Airo national research journal

Reduction

In this section we suppose a fixed monomial ordering, which will not be defined explicitly.

Given two polynomials f and g, one says that f is reducible by g if some monomial m in f is a multiple of the leading monomial lm(g) of g. If m happens to be the leading monomial of f then one says that f is lead-reducibleby g. If c is the coefficient of m in f and m = q lm(g), the one-step reduction of f by g is the operation that associates to f the polynomial

The main properties of this operation are that the resulting polynomial does not contain the monomial m and that the monomials greater than m (for the monomial ordering) remain unchanged. This operation is not, in general, uniquely defined; if several monomials in f are multiples of lm(g) one may choose arbitrarily the one that is reduced. In practice, it is better to choose the greatest one for the monomial ordering, because otherwise subsequent reductions could reintroduce the monomial that has just been removed.

Given a finite set G of polynomials, one says that f is reducible or lead-reducible by G if it is reducible or lead-reducible, respectively, by an element of G. If it is the case, then one defines  . The (complete) reduction of f by G consists in applying iteratively this operator   until getting a polynomial  , which is irreducible by G. It is called a normal form of f by G. In general this form is not uniquely defined (this is not a canonical form); this non-uniqueness is the starting point of Gröbner basis theory.

For Gröbner basis computations, except at the end, it is not necessary to do a complete reduction: a lead-reduction is sufficient, which saves a large amount of computation.

 

Research Methodology

The theory of Gröbner bases has been extended by many authors in various directions. It has been generalized to other structures such as polynomials over principal ideal rings or polynomial rings, and also some classes of non-commutative rings and algebras, like Ore algebras.

Polynomial Algebra

Certain sets of polynomials have special algebraic structure, they may be rings, fields or ideals. Algebraic properties associated to these structures play a very im-portant role in solving computational tasks involving polynomials. In this chapter, we will discuss various aspects of polynomials which will play a fundamental role in our later discussion. We will define ideals over polynomial rings and will give brief summary on Groebner bases for ideals and modules. Most of the material given in this section is taken from [1, 18].

  A monomial, in n indeterminates x₁, . . . , x_n, is a product of the form x^u₁¹ x^u₂² · · · x^u_nⁿ , where the u_i are non-negative integers, and u = (u₁, . . . , u_n). The total degree of this monomial is the sum |u| = u₁ + · · · + u_n.

  2.1.1 Definition [Polynomial] A polynomial f in x₁, x₂, . . . , x_n with the coeffi-cients in K (where K is any field) is a finite linear combination of the monomials, written as

f = cuxu, cu ∈ K,

(2.1)

 

u

 

c_u is called the coefficient of the monomial x^u = x^u₁¹ · · · x^u_nⁿ . If c_u , 0 then we call c_ux_u a term of f . The set of all polynomials in x₁, x₂, . . . , x_n with coefficients in

is denoted by K[x] = K[x₁, . . . , x_n]. These polynomials in n variables, over a field K, together with operations of addition and multiplication, satisfy all axioms of ring, and so form commutative polynomial ring.

 

Objective

This research work is elaborating the methodology of Grobner and its application in various mathematical aspects.

The main objective is to determine the more and more implementation of the theory of Grobner in the world of Mathematics

Through this work we are to know the following implementations

         CoCoA free computer algebra system for computing Gröbner bases.

         GAP free computer algebra system that can perform Gröbner bases calculations.

         FGb, Faugère's own implementation of his F4 algorithm, available as a Maple library.^[5] To the date, as of 2014, it is, with Magma, the fastest implementation for rational coefficients and coefficients in a finite field of prime order

         Macaulay 2 free software for doing polynomial computations, particularly Gröbner bases calculations.

         Magma has a very fast implementation of the Faugère's F4 algorithm.^[6]

         Maple has implementations of the Buchberger and Faugère F4 algorithms, as well as Gröbner trace

         Mathematica includes an implementation of the Buchberger algorithm, with performance-improving techniques such as the Gröbner walk, Gröbner trace, and an improvement for toric bases

         SINGULAR free software for computing Gröbner bases

         Sage provides a unified interface to several computer algebra systems (including SINGULAR and Macaulay), and includes a few Gröbner basis algorithms of its own.

