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Do the symmetries of product spaces hold the key to unification?

We explore the symmetry groups of changes of basis on tangent spaces and show that these seem to point towards a new kind of Kaluza-Klein theory.

These transformations, for any pseudo-Riemannian spacetime, form a general linear group. This has a pseudo-orthogonal subgroup which preserves pseudo-orthonormality – that is, it maps frame bases to other frame bases.

On product spaces, there is a class of coordinate system which respects the factor spaces. In such coordinate systems, the full group of general linear symmetries is non-linearly realised, with only the transformations in the tangent spaces to the factor spaces realised linearly. Higher-dimensional tensors decompose into multiplets for the factor spaces. The metric takes a block diagonal form.

For a Cartesian product space, each block of the metric varies only with the coordinates for the corresponding factor space, but this is not true for more general product spaces. We focus on spaces for which the blocks are independent of the coordinates on the compact factor spaces (a higher-dimensional version of Kaluza’s “cylinder condition”), but all of them vary with the familiar four dimensions of spacetime. We show that in such a case, the covariant derivatives of vectors take exactly the right form for coupling to gravity and internal symmetries. We illustrate this for U(1) and SU(2). The gauge fields are components of the Levi-Civita spin connection (components which are absent in a Cartesian product space) and their field strengths are components of the Riemann curvature tensor in particular coordinates.

This model appears to exploit a loophole in O’Raifeartaigh’s no-go theorem. This is because transformations which appear as translations in the decompactification limit are realised as additional rotational transformations on the compact space.

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On the chirality of drugs and biomacromolecule structures

As a phenomenon associated with the homochirality of monomers that make up the essential classes of biological macromolecules, symmetry is of fundamental importance not only for all molecular biology as a systemic factor of its organization but also for pharmacology as well as a systemic factor of drug stereospecificity. Enantiomers, including pharmaceuticals, may exhibit utterly different chemical specificity in processes involving chiral compounds, as well as different bioactivity. It is crucial to consider the peculiarities of interaction of enantiomers with asymmetric compounds of the organism when creating drugs since it may turn out that just one drug form has a therapeutic effect. At the same time, the other could be less active or even cause severe side effects, being toxic.

More than half of the drugs currently in use are chiral, and most of the last ones are marketed as racemates. More than half of the drugs developed in recent years consist of chiral molecules. Chiral drugs are used in the treatment of a wide range of diseases, including cardiovascular and gastrointestinal. Obtaining optically pure forms of the substance is a complicated and expensive task, but their use in many cases could reduce the dosage and the number of side effects. The therapeutic activity of enantiomers, their pharmacokinetics, and pharmacodynamics are currently intensively studied. However, the physical nature of the differences in the therapeutic effects of enantiomers has not yet totally been established.

The stereospecific drug-target interaction should be considered more broadly than only local complementarity. Here, to develop the concept of chirality's role in the structure formation of biological macromolecules, we discuss the bioactivity of chiral drugs and make assumptions about the possible relationship between the drug chirality and the drug effect on a specific chiral molecular target.

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Symmetric features of structure formation in molecular biological and model chiral systems

One of the most actual fundamental problems of natural sciences is the problem of protein folding. Elucidation of the mechanisms of the "correct" folding of the polypeptide chain into the native conformation is not so much a biological problem as a physical one. The structure of a protein determines its function and role in the biosystem. However, despite the abundant data on the different proteins' functions, some of them remain unclear.

In this work, we present the analysis of the symmetry characteristics of the protein structures of the following classes: hydrolases, isomerases, oxidoreductases, chaperones, structural proteins, viral proteins, electron transport proteins, exocytosis/endocytosis proteins. We analyzed the arrangement of secondary structures (α-helices, β-sheets, 3_10-helices, irregular structures) and coiled coil structures in polypeptide chains. Common and individual features of studied protein classes are revealed.

