This paper demonstrates analytically how a nature reserve may protect the total population, realize
maximum sustainable yield (MSY), maximum economic yield (MEY) and consumer surplus (CS)
and how this depends on biological growth, migration, reserve size and economic parameters. The
pre-reserve population is assumed to follow the logistic growth law and two post-reserve growth
models are discussed. For Model A, the post-reserve growth has a common carrying capacity as in
the pre-reserve case. In Model B, each sub-population has its own carrying capacity proportionate to
its distribution area. Population protection against extinction is assured against low cost harvesting,
including zero cost, when relative reserve size is greater than relative migration. Reserve size may be
tuned to realize MSY in Model A, but not in Model B. MEY can not be realized in any of the two
models, but generally economic yield is greater in Model A than B. CS is greater with a reserve than
without.
We give a simple criterion for the cyclicity of the m-torsion subgroup
of the group of rational points on an elliptic curve defined over a finite field of
characteristic larger than 3 for m = 2, 3, 4, 6, 12.
We find an invariant characterization of planar webs of maximum rank.
For 4-webs, we prove that a planar 4-web is of maximum rank three if and
only if it is linearizable and its curvature vanishes. This result leads to
the direct web-theoretical proof of the Poincar´e’s theorem: a planar 4-
web of maximum rank is linearizable. We also find an invariant intrinsic
characterization of planar 4-webs of rank two and one and prove that in
general such webs are not linearizable. This solves the Blaschke problem
“to find invariant conditions for a planar 4-web to be of rank 1 or 2 or 3”.
Finally, we find invariant characterization of planar 5-webs of maximum
rank and prove than in general such webs are not linearizable.
We discuss the dimensional characterization of the solutions
space of a formally integrable system of partial differential equations and
provide certain formulas for calculations of these dimensional quantities.
All points on the surface of the Earth are moving. To define the velocity of a given
point, we can place a GPS receiver there and measure the coordinates every day. After
collecting enough data, we can generate a time series of three coordinates, North, East
and Height directions. The most used technique to determine such displacements, is the linear model.
The main objective of this thesis is to show how to estimate the velocity of a given
point, using statistical methods to improve the results.
The improvement of the site velocity achieved by exluding all signals that are not tec- tonic origine (seasonal variations, spacially correlated noise reduction ).
Time series for all directions contain gaps(missing data), outliers, offsets and various
data length.
The data discontinuities are detected and corrected by a simple algorithm, based on
binary search to detect the time of abruption. The outliers are eliminated by using robust
estimation techniques. Simulation is used to fill the gaps.
The data obtained from permanent GPS-stations in Norway and some other European countries are unevenly sampled. We therefore use the Lomb-Scargle method to perform spectral analysis. This allows us to detect annual and interannual variations.
The methods of Principal Components (also known as Empirical Orthogonal Functions,
or EOF) and Factor Analysis are used to correct for common fluctuations. We use data
from 8 permanent GPS-stations (SATREF) in these investigations.
The use of elliptic curves in cryptography was suggested independently by Neal Koblitz and Victor Miller in 1985. Being a relatively new field, there is still a lot of ongoing research on the subject, but elliptic curve cryptography, or ECC for short, has already been implemented in real-life applications. Its strength was proved in 2003 when the U.S. National Security Agency adopted ECC for protecting information classified as mission-critical by the U.S. government. The security of public-key cryptographic systems that can be considered secure, efficient, and commercially viable is directly tied to the relative hardness of their underlying mathematical problems. In the case of ECC, this mathematical problem is to solve the discrete logarithm problem over elliptic curves, or ECDLP for short. Because the best-known way to solve ECDLP is fully exponential, we can use substantially smaller key sizes to obtain equivalent strengths compared to other systems. Hence, ECC provides the most security per bit of any public-key scheme known.
In this thesis we have focused on presenting the known attacks on the ECDLP. We started by introducing some basic facts from the theory of elliptic curves. In the rest of the thesis we have described, analyzed and presented running time estimates of attacks on the ECDLP. This included a presentation of attacks which are specially designed to exploit weaknesses in the structure of some classes of elliptic curves. We have also presented attacks which can be used to solve the ECDLP over general elliptic curves. This included Pollard’s rho and lambda algorithms, where the former was used for solving the ECDLP challenges set by the Certicom company.
