Narrow Results

We prove a separation between two models of the reconfigurable mesh (rmesh), the cross-over model and the non-cross-over model. Specifically, for any constant k ≥ 1, we show that in the non-cross-over model, a k × n rmesh requires Ω(log n) time to compute the parity of n bits stored one per column, while in the cross-over model, when k ≥ 3, a k × n rmesh can compute the parity in constant time. This is the first separation demonstrated between the cross-over and non-cross-over model. The lower bound does not rely on the bandwidth constraints of the mesh and does not restrict the instruction sets of the processors. Moreover, it is the first lower bound for the rmesh which requires only binary inputs.

We show that the large, direct CP-violation parameter A_ππ=-C_ππ, reported by the BELLE collaboration in the decays , implies an unusual situation in which the presence of a very large difference between two strong-interaction phases (~ -110°) plays an essential role. We make the demonstration within a model of strong, two-body quasi-elastic interactions between physical hadrons. The model can accommodate a large difference between two strong-interaction phases, for which it provides a natural enhancement.

We first briefly review tests on CPT invariance based on the consequences of the CPT theorem and then present some possible CPT tests due to exotic models in which some of the CPT conditions are lost, such as those without hermiticity.

Using the functional analysis method, we present the exact solutions of the double ring-shaped oscillator (DRSO) potential with certain parity in the cylindrical coordinates. Such a quantum system is separated to two differential equations, i.e. a one-dimensional harmonic oscillator plus an inverse square term and a two-dimensional harmonic oscillator plus an inverse square term. The key point is how to find the adapted symmetrical solutions of the one-dimensional harmonic oscillator plus an inverse square term at the singular point $z=0$. The obtained results are compared with those in the spherical coordinates. We also explore intimate connections $E_T(i\rho )=-E_T(\rho )$ and $E_z(iz)=-E_z(z)$ by substituting $\rho \to i\rho$ and $z\to iz$.

A right–right–left extension of the Standard Model is proposed. In this model, SM gauge group $\mathrm{\text{SU}}{(2)}_L\otimes \mathrm{\text{U}}{(1)}_Y$ is extended to $\mathrm{\text{SU}}{(2)}_L\otimes \mathrm{\text{SU}}{(2)}_R\otimes \mathrm{\text{SU}}{(2)}_R^{\prime }\otimes \mathrm{\text{SU}}{(2)}_L^{\prime }\otimes \mathrm{\text{U}}{(1)}_Y$. The gauge symmetries $\mathrm{\text{SU}}{(2)}_R^{\prime }$, $\mathrm{\text{SU}}{(2)}_L^{\prime }$ are the mirror counterparts of the $\mathrm{\text{SU}}{(2)}_L$ and $\mathrm{\text{SU}}{(2)}_R$, respectively. Parity is spontaneously broken when the scalar Higgs fields acquire vacuum expectation values (VEVs) in a certain pattern. Parity is restored at the scale of $\mathrm{\text{SU}}{(2)}_L^{\prime }$. The gauge sector has a unique pattern. The scalar sector of the model is optimum, elegant and unique.

In this paper, we consider the non-commutative quantum mechanics (NCQM) with parity (or space reflection) in two dimensions. Using the parity operators R_i, we construct the deformed Heisenberg algebra with parity in the non-commutative plane. We use this algebra to discuss the isotropic harmonic Hamiltonian with parity.

A left-right symmetric mirror model restoring parity at a high scale in a way such that the mirror fermions and mirror gauge sector simultaneously could exist at TeV scale is discussed. We also provide an ultraviolet completion of the model with vector-like fermions, and discuss some theoretical and phenomenological implications of this model.

We examine the basic matrix equation for a spin-2 field in spherical coordinates and a tetrad of Minkowski spacetime. The technique of Wigner D-functions is used to separate the variables. The operators of energy, the square and the third projection of the total angular momentum, and the space reflection operators are then diagonalized on the solutions. The separation of the variables is done, the problem reduces to 11 radial second-order differential equations for 11 variables related to the scalar and symmetric tensors. We then consider the case of a massless field. The four gauge solutions for the massless spin-2 field have been found in explicit form. These gauge solutions do not contribute to physically observable quantities such as the energy–momentum tensor. They are derived according to Pauli–Fierz theory from four independent spherical solutions for a spin-1 massless particle. We have also constructed two types of solutions, related to opposite parities, which are not the gauge solutions. These solutions describe physically observable states with spherical symmetry for the spin-2 massless field.

Yang's theorem states that an initial J = 1 state cannot decay into two photons. Because of this result some reactions relating to elementary particles or atomic transitions can be ruled out. The theorem is not valid in the presence of background electric or magnetic fields. In this work we show that the decay of a J = 1 particle into two photons is permitted by the Bose symmetry and rotational invariance when the background of the decay process is not pure vacuum but contains an external classical magnetic/electric field. We also discuss constraints on these amplitudes from the CP invariance.

