K. Eswaran

• Ph.D.
• Professor at Sreenidhi Institute of Science & Technology

As is well-known, the celebrated Riemann Hypothesis (RH) is the prediction that all the non-trivial zeros of the zeta function ζ(s) lie on a vertical line in the complex s-plane at Re(s) = 1/2. Very many efforts to prove this statement have been directed to investigating the analytic properties of the zeta function, however all these efforts have n...

The Riemann Hypothesis is a celebrated unsolved problem which concerns the zeta function and its zeros. In this brief note describes the methodology adopted by the author. This paper has been peer reviewed by an Expert Committee

In this note I will try to describe in the simplest possible terms the method I adopted to prove the Riemann Hypothesis. My purpose is to help the reader not only to understand the proof but also enable him/her to verify the work (if necessary) for his/her own satisfaction. As far as clearly spelling out the outline of the logic of my proof of RH,...

In this brief paper we show, by using the methods of our earlier work, that it is possible to prove the Mertens Function Equivalent statement of the Riemann Hypothesis .

In this paper, drawing on the methods I previously developed for the proof of RH, I offer two different proofs of the Generalized Riemann Hypothesis. The first is based on the Summatory Liouville function and the second [in the Appendix] on the Mertens function involving the Mobius coefficients. In my work 2018 [1], I briefly made the point that my...

Resume and CV of K. Eswaran

This writeup explains, in brief, our Non-Iterative Neural Algorithms to a non-specialist AI researcher. All these algorithms are based on the method of separation of high dimensioned sample points (data) by hyperplanes, namely the KE's Sieve algorithm which is first explained. Then we describe how one can then develop a non-iterative algorithm for...

An invention based on a new algorithm called “Neural Inverse” is presented. This invention is based on a mathematical discovery made earlier by the inventor, concerning the geometrical properties and behaviour of hyper planes in high n-dimensional spaces. The invention is useful for making an Artificial Neural Networks (ANN) System which can work a...

This note contains the consolidated answers to the queries on my proofs of the Riemann Hypothesis and the Generalized Riemann hypothesis.

An invention which classifies data and patterns in a non-iterative manner by using artificial neural networks (ANN) is presented. It follows up from a mathematical discovery made earlier by the inventor, concerning the geometrical properties and behavior of hyper-planes in high n-dimensional spaces. It builds on the algorithms and methods described...

A problem of great interest in the field of computer vision is predicting frames of a video sequence. The success of deep learning in computer vision has led to the emergence of deep-learning-based video prediction as an exciting new study area. The frame prediction is very useful in many applications, such as robot navigation and autonomous vehicl...

The difficulty of predicting video frames has drawn a lot of attention since it is important for numerous computer vision applications, including robotics and driverless cars. Many applications, including video categorization, activity identification, and video summarizing, struggle with the difficulty of detecting frames in videos. Some of the exi...

Cancer remains a defining illness of our generation as it evades both immunity and drugs. Early diagnosis is crucial in the battle against cancer. Thanks to the advent of next generation sequencing technologies (NGS) followed by the omics revolution, genetic information can now be explored for applications towards early diagnosis [1]. Recent studie...

Chronic liver syndrome is a major cause of death and disease worldwide. It ensues all over the world regardless of age, gender, area or race. Cirrhosis is an final consequence of a lot of liver sicknesses characterised with fibrosis and architectural alteration of the liver with the development of the liver with the formation of renewing swellings...

Liver disease is treated as one of the most serious areas of concern in medicine not only in India but also across the world. To improve the competence of diagnosis at the preliminary level of disease accurately, we have taken the help of machine learning algorithms which is part of Artificial intelligence. In this paper, Different Machine Learning...

Liver disease is treated as one of the most serious areas of concern in medicine not only in India but also across the world. To improve the competence of diagnosis at the preliminary level of disease accurately, we have taken the help of machine learning algorithms which is part of Artificial intelligence. In this paper, Different Machine Learning...

