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India has a diverse climate and topography; different kinds of forests, with a minimum of ten distinct bio-geographical areas, there are many as well as three biodiversity hotspots on the planet's surface. Because other native habitats have lost their native status, the forest currently houses the majority of terrestrial species. There is a vast network of protected areas in place, including 514 wildlife sanctuaries, there are 99 national parks in the United States, 18 biosphere reserves are included in the project, and several holy groves managed by native groups. Due to increased people in the world demands, land use changes, and the introduction of invasive alien species, forest degradation and biodiversity loss persist despite a favorable forest policy and a strong regulatory framework. The extent and loss of biodiversity must be constantly monitored, and individuals must be encouraged to join in large-scale biodiversity conservation rehabilitation.

India has a diverse climate and topography; different kinds of forests, with a minimum of ten distinct bio-geographical areas, there are many as well as three biodiversity hotspots on the planet's surface. Because other native habitats have lost their native status, the forest currently houses the majority of terrestrial species. There is a vast network of protected areas in place, including 514 wildlife sanctuaries, there are 99 national parks in the United States, 18 biosphere reserves are included in the project, and several holy groves managed by native groups. Due to increased people in the world demands, land use changes, and the introduction of invasive alien species, forest degradation and biodiversity loss persist despite a favorable forest policy and a strong regulatory framework. The extent and loss of biodiversity must be constantly monitored, and individuals must be encouraged to join in large-scale biodiversity conservation rehabilitation.

India has a diverse climate and topography; different kinds of forests, with a minimum of ten distinct bio-geographical areas, there are many as well as three biodiversity hotspots on the planet's surface. Because other native habitats have lost their native status, the forest currently houses the majority of terrestrial species. There is a vast network of protected areas in place, including 514 wildlife sanctuaries, there are 99 national parks in the United States, 18 biosphere reserves are included in the project, and several holy groves managed by native groups. Due to increased people in the world demands, land use changes, and the introduction of invasive alien species, forest degradation and biodiversity loss persist despite a favorable forest policy and a strong regulatory framework. The extent and loss of biodiversity must be constantly monitored, and individuals must be encouraged to join in large-scale biodiversity conservation rehabilitation.

India has a diverse climate and topography; different kinds of forests, with a minimum of ten distinct bio-geographical areas, there are many as well as three biodiversity hotspots on the planet's surface. Because other native habitats have lost their native status, the forest currently houses the majority of terrestrial species. There is a vast network of protected areas in place, including 514 wildlife sanctuaries, there are 99 national parks in the United States, 18 biosphere reserves are included in the project, and several holy groves managed by native groups. Due to increased people in the world demands, land use changes, and the introduction of invasive alien species, forest degradation and biodiversity loss persist despite a favorable forest policy and a strong regulatory framework. The extent and loss of biodiversity must be constantly monitored, and individuals must be encouraged to join in large-scale biodiversity conservation rehabilitation.

India has a diverse climate and topography; different kinds of forests, with a minimum of ten distinct bio-geographical areas, there are many as well as three biodiversity hotspots on the planet's surface. Because other native habitats have lost their native status, the forest currently houses the majority of terrestrial species. There is a vast network of protected areas in place, including 514 wildlife sanctuaries, there are 99 national parks in the United States, 18 biosphere reserves are included in the project, and several holy groves managed by native groups. Due to increased people in the world demands, land use changes, and the introduction of invasive alien species, forest degradation and biodiversity loss persist despite a favorable forest policy and a strong regulatory framework. The extent and loss of biodiversity must be constantly monitored, and individuals must be encouraged to join in large-scale biodiversity conservation rehabilitation.

India has a diverse climate and topography; different kinds of forests, with a minimum of ten distinct bio-geographical areas, there are many as well as three biodiversity hotspots on the planet's surface. Because other native habitats have lost their native status, the forest currently houses the majority of terrestrial species. There is a vast network of protected areas in place, including 514 wildlife sanctuaries, there are 99 national parks in the United States, 18 biosphere reserves are included in the project, and several holy groves managed by native groups. Due to increased people in the world demands, land use changes, and the introduction of invasive alien species, forest degradation and biodiversity loss persist despite a favorable forest policy and a strong regulatory framework. The extent and loss of biodiversity must be constantly monitored, and individuals must be encouraged to join in large-scale biodiversity conservation rehabilitation.