         SymPy Python computer algebra system uses Gröbner bases to solve polynomial systems

Coding theory plays an important role in efficient transmission of data over noisy communication channels.

         Based on experimentation and comparison of timings with other methods reported, we state a tentative conclusion. The methods of this paper are viable and effective when the problem at hand is unperturbed from an exactly solvable variant. They often give good results when the problem is overdetermined, provided the noise is modest relative to an exactly solvable nearby problem, and the scale of coefficients does not vary too much. In other situations it is not clear whether our methods can be adapted so readily.

         While most examples covered seem to work efficiently and give reasonable results, it remains an open question as to how competitive these methods are in regard to speed and quality of results, as compared to other approaches. An advantage to Gröbner bases is that polynomial algebra is carried out in a sparse setting; many methods based on linear algebra require dense matrix manipulation. The examples presented offer evidence that, when working with input of modest degree, Gröbner bases methods are viable. That the coding is simple makes them all the more attractive.

Hypothesis

The Gröbner bases method is a powerful tool in symbolic and algebraic computing, which is currently not yet fully utilized in SymPy. Also implementation of Buchberger’s algorithm is quite limited at the moment. However, as we showed in this chapter, SymPy can be used for solving practical problems in symbolic mathematics, specifically problems which involve solving systems of polynomials. We hope that, in foreseeable future, improved algorithms for computing Gröbner bases will be implemented, so that SymPy will be able to tackle more complex problems.

Adams, W. W. and Loustaunau, P. An Introduction to Gröbner Bases. Providence, RI: Amer. Math. Soc., 1994.

Becker, T. and Weispfenning, V. Gröbner Bases: A Computational Approach to Commutative Algebra. New York: Springer-Verlag, 1993.

Boege, W.; Gebauer, R.; and Kredel, H. "Some Examples for Solving Systems of Algebraic Equations by Calculating Gröbner Bases." J. Symb. Comput. 1, 83-98, 1986.

Buchberger, B. "Gröbner Bases: An Algorithmic Method in Polynomial Ideal Theory." Ch. 6 in Multidimensional Systems Theory(Ed. N. K. Bose). New York: van Nostrand Reinhold, 1982.

Buchberger, B. "A Criterion for Detecting Unnecessary Reductions in the Construction of Groebner Bases." Proceedings of the International Symposium on Symbolic and Algebraic Computation. pp. 3-21, June 1979.

Buchberger, B. "Groebner Bases: A Short Introduction for Systems Theorists." http://www.risc.uni-linz.ac.at/people/buchberg/papers/2001-02-19-A.pdf.

Buchberger, B. and Zapletal, A. "Gröbner Bases .. Bibliography." http://www.ricam.oeaw.ac.at/Groebner-Bases-Bibliography/.

Cox, D.; Little, J.; and O'Shea, D. Ideals, Varieties, and Algorithms: An Introduction to Algebraic Geometry and Commutative Algebra, 2nd ed. New York: Springer-Verlag, 1996.

Eisenbud, D. Commutative Algebra with a View toward Algebraic Geometry. New York: Springer-Verlag, 1995.

Faugere, J. C.; Gianni, P.; Lazard, D.; and Mora, T. "Efficient Computation of Zero-Dimensional Groebner Bases by Change of Ordering." J. Symb. Comput. 16, 329-344, 1993.

Giovini, A.; Mora, T.; Niesi, G.; Robbiano, L.; and Traverso, C. "One Sugar Cube, Please?, or Selection Strategies in the Buchberger Algorithm." Proceedings of the International Symposium on Symbolic and Algebraic Computation. pp. 49-54, June 1991.

Harris, J. "Rearranging Expressions by Patterns." Mathematica J.

