Narrow Results

Using the Calabi–Yau technique to solve Monge-Ampère equations, we translate a result of T. Fujita on approximate Zariski decompositions into an analytic setting and combine this to the holomorphic Morse inequalities in order to express the volume of a line bundle as the maximum of the mean curvatures of all the singular Hermitian metrics on it, with a way to pick an element at which the maximum is reached and satisfying a singular Monge–Ampère equation. This enables us to introduce the volume of any (1,1)-class on a compact Kähler manifold, and Fujita's theorem is then extended to this context.

We establish some comparison theorems on Finsler manifolds with curvature quadratic decay. As their applications, we obtain some optimal Cheeger–Gromov–Taylor type compact theorems, volume growth and Mckean type estimate for the first Dirichlet eigenvalue for such manifolds. Although we present the results for Finsler manifolds, they are all new results for Riemannian manifolds.

We calculate here the Raman frequencies of some lattice modes as a function of pressure at constant temperatures for the solid and liquid phases of benzene. The observed data for the molar volume from literature is used to calculate the Raman frequencies through the mode Grüneisen parameter in benzene.

Our calculated frequencies are in good agreement with the observed data when the mode Grüneisen parameter is taken as a constant at one particular pressure in solid benzene.

It is shown here that the Raman frequencies can be calculated from the volume data, as demonstrated for benzene here.

We investigate the complexity of counting the number of integer points in tropical polytopes, and the complexity of calculating their volume. We study the tropical analogue of the outer parallel body and establish bounds for its volume. We deduce that there is no approximation algorithm of factor $\alpha =2^{\mathrm{\text{poly}}(m,n)}$ for the volume of a tropical polytope given by $n$ for the volume of a tropical polytope given by $n$ vertices in a space of dimension $m$, unless P$=$NP. Neither is there such an approximation algorithm for counting the number of integer points in tropical polytopes described by vertices. It follows that approximating these values for tropical polytopes is more difficult than for classical polytopes. Our proofs use a reduction from the problem of calculating the tropical rank.

By obtaining surgery descriptions of knots which lie on the genus one fiber of the trefoil or figure eight knot, we show that these include hyperbolic knots with arbitrarily large volume. These knots admit lens space surgeries and form two families of Berge knots. By way of tangle descriptions we also obtain surgery descriptions for these knots on minimally twisted chain links.

In this paper, we give a formula of the complex volumes of hyperbolic knot complements, which is related to the volume conjecture for hyperbolic knots.

Agol has conjectured that minimally twisted n-chain links are the smallest volume hyperbolic manifolds with n cusps, for n ≤ 10. In his thesis, Venzke mentions that these cannot be smallest volume for n ≥ 11, but does not provide a proof. In this paper, we give a proof of Venzke's statement for a number of cases. For n ≥ 60 we use a formula from work of Futer, Kalfagianni and Purcell to obtain a lower bound for volume. The proof for n between 12 and 25 inclusive uses a rigorous computer computation that follows methods of Moser and Milley. Finally, we prove that the n-chain link with 2m or 2m + 1 half-twists cannot be the minimal volume hyperbolic manifold with n cusps, provided n ≥ 60 or |m| ≥ 8, and we give computational data indicating this remains true for smaller n and |m|.

We calculate the volumes of the hyperbolic twist knot cone-manifolds using the Schläfli formula. Even though general ideas for calculating the volumes of cone-manifolds are around, since there is no concrete calculation written, we present here the concrete calculations. We express the length of the singular locus in terms of the distance between the two axes fixed by two generators. In this way the calculation becomes easier than using the singular locus directly. The volumes of the hyperbolic twist knot cone-manifolds simpler than Stevedore's knot are known. As an application, we give the volumes of the cyclic coverings over the hyperbolic twist knots.

This note is dedicated to the memory of Slavik Jablan, and the last work we had together on the computation and properties of Cherns–Simons invariant of knots and links.

Let $C(2n,3)$ be the family of two bridge knots of slope $(4n+1)/(6n+1)$. We calculate the volumes of the $C(2n,3)$ cone-manifolds using the Schläfli formula. We present the concrete and explicit formula of them. We apply the general instructions of Hilden, Lozano and Montesinos-Amilibia and extend the Ham, Mednykh and Petrov’s methods. As an application, we give the volumes of the cyclic coverings over those knots. For the fundamental group of $C(2n,3)$, we take and tailor Hoste and Shanahan’s. As a byproduct, we give an affirmative answer for their question whether their presentation is actually derived from Schubert’s canonical two-bridge diagram or not.

We give explicit formulas for the volumes of hyperbolic cone-manifolds of double twist knots, a class of two-bridge knots which includes twist knots and two-bridge knots with Conway notation $C(2n,3)$. We also study the Riley polynomial of a class of one-relator groups which includes two-bridge knot groups.

