Variety Membership Testing in Algebraic Complexity Theory

In this thesis, we study some of the central problems in algebraic complexity theory through the lens of the variety membership testing problem. In the first part, we investigate whether separations between algebraic complexity classes can be phrased as instances of the variety membership testing problem. For this, we compare some complexity classes with their closures. We show that monotone commutative single-(source, sink) ABPs are closed. Further, we prove that multi-(source, sink) ABPs are not closed in both the monotone commutative and the noncommutative settings. However, the corresponding complexity classes are closed in all these settings. Next, we observe a separation between the complexity class VQP and the closure of VNP. In the second part, we cover the blackbox polynomial identity testing (PIT) problem, and the rank computation problem of symbolic matrices, both phrasable as instances of the variety membership testing problem. For the blackbox PIT, we give a randomized polynomial time algorithm that uses the number of random bits that matches the information-theoretic lower bound, differing from it only in the lower order terms. For the rank computation problem, we give a deterministic polynomial time approximation scheme (PTAS) when the degrees of the entries of the matrices are bounded by a constant. Finally, we show NP-hardness of two problems on 3-tensors, both of which are instances of the variety membership testing problem. The first problem is the orbit closure containment problem for the action of GLk x GLm x GLn on 3-tensors, while the second problem is to decide whether the slice rank of a given 3-tensor is at most r

Indian Life in Transition: A Critical Study of A Tiger At Twilight

In A Tiger At Twilight (1991) the erstwhile raja of Samargarh returns to his abandoned palace in Nijanpur, after years of self-exile, with his sick daughter and his supposed half-sister, and immediately assumes the responsibility of killing a man-eating tiger. Assisting him are a few noted men of the valley including Dev, the owner and manager of a resort. But as the hunt intensifies Dev realizes that things are not as they seem: Heera, the raja’s sister, has an inexplicable power over the men in the hunting party and a strange connection with the tiger. As the men get closer to killing the beast, bizarre things begin to happen, hinting at the influence of the supernatural

Polynomial Identity Testing for Low Degree Polynomials with Optimal Randomness

We give a randomized polynomial time algorithm for polynomial identity testing for the class of n-variate poynomials of degree bounded by d over a field ?, in the blackbox setting. Our algorithm works for every field ? with | ? | ? d+1, and uses only d log n + log (1/ ?) + O(d log log n) random bits to achieve a success probability 1 - ? for some ? > 0. In the low degree regime that is d ? n, it hits the information theoretic lower bound and differs from it only in the lower order terms. Previous best known algorithms achieve the number of random bits (Guruswami-Xing, CCC\u2714 and Bshouty, ITCS\u2714) that are constant factor away from our bound. Like Bshouty, we use Sidon sets for our algorithm. However, we use a new construction of Sidon sets to achieve the improved bound. We also collect two simple constructions of hitting sets with information theoretically optimal size against the class of n-variate, degree d polynomials. Our contribution is that we give new, very simple proofs for both the constructions

Spiritual Strain in Chasing The Rainbow: Growing Up in An Indian Village

Chasing the Rainbow: Growing up in an Indian Village is a collection of short stories written by Manoj Das. It was published in 2004. It is part memoir, part social history of a vanishing ethos. This genre, in particular, seems to be more suitable for the mirroring the Indian life since its writer is free to choose any one part of life and deals it with utmost care and sincerity which it requires. Manoj Das is one of the greatest short story writers of India. Ruskin Bond, an acclaimed short story writer and novelist, remarks, “There are only a few good storytellers left in the world today and one of them is Manoj Das” (blurb on Das’s The Escapist)

Nonlinear viscoelastic response of a thermodynamically metastable polymer melt

Ultra High Molecular Weight Polyethylene (UHMw-PE) is an engineering polymer that is widely used in demanding applications because of its un-paralleled properties such as high abrasion resistance, high-modulus and high-strength tapes and fibres, biaxial films etc. In common practice, to achieve the uniaxial and the biaxial products, the solution processing route is adopted to reduce the number of entanglements per chain, such as found in Dyneema(R) from DSM(R). Another elegant route to reduce the number of entanglements to ease solid-state processing is through controlled polymerisation using a single-site catalytic system. In this theses, how different polymerisation condition, such as temperature and time control molecular weight and the resultant entangled state in synthesised disentangled UHMw-PE is addressed. Linear dynamic melt rheology is used to follow entanglement formation in an initially disentangled melt. With the help of rheological studies, heterogeneity in the distribution of entanglements along the chain length and the crystal morphology produced during polymerisation is considered. For the understanding of influence of large shear flow on melt dynamics large amplitude oscillatory shear (LAOS) is used and the non-linear viscoelastic regime is explored. A remarkable feature of overshoot in loss (viscous) modulus with increasing deformation (strain) in UHMw-PE melt in the LAOS is observed. This observation is characteristic of colloidal systems. The role of entanglement density in the amorphous region of the synthesised disentangled UHMw-PE (semi-crystalline polymers) on the melting and crystallisation is presented. To understand the effect of topological differences on melting behaviour, nascent entangled, nascent disentangled and melt-crystallised samples have been used. The role of superheating on the melting process is also addressed. Preliminary results on characteristic melting time of a crystal using TM-DSC are also presented

