Graham Bell, who became famous for the invention of the telephone and who built kites that were big enough to lift people into the air. Another engineering genius was Vladimir Grgorevic Suchov, whose genius can definitely be compared to Gustav Eiffel. In 1919 he designed towers up to 350 m high on the principle of hyperbolic paraboloids. The lightness of his constructions is seldom achieved even today, in Moscow there is a television tower with a height of 160 m. His tent constructions with suspended steel grids can be seen as the forerunners of the Olympic roof in Munich or the new Center Pompidou Metz. In , Suchov realized the first double-curved lattice shells on the floor plan of rectangular halls as early as the 19th century. Most of the structures still in existence are massively endangered by corrosion and destruction; current rescue operations are trying to preserve this legacy."

"The hyperbolic geometry has advantage of a negative which makes it superior in stability against external pressures..."

"The... structure is made of high-strength [cement,] Reinforced... Concrete... in the form of hyperbolic thin shell standing on diagonal, meridional, or vertical supporting columns and radial supports. The shell is sufficiently stiffened by upper and lower edge members."

"[T]o achieve sufficient resistance against instability, large cooling tower shells may be stiffened by additional internal or external rings which may also be used as repair or rehabilitation..."

"Deformation response and ultimate strength of RC shell structures are governed predominantly by material response of concrete and reinforcing steel, tensile cracking of concrete, [and] bond between concrete and steel.... Softening response of concrete due to quasi-brittle cracking in tension also... influences the nonlinear response by inducing loss of strength and stiffness... Due to all [of] these, analysis requires attention for realistic modeling of the layer of shell concrete confined between the reinforcement layers. ...[O]ne of the most challenging areas... is the modeling techniques using the layered elements."

"Shukhov's water tower['s]... double curved surface... was generated by a mesh of straight members overlapping in contrary directions... supported by horizontal rings. While... constructed from steel... Shukhov's 1896 patent application... initially mentions straight wooden beams as a material option. ...[T]he ...application describes ...being able to resist extreme forces while using very little material. As a result, Shukhov's... design was used extensively throughout Russia in the first half of the twentieth century."

"Wavelets are everywhere nowadays. Whether in signal or image processing, in astronomy, in fluid dynamics (turbulence), or in condensed matter physics, wavelets have found applications in almost every corner of physics. Furthermore, wavelet methods have become standard fare in applied mathematics, numerical analysis, and approximation theory."

"On the one hand, the concept of wavelets can be viewed as a synthesis of ideas which originated during the last twenty or thirty years in engineering (subbing coding), physics (coherent states, renormalization group), and pure mathematics (study of Calderón-Zygmund operators). As a conseuqence of these interdiscplinary origins, wavelets appeal to scientists and engineers of many different backgrounds. On the other hand, wavelets are a fairly simple mathematical tool with a great variety of possible applications."

"Wavelets were developed independently by mathematicians, quantum physicists, electrical engineers and geologists, but collaborations among these fields during the last decade have led to new and varied applications. What are wavelets, and why might they be useful to you? The fundamental idea behind wavelets is to analyze according to scale. Indeed, some researchers feel that using wavelets means adopting a whole new mind-set or perspective in processing data. Wavelets are functions that satisfy certain mathematical requirements and are used in representing data or other functions."

"Wavelet theory is nowadays a very active field of approximation theory with a wide impact on signal analysis, high-performance imaging applications, and adaptive transversal filter theory. It is concerned with the modeling of univariate and multivariate signals with a set of specific signals. The specific signals are just the wavelets. Families of wavelets are used to approximate a given signal (with respect to the L2 norm, say), and each element in the wavelet set is constructed from the same original window, the mother wavelet."

"Wavelets were introduced at the beginning of the 'eighties by J. Morlet, a French geophysicist at Elf-Aquitaine, as a tool for signal analysis in view of applications for the analysis of seismic data. The numerical success of Morlet prompted A. Grossmann to make a more detailed study of the wavelet transform, which resulted in a paper giving the mathematical foundations (see Grossmann & Morlet ..., where the title of the paper still shows the name wavelets of constant shape. In 1985, the harmonic analyst Y. Meyer became aware of this theory and he recognised many classical results inside it. Meyer pointed out to Grossmann and Morlet that there was a connection between their signal analysis methods and existing, powerful techniques in the mathematical study of singular integral operators. Then Ingrid Daubechies became involved, and all this resulted in the first construction of a special type of frames (see Daubechies, Grossmann & Meyer ..), (the concept frame generalizes the concept basis in a Hilbert space). It was also the start of a cross-fertilization between the signal analysis applications and the purely mathematical aspects of techniques based on dilations and translations."

