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Recommended for you. One considers the single unit interval 0, 1 or real numbers. The restriction of f to Ki is modeled by a polynomial of degree 3: The result is the following system of equations: Then there is a polynomial spline f: The situation is that of one-dimensional splines. One is interested More precisely, and more generally, if P0 , P1 ,. Pd is any sequence of control points in Rm , then the unique polynomial function f: Let us restate the formula in more geometric Splines terms.
The reason that we want a curve to be contained in a simplex is that this guarantees a certain degree of predictable behavior. In practice, a curve satisfying this condition will be contained in the convex hull of its control points and will not wildly run around as may do Lagrange polynomials of higher degrees.
Recall that the Lagrange curve L traverses all basis points, with the additional condition that the order in which the curve passes through the points must be e0 , e1 ,. Instead a geometrically very useful condition is required: This goal is met by an ingenious interpretation of the following observation. Here is the analogue of the property for Lagrange polynomials: The curve is completely contained in the simplex.
Pd is the composed map B P0 , P1 ,. Denote by B[P0 , P1 ,. Pd in Rn. Corollary Let B P0 , P1 ,. Pd ] is contained in the convex hull of the control points. Concerning Bernstein polynomials, we have these formulas: Then we have B P0 , P1 ,. Proof The algorithm follows from the recursive formulas for Bernstein polynomials in proposition , we omit it and refer to .
In this schema a value at the tail of an arrow is multiplied by the factor above the arrow. The value at the head of the arrow is the sum of two of those scaled values: We present here an elegant method for combining a number of curves to such shapes of higher complexity. The method is part of a general theory of mathematical structures, called tensor products.
Assuming such a factorization, the members of the canonical basis e0 , e1 ,. We shall see that this is in fact a very practical notation for higher-dimensional spline theory. The tensor product is used as follows: This map is linear in each argument xi. Proposition Let d1 , d2. Exercise Give a proof of proposition Then the direct product of these data yields this application: Such surfaces are used as functions on the rectangular units of partitions for two-dimensional splines. For example, in automobile industry and similar applications, this construction is essential.
The construction described above is the composition of 3 functions: Like the Bernstein curve, the B-spline curve is of the type discussed above, i. The functions of order 0 are the characteristic functions of the closedopen knot intervals, i. We omit the details here and refer to . A lot of information on curves and surfaces can be found in a standard text on computer graphics, see . Nonetheless, it received the mathematics prize in and thus became a starting point of a big mathematical theory, which has an incredible omnipresence in modern natural science.
Fourier was not only a mathematical revolutionary, he was also politically active and even was put to jail as a terrorist during the French revolution.
However, mathematically speaking, this basis is far from trivial. The Fourier basis is at the same time extraordinary since it is of a striking simplicity: But the latter is a complex mathematical construction.
Moreover, the recent development of wavelets is based on Fourier transforms. We append a short exposition of Fourier transforms, because this theory is needed to deal with the theory of wavelets, which we present in chapter A function f: A periodic function with period p. For the following proposition, recall from remark 32 that a function f: Proposition For a p-periodic function f: Proof We use the criterion in proposition of integrability of a function.
Observe that our function has complex values, and therefore the set of noncontinuity is just the union of the sets of non-continuity for the real part and the imaginary part of the function.
Now, ii is a special case of i. Proof Suppose that f is continuous. The theorem of change of variable is also valid for integrable functions, see , chapter 3. Therefore, p R, C is called the C-algebra of integrable p-periodic functions. It is closed under sums and products of functions and contains C, therefore it is called the C-algebra of piecewise smooth p-periodic functions. Three curves in P Cp1 R, R: The Fourier theory deals with the description of the algebra P Cp1 R, C in terms of special orthonormal C-vector space bases.
Later we shall present more general, but mathematically straightforward, statements for arbitrary periods.
If N1 N1. Here are the important properties of this generalized bilinear form: Proof Except for statement vii , this sorite is straightforward, in particular, the Schwarz inequality is proved by the same method as the Schwarz inequality in sorite Evidently, orthogonality is a sym1 metric relation.
