On Exact and Approximate Solutions for Hard Problems: An Alternative Look

We discuss in an informal, general audience style the da Costa-Doria conjecture about the independence of the P = NP hypothesis and try to briefly assess its impact on practical situations in economics. The paper concludes with a discussion of the Coppe-Cosenza procedure, which is an approximate, partly heuristic algorithm for allocation problems.P vs. NP , allocation problem, assignment problem, traveling salesman, exact solution for NP problems, approximate solutions for NP problems, undecidability, incompleteness

Effect of the boundary condition on the vortex patterns in mesoscopic three-dimensional superconductors - disk and sphere

The vortex state of mesoscopic three-dimensional superconductors is determined using a minimization procedure of the Ginzburg-Landau free energy. We obtain the vortex pattern for a mesoscopic superconducting sphere and find that vortex lines are naturally bent and are closest to each other at the equatorial plane. For a superconducting disk with finite height, and under an applied magnetic field perpendicular to its major surface, we find that our method gives results consistent with previous calculations. The matching fields, the magnetization and $H_{c3}$, are obtained for models that differ according to their boundary properties. A change of the Ginzburg-Landau parameters near the surface can substantially enhance $H_{c3}$ as shown here.Comment: 7 pages, 4 figures (low resolution

The Average Kinetic Energy of the Superconducting State

Isothermal magnetization curves are plotted as the magnetization times the magnetic induction, $4 \pi M \cdot B$, versus the applied field, H. We show here that this new curve is the average kinetic energy of the superconducting state versus the applied field, for type-II superconductors with a high Ginzburg-Landau parameter $\kappa$. The maximum of $4 \pi M \cdot B$ occurs at a field, $H^{*}$, directly related to the upper critical field, $H_{c2}$, suggesting that $H_{c2}(T)$ may be extracted from such plots even in cases when it is too high for direct measurement. We obtain these plots both theoretically, from the Ginzburg-Landau theory, and experimentally, using a Niobium sample with $T_c = 8.5 K$, and compare them.Comment: 11 pages, 9 postscript figure

Yang-Mills Families

The Yang-Mills theory structure is based on group theory. It rules the symmetry relationship where the number of potential fields transforming under a same group must be equal to the number of group generators. It defines the group valued expression  from where the corresponding non-abelian symmetry properties are derived. Nevertheless based on different origins as Kaluza-Klein, fibre bundles, supersymmetry, s-model , BRST and anti-BRST algorithm, counting degrees of freedom leads to a Yang-Mills extension under the existence of different potential fields rotating under a same single group. They establish for SU(N) the relationship  where  and  is a flavor index, . Physically, it says that different Yang-Mills families can share a common symmetry group. They develop a whole non-abelian gauge theory. The effort in this work is to explore such non-abelian extension. First, to build up it on the so-called  constructor basis where gauge symmetry is more available for expressing the corresponding fields strengths, Lagrangian and classical equations. After that, given that the physical fields are those associated to the poles of two-point Green functions, one derives the physical Lagrangian L  written in the physical basis . A new physical Lagrangian associated to  symmetry is generated. The meaning of Yang-Mills families appears. A symmetry of difference is realized. Where every quanta is distinguished from each other. It yields a quanta diversity associated to corresponding Yang-Mills families. There are N-spin 1 and N-spin 0 quanta separated by different quantum numbers through a whole N-dynamics. An extension to QCD becomes possible

New Faraday lines through Four Bosons EM

Field physics was founded by Faraday introducing magnetic fields (1831), electric fields (1837) and light as an EM wave (1846), initiating the process where nature is made by matter and fields. Consider that, ordinary space is full of fields. The Faraday view is basis for modern quantum field theory. The concept of fields set up a physicality in development. Physics would like to know how far matter is created by fields. Generate matter from nonlinear fields. Faraday lines of force relating physical entities as electric charge and mass depending on fields. Our purpose is on Faraday lines for nonlinear abelian electromagnetism. Introduce the Four Bosons EM. The phenomenology of a generic charge $\{+,0,-\}$ transmitted by four bosons $\{A_{\mu}, U_{\mu}, V_{\mu}^{\pm}\}$. Nonlinear equations constituted. New Faraday lines were introduced. The potentials fields of physics are developed. Granular and collective fields strengths expressed. Four types of fields charges are derived. They are electric charge, modulated, neutral, Bianchi. This work introduces a systematic procedure of associative physics. Mass and charge are generated due to the four fields interrelationships. Masses are derived without spontaneous symmetry breaking. It is obtained naturally from gauge symmetry, London, and mixing terms. Electric charge is written by fields through the Noether theorem. EM interactions not necessarily coupled with electric charge are proposed. An enlargement of EM energy is derived.Comment: 20 pages, 0 figure

