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This paper investigates the analytic properties of the Liouville function's Dirichlet series that obtains from the function F(s)= zeta(2s)/zeta(s), where s is a complex variable and zeta(s) is the Riemann zeta function. The paper employs a novel method of summing the series by casting it as an infinite number of sums over sub-series that exhibit a certain symmetry and rapid convergence. In this procedure, which heavily invokes the prime factorization theorem, each sub-series has the property that it oscillates in a predictable fashion, rendering the analytic properties of the Dirichlet series determinable. With this method, the paper demonstrates that, for every integer with an even number of primes in its factorization, there is another integer that has an odd number of primes (multiplicity counted) in its factorization. Furthermore, by showing that a sufficient condition derived by Littlewood (1912) is satisfied, the paper demonstrates that the function F(s) is analytic over the two half-planes Re(s) > 1/2 and Re(s)<1/2. This establishes that the nontrivial zeros of the Riemann zeta function can only occur on the critical line Re(s)=1/2., Comment: 40 pages, 1 figure. In this Version, some typos have been removed and the Appendix 5 is slightly modified and a new Appendix 6 has been added

As machine learning is applied to an increasing variety of complex problems, which are defined by high dimensional and complex data sets, the necessity for task oriented feature learning grows in importance. With the advancement of Deep Learning algorithms, various successful feature learning techniques have evolved. In this paper, we present a novel way of learning discriminative features by training Deep Neural Nets which have Encoder or Decoder type architecture similar to an Autoencoder. We demonstrate that our approach can learn discriminative features which can perform better at pattern classification tasks when the number of training samples is relatively small in size., Comment: 12 pages, 8 figures and 8 tables

Recently an algorithm, was discovered, which separates points in n-dimension by planes in such a manner that no two points are left un-separated by at least one plane{[}1-3{]}. By using this new algorithm we show that there are two ways of classification by a neural network, for a large dimension feature space, both of which are non-iterative and deterministic. To demonstrate the power of both these methods we apply them exhaustively to the classical pattern recognition problem: The Fisher-Anderson's, IRIS flower data set and present the results. It is expected these methods will now be widely used for the training of neural networks for Deep Learning not only because of their non-iterative and deterministic nature but also because of their efficiency and speed and will supersede other classification methods which are iterative in nature and rely on error minimization., Comment: 18 pages, 5 figures

We show that if you represent all primes with less than n-digits as points in n-dimensional space, then they can be stored and retrieved conveniently using n-dimensional geometry. Also once you have calculated all the prime numbers less than n digits, it is very easy to find out if a given number having less than n-digits is or is not a prime. We do this by separating all the primes which are represented by points in n-dimension space by planes. It so turns out that the number of planes q, required to separate all the points represented by primes less than n-digit, are very few in number. Thus we obtain a very efficient storage and retrieval system in n-dimensional space. In addition the storage and retieval repository has the property that when new primes are added there is no need to start all over, we can begin where we last left off and add the new primes in the repository and add new planes that separate them as and when necessary. Also we can arrange matters such that the repository can begin to accept larger primes which has more digits say n' where n' > n. The algorithm does not make use of any property of prime numbers or of integers in general,except for the fact that any n-digit integer can be represented as a point in n-dimension space. Therefore the method can serve to be a storage and retrieval repository of any set of given integers, in practical cases they can represent information. Thus the algorithm can be used to devise a very efficient storage and retrieval system for large amounts of digital data., Comment: 15 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1509.08742

Given a set of N points, we have discovered an algorithm that can separate these points from one another by n-dimensional planes. Each point is chosen at random and put into a set S and planes which separate them are determined and put into S. The algorithm gives a method of choosing points and planes which separate them, till all the points are separated. A proof is provided with a worked example. The algorithm is non iterative and always halts successfully and the algorithm strictly follows Shannon's principle of making optimal use of information as it advances stage by stage. It also has a restart facility and can take care of new points from where it left off.At some later stage if the dimension of the data is increased from n to n+r, the algorithm can still continue from where it left off, after some simple adjustments, and tackle the new data points which are of a higher dimension. and separate them. The computational complexity is O(n.N log(N)) + O(n3 log(N)), where N is the given number of points and n3 is the cube of n - the dimension of space. The algorithm is made possible because a new concept called Orientation Vector is used. This vector is a Hamming vector and is associated with each point and has been so devised that it has all the information necessary to ascertain if two points are separate or not when among a collection of planes.Its application to data retrieval problems in very large medical data bases is also given., Comment: 36 pages, 12 figures

In this paper we introduce a new method which employs the concept of "Orientation Vectors" to train a feed forward neural network and suitable for problems where large dimensions are involved and the clusters are characteristically sparse. The new method is not NP hard as the problem size increases. We `derive' the method by starting from Kolmogrov's method and then relax some of the stringent conditions. We show for most classification problems three layers are sufficient and the network size depends on the number of clusters. We prove as the number of clusters increase from N to N+dN the number of processing elements in the first layer only increases by d(logN), and are proportional to the number of classes, and the method is not NP hard. Many examples are solved to demonstrate that the method of Orientation Vectors requires much less computational effort than Radial Basis Function methods and other techniques wherein distance computations are required, in fact the present method increases logarithmically with problem size compared to the Radial Basis Function method and the other methods which depend on distance computations e.g statistical methods where probabilistic distances are calculated. A practical method of applying the concept of Occum's razor to choose between two architectures which solve the same classification problem has been described. The ramifications of the above findings on the field of Deep Learning have also been briefly investigated and we have found that it directly leads to the existence of certain types of NN architectures which can be used as a "mapping engine", which has the property of "invertibility", thus improving the prospect of their deployment for solving problems involving Deep Learning and hierarchical classification. The latter possibility has a lot of future scope in the areas of machine learning and cloud computing., Comment: 15 pages 13 figures

The problem of optimizing a linear objective function,given a number of linear constraints has been a long standing problem ever since the times of Kantorovich, Dantzig and von Neuman. These developments have been followed by a different approach pioneered by Khachiyan and Karmarkar. In this paper we present an entirely new method for solving an old optimization problem in a novel manner, a technique that reduces the dimension of the problem step by step and interestingly is recursive. A theorem which proves the correctness of the approach is given. The method can be extended to other types of optimization problems in convex space, e.g. for solving a linear optimization problem subject to nonlinear constraints in a convex region.

