Nontrivial Cycles Research Articles

ECONOMICAL TORIC SPINES VIA CHEEGER'S INEQUALITY

Let [Formula: see text] denote the graph whose set of vertices is {0,…,m - 1}d, where two distinct vertices are adjacent if and only if they are either equal or adjacent in the m-cycle Cm in each coordinate. Let [Formula: see text] denote the graph on the same set of vertices in which two vertices are adjacent if and only if they are adjacent in one coordinate in Cm and equal in all others. Both graphs can be viewed as graphs of the d-dimensional torus. We prove that one can delete [Formula: see text] vertices of G1 so that no topologically nontrivial cycles remain. This improves an O(d log 2 (3/2)md - 1) estimate of Bollobás, Kindler, Leader and O'Donnell. We also give a short proof of a result implicit in a recent paper of Raz: one can delete an [Formula: see text] fraction of the edges of G∞ so that no topologically nontrivial cycles remain in this graph. Our technique also yields a short proof of a recent result of Kindler, O'Donnell, Rao and Wigderson; there is a subset of the continuous d-dimensional torus of surface area [Formula: see text] that intersects all nontrivial cycles. All proofs are based on the same general idea: the consideration of random shifts of a body with small boundary and no nontrivial cycles, whose existence is proved by applying the isoperimetric inequality of Cheeger or its vertex or edge discrete analogues.

A nontrivial algebraic cycle in the Jacobian variety of the Fermat sextic

We compute some value of the harmonic volume for the Fermat sextic. Using this computation, we prove that some special algebraic cycle in the Jacobian variety of the Fermat sextic is not algebraically equivalent to zero.

Superpotentials from stringy instantons without orientifolds

In this paper we show that it is possible to derive non-perturbative superpotential terms from a stringy instanton without introducing orientifold planes. The instanton is realized by a Euclidean D brane wrapping a non-trivial cycle upon which we also wrap a single space-filling D brane. The standard problem of unwanted neutral fermionic zero modes is evaded by the appearance of couplings to charged bosonic zero modes in the instanton moduli action. Since the Euclidean D brane wraps a cycle which is not associated to any low energy gauge dynamics, it can not be interpreted as an ordinary gauge instanton, but rather as a stringy one. By considering such a brane configuration at an orbifold singularity, we can explicitly evaluate the instanton moduli space integral and find a holomorphic superpotential term with the structure of a baryonic mass term.

Homology and K-theory methods for classes of branes wrapping nontrivial cycles

We apply some methods of homology and K-theory to special classes of branes wrapping homologically nontrivial cycles. We treat the classification of four-geometries in terms of compact stabilizers (by analogy with Thurston's classification of three-geometries) and derive the K-amenability of Lie groups associated with locally symmetric spaces listed in this case. More complicated examples of T-duality and topology change from fluxes are also considered. We analyse D-branes and fluxes in type II string theory on with torsion H-flux and demonstrate in detail the conjectured T-duality to with no flux. In the simple case of , T-dualizing the circles reduces to the duality between with H-flux and with no flux.

Global dynamics of a Predator-Prey model with Hassell-Varley Type functional response

Predator-prey models with Hassell-Varley type functional response are appropriate for interactions where predators form groups and have applications in biological control. Here we present a systematic global qualitative analysis to a general predator-prey model with Hassell-Varley type functional response. We show that the predator free equilibrium is a global attractor only when the predator death rate is greater than its growth ability. The positive equilibrium exists if the above relation reverses. In cases of practical interest, we show that the local stability of the positive steady state implies its global stability with respect to positive solutions. For terrestrial predators that form a fixed number of tight groups, we show that the existence of an unstable positive equilibrium in the predator-prey model implies the existence of an unique nontrivial positive limit cycle.