We construct a D₄ flavor model based on SU(3)_C ⊗SU(3)_L ⊗U(1)_X gauge symmetry responsible for fermion masses and mixings. The neutrinos get small masses from antisextets which are in a singlet and a doublet under D₄. If the D₄ symmetry is violated as perturbation by a Higgs triplet under SU(3)_L and lying in of D₄, the corresponding neutrino mass mixing matrix gets the most general form. In this case, the model can fit the experimental data in 2012 on neutrino masses and mixing. Our results show that the neutrino masses are naturally small and a little deviation from the tribimaximal neutrino mixing form can be realized. The quark masses and mixing matrix are also discussed. In the model under consideration, the CKM matrix can be different from the unit matrix. The scalar potential of the model is more simpler than those of the model based on S₃ and S₄. Assignation of VEVs to antisextets leads to the mixing of the new gauge bosons and those in the Standard Model. The mixing in the charged gauge bosons as well as the neutral gauge boson is considered.

We investigate an alternative CPT-odd Lorentz-breaking QED which includes the Carroll–Field–Jackiw (CFJ) term of the Standard Model Extension (SME), writing the gauge sector in the action in a Palatini-like form, in which the vectorial field and the field-strength tensor are treated as independent entities. Interestingly, this naturally induces a Lorentz-violating mass term in the classical action. We study physical consistency aspects of the model both at classical and quantum levels.

In this paper, we study the quantization of Dirac field theory in the $\kappa$-deformed space–time. We adopt a quantization method that uses only equations of motion for quantizing the field. Starting from $\kappa$-deformed Dirac equation, valid up to first order in the deformation parameter $a$, we derive deformed unequal time anticommutation relation between deformed field and its adjoint, leading to undeformed oscillator algebra. Exploiting the freedom of imposing a deformed unequal time anticommutation relations between $\kappa$-deformed spinor and its adjoint, we also derive a deformed oscillator algebra. We show that deformed number operator is the conserved charge corresponding to global phase transformation symmetry. We construct the $\kappa$-deformed conserved currents, valid up to first order in $a$, corresponding to parity and time-reversal symmetries of $\kappa$-deformed Dirac equation also. We show that these conserved currents and charges have a mass-dependent correction, valid up to first order in $a$. This novel feature is expected to have experimental significance in particle physics. We also show that it is not possible to construct a conserved current associated with charge conjugation, showing that the Dirac particle and its antiparticle satisfy different equations in $\kappa$ space–time.

Comparisons of the positive and negative halves of the distributions of parity-odd event variables in particle-physics experimental data can provide sensitivity to sources of nonstandard parity violation. Such techniques benefit from lacking first-order dependence on simulations or theoretical models, but have hitherto lacked systematic means of enumerating all discoverable signals. To address that issue this paper seeks to construct sets of parity-odd event variables which may be proved to be able to reveal the existence of any Lorentz-invariant source of nonstandard parity violation which could be visible in data consisting of groups of real nonspace-like four-momenta exhibiting certain permutation symmetries.

This paper proves, via an analytical approach, that 170 (out of 256) Boolean CA rules in a one-dimensional cellular automata (CA) are time-reversible in a generalized sense. The dynamics on each attractor of a time-reversible rule N is exactly mirrored, in both space and time, by its bilateral twin ruleN^†. In particular, all 69 period-1 rules, 17 (out of 25) period-2 rules, and 84 (out of 112) Bernoulli rules are time-reversible.

The remaining 86 CA rules are time-irreversible in the sense that N and N^† mirror their dynamics only in space, but not in time. In this case, each attractor of N defines a unique arrow of time.

A simple "time-reversal test" is given for testing whether an attractor of a CA rule is time-reversible or time-irreversible. For a time-reversible attractor of a CA rule N the past can be uniquely recovered from the future of N^†, and vice versa. This remarkable property provides 170 concrete examples of CA time machines where time travel can be routinely achieved by merely hopping from one attractor to its bilateral twin attractor, and vice versa. Moreover, the time-reversal property of some local rules can be programmed to mimic the matter–antimatter "annihilation" or "pair-production" phenomenon from high-energy physics, as well as to mimic the "contraction" or "expansion" scenarios associated with the Big Bang from cosmology.

Unlike the conventional laws of physics, which are based on a unique universe, most CA rules have multiple universes (i.e. attractors), each blessed with its own laws. Moreover, some CA rules are endowed with both time-reversible attractors and time-irreversible attractors.