Landau and Ott scaling for the kinetic energy density and the low TcT_c conventional superconductors, Li2Pd3BLi_{2}Pd_{3}B and Nb

The scaling approach recently proposed by Landau and Ott for isothermal magnetization curves is extended to the average kinetic energy density of the condensate. Two low $T_c$ superconductors, Nb and $Li_{2}Pd_{3}B$ are studied and their isothermal reversible magnetization shown to display Landau and Ott scaling. Good agreement is obtained for the upper critical field $H_{c2}(T)$, determined from the Abrikosov approximation for the reversible region (standard linear extrapolation of the magnetization curve), and from the maximum of the kinetic energy curves. For the full range of data, which includes the irreversible region, the isothermal $d.M.B/H^2$ curves for $Li_2Pd_3B$ show an impressive collapse into a single curve over the entire range of field measurements. The Nb isothermal $d.M.B/H^2$ curves exhibit the interesting feature of a constant and temperature independent minimum value

Electric Charge Mutation by Four Vector Bosons

A general theory of electric charge is proposed. It is based on two phenomenologies. Electric charge mutation and conservation law. Three charges $\{ +, - ,0\}$ transformations physics succeeds. Quantum field theory underlies corresponding creations and annihilation. A potential field's quadruplet is ruled. Microscopic electromagnetism is processed by four vectors bosons intermediations. The electromagnetism closure is accomplished. The quadruplet $A_{\mu I} \equiv \{ A_\mu, U_\mu, V_\mu^\pm\}$ completeness introduces the most generic EM energy flux between electric charges. Charge mutation includes that besides usual photon, EM phenomena is enlarged by massive and charged photons. Charge conservation associates these four vector fields. Electric charge symmetry, extends EM for an abelian symmetry $U_{Q} \equiv U(1) \times SO(2)_{global}$. A new EM Lagrangian beyond Maxwell results. A symmetry equation for electric charge is established through Noether theorem. The electric charge transfer physics extends the EM phenomenon. Nonlinear Electromagnetic fields modified electric charge symmetry, new EM regimes. Potential fields become a physical entity producing conglomerates, collective fields, mass, sources, charges, monopoles, forces. EM features ruled from an extended electric charge abelian symmetry. Systemic, nonlinear, neutral, spintronics, photonics, electroweak EM regimes are constituted.Comment: 45 pages, not figur

Vanishing of the upper critical field in Bi_2Sr_2CaCu_2O_{8+\delta} from Landau-Ott scaling

We apply Landau-Ott scaling to the reversible magnetization data of Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ published by Y. Wang et al. [\emph{Phys. Rev. Lett. \textbf{95} 247002 (2005)}] and find that the extrapolation of the Landau-Ott upper critical field line vanishes at a critical temperature parameter, T^*_c, a few degrees above the zero resistivity critical temperature, T_c. Only isothermal curves below and near to T_c were used to determine this transition temperature. This temperature is associated to the disappearance of the mixed state instead of a complete suppression of superconductivity in the sample.Comment: 3 figure

Using imprecise continuous time Markov chains for assessing the reliability of power networks with common cause failure and non-immediate repair.

We explore how imprecise continuous time Markov chains can improve traditional reliability models based on precise continuous time Markov chains. Specifically, we analyse the reliability of power networks under very weak statistical assumptions, explicitly accounting for non-stationary failure and repair rates and the limited accuracy by which common cause failure rates can be estimated. Bounds on typical quantities of interest are derived, namely the expected time spent in system failure state, as well as the expected number of transitions to that state. A worked numerical example demonstrates the theoretical techniques described. Interestingly, the number of iterations required for convergence is observed to be much lower than current theoretical bounds