Given a Boolean function f:{ -1,1}n → {-1,1}, define the Fourier distribution to be the distribution on subsets of [n], where each S ⊆ [n] is sampled with probability f (S)2. The Fourier Entropy-influence (FEI) conjecture of Friedgut and Kalai [28] seeks to relate two fundamental measures associated with the Fourier distribution: does there exist a universal constant C > 0 such that H(f2) ≤ C · Inf (f), where H(f2) is the Shannon entropy of the Fourier distribution of f and Inf(f) is the total influence of f? In this article, we present three new contributions toward the FEI conjecture: (1) Our first contribution shows that H(f2) ≤ 2 · aUC⊕(f), where a UC⊕(f) is the average unambiguous parity-certificate complexity of f. This improves upon several bounds shown by Chakraborty et al. [20]. We further improve this bound for unambiguous DNFs. We also discuss how our work makes Mansour's conjecture for DNFs a natural next step toward resolution of the FEI conjecture. (2) We next consider the weaker Fourier Min-entropy-influence (FMEI) conjecture posed by O'Donnell and others [50, 53], which asks if H∞ (f2) ≤ C · Inf(f), where H ∞ (f2) is the min-entropy of the Fourier distribution. We show H∞(f2) ≤ 2·Cmin⊕(f), where Cmin⊕(f) is the minimum parity-certificate complexity of f. We also show that for all ϵ≥0, we have H∞(f2) ≤ 2 log (||f||1,ϵ/(1-ϵ)), where ||f||1,ϵ is the approximate spectral norm of f.

Given a Boolean function f : {−1, 1}n → {−1, 1}, define the Fourier distribution to be the distribution on subsets of [n], where each S ⊆ [n] is sampled with probability fb(S)2. The Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai [24] seeks to relate two fundamental measures associated with the Fourier distribution: does there exist a universal constant C > 0 such that H(f-2) ≤ C · Inf(f), where H(f-2) is the Shannon entropy of the Fourier distribution of f and Inf(f) is the total influence of f? In this paper we present three new contributions towards the FEI conjecture: (i) Our first contribution shows that H(f-2) ≤ 2 · aUC (f), where aUC (f) is the average unambiguous parity-certificate complexity of f. This improves upon several bounds shown by Chakraborty et al. [16]. We further improve this bound for unambiguous DNFs. (ii) We next consider the weaker Fourier Min-entropy-Influence (FMEI) conjecture posed by O'Donnell and others [43, 40] which asks if H∞(f-2) ≤ C · Inf(f), where H∞(f-2) is the min-entropy of the Fourier distribution. We show H∞(f-2) ≤ 2 · C min(f), where C min(f) is the minimum parity certificate complexity of f. We also show that for all ε ≥ 0, we have H∞(f-2) ≤ 2 log(kfbk1,ε/(1 − ε)), where kfbk1,ε is the approximate spectral norm of f. As a corollary, we verify the FMEI conjecture for the class of read-k DNFs (for constant k). (iii) Our third contribution is to better understand implications of the FEI conjecture for the structure of polynomials that 1/3-approximate a Boolean function on the Boolean cube. We pose a conjecture: no flat polynomial (whose non-zero Fourier coefficients have the same magnitude) of degree d and sparsity 2ω(d) can 1/3-approximate a Boolean function. This conjecture is known to be true assuming FEI and we prove the conjecture unconditionally (i.e., without assuming the FEI conjecture) for a class of polynomials. We discuss an intriguing connection between our conjecture and the constant for the Bohne

We study how efficiently a k-element set S ? [n] can be learned from a uniform superposition |Si of its elements. One can think of |Si = Pi?S |ii/p|S| as the quantum version of a uniformly random sample over S, as in the classical analysis of the “coupon collector problem.” We show that if k is close to n, then we can learn S using asymptotically fewer quantum samples than random samples. In particular, if there are n - k = O(1) missing elements then O(k) copies of |Si suffice, in contrast to the T(k log k) random samples needed by a classical coupon collector. On the other hand, if n - k = ?(k), then ?(k log k) quantum samples are necessary. More generally, we give tight bounds on the number of quantum samples needed for every k and n, and we give efficient quantum learning algorithms. We also give tight bounds in the model where we can additionally reflect through |Si. Finally, we relate coupon collection to a known example separating proper and improper PAC learning that turns out to show no separation in the quantum case.