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 $C(2n,4)$.

A ternary, integral quadratic form is called unilateral if its associated orbifold $Q_f$ is orientable. Examples of unilateral forms $f$ and their associated orbifolds $Q_f$ are given. The examples are selected to give credit to the conjecture that every form with square-free, prime determinant has a unilateral $B$-cover.

An example of an integral ternary quadratic form $f$ such that its associated orbifold $Q_f$ is a manifold is given. Hence, the title is proved.

Cho and Murakami defined the potential function for a link $L$ in $S^3$ whose critical point, slightly different from the usual sense, corresponds to a boundary-parabolic representation $\rho :{\pi }_1(S^3\setminus L)\to {\mathrm{\text{PSL}}}_2(ℂ)$. They also showed that the volume and Chern–Simons invariant of $\rho$ can be computed from the potential function with its partial derivatives. In this paper, we extend the potential function to a representation that is not necessarily boundary-parabolic. We show that under a mild assumption it leads us to a combinatorial formula for computing the volume and Chern–Simons invariant of a ${\mathrm{\text{PSL}}}_2(ℂ)$-representation of a closed 3-manifold.

We provide an elementary polyhedral approach to study and deduce results about the arithmeticity and commensurability of an infinite family of hyperbolic link complements $M_n$ for $n\ge 3$. The manifold $M_n$ is the complement of ${𝕊}^3$ by the ($2n$)-link chain ${{𝒟}}_{2n}$ and has $2n$ cusps. We show that $M_n$ is closely related to a hyperbolic Coxeter orbifold that is commensurable to an orbifold with a single cusp. Vinberg’s arithmeticity criterion and certain cusp density and volume computations allow us to reproduce some of the main results in [20] and [18] about $M_n$ in a comparatively elementary and direct way.

As a by-product, we give a rigorous proof of Thurston’s volume formula for $M_n$ and deduce that, for $n\ge 6$, the volume of $M_n$ is strictly bigger than the volume of the ($2n-1$)-cyclic cover over one component of the Whitehead link.

In this paper, we propose a technique for intuitive, interactive modelling of 3D shapes. The technique is based on the Level–Set Method which has the virtue of easily handling changes to the topology of the represented solid. Furthermore, this method also leads to sculpting operations that are very simple and intuitive from a user perspective. A final virtue is that the LSM makes it easy to maintain a distance field representation of the represented solid. This has a number of benefits such as simplification of the rendering scheme and the curvature computation. A number of LSM speed functions which are suitable for shape modelling are proposed. However, normally these would result in tools that would affect the entire model. To facilitate local changes to the model, we introduce a windowing scheme which constrains the LSM to affect only a small part of the model. The LSM based sculpting tools have been incorporated in our sculpting system which also includes facilities for volumetric CSG and several techniques for visualization.

This paper reports an empirical analysis of the relationship between return autocorrelation, trading volume and volatility, following the seminal paper by Campbell, Grossman and Wang (1992) using data for A shares traded on the Shanghai and Shenzhen stock exchanges for the period 1992–2002. Campbell et al. argue that autocorrelation of returns will be negatively related to trading volume given that market makers will need to be rewarded with higher returns for accommodating noise traders. For our full sample we find remarkably consistent support for the CGW hypothesis and results — return autocorrelations are negatively but non-linearly related to lagged trading volume and less strongly to volatility. These results are quite robust with respect to different messures of volume and volatility. We argue that this is a striking result in view of the substantial differences between the US market in the 1960s, 1970s and 1980s and the Chinese market of the 1990s. The relationship proves to be unstable over short sub-periods although whether this is due to the relatively short sample we use or to the inherent instability of the Chinese market in its first decade of operation will not be clear until much longer data sets are available for Chinese stock prices.

We investigate various properties of the sublevel set G = {x : g(x) ≤ 1} and the integration of h on this sublevel set when g and h are positively homogeneous functions (and in particular homogeneous polynomials). For instance, the latter integral reduces to integrating hexp(-g) on the whole space ℝⁿ (a nonGaussian integral) and when g is a polynomial, then the volume of G is a convex function of the coefficients of g. We also provide a numerical approximation scheme to compute the volume of G or integrate h on G (or, equivalently to approximate the associated nonGaussian integral). We also show that finding the sublevel set {x : g(x) ≤ 1} of minimum volume that contains some given subset K is a (hard) convex optimization problem for which we also propose two convergent numerical schemes. Finally, we provide a Gaussian-like property of nonGaussian integrals for homogeneous polynomials that are sums of squares and critical points of a specific function.

It is proved that the volume of spherical or hyperbolic simplices, when considered as a function of the dihedral angles, can be extended continuously to degenerated simplices. This verifies affirmatively a conjecture of John Minor.