Of late I have been asked a variety of questions about my RH proof. In this note I answer a few I think are important, or else are so subtle that to answer them for oneself a reader would require a concentrated study of my papers, and also some broad questions asked by students.

The novel coronavirus (COVID-19) pandemic is pressurizing the healthcare systems across the globe and few of them are on the verge of failing. The detection of this virus as early as possible will help in contaminating the spread of it as the virus is mutating itself as fast as possible and currently there are about 4,300 strains of the virus accor...

In the modern era, people are affected by different types of health issues; one of them is Malaria which may be incipient among all aged people. Malaria is a major disease which may be infected by a female mosquito bite and it is spread from one person to another through mosquitoes. The traditional mechanism to detect the malaria disease is visuall...

The novel coronavirus (COVID-19) pandemic is pressurizing the healthcare systems across the globe and few of them are on the verge of failing. The detection of this virus as early as possible will help in contaminating the spread of it as the virus is mutating itself as fast as possible and currently there are about 4,300 strains of the virus accor...

Very Brief Abstract : As is well-known, the celebrated Riemann Hypothesis (RH) is the prediction that all the non-trivial zeros of the zeta function ζ ( s ) lie on a vertical line in the complex s-plane at Re( s ) =1/2 (the critical line). This has been an unsolved conundrum for the last 160 years. In this lecture a new approach from an entirely d...

In today's modern lifestyle people are effecting by different health issues, one among them is heart disease which may be incipient from a very early age. Cardiovascular disease remains as the number one cause of death globally. The main objective of this paper is to identify the presence or absence of heart disease for an individual. In the health...

A room in a smart home is fixed with environmental sensors for sensing of the indoor air quality. Environmental sensors can be any sensor from simple air temperature sensor to an indoor air quality measurement system, which holds different types of sensors or a networked sensor. Purpose of these sensors is to determine the indoor air quality and th...

Marking the attendance in schools and colleges is a key activity by the teachers. They face problems when there are a large number of students in a class. These difficulties can be overcome by using computer aided face recognition techniques. In this paper we use the KNN algorithm and compare our results with the LDA (linear discriminant analysis).

This brief Note describes the scheme and the essential steps in the proof of the Riemann Hypothesis as given by K.Eswaran in his Main Paper.: “The Final and Exhaustive Proof of The Riemann Hypothesis from First Principles”. See Ref [1] (in Reference List below). This write up also provides an answer to queries raised by readers and other research...

An invited Lecture was given by K. Eswaran, concerning the Riemann Hypothesis and his proof, at the Ramanujan Research Centre, Government Arts College, Kumabkonam, in the One Day National Seminar on Current Trends in Mathematical Sciences; 1st March 2019. On the evening of the same day, Eswaran delivered a slightly shorter version of the above Lect...

Diabetes mellitus commonly referred as diabetes, is a complex condition which impairs the body's ability to produce or respond to insulin, leading to high blood sugar levels. The diagnosis of diabetes is of great importance due to its severe long-term complications like cardiovascular disease, stroke, chronic kidney disease and damage to the eyes....

Breast cancer is a disease which occurs when the cells in the breast grow out of control due to the changes in the genes called mutations. These abnormal cells get accumulated and eventually form a tumor or can be felt as a lump. The main factors causing breast cancer are advancing age and family history. So, earlier detection of breast cancer is n...

In this brief note, I wish to state that the method of analysis which was described in detail in my other papers, (e.g. see [1]), can also be used to investigate the position of zeros in other kinds of zeta functions. As an example if we consider the Dirichlet L-function.

This is a Lecture which deals with my proof of the Riemann Hypothesis from first principles. Duration of the lecture One hour and forty minutes. Delivered on 19th November 2018 at The Institute of Engineers India, Khairatabad, Hyderabad India

In this brief note we examine the contribution to the summatory Liouville series L(N), by the first finite N0, (finite) terms, and show that as N tends to infinity they do not prevent the L(N), from becoming a random-walk as N → ∞. This note is a supplement material to the paper: "The Final and Exhaustive Proof of the Riemann Hypothesis from First...

