G. Polya, Mathematics and Plausible Reasoning. Volume I: Induction and Analogy Mathematics. XVI + S. m. zahlr. Abb. Volume II: Patterns of Plausible . View eBook Mathematics And Plausible Reasoning, Volume 1 (Princeton Paperback) By G. Polya KINDLE PDF EBOOK EPUB. Review eBook Mathematics And Plausible Reasoning, Volume 1 (Princeton Paperback) By G. Polya EBOOK EPUB KINDLE PDF.

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download Mathematics and Plausible Reasoning [Two Volumes in One] on site. com ✓ FREE SHIPPING on qualified orders. "Polya does a masterful job of showing just how plausible reasoning is used in mathematics The material in both volumes is fresh and highly original; the . G. Polya, Mathematics and plausible reasoning. (Preface, Vol I). • Strictly speaking, all our knowledge outside mathematics and demonstrative logic (which .

Since then it has been a separate branch of applied mathematics. Tensors are multi-linear maps. That is to say, they are linear in many variables and are functions of them just like vectors are also functions. They are functions of coordinates variable. As an example, metric tensor is a function of coordinates. Riemann tensor is also function of coordinates. Tensors transform in special way under change of coordinates. Application of calculus to tensor gives different resutls than scalars and vectors. Tensor calculus not only involves calculus but also linear algebra. Tensor field is tensor valued function defined on a general manifold. Manifold is a generalized space of which Euclidean space is a special case. In each point in a manifold tensor field assigns a tensor quantity like vector field assigns a vector at each point in space.

Plausible reasoning Plausible Reasoning PR is a weak inferencing approach, used when a deterministic answer to a question is unavailable. In other words, to cope with missing knowledge, plausible reasoning is conducted on the available evidence and experiments; which are typically empirical, inexact, and uncertain. In this regard, the result of plausible reasoning is not a conclusive answer but rather the best-effort answer in light of what is known so far [ 5 , 6 ].

Based on findings from argumentation studies on the crossroads of philosophy, reasoning, and logic, Tindale [ 7 ] and Walton et al. Summarizing these characteristics, plausible reasoning can be recognized as a method which is [ 8 — 11 ]: Non-demonstrative: non-demonstrative reasoning depends on knowledge discovery, making hypotheses, and learning new concepts.

Typical examples include physician diagnosis, economical statistical evidence, or conclusions of scientific research. On the other hand, demonstrative reasoning i. Ampliative: non-ampliative reasoning, like deduction, explicates and instantiates what was already expressed in the captured domain-specific knowledge e. Instead, ampliative reasoning generates inferences that go beyond what is contained in the captured knowledge. Non-monotonic: implies that the validity of a plausible hypothesis can be detracted by a new piece of information.

Subjective: a plausible argument is an expression of beliefs, opinions, personal preferences, values, feelings, and judgments. Further, plausible reasoning leverages a set of recurring patterns from human reasoning processes employed to infer answers; which do not necessarily occur in formal logic or non-classical logics such as fuzzy logic [ 12 , 13 ]. Firstly, plausible reasoning assumes that a large amount of human knowledge is stored in a hierarchical format, which is constantly being updated, modified, and extended.

A first category of plausible patterns leverages this hierarchical nature, and includes generalization, specialization, similarity, and dissimilarity [ 14 , 15 ]. A second category of plausible patterns are based on non-hierarchical relations between concepts, namely partial order relations, and includes interpolation and a fortiori. Table 1 Open in a separate window Leveraging the relations between concepts and entities, all these patterns manifest new knowledge ampliative that is a result of exploring the plausible closure non-demonstrative of the KB.

Further, these manifestations are non-monotonic, since their validity can be detracted by new, explicit knowledge; and subjective, since their truthfulness strongly depends on the experiences and background knowledge captured in the KB. While these patterns can be used individually, plausible reasoning techniques such as inductive generalization, analogical reasoning, and abduction, are inspired by these patterns to perform more complex reasoning.

In this paper, we focus on inductive generalization and analogical reasoning, which we summarize below. There were difficulties , which the progress of physics made evident. So long as the earth could be regarded motionless , axes fixed relatively to the earth and clocks which remains on the surface of the earth seem to suffice. It was possible to disregard the fact that nobody is quite rigid and no clock is quite accurate , because the system of physical laws suggested by the choice of most rigid bodies and most accurate clock can be used to estimate the departure from these instruments from strict constancy , and the results were on the whole self-consistent.

But in astronomical problems, including that of the tides , the earth could not be treated as fixed. It was necessary to Newtonian dynamics that the axes should not have any acceleration, but it resulted from law of gravity that any material axes must have some acceleration.

The axes, thus, become ideal structures in absolute space; actual measurement carried out by any rigid road could only approximate to the results which would have if we used any unaccelerated axes. This difficulty was not the most serious ; the worst trouble was concerned with absolute acceleration.

Then came the experimental discovery of the facts which led to the special theory of relativity: the variation of length and mass with velocity, and the constancy of the velocity of light in vacua no matter what body was used to define the co-ordinates. This set of difficulties was solved by special theory of relativity, which showed that equivalent results come from employing as reference-body any set of bodies in uniform rectilinear motion.

The tensor above is a contra -variant rank two tensor. It has special transformation properties as given in the equation. The differentials in the denominator and numerator are reversed compared to that in the covariant one described before. We can also form a mixed tensor with both covariant and contravariant components. Different component in such tensor will transform differently. An example is given below: There is no physical distinction between covariant and contravariant tensors.

