Binary Quadratic Research Articles

Finding Debt Cycles: QUBO Formulations for the Maximum Weighted Cycle Problem Solved Using Quantum Annealing

The problem of finding the maximum weighted cycle in a directed graph map to solve optimization problems is NP-hard, implying that approaches in classical computing are inefficient. Here, Quantum computing might be a promising alternative. Many current approaches to the quantum computer are based on a Quadratic Unconstrained Binary Optimization (QUBO) problem formulation. This paper develops four different QUBO approaches to this problem. The first two take the starting vertex and the number of vertices used in the cycle as given, while the latter two loosen the second assumption of knowing the size of the cycle. A QUBO formulation is derived for each approach. Further, the number of binary variables required to encode the maximum weighted cycle problem with one or both assumptions for the respective approach is made explicit. The problem is motivated by finding the maximum weighted debt cycle in a debt graph. This paper compares classical computing versus currently available (hybrid) quantum computing approaches for various debt graphs. For the classical part, it investigated the Depth-First-Search (DFS) method and Simulated Annealing. For the (hybrid) quantum approaches, a direct embedding on the quantum annealer and two types of quantum hybrid solvers were utilized. Simulated Annealing and the usage of the hybrid CQM (Constrained Quadratic Model) had promising functionality. The DFS method, direct QPU, and hybrid BQM (Binary Quadratic Model), on the other hand, performed less due to memory issues, surpassing the limit of decision variables and finding the right penalty values, respectively.

A Convex Reformulation and an Outer Approximation for a Large Class of Binary Quadratic Programs

Binary Quadratic Program with Variable Partitioning Constraints The binary quadratic program with variable partitioning constraints is a very general class of optimization problems that is very difficult to solve because of the nonconvexity and integrality of the variables and is ubiquitous, among others, in network design, computer vision, and transportation and logistics. In their article, “A convex reformulation and an outer approximation for a large class of binary quadratic programs,” Rostami et al. show how to transform such a nonconvex challenging problem into a convex bilinear program with decomposable structure. The authors develop a branch-and-cut algorithm based on outer approximation cuts, in which the cuts are generated on the fly by efficiently solving separation subproblems. The results of their computational experiments on different problems confirm the efficacy of the solution methods in solving large-scale problem instances.

Query batching optimization in database systems

Techniques based on sharing data and computation among queries have been an active research topic in database systems. While work in this area developed algorithms and systems that are shown to be effective, there is a lack of rigorous modeling and theoretical study for query processing and optimization. Query batching in database systems has strong resemblance to order batching in the warehousing context. While the latter is a well-studied problem, the literature on optimization techniques for query batching problem is quite scarce/nonexistent. In this study, we develop a Mixed Binary Quadratic Program (MBQP) to optimize query-batching in a database system. This model uses the coefficients of a linear regression on the query retrieval time, trained by a large set of randomly generated query sets. We also propose two heuristics, the so-called restricted-cardinality search methods I and II, for solving the proposed MBQP. To demonstrate the effectiveness of our proposed techniques, we conduct a comprehensive computational study over randomly generated instances of three well-known database benchmarks. The computational results show that when the proposed MBQP is solved using the designed heuristics, an improvement of up to 61.8% in retrieving time is achieved.

ON BINARY QUADRATIC EQUATION y2 = 35x2 + 29

ON BINARY QUADRATIC EQUATION y2 = 35x2 + 29

Observation on the Binary Quadratic Equation

Observation on the Binary Quadratic Equation

On the Binary Quadratic Diophantine Equation y2=80x2-16

On the Binary Quadratic Diophantine Equation y2=80x2-16

Conjugacy Classes and Binary Quadratic Forms for the Hecke Groups

Abstract.In this paper we give a lower bound with respect to block length for the trace of non-elliptic conjugacy classes of the Hecke groups. One consequence of our bound is that there are finitely many conjugacy classes of a given trace in anyHecke group. We show that another consequence of our bound is that class numbers are finite for related hyperbolic ℤ[λ]-binary quadratic forms. We give canonical class representatives and calculate class numbers for some classes of hyperbolic ℤ[λ]-binary quadratic forms.

Decoding of the (31, 16, 7) quadratic residue code

An algebraic decoding algorithm is proposed to correct all error patterns of up to three errors in the binary (31, 16, 7) Quadratic Residue (QR) code with reducible generator polynomial. The decoding technique, a modification of the decoding algorithm given by Reed et al., is based on the application of the decoding algorithm proposed by Truong et al. The computation of all syndromes is done in a small field, namely, GF(25). Thus, the computational complexity can be reduced. A full simulation shows that this novel decoding method is superior to the algebraic decoding algorithm given by Reed et al.

Lower Bounds for the Principal Genus of Definite Binary Quadratic Forms

AbstractWe apply Tatuzawa's version of Siegel's theorem to derive two lower bounds on the size of the principal genus of positive definite binary quadratic forms.