In this paper it is demonstrated that that the probability that an integer has an even number of prime factors (multiplicity included) is equal to the probability that it has an odd number of prime factors. This result, which was already proved alternatively in [5], is closely connected to the Riemann Hypothesis (RH) and provides a route to the pro...

This note is for those who are looking for a reading guide that logically links the key steps, in the paper: "A Rigorous Proof of the Riemann Hypothesis from First Principles" by K.Eswaran. It is strongly suggested that the Abstract and the Extended Abstract (which reveals the plan) of the above paper, should be first read before perusal of this Gu...

Transient analysis of cooling water system pipeline layout of a 520 MW thermal power plant is undertaken for different possible events of operation of valves and cooling water pumps. A water hammer analysis program has been developed and different natures of valve opening and closure times were attempted, to mitigate the effects of high/low transie...

The Latest and FINAL VERSION of this paper has been uploaded on May 9 2018 in Research Gate. IT IS TITLED: The Final and Exhaustive Proof of the Riemann Hypothesis from First Principles by K.Eswaran PLEASE READ THE ABOVE LATEST AND FINAL VERSION. ***Please see Corrigendum at the end of this description.**** ABSTRACT :As is well known the celebrate...

This is a Invited talk on AI and new algorithms delivered in the IEEE Students’ Congress Hyderabad on 4th November 2017

This paper demonstrates the results of a very fast recently discovered, algorithm which was deployed to classify liver function data. This data was from a hospital in Hyderabad and the prediction of the algorithm was very accurate and it can be used for rapid initial diagnosis.

Activity recognition is one of the most significant technologies in many applications such as medical research, human survey system and it is a trending research topic in healthcare and smart homes. The Data set[4] was taken from an online repository named UCI, in which experiments have been carried out with a batch of 30 volunteers within an age o...

A person easily cannot identify a diseased tree by mere eyesight. Different trees are exposed to different diseases. In this paper, our aim is to solve this problem using our algorithm. To serve this purpose, a data set was taken from an online repository named UCI. It is Wilt data set that consists of some diseased trees along with other land cove...

Identifying the gamma ray events from background hadron signals in gamma-ray Cherenkov telescope is the one of the important issues to progress detector technology in astronomy. The data set is MAGIC, a Cherenkov telescope experiment taken from an online machine learning site called UCI. The data has got two classes, i.e., the image originating eit...

In this paper and the next an entirely novel method of supervised neural learning which is non-iterative is described. The process is as follows: every data point which may be n-dimensional is first separated from every other point by hyper-planes, so that no two points are un-separated by at least one hyper plane. This is always possible in high d...

As described in Part I of this two paper series, an algorithm was recently 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. By using this new algorithm it can be shown that there are two ways of classification by a neural network, for a large dimension featu...

This paper demonstrates the results of a very fast recently discovered, algorithm which was deployed to classify liver function data. This data was from a hospital in Hyderabad and the prediction of the algorithm was very accurate and it can be used for rapid initial diagnosis.

A person easily cannot identify a diseased tree by mere eyesight. Different trees are exposed to different diseases. In this paper, our aim is to solve this problem using our algorithm. To serve this purpose, a dataset was taken from an online repository named UCI. It is Wilt dataset that consists of some diseased trees along with other land cover...

A system and method which enables a robotic device to perform an intelligent learning and memory activity to provide a systematic methodology which is used to recall and play back by learning past events recorded in video and recall and play back any given event just like a human being, is provided. This enables quick recalling and recollecting of...

K.Eswaran's Detailed Biodata, 2017

This is a Road Map of the two papers connected with coin tosses and the proof of the Riemann Hypothesis by the above Author.

This paper (is the Latest version) 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 ove...

