Explicit volume formula for a hyperbolic tetrahedron in terms of edge lengths

We consider a compact hyperbolic tetrahedron of a general type. It is a convex hull of four points called vertices in the hyperbolic space [Formula: see text]. It can be determined by the set of six edge lengths up to isometry. For further considerations, we use the notion of edge matrix of the tetrahedron formed by hyperbolic cosines of its edge lengths. We establish necessary and sufficient conditions for the existence of a tetrahedron in [Formula: see text]. Then we find relations between their dihedral angles and edge lengths in the form of a cosine rule. Finally, we obtain exact integral formula expressing the volume of a hyperbolic tetrahedron in terms of the edge lengths. The latter volume formula can be regarded as a new version of classical Sforza’s formula for the volume of a tetrahedron but in terms of the edge matrix instead of the Gram matrix.

The volume of a compact hyperbolic antiprism

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216518420105 ◽

2018 ◽

Vol 27 (13) ◽

pp. 1842010

Author(s):

Nikolay Abrosimov ◽

Bao Vuong

Keyword(s):

Symmetry Group ◽

Hyperbolic Space ◽

Convex Polyhedron ◽

Rotational Symmetry ◽

Sufficient Conditions ◽

Dihedral Angles ◽

Integral Formulas ◽

We consider a compact hyperbolic antiprism. It is a convex polyhedron with [Formula: see text] vertices in the hyperbolic space [Formula: see text]. This polyhedron has a symmetry group [Formula: see text] generated by a mirror-rotational symmetry of order [Formula: see text], i.e. rotation to the angle [Formula: see text] followed by a reflection. We establish necessary and sufficient conditions for the existence of such polyhedra in [Formula: see text]. Then we find relations between their dihedral angles and edge lengths in the form of a cosine rule. Finally, we obtain exact integral formulas expressing the volume of a hyperbolic antiprism in terms of the edge lengths.

RIEMANNIAN GEOMETRY OF HARTOGS DOMAINS

International Journal of Mathematics ◽

10.1142/s0129167x09005236 ◽

2009 ◽

Vol 20 (02) ◽

pp. 139-148 ◽

Cited By ~ 4

Author(s):

ANTONIO J. DI SCALA ◽

ANDREA LOI ◽

FABIO ZUDDAS

Keyword(s):

Hyperbolic Space ◽

Open Subset ◽

Riemannian Geometry ◽

Sufficient Conditions ◽

Complex Hyperbolic Space ◽

Hartogs Domain ◽

Straight Line ◽

Hartogs Domains ◽

Geodesically Complete ◽

Let DF = {(z0, z) ∈ ℂn | |z0|2 < b, ||z||2 < F(|z0|2)} be a strongly pseudoconvex Hartogs domain endowed with the Kähler metric gF associated to the Kähler form [Formula: see text]. This paper contains several results on the Riemannian geometry of these domains. These are summarized in Theorems 1.1–1.3. In the first one we prove that if DF admits a non-special geodesic (see definition below) through the origin whose trace is a straight line then DF is holomorphically isometric to an open subset of the complex hyperbolic space. In the second theorem we prove that all the geodesics through the origin of DF do not self-intersect, we find necessary and sufficient conditions on F for DF to be geodesically complete and we prove that DF is locally irreducible as a Riemannian manifold. Finally, in Theorem 1.3, we compare the Bergman metric gB and the metric gF in a bounded Hartogs domain and we prove that if gB is a multiple of gF, namely gB = λ gF, for some λ ∈ ℝ+, then DF is holomorphically isometric to an open subset of the complex hyperbolic space.

The moments of the number of exits from a simply connected region

Advances in Applied Probability ◽

10.1239/aap/1035227998 ◽

1998 ◽

Vol 30 (1) ◽

pp. 167-180 ◽

Cited By ~ 2

Author(s):

Robert Illsley

Keyword(s):

Integral Formula ◽

Sufficient Conditions ◽

Moment Matrix ◽

Expected Number ◽

Factorial Moments ◽

Gaussian Case ◽

Simply Connected Region ◽

Necessary And Sufficient ◽

Simply Connected ◽

Stationary Vector

We generalise the work of Cramér and Leadbetter, Ylvisaker and Ito on the level crossings of a stationary Gaussian process to multivariate processes. Necessary and sufficient conditions for the existence of the expected number of crossings E(C) of the boundary of a region of ℝp by a stationary vector stochastic process are obtained, along with an explicit formula for E(C) in the Gaussian case. A rigorous proof of Belyaev's integral formula for the factorial moments of the number of exits from a region of ℝp is given for a class of processes which includes Gaussian processes having a finite second order spectral moment matrix. Applications to χ2 processes are briefly considered.