We insert it for reasons of normalization. For the following orthogonality relations, we need this exercise about primitive functions. Proof These equations are direct applications of the orthogonality formulas in exercise We need the following auxiliary construction: What values at those points can we derive from the information given by Fourier series information at those points? By virtue of integrating over the whole period, we can evidently not expect to obtain the delicate information about the behavior of f at such critical points.
But we have this information: Proof We shall not give a complete proof, but describe essential steps thereof, which can be stated in a compact way.
Then it is shown that the remainder has this form: We may clearly suppose that g is continuous and then treat the general case by adding up the continuous parts of g. The special role of FN f is made evident by this result: This condition is equivalent to an and bn being real, as can be seen from the transformation formulas given above. It is apparent that f is an approximation of a sawtooth curve. This technique of proceeding from amplitude and phase spectra to the corresponding function is called Fourier synthesis.
Then we have the representation Amplitude spectrum An n. The wobbling at the ridges is an indicator of the discontinuities of the approximated function f. As terms of higher order are added, the wobbling gets more intensive. In digital signal processing DSP , where the focus is on time-dependent functions, Fourier analysis is also known as passing from the time domain to the frequency domain.
Approximations to the rectangle function f: How are such obstructions overcome? We give a short overview of this theory in section For example, the popular MP3 audio compression algorithms are based on Fourier theory. The concrete situation is this: Of course, the measured function is by no means periodic, i. The idea therefore is to interpret the measured values as if they they were derived from a periodic function, and then apply the Fourier method to these data. As we shall apply this setup to the FFT algorithm, we need to have a very special distribution of the arguments from A as follows.
The above formula looks as if it contained Fourier Theory positive powers only. This frequency is also called the Nyquist frequency. The remainder of this discussion is devoted to the solution of the above formula, i. We also work on ZN concerning the given function f , i. The following corollary yields the representation of f at N given arguments in terms of the given basis as required above. The function f from example black , and its periodic continuation gray. In order to describe the notion of calculation speed in a precise way, one introduces the Landau symbols: In particular, The proof relies on a lemma which manages the recursive step from Fourier N to Fourier 2N.
But the lemma is more than an auxiliary result, its very content is implemented in the calculation routines, so we should make it explicit in its own right, instead of hiding it in a proof section.
Exercise Give a proof of the preceding formula. We shall therefore not include the proofs of the results of this section. The following statements should be understood in this sense of considering the classes of functions in Lr R.
The interest in these functions resides in the following proposition: Then the Fourier transform of f is the function F f: A step function of width 2b. See [3, p. The sinc function. Geoexploration, Types of wavelets: This idea is however only of theoretical use and must be adapted to a calculus which can be handled by computers.
For such a space, one extends the concept of a basis from linear algebra to a Schauder basis. Proposition Suppose that we are given an orthonormal Schauder basis ei i of L2 R.
On the other hand, the condition i ci 2 A last fact, concerning Hilbert subspaces, must be mentioned. Here is the crucial fact about orthogonal Hilbert subspaces: But the sequence si i is Cauchy.
The formula is as follows. The following corollary is immediate from the isometric property of deformations: In fact, we have this two-scale relation: Claim ii is a standard result in functional analysis, we do not prove it here.
Proposition The system of deformed Meyer wavelets is orthonormal. The proof technique of this proposition is far beyond our modest context. See  for a proof. However, the essential technique can also be traced for the Haar wavelet. This is what we shall discuss now. We need the proof idea for the fast wavelet transform, so let us make it explicit here: Why is it also a basis of W0?
A Comprehensive Textbook of Classical Mathematics
This follows from lemma and proposition We know from proposition that the decreasing chain. The general step is this: Let us recapitulate the MSA described here with respect to the chain. We are The function f0 which is to be transformed one may think of f0 as being a sequence of samples read from a CD, for instance. Figures Haar wavelets exemplify in a simple way the principles behind wavelet approximation.