 

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Womanhood in the light of The Life Divine

 By  Umesh Prasad Research Scholar English

 Jabalpur Madhya Pradesh India 

The call of womanhood is deep, challenging and broad. We at Ladies against Feminism desire to raise high the Standard of godly womanhood and proclaim it beautiful -- not just with our words, but with our very lives. From comforting the dying to nurturing the weak to praising the strong, we glory in our role as women and find it rich beyond measure. We invite you to explore this marvelous calling with us, whatever your background or beliefs. Feminism has had its time on history’s stage. We now call for it to step down and face the ruin and despair it has wrought and repent of its wrongs. One lifetime is given us to make things beautiful, wholesome, nourishing, glad-hearted and noble. May we use our time wisely!¹

The Life Divine explores for the Modern mind the great streams of Indian metaphysical thought, reconciling the truths behind each and from this synthesis extends in terms of consciousness the concept of evolution. The unfolding of Earth's and men spiritual destiny is illuminated, pointing the way to a Divine Life on Earth.

The Life Divine is no mere call to a life of piety, asceticism or outward religious fervor. It is a call to bring the Divine as a force of higher consciousness into all that we are and do, both individually and as a species. The Life Divine unfolds a panoramic exploration of consciousness from the Absolute (Brahman), to the Cosmic Creator (Ishvara), to the individual soul (Jivatman), and all the realms of existence, manifest and unmanifest, known and unknown. There are few books that cover such an expanse and with such depth, direct knowledge and clarity. For those who want to widen their horizons and extend their awareness into the realms of higher consciousness, there is perhaps no other book that is as complete, comprehensive and challenging. Reading it requires both concentration and meditation of a very high order, but brings great riches of inner insight in return.

As someone who has studied the main religious traditions of the world, and has written extensively on the traditions of India, this book has remained with me as life time companion. I recommend it to all those who are looking at the spiritual life as a quest for a higher consciousness and grace that can transform all that we do. One can continue to delve into the book for new wisdom and insight year after year. The Life Divine teaches us in depth about the great spiritual traditions of India, Veda, Vedanta, Samkhya, Yoga, Tantra and Buddhism, but from a view of practice and realization, and a seeking for the universal truth behind all these great teachings.
In The Life Divine, Sri Aurobindo explains that there are two primary means by which we connect to and are transformed by the Spirit.

First, he informs us that in the depths of our being there is a True Self and Spirit -- an evolving Soul that he calls the "Psychic Being" -- which is the primary lever and means of our evolutionary change. By connecting to our Evolving Soul, we perceive our ultimate purpose in life, open to the vast array of spiritual insights, forces, and powers in the universe; feel a sense of oneness with everything and all around us; and as a result experience ultimate joy and delight in being alive.

Second, he reveals that around us there is also a descending spiritual Force -- a Supramental truth consciousness -- that we can open to, allowing Its infinite truth power to enter our being, and uplift, perfect, and transform it. It turns out that this supramental truth consciousness is not only the descending spiritual power that can hasten our progress, evolution, and transformation, but it is in fact the very force and power that enabled the universe to emerge from a Divine Source.

As we take up the call to discover our higher nature; as we come in touch with our evolving soul; as we open and surrender to the descending supramental power, we begin to make the decisive evolutionary and transformational change. Through this reversal of consciousness, our old nature is transmuted into a new supernature. Not only are our mental, vital, and physical parts uplifted, and our capacities dramatically enhanced, but we also feel a deep connection and oneness with the world around us. In addition, we begin to feel a profound connection with the cosmic and transcendent spiritual reality, which helps us perceive life's true purpose, and our own purpose as instruments of the Higher Power. Finally, out of all of these inner and outer experiences, we feel a deep and abiding Joy and Delight in being alive.²

As individuals make this decisive change in their nature, as they infuse all parts of their being with this higher consciousness, there then gradually emerges a new type of Person -- a Gnostic, Supramental being, whose existence is organized around the power of the Spirit. As a number of such individuals make this decisive change, a new human social existence begins to emerge, culminating in a Divine Life on earth.

This is the powerful vision that Sri Aurobindo has offered the world. This is the revelation of a future existence for the individual and the society that he has captured so eloquently in his metaphysical treatise .