This paper investigates the analytic properties of the Liouville function’s Dirichlet series that obtains from the function F(s) ≡ ζ(2s)/ζ(s), where s is a complex variable and ζ(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...

In a previous paper entitled The Dirichlet Series for the Liouville Function and the Riemann Hypothesis, the author had proved the validity of the Riemann Hypothesis (RH). In this technical note, we perform a numerical analysis and provide supporting empirical evidence that is consistent with the formal theorems that were key to establishing the co...

This paper investigates the analytic properties of the Liouville function's Dirichlet series that obtains from the function F (s) ≡ ζ(2s)/ζ(s), where s is a complex variable and ζ(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...

This paper investigates the analytic properties of the Liouville function’s Dirichlet series that obtains from the function F(s) ≡ ζ(2s)/ζ(s), where s is a complex variable and ζ(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...

This paper (Final Version) investigates the analytic properties of the Liouville function’s Dirichlet series that obtains from the function F(s) ≡ ζ(2s)/ζ(s), where s is a complex variable and ζ(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 ex...

This paper investigates the analytic properties of the Liouville function's Dirichlet series that obtains from the function F (s) ≡ ζ(2s)/ζ(s), where s is a complex variable and ζ(s) is the Riemann zeta function. The paper employs a novel method of summing the series by casting it as an infinite number of partial sums over sub-series that exhibit a...

The present invention discloses a new methodology which can be used to classify specific problems and describes a product such as the “classification engine” which is implementable in hardware for specific problems. The specific problems could be such as Face Recognition Systems, Disease diagnostic systems, Robotic Inspection systems for use in the...

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 nov...

Recently an algorithm, was discovered, which separates points in n-dimension by planes in such a manner that no two points are left unseparated 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 de...

What the Algorithm Does: Imagine we are given a set G of N points in n-dimensional space. Basically the algorithm finds planes, in n-dimension space such that they can separate all the N points, in such a manner that every point is separated from every other point by at least one plane. The output of the algorithm is a Set S containing the points a...

Since submitting the above paper a computer program that solves a non-trivial problem by using the Algorithm has been written. It generated 2000 points (in 15-dimension space) randomly and applied the algorithm to show that only 22 planes can separate each of the 2000 points. The time taken to run the program was almost instantaneous in a Toshiba l...

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 se...

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 d...

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...

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 ol...

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...

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 John von Neumann. These developments have been followed by a different approach pioneered by Khachiyan and Karmarkar. In this paper we attempt a new approach for solving an old...

In this paper we build upon the Mirroring theorem introduced in [15] as a new method of unsupervised hierarchical pattern classification. The Mirroring theorem affirms that "given a collection of samples with enough information in it such that it can be classified into classes and sub-classes then 1. There exists a mapping which classifies and sub-...

In this paper, we develop a hierarchical classifier (an invertedtree-like structure) consisting of an organized set of “blocks” eachof which is actually a module that performs a feature extractionand an associated classification. We build each of such blocks bycoupling a Mirroring Neural Network (MNN) with a clustering(algorithm) wherein the functi...

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 sub-classes then (i) There exists a mapping which classifies and subclassifies these samples (ii) There exists a hierarchical classifier which can be cons...

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 wi...

- 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 transfo...

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 dat...

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...

The manner in which the extension of a small wave packet moving in a circular orbit in a central potential V(r) changes with time is examined. It is shown that in general, the extension in the orbital plane increases indefinitely at large times unless rV"-V'=0 at the radius of the orbit. The question whether there exist any special 'coherent' wave...

The quantum mechanical analogue of a classical nonlinear system is shown to be exactly solvable and its energy levels and eigenfunctions are obtained completely. The symmetric version (k0=0) of this model is the SU(2)(X)SU(2) chiral invariant Lagrangian in the Gasiorowicz-Geffen coordinates. The radial part of the classical equation of motion (in b...

