Quadratic Number Research Articles

Mobius Group Generated by Two Elements of Order 2, 4, and Reduced Quadratic Irrational Numbers

The construction of circuits for the evolution of orbits and reduced quadratic irrational numbers under the action of Mobius groups have many applications like in construction of substitution box (s-box), strong-substitution box (s.s-box), image processing, data encryption, in interest for security experts, and other fields of sciences. In this paper, we investigate the behavior of reduced quadratic irrational numbers (RQINs) in the coset diagrams of the set Q ′ ′ m = η / s : η ∈ Q ∗ m , s = 1 , 2 under the action of group H = < x ′ , y ′ : x ′ 2 = y ′ 4 = 1 > , where m is square free integer and Q ∗ m = a ′ + m / c ′ , a ′ , a ′ 2 − m / c ′ c ′ = 1 , c ′ ≠ 0 . We discuss the type and reduced cardinality of the orbit Q ′ ′ p . By using the notion of congruence, we give the general form of reduced numbers (RNs) in particular orbits under certain conditions on prime p . Further, we classify that for a reduced number r whether − r , r ¯ , − r ¯ lying in orbit or not. AMS Mathematics subject classification (2010): 05C25, 20G401.

Speeding Up Fermat’s Factoring Method using Precomputation

The security of many public-key cryptosystems and protocols relies on the difficulty of factoring a large positive integer n into prime factors. The Fermat factoring method is a core of some modern and important factorization methods, such as the quadratic sieve and number field sieve methods. It factors a composite integer n=pq in polynomial time if the difference between the prime factors is equal to ∆=p-q≤n^(0.25) , where p>q. The execution time of the Fermat factoring method increases rapidly as ∆ increases. One of the improvements to the Fermat factoring method is based on studying the possible values of (n mod 20). In this paper, we introduce an efficient algorithm to factorize a large integer based on the possible values of (n mod 20) and a precomputation strategy. The experimental results, on different sizes of n and ∆, demonstrate that our proposed algorithm is faster than the previous improvements of the Fermat factoring method by at least 48%.

Anharmonicity-induced excited-state quantum phase transition in the symmetric phase of the two-dimensional limit of the vibron model

In most cases, excited state quantum phase transitions can be associated with the existence of critical points (local extrema or saddle points) in a system's classical limit energy functional. However, an excited-state quantum phase transition might also stem from the lowering of the asymptotic energy of the corresponding energy functional. One such example occurs in the 2D limit of the vibron model, once an anharmonic term in the form of a quadratic bosonic number operator is added to the Hamiltonian. The study of this case in the broken-symmetry phase was presented in Phys. Rev. A. 81 050101 (2010). In the present work, we delve further into the nature of this excited-state quantum phase transition and we characterize it in the, previously overlooked, symmetric phase of the model making use of quantities such as the effective frequency, the expected value of the quantum number operator, the participation ratio, the density of states, and the quantum fidelity susceptibility. In addition to this, we extend the usage of the quasilinearity parameter, introduced in molecular physics, to characterize the phases in the spectrum of the anharmonic 2D limit of the vibron model and a down-to-earth analysis has been included with the characterization of the critical energies for the linear isomers HCN/HNC.

The monodromy of generalized Kummer varieties and algebraic cycles on their intermediate Jacobians

We compute the subgroup of the monodromy group of a generalized Kummer variety associated to equivalences of derived categories of abelian surfaces. The result was previously announced in arXiv:1201.0031. Mongardi showed that the subgroup constructed here is in fact the whole monodromy group. As an application we prove the Hodge conjecture for the generic abelian fourfold of Weil type with complex multiplication by an arbitrary imaginary quadratic number field K, but only for polarizations with discriminant 1. The latter result is inspired by a recent observation of O'Grady that the third intermediate Jacobians of smooth projective varieties of generalized Kummer deformation type form complete families of abelian fourfolds of Weil type. Finally, we prove the surjectivity of the Abel-Jacobi map from the Chow group of co-dimension two algebraic cycles homologous to zero on every projective irreducible holomorphic symplectic manifold Y of Kummer type onto the third intermediate Jacobian of Y, as predicted by the generalized Hodge Conjecture.