We study how efficiently a k-element set S ? [n] can be learned from a uniform superposition |Si of its elements. One can think of |Si = Pi?S |ii/p|S| as the quantum version of a uniformly random sample over S, as in the classical analysis of the “coupon collector problem.” We show that if k is close to n, then we can learn S using asymptotically fewer quantum samples than random samples. In particular, if there are n - k = O(1) missing elements then O(k) copies of |Si suffice, in contrast to the T(k log k) random samples needed by a classical coupon collector. On the other hand, if n - k = ?(k), then ?(k log k) quantum samples are necessary. More generally, we give tight bounds on the number of quantum samples needed for every k and n, and we give efficient quantum learning algorithms. We also give tight bounds in the model where we can additionally reflect through |Si. Finally, we relate coupon collection to a known example separating proper and improper PAC learning that turns out to show no separation in the quantum case.

The International Telecommunication Union (ITU) is currently deliberating on the finalization of candidate radio interface technologies (RITs) for IMT-2020 (International Mobile Telecommunications) suitability. The candidate technologies are currently being evaluated and after a couple of ITU-Radiocommunication sector (ITU-R) working party (WP) meetings, they will become official. Although, products based on the candidate technology from 3GPP (5G new radio (NR)) is already commercial in several operator networks, the ITU is yet to officially declare it as IMT-2020 qualified. Along with evaluation of the 3GPP 5G NR specifications, our group has evaluated many other proponent technologies. 3GPP entire specifications were examined and evaluated through simulation using Matlab and using own developed simulator which is based on the Go-language. The simulator can evaluate complete 5G NR performance using the IMT-2020 evaluation framework. In this work, we are presenting latency parameters which has shown some minor differences from the 3GPP report. Especially, for time division duplexing (TDD) mode of operation, the differences are observed. It might be possible that the differences are due to assumptions made outside the scope of the evaluation. However, we considered the worst case parameter. Although, the report is submitted to ITU but it is also important for the research community to understand why the differences and what were the assumptions in scenario for which differences are observed., Comment: Accepted in 2020 IEEE 3rd 5G World Forum (WF-5G) Conference

We consider a generic framework of optimization algorithms based on gradient descent. We develop a quantum algorithm that computes the gradient of a multi-variate realvalued function f : ℝd → ℝ by evaluating it at only a logarithmic number of times in superposition. Our algorithm is an improved version of Jordan's gradient computation algorithm [28], providing an approximation of the gradient ▽f with quadratically better dependence on the evaluation accuracy of f, for an important class of smooth functions. Furthermore, we show that objective functions arising from variational quantum circuits usually satisfy the necessary smoothness conditions, hence our algorithm provides a quadratic improvement in the complexity of computing their gradient. We also show that in a continuous phase-query model, our gradient computation algorithm has optimal query complexity up to poly-logarithmic factors, for a particular class of smooth functions. Moreover, we show that for low-degree multivariate polynomials our algorithm can provide exponential speedups compared to Jordan's algorithm in terms of the dimension d.One of the technical challenges in applying our gradient computation procedure for quantum optimization problems is the need to convert between a probability oracle (which is common in quantum optimization procedures) and a phase oracle (which is common in quantum algorithms) of the objective function f. We provide efficient subroutines to perform this delicate interconversion between the two types of oracles incurring only a logarithmic overhead, which might be of independent interest. Finally, using these tools we improve the runtime of prior approaches for training quantum auto-encoders, variational quantum eigensolvers (VQE), and quantum approximate optimization algorithms (QAOA).

We prove a characterization of t-query quantum algorithms in terms of the unit ball of a space of degree-(2t) polynomials. Based on this, we obtain a refined notion of approximate polynomial degree that equals the quantum query complexity, answering a question of Aaronson et al. ["Polynomials, Quantum Query Complexity, and Grothendieck's Inequality," in Proceedings of the 31st Conference on Computational Complexity, CCC 2016, Schloss Dagstuh, 2016, pp. 25:1--25:19]. Our proof is based on a fundamental result of Christensen and Sinclair [J. Funct. Anal., 72 (1987), pp. 151--181] that generalizes the well-known Stinespring representation for quantum channels to multilinear forms. Using our characterization, we show that many polynomials of degree four are far from those coming from two-query quantum algorithms. We also give a simple and short proof of one of the results of Aaronson et al. showing an equivalence between one-query quantum algorithms and bounded quadratic polynomials.