Using an analytical approach, the time-τ return map of each Bernoulli στ-shift attractor of all 112 Bernoulli rules are shown to obey an ultra-compact formula in closed form, namely,.

or its inverse map.

These maps completely characterize the time-asymptotic (steady state) behavior of the nonlinear dynamics on the attractors. In-depth analysis of all but 18 global equivalence classes of CA rules have been derived, along with their basins of attraction, which characterize their transient regimes.

Above all, this paper provides a rigorous nonlinear dynamics foundation for a paradigm shift from an empirical-based approach à la Wolfram to an attractor-based analytical theory of cellular automata.

Free knots are a simplification of virtual knots obtained by forgetting arrow/sign information at classical crossings. First non-trivial examples of free knots were constructed recently by the second named author.

By using parity considerations, we construct invariants of free knots valued in certain groups. These groups have a simple combinatorial description, the first one being the infinite dihedral group.

The present paper produces examples of Gauss diagram formulae for virtual knot invariants which have no analogue in the classical knot case. These combinatorial formulae contain additional information about how a subdiagram is embedded in a virtual knot diagram. The additional information comes from the second author's recently discovered notion of parity. For a parity of flat virtual knots, the new combinatorial formulae are Kauffman finite-type invariants. However, many of the combinatorial formulae possess exotic properties. It is shown that there exists an integer-valued virtualization invariant combinatorial formula of order n for every n (i.e. it is stable under the map which changes the direction of one arrow but preserves the sign). Hence, it is not of Goussarov–Polyak–Viro finite-type. Moreover, every homogeneous Polyak–Viro combinatorial formula admits a decomposition into an "even" part and an "odd" part. For the Gaussian parity, neither part of the formula is of GPV finite-type when it is non-constant on the set of classical knots. In addition, eleven new non-trivial combinatorial formulae of order 2 are presented which are not of GPV finite-type.

The aim of the present paper is to prove that the minimal number of virtual crossings for some families of virtual knots grows quadratically with respect to the minimal number of classical crossings. All previously known estimates for virtual crossing number ([D. M. Afanasiev, Refining the invariants of virtual knots by using parity, Sb. Math.201(6) (2010) 3–18; H. A. Dye and L. H. Kauffman, Virtual crossing numbers and the arrow polynomial, in The Mathematics of Knots. Theory and Applications, eds. M. Banagl and D. Vogel (Springer-Verlag, 2010); S. Satoh and Y. Tomiyama, On the crossing number of a virtual knot, Proc. Amer. Math. Soc.140 (2012) 367–376] and so on) were principally no more than linear in the number of classical crossings (or, what is the same, in the number of edges of a virtual knot diagram) and no virtual knot was found with virtual crossing number greater than the classical crossing number.

We investigate an application of crossing parity for the bracket expansion of the Jones polynomial for virtual knots. In addition we consider an application of parity for the arrow polynomial as well as for the categorifications of both polynomials. We present a number of examples found through our calculations. We provide tables of calculations for these invariants on virtual knots with at most four real crossings.

In this paper, we construct an invariant for virtual knots in the thickened sphere S_g with g handles; this invariant is a Laurent polynomial in 2g + 3 variables. To this end, we use a modification of the Wirtinger presentation of the knot group and the concept of parity introduced by Manturov. By using this invariant, one can prove that the knots shown in Fig. 1 are not equivalent [S. A. Grishanov, V. R. Meshkov and V. A. Vassiliev, Recognizing textile structures by finite type knot invariants, J. Knot Theory Ramifications18(2) (2009) 209–235]. Section 4 of the paper is devoted to an enhancement of the invariant (construction of the invariant module) by using the parity hierarchy concept suggested by Manturov. Namely, we discriminate between odd crossings and two types of even crossings; the latter two types depend on whether an even crossing remains even/odd after all odd crossings of the diagram are removed. The construction of the invariant also works for virtual knots.

In this paper, we define some polynomial invariants for virtual knots and links. In the first part we use Manturov's parity axioms [Parity in knot theory, Sb. Math.201 (2010) 693–733] to obtain a new polynomial invariant of virtual knots. This invariant can be regarded as a generalization of the odd writhe polynomial defined by the first author in [A polynomial invariant of virtual knots, preprint (2012), arXiv:math.GT/1202.3850v1]. The relation between this new polynomial invariant and the affine index polynomial [An affine index polynomial invariant of virtual knots, J. Knot Theory Ramification22 (2013) 1340007; A linking number definition of the affine index polynomial and applications, preprint (2012), arXiv:1211.1747v1] is discussed. In the second part we introduce a polynomial invariant for long flat virtual knots. In the third part we define a polynomial invariant for 2-component virtual links. This polynomial invariant can be regarded as a generalization of the linking number.