The Density of Fan-Planar Graphs

A topological drawing of a graph is fan-planar if for each edge $e$ the edges crossing $e$ form a star and no endpoint of $e$ is enclosed by $e$ and its crossing edges. A fan-planar graph is a graph admitting such a drawing. Equivalently, this can be formulated by three forbidden patterns, one of which is the configuration where $e$ is crossed by two independent edges and the other two where $e$ is crossed by two incident edges in a way that encloses some endpoint of $e$. A topological drawing is simple if any two edges have at most one point in common.
 Fan-planar graphs are a new member in the ever-growing list of topological graphs defined by forbidden intersection patterns, such as planar graphs and their generalizations, Turán-graphs and Conway's thrackle conjecture. Hence fan-planar graphs fall into an important field in combinatorial geometry with applications in various areas of discrete mathematics.
 As every $1$-planar graph is fan-planar and every fan-planar graph is $3$-quasiplanar, they also fit perfectly in a recent series of works on nearly-planar graphs from the areas of graph drawing and combinatorial embeddings.
 In this paper we show that every fan-planar graph on $n$ vertices has at most $5n-10$ edges, even though a fan-planar drawing may have a quadratic number of crossings. Our bound, which is tight for every $n \geq 20$, indicates how nicely fan-planar graphs fit in the row with planar graphs ($3n-6$ edges) and $1$-planar graphs ($4n-8$ edges). With this, fan-planar graphs form an inclusion-wise largest non-trivial class of topological graphs defined by forbidden patterns, for which the maximum number of edges on $n$ vertices is known exactly.
 We demonstrate that maximum fan-planar graphs carry a rich structure, which makes this class attractive for many algorithms commonly used in graph drawing. Finally, we discuss possible extensions and generalizations of these new concepts.

Chromatic Selmer groups and arithmetic invariants of elliptic curves

Chromatic Selmer groups are modified Selmer groups with local information for supersingular primes p. We sketch their role in establishing the p-primary part of the Birch–Swinnerton-Dyer formula in Sections 2–5, and then study the growth of the Mordell–Weil rank along the ℤ p 2 -extension of a quadratic imaginary number field in which p splits in Section 6.

Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations

Self-similar sets with the open set condition, the linear objects of fractal geometry, have been considered mainly for crystallographic data. Here we introduce new symmetry classes in the plane, based on rotation by irrational angles. Examples without characteristic directions, with strong connectedness and small complexity, were found in a computer-assisted search. They are surprising since the rotations are given by rational matrices, and the proof of the open set condition usually requires integer data. We develop a classification of self-similar sets by symmetry class and algebraic numbers. Examples are given for various quadratic number fields.

Minimizing the Machine Processing Time in a Flow Shop Scheduling Problem under Piecewise Quadratic Fuzzy Numbers

The piecewise quadratic fuzzy number (PQFN) can signify uncertain information that exists in scientific, technological, and engineering fields. Hence, it is a useful tool for describing information in scheduling problems. This study examines structured n‐job flow shop scheduling with fuzzy piecewise quadratic processing times and three machines. Close interval approximation of PQFNs is also offered as one of the most effective approximate intervals. Furthermore, the leasing cost of equipment is minimized with the use of a fuzzy style and an inventive algorithm. To demonstrate how the proposed framework can be used, a numerical illustration is provided.

Comparing the number of ideals in quadratic number fields

<abstract><p>Denote by $ a_{K}(n) $ the number of integral ideals in $ K $ with norm $ n $, where $ K $ is a algebraic number field of degree $ m $ over the rational field $ \mathcal{Q} $. Let $ p $ be a prime number. In this paper, we prove that, for two distinct quadratic number fields $ K_i = \mathcal{Q}(\sqrt{d_i}), \ i = 1, 2 $, the sets both</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \{p\ |\ a_{K_1}(p)< a_{K_2}(p)\} \text{ and } \{p\ |\ a_{K_1}(p^2)< a_{K_2}(p^2)\} $\end{document} </tex-math></disp-formula></p> <p>have analytic density $ 1/4 $, respectively.</p></abstract>

Quaternion Integers Based Higher Length Cyclic Codes and Their Decoding Algorithm

The decoding algorithm for the correction of errors of arbitrary Mannheim weight has discussed for Lattice constellations and codes from quadratic number fields. Following these lines, the decoding algorithms for the correction of errors of length cyclic codes over quaternion integers of Quaternion Mannheim weight one up to two coordinates have considered. In continuation, the case of cyclic codes of lengths and has studied to improve the error correction efficiency. In this study, we present the decoding of cyclic codes of length and length 2 (where is prime integer and is Euler phi function) over Hamilton Quaternion integers of Quaternion Mannheim weight for the correction of errors. Furthermore, the error correction capability and code rate tradeoff of these codes are also discussed. Thus, an increase in the length of the cyclic code is achieved along with its better code rate and an adequate error correction capability.