We prove a characterization of t-query quantum algorithms in terms of the unit ball of a space of degree-(2t) polynomials. Based on this, we obtain a refined notion of approximate polynomial degree that equals the quantum query complexity, answering a question of Aaronson et al. ["Polynomials, Quantum Query Complexity, and Grothendieck's Inequality," in Proceedings of the 31st Conference on Computational Complexity, CCC 2016, Schloss Dagstuh, 2016, pp. 25:1--25:19]. Our proof is based on a fundamental result of Christensen and Sinclair [J. Funct. Anal., 72 (1987), pp. 151--181] that generalizes the well-known Stinespring representation for quantum channels to multilinear forms. Using our characterization, we show that many polynomials of degree four are far from those coming from two-query quantum algorithms. We also give a simple and short proof of one of the results of Aaronson et al. showing an equivalence between one-query quantum algorithms and bounded quadratic polynomials.

In robotics, mechanized and computer simulation for accurate and fast crash detection between general geometric models is a fundamental problem. The explanation of this problem will gravely improve driver safety and traffic efficiency, vehicular ad hoc networks (VANETs) have been employed in many scenarios to provide road safety and for convenient travel of the people. They offer self-organizing decentralized environments to disseminate traffic data, vehicle information and hazardous events. In order to avoid accidents during roadway travels, which are a major burden to the society, the data, such as traffic data, vehicle data and the road condition, play a critical role. VANET is employed for disseminating the data. Still the scalability issues occur when the communication happens under high-traffic regime where the vehicle density is high. The data redundancy and packet collisions may be high which cause broadcast storm problems. Here the traffic regime in the current state is obtained from the speed of the vehicle. Thus the data reduction is obtained. In order to suppress the redundant broadcast D-SAC data, dissemination protocol is presented in this paper. Here the data are classified according to its criticality and the probability is determined. The performance of the D-SAC protocol is verified through conventional methods with simulation.

In learning theory, the VC dimension of a concept class C is the most common way to measure its “richness.” A fundamental result says that the number of examples needed to learn an unknown target concept c∈C under an unknown distribution D, is tightly determined by the VC dimension d of the concept class C. Specifically, in the PAC model Θ(dϵ+log(1/δ)ϵ) examples are necessary and sufficient for a learner to output, with probability 1−δ, a hypothesis h that is ϵ-close to the target concept c (measured under D). In the related agnostic model, where the samples need not come from a c∈C, we know that Θ(dϵ2+log(1/δ)ϵ2) examples are necessary and sufficient to output an hypothesis h∈C whose error is at most ϵ worse than the error of the best concept in C. Here we analyze quantum sample complexity, where each example is a coherent quantum state. This model was introduced by Bshouty and Jackson (1999), who showed that quantum examples are more powerful than classical examples in some fixed-distribution settings. However, Atıcı and Servedio (2005), improved by Zhang (2010), showed that in the PAC setting (where the learner has to succeed for every distribution), quantum examples cannot be much more powerful: the required number of quantum examples is Ω(d1−ηϵ+d+log(1/δ)ϵ) for arbitrarily small constant η>0. Our main result is that quantum and classical sample complexity are in fact equal up to constant factors in both the PAC and agnostic models. We give two proof approaches. The first is a fairly simple information-theoretic argument that yields the above two classical bounds and yields the same bounds for quantum sample complexity up to a log(d/ϵ) factor. We then give a second approach that avoids the log-factor loss, based on analyzing the behavior of the “Pretty Good Measurement” on the quantum state-identification problems that correspond to learning. This shows classical and quantum sample complexity are equal up to constant factors for every concept class C.