"A Haar wavelet is the simplest type of wavelet. In discrete form, Haar wavelets are related to a mathematical operation called the Haar transform. The Haar transform serves as a prototype for all other wavelet transforms."

"What is the role of counterexamples in mathematics? (Are there any in Euclid?)"

"Counterexample philosophy is a distinctive pattern of argumentation philosophers since Plato have employed when attempting to hone their conceptual tools."

"Whenever the bigger theorems are stated and proven, Landau usually shows that all the hypotheses are needed by dropping each one and giving a counterexample. In some cases the counterexamples are very elaborate, such as van der Waerden’s continuous, nowhere differentiable function, and a continuous function whose Fourier series diverges."

"Occasionally, one individual may come up with a "proof," and another with a "counterexample." Since a valid proof and counterexample cannot peacefully coexist, either the proof has some logical or mathematical flaw, or the counterexample does not faithfully represent the conditions involved, or perhaps both. This is another reason why it is so important to have good command of the underlying logic."

"The idea that theorems follow from the postulates does not correspond to simple observation. If the Pythagorean theorem were found to not follow from the postulates, we would again search for a way to alter the postulates until it was true. Euclid's postulates came from the Pythagorean theorem, not the other way around."

"The requisites for the axioms are various. They should be simple, in the sense that each axiom should enumerate one and only one statement. The total number of axioms should be few. A set of axioms must be consistent, that is to say, it must not be possible to deduce the contradictory of any axiom from the other axioms. According to the logical 'Law of Contradiction,' a set of entities cannot satisfy inconsistent axioms. Thus the existence theorem for a set of axioms proves their consistency. Seemingly this is the only possible method of proof of consistency."

"A recurring concern has been whether set theory, which speaks of infinite sets, refers to an existing reality, and if so how does one ‘know’ which axioms to accept. It is here that the greatest disparity of opinion exists (and the greatest possibility of using different consistent axiom systems)."

"Richard Borcherds, (quote at 21:37 of 1:36:06 in video)"

"We now know that equations of degree 3 are in some kind of border area between equations of lower degree (easy) and equations of higher degree (very hard)."

"The equations of conic sections involve at most the square of x and y, never the third or higher powers. In contrast, the equation of an elliptic curve has an x^3-term; a typical example is y^2 = x^3 - x."

"For about 1500 years, from the time of Diophantus to Newton, elliptic curves were known only as curves defined by certain cubic equations. This put them just a step beyond the conic sections, and some of their geometric and arithmetic properties can in fact be viewed as generalisations of properties of conics."

"Just as Weil's conjectures were about counting solutions to equations in a situation where the number of solutions is known to be finite, the BSD conjecture concerns the simplest class of polynomial equations—elliptic curves—for which there is no simple way to decide whether the number of solutions is finite or infinite."

"I shall not now speak of the knowledge of the Hindus … of their suitable discoveries in the science of astronomy—discoveries even more ingenious than those of the Greeks and Babylonians, of their rational system of mathematics, or of their method of calculation which no words can praise strongly enough; I mean the system using nine symbols."

"There is not, and cannot be, number as such. There are several number worlds as there are several Cultures. We find an Indian, an Arabian, a Classical, a Western type of mathematical thought and, corresponding with each, a type of number — each type fundamentally peculiar and unique, an expression of a specific world feeling, a symbol having a specific validity which is even capable of scientific definition, a principle of ordering the Become which reflects the central essence of one and only one soul, viz., the soul of that particular Culture. Consequently, there are more mathematics than one. ... and so it is understandable that even negative numbers, which to us offer no conceptual difficulty, were impossible in the Classical mathematic, let alone zero as a number, that refined creation of a wonderful abstractive power which, for the Indian soul that conceived it as base for a positional numeration, was nothing more nor less than the key to the meaning of existence. Negative magnitudes have no existence.... But when we are told that probably (it is at best a doubtful venture to meditate upon so alien an expression of Being) the Indians conceived numbers which according to our ideas possessed neither value nor magnitude nor relativity, and which only became positive and negative, great or small units in virtue of position, we have to admit that it is impossible for us exactly to re-experience what spiritually underlies this kind of number. For us, 3 is always something, be it positive or negative; for the Greeks it was unconditionally a positive magnitude, +3; but for the Indian it indicates a possibility without existence, to which the word “something” is not yet applicable, outside both existence and non-existence which are properties to be introduced into it. +3, -3, 1/3, are thus emanating actualities of subordinate rank which reside in the mysterious substance (3) in some way that is entirely hidden from us. It takes a Brahmanic soul to perceive these numbers as self-evident, as ideal emblems of a self-complete world form; to us they are as unintelligible as is the Brahman Nirvana, for which, as lying beyond life and death, sleep and waking, passion, compassion and dispassion and yet somehow actual, words entirely fail us. Only this spirituality could originate the grand conception of nothingness as a true number, zero, and even then this zero is the Indian zero for which existent and non-existent are equally external designations."

