Suppose g > 2 is an odd integer. For real number X > 2, define SgðXÞ the number of squarefree integers d4X with the class number of the real quadratic field Qð ffiffiffi d p Þ being divisible by g. By constructing the discriminants based on the work of Yamamoto, we prove that a lower bound SgðXÞ4X 1=g e holds for any fixed e > 0, which improves a result of Ram Murty. # 2002 Elsevier Science (USA)

A square-ordered field, also called Hilbert field of type (A), is understood to be an ordered field all of whose positive elements are squares. The problem of classifying, up to isomorphism, all 4-dimensional quadratic division algebras over a square-ordered field k is shown to be equivalent to the problem of finding normal forms for all pairs (X, Y ) of 3× 3-matrices over k, X being antisymmet...

For a quadric Q over a real field k, we investigate whether finiteness of the Pythagoras number of the function field k(Q) implies the existence of a uniform bound on the Pythagoras numbers of all finite extensions of k. We give a positive answer if the quadratic form that defines Q is weakly isotropic. In the case where Q is a conic, we show that the Pythagoras number of k(Q) is 2 only if k is...

In this article we present a method to define algebraic structure (field operations) on a representation of real numbers by coinductive streams. The field operations will be given in two algorithms (homographic and quadratic algorithm) that operate on streams of Möbius maps. The algorithms can be seen as coalgebra maps on the coalgebra of streams and hence they will be formalised as general cor...

In this paper we intend to examine the application of Kullback-Leibler, Hellinger and LINEX loss function in Dynamic Linear Model using the real price of oil for 106 years of data from 1913 to 2018 concerning the asymmetric problem in filtering and forecasting. We use DLM form of the basic Hoteling Model under Quadratic loss function, Kullback-Leibler, Hellinger and LINEX trying to address the ...

In this paper we discuss image registration techniques with a focus on volume preserving constraints. These constraints can reduce the non-uniqueness of the registration problem significantly. Our implementation is based on a constrained optimization formulation. To solve the problem we use a variant of the Sequential Quadratic Programming method. Moreover, we present results on synthetic as we...

Hilbert’s law of quadratic reciprocity, as reformulated by Hasse, states that for a quaternion algebra D over a number field K, the number of ramified places of D is even. In general a number field has three kinds of places: finite, real, and complex. The complex places are never ramified. If D is assumed unramified at all finite places, it follows that D is ramified at an even number of real p...

This paper studies linear quadratic games with set up costs monotonic on the number of active players, namely, players whose action is non-zero. Such games arise naturally in joint replenishment inventory systems. Building upon a preliminary analysis of the properties of the best response strategies and Nash equilibria for the given game, the main contribution is the study of the same game unde...

We present improvements to the index-calculus algorithm for the computation of the ideal class group and regulator of a real quadratic field. Our improvements consist of applying the double large prime strategy, an improved structured Gaussian elimination strategy, and the use of Bernstein’s batch smoothness algorithm. We achieve a significant speed-up and are able to compute the ideal class gr...

We present an effective and practical algorithm for computing Gross-Stark units over a real quadratic base field F. Our algorithm allows us to explicitly construct certain relative abelian extensions of F where these units lie, using only information from the base field. These units were recently proved to always exist within the correct extension fields of F by Dasgupta, Darmon, and Pollack, w...