Narrow Results

An explicit formula for the $A$-polynomial of the knot with Conway’s notation $C(2n,3)$ is obtained from the explicit Riley–Mednykh polynomial of it.

This paper extends the work by Mednykh and Rasskazov presented in [On the structure of the canonical fundamental set for the 2-bridge link orbifolds, Universität Bielefeld, Sonderforschungsbereich 343, Discrete Structuren in der Mathematik, Preprint (1988), pp. 98–062, www.mathematik.uni-bielefeld.de/sfb343/preprints/pr98062.ps.gz]. By using their approach, we derive the Riley–Mednykh polynomial for a family of $2$-bridge knot orbifolds. As a result, we obtain explicit formulae for the volumes and Chern–Simons invariants of orbifolds and cone-manifolds on the knot with Conway’s notation $C(2n,4)$.

We hereby present an explicit formula for European options on coupon bearing bonds in the Heath–Jarrow–Morton one factor model with non-stochastic volatility. The formula extends the Jamshidian formula for zero-coupon bonds for special form of volatility. Moreover we present a formula for zero-coupon bonds without condition on the volatility. We provide also an explicit way to compute the hedging ratio (Δ) in order to hedge the options individually.

We derive explicit formulae for spherically symmetric solutions to the system of multidimensional zero-pressure gas dynamics and its adhesion approximation. The asymptotic behavior of the explicit solutions of the adhesion approximation is studied here. We observe that the radial components of the velocity and density satisfy a simpler equation, which enables us to get explicit formulae for different types of domains and study its asymptotic behavior. A class of solutions to the inviscid system with conditions on the mass instead of conditions at origin is also analyzed here.

The optimal control problem for Burgers equation was first considered by Castro, Palacios and Zuazua. They proved the existence of a solution and proposed a numerical scheme to capture an optimal solution via the method of "alternate decent direction". In this paper, we introduce a new strategy for the optimal control problem for scalar conservation laws with convex flux. We propose a new cost function and by the Lax–Oleinik explicit formula for entropy solutions, the nonlinear problem is converted to a linear problem. Exploiting this property, we prove the existence of an optimal solution and, by a backward construction, we give an algorithm to capture an optimal solution.

We prove a formula for the limit of logarithmic derivatives of zeta functions in families of global fields with an explicit error term. This can be regarded as a rather far reaching generalization of the explicit Brauer–Siegel theorem both for number fields and function fields.

We prove some results concerning the distribution of primes assuming the Riemann hypothesis. First, we prove the explicit result that there exists a prime in the interval for all x ≥ 2; this improves a result of Ramaré and Saouter. We then show that the constant 4/π may be reduced to (1 + ϵ) provided that x is taken to be sufficiently large. From this, we get an immediate estimate for a well-known theorem of Cramér, in that we show the number of primes in the interval is greater than for c = 3 + ϵ and all sufficiently large values of x.

We provide a lightweight algorithm to express each of the elementary symmetric polynomials as a linear combination of (products of) power sum symmetric polynomials and, also, the power sums purely in terms of the elementary symmetric polynomials. Our method does not use Newton’s identities, which give such relations only implicitly. Each identity in the sequence is generated through a single differentiation.