Polyhedral Results for 1-Restricted Simple 2-Matchings

A simple 2-matching in a graph is a subgraph whose connected components are nontrivial paths and cycles. A simple 2-matching is called 1-restricted if each connected component has two or more edges. In this paper we consider the problem of finding maximum weight 1-restricted simple 2-matchings (which is a relaxation of the traveling salesman problem). We present an integer programming formulation for this problem, characterize the extreme points of the linear programming relaxation, and characterize the graphs for which the linear programming relaxation has all integral extreme points. We show how to recognize these graphs in polynomial time. We also introduce a new class of blossom-type inequalities that tighten the general linear programming relaxation. A complete description of the convex hull of 1-restricted simple 2-matchings is unknown.

Qtorus inN=2supersymmetric QED

We construct ``flying saucer'' solitons in supersymmetric $\mathcal{N}=2$ gauge theory, which is known to support Bogomol'nyi-Prasad-Sommerfield domain walls with a U(1) gauge field localized on its worldvolume. We demonstrate that this model supports exotic particlelike solitons with the shape of a torus. $Q$ tori, and also similar solitons of higher genera, are obtained by folding the domain wall into an appropriate surface. Nontrivial cycles on the domain wall worldvolume (handles) are stabilized by crossed electric and magnetic fields inside the folded domain wall. Three distinct frameworks are used to prove the existence of these flying saucer solitons and study their properties: the worldvolume description (including the Dirac-Born-Infeld action), the bulk-theory description in the sigma-model limit, and the bulk-theory description in the thin-edge approximation. In the sigma-model framework the $Q$ torus is shown to be related to the Hopf Skyrmion studied previously.

Tangency properties of a pentagonal tiling generated by a piecewise isometry

Piecewise isometries (PWIs) are known to have dynamical properties that generate interesting geometric planar packings. We analyze a particular PWI introduced by Goetz that generates a packing by periodically coded cells, each of which is a pentagon. Our main result is that the tangency graph associated with this packing is a forest (i.e., has no nontrivial cycles). We show, however, that this is not a general property of PWIs by giving an example that has an infinite number of cycles in the tangency graph of its periodically coded cells.

A nontrivial algebraic cycle in the Jacobian variety of the Klein quartic

We prove some value of the harmonic volume for the Klein quartic C is nonzero modulo $$\frac{1}{2}{\mathbb{Z}}$$ , using special values of the generalized hypergeometric function 3 F 2. This result tells us the algebraic cycle C − C − is not algebraically equivalent to zero in the Jacobian variety J(C).

Eliminating Cycles in the Discrete Torus

In this paper we consider the following question: how many vertices of the discrete torus must be deleted so that no topologically nontrivial cycles remain? We look at two different edge structures for the discrete torus. For (ℤmd)1, where two vertices in ℤm are connected if their l1 distance is 1, we show a nontrivial upper bound of $d^{\log_{2}(3/2)}m^{d-1}\approx d^{0.6}m^{d-1}$ on the number of vertices that must be deleted. For (ℤmd)∞, where two vertices are connected if their l∞ distance is 1, Saks et al. (Combinatorica 24(3):525–530, 2004) already gave a nearly tight lower bound of d(m−1) d−1 using arguments involving linear algebra. We give a more elementary proof which improves the bound to md−(m−1)d, which is precisely tight.

The Abel–Jacobi map for higher Chow groups, II

We explicitly describe cycle-class maps c_H from motivic cohomology to absolute Hodge cohomology, for smooth quasi-projective and (some) proper singular varieties, and compute special cases of the latter. For smooth projective varieties, we also study Hodge-theoretically defined "higher Abel-Jacobi maps" on the kernel of c_H; this leads to new results on nontrivial indecomposable higher Chow cycles in the regulator kernel.

Baryon currents in QCD with compact dimensions

On a compact space with nontrivial cycles, for sufficiently small values of the radii of the compact dimensions, SU(N) gauge theories coupled with fermions in the fundamental representation spontaneously break charge conjugation, time reversal, and parity. We show at one loop in perturbation theory that a physical signature for this phenomenon is a nonzero baryonic current wrapping around the compact directions. The persistence of this current beyond the perturbative regime is checked by lattice simulations.

