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On the positive region of \pi(x) - li(x)
Get PDFThe difference \pi(x) - li(x) has been the subject of lively interest since Littlewood's theorem (1914) that \pi(x) - li(x) changes sign infinitely often. The issue is to find an upper bound for the first crossover. Two papers on this issue were published in July 2010: Chao-Plymen, Int. J. Number Theory 6 (2010) 681 - 690, and Saouter-Demichel, Math. Comp. 79 (2010) 2395 - 2405. This double project includes a complete proof of Lehman's theorem, in which two crucial constants are reduced. A further improvement on the Saouter-Demichel article leads to some new theorems
Geometric structure in the tempered dual of the p-adic group SL(4)
No full textWe exhibit a definite geometric structure in the tempered dual of the p-adic group SL(4).
Especially interesting is the case of SL(4,Q2), when we reveal a tetrahedron of reducibility in the tempered dual. This conforms to a recent geometric conjecture.
The L-packets in this article all conform to the L-packet conjecture in http://eprints.ma.man.ac.uk/1504
Hedging Strategies : Complete and Incomplete Systems of Markets
Get PDFWe are motivated by the latest statistical facts that weather directly affects about 20% of the U.S. economy and, as a result energy companies experience enormous potential losses due to weather that is colder or warmer than expected for a certain period of a year. Incompleteness and illiquidity of markets renders hedging the exposure using energy as the underlying asset impossible. We attempt to price and hedge a written European call option with an asset that is highly correlated with the underlying asset; still, a significant amount of the total risk cannot be diversified. Yet, our analysis begins by considering hedging in a complete markets system that can be utilised as a theoretical point of reference, relative to which we can assess incompleteness. The Black-Scholes Model is introduced and the Monte Carlo approach is used to investigate the effects of three hedging strategies adopted; Delta hedging, Static hedging and a Stop-Loss strategy. Next, an incomplete system of markets is assumed and the Minimal Variance approach is demonstrated. This approach results in a non-linear PDE for the option price. We use the actuarial standard deviation principle to modify the PDE to account for the unhedgeable risk. Based on the derived PDE, two additional hedging schemes are examined: the Delta hedging and the Stop-loss hedging. We set up a risk-free bond to keep track of any money injected or removed from the portfolios and provide comparisons between the hedging schemes, based on the Profit/Loss distributions and their main statistical features, obtained at expiry
FORWARD AND INVERSE PROBLEM FOR NEMATIC LIQUID CRYSTALS
Get PDFThis thesis starts with an introduction to liquid crystal properties, which are
needed to proceed with this research. From the dielectric tensor which appears
in the Maxwell equations, we were able to obtain a relationship between the
elements on the main diagonal of the dielectric tensor. This relationship has
been discussed and illustrated with some examples for both positive and negative
birefringence.
By introducing a constrain on the Berreman model, we were able to derive
a 2 × 2 differential equation in matrix form which works for both normal and
oblique incident. This equation gives us a simple and intuitive means to analyze
the evolution of light through all sorts of media i.e. isotropic, anisotropic with
a fixed transmission axis and anisotropic with a twisted transmission axis of
anisotropy.
One of the objectives of this research was to find the right technique to solve
the 2 × 2 dynamic equation. Fortunately, the classic Floquet’s theory guarantees
the existence of the solution and it gives some of its characteristics. In fact, we
were able to solve the 2×2 Schrödinger equation by a new method which we called
it in this thesis a rotational frame method. The obtained solution is consistent
with Floquet’s theory and agrees totally with the Jones solutions. Also, this
solution allows us to test the Berreman approximation.Finally, in this research we were able to encode the orientation of the optical
axis inside a liquid crystal sample, into the potential of the Schrödinger equation.