The moments of the number of exits from a simply connected region

Advances in Applied Probability ◽

10.1017/s0001867800008144 ◽

1998 ◽

Vol 30 (01) ◽

pp. 167-180 ◽

Cited By ~ 1

Author(s):

Robert Illsley

Keyword(s):

Integral Formula ◽

Sufficient Conditions ◽

Moment Matrix ◽

Expected Number ◽

Factorial Moments ◽

Gaussian Case ◽

Simply Connected Region ◽

Necessary And Sufficient ◽

Simply Connected ◽

Stationary Vector

We generalise the work of Cramér and Leadbetter, Ylvisaker and Ito on the level crossings of a stationary Gaussian process to multivariate processes. Necessary and sufficient conditions for the existence of the expected number of crossings E(C) of the boundary of a region of ℝ p by a stationary vector stochastic process are obtained, along with an explicit formula for E(C) in the Gaussian case. A rigorous proof of Belyaev's integral formula for the factorial moments of the number of exits from a region of ℝ p is given for a class of processes which includes Gaussian processes having a finite second order spectral moment matrix. Applications to χ2 processes are briefly considered.

Pluricanonical systems for 3-folds and 4-folds of general type

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004111000648 ◽

2011 ◽

Vol 152 (1) ◽

pp. 9-34 ◽

Cited By ~ 3

Author(s):

LORENZO DI BIAGIO

Keyword(s):

Lower Bounds ◽

Large Volume ◽

General Type ◽

Sufficient Conditions ◽

Canonical Map ◽

Pluricanonical Maps ◽

AbstractWe explicitly find lower bounds on the volume of threefolds and fourfolds of general type in order to have non-vanishing of pluricanonical systems and birationality of pluricanonical maps. In the case of threefolds of large volume, we also give necessary and sufficient conditions for the fourth canonical map to be birational.

Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334) ◽

Necessary and sufficient conditions for robust stability of the convex-hull of matrices with rank-one uncertainty

10.1109/acc.2000.877036 ◽

2000 ◽

Author(s):

J. Collado ◽

R. Lozano

Keyword(s):

Convex Hull ◽

Robust Stability ◽

Sufficient Conditions ◽

Rank One ◽

REMARKS ON SELF-AFFINE FRACTALS WITH POLYTOPE CONVEX HULLS

Fractals ◽

10.1142/s0218348x1000510x ◽

2010 ◽

Vol 18 (04) ◽

pp. 483-498 ◽

Cited By ~ 3

Author(s):

IBRAHIM KIRAT ◽

ILKER KOCYIGIT

Keyword(s):

Convex Hull ◽

Lebesgue Measure ◽

Spectral Radius ◽

Sufficient Conditions ◽

Convex Hulls ◽

Compact Set ◽

Joint Spectral Radius ◽

Digit Set ◽

Necessary And Sufficient ◽

Digit Expansions

Suppose that the set [Formula: see text] of real n × n matrices has joint spectral radius less than 1. Then for any digit set D = {d1, …, dq} ⊂ ℝn, there exists a unique non-empty compact set [Formula: see text] satisfying [Formula: see text], which is typically a fractal set. We use the infinite digit expansions of the points of F to give simple necessary and sufficient conditions for the convex hull of F to be a polytope. Additionally, we present a technique to determine the vertices of such polytopes. These answer some of the related questions of Strichartz and Wang, and also enable us to approximate the Lebesgue measure of such self-affine sets. To show the use of our results, we also give several examples including the Levy dragon and the Heighway dragon.

The extinction time of a general birth and death process with catastrophes

Journal of Applied Probability ◽

10.1017/s0021900200116031 ◽

1986 ◽

Vol 23 (04) ◽

pp. 851-858 ◽

Cited By ~ 1

Author(s):

P. J. Brockwell

Keyword(s):

Laplace Transform ◽

Size Distribution ◽

Sufficient Conditions ◽

Death Process ◽

Extinction Time ◽

Birth And Death Process ◽

Different Types ◽

The Laplace Transform ◽

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.