Calabi–Yau 3-folds from 2-folds

We consider type IIA string theory on a Calabi–Yau 2-fold with D6-branes wrapping 2-cycles in the 2-fold. We find a complete set of conditions on the supergravity solution for any given wrapped brane configuration in terms of SU(2) structures. We reduce the problem of finding a supergravity solution for the wrapped branes to finding a harmonic function on R × CY2. We then lift this solution to eleven dimensions as a product of R(4,1) and a Calabi–Yau 3-fold. We show how the metric on the 3-fold is determined in terms of the wrapped brane solution. We write down the distinguished (3, 0) form and the Kähler form of the 3-fold in terms of structures defined on the base 2D complex manifold. We discuss the topology of the 3-fold in terms of the D6-branes and the underlying 2-fold. We show that in addition to the non-trivial cycles inherited from the underlying 2-fold there are N − 1 new 2-cycles. We construct closed (1, 1) forms corresponding to these new cycles. We also display some explicit examples. One of our examples is that of D6-branes wrapping the 2-cycle in an A1 ALE space, the resulting 3-fold has h(1,1) = N, where N is the number of D6-branes.

Multilinear forms and graded algebras

In this paper we investigate the class of the connected graded algebras which are finitely generated in degree 1, which are finitely presented with relations of degrees greater or equal to 2 and which are of finite global dimension D and Gorenstein. For D greater or equal to 4 we add the condition that these algebras are homogeneous and Koszul. It is shown that each such algebra is completely characterized by a multilinear form satisfying a twisted cyclicity condition and some other nondegeneracy conditions depending on the global dimension D. This multilinear form plays the role of a volume form and canonically identifies in the quadratic case to a nontrivial Hochschild cycle of maximal degree. Several examples including the Yang–Mills algebra and the extended 4-dimensional Sklyanin algebra are analyzed in this context. Actions of quantum groups are also investigated.

Periodic points and stability in Clark's delayed recruitment model

We provide further insight on the dynamics of Clark's delayed recruitment equation, depending on the relevant parameters involved in the model. We pay special attention to the stability and bifurcations from the positive equilibrium, and to the existence and attraction properties of nontrivial cycles. A detailed analysis is worked out for a three-dimensional example.

The sufficiency of arithmetic progressions for the 3𝑥+1 Conjecture

Define T : Z + → Z + T:\mathbb {Z} ^{+}\rightarrow \mathbb {Z} ^{+} by T ( x ) = ( 3 x + 1 ) / 2 T\left ( x\right ) =\left ( 3x+1\right ) /2 if x x is odd and T ( x ) = x / 2 T\left ( x\right ) =x/2 if x x is even. The 3 x + 1 3x+1 Conjecture states that the T T -orbit of every positive integer contains 1 1 . A set of positive integers is said to be sufficient if the T T -orbit of every positive integer intersects the T T -orbit of an element of that set. Thus to prove the 3 x + 1 3x+1 Conjecture it suffices to prove it on some sufficient set. Andaloro proved that the sets 1 + 2 n N 1+2^{n} \mathbb {N} are sufficient for n ≤ 4 n\leq 4 and asked if 1 + 2 n N 1+2^{n}\mathbb {N} is also sufficient for larger values of n n . We answer this question in the affirmative by proving the stronger result that A + B N A+B\mathbb {N} is sufficient for any nonnegative integers A A and B B with B ≠ 0 , B\neq 0, i.e. every nonconstant arithmetic sequence forms a sufficient set. We then prove analagous results for the Divergent Orbits Conjecture and Nontrivial Cycles Conjecture.