Another aspect of this work is related to the symmetric mechanical features of structure formation. The mechanical properties of α-helices during the formation of coiled coil superhelices are considered, taking into account the chirality of the initial structures. The coiled coil structure was described geometrically in 1953 by F. Crick, and later this description was improved and refined. In this work, we calculated the moduli of tension and force arising during the formation of right-handed and left-handed superhelices from right-handed α-helices in the model system. Thus, it was shown that during the formation of a right-handed supercoil from right-handed α-helices, the modulus of strength and tension is larger than during the formation of a left-handed supercoil from right-handed α-helices. Mechanically, it is easier to form a left-handed supercoil from right-handed α-helices, which is observed in native structures.

The results of this study can be used to develop the concept of a protein as a molecular machine.

  • Open access
  • 48 Reads
Subpolygroup commutativity degrees of finite polygroups

Cayley's theorem states that every group G is isomorphic to a subgroup of the symmetric group acting on G. A hypergroup represents a natural generalization of a group and it was introduced by Marty in 1934. A special class of hypergroups, known as polygroups (or quasi-canonical hypergroups), was introduced and studied by Comer in 1984. There are many ways to construct polygroups. One of these ways is the polygroup associated to a group G. On the other hand the commutativity degree in groups have been of great importance for many algebraists and many related work was done. In this paper and inspired by the work related to subgroup commutativity degree of a finite group, we introduce the concept of subpolygroup commutativity degree sd(P) of a finite polygroup P. This quantity measures the probability of two random subpolygroups of P commuting. By finding a relation between the lattice of a group and the lattice of its associated polygroup, we find explicit formulas for the subpolygroup commutativity degree of polygroups associated to finite groups such as the dihedral group, quasi- dihedral group, and generalized quaternion group.

  • Open access
  • 112 Reads
Generalized scalar-tensor gravity motivated inflationary models

The models of cosmological inflation based on generalized scalar-tensor theories of gravity with specific connection between cosmological dynamics and coupling function are considered. In framework of this model the correspondence between well known types of the scalar-tensor gravity theories and physical motivated potentials of a scalar field is discussed. Methods for constructing exact and approximate cosmological solutions are considered. It is shown that these models correspond to the observational constraints on the values of the cosmological perturbations parameters for arbitrary potential of a scalar field and arbitrary coupling function and leads to exactly scale-invariant spectrum of relic gravitational waves. The specific model of cosmological inflation in Einstein frame corresponding to the models under consideration is determined as exponential power-law inflation. It is shown that the values of the parameters of cosmological perturbations coincide in Jordan and Einstein frames in these models for the case of the slow-roll approximation. The possibility of direct detection of the high-frequency relic gravitational waves at the present time predicted in such a models is discussed as well.

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Formation and detection of Majorana modes in quantum spin Hall trenches

We propose a novel realization for a topologically superconducting phase hosting Majorana zero modes on the basis of quantum spin Hall systems. Remarkably, our proposal is completely free of ferromagnets. Instead, we confine helical edge states around a narrow defect line of finite length in a two-dimensional topological insulator. We demonstrate the formation of a new topological regime, hosting protected Majorana modes in the presence of s-wave superconductivity and Zeeman coupling. Interestingly, when the system is weakly tunnel coupled to helical edge state reservoirs, a particular transport signature is associated with the presence of a non-Abelian Majorana zero mode.

The experimental relevance of the setup is due to the fact that very recently quantum point contacts between helical edges have been realized. The transport properties of such structures will also be briefly addressed. Particular attention will be devoted to the so called 0.5 anomaly characterizing the linear conductance. The relevance of electron-electron interactions will hence be addressed

  • Open access
  • 49 Reads
The Bell theorem revisited: geometric phases in gauge theories

The Bell theorem stands as an insuperable roadblock in the path to a very desired intuitive solution of the EPR paradox and, hence, it lies at the core of the current lack of a clear interpretation of the quantum formalism. The theorem states through an experimentally testable inequality that the predictions of quantum mechanics for the Bell polarization states of two entangled particles cannot be reproduced by any statistical model of hidden variables that shares certain intuitive features. In this paper we show, however, that the proof of the Bell theorem involves a subtle, though crucial, assumption that is not required by fundamental physical principles and, hence, it is not necessarily fulfilled in the experimental setup that tests the inequality. Indeed, this assumption can neither be properly implemented within the standard framework of quantum mechanics. Namely, the proof of the theorem assumes that there exists a preferred absolute frame of reference, supposedly provided by the lab, which enables to compare the orientation of the polarization measurement devices for successive realizations of the experiment and, hence, to define jointly their response functions over the space of hypothetical hidden configurations for all their possible alternative settings. We notice, however, that the preferred frame of reference required by the proof of the Bell theorem cannot exist in models in which the gauge symmetry of the experimental setup under global rigid rotations of the two detectors is spontaneously broken by the hidden configurations of the pair of entangled particles and a non-zero geometric phase appears under some cyclic gauge symmetry transformations. Following this observation, we build an explicitly local model of hidden variables that reproduces the predictions of quantum mechanics for the Bell states.

