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 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.

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.

Several identities similar to the Schläfli formula are established for tetrahedra in a space of constant curvature.

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 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.

I use five separate measures of deviation from Put-Call Parity of options on a stock without splits or dividends as separate negative measures for market efficiency. I rely upon the theory of trading volume as a function of short sales costs, etc., and that of market efficiency as a function of trading volume, etc. derived by Bhattacharya (2019). I use Three-Stage Least Squares (3SLS) to estimate this structural system, separately for Nasdaq and non-Nasdaq U.S. stocks. I find, contrary to much previous theoretical and empirical work, that the impact of short sales costs & constraints on market efficiency is not significantly negative and that the impact of trading volume on market efficiency is not significantly positive, and my results are robust to various econometric specifications and financial economic assumptions.

We express the volume of a simplex in spherical or hyperbolic spece by iterated integrals of differential forms following Schläfli and Aomoto. We study analytic properties of the volume function and describe the differential equation satisfied by this function.

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.

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.

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.

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.

We investigate geometrical properties and inequalities satisfied by the complex difference body, in the sense of studying which of the classical ones for the difference body have an analog in the complex framework. Among others we give an equivalent expression for the support function of the complex difference body and prove that, unlike the classical case, the dimension of the complex difference body depends on the position of the body with respect to the complex structure of the vector space. We use spherical harmonics to characterize the bodies for which the complex difference body is a ball, we prove that it is a polytope if and only if the two bodies involved in the construction are polytopes and provide several inequalities for classical magnitudes of the complex difference body, as volume, quermassintegrals and diameter, in terms of the corresponding ones for the involved bodies.

Teissier has proven remarkable inequalities for intersection numbers s_i = (ℒⁱ ⋅ ℳ^d-i) of a pair of nef line bundles ℒ, ℳ on a d-dimensional complete algebraic variety over a field. He asks if two nef and big line bundles are numerically proportional if the inequalities are all equalities. In this paper, we show that this is true in the most general possible situation, for nef and big line bundles on a proper irreducible scheme over an arbitrary field k. Boucksom, Favre and Jonsson have recently established this result on a complete variety X over an algebraically closed field of characteristic zero. Their proof involves an ingenious extension of the intersection theory on a variety to its Zariski Riemann Manifold. This extension requires the existence of a direct system of nonsingular varieties dominating X. We make use of a simpler intersection theory which does not require resolution of singularities, and extend volume to an arbitrary field and prove its continuous differentiability, extending results of Boucksom, Favre and Jonsson, and of Lazarsfeld and Mustaţă. A goal in this paper is to provide a manuscript which will be accessible to many readers. As such, subtle topological arguments which are required to give a complete proof in [S. Bouksom et al., J. Algebraic Geometry18 (2009) 279–308] have been written out in this manuscript, in the context of our intersection theory, and over arbitrary varieties.

The aim of this paper is to give fine asymptotics for random variables with moments of Gamma type. Among the examples, we consider random determinants of Laguerre and Jacobi beta ensembles with varying dimensions (the number of observed variables and the number of measurements vary and may be different). In addition to the Dyson threefold way of classical random matrix models (GOE, GUE, GSE), we study random determinants of random matrices of the so-called tenfold way, including the Bogoliubov–de Gennes and chiral ensembles from mesoscopic physics. We show that fixed-trace matrix ensembles can be analyzed as well. Finally, we add fine asymptotics for the $p(n)$-dimensional volume of the simplex with $p(n)+1$ points in ${ℝ}^n$ distributed according to special distributions, which is strongly correlated to Gram matrix ensembles. We use the framework of mod-$\phi$ convergence to obtain extended limit theorems, Berry–Esseen bounds, precise moderate deviations, large and moderate deviation principles as well as local limit theorems. The work is especially based on the recent work of Dal Borgo et al. [Mod-Gaussian convergence for random determinants, Ann. Henri Poincaré (2018)].