In this thesis we study the correspondence between categorical notions and bialgebra notions, and make a kind of dictionary and grammar book for translation between these notions. We will show how to obtain an antipode, and how to define braidings and quantizations. The construction is done in two ways. First we use the properties of a bialgebra to define a monoidal structure on (co)modules over this bialgebra. Then we go from a (strict) monoidal category and use a certain monoidal functor from this category to reconstruct bialgebra and (co)module structures. We will show that these constructions in a sense are inverse to each other. In some cases the correspondence is 1-1, and in the final Part we conjecture when this is the case for the category of comodules that are finitely generated and projective over the base ring k. We also briefly discuss how to transfer the results to non-strict categories.
A criterion in terms of differential invariants for a metric on a surface
to be Liouville is established. Moreover, in this paper we completely solve
in invariant terms the local mobility problem of a 2D metric, considered
by Darboux: How many quadratic in momenta integrals does the geodesic
flow of a given metric possess? The method is also applied to recognition
of other polynomial integrals of geodesic flows.
Description:
Dette er forfatternes aksepterte versjon.
This is the author’s final accepted manuscript.
Differential invariants of a (pseudo)group action can vary when restricted
to invariant submanifolds (differential equations). The algebra
is still governed by the Lie-Tresse theorem, but may change a lot. We
describe in details the case of the motion group O(n) ⋉ R^{n} acting on the
full (unconstraint) jet-space as well as on some invariant equations.
In this note we discuss some formal properties of universal linearization operator, relate this to brackets of non-linear differential operators and discuss application to the calculus of auxiliary integrals, used in compatibility reductions of PDEs.
Many methods for reducing and simplifying differential equations are known. They provide
various generalizations of the original symmetry approach of Sophus Lie. Plenty of relations
between them have been noticed and in this note a unifying approach will be discussed.
It is rather close to the differential constraint method, but we make this rigorous basing on
recent advances in compatibility theory of non-linear overdetermined systems and homological
methods for PDEs.
Description:
Dette er forfatternes aksepterte versjon.
This is the author’s final accepted manuscript.
In this paper we describe the algebra of differential invariants for GL(n,C)-structures. This leads to classification of almost complex structures
of general positions. The invariants are applied to the existence
problem of higher-dimensional pseudoholomorphic submanifolds.
If E is an elliptic curve, then the Galois group of the extension generated by the n-torsion points acts on these points. We prove a quadratic reciprocity law involving this group action. This law is an extension of the usual quadratic reciprocity law.
Description:
This is a preprint of an article to
be published in Acta Scientiarum Mathematicarum
Waves that are reflected and refracted by material bodies also transfer momentum to these bodies. This means that the wave field induces a force on the bodies, and multiple reflections between bodies induce forces between them.
Light is an electromagnetic wave phenomenon, and the waves carry energy and momentum. Hence, any object that is scattering and refracting light is also acted upon by a light induced force. This force is a tiny force and is usually ignored, but if the objects are small enough the force induced by the light field would dominate all other forces. Due to this it is possible to manipulate small objects using light from a laser.
This thesis is based on an experiment on optical binding of two dielectric spheres, where the spheres were small enough to make the force induced by the light field the dominating force. In the experiment bistability and hysteresis in the equilibrium separations of the optically bound dielectric spheres were observed in one dimension. In this thesis the experiment will be modeled with a simplified setup, and the goal is to see if it is possible to find bistability in two dimensions also. Numerical approximations are used to calculate the wave field, and from this the force on the objects can be found.
Description:
Appendix A to D have been removed from the published pdf at the request of the author.
In this thises we consider the Lie algebra that corresponds to the Lie
pseudogroup of all conformal transformations on the plane. This conformal Lie algebra is canonically represented as a Lie algebra of vector fields on R^2. We will find all possible representations of vector fields in R^3=J^0R^2 which projects to the canonical representation and find the algebra of scalar differential invariants for each these representations of the conformal Lie algebra in J^0R^2.
We establish an efficient compatibility criterion for a system of generalized
complete intersection type in terms of certain multi-brackets of
differential operators. These multi-brackets generalize the higher Jacobi-
Mayer brackets, important in the study of evolutionary equations and the
integrability problem. We also calculate Spencer δ-cohomology of generalized
complete intersections and evaluate the formal functional dimension
of the solutions space. The results are applied to establish new integration
methods and solve several differential-geometric problems.