Operational Analysis of Private Cloud using Eucalyptus

Distributed environment is an invoking idea in computer field since it gave permissions that the assets to be purveyed according to the client needs 1 The paper addresses the system of arrangement of a private cloud in improving the practical furthest reaches of cloud processing at compelled states of arrangement It is the review of all previous research based on Private Cloud using Eucalyptus It gives benefits on virtual machines where the client impart assets programming and different gadgets on interest Cloud administrations are backed with proprietor and Open Source Systems OSS As Restrictive items remain exceptionally costly clients unable to permitted test on their item and protection is a significant affair in it Cloud registering frameworks in a broad sense give access to expansive pools of information and computational assets through a mixed bag of interfaces These sorts of frameworks offer another programming focus for versatile application engineers and have picked up ubiquity over the recent years Then again most distributed computing frameworks in operation today are exclusive depend upon base that is undetectable to the research group or are not unequivocally intended to be instrumented and adjusted by frameworks specialist

Greedy Strikes Again: A Deterministic PTAS for Commutative Rank of Matrix Spaces

We consider the problem of commutative rank computation of a given matrix space. A matrix space is a (linear) subspace of the (linear) space of n x n matrices over a given field. The problem is fundamental, as it generalizes several computational problems from algebra and combinatorics. For instance, checking if the commutative rank of the space is n, subsumes problems such as testing perfect matching in graphs and identity testing of algebraic branching programs. An efficient deterministic computation of the commutative rank is a major open problem, although there is a simple and efficient randomized algorithm for it. Recently, there has been a series of results on computing the non-commutative rank of matrix spaces in deterministic polynomial time. Since the non-commutative rank of any matrix space is at most twice the commutative rank, one immediately gets a deterministic 1/2-approximation algorithm for the computation of the commutative rank. This leads to a natural question of whether this approximation ratio can be improved. In this paper, we answer this question affirmatively. We present a deterministic Polynomial-time approximation scheme (PTAS) for computing the commutative rank of a given matrix space B. More specifically, given a matrix space and a rational number e > 0, we give an algorithm, that runs in time O(n^(4 + 3/e)) and computes a matrix A in the given matrix space B such that the rank of A is at least (1-e) times the commutative rank of B. The algorithm is the natural greedy algorithm. It always takes the first set of k matrices that will increase the rank of the matrix constructed so far until it does not find any improvement, where the size of the set k depends on e

Algebraic Independence over Positive Characteristic: New Criterion and Applications to Locally Low Algebraic Rank Circuits

The motivation for this work comes from two problems--test algebraic independence of arithmetic circuits over a field of small characteristic, and generalize the structural property of algebraic dependence used by (Kumar, Saraf CCC\u2716) to arbitrary fields. It is known that in the case of zero, or large characteristic, using a classical criterion based on the Jacobian, we get a randomized poly-time algorithm to test algebraic independence. Over small characteristic, the Jacobian criterion fails and there is no subexponential time algorithm known. This problem could well be conjectured to be in RP, but the current best algorithm puts it in NP^#P (Mittmann, Saxena, Scheiblechner Trans.AMS\u2714). Currently, even the case of two bivariate circuits over F_2 is open. We come up with a natural generalization of Jacobian criterion, that works over all characteristic. The new criterion is efficient if the underlying inseparable degree is promised to be a constant. This is a modest step towards the open question of fast independence testing, over finite fields, posed in (Dvir, Gabizon, Wigderson FOCS\u2707). In a set of linearly dependent polynomials, any polynomial can be written as a linear combination of the polynomials forming a basis. The analogous property for algebraic dependence is false, but a property approximately in that spirit is named as ``functional dependence\u27\u27 in (Kumar, Saraf CCC\u2716) and proved for zero or large characteristic. We show that functional dependence holds for arbitrary fields, thereby answering the open questions in (Kumar, Saraf CCC\u2716). Following them we use the functional dependence lemma to prove the first exponential lower bound for locally low algebraic rank circuits for arbitrary fields (a model that strongly generalizes homogeneous depth-4 circuits). We also recover their quasipoly-time hitting-set for such models, for fields of characteristic smaller than the ones known before. Our results show that approximate functional dependence is indeed a more fundamental concept than the Jacobian as it is field independent. We achieve the former by first picking a ``good\u27\u27 transcendence basis, then translating the circuits by new variables, and finally approximating them by truncating higher degree monomials. We give a tight analysis of the ``degree\u27\u27 of approximation needed in the criterion. To get the locally low algebraic rank circuit applications we follow the known shifted partial derivative based methods