However, Haar approximations of continuous functions are These geometric objects have been unleashed to science and popularized by the mathematician Benoit Mandelbrot around , mainly through his book The Fractal Geometry of Nature , following a preliminary study of the Julia set. This time, however, nature was not to be restricted to planetary trajectories cast into quadratic equations by classical geometry. This is perhaps the reason why this kind of geometry has attracted the interest of scientists and artists alike.
Mathematically, we encounter a much more prosaic scenery. We want to present a thoroughly mathematical setup of the basic ideas of a particular kind of fractals.
However, in this theory, the points being mapped by contractions are not original geometric points, but big objects, i. A review of section A typical continuous map f: In this chapter, X, d will always denote a complete metric space.
Recall the three-fold characterization of a compact set in Rn given in proposition This means that there is countable set of open sets, such that every every open set is a union of sets from this basis.
Reread the proof, and everything will work mutatis mutandis. The metric on H is given as follows. Analogously, the function d? Exercise Verify the claims concerning the continuity of the distance functions dx and d? To prove the triangle inequality, take three compact sets A, B and C. The completeness of H X requires a long technical proof, which we must omit here.
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Refer to , section 2. We have an injective isometry iX: Moreover, we know from lemma that for any continuous map f: This refers to the commutation The systematic background of this type of commutativity has been dealt with in the chapter 36 on category theory under the title of natural transformations.
In particular, H: Proof Let f: We show that H f: Proof We already know that isometries are continuous. On the other hand, let f: We then write iX: More generally, if f: Clearly, morphisms of contractions can be composed, composition Proposition If f: However, there is a dramatic generalization when we consider the set of all contractions Contra H X and its subset H Contra X.
Lemma The set Contra X is closed under composition, i. We have the following structure1 on Contra H X: Showing properties i through iv reduces to straightforward set-theoretic calculations. Example The Sierpinski carpet is very similar to the diagonal from the previous example. Example The famous Koch curve is constructed using the same procedure as the previous two examples. To make things more clear, the transformation is split into four parts: A scaling by 1 3.
The approximated attractor of the Sierpinski fractal. To make the four parts of the transformation more Fractals easily recognizable, their respective results are set in black for Koch1 , dark gray for Koch2 , middle gray for Koch3 and light gray for Koch4.
The latter means that there are two positive constants u and v such that Exercise Prove that equivalence among metrics is in fact an equivalence relation. A classical method for determining such equivalence classes is the exhibition of numerical invariants, i. The concept of a geometric dimension is motivated by two observations: To begin with, when we have a curved line A in Rn , the dimension should be 1.
The ingenious idea in the concept of dimension, which we shall now describe, is that both observations can be merged into one and the same criterion.
For example, if one covers a square S by adjacent 1 small squares Si having side length equal to, say, 10 of the side length of 2 S, then one needs 10 such squares, whereas if one covers an interval I by 1 adjacent intervals of length 10 of the length of I, one needs of them. So the growth of the number of covering standard charts is an indicator of the dimension: It describes geometric forms having fractional dimension.
Since K is compact, this number always exists. If, instead, we had taken closed cubes, the number would change. The length of the unit interval is, of course, 1. Draw a few iterations to get an approximation of Fix Gasket , starting with a triangle of side length 1. Use the box counting theorem to determine dim Fix Gasket.
Exercise All examples so far have been about compact subsets of R2. Suppose that we have a fractal isomorphism f: Proof The proof of this theorem is very technical, we have to omit it here and again refer to , section 5. This is not the Borg spaceship, but an approximation of Fix Sponge. A last example will close this chapter and illustrate how fractals can be used to model forms that occur in nature.
The Koch curve K a , and its transformation T K b. The approximated attractor Fix Fern. The processing performed by formal neurons consists of several stages: This result is changed i. The inner workings of a formal neuron. Since their introduction in the s neural networks have been used in research and industry, especially for AI and robotics. The mathematical model we pursue in this chapter deals with Here is the basic framework of data streams. We use the notation pi: Depending on the given context we shall make use of the appropriate view without special emphasis.