"I will omit all discussion of the science of the Hindus, a people not the same as the Syrians, their subtle discoveries in the science of astronomy, discoveries that are more ingenious than those of the Greeks and the Babylonians; their computing that surpasses description. I wish only to say that this computation is done by means of nine signs. If those who believe because they speak Greek, that they have reaced the limits of science should know these things, they would be convinced that there are also others who know something."

"The Hindus no less than the Greeks have shared in the work of constructing scientific concepts and methods in the investigation of physical phenomena , as well as of building up a body of positive knowledge which has been applied to industrial technique; and Hindu scientific ideas and methodology (e.g. the inductive method or methods of algebraic analysis) have deeply influenced the course of natural philosophy in Asia—in the East as well as the West—in China and Japan, as well as in the Saracen empire."

"However, Seidenberg was told by the indologists that these Sutras, or any Vedic text for that matter, were definitely written later than 1700 BC. But mathematical data cannot be manipulated just like that, and Seidenberg remained convinced of his case: “Whatever the difficulty there may be [concerning chronology], it is small in comparison with the difficulty of deriving the Vedic ritual application of the theorem from Babylonia. (The reverse derivation is easy)… the application involves geometric algebra, and there is no evidence of geometric algebra from Babylonia. And the geometry of Babylonia is already secondary whereas in India it is primary.” [To satisfy the indologists, he said that the Shulba Sutra had conserved an older tradition, and that it is from this one that the Babylonians had learned their mathematics:] “Hence we do not hesitate to place the Vedic (…) rituals, or more exactly, rituals exactly like them, far back of 1700 BC. (…) elements of geometry found in Egypt and Babylonia stem from a ritual system of the kind described in the Sulvasutras.”"

"Like the crests on the heads of peacocks, like the gems on the hoods of the cobras, mathematics is at the top of the Vedanga sastras."

"We know that the trigonometric sine is not mentioned by Greek mathematicians and astronomers, that it was used in India from the Gupta period onwards (third century).... The only conclusion possible is that the use of sines is an Indian development and not a Greek one. But Tannery, persuaded that the Indians could not have made any mathematical inventions, preferred to assume that the sine was a Greek idea not adopted by Hipparchus, who gave only a cable of chords. For Tannery, the fact that the Indians knew of sines was sufficient proof that they must have heard about them from the Greeks."

"I once heard, and I think it is true, that only one man in the world—some Indian mathematician—understood the mathematics of string theory in eleven-dimensional space, and he dreamed it."

"It is more likely that Pythagoras was influenced by India than by Egypt. Almost all the theories, religions, philosophical and mathematical taught by the Pythagoreans, were known in India in the sixth century B.C., and the Pythagoreans, like the Jains and the Buddhists, refrained from the destruction of life and eating meat and regarded certain vegetables such as beans as taboo" "It seems that the so-called Pythagorean theorem of the quadrature of the hypotenuse was already known to the Indians in the older Vedic times, and thus before Pythagoras."

"However, it is not unlikely that the Arabs, who received from the Indians the numeral figures (which the Greeks knew not), did from them also receive the use of them, and many profound speculations concerning them, which neither Latins nor Greeks know, till that now of late we have learned them from thence. From the Indians also they might learn their algebra, rather than from Diophantus."

"Many of the key results of mathematics, some of which are at the very “core” of modern day mathematics, are of Indian origin. ...‘I wish to conclude initially by simply saying that the work of Indian mathematicians has been severely neglected by western historians.’...‘it is likely European scholars are resistant due to the way in which the inclusion of non-European, including Indian, contributions shakes up views that have been held for hundredsof years, and challenges the very foundations of the Eurocentric ideology."

"C'est de l'Inde que nous vient l'ingénieuse méthode d'exprimer tous les nombres avec dix caractères, en leur donnant à la fois, une valeur absolue et une valeur de position; idée fine et importante, qui nous paraît maîntenant si simple, que nous en sentons à peine, le mérite. Mais cette simplicité même, et l'extrême facilité qui en résulte pour tous les calculs, placent notre système d'arithmétique au premier rang des inventions utiles; et l'on appréciera la difficulté d'y parvenir, si l'on considère qu'il a échappé au génie d'Archimède et d'Apollonius, deux des plus grands hommes dont l'antiquité s'honore."