Toric Surfaces Research Articles

Paraxial approximation of the electrostatic potential of a charged nonconducting torus

In an electrostatics it is possible to allocate two kinds of problems: the field determination at an unknown location of the charges, but the given electric potential at the boundaries of the considered region, and the calculation of the potential and components of the electric field strength in a region free of charges from the known spatially-limited electric charge distribution. In this paper, preference has been given to the second formulation, which has been added by the principle of superposition of electric fields. Further, we consider the problem of the distribution of the electrostatic potential over a thin-walled torus, uniformly charged along the surface. The condition of constancy of the surface charge density is realized only when using a torus with dielectric walls. To find the distribution of the electrostatic potential around a non-conducting torus uniformly charged along the surface, the Poisson equation was considered, which solution was represented as a surface integral. In toroidal coordinates, an expression for the electrostatic potential is obtained, which is calculated in terms of the complete elliptic integral of the first kind. The torus potential is explored on presence of a local extreme. The functions of cylindrical coordinates (paraxial approximation) are constructed, approximating the potential and field intensity in the outer region of the torus, and it is proved that the potential has a saddle shape in the region close to the centre. The spatial distribution of the electrostatic potential in the outer region of a uniformly charged along the torus surface has been visualized, using Matlab environment.

A simulation algorithm for Brownian dynamics on complex curved surfaces.

Brownian dynamics of colloidal particles on complex curved surfaces has found important applications in diverse physical, chemical, and biological processes. However, most Brownian dynamics simulation algorithms focus on relatively simple curved surfaces that can be analytically parameterized. In this work, we develop an algorithm to enable Brownian dynamics simulation on extremely complex curved surfaces. We approximate complex curved surfaces with triangle mesh surfaces and employ a novel scheme to perform particle simulation on these triangle mesh surfaces. Our algorithm computes forces and velocities of particles in global coordinates but updates their positions in local coordinates, which combines the strengths from both global and local simulation schemes. We benchmark the proposed algorithm with theory and then simulate Brownian dynamics of both single and multiple particles on torus and knot surfaces. The results show that our method captures well diffusion, transport, and crystallization of colloidal particles on complex surfaces with nontrivial topology. This study offers an efficient strategy for elucidating the impact of curvature, geometry, and topology on particle dynamics and microstructure formation in complex environments.

Profile of and Variability in Double-Peaked Balmer Emission Lines in 3C 445

Abstract We extract the multiple-epoch Balmer-line profiles of the heavily obscured quasar 3C 445 from the spectral curves in the literature, and analyze the emission-line profiles of the Hα and Hβ lines and the profile variability in the Hα line in the large time interval of more than three decades. The profile comparison between the Hα and Hβ lines shows that both Balmer lines share the profile with the same form, while the blue system of the Hβ line is seriously weaker than that of the Hα line. Moreover, the blue system of the Hα line suddenly disappeared completely and then did not appear again, however the other two components did not exhibit significant variation in the velocity or the amplitude. These findings suggest that the blue system of 3C 445, as with SDSS J153636.22+044127.0 and its analogs, is probable the result of the shock-heated outflowing gases. The observation angle of almost edge-on which the previous studies suggested can easily produce the high-speed and high-temperature shock in the collision between the massive outflow and the inner surface of the dusty torus.

Multibody dynamics simulation of thin-walled four-point contact ball bearing with interactions of balls, ring raceways and crown-type cage

The dynamic characteristics of thin-walled four-point contact ball bearing with crown-type cage are important to the dynamic performance and motion accuracy of an industrial robot. Considering multi-clearances, dynamic contact and impact relationships of the ball, ring raceway and crown-type cage, a general methodology for dynamic simulation analysis of the bearing is investigated in the proposed work. In accordance with the geometry of torus, the geometric equation of accurate ring raceway is derived and integrated into the three-dimensional ring raceway using user’s subroutines. The parameterized and assembled three-dimensional model of the bearing is established using ADAMS’s macro-programs. Applying a penalty formulation and a unilateral nonlinear spring–damper model to the bearing, the internal contact interaction is represented as the compliant contact force model using IMPACT function. The multibody contact dynamic models of the bearing are solved by HHT algorithm with ADAMS/Solver. The dynamic results of the contact force, impact force and motion stability of the bearing are discussed under the condition of different loads. The static load distribution and cage’s angular velocity of simulation model are verified by the theoretical values. The motion trajectory of outer ring’s center is circular with a whirling motion. The sphere-to-partial torus surface contacts (ball–racetrack contact) are always four contact points in the load zone of the bearing. Applying pure radial load or rotating radial load, the impact force of ball-to-cage small pocket contact is much larger than that of radial and axial load combination in the non-load zone of the bearing. As a result of the large impact force of ball-to-cage small pocket contact, the angular velocities of the ball and cage are varying greatly in the non-load zone. The impact force of ball-to-cage big pocket contact is very small. The angular velocity of the ball is always that of pure rolling in the load zone and varying slightly in the non-load zone. The new method can be applied to investigate dynamic analysis and design of high-precision industrial robots with multi-clearances, multi-ball bearings under complex, time-varying working conditions.