The common construction is as follows. One is given a function f: Clearly such a function commutes with every shift operator the reader should verify this as an exercise. For example, if? We will also deal with other more general functions h: Consider the way a neuron processes its input. All the signals that ever reach the neuron as inputs or leave the neuron can be interpreted as n-streams. If the weights can change over time as is the case in Hebbian learning, they must be described as n-streams, too.
The weighted sum calculated from the inputs is now the stream that is the result of the scalar product of the input and weight n-streams. A pair w, x of weight n-stream w and input nstream x is mapped to the output stream o a w, x.
Dn Here the identity Id and the projection pr 1 simply ensure that the the n-stream of weight represented in the upper line is identical to the one in the lower line.
This is called the state space of the formal neuron N. Its image a 0 , and the image o a 0 , together with the image a 0 under the identity on D concludes the description of the unique limit element. Therefore, the only output element in the axon of such a neuron is the element o a 0. Therefore a zero-dimensional neuron is also called an input neuron, since its state space is a singleton, yielding a single axonal output stream o.
An input neuron typically represents the sensorial input for the Neural Networks Fig. The diagonal arrows represent the time-shift operator: The calculation of the weighted sum is symbolized by the pair of curved arrows going to the right.
The dashed arrows represent the Hebb learning function, which uses the weights from the previous time step and the current result of the activation function to change the current weights.
Example Neurons may also just pipe information without further changes. What does the state space look like? Then it consists of all these elements at the places of our diagram: We simply calculate the result for all four input combinations: Exercise Show that the negation NOT: In general, it can be shown that of all possible functions f: The solutions of this equation are described as follows.
The latter is the kernel of the non-zero linear form w,? We can This is clear from the fact that w,? The vector w is normal to the plane. As an application one recognizes immediately that XOR is not representable by a perceptron. But there is more to be done in our conceptualization in order to solve the XOR problem and other problems related to the representation of logical functions by use of perceptrons. Let N be the set of all neurons. Denote by E the set of all elementary morphisms.
Then the neural category CN is the path category of the directed graph EN: Except when explicitly stressed, neural networks are assumed to be elementary in the sense that its morphisms are all elementary. This means that we are given a morphism D: An output neuron in a neural network is a neuron which is not the domain of a morphism of D.
Observe that there is no elementary morphism i: An elementary morphism i: Graphical representation of a neural network showing how the axon output serves as dendritic input for one or more other neurons. More generally, if we are given any morphism i1 i2 ik f: The subsubspace of SD with w. If in a neural network D, a neuron has an input index which is not connected to the axonal output of another neuron, it is called a free input.
These constructions deserve and need a number of comments and illustrations. Remark 34 To begin with, if a general neural network is given, it may happen that one would like to have the output of a neuron Ni which is not an output neuron. This is easily realized by adding a This is easily settled in the following exercise The concatenation Thru0: These types of neural morphisms play the role of stream pipes without further functionality, except for a controlled time shift.
Without loss of generality, we may Neural Networks even suppose that X and Y have no points on H otherwise, shift H a little away from X. By the previous remark 35, we may suppose that the threshold vanishes, i.
So we are led to this problem: The point is that one has to know that such a vector exists. This is the perceptron learning algorithm. The remarkable thing is that it is itself a neuron N, more precisely: Now, take an increasing sequence of indexes t0 Here is one construction. Then we consider this neural network XOR: Therefore, one is interested in the construction of neural networks in order to represent logical functions f: The above functionality can however be achieved by a slightly simpler neural network.
To this end, we integrate the negation in the conjunction through this change of weights and thresholds: It is a network which is of a type called multi-layered perceptron. The other layers are called the hidden layers. A neural network whose digraph is cyclic is called recurrent. The digraph of D is a complete digraph with m vertexes, i. The following proposition shows that 1-layer perceptrons can be used to represent any logical function. Proposition Any logical function f: For proofs of these propositions, see .