Algebraic decoding of the (41, 21, 9) Quadratic Residue code

In this paper, an algebraic decoding algorithm is proposed to correct all patterns of four or fewer errors in the binary (41, 21, 9) Quadratic Residue (QR) code. The technique needed here to decode the (41, 21, 9) QR code is different from the algorithms developed in [I.S. Reed, T.K. Truong, X. Chen, X. Yin, The algebraic decoding of the (41, 21, 9) Quadratic Residue code, IEEE Transactions on Information Theory 38 (1992 ) 974–986]. This proposed algorithm does not require to solve certain quadratic, cubic, and quartic equations and does not need to use any memory to store the five large tables of the fundamental parameters in GF(2 20) to decode this QR code. By the modification of the technique developed in [R. He, I.S. Reed, T.K. Truong, X. Chen, Decoding the (47, 24, 11) Quadratic Residue code, IEEE Transactions on Information Theory 47 (2001) 1181–1186], one can express the unknown syndromes as functions of the known syndromes. With the appearance of known syndromes, one can solve Newton’s identities to obtain the coefficients of the error-locator polynomials. Besides, the conditions for different number of errors of the received words will be derived. Computer simulations show that the proposed decoding algorithm requires about 22% less execution time than the syndrome decoding algorithm. Therefore, this proposed decoding scheme developed here is more efficient to implement and can shorten the decoding time.

On Trigroups and Semigroups of Binary Quadratic Forms Values and of their Associated Linear Operators

On Trigroups and Semigroups of Binary Quadratic Forms Values and of their Associated Linear Operators

The Weight Distributions of Some Binary Quadratic Residue Codes

The weight distributions of binary quadratic residue codes C can be computed from the weight distribution of a subset of C containing one-fourth (resp., one-eighth) of the codewords in C when the length of the code is congruent to 1 (resp., -1) modulo 8. An algorithm to determine the weight distributions of binary cyclic codes is given. As a consequence, the weight distributions of (73,37,13), (89,45,17), and (97,49,15) quadratic residue codes are determined precisely.

Class Numbers of Indefinite Binary Quadratic Forms and the Residual Indices of Integers Modulo p

Let h(d) be the class number of properly equivalent primitive binary quadratic forms ax2+bxy+cy2 with discriminant d=b2-4ac. The behavior of h(5p2), where p runs over primes, is studied. It is easy to show that there are few discriminants of the form 5p2 with large class numbers. In fact, one has the estimate $$\# \{ p \leqslant x|h(5p^2 ) >x^{1 - \delta } \} \ll x^{2\delta } ,$$ where δ is an arbitrary constant number in (0;1/2). Assume that α(x) is a positive function monotonically increasing for x→∞ and α(x)→∞. If $$\alpha (x) \leqslant (\log x)(\log \log x)^{ - 3} $$ , then (assuming the validity of the extended Riemann hypothesis for certain Dedekind zeta-functions) it is proved that $$\# \left\{ {p \leqslant x|\left( {\frac{5}{p}} \right) = 1,h(5p^2 ) >\alpha (x)} \right\} \asymp \frac{{\pi (x)}}{{\alpha (x)}}.$$ It is also proved that for an infinite set of p with $$\left( {\frac{5}{p}} \right) = 1$$ one has the inequality $$h(5p^2 ) \geqslant \frac{{\log \log p}}{{\log _k p}},$$ where log k p is the k-fold iterated logarithm (k is an arbitrary integer, k≥3). Results on mean values of h(5p 2 ) are also obtained. Similar facts are true for the residual indices of an integer a≥2 modulo p: $$r(a,p) = \frac{{p - 1}}{{o(a,p)}},$$ where o(a,p) is the order of a modulo p. Bibliography: 13 titles.

On the Class Numbers of Indefinite Binary Quadratic Forms with Discriminant dp2

A number of results on the average values of the class numbers of indefinite binary quadratic forms with discriminants divisible by a large square are proved. The main result is as follows. Let d = 4n 2 + 1. Then $$\sum\limits_{1 \leqslant n \leqslant X} {\frac{1}{{h\left( d \right)}}} \sum\limits_{2X \leqslant p \leqslant 3X} {h\left( {dp^2 } \right) = O\left( {X^2 } \right)} ,$$ where h(d) is the class number for the discriminant d and \(\sum '\) means that the summation is performed over the square-free d only. Bibliography: 9 titles.

The Diophantine Equation xy + yz + zx = n and Indecomposable Binary Quadratic Forms

There are 18 (and possibly 19) integers that are not of the form xy+yz+xz with positive integers x, y, z. The same 18 integers appear as exceptional discriminants for which no indecomposable positive definite binary quadratic form exists. We show that the two problems are equivalent.

