Modeling Alcohol Concentration in Blood via a Fractional Context

We use a conformable fractional derivative G T α through two kernels T ( t , α ) = e ( α − 1 ) t and T ( t , α ) = t 1 − α in order to model the alcohol concentration in blood; we also work with the conformable Gaussian differential equation (CGDE) of this model, to evaluate how the curve associated with such a system adjusts to the data corresponding to the blood alcohol concentration. As a practical application, using the symmetry of the solution associated with the CGDE, we show the advantage of our conformable approaches with respect to the usual ordinary derivative.

A functional equation related to generalized entropies and the modular group

We solve a functional equation connected to the algebraic characterization of generalized information functions. To prove the symmetry of the solution, we study a related system of functional equations, which involves two homographies. These transformations generate the modular group, and this fact plays a crucial role in solving the system. The method suggests a more general relation between conditional probabilities and arithmetic.

Nonlocal Symmetries for Time-Dependent Order Differential Equations

A new type of ordinary differential equation is introduced and discussed: time-dependent order ordinary differential equations. These equations are solved via fractional calculus by transforming them into Volterra integral equations of second kind with singular integrable kernel. The solutions of the time-dependent order differential equation represent deformations of the solutions of the classical (integer order) differential equations, mapping them into one-another as limiting cases. This equation can also move, remove or generate singularities without involving variable coefficients. An interesting symmetry of the solution in relation to the Riemann zeta function and Harmonic numbers is observed.

Nonlocal Symmetries for Time-Dependent Order Differential Equations

A new type of ordinary differential equation is introduced and discussed, namely, the time-dependent order ordinary differential equations. These equations can be solved via fractional calculus and are mapped into Volterra integral equations of second kind with singular integrable kernel. The solutions of the time-dependent order differential equations smoothly deforms solutions of the classical integer order ordinary differential equations into one-another, and can generate or remove singularities. An interesting symmetry of the solution in relation to the Riemann zeta function and Harmonic numbers was also proved.

Symmetric Time-Reversible Flows with a Strange Attractor

A symmetric chaotic flow is time-reversible if the equations governing the flow are unchanged under the transformation t → -t except for a change in sign of one or more of the state space variables. The most obvious solution is symmetric and the same in both forward and reversed time and thus cannot be dissipative. However, it is possible for the symmetry of the solution to be broken, resulting in an attractor in forward time and a symmetric repellor in reversed time. This paper describes the simplest three-dimensional examples of such systems with polynomial nonlinearities and a strange (chaotic) attractor. Some of these systems have the unusual property of allowing the strange attractor to coexist with a set of nested symmetric invariant tori.

Likelihood-based molecular-replacement solution for a highly pathological crystal with tetartohedral twinning and sevenfold translational noncrystallographic symmetry

Translational noncrystallographic symmetry (tNCS) is a pathology of protein crystals in which multiple copies of a molecule or assembly are found in similar orientations. Structure solution is problematic because this breaks the assumptions used in current likelihood-based methods. To cope with such cases, new likelihood approaches have been developed and implemented inPhaserto account for the statistical effects of tNCS in molecular replacement. Using these new approaches, it was possible to solve the crystal structure of a protein exhibiting an extreme form of this pathology with seven tetrameric assemblies arrayed along thecaxis. To resolve space-group ambiguities caused by tetartohedral twinning, the structure was initially solved by placing 56 copies of the monomer in space groupP1 and using the symmetry of the solution to define the true space group,C2. The resulting structure of Hyp-1, a pathogenesis-related class 10 (PR-10) protein from the medicinal herb St John's wort, reveals the binding modes of the fluorescent probe 8-anilino-1-naphthalene sulfonate (ANS), providing insight into the function of the protein in binding or storing hydrophobic ligands.

Numerical Study of Oscillatory Flow Past Four Cylinders in Square Arrangement for Pitch Ratio Equal to 4

The results of a numerical study of the viscous oscillating flow around four circular cylinders are presented herein, for a constant frequency parameter, β, equal to 50, and Keulegan-Carpenter numbers, KC, ranging between 0.2 and 10. The cylinders were placed on the vertices of a square, whose two sides were perpendicular and two parallel to the oncoming flow, for a pitch ratio, P/D, equal to 4. The finite-element method was employed for the solution of the Navier-Stokes equations, in the formulation where the stream function and the vorticity are the field variables. The streamlines and the vorticity contours generated from the solution were used for the flow visualization. When the Keulegan-Carpenter number is lower than 4, the flow remains symmetrical with respect to the horizontal axis of symmetry of the solution domain and periodic at consecutive cycles. As KC increases to 4 the flow becomes aperiodic in different cycles, although symmetry with respect to the horizontal central line of the domain is preserved. For KC equal to 5 asymmetries appear intermittently in the flow, which are eventually amplified as KC increases still further. These asymmetries, in association with the aperiodicity at different cycles, lead to an almost chaotic configuration, as KC grows larger. For characteristic cases the flow pattern and the traces of the hydrodynamic forces are presented. In addition, the mean and r.m.s. values of the in-line and transverse forces and the hydrodynamic coefficients of the inline force were evaluated for the entire range of Keulegan-Carpenter numbers examined.

Numerical Study of Oscillatory Flow Past a Pair of Cylinders in a Side-by-Side Arrangement

The results of a numerical study of the viscous oscillating flow around a pair of circular cylinders are presented herein, for a constant frequency parameter, β, equal to 50, and Keulegan-Carpenter numbers, KC, ranging between 0.2 and 10. The cylinders were placed side-by-side to the oncoming flow, for a pitch to diameter ratio, P/D, equal to 2. The finite-element method was employed for the solution of the Navier-Stokes equations, in the formulation where the stream function and the vorticity are the field variables. The vorticity contours generated from the solution were used mainly for the flow visualization, whereas the stream-lines and isobars are shown in some cases. At low values of the Keulegan-Carpenter number the flow remains symmetrical with respect to the horizontal axis of symmetry of the solution domain. As the Keulegan-Carpenter number is increased asymmetries appear in the flow, which are eventually amplified and lead finally to more complicated vortex-shedding patterns. These asymmetries generate an aperiodic flow configuration at consecutive cycles, which becomes almost chaotic as KC grows larger. For the various Keulegan-Carpenter numbers examined the time-histories of the hydrodynamic forces are presented, and the r.m.s. values of the hydrodynamic forces and the coefficients of the in-line force were evaluated.