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Newton polygons and the constant associated with the Prouhet–Tarry–Escott problem
No full textIn a 2017 article Filaseta and Markovich obtained new information on the lower bounds of 2-adic valuation of certain constants Cn associated with the Prouhet–Tarry–Escott (PTE) problem for the cases n = 8 and n = 9 by using the classical theory of Newton polygons, and also pointed out that it would be of interest to obtain improved lower bounds in the cases when 10 ≤ n ≤ 12. In the present article, we obtain new 2-adic information on the lower bounds of Cn for the cases n = 10 and n = 12
On motives of parabolic Higgs bundles and parabolic connections
No full textLet X be a compact Riemann surface of genus g ≥ 2 and let D ⊂ X be a fixed finite subset. We considered the moduli spaces of parabolic Higgs bundles and of parabolic connections over X with the parabolic structure over D. For generic weights, we showed that these two moduli spaces have equal Grothendieck motivic classes and their E-polynomials are the same. We also show that the Voevodsky and Chow motives of these two moduli spaces are also equal. We showed that the Grothendieck motivic classes and the E-polynomials of parabolic Higgs moduli and of parabolic Hodge moduli are closely related. Finally, we considered the moduli spaces with fixed determinants and showed that the above results also hold for the fixed determinant case
On the monogenity and Galois group of certain classes of polynomials
No full textWe say a monic polynomial g(x) ϵ ℤ[x] of degree n is monogenic if g(x) is irreducible over ℚ and {1, θ, ..., θn-1} is a basis for the ring ℤK of integers of number field K = ℚ(θ), where θ is a root of g(x). Let f(x)=xn+cσi=1n(ax)n-iϵZ[x]andF(x)=xn+cσi=1nai-1xn-iϵZ[x] be irreducible polynomials having degree n ≥ 3. In this paper, we provide necessary and sufficient conditions involving only a, c, n for the polynomials f(x) and F(x) to be monogenic. As an application, we also provide a class of polynomials having a non square-free discriminant and Galois group Sn, the symmetric group on n letters
PAIRS OF PROJECTIONS AND COMMUTING ISOMETRIES
No full textThe non-zero part of compact defect operators of Berger–Coburn–Lebow pairs (BCL pairs) of isometries are diagonal operators of the form (Formula presented.). We discuss the question of constructing an irreducible BCL pair from a diagonal operator of the above type. The answer is sometimes yes, sometimes no. This also partially addresses the question of He, Qin, and Yang. Our explicit constructions of BCL pairs yield concrete examples of pairs of commuting isometries
Patterns of Income Growth in an Eastern Uttar Pradesh Village, 2006–23
No full textDrawing on longitudinal data from a Dalit-majority village in eastern Uttar Pradesh, change in household incomes is quantified and sources of income growth are identified. While household income has more than doubled on average, the contribution of the agriculture sector has shrunk. Crop production is not a dynamic sector of the rural economy. Crop cultivation is largely undertaken by Scheduled Caste and Other Backward Class households on small plots of leased-in land in order to acquire foodgrain for home consumption. The rise in income across all caste groups is driven by earnings from outside the village, from non-agricultural wage employment, salaries, and from remittances of migrants
Prediction of number of rainy days over different monsoon regions in India
No full textIndian monsoon rainfall is an extremely important affair in socio-economic well-being of the country. Recently, climatic changes have introduced significant uncertainty and irregularity in the Indian Summer Monsoon (ISM) cycle. Such unpredictability has been evidenced in all the elements of monsoon, e.g., onset, intensity, and regularity. Rain amount has been the eye of all predictions for obvious reasons. But, in the recent scenario, other components of the monsoon process have also become extremely crucial to be forecasted. The current work has regionalized Indian subcontinent using a rough-fuzzy c-means algorithm and proposed five updated monsoon zones. Four deep learning networks using Bi-LSTM architecture for four of the resulting regions have been developed for prediction of the number of rainy days one month ahead of time with a spatial resolution of 10 × 10. The model accuracy is found to be 67.48%, 92.80%, 70.72% and 88.75%. A stronger model (architecture) has then been developed by ensembling the said four models whose validity is tested on the fifth region (monsoon zone) with an accuracy of 79.80%. The spatial matching between the actual and predicted number of monsoon rainy days has been found to be good. The certainty values of prediction are determined by rough set-based decision rules. Such studies on prediction of ISM elements using deep learning methods can be of compelling interest in tropical meteorology in near future
Quantized Redshift and its Significance for Recent Observation
