In 1924, Helmut Hasse established a local-to-global principle for representations of rational quadratic forms. Unfortunately, an analogous local-to-global principle does not hold for representations over the integers. A quadratic polynomial is called regular if such a principle exists; that is, if it represents all the integers which are represented locally by the polynomial itself over Zp for ...

Fins. I and z: FIG. I (left). Calculated mean quadratic elongation (h)-bond angle variance (~) distributions for four ideal types of distortion in coordination octahedra arising from the progressive shortening of the octahedral edges emphasized in insets a, b, c, and d: RGR, trend line for observed data (Robinson et al., I97I). FIG. 2 (right). Calculated mean quadratic elongation (;~)-bond angl...

Let $\mathbb{k}$ be an algebraically closed field. We classify all of the quadratic twisted tensor products $A \otimes_{\tau} B$ in cases where $(A, B) = (\mathbb{k}[x], \mathbb{k}[y])$ and (\mathbb{k}[x, y], \mathbb{k}[z])$. determine when a product this form is Koszul, it Artin-Schelter regular.

We present a minimal approach to the generation of baryon ($B$) asymmetry Universe, in which is generated complex inflaton condensate via $B$-violating quadratic potential terms and Affleck-Dine mechanism. show that mass create an oscillating at late times. The final transferred Standard Model sector reheating naturally reduced magnitude observed $B$ by effect averaging over oscillations. This ...

Rational Bezier and B-spline representations of circles have been heavily publicized. However, all the literature assumes the rational Bezier segments in the homogeneous space are both planar and (equivalent to) quadratic. This creates the illusion that circles can only be achieved by planar and quadratic curves. In this paper we show circles that are formed by higher order rational Bezier curv...

We conjecture that splines of degree n can interpolate points on a smooth curve in R m with order of contact k ? 1 = n ? 1 + b(n ? 1)=(m ? 1)c at every n-th knot. Moreover, this Geometric Hermite Interpolant (GHI) has the optimal approximation order k + 1. We give a proof of this conjecture for planar quadratic spline curves and describe a simple construction of curvature continuous quadratic s...

For infinitely many primes p = 4k+1 we give a slightly improved upper bound for the maximal cardinality of a set B ⊂ Zp such that the difference set B−B contains only quadratic residues. Namely, instead of the ”trivial” bound |B| ≤ √p we prove |B| ≤ √ p − 1, under suitable conditions on p. The new bound is valid for approximately three quarters of the primes p = 4k + 1.

This paper discusses a Cooperative Path Planning (CPP) design methodology for multi-vehicle systems and a Nonlinear Trajectory Generation (NTG) algorithm. Three scenarios of multi-vehicle tasking are proposed at the CPP framework. The NTG algorithm is, then, used to generate realtime trajectory for desired vehicle activities. Given system dynamics and constraints, the NTG algorithm first finds ...

(here b and a run over the quadratic nonresidues and quadratic residues, respectively, that lie between 0 and p) is a unit of the real quadratic field R(y/p), and that 77> 1. The fact that -n > 1 is usually deduced from the theory of the classnumber of quadratic fields. We present a short proof independent of the theory of the class-number. As in the paper of Chowla and Mordell [Note on the non...