Most notably, the Life Divine outlines the spiritual purpose of the soul and of our human lives. It charts a way to a future in which we can go beyond our current mentality of ego and strife to a world of Divine peace, bliss and knowledge. It charts the transformation of our species from a confused adolescence to the maturity of wisdom and grace. Sri Aurobindo shows how the Divine Shakti can descend into our minds and lift us to a higher level of intelligence as our natural state of existence. The book is perhaps the best study of the spiritual evolution of humanity, the evolution of consciousness in man and nature, which is available.

Ref:-
[1] Tudor, Tasha and Richard Brown. The Private World of Tasha Tudor. New York: Little, Brown and Company, 1992. Page 104.

[2] The Life Divine, Lotus Press, Twin Lakes, Wisconsin ISBN 0-941524-61-2

 

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Introduction

Cloud computing relies on sharing of resources to achieve coherence and economies of scale, similar to a utility (like the electricity grid) over a network. At the foundation of cloud computing is the broader concept of converged infrastructure and shared services.

Cloud computing, just "the cloud", also focuses on maximizing the effectiveness of the shared resources. Cloud resources are usually not only shared by multiple users but are also dynamically reallocated per demand. This can work for allocating resources to users. For example, a cloud computer facility that serves European users during European business hours with a specific application (e.g., email) may reallocate the same resources to serve North American users during North America's business hours with a different application (e.g., a web server). This approach should maximize the use of computing power thus reducing environmental damage as well since less power, air conditioning, rackspace, etc. are required for a variety of functions. With cloud computing, multiple users can access a single server to retrieve and update their data without purchasing licenses for different applications.

Many of today’s Information Technology (IT) applications rely on access to state-of-the-art computing facilities. For instance, as business decisions are increasingly driven by (data) analytics, the practice of operations research and business analytics becomes inherently intertwined with the management of IT resources. In response to the resulting demand for flexible computing resources, cloud computing has taken the IT industry by storm over the past few years. According to the National Institute of Standards and Technology (NIST), cloud computing is a model for enabling convenient, on-demand network access to a shared pool of configurable computing resources (for example, networks, servers, storage, applications, and services) that can be rapidly provisioned and released with minimal management effort or service-provider interaction. Cloud computing is a service where computing is provided as a commodity, much akin to electricity or cable television. Thus, cloud computing is not about a specific technology; rather it is a step in the commoditization of IT enabled by technological advances.

 Cloud Computing

 A key differentiating element of a successful information technology (IT) is its ability to become

a true, valuable, and economical contributor to cyber infrastructure. “Cloud” computing embraces cyber infrastructure, and builds upon decades of research in virtualization, distributed

computing, “grid computing”, utility computing, and, more recently, networking, web and software services. It implies a service oriented architecture, reduced information technology

overhead for the end-user, greater flexibility, reduced total cost of ownership, on demand services and many other things.

Cloud computing is a paradigm that focuses on sharing data and computations over a scalable network of nodes. Examples of such nodes include end user computers, data centers, and Web Services. We term such a network of nodes as a cloud. An application based on such clouds is taken as a cloud application. Basically cloud is a metaphor for internet and is an abstraction for the complex infrastructure it conceals.

Advantages

(i) Reduced Cost: Cloud technology is paid incrementally (you pay only for what you need), saving organizations money in the short run. Money saved can be used for other important resources.

(iii) Highly Automated: IT personnel not needed to keep software up to date as maintenance is the job of the service provider on the cloud.

(iv) More Mobility: Employees can access information wherever they are, rather than having to remain at their desks.

(v) Allows IT to Shift Focus: No longer having to worry about constant server updates and other computing issues, government organizations will be free to concentrate on innovation.

 Drawbacks

 The interesting thing about cloud computing is that we've redefined cloud computing to include everything that we already do. One reason you should not use web applications to do your computing is that you lose control. It's just as bad as using a proprietary program. But certainly shifting to cloud computing has other problems including:

 (i) Security: Is there a security standard?

(ii) Reliance on 3rd Party: Control over own data is lost in the hands of an “difficult-to-trust” provider

(iii) Cost of transition: Is it feasible for me to move from the existing architecture of my data center to the architecture of the cloud?

