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Journal of Physical Mathematics

1. Foundations of a Unified Physics

IB, Pestov
Since space is the primary concept of science, we put forward an idea that regularities of unified physics are concealed in a simple relation: everything in the concept of space and the concept of space in everything. With this hypothesis as a ground, a conceptual structure of a unified geometrical theory of fundamental interactions is created and deductive derivation of its main equations is produced. The formulated theory provides opportunity to understand the origin and nature of physical fields and local internal symmetry; time and energy; spin and charge; confinement and quark-lepton symmetry; dark energy and dark matter, giving by...

2. New Analytical Formulae to Calibrate HPGe Well-type Detectors Efficiency and to Calculate Summing Corrections

MI, Abbas
Direct mathematical method is used in this paper to calculate the photo peak efficiency of a system containing a radioactive source placed in the well of a well type detector. Attenuation of the source and detector’s walls are included. In addition summing corrections are also included. The comparison between published experimental values and present calculated values are given. This work proved quite success in predicting the photo peak efficiency of certain detectors even for a radionuclide emitting rays with different energies.

3. Introduction to Non Commutative Algebraic Geometry

Siqveland, Arvid
Ordinary commutative algebraic geometry is based on commutative polynomial algebras over an algebraically closed field k. Here we make a natural generalization to matrix polynomial k-algebras which are non-commutative coordinate rings of non-commutative varieties.

4. Physical Interpretation of Noncommutative Algebraic Varieties

A, Siqveland
The theory of algebraic varieties gives an algebraic interpretation of differential geometry, thus of our physical world. To treat, among other physical properties, the theory of entanglement, we need to generalize the space parametrizing the objects of physics. We do this by introducing noncommutative varieties.

S, Mukherjee

6. Quaternionic Dirac Scattering

S, De Leo; G, Ducati; S, Giardino
The scattering of a Dirac particle has been studied for a quaternionic potential step. In the potential region an additional diffusion solution is obtained. The quaternionic solution which generalizes the complex one presents an amplification of the reflection and transmission rates. A detailed analysis of the quaternionic spinorial velocities shed new light on the additional solution. For pure quaternionic potentials, the interesting and surprising result of total transmission is found. This suggests that the presence of pure quaternionic potentials cannot be seen by analyzing the reflection or transmission rates. It has been observed by measuring the mean value of some...

7. What Mathematics is the Most Fundamental?

FM, Lev
Standard mathematics (involving such notions as infinitely small/large and continuity) is usually treated as fundamental while finite mathematics is treated as inferior which is used only in special applications. In the first part of this note we argue that the situation is the opposite: standard mathematics is only a degenerate case of finite one in the formal limit when the characteristic of the ring or field used in finite mathematics goes to infinity, and finite mathematics is more pertinent for describing nature than standard one. In the second part we argue that foundation of finite mathematics is natural while foundational...

8. The Entropy Production of a Nonequilibrium Open System

Yu, Ming Bao
A nonequilibrium open system is studied in the projection operator formalism. The environment may linearly deviate from its initial state under the reaction from the open system. If the relevant statistical operator of the system is a generalized canonical one, the transport equation, the second kind of fluctuation-dissipation theorem and the entropy production rate of the open system can be derived and expressed in terms of correlation functions of fluctuations of random forces and interaction random forces.

9. Editorial for Journal of Physical Mathematics

T, Kohno; E, Paal; AA, Voronov

10. L P Donoho-Stark Uncertainty Principles for the Dunkl Transform on ${\mathbb{R}^{\text{d}}}$

Soltani, Fethi
In the Dunkl setting, we establish three continuous uncertainty principles of concentration type, where the sets of concentration are not intervals. The first and the second uncertainty principles are $L^p$ versions and depend on the sets of concentration T and W, and on the time function $f$. The time-limiting operators and the Dunkl integral operators play an important role to prove the main results presented in this paper. However, the third uncertainty principle is also $L^p$ version depends on the sets of concentration and he is independent on the band limited function $f$. These uncertainty principles generalize the results obtained for the...

11. Recent Experimental Results Claim for a Rigorous Understanding of Nuclear Forces

dos Anjos, Petrus HR

12. Puiseux Series Expansions for the Eigenvalues of Transfer Matrices and Partition Functions from the Newton Polygon Method for Nanotubes and Ribbons

Schmidt, Jeffrey R
For certain classes of lattice models of nanosystems the eigenvalues of the row-to-row transfer matrix and the components of the corner transfer matrix truncations are algebraic functions of the fugacity and of Boltzmann weights. Such functions can be expanded in Puiseux series using techniques from algebraic geometry. Each successive term in the expansions in powers a Boltzmann weight is obtained exactly without modifying previous terms. We are able to obtain useful analytical expressions for any thermodynamic function for these systems from the series in circumstances in which no exact solutions can be found.