Greenberg’s conjecture for real quadratic fields and the cyclotomic ℤ₂-extensions

Let A n \mathcal {A}_n be the 2 2 -part of the ideal class group of the n n -th layer of the cyclotomic Z 2 \mathbb {Z}_2 -extension of a real quadratic number field F F . The cardinality of A n \mathcal {A}_n is related to the index of cyclotomic units in the full group of units. We present a method to study the latter index. As an application we show that the sequence of the A n \mathcal {A}_n ’s stabilizes for the real fields F = Q ( f ) F=\mathbb {Q}(\sqrt {f}) for any integer 0 > f > 10000 0>f>10000 . Equivalently Greenberg’s conjecture holds for those fields.

Influence of quadrat characteristics on the evolution of the dispersion effect for fiber–water dispersions

Dispersing fibers in a water dispersion is an important issue for many fiber-based materials that significantly affects the mechanical and many other properties of materials. However, the measurement and assessment of the dispersion effect remains a significant challenge. In this study, we presented an image analysis method based on quadrat analysis from ecology and geography, transforming the issue of the dispersion effect into the statistics of point distribution. Furthermore, we changed the type of sampling and adjusted the shape, size and numbers of each quadrat to investigate its influences on the evaluation results. Our results showed that the area of one quadrat had a more obvious effect on the evaluation results compared to the number of quadrats. In addition, having a quadrat of an optimum shape enlarged the difference in various dispersion effects; the results of a square quadrat exhibited stably in both complete coverage and random sampling. Quadrat analysis realizes good measurement of dispersion states as a result of image processing and offers an assessment of the dispersion effect in a fiber–water dispersion.

Composition of binary quadratic forms over number fields

AbstractIn this article, the standard correspondence between the ideal class group of a quadratic number field and the equivalence classes of binary quadratic forms of given discriminant is generalized to any base number field of narrow class number one. The article contains an explicit description of the correspondence. In the case of totally negative discriminants, equivalent conditions are given for a binary quadratic form to be totally positive definite.

On the Distribution of αp Modulo One in Quadratic Number Fields

Abstract We investigate the distribution of αp modulo one in quadratic number fields 𝕂 with class number one, where p is restricted to prime elements in the ring of integers of 𝕂. Here we improve the relevant exponent 1/4 obtained by the first- and third-named authors for imaginary quadratic number fields [On the distribution of αp modulo one in imaginary quadratic number fields with class number one, J. Théor. Nombres Bordx. 32 (2020), no. 3, 719–760]) and by the first- and second-named authors for real quadratic number fields [Diophantine approximation with prime restriction in real quadratic number fields, Math. Z. (2021)] to 7/22. This generalizes a result of Harman [Diophantine approximation with Gaussian primes, Q. J. Math. 70 (2019), no. 4, 1505–1519] who obtained the same exponent 7/22 for ℚ (i) by extending his method which gave this exponent for ℚ [On the distribution of αp modulo one. II, Proc. London Math. Soc. 72, (1996), no. 3, 241–260]. Our proof is based on an extension of Harman’s sieve method to arbitrary number fields. Moreover, we need an asymptotic evaluation of certain smooth sums over prime ideals appearing in the above-mentioned work by the first- and second-named authors, for which we use analytic properties of Hecke L-functions with Größencharacters.

Determination of all imaginary quadratic fields for which their Hilbert 2-class fields have 2-class groups of rank 2

We determine those imaginary quadratic number fields whose 2-class groups have rank 3 and 4-rank ≤2 and such that the 2-class groups of their Hilbert 2-class fields have rank 2. This result, along with previous work, gives a complete determination of all complex quadratic number fields for which the 2-class groups of their 2-class fields have rank at most 2.

An approach for computing generators of class fields of imaginary quadratic number fields using the Schwarzian derivative