Discrete symmetries in dimer diagrams

We apply dimer diagram techniques to uncover discrete global symmetries in the fields theories on D3-branes at singularities given by general orbifolds of general toric Calabi-Yau threefold singularities. The discrete symmetries are discrete Heisenberg groups, with two ZN generators A, B with commutation AB = C BA, with C a central element. This fully generalizes earlier observations in particular orbifolds of C3, the conifold and Yp,q . The solution for any orbifold of a given parent theory follows from a universal structure in the infinite dimer in R2 giving the covering space of the unit cell of the parent theory before orbifolding. The generator A is realized as a shift in the dimer diagram, associated to the orbifold quantum symmetry; the action of B is determined by equations describing a 1-form in the dimer graph in the unit cell of the parent theory with twisted boundary conditions; finally, C is an element of the (mesonic and baryonic) non-anomalous U (1) symmetries, determined by geometric identities involving the elements of the dimer graph of the parent theory. These discrete global symmetries of the quiver gauge theories are holographically dual to discrete gauge symmetries from torsion cycles in the horizon, as we also briefly discuss. Our findings allow to easily construct the discrete symmetries for infinite classes of orbifolds. We provide explicit examples by constructing the discrete symmetries for the infinite classes of general orbifolds of C3, conifold, and complex cones over the toric del Pezzo surfaces, dP1, dP2 and dP3.

Toric log del Pezzo surfaces with one singularity

AbstractThis paper focuses on the classification (up to isomorphism) of all toric log Del Pezzo surfaces with exactly one singularity, and on the description of how they are embedded as intersections of finitely many quadrics into suitable projective spaces.

Fine structure of Jackiw-Teitelboim quantum gravity

We investigate structural aspects of JT gravity through its BF description. In particular, we provide evidence that JT gravity should be thought of as (a coset of) the noncompact subsemigroup SL+(2, ℝ) BF theory. We highlight physical implications, including the famous Plancherel measure sinh 2π sqrt{E} . Exploiting this perspective, we investigate JT gravity on more generic manifolds with emphasis on the edge degrees of freedom on entangling surfaces and factorization. It is found that the one-sided JT gravity degrees of freedom are described not just by a Schwarzian on the asymptotic boundary, but also include frozen SL+(2, ℝ) degrees of freedom on the horizon, identifiable as JT gravity black hole states. Configurations with two asymptotic boundaries are linked to 2d Liouville CFT on the torus surface.

Computing graded Betti tables of toric surfaces

We present various facts on the graded Betti table of a projectively embedded toric surface, expressed in terms of the combinatorics of its defining lattice polygon. These facts include explicit formulas for a number of entries, as well as a lower bound on the length of the quadratic strand that we conjecture to be sharp (and prove to be so in several special cases). We also present an algorithm for determining the graded Betti table of a given toric surface by explicitly computing its Koszul cohomology and report on an implementation in SageMath. It works well for ambient projective spaces of dimension up to roughly 25 25 , depending on the concrete combinatorics, although the current implementation runs in finite characteristic only. As a main application we obtain the graded Betti table of the Veronese surface ν 6 ( P 2 ) ⊆ P 27 \nu _6(\mathbb {P}^2) \subseteq \mathbb {P}^{27} in characteristic 40 009 40\,009 . This allows us to formulate precise conjectures predicting what certain entries look like in the case of an arbitrary Veronese surface ν d ( P 2 ) \nu _d(\mathbb {P}^2) .

Constructing non-Mori Dream Spaces from negative curves

We study blowups of weighted projective planes at a general point, and more generally blowups of toric surfaces of Picard number one. Based on the positive characteristic methods of Kurano and Nishida, we give a general method for constructing examples of Mori Dream Spaces and non-Mori Dream Spaces among such blowups. Compared to previous constructions, this method uses the geometric properties of the varieties and applies to a number of cases. We use it to fully classify the examples coming from two families of negative curves.

On derived categories of arithmetic toric varieties

We begin a systematic investigation of derived categories of smooth projective toric varieties defined over an arbitrary base field. We show that, in many cases, toric varieties admit full exceptional collections. Examples include all toric surfaces, all toric Fano 3-folds, some toric Fano 4-folds, the generalized del Pezzo varieties of Voskresenskii and Klyachko, and toric varieties associated to Weyl fans of type $A$. Our main technical tool is a completely general Galois descent result for exceptional collections of objects on (possibly non-toric) varieties over non-closed fields.

Quantum-entangled superluminal double-helix photon produces a relativistic superluminal quantum-vortex zitterbewegung electron and positron

Two quantum-entangled spin-½ half-photons compose a double-helix photon model. The photon model is composed of a pair of superluminal helically-moving opposite electric charges of magnitude Q = e sqrt(2/alpha) = 16.6 e where alpha is the fine structure constant. During electron-positron pair production these charged half-photons curl up their single-helical trajectories to form geometrically-compatible relativistic zitterbewegung-frequency superluminal quantum-vortex electron and positron models. The superluminal charged energy quantum composing the resting electron and positron models moves on the surface of a mathematical horn torus.