No full textWith the recent observational evidence in extragalactic astronomy, the interpretation of the nature of quasar redshift continues to be a research interest. Very high redshifts are being detected for extragalactic objects that are presumably very distant and young while also exhibiting properties that are characteristic of a more mature galaxy such as ours. According to Halton Arp and Geoffrey Burbidge, redshift disparities consist of an intrinsic component and are related to an evolutionary process. Karlsson observed redshift periodicity at integer multiples of 0.089 in log scale and Burbidge observed redshift periodicity at integer multiples of 0.061 in linear scale. Since Singular Value Decomposition based periodicity estimation is known to be superior for noisy data sets, especially when the data contain multiple harmonics and overtones, mainly irregular in nature, we have chosen it to be our primary tool for analysis of the quasar-galaxy pair redshift data. We have observed a fundamental periodicity of 0.051 with a confidence interval of 95% in linear scale with the site-available Sloan Digital Sky Survey Data Release 7 (SDSS DR7) quasar-galaxy pair data set. We have independently generated quasar-galaxy pair data sets from both 2dF and SDSS and found fundamental periodicities of 0.077 and 0.089, respectively, in log scale with a confidence interval of 95%
QUANTUM-SAFE IDENTITY-BASED BROADCAST ENCRYPTION WITH PROVABLE SECURITY FROM MULTIVARIATE CRYPTOGRAPHY
No full textIdentity-Based Broadcast Encryption (IBBE) is a novel concept that can efficiently and securely transmit confidential content to a group of authorized users without the traditional Public-Key Infrastructure (PKI). After carefully exploring these areas, we have observed that none of the existing works have adopted the quantum-attack resistant cryptographic machinery Multivariate Public-Key Cryptography (MPKC) with provable security. We are the first to design a quantum-safe IBBE that solely relies on the MPKC framework. Our proposed protocol has achieved O(n)-size communication bandwidth and n3 · O( max{N, δ4})-size overhead storage without any security breach. Here, n is the number of variables for each multivariate polynomial, N represents the total number of system users, and δ denotes a positive fixed-length. More positively, our design has achieved the adaptive INDistinguishable Chosen-Ciphertext Attack (IND-CCA) security in the Random Oracle Model (ROM) under the hardness of standard Multivariate Quadratic (MQ) problem. We emphasize that our system can also be immune against collusion attacks where several users come together to create an illicit decryption box
Rank-preserving multidimensional mechanisms: An equivalence between identical-object and heterogeneous-object models
No full textWe show that the mechanism-design problem for a monopolist selling multiple, heterogeneous objects to a buyer with ex ante symmetric and additive values is equivalent to the mechanism-design problem for a monopolist selling identical objects to a buyer with decreasing marginal values. We derive three new results for the identical-objects model: (i) a new condition for revenue monotonicity of stochastic mechanisms, (ii) a sufficient condition on priors, such that prices in optimal deterministic mechanism are not increasing, and (iii) a simplification of incentive constraints for deterministic mechanisms. We use the equivalence to establish corresponding results in the heterogeneous-objects model
Studies in Boolean Function Analysis
No full textBoolean functions are important in both theoretical computer science and cryptography. Over the past few decades, significant advancements have been made in this area. The Walsh transform, a variant of the Fourier transform applied to the function , where is a Boolean function, is a vital tool for studying Boolean functions in both fields. The Walsh/Fourier coefficients of a Boolean function offer insights into its properties, and many concepts in both areas can be interpreted in terms of these coefficients. Hence, it is reasonable to assume that analyzing Boolean functions from both perspectives is interconnected, and results from one area can be applied to the other to obtain new outcomes or improve established proof techniques. However, surprisingly, the theory of Boolean functions developed almost parallel in these two fields. The objective of this thesis is to investigate and establish connections between various concepts of Boolean functions used in theoretical computer science and cryptography. Through our research, we have solved several existing problems, introduced new ones, and obtained results related to these problems. Furthermore, we have developed new concepts in Boolean function analysis and their applications that are pertinent in both theoretical computer science and cryptography. In the course of our research, we have shown a general counterexample to the ``Majority is Least Stable\u27\u27 conjecture, which was previously shown only for . We have also proposed the first-ever lower bound for the ``Fourier min-entropy/influence conjecture\u27\u27 in this thesis. Additionally, we utilized programming techniques to explore and unveil some intriguing counting results associated with unate functions and Dedekind numbers