In this paper we consider a classical treatment of a very dense collection of photons forming a self-sustained globe under its own gravitational influence. We call this a "photonic globe" We show that such a dense photonic globe will have a radius closely corresponding to the Schwarzschild radius. Thus lending substance to the conjuncture that the region within the Schwarzschild radius of a black hole contains only pure radiation. As an application example, we consider the case of a very large photonic globe whose radius corresponds to the radius of the universe and containing radiation of the frequency of the microwave background (160.2 GHZ). It so turns out that such a photonic globe has an average density which closely corresponds to the observed average density of our universe., Comment: 3 pages

In this paper, we prove a crucial theorem called Mirroring Theorem which affirms that given a collection of samples with enough information in it such that it can be classified into classes and subclasses then (i) There exists a mapping which classifies and subclassifies these samples (ii) There exists a hierarchical classifier which can be constructed by using Mirroring Neural Networks (MNNs) in combination with a clustering algorithm that can approximate this mapping. Thus, the proof of the Mirroring theorem provides a theoretical basis for the existence and a practical feasibility of constructing hierarchical classifiers, given the maps. Our proposed Mirroring Theorem can also be considered as an extension to Kolmogrovs theorem in providing a realistic solution for unsupervised classification. The techniques we develop, are general in nature and have led to the construction of learning machines which are (i) tree like in structure, (ii) modular (iii) with each module running on a common algorithm (tandem algorithm) and (iv) selfsupervised. We have actually built the architecture, developed the tandem algorithm of such a hierarchical classifier and demonstrated it on an example problem., Comment: 10 pages IEEE format, International Journal of Computer Science and Information Security, IJCSIS 2009, ISSN 1947 5500, Impact Factor 0.423, http://sites.google.com/site/ijcsis/

- In this paper we present a method to compute the coefficients of the fractional Fourier transform (FrFT) on a quantum computer using quantum gates of polynomial complexity of the order O(n^3). The FrFt, a generalization of the DFT, has wide applications in signal processing and is particularly useful to implement the Pseudopolar and Radon transforms. Even though the FrFT is a non-unitary operation, to develop its quantum counterpart, we develop a unitary operator called the quantum Pseudo-fraction Fourier Transform (QPFrFT) in a higher-dimensional Hilbert space, in order to computer the coefficients of the FrFT. In this process we develop a unitary operator denoted U by which is an essential step to implement the QPFrFT. We then show the application of the operator U in the problem of quantum phase estimation.

In this paper, we present a new kind of learning implementation to recognize the patterns using the concept of Mirroring Neural Network (MNN) which can extract information from distinct sensory input patterns and perform pattern recognition tasks. It is also capable of being used as an advanced associative memory wherein image data is associated with voice inputs in an unsupervised manner. Since the architecture is hierarchical and modular it has the potential of being used to devise learning engines of ever increasing complexity.

This paper proposes an unsupervised learning technique by using Multi-layer Mirroring Neural Network and Forgy's clustering algorithm. Multi-layer Mirroring Neural Network is a neural network that can be trained with generalized data inputs (different categories of image patterns) to perform non-linear dimensionality reduction and the resultant low-dimensional code is used for unsupervised pattern classification using Forgy's algorithm. By adapting the non-linear activation function (modified sigmoidal function) and initializing the weights and bias terms to small random values, mirroring of the input pattern is initiated. In training, the weights and bias terms are changed in such a way that the input presented is reproduced at the output by back propagating the error. The mirroring neural network is capable of reducing the input vector to a great degree (approximately 1/30th the original size) and also able to reconstruct the input pattern at the output layer from this reduced code units. The feature set (output of central hidden layer) extracted from this network is fed to Forgy's algorithm, which classify input data patterns into distinguishable classes. In the implementation of Forgy's algorithm, initial seed points are selected in such a way that they are distant enough to be perfectly grouped into different categories. Thus a new method of unsupervised learning is formulated and demonstrated in this paper. This method gave impressive results when applied to classification of different image patterns., Comment: Presented in IEEE International Conference on Advances in Computer Vision and Information Technology (ACVIT-07), Nov. 28-30 2007

In this paper, we present a Mirroring Neural Network architecture to perform non-linear dimensionality reduction and Object Recognition using a reduced lowdimensional characteristic vector. In addition to dimensionality reduction, the network also reconstructs (mirrors) the original high-dimensional input vector from the reduced low-dimensional data. The Mirroring Neural Network architecture has more number of processing elements (adalines) in the outer layers and the least number of elements in the central layer to form a converging-diverging shape in its configuration. Since this network is able to reconstruct the original image from the output of the innermost layer (which contains all the information about the input pattern), these outputs can be used as object signature to classify patterns. The network is trained to minimize the discrepancy between actual output and the input by back propagating the mean squared error from the output layer to the input layer. After successfully training the network, it can reduce the dimension of input vectors and mirror the patterns fed to it. The Mirroring Neural Network architecture gave very good results on various test patterns., Comment: Presented in IEEE International Conference on Advances in Computer Vision and Information Technology (ACVIT-07), Nov. 28-30 2007