(iv) Uncertainty of benefits: Are there any long term benefits?

 Conclusion

 Cloud computing is an emerging computing paradigm that is increasingly popular. Leaders in the industry, such as Microsoft, Google, and IBM, have provided their initiatives in promoting cloud computing. However, the public literature that discusses the research issues in cloud computing are still inadequate. In a study of the research literature surrounding cloud computing, I found that there is a distinct focus on the needs of the scientific computing community. Big IT companies are also building their own version of cloud. But still there are many question have left without an answer and indeed the most important one is security. One of the other aspects of the cloud which is left is the social aspect of it. The Cloud is going to happen but which services should be offered on the cloud and for whom. What happens if smaller IT companies start to offer their services on the cloud and no one uses them?! I believe that everything eventually can move to the Cloud. The question is if users are ready for that and if it’s the right move and this need must be addressed.

 References

AIEER 2009 Volume I

AIEER 2010 Volume III

AIEER 2011 Volume V

AIEER 2012 Volume VII

AIEER 2013 Volume VIII

 [1] Armbrust, M. Fox, A., Friffith, R., Joseph, A. D., Katz, R., Konwinskii, A., Lee, G., Patterson, D.,Rabkin, A., Stoica, I., Zaharia, M. (2010). “A View of Cloud Computing”, Communications of theACM, 53(4).

 [2] D.E. ATKINS ET AL., “Revolutionizing Science and Engineering Through Cyberinfrastructure: Report of theNational Science FoundationBlue-ribbonAdvisory Panel on Cyberinfrastructure”, NSF, Report of theNational Science FoundationBlue-ribbonAdvisory

Panel on Cyberinfrastructure, January 2003,

http://www.nsf.gov/od/oci/reports/atkins.pdf

 [3] Gentry, C. (2010). “Computing Arbitrary Functions of Encrypted Data”, Communications of theACM, 53(3).

 [4] Mell, P., Grance., T. (2011). “The NIST Definition of Cloud Computing”, Special publication 800-145, National Institute of Standards and Technology.

 [5] Lijun Mei, W.K. Chan, T.H. Tse, “A Tale of Clouds: Paradigm Comparisons and Some Thoughts on Research Issues”, To appear in Proceedings of the 2008 IEEE Asia-Pacific Services Computing Conference (APSCC 2008), IEEE Computer Society Press, Los Alamitos, CA

 

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 Organic Pollutants from Yamuna River

 Submitted by SAT PAUL Research Scholar ZOOLOGY

 ______________________________________Introduction

The research work is being processed to find out the data related to the role of Micro-bacteria in cleaning of organic pollutants from Yamuna. River Yamuna is one of the most polluted rivers of the India. It originates from Yamunotri glaciers in the lower Himalayas at an elevation of approximately 6387 meters. The barrages formed on the river are playing a major role in escalating the river pollution. River can be divided into five segments on the bases of hydrological and ecological conditions. Water quality of only one segment (Himalayan segment) meets the river water quality standards. Normally no water is allowed to flow downstream of the Himalayan segment (Tejewala barrage) especially in the summer and winter seasons to fulfill the demand of water of the surrounding area. Whatever water flows in the downstream of the Tajewala barrage is the untreated or partially treated domestic and Industrial wastewater contributed through various drains. The discharge of untreated domestic and industrial effluents have severely affected the quality of Yamuna River and now it falls under the category E, which makes it fit only for recreation and industrial cooling, completely ruling out the possibility for underwater life and domestic supply. Almost every year mass death of fishes is reported. Pollution levels in the Yamuna River have risen. Biochemical oxygen demand (BOD) load has increased by 2.5 times between 1980 and 2005: From 117 tonnes per day (TDP) in 1980 to 276 TDP in 2005. The Yamuna has been reduced to a small stream, draining industrial effluents, sewage, dirt and other toxic substances. There is an urgent need to take stringent measures to alleviate these pollution loads and save an ailing river.