Class Numbers of Indefinite Binary Quadratic Forms

Let \(h(d)\) be the class number of properly equivalent primitive binary quadratic forms \(ax^2 + bxy + cy^2\) of discriminant \(d = b^2 a - 4ac\). The case of indefinite forms \((d < 0)\) is considered. Assuming that the extended Riemann hypothesis for some fields of algebraic numbers holds, the following results are proved. 1. Let \(\alpha (x)\) be an arbitrarily slow monotonically increasing function such that \(\alpha (x) \to \infty\). Then $$\# \left\{ {p \leqslant \left. x \right|{\text{ }}\left( {\frac{{\text{5}}}{p}} \right) = 1,h(5p^2 ) >(\log p)^{\alpha (p)} } \right\} = o(\pi (x)),$$ where \(\pi (x) = \# \{ p \leqslant x\}\). 2. Let F be an arbitrary sufficiently large positive constant. Then for \(x >x_F\), the relation $$\# \left\{ {p \leqslant \left. x \right|{\text{ }}\left( {\frac{{\text{5}}}{p}} \right) = 1,h(5p^2 ) >F} \right\} \asymp \frac{{\pi (x)}}{F}$$ holds. 3. The relation $$\# \left\{ {p \leqslant \left. x \right|{\text{ }}\left( {\frac{{\text{5}}}{p}} \right) = 1,h(5p^2 ) = 2} \right\} \sim \frac{9}{{19}}A\pi (x)$$ holds, where A is Artin's constant. Hence, for the majority of discriminants of the form \(d = 5p^2\), where \({\left( {\frac{{\text{5}}}{p}} \right) = 1}\) , the class numbers are small. This is consistent with the Gauss conjecture concerning the behavior of \(h(d)\) for the majority of discriminants \(d >0\) in the general case. Bibliography: 22 titles.

True Dimension of Some Binary Quadratic Trace Goppa Codes

We compute in this paper the true dimension over $$\mathbb{F}_2$$ of Goppa Codes Γ(L, g) defined by the polynomial $$g(z) = {\text{Tr}}_{\mathbb{F}_{2^{2s} } :\mathbb{F}_{2^s } } (z)$$ proving, this way, a conjecture stated in [14,16].

Bijective Arithmetic Codings of Hyperbolic Automorphisms of the 2-Torus, and Binary Quadratic Forms

We study the arithmetic codings of hyperbolic automorphisms of the 2-torus, i.e., the continuous mappings acting from a certain symbolic space of sequences with a finite alphabet endowed with an appropriate structure of the additive group onto the torus which preserve this structure and turn the two-sided shift into a given automorphism of the torus. This group is uniquely defined by an automorphism, and such an arithmetic coding is a homomorphism of that group onto \Bbb T^2. The necessary and sufficient condition of the existence of a bijective arithmetic coding is obtaineds it is formulated in terms of a certain binary quadratic form constructed by means of a given automorphism. Furthermore, we describe all bijective arithmetic codings in terms of the Dirichlet group of the corresponding quadratic field. The minimum of that quadratic form over the nonzero elements of the lattice coincides with the minimal possible order of the kernel of a homomorphism described above.

Zeta Functions for Equivalence Classes of Binary Quadratic Forms

Proceedings of the London Mathematical SocietyVolume s3-69, Issue 2 p. 250-276 Articles Zeta Functions for Equivalence Classes of Binary Quadratic Forms Pia Bauer, Pia Bauer DMI École Normale Supérieure, 45 rue d'Ulm 75230, Paris Cedex 05, FranceSearch for more papers by this author Pia Bauer, Pia Bauer DMI École Normale Supérieure, 45 rue d'Ulm 75230, Paris Cedex 05, FranceSearch for more papers by this author First published: September 1994 https://doi.org/10.1112/plms/s3-69.2.250Citations: 1AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volumes3-69, Issue2September 1994Pages 250-276 RelatedInformation

Some aspects of effectively constructive mathematics that are relevant to the foundations of neoclassical mathematical economics and theory of games

This paper discusses certain key issues of the effectiveness of certain mathematical constructions within mathematical economics and the theory of games. By effective, we mean recursive constructions and, more generally, constructions that are recursively enumerable. This is a more precise notion than the unspecified concept of construction usually found in constructive analysis (e.g. Bishop's formulation). The advantage to our commitment to recursive constructions is that with the precision of the interpretation of effective, qualitative and quantitative measures of complexity can be obtained which are usually left unspecified in the latter frameworks. Among the results presented is a proof that for the case of totally discrete Hurwiczian resource allocation mechanisms of the form: π=〈E,M,A〈h,g〉〉 having binary quadratics as outcome and performance criteria, there is an R.E. (recursively enumerable) class α={ π j } j< w of such mechanisms whose realization is uniformaly sub-recursive in NP-complete complexity. The complexity of this class is uniformly less than a class of p-stage multilinear programmes: L={〈{A i j} i∈I,{x i j} i∈I,b,{c i j} i∈I〉} j∈ j . The level of complexity for the Hurwiczian mechanisms π = 〈 E, M, A〈 h, g〉〉 that are binary quadratic as stipulate is bounded by the complexity of the following ordinary problem: If 〈 a 1,…, a n 〉 is an arbitrary sequence of integers, is 〈 a 1ϑ,…, a n ϑ〉 a solution to the integral equation: ∫ 0 2π π j=1 n (cos(a jθ)) dθ 2 .