In contrast to proposition , this algorithm starts with a given neural network and does not vary its vertexes, but only the weights. It has the property that its derivative is expressed in terms of the function x itself, i. We also assume that D is saturated and one-dimensional. To The correction of w is obtained from the map ET: The back-propagation algorithm manages to calculate the correction term dET w t by means of a recursive calculation of correction terms related to the output layer, and then going back to 2 Observe that because of the acyclicity of multi-layered perceptrons, the output is always uniquely determined by the input for given weights.
ET visualized as a hilly landscape. Peaks represent minimal ET values. On the right side, gradient vectors indicate the fastest path to a peak.
In order to ease notation, we temporarily omit the time reference in the following calculations. Denote by i hm. Then the gradik wjk i vk j the scalar product m ent component for the variable wjl , i. As already mentioned, there is no guarantee that this algorithm eventually yields a stable result and that this result is in fact a local minimum. So the previous back-propagation method is used for learning.
This type of learning is called supervised learning since the change of the weights follows external target, i. For unsupervised learning, the system evolves independently of external targets in the sense of what is called self-organization. This is far from what is happening in a real environment.
Let us indicate the essential steps needed for rendering the system autonomous. There are several partial problems to solve. The training set T should also be realized as a temporal construction in the temporal stream of our system.
Then, each such training segment is juxtaposed to the previous segment, and at the end of all segments, the calculation of the total change the gradient is performed by a Hebb function referring to all previous weight data from the totality of training segments.
An alternative would be to extend the input stream by sending the desired outputs along with the inputs, so they are available at the output layer to be used for changing the weights.
Closing this chapter on neural networks, we should add a remarkable theorem by George Cybenko . The question is how well any function may be approximated by a multi-layered network. Here is an answer: Proper statistics in the sense of inductive statistics, i.
We shall however give a short sketch of the ideas in inductive statistics, in particular the maximum-likelihood method for guessing probability distribution parameters.
Here, an event is any subset of faces, describing a property of faces, e. Example We are checking n devices for their operativeness. An outcome 1 Henceforth, we assume that dice are of cubic form, their faces bearing the numbers 1 to 6. Often, if the involved event spaces A and C are clear, one also writes f: Exercise Show that a continuous map f: It follows that a map f: We denote by VX the image set of X, and call it the set of values of X.
The last example gives rise to an n-dimensional generalization of random variables and leads to the construction of one-dimensional variables deduced from n-dimensional variables.
Xn of random variables. Equivalently, by the universal property of n-dimensional Borel sets discussed in example , X is a morphism X: An important linear combination of random variables X1 , X2 ,.
In fact, it is less interesting to know an event in the dice event space than the frequency it has when throwing a dice for a number of times. This leads to the second pillar of probability theory: Only the combination of random variables and probability functions generates relevant probabilistic statements. Here is the axiomatic setup introduced by Andrei Nikolaevich Kolmogorov in . Then a probability measure P for A is a map P: Sorite Let P: Then we have these properties: This adds up to the correct formula for n.
Verify that this probability measure conforms to the Kolmogorov axioms. Example Let us observe a harddisk regarding its reliability.
Probabilities are often considered as relative values with respect to a given event B, in the sense that one pretends that B is the certain event.
Imagine that chemist analyzes a sample x. He already knows, from previous tests, that it contains traces of a substance a with probability 23 , i. From long experience, he knows that if a sample contains a, then the indi4 4 cator colors green with probability 5 , i.
With all this information, the chemist wants to know the probability that, given that the indicator turns green, the sample contains traces of a, i. The Bayes formula yields The chapter also explains when derivatives are primarily used.
Hit a particularly tricky question? There are several partial problems to solve. Descriptive Theory of Sets. In this context, not every subset of R is equally interesting. Clearly, by construction, f is a morphism of cones f: Here is the operational restatement of what basic data a diagram in a category really requires.
Here, we use a more general argument: Because the number of characters on a page is large compared to the number of expected misprints, X can be modeled using the Poisson distribution P4.