They both contain the same physical content. It is only the interpretation that are different. As the transformation is defined as the infinitesimal displacement , we need not worry about curvature of space. In case of two displacement, the difference can be spotted easily. When the axes do not intersect orthogonally, there is only two way to project the vector onto the axes : either parallel or vertical. Covariant and contravariant component of a vectore can be geometrically interpreted as the component that is parallel projection to axis and component that is perpendicular projection to axis.

When the axes are orthogonal, contra-variant and covariant projections becomes identical. The above tensor is type 1,2 tensor. So why are tensors so important in physics or mathematics? It is actually an entity studied under linear algebra. In linear algebra , it can have any rank and thus can be more generalized form of vectors or scalars.

In physics , it has became a necessity to use tensor. Many physical quantities needs more than one direction to be specified. For example stress is a tensor quantity which has two directions : one is normal to a surface and other is tangential to the surface. The former is called normal stress and the latter is called shear stress.

Mechanical engineers are used to it more than others. It is this tensor that attracted Einstein to use it in his theory. He just included a time dimension to it and make it relativistic. We will come to that when discussing the theory of relativity in more details.

JS" The notion of covariant derivative can be developed by taking the derivative of vector function which takes one or more value as argument and returns a vector. The covariant derivative is the summation of usual derivative and a Christoffel symbol coupled term.

If we assign a vector to each direction in space we get a tensor of second order. Such tensor's components are labelled by pair i,j.

Stress is a second rank tensor. In three dimensions it has nine components which Einstein generalized in four dimensions including time. The generalized tensor is known as Stress-energy tensor , which acts as a source of graviational field. The tensor relates a unit-length direction vector n to the stress vector T n across an imaginary surface perpendicular to n: Tensor product is defined as the bilinear map that takes elements from two vector spaces and produces element of a third vector space.

It does, in this process, generalises outer product of vectors. Tensor product can be formed with a vector space and its dual as well. In that case we get what is called mixed tensors each of which is itself an element of another vector space. So tensor product is generally , a composition of vector spaces over a field F. In short, it maps cartesian product of two vector space to another vector space, that generalizes vector outer product.

Outer product of two vectors of rank n and m is the matrix nxm. It is often called permutation symbol, alternative symbol. It is anti-symmetric tensor. It can be generalized in higher dimensions also. Invariant theory Tensors and tensor equations are invariant quantity.

Vector and tensor , as we have seen do not change when we change coordinates or coordinate system. The components of tensors change but a linear combination of them remain the same. It is the spacetime interval which remain invariant or same no matter whichever coordinates are used.

Generalizing divergence of a vector, we can form divergence of any tensor using the covariant derivative of it. We can get Einstein's tensor from Riemann's curvature tensor.

This is known as the "fundamental theorem of mechanics". The mass and momentum copnservation law, on the other hand, says that divergence of Stress-energy tensor is identically zero.

General relativity generalizes special relativity and Newton's law of universal gravitation, affording a unified description of gravity as a geometric property of space and time, or spacetime.

In particular, the curvature of spacetime is directly connected to the energy and momentum of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of partial differential equations.

Without the method of tensor calculus GR formulation would not have been possible Some predictions of general relativity differ significantly from those of classical physics, especially concerning the passage of time, the geometry of space, the motion of bodies in free fall, and the propagation of light. Examples of such differences include gravitational time dilation, gravitational lensing, the gravitational redshift of light, and the gravitational time delay.

The predictions of general relativity in relation to classical physics have been confirmed in all observations and experiments to date. Although general relativity is not the only relativistic theory of gravity, it is the simplest theory that is consistent with experimental data.

However, unanswered questions remain, the most fundamental being how general relativity can be reconciled with the laws of quantum physics to produce a complete and self-consistent theory of quantum gravity.

Einstein's theory has important astrophysical implications. Garden of Cosmic Delights Hartle and Hawking saw a lot of each other from the s on, typically when they met in Cambridge for long periods of collaboration. In , Albert Einstein discovered that concentrations of matter or energy warp the fabric of space-time, causing gravity.

Hawking and Hartle were thus led to ponder the possibility that the universe began as pure space, rather than dynamical space-time. And this led them to the shuttlecock geometry. In the s, Feynman devised a scheme for calculating the most likely outcomes of quantum mechanical events. To predict, say, the likeliest outcomes of a particle collision, Feynman found that you could sum up all possible paths that the colliding particles could take, weighting straightforward paths more than convoluted ones in the sum.

Likewise, Hartle and Hawking expressed the wave function of the universe — which describes its likely states — as the sum of all possible ways that it might have smoothly expanded from a point. If the weighted sum of all possible expansion histories yields some other kind of universe as the likeliest outcome, the no-boundary proposal fails.

The problem is that the path integral over all possible expansion histories is far too complicated to calculate exactly. Countless different shapes and sizes of universes are possible, and each can be a messy affair. Even the minisuperspace calculation is hard to solve exactly, but physicists know there are two possible expansion histories that potentially dominate the calculation.

These rival universe shapes anchor the two sides of the current debate. Weirder expansion histories, like football-shaped universes or caterpillar-like ones, mostly cancel out in the quantum calculation. One of the two classical solutions resembles our universe.

As in the real universe, density differences between regions form a bell curve around zero. If this possible solution does indeed dominate the wave function for minisuperspace, it becomes plausible to imagine that a far more detailed and exact version of the no-boundary wave function might serve as a viable cosmological model of the real universe. The other potentially dominant universe shape is nothing like reality.

As it widens, the energy infusing it varies more and more extremely, creating enormous density differences from one place to the next that gravity steadily worsens. Density variations form an inverted bell curve, where differences between regions approach not zero, but infinity. If this is the dominant term in the no-boundary wave function for minisuperspace, then the Hartle-Hawking proposal would seem to be wrong.

The two dominant expansion histories present a choice in how the path integral should be done.