In a finite projective plane PG(2; q), a set K of k points is a (k; n)-arc for 2 ≤ n ≤ q - 1 if the following two properties hold:
1. Every line intersects K in at most n points.
2. There exists a line which intersects K in exactly n points.
Algebraic curves of degree n give examples of (k; n)-arc; the parameter n is called the degree of the arc. In PG(2; q); the problem of finding mn(2; q) and tn(2; q) (the maximum and the minimum value of k for which a complete (k; n)-arc exists) and the problem of classifying such arcs up to projective equivalence, are crucial problems in finite geometry. One of the important application of these arcs in coding theory are projective codes that cannot be extended to larger codes.
The aim of this project is to classify (k; n)-arcs if possible for 3 ≤ n ≤ 5 and to construct large arcs in PG(2; 11): Algebraic and new combinatorial methods are used to perform the classification and the construction of such arcs with different degrees. Those procedures are implemented using different open-source software packages such as GAP [35] and Orbiter [10].
We were successful in obtaining new isomorphism types of (k; 5)-arcs for k = 5,…, 13 in PG(2; 11): We have also developed a new classification algorithm for cubic curves in small projective planes. Moreover, a new upper bound is proved for the number of 5-secants of (45; 5)-arc. In addition to proving our new lower bound for the complete (k; 5)-arc in PG(2; 11): The non existence of (44; 5)-arc and (45; 5)-arc is formulated as a new conjecture for q = 11: Using an arc of degree 2 and exploiting the complement relation between arcs and blocking sets we find new 134 isomorphism types of (77; 8)-arcs in PG(2; 11):
Combinatorial formulas for the moments of the Brownian motion on classical compact Lie groups are obtained. These expressions are deformations of formulas of B. Collins and P. Śniady for moments of the Haar measure and yield a proof of the First Fundamental Theorem of invariant theory and of classical Schur–Weyl dualities based on stochastic calculus.
This paper characterizes the Kalai-Smorodinsky bargaining solution when firms and unions negotiate over wages alone, and firms set the level of employment in order to maximize profits given the agreed wage. The Kalai-Smorodinsky solution is analysed for the case that the wage elasticity of employment and the union's risk aversion are both constant. In this case there is a simple relationship between the Kalai-Smorodinsky and the Nash solutions.
This paper contains computer pictures of generalised
Mandelbrot and Mandelbar sets, and their associated
Julia sets, from which it is evident that their symmetry
groups possess an elegant and simple structure. We
show that (i) the Mandelbrot set M(p) generated by the
iteration zt+ ~ = ztp + c remains invariant under the
symmetry transforms of the dihedral group Dp_ 1 (i.e.,
these are isomet-ries of M(p)); (ii) the Mandelbar set
M(p) is invariant under the isometries inDp + 1; and (iii)
the Julia sets of points inside M(p) (or M(p)) are invariant
under the isometries in either Dp or just the cyclic
group Cp, depending on whether the seed point is on
or off a symmetry axis of the parent Mandelbrot (or
Mandelbar) set. The proofs are relatively easy, but
showing that there are no other isometries of these sets
is not so straightforward. As is often the case in the
theory of chaos, what is obvious geometrically is difficult
to prove analytically. For the generalised Mandelbrot
and Mandelbar sets with even p we have in fact
proved that the dihedral symmetry transforms are the
only isometries of these sets, but the method does not
appear to be applicable to odd p, or to the Julia sets.
Using an elementary counting procedure on biquadratic polynomials over Zp it is shown that the probability distribution of odd, unramified rational primes according to decomposition type in a fixed dihedral numberfield is identical to the probility of separable quartic polynomials (mod p) whose roots generate numberfields with normal closure having Galois group isomorphic to D4, as p → ∞. This verifies a conjecture about a converse to the Tschebotarev density theorem. Further evidence in support of this conjecture is provided in quadratic and coubic numberfields.
A dihedral number field is a non-normal quartic field K which possesses a quadratic subfield k. That is, for some integer α of k. Integral bases of these fields were known by Sommer (1907), but the form in which they were known was of little use for computational purposes. In this paper we construct integral bases of those dihedral fields with quadratic subfield of the form , d 1 (mod 8), for which only rational quantities need be determined. Although the general theory may easily be generalized to the case d ≡ 1 (mod 8), the actual determination of integral bases in this case is left to a later paper.
This paper introduces a method for simulating multivariate samples that have exact means, covariances, skewness and kurtosis. We introduce a new class of rectangular orthogonal matrix which is fundamental to the methodology and we call these matrices L matrices. They may be deterministic, parametric or data specific in nature. The target moments determine the L matrix then infinitely many random samples with the same exact moments may be generated by multiplying the L matrix by arbitrary random orthogonal matrices. This methodology is thus termed “ROM simulation”. Considering certain elementary types of random orthogonal matrices we demonstrate that they generate samples with different characteristics. ROM simulation has applications to many problems that are resolved using standard Monte Carlo methods. But no parametric assumptions are required (unless parametric L matrices are used) so there is no sampling error caused by the discrete approximation of a continuous distribution, which is a major source of error in standard Monte Carlo simulations. For illustration, we apply ROM simulation to determine the value-at-risk of a stock portfolio.
Random orthogonal matrix (ROM) simulation is a very fast procedure for generating multivariate random samples that always have exactly the same mean, covariance and Mardia multivariate skewness and kurtosis. This paper investigates how the properties of parametric, data-specific and deterministic ROM simulations are influenced by the choice of orthogonal matrix. Specifically, we consider how cyclic and general permutation matrices alter their time-series properties, and how three classes of rotation matrices – upper Hessenberg, Cayley, and exponential – influence both the unconditional moments of the marginal distributions and the behaviour of skewness when samples are concatenated. We also perform an experiment which demonstrates that parametric ROM simulation can be hundreds of times faster than equivalent Monte Carlo simulation.
The current paper presents a numerical technique in solving the 3D heat conduction equation. The Finite Volume method is used in the discretisation scheme. Gauss's theorem has also been employed for solving the integral parts of the general heat conduction equation in solving problems of steady and unsteady states. The proposed technique is applicable to unstructured (tetrahedral) elements for dealing with domains of complex geometries. The validation cases of the developed, FORTRAN based, heat conduction code in 1D, 2D and 3D representations have been reviewed with a grid independence check. Comparisons to the available exact solution and a commercial software solver are attached to the manuscript.