___________________________________________Objective

The main objective of this work is to collect the data to understand the role of micro-bacteria in cleaning organic pollutants from Yamuna. As the impurities of Yamuna is well known to us and it’s process of cleaning is also popular in many area, therefore the basics of purification and mainly the importance of micro-bacteria are to be known. For many government and non-government organizations this work is behaving like a leaning chapter for the development and purification of Yamuna in the near future.

Review of literature

The hymn also praises her for being the source of all spiritual abilities. And while the Ganges is considered an epitome of asceticism and higher knowledge and can grant us Moksha or liberation, it is Yamuna, who, being a holder of infinite love and compassion, can grant us freedom, even from death, the realm of her elder brother. She rushes down the Kalinda Mountain, and verily describes her as the daughter of Kalinda, giving her another name, Kalindi, the backdrop of Krishna Leela. The text also talks about her water being of the colour of Lord Krishna, which is dark (Shyam).

It is mention as Iomanes (Ioames) in the surveys of Seleucus I Nicator, an officer of Alexander the Great and one of the Diadochi, who visited India in 305 BC, later Megasthenes, a Greek traveller and geographer, visited India, sometimes before 288 BC, the date of Chandragupta's death, also mention the river in his text Indica, where he described the region around it as the land of Surasena.^[14] In Mahabharata, Indraprastha, the capital of Pandavas was also situated on the banks of Yamuna, it is considered to the modern day city of Delhi.

Geological evidence indicates that in the distant past the Yamuna was a tributary of the Ghaggar River (also known as the Vedic Sarasvati River), but that it later changed its course eastward due to a tectonic event, becoming a tributary of the Ganges. This may have led to the Sarasvati River drying up, the end of manyHarappan civilization settlements, and creation of the Thar desert. However, recent geological research suggests that the diversion of the Yamuna to the Ganges may have occurred during the Pleistocene, and thus could not be connected to the decline of the Harappan civilization in the region.

The importance of the Ganges–Yamuna river basin, and the Doab region as traditional the seat of power, can be derived from the fact, in much of early history of India, most of great empires, which ruled over majority of India, until the Chalukyas King, Vinayaditya, were based in the highly fertile Ganges–Yamuna basin, including the Magadha (ca 600 BC), Maurya Empire (321–185 BC), Sunga Empire (185–73 BCE), Kushan Empire (1st–3rd centuries CE), Gupta Empire (280–550 CE), and many had their capitals here, in cities like Pataliputra orMathura. The government of India over the next five years has prepared plans to rebuild and repair the sewage system and the drains that empty into the river. To address river pollution, certain measures of river cleaning have been taken by the Government's Ministry of Environment and Forests in twelve towns of Haryana, eight towns of Uttar Pradesh, and Delhi, under the Yamuna Action Plan (YAP) which has been implemented since 1993 by the National River Conservation Directorate (NRCD) of the Ministry of Environment and Forests. The Japan Bank for International Cooperation is participating in the Yamuna Action Plan in 15 of the above 21 towns (excluding 6 towns of Haryana included later on the direction of Supreme Court of India) with soft loan assistance of 17.773 billion Japanese Yen (equivalent to about Rs. 700 crore INR) while the Government of India is providing the funds for the remaining 6 towns added later. In 2007 the Indian government's plans to repair sewage lines were predicted to improve the water quality of the river 90% by the year 2010.

However in 2009, the Union government admitted to the Lok Sabha (Indian Parliament), the failure of the Ganga Action Plan (GAP) and the Yamuna Action Plan (YAP), saying that "rivers Ganga and Yamuna are no cleaner now than two decades ago" despite spending over Rs 1,700 crore to control pollution. According to a CSEofficial, these plans adopted the Thames model, based on a centralized sewage treatment system. This meant that huge sum of money and a 24-hr power supply were needed to manage the treatment plants, while only an 8-hr power supply was available, contributing to the failure of both river plans.^[34] In August 2009, the Delhi Jal Board (DJB) initiated its plan for resuscitating the Yamuna’s 22 km stretch in Delhi by constructing interceptor sewers, at the cost of about Rs 1,800 crore.

  Research Methodology

The processes below are the ones commonly used in water purification. Some or most may not be used depending on the scale of the plant and quality of the raw (source) water.