We introduce the Arithmetic Site: an algebraic geometric space deeply related to the non-commutative geometric approach to the Riemann Hypothesis. We prove that the non-commutative space quotient of the adèle class space of the field of rational numbers by the maximal compact subgroup of the idèle class group, which we had previously shown to yield the correct counting function to obtain the co...

In this paper a new aggregation operator for a class of LR Fuzzy Numbers i.e., Triangular Fuzzy Numbers (TFNs) in which the Land Rapex angles of the piecewise continuous linear membership function of the composite or resultant or aggregate TFN are the arithmetic means of the corresponding Land Rapex angles of the individual TFNs is proposed. The Land Rapex angles have been treated independently...

The arithmetic mean is the mean for addition and the geometric mean is that for multiplication. Then what kind of binary operation is associated with the arithmetic-geometric mean (AGM) due to C. F. Gauss? If it is possible to construct an arithmetic operation such that AGM is the mean for this operation, it can be regarded as an intermediate operation between addition and multiplication in vie...

The leaf area index (LAI) is a crucial variable in climate, ecological, and land surface modeling. However, the estimation of LAI from coarse-resolution remote sensing data can be affected by spatial scaling bias, which arises nonlinearity retrieval models heterogeneity surface. This study provides an algorithm named Arithmetic Mean Geometric (AM–GM) to correct bias. It established based on neg...

A keystone in the classical theory of diophantine approximation is the construction of an auxilliary polynomial. The polynomial is constructed so that it is forced (for arithmetic reasons) to vanish at certain approximating points and this contradicts an upper bound on the order of vanishing obtained by other (usually geometric) techniques; the contradiction then allows one to prove finiteness ...

We construct a martingale which has the same marginals as the arithmetic average of geometric Brownian motion. This provides a short proof of the recent result due to P. Carr et al [7] that the arithmetic average of geometric Brownian motion is increasing in the convex order. The Brownian sheet plays an essential role in the construction. Our method may also be applied when the Brownian motion ...

In this paper, we consider an arithmetic Hodge index theorem for a family of semi-stable curves, generalizing Faltings-Hriljac’s arithmetic Hodge index theorem for an arithmetic surface.

We prove the arithmetic Hodge index and hard Lefschetz conjectures for the Grassmannian G = G(2, N) parametrizing lines in projective space, for the natural arithmetic Lefschetz operator defined via the Plücker embedding of G in projective space. The analysis of the Hodge index inequality involves estimates on values of certain Racah polynomials. 0. Introduction Let X be an arithmetic variety, ...

Two inequalities for weighted sums of powers are established. Applications to Jacobian elliptic functions and Legendre’s elliptic integrals of the first kind are presented. Some known and new inequalities for circular and hyperbolic functions are obtained. Applications to certain iterative means including Gauss’ arithmetic-geometric mean and Schwab-Borchardt mean are included. Mathematics subje...

We construct a martingale which has the same marginals as the arithmetic average of geometric Brownian motion. This provides a short proof of the recent result due to P. Carr et al [5] that the arithmetic average of geometric Brownian motion is increasing in the convex order. The Brownian sheet plays an essential role in the construction. Our method may also be applied when the Brownian motion ...