The Experts below are selected from a list of 27 Experts worldwide ranked by ideXlab platform
Harpreet Singh Bedi - One of the best experts on this subject based on the ideXlab platform.
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formal schemes of Rational Degree
Indian Journal of Pure & Applied Mathematics, 2021Co-Authors: Harpreet Singh BediAbstract:Formal schemes in algebraic geometry consist of power series with integer Degree, this idea can be naturally carried over to power series with Rational Degree. In this paper formal schemes with Rational Degree are constructed. These schemes are non noetherian and thus require slight modification of the standard approach. The first part of the paper constructs non noetherian continuous valuation rings from discrete valuation rings, and these rings are reffered to as ‘eka’ (one in Hindi) rings. These new rings are designed to carry the properties of discrete valuation to continuous valuation faithfully. The second part of the paper constructs non noetherian formal schemes with Rational Degree and shows their admissibility. The corresponding flatness and coherence is proved. Finally line bundles of Rational Degree are constructed and their Cech cohomology computed.
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formal schemes of Rational Degree
arXiv: Algebraic Geometry, 2019Co-Authors: Harpreet Singh BediAbstract:Non notherian Formal schemes of perfectoid type (for example $\mathbb{Z}_p[p^{1/p^\infty}]\langle X^{1/p^\infty} \rangle$ along with its multivariate version) with Rational Degree are constructed and are shown to be admissible. These formal schemes are a Rational Degree avatar of Tate affinoid algebras and come equipped with non Notherian rings. The corresponding notion of topologically finite presentation are defined and Gabber's Lemma, admissible blow ups (Raynaud's approach) are shown to hold under certain assumptions. A new notion of rings called eka$^d$ are introduced, which recover most examples of perfectoid affinoid algebras, without resorting to Huber's construction, Witt vectors or Frobenius. This version fixes some errors in the last version
Chunlong Xiong - One of the best experts on this subject based on the ideXlab platform.
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a methodology for evaluating micro surfacing treatment on asphalt pavement based on grey system models and grey Rational Degree theory
Construction and Building Materials, 2017Co-Authors: Xiaoning Zhang, Chunlong XiongAbstract:Abstract This paper presents and demonstrates a methodology for evaluating a micro-surfacing treatment on asphalt pavement based on the grey system model and grey relational Degree theory. Firstly, over 5,375,000 data points from the Guangdong Province are collected and processed using the Pauta criterion and short-term trends of selected performance indicators including pavement surface condition index (PCI), riding quality index (RQI), rut depth index (RDI), and skid resistance index (SRI) are analysed. Grey models of different types of indicators are then established for predicting long-term performance. A new PQI model was developed using the optimum predicted long-term performance with the application of the grey Rational Degree theory. Finally, the service life of the micro-surfacing treatment was determined using the new PQI model. Results demonstrate a representative service life over 4.5 years for the micro-surfacing treatment with a critical PQI value of 80. The proposed methodology validated the micro-surfacing treatment as a valuable assessment system that can be successfully applied with or without other pavement maintenance treatments.
Xiaoning Zhang - One of the best experts on this subject based on the ideXlab platform.
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a methodology for evaluating micro surfacing treatment on asphalt pavement based on grey system models and grey Rational Degree theory
Construction and Building Materials, 2017Co-Authors: Xiaoning Zhang, Chunlong XiongAbstract:Abstract This paper presents and demonstrates a methodology for evaluating a micro-surfacing treatment on asphalt pavement based on the grey system model and grey relational Degree theory. Firstly, over 5,375,000 data points from the Guangdong Province are collected and processed using the Pauta criterion and short-term trends of selected performance indicators including pavement surface condition index (PCI), riding quality index (RQI), rut depth index (RDI), and skid resistance index (SRI) are analysed. Grey models of different types of indicators are then established for predicting long-term performance. A new PQI model was developed using the optimum predicted long-term performance with the application of the grey Rational Degree theory. Finally, the service life of the micro-surfacing treatment was determined using the new PQI model. Results demonstrate a representative service life over 4.5 years for the micro-surfacing treatment with a critical PQI value of 80. The proposed methodology validated the micro-surfacing treatment as a valuable assessment system that can be successfully applied with or without other pavement maintenance treatments.
Bedi, Harpreet Singh - One of the best experts on this subject based on the ideXlab platform.
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Rational Degree Algebraic Geometry
2020Co-Authors: Bedi, Harpreet SinghAbstract:Elementary Algebraic Geometry can be described as study of zeros of polynomials with integer Degrees, this idea can be naturally carried over to `polynomials' with Rational Degree. This paper explores affine varieties, tangent space and projective space for such polynomials and notes the differences and similarities between Rational and integer Degrees. The line bundles $\mathcal{O}(n),n\in\mathbb{Q}$ are also constructed and their \v{C}ech cohomology computed.Comment: Comments welcom
Lovett S - One of the best experts on this subject based on the ideXlab platform.
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Algebraic attacks against random local functions and their countermeasures
eScholarship University of California, 2018Co-Authors: Applebaum B, Lovett SAbstract:© 2018 Society for Industrial and Applied Mathematics. Suppose that you have n truly random bits x = (x1, . . ., xn) and you wish to use them to generate m n pseudorandom bits y = (y1, . . ., ym) using a local mapping, i.e., each yi should depend on at most d = O(1) bits of x. In the polynomial regime of m = ns, s > 1, the only known solution, originating from [Goldreich, Electronic Colloquium on Computational Complexity (ECCC), 2000], is based on random local functions: Compute yi by applying some fixed (public) d-ary predicate P to a random (public) tuple of distinct input indices (xi1 , . . ., xid). Our goal in this paper is to understand, for any value of s, how the pseudorandomness of the resulting sequence depends on the choice of the underlying predicate. We derive the following results: (1) We show that pseudorandomness against F2-linear adversaries (i.e., the distribution y has small bias) is achieved if the predicate is (a) k = Ω(s)-resilient, i.e., uncorrelated with any k-subset of its inputs, and (b) has algebraic Degree of Ω(s) even after fixing Ω(s) of its inputs. We also show that these requirements are necessary, and so they form a tight characterization (up to constants) of security against linear attacks. Our positive result shows that a d-local small-biased generator can have output length of nΩ(d), answering an open question of Mossel, Shpilka, and Trevisan [Proceedings of FOCS, 2003]. Our negative result shows that a candidate for a pseudorandom generator proposed by Applebaum [Comput. Complexity, 25 (2016), pp. 667–722] and by O’Donnell and Witmer [Proceedings of CCC 2014] is insecure. We use similar techniques to refute a conjecture of Feldman, Perkins, and Vempala [Proceedings of STOC 2015] regarding the hardness of planted constraint satisfaction problems. (2) Motivated by the cryptanalysis literature, we consider security against algebraic attacks. We provide the first theoretical treatment of such attacks by formalizing a general notion of algebraic inversion and distinguishing attacks based on the polynomial calculus proof system. We show that algebraic attacks succeed if and only if the predicate P has Rational Degree e = Θ(s), where the Rational Degree of a predicate P is the smallest integer e for which there exist Degree e polynomials Q, R, not both zero, such that PQ = R. As a corollary, we obtain the first example of a predicate P for which the generated sequence y passes all linear tests but fails to pass some polynomial-time computable test, answering an open question posed by Applebaum [Comput. Complexity, 25 (2016), pp. 667–722]