Bifurcation analysis of an automatic dynamic balancing mechanism for eccentric rotors

We present a nonlinear bifurcation analysis of the dynamics of an automatic dynamic balancing mechanism for rotating machines. The principle of operation is to deploy two or more masses that are free to travel around a race at a fixed distance from the hub and, subsequently, balance any eccentricity in the rotor. Mathematically, we start from a Lagrangian description of the system. It is then shown how under isotropic conditions a change of coordinates into a rotating frame turns the problem into a regular autonomous dynamical system, amenable to a full nonlinear bifurcation analysis. Using numerical continuation techniques, curves are traced of steady states, limit cycles and their bifurcations as parameters are varied. These results are augmented by simulations of the system trajectories in phase space. Taking the case of a balancer with two free masses, broad trends are revealed on the existence of a stable, dynamically balanced steady-state solution for specific rotation speeds and eccentricities. However, the analysis also reveals other potentially attracting states—non-trivial steady states, limit cycles, and chaotic motion—which are not in balance. The transient effects which lead to these competing states, which in some cases coexist, are investigated.

Cycloops: dark matter or a monopole problem for brane inflation?

We consider cosmic loop production by long string interactions in cosmological models with compact extra dimensions. In the case that the compact manifold is not simply connected, we focus on the possibility of loops wrapping around non-trivial cycles and becoming topologically trapped. Such loops, denoted cycloops, behave like matter in the radiation era, posing a potential monopole problem. We calculate the number distribution and the energy density of these objects as functions of cosmic time and use them to study cosmological constraints imposed on simple brane inflation models. For typical choices of parameters we find that to avoid cycloop domination before the matter-radiation transition, the strings must be unacceptably light, namely $G\mu<10^{-18}$, unless some mechanism to dilute the cycloops is provided. By exploring the full parameter space however, we are able to find models with $G\mu \sim 10^{-14}$, which is marginally consistent with brane inflation. In such models the cycloop could provide an interesting dark matter candidate.

Analytical results for the statistical distribution related to a memoryless deterministic walk: Dimensionality effect and mean-field models

Consider a medium characterized by N points whose coordinates are randomly generated by a uniform distribution along the edges of a unitary d-dimensional hypercube. A walker leaves from each point of this disordered medium and moves according to the deterministic rule to go to the nearest point which has not been visited in the preceding mu steps (deterministic tourist walk). Each trajectory generated by this dynamics has an initial nonperiodic part of t steps (transient) and a final periodic part of p steps (attractor). The neighborhood rank probabilities are parametrized by the normalized incomplete beta function Id= I1/4 [1/2, (d+1) /2] . The joint distribution S(N) (mu,d) (t,p) is relevant, and the marginal distributions previously studied are particular cases. We show that, for the memory-less deterministic tourist walk in the euclidean space, this distribution is Sinfinity(1,d) (t,p) = [Gamma (1+ I(-1)(d)) (t+ I(-1)(d) ) /Gamma(t+p+ I(-1)(d)) ] delta(p,2), where t=0, 1,2, ... infinity, Gamma(z) is the gamma function and delta(i,j) is the Kronecker delta. The mean-field models are the random link models, which correspond to d-->infinity, and the random map model which, even for mu=0 , presents nontrivial cycle distribution [ S(N)(0,rm) (p) proportional to p(-1) ] : S(N)(0,rm) (t,p) =Gamma(N)/ {Gamma[N+1- (t+p) ] N( t+p)}. The fundamental quantities are the number of explored points n(e)=t+p and Id. Although the obtained distributions are simple, they do not follow straightforwardly and they have been validated by numerical experiments.

Nonlinear Vibration of Parametrically Excited, Viscoelastic, Axially Moving Strings

The dynamic response of parametrically excited, axially moving viscoelastic belts is investigated in this paper. Results are compared to previous work in which the partial, not material, time derivative was used in the viscoelastic constitutive relation. It is found that this added “steady state” dissipation greatly affects both the existence and amplitudes of nontrivial limit cycles. The discrepancy increases with increasing translation speed. To limit the comparison to the additional physics included in the model, the solution procedure of Zhang and Zu [1,2], who applied the method of multiple scales to the governing equations prior to discretization, is retained. The excitation here is provided by physically stretching the belt. In this case, viscoelastic behavior and excitation frequency also affects the amplitude of the tension fluctuations.