We consider the problem of a particular kind of quantum correlation that arises in some two-party games. In these games, one player is presented with a question they must answer, yielding an outcome of either “win” or “lose”. Molina and Watrous [30] studied such a game that exhibited a perfect form of hedging, where the risk of losing a first game can completely o set the corresponding risk for a second game. This is a non-classical quantum phenomenon, and establishes the impossibility of performing strong error-reduction for quantum interactive proof systems by parallel repetition, unlike for classical interactive proof systems. We take a step in this article towards a better understanding of the hedging phenomenon by giving a complete characterization of when perfect hedging is possible for a natural generalization of the game in [30]. Exploring in a di erent direction the subject of quantum hedging, and motivated by implementation concerns regarding loss-tolerance, we also consider a variation of the protocol where the player who receives the question can choose to restart the game rather than return an answer. We show that in this setting there is no possible hedging for any game played with state spaces corresponding to finite-dimensional complex Euclidean spaces.

We prove a characterization of quantum query algorithms in terms of polynomials satisfying a certain (completely bounded) norm constraint. Based on this, we obtain a refined notion of approximate polynomial degree that equals the quantum query complexity, answering a question of Aaronson et al. (CCC’16). Using this characterization, we show that many polynomials of degree at least 4 are far from those coming from quantum query algorithms. Our proof is based on a fundamental result of Christensen and Sinclair (J. Funct. Anal., 1987) that generalizes the well-known Stinespring representation for quantum channels to multilinear forms. We also give a simple and short proof of one of the results of Aaronson et al. showing an equivalence between one-query quantum algorithms and bounded quadratic polynomials.

We prove a characterization of quantum query algorithms in terms of polynomials satisfying a certain (completely bounded) norm constraint. Based on this, we obtain a refined notion of approximate polynomial degree that equals the quantum query complexity, answering a question of Aaronson et al. (CCC’16). Using this characterization, we show that many polynomials of degree at least 4 are far from those coming from quantum query algorithms. Our proof is based on a fundamental result of Christensen and Sinclair (J. Funct. Anal., 1987) that generalizes the well-known Stinespring representation for quantum channels to multilinear forms. We also give a simple and short proof of one of the results of Aaronson et al. showing an equivalence between one-query quantum algorithms and bounded quadratic polynomials.

We consider the problem of a particular kind of quantum correlation that arises in some two-party games. In these games, one player is presented with a question they must answer, yielding an outcome of either “win” or “lose”. Molina and Watrous [30] studied such a game that exhibited a perfect form of hedging, where the risk of losing a first game can completely o set the corresponding risk for a second game. This is a non-classical quantum phenomenon, and establishes the impossibility of performing strong error-reduction for quantum interactive proof systems by parallel repetition, unlike for classical interactive proof systems. We take a step in this article towards a better understanding of the hedging phenomenon by giving a complete characterization of when perfect hedging is possible for a natural generalization of the game in [30]. Exploring in a di erent direction the subject of quantum hedging, and motivated by implementation concerns regarding loss-tolerance, we also consider a variation of the protocol where the player who receives the question can choose to restart the game rather than return an answer. We show that in this setting there is no possible hedging for any game played with state spaces corresponding to finite-dimensional complex Euclidean spaces.

We prove a characterization of t-query quantum algorithms in terms of the unit ball of a space of degree-2t polynomials. Based on this, we obtain a refined notion of approximate polynomial degree that equals the quantum query complexity, answering a question of Aaronson et al. (CCC’16). Our proof is based on a fundamental result of Christensen and Sinclair (J. Funct. Anal., 1987) that generalizes the well-known Stinespring representation for quantum channels to multilinear forms. Using our characterization, we show that many polynomials of degree four are far from those coming from two-query quantum algorithms. We also give a simple and short proof of one of the results of Aaronson et al. showing an equivalence between one-query quantum algorithms and bounded quadratic polynomials.

We prove a characterization of t-query quantum algorithms in terms of the unit ball of a space of degree-2t polynomials. Based on this, we obtain a refined notion of approximate polynomial degree that equals the quantum query complexity, answering a question of Aaronson et al. (CCC’16). Our proof is based on a fundamental result of Christensen and Sinclair (J. Funct. Anal., 1987) that generalizes the well-known Stinespring representation for quantum channels to multilinear forms. Using our characterization, we show that many polynomials of degree four are far from those coming from two-query quantum algorithms. We also give a simple and short proof of one of the results of Aaronson et al. showing an equivalence between one-query quantum algorithms and bounded quadratic polynomials.

This paper surveys quantum learning theory: the theoretical aspects of machine learning using quantum computers. We describe the main results known for three models of learn- ing: exact learning from membership queries, and Probably Approximately Correct (PAC) and agnostic learning from classical or quantum examples.