Gas Insulated substations (GIS) up to 500kV class have been widely accepted over conventional air insulated substation due to several advantages. However, the presence of floating metal particles and protrusions within the GIS at various locations could seriously affect the performance. The paper describes the method of detection of partial dischar...

In modern day production, tool condition monitoring systems are needed to get better quality of jobs and to ensure reduction in the downtime of machine tools due to catastrophic tool failures. Tool condition monitors alert the operator about excessive tool wear and stop the machine in case of an impending breakage or collision of tool. A tool condi...

This paper is concerned with the problem of separation of data, by a Neural based computer recognition system. To this end certain types of data which are "tricky" are studied in order to see if they can be separated (ie. classified) by a neural network or by a Kohonen based classifier. It is shown that there exist data which cannot simply be separ...

Pressure surges in pipelines are caused due to different events either planned or accidental. It is essential to determine the magnitude and frequency of pressures and forces triggered due to these transients to estimate the stresses and vibration levels in the pipeline networks. In this paper an effort is made to study these transients in incompre...

Artificial Neural Networks play a pivotal role in the branch of Artificial Intelligence. They can be trained efficiently for a variety of tasks using different methods, of which the Back Propagation method is one among them. The paper studies the choosing of various design parameters of a neural network for the Back Propagation method. The study sh...

Pressure transients in pipelines are triggered by various events, some of which are accidental and others form a part of operational and safety procedures. Nevertheless it is important to determine beforehand the pressure increases, the velocity changes and the forces that these transients cause, so that the consequent vibrations and stresses can b...

The effect of a small crack on the SIF at the tip of a neighbouring main crack is studied using the finite element method. Both collinear and stacked parallel arrangements are examined. Two extreme orientations of cracks with respect to the interface are considered. Very good agreement is obtained with the available analytical solution in the case...

The possibility of using a compact tension type of specimen geometry for the measurement of interface fracture toughness for sandwich construction has been examined. Three material pairs have been considered. The fracture toughness data and its variation with the load phase angle are presented. The possibility of specifying the condition for the on...

The effect of elastic mismatch and the thickness of the adhesive on the stress intensity factor at the tip of a crack lying in the adhesive is studied using an epoxy layer sandwiched between two identical substrates. Results show that there can be shielding at the main crack tip caused by a material mismatch whose magnitude will depend on the adhes...

A specification of the curvatures at all points determines the surface of an object uniquely except for a translation and a rotation. The aim of this study is to determine the curvatures at different points on various objects in the scene. This is done by a lighting arrangement, wherein the surface of objects are illuminated by a source of light be...

It is shown that there exists a category of two-dimensional diffraction problems, which can be put into a `standard form' of dual integral equations. These diffraction problems include: diffraction of electromagnetic waves by a finite strip, a finite slit, the diffraction of scalar or vector elastic waves by a rigid strip or crack, etc. A general m...

The paper deals with, briefly, the various image processing techniques such as image segmentation, image enhancement and classification for automatically performing the task of detection of defects in welds and castings. The advantages that accrue from such techniques are discussed.

A knowledge of the vibration characteristics of discs of steam turbines is a prerequisite for a successful design of the turbine. As a matter of fact, considerable research effort, both theoretical and experimental, has been directed towards understanding the dynamic behaviour of blades and discs taken singly and jointly. This paper presents a theo...

In this paper the simultaneous uncertainties of the generalized cosine and sine operators are studied. We obtain states which minimize (ΔC)2 (ΔS)2/< -i[C, S]>2; these states are characterized by a complex number λ and form an over-complete set within an ellipse in the λ-plane. In questo articolo si studiano le incertezze simultanee degli operatori...

Summary It is shown that there exists for the quantum harmonic oscillator a large class of « semi-coherent » states, which represent harmonically oscillating wave packets whose « spread » or « size » remains constant. These states are different from the coherent states in not having the minimum value 1/2ħ for the uncertainty product Δx·Δp.