13. A Remark on the Hopf invariant for Spherical 4-braids

PM, Akhmet'ev
An approach by J.Wu describes homotopy groups $\pi_{\mathrm{n}}(\mathrm{S^2})$ of the standard 2-sphere as isotopy classes of spherical n+1-strand Brunnian braids. The case n=3 is investigated for applications.

14. A Modified N=2 Extended Supersymmetry

N, Djeghloul; M, Tahiri
A modification of the usual extended N=2 super symmetry algebra implementing the two dimensional permutation group is performed. It is shown that one can found a multiplet that forms an off-shell realization of this alternative extension of standard super symmetry.

15. The Low Lying Energy-Momentum Spectrum for the Lattice Four-Fermi Model

dos Anjos, Petrus H. R.; da Veiga, Paulo A. Faria
We obtain the low-lying energy-momentum spectrum for the imaginary-time lattice four-Fermi or Gross-Neveu model in $d+1$ space-time dimensions ($d=1,2,3$) and with $N$-component fermions. Let $0<\kappa\ll 0$ be the hopping parameter, $\lambda>0$ the four-fermion coupling, $m>0$ the bare fermion mass and take $s\times s$ spin matrices ($s=2,4$). Our analysis of the one and the two-particle spectrum is based on spectral representation for suitable two- and four-fermion correlations. The one-particle energy-momentum spectrum is obtained rigorously and is manifested by $\frac{sN}{2}$ isolated and identical dispersion curves, and the mass of particles has asymptotic value order $-\ln\kappa$. The existence of two-particle bound states above or below the two-particle band depends on whether Gaussian domination...

16. Extended Lie Derivatives and a New Formulation of $D=11$ Supergravity

Castellani, Leonardo
Introducing an extended Lie derivative along the dual of $A$, the $3$-form field of $D=11$ supergravity, the full diffeomorphism algebra of $D=11$ supergravity is presented. This algebra suggests a new formulation of the theory, where the $3$-form field $A$ is replaced by bivector $B^{ab}$, bispinor $B^{\alpha\beta}$, and spinor-vector $\eta^{a\beta}$ $1$-forms. Only the bivector $1$-form $B^{ab}$ is propagating, and carries the same degrees of freedom of the $3$-form in the usual formulation, its curl $\mathcal{D}_{[\mu} B^{ab}_{\nu]}$ being related to the $F_{\mu \nu a b}$ curl of the $3$-form. The other $1$-forms are auxiliary, and the transformation rules on all the fields close on the equations of motion of $D=11$ supergravity.

17. Lie Derivatives along Antisymmetric Tensors, and the M-Theory Superalgebra

Castellani, Leonardo
Free differential algebras (FDAs) provide an algebraic setting for field theories with antisymmetric tensors. The "presentation'' of FDAs generalizes the Cartan-Maurer equations of ordinary Lie algebras, by incorporating $p$-form potentials. An extended Lie derivative along antisymmetric tensor fields can be defined and used to recover a Lie algebra dual to the FDA that encodes all the symmetries of the theory including those gauged by the $p$-forms. The general method is applied to the FDA of $D=11$ supergravity: the resulting dual Lie superalgebra contains the M-theory supersymmetry anticommutators in presence of $2$-branes.

18. Further Developments for the Auxiliary Field Method

Semay, Claude; Buisseret, Fabien; Silvestre-Brac, Bernard
The auxiliary field method is a technique to obtain approximate closed formulae for the solutions of both nonrelativistic and semirelativistic eigenequations in quantum mechanics. For a many-body Hamiltonian describing identical particles, it is shown that the approximate eigenvalues can be written as the sum of the kinetic operator evaluated at a mean momentum $p_0$ and of the potential energy computed at a mean distance $r_0$. The quantities $p_0$ and $r_0$ are linked by a simple relation depending on the quantum numbers of the state considered and are determined by an equation which is linked to the generalized virial theorem. The (anti)variational character of the method is discussed, as...

19. Histories Distorted by Partial Isometries

Cho, Ilwoo
In quantum dynamical systems, a history is defined by a pair $(M,\gamma)$, consisting of a type $I$ factor $M$, acting on a Hilbert space $H$, and an $E_0$-group $\gamma = (\gamma_t)_{t\in \Bbb{R}}$, satisfying certain additional conditions. In this paper, we distort a given history $(M,\gamma)$, by a finite family $\mathcal{G}$ of partial isometries on $H$. In particular, such a distortion is dictated by the combinatorial relation on the family $\mathcal{G}$. Two main purposes of this paper are (i) to show the existence of distortions on histories, and (ii) to consider how distortions work. We can understand Sections 3, 4 and 5 as the proof of the existence of distortions...

20. Operational Methods and Lorentz-Type Equations of Motion

Babusci, Danilo; Dattoli, Giuseppe; Sabia, Elio
We propose an operational method for the solution of differential equations involving vector products. The technique we propose is based on the use of the evolution operator, defined in such a way that the wealth of techniques developed within the context of quantum mechanics can also be exploited for classical problems. We discuss the application of the method to the solution of the Lorentz-type equations.

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