The absolute separability problem asks for a characterization of the quantum states ρ∈Mm⊗Mn with the property that UρU† is separable for all unitary matrices U. We investigate whether or not it is the case that ρ is absolutely separable if and only if UρU† has positive partial transpose for all unitary matrices U. In particular, we develop an easy-to-use method for showing that an entanglement witness or positive map is unable to detect entanglement in any such state, and we apply our method to many well-known separability criteria, including the range criterion, the realignment criterion, the Choi map and its generalizations, and the Breuer-Hall map. We also show that these two properties coincide for the family of isotropic states, and several eigenvalue results for entanglement witnesses are proved along the way that are of independent interest.

This paper attempts to trace the development of an unusual and skilled class of alloys, of binary high-tin bronze (i.e. alloys of only copper with a higher percentage of tin), which are found from surprisingly early contexts from Indian antiquity. In particular, the deliberate use was made of binary beta bronze with around 22–24 % tin, specifically exploiting the properties of higher hot-forgability of bronze of this composition due to the formation of the high temperature beta intermetallic compound phase of 22.9 % tin. Quenching resulted in the retention of the beta phase, yielding a musical alloy with golden lustre and improved tensile strength as compared to the as-cast state. Examples of hot forged and quenched high-tin beta bronzes studied by the author from the South Indian Iron Age and megalithic cultures from Tamil Nadu and Maharashtra and Gandharan Grave Culture of Taxila are summarised here ranking amongst the earliest and most finely wrought such finds. There are technological and morphological similarities to surviving high-tin bronze crafts practices documented by the author in Kerala since 1990. Since the 1990’s she has also documented the making of high-tin delta bronze mirrors at Aranmula with a composition closer to the pure delta phase of 32.6 % tin, which instead exploited the specular properties this alloy while managing its brittleness. Although it is difficult to speculate about origins, a long standing practice of using binary tin–bronzes (i.e. only copper–tin alloys) can be detected going back to Harappan bronzes which also seem to be predominantly binary bronzes with not much lead added to them. Though most of these seem to be low-tin bronze, the presence of a couple with higher tin of about 20 % is also notable in terms of the above discussion.

Consumers make many decisions in everyday life involving finances, food, and health. It is known from behavioral economics research that people are often driven by short-term gratification, that is, people tend to choose the immediate, albeit smaller reward. But choosing the delayed reward, that is, delaying the gratification, can actually be beneficial. How can we motivate consumers to resist the "now" and invest in their future, leading to sustainable or healthy habits? We review recent developments from behavioral and neuroimaging studies that are relevant for understanding consumer decisions. Further, we present results from our field research that examined whether we can increase the perceived value of a (delayed) environmental benefit using tailored communication, that is, change the way it is framed. More specifically, we investigated whether we can boost the value of an abstract, long-term "green" claim of a product by expressing it as a concrete, short-term benefit. This is a new application area for behavioral economics.

Consumers make many decisions in everyday life involving finances, food, and health. It is known from behavioral economics research that people are often driven by short-term gratification, that is, people tend to choose the immediate, albeit smaller reward. But choosing the delayed reward, that is, delaying the gratification, can actually be beneficial. How can we motivate consumers to resist the "now" and invest in their future, leading to sustainable or healthy habits? We review recent developments from behavioral and neuroimaging studies that are relevant for understanding consumer decisions. Further, we present results from our field research that examined whether we can increase the perceived value of a (delayed) environmental benefit using tailored communication, that is, change the way it is framed. More specifically, we investigated whether we can boost the value of an abstract, long-term "green" claim of a product by expressing it as a concrete, short-term benefit. This is a new application area for behavioral economics.

This paper explores the origins of the Nataraja cult in the state of Tamil Nadu in southern India. The trajectory of the dancing Shiva is traced: from the processional worship of metal icons outside the sanctum to the cultic elevation of the Nataraja bronze into the sanctum at Chidambaram. Archaeometallurgical, iconographic and literary evidence discussed shows that the Nataraja bronze, depicting Shiva's anandatandava or 'dance of bliss', was a Pallava innovation (seventh to mid-ninth century), rather than tenth-century Chola as widely believed. That this formulation was informed of 'cosmic' or metaphysical connotations is also argued on the basis of the testimony of the hymns of Tamil saints.