Real inflection points of real hyperelliptic curves

Given a real hyperelliptic algebraic curve $X$ with non-empty real part and a real effective divisor $\mc{D}$ arising via pullback from $\mathbb{P}^1$ under the hyperelliptic structure map, we study the real inflection points of the associated complete real linear series $|\mc{D}|$ on $X$. To do so we use Viro's patchworking of real plane curves, recast in the context of some Berkovich spaces studied by M. Jonsson. Our method gives a simpler and more explicit alternative to limit linear series on metrized complexes of curves, as developed by O. Amini and M. Baker, for curves embedded in toric surfaces.

Higher Nash blowups of the A3-singularity

We show that the n-th Nash blowup of the toric surface singularity of type A3 is singular for any . It has been known that the normalization of the n-th Nash blowup of an affine normal toric variety is the toric variety associated to the Gröbner fan of a certain ideal. In our case, we prove that the Gröbner fan contains a certain non-regular cone for any . Thus we conclude that the normalizations are singular, and so are the Nash blowups.

Description of aspheric surfaces

Abstract Aspheric surfaces, in particular rotationally invariant surfaces, can be described according to the ISO standard 10110 Part 12 as sagitta functions of the surface coordinates. Usually, such functions are standardized as a combination of conic terms and power series or orthogonal polynomials. Similar functions are applied for surface forms, which are not rotationally invariant as cylindric and toric surfaces. In the following, different forms of describing aspheric surfaces as given in the standard as well as other forms will be presented and compared in an overview, and their special features will be discussed.

Weierstrass Semigroups Satisfying MP Equalities and Curves on Toric Surfaces

A numerical semigroup H is said to be cyclic if it is the Weierstrass semigroup of a total ramification point of some cyclic covering of the projective line. In this case, the elements of the standard basis of H satisfy numerical conditions that we have chosen to term MP equalities, after Morrison and Pinkham who first proved them. The converse is not true, as there are semigroups satisfying the MP equalities that are not cyclic. In this paper, we consider the situation of a smooth curve C lying on a smooth compact toric surface S acted on by the torus T which is a dense open subset of S. We prove that the Weierstrass semigroup of a total ramification point of the restriction to C of a toric fibration of S, which lies on some T-invariant divisor, is cyclic if and only if it satisfies the MP equalities.

Canonical syzygies of smooth curves on toric surfaces

In a first part of this paper, we prove constancy of the canonical graded Betti table among the smooth curves in linear systems on Gorenstein weak Fano toric surfaces. In a second part, we show that Green's canonical syzygy conjecture holds for all smooth curves of genus at most 32 or Clifford index at most 6 on arbitrary toric surfaces. Conversely we use known results on Green's conjecture (due to Lelli-Chiesa) to obtain new facts about graded Betti tables of projectively embedded toric surfaces.

Refined Descendant Invariants of Toric Surfaces

We construct refined tropical enumerative genus zero invariants of toric surfaces that specialize to the tropical descendant genus zero invariants introduced by Markwig and Rau when the quantum parameter tends to $1$. In the case of trivalent tropical curves our invariants turn to be the Goettsche-Schroeter refined broccoli invariants. We show that this is the only possible refinement of the Markwig-Rau descendant invariants that generalizes the Goettsche-Schroeter refined broccoli invariants. We discuss also the computational aspect (a lattice path algorithm) and exhibit some examples.

Blow-Ups of Toric Surfaces and the Mori Dream Space Property

We study the question of whether the blow-ups of toric surfaces of Picard number one at the identity point of the torus are Mori dream spaces. For some of these toric surfaces, the question whether the blow-up is a Mori dream space is equivalent to countably many planar interpolation problems. We state a conjecture which generalizes a theorem of González and Karu. We give new examples of Mori dream spaces and not Mori dream spaces among these blow-ups.

Mirror symmetry and elliptic Calabi-Yau manifolds

We find that for many Calabi-Yau threefolds with elliptic or genus one fibrations mirror symmetry factorizes between the fiber and the base of the fibration. In the simplest examples, the generic CY elliptic fibration over any toric base surface B that supports an elliptic Calabi-Yau threefold has a mirror that is an elliptic fibration over a dual toric base surface tilde{B} that is related through toric geometry to the line bundle −6KB. The Kreuzer-Skarke database includes all these examples and gives a wide range of other more complicated constructions where mirror symmetry also factorizes. Since recent evidence suggests that most Calabi-Yau threefolds are elliptic or genus one fibered, this points to a new way of understanding mirror symmetry that may apply to a large fraction of smooth Calabi-Yau threefolds. The factorization structure identified here can also apply for CalabiYau manifolds of higher dimension.

Recursive representations of arbitrary Virasoro conformal blocks

We derive recursive representations in the internal weights of N -point Virasoro conformal blocks in the sphere linear channel and the torus necklace channel, and recursive representations in the central charge of arbitrary Virasoro conformal blocks on the sphere, the torus, and higher genus Riemann surfaces in the plumbing frame.