  • Open access
  • 44 Reads
Non-Markovian rate process of TRP ion channel activity

The Markov state model (MSM) is a popular, simple, phenomenological theoretical tool for describing the hierarchy of time scales involved in protein functional processes, e.g. ion channel gating. A MSM is a particular case of the general non-Markovian model, where the transition from one state to another depends only on the relevant transition rate from one state to the next, and does not include the history of state occupancy within the system, i.e., it only includes reversible, non-dissipative processes. Therefore, it requires knowledge of the structural details of the conformational changes of the channel and, it is not predictive when those details are not known. In the case of ion channels, this simple description fails in real (non-equilibrium) situations, such as local temperature changes or energy losses during channel gating. Overcoming these limitations is key to going beyond a mere phenomenological description, towards a better understanding of the physics underlying the channel dynamics. Here, we show that it is possible to use non-Markovian equations (i.e. offer a general description that includes MSM as a particular case) to develop a relatively simple analytical model that successfully predicts experimental data on the thermodynamic activity out of equilibrium of the temperature-sensitive channels TRPV1 and TRPM8. Our model is able to predict asymmetrical opening and closing rates, infinite processes, “new state creation” and temperature changes throughout the process.

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  • 73 Reads
Optimal Ellipsoid Approximations in Control Theory
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Minimum volume ellipsoids containing a given set arise often in control theory observed as a problem that solves differential equation with inputs and outputs. Generally, the problem is described by a differential inclusion $\dot x\in F(x(t),t)$, where $F$ is a set valued function on $\mathbb{R}^n\times \mathbb{R}_+$. An ellipsoid is given itself with a symmetric, positive definite matrix $Q$ such that $\mathcal{E}=\{\xi\in\mathbb{R}^n, (\xi-\xi_0)^TQ^{-1}(\xi-\xi_0)\leq 1\}$.

If a linear differential inclusion is given by $\dot x\in \Omega x$, $x(0)=x_0$, and with $\Omega\subseteq\mathbb{R}^{n\times n}$, then sufficient condition for the system stability is to find a positive definite symmetric matrix $P$ such that the quadratic function $V(\xi)=\xi^TP\xi$ decreases along every nonzero state trajectory.

Specific linear differential inclusions are described, such as the linear time-invariant, Polytopic, norm-bound with additional output that affects the additional input in bounded measure or diagonal norm-bound bounds of input and output functions are given componentwise.

In our work we interpreted stability conditions of above systems in terms of ellipsoid that is invariant to a solution of a differential inclusions: if $x(t_0)\in\mathcal{E}$ then $x(t)\in\mathcal{E}$ for every $t\geq t_0$.

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  • 89 Reads
The Energetics of Quantum Vacuum Friction

We consider a neutral but polarizable particle moving through a quantum thermal vacuum. The balance of energy and momentum causes the particle to radiate and generates a frictional force on the particle. If the particle is like an atom with no intrinsic dissipation, the friction entirely originates in the fluctuations of the electromagnetic field. Alternatively, but equivalently, we can think of the particle as having dipole fluctuations, corresponding to a definite temperature different from that of the thermal background. The friction we are investigating is the relativistic generalization of the Einstein-Hopf effect. The frictional force and power may be interpreted through a Doppler distortion of the spectral density of the radiation. We find a number of interesting features depending on the velocity and the properties of the particle. The forces are small, but it may prove possible to observe such effects, either through deceleration or through measuring the temperature of the atom, for high velocities. The formalism we have developed may be readily applied to more complicated scenarios, such as a particle moving parallel to a dielectric or metal surface.