Let N N be one of the 38 38 distinct square-free integers such that the arithmetic group Γ 0 ( N ) + \Gamma _0(N)^+ has genus one. We constructed canonical generators x N x_N and y N y_N for the associated function field (see Jorgenson, L. Smajlović, and H. Then [Exp. Math. 25 (2016), pp. 295–319]). In this article we study the Schwarzian derivative of x N x_N , which we express as a polynomial in y N y_N with coefficients that are rational functions in x N x_N . As a corollary, we prove that for any point e e in the upper half-plane which is fixed by an element of Γ 0 ( N ) + \Gamma _0(N)^+ , one can explicitly evaluate x N ( e ) x_N(e) and y N ( e ) y_N(e) . As it turns out, each value x N ( e ) x_N(e) and y N ( e ) y_N(e) is an algebraic integer which we are able to understand in the context of explicit class field theory. When combined with our previous article (see Jorgenson, L. Smajlović, and H. Then [Exp. Math. 29 (2020), pp. 1–27]), we now have a complete investigation of x N ( τ ) x_N(\tau ) and y N ( τ ) y_N(\tau ) at any CM point τ \tau , including elliptic points, for any genus one group Γ 0 ( N ) + \Gamma _0(N)^+ . Furthermore, the present article when combined with the two aforementioned papers leads to a procedure which we expect to yield generators of class fields, and certain subfields, using the Schwarzian derivative and which does not use either modular polynomials or Shimura reciprocity.

Smaller Cuts, Higher Lower Bounds

This article proves strong lower bounds for distributed computing in the congest model, by presenting the bit-gadget : a new technique for constructing graphs with small cuts. The contribution of bit-gadgets is twofold. First, developing careful sparse graph constructions with small cuts extends known techniques to show a near-linear lower bound for computing the diameter, a result previously known only for dense graphs. Moreover, the sparseness of the construction plays a crucial role in applying it to approximations of various distance computation problems, drastically improving over what can be obtained when using dense graphs. Second, small cuts are essential for proving super-linear lower bounds, none of which were known prior to this work. In fact, they allow us to show near-quadratic lower bounds for several problems, such as exact minimum vertex cover or maximum independent set, as well as for coloring a graph with its chromatic number. Such strong lower bounds are not limited to NP-hard problems, as given by two simple graph problems in P, which are shown to require a quadratic and near-quadratic number of rounds. All of the above are optimal up to logarithmic factors. In addition, in this context, the complexity of the all-pairs-shortest-paths problem is discussed. Finally, it is shown that graph constructions for congest lower bounds translate to lower bounds for the semi-streaming model, despite being very different in its nature.

A recurring population of the sea hare Bursatella hirsuta (Gastropoda: Aplysiidae) at Rottnest Island, Western Australia

ABSTRACT In many, but not all, years a recurring population of the marine aplysiid Bursatella hirsuta occurs on an intertidal limestone platform stretching between Little Armstrong Bay and North Point at Rottnest Island, Western Australia. Using a transect/quadrat method, we measured densities of B. hirsuta during the austral summer of 2020/2021. No individuals were present in December 2020, but a population with a mean density of 16.5 ± 1.6 (SE) inds m−2 was present on 17 January 2021. Density was low in bare sand (2.6 ± 3.2 inds m−2) and in the seagrass Amphibolis antarctica (2.0 ± 1.4 inds m−2), high in mixed algae, mostly Phaeophyceae (18.2 ± 1.8 inds m−2) and greatest (27.2 ± 14.3 inds m−2) in a small number of quadrats with a mixture of sand and algae or sand and A. antarctica. The population was estimated at >600,000 individuals. The species was present in February but had disappeared by late March 2021. The population at Little Armstrong Bay and North Point provides a fertile opportunity for developing a better understanding of the biology of B. hirsuta and broader questions of boom-and-bust populations.

Solving Discounting Problem Using Piece-Wise Quadratic Fuzzy Numbers

The discounting problem is one of the important aspects in investment, portfolio selection, purchasing with credit, and many other financial operations. In this paper, a discounting problem using piecewise quadratic fuzzy numbers is proposed. The implementation of piecewise quadratic fuzzy numbers is described based on such operations. Fuzzy arithmetic and interval number arithmetic are used for computation. The close interval approximation of piecewise quadratic fuzzy numbers is used for solving the proposed discounting problem. This research article addresses the discounted investment for Year 1, Year 2, and Year 3. Additionally, the authors determine the cumulative discounted investment for different values of the parameter α ranging from 0 to 1. A discounting problem using piecewise quadratic fuzzy numbers is solved as a numerical example to illustrate the proposed procedure.

The Hecke Group H λ 4 Acting on Imaginary Quadratic Number Fields

Let H λ 4 be the Hecke group x , y : x 2 = y 4 = 1 and, for a square-free positive integer n , consider the subset ℚ ∗ − n = a + − n / c | a , b = a 2 + n / c ∈ ℤ , c ∈ 2 ℤ of the quadratic imaginary number field ℚ − n . Following a line of research in the relevant literature, we study the properties of the action of H λ 4 on ℚ ∗ − n . In particular, we calculate the number of orbits arising from this action for every such n . Some illustrative examples are also given.