As a consequence of that, solving the inverse problem of the Schrödinger equation
that is recovering the potential, is indeed recovering the orientation of the director
inside the sample. The Berreman inverse problem and its corresponding linearized
problem has been considered in this thesis. In these sections, we give a rigorous
derivation for the Fréchet derivative
Local presentability of categories of sheaves of modules
Get PDFWe show that the category of modules over a ring in a Grothendieck topos is monadic, and as a consequence, the category of modules over a ring in a locally finitely presentable topos is locally finitely presentable
Witten-Hodge theory for manifolds with boundary
Get PDFWe consider a compact, oriented, smooth Riemannian manifold (with or without boundary) and we suppose is a torus acting by isometries on . Given in the Lie algebra and corresponding vector field on , one defines Witten's inhomogeneous operator \d_{X_M} = \d+\iota_{X_M}: \Omega_G^\pm \to\Omega_G^\mp (even/odd invariant forms on ). Witten \cite{Witten} showed that the resulting cohomology classes have -harmonic representatives (forms in the null space of \Delta_{X_M} = (\d_{X_M}+\delta_{X_M})^2), and the cohomology groups are isomorphic to the ordinary de Rham cohomology groups of the fixed point set. Our principal purpose is to extend these results to manifolds with boundary. In particular, we define relative (to the boundary) and absolute versions of the -cohomology and show the classes have representative -harmonic fields with appropriate boundary conditions. To do this we present the relevant version of the Hodge-Morrey-Friedrichs decomposition theorem for invariant forms in terms of the operator \d_{X_M} and its adjoint ; the proof involves showing that certain boundary value problems are elliptic. We also elucidate the connection between the -cohomology groups and the relative and absolute equivariant cohomology, following work of Atiyah and Bott \cite{AB}. This connection is then exploited to show that every harmonic field with appropriate boundary conditions on has a uniqe extension to an -harmonic field on , with corresponding boundary conditions
A posteriori error bounds for discrete balanced truncation
Get PDFBalanced truncation of discrete linear time-invariant systems is an automatic method once an error tolerance is specified and yields an a priori error bound, which is why it is widely used in engineering for simulation and control. We present some new insight into this method. We derive a discrete version of Antoulas's -norm error formula \cite[p.218]{Ant05} and show how to adapt it to some special cases. This error bound is an a posteriori computable upper bound for the -norm of the error system defined as the system whose transfer function corresponds to the difference between the transfer function of the original system and the transfer function of the reduced system. The main advantage of our results is that we use the information already available in the balanced truncation algorithm in order to compute the -norm instead of computing one gramian of the corresponding error system. There is always a computational restriction on solving high-dimensional Stein equations for gramians. The a posteriori bound gives insight into the quality of the reduced system and can be used to solve many problems accompanying the order reduction operation
Parameterising Structure Preserving Transformations Connecting Quadratic Matrix Polynomials
Get PDFStructure Preserving Transformations in the present definition provide a formal means by which every quadratic system isospectral to a given quadratic system may be determined. This paper presents a parameterisation for these structure preserving transformations which does not require that any of the coefficient matrices is non-singular and which provides a straightforward means to preserve each one of 8 classes of symmetry possible in a quadratic matrix polynomial
Dynamics of poles with position-dependent strengths and its optical analogues
Get PDFDynamics of point vortices is generalized in two ways: first by making the strengths complex, which allows for sources and sinks in superposition with the usual vortices, second by making them functions of position. These generalizations lead to a rich dynamical system, which is nonlinear and yet has enough conservation laws coming from a Hamiltonian-like formalism. We then discover that in this system the motion of a pair mimics the behavior of rays in geometric optics. We describe several exact solutions with optical analogues, notably Snell's law and the law of reflection off a mirror, and perform numerical experiments illustrating some striking behavior
Error estimation and stabilization for low order finite elements
Get PDFThis thesis covers three topics�a posteriori error estimation, mixed finite element ap- proximations for anisotropic meshes and the solution of the time-dependent Navier-Stokes equations using a stabilized Q1 � P0 approximation.
First, we find effective error estimators for (bi-)quadratic approximations for the dif- fusion problem, and (bi-)quadratic velocity and (bi-)linear pressure mixed approximations for incompressible flow problems. The efficiency and reliability of the error estimators are established in the case of the Stokes problem.
Second, since standard inf-sup stable mixed approximations typically become unstable for anisotropic meshes, we devote our attention to a stabilized Q1�P0 approximation, which is introduced by Kechkar and Silvester [Math. Comp., 58, 1�10, 1992]. We establish a robust a priori error bound for this stabilized Q1 � P0 approximation for anisotropic meshes.
Finally, the stabilized Q1 � P0 approximation is applied to solving time dependent in- compressible flow problems with an adaptive time stepping method introduced by Kay et al. [SIAM J. Sci. Comput., 32, 111�128, 2010]. The main contribution of this part is to find the optimal stabilization parameter, which is eventually shown to be inversely proportional to the Reynolds number of the flow