On the Nonlinear Dynamical Equation in the p-adic String Theory

In this work nonlinear pseudo-differential equations with the infinite number of derivatives are studied. These equations form a new class of equations which initially appeared in p-adic string theory. These equations are of much interest in mathematical physics and its applications in particular in string theory and cosmology. In the present work a systematical mathematical investigation of the properties of these equations is performed. The main theorem of uniqueness in some algebra of tempored distributions is proved. Boundary problems for bounded solutions are studied, the existence of a space-homogenous solution for odd p is proved. For even p it is proved that there is no continuous solutions and it is pointed to the possibility of existence of discontinuous solutions. Multidimensional equation is also considered and its soliton and q-brane solutions are discussed.Comment: LaTex, 18 page

QCD-Thermodynamics using 5-dim Gravity

We calculate the critical temperature and free energy of the gluon plasma using the dilaton potential arXiv:0911.0627[hep-ph] in the gravity theory of AdS/QCD. The finite temperature observables are calculated in two ways: first, from the Page-Hawking computation of the free energy, and secondly using the Bekenstein-Hawking proportionality of the entropy with the area of the horizon. Renormalization is well defined, because the T=0 theory has asymptotic freedom. We further investigate the change of the critical temperature with the number of flavours induced by the change of the running coupling constant in the quenched theory. The finite temperature behaviour of the speed of sound, spatial string tension and vacuum expectation value of the Polyakov loop follow from the corresponding string theory in AdS_5.Comment: 38 pages, 12 figure

Nonlinear equations for p-adic open, closed, and open-closed strings

We investigate the structure of solutions of boundary value problems for a one-dimensional nonlinear system of pseudodifferential equations describing the dynamics (rolling) of p-adic open, closed, and open-closed strings for a scalar tachyon field using the method of successive approximations. For an open-closed string, we prove that the method converges for odd values of p of the form p=4n+1 under the condition that the solution for the closed string is known. For p=2, we discuss the questions of the existence and the nonexistence of solutions of boundary value problems and indicate the possibility of discontinuous solutions appearing.Comment: 16 pages, 3 figure

Some Remarks on Producing Hopf Algebras

We report some observations concerning two well-known approaches to construction of quantum groups. Thus, starting from a bialgebra of inhomogeneous type and imposing quadratic, cubic or quartic commutation relations on a subset of its generators we come, in each case, to a q-deformed universal enveloping algebra of a certain simple Lie algebra. An interesting correlation between the order of initial commutation relations and the Cartan matrix of the resulting algebra is observed. Another example demonstrates that the bialgebra structure of sl_q(2) can be completely determined by requiring the q-oscillator algebra to be its covariant comodule, in analogy with Manin's approach to define SL_q(2) as a symmetry algebra of the bosonic and fermionic quantum planes.Comment: 6 pages, LATEX, no figures, Contribution to the Proceedings of the 4th Colloquium "Quantum Groups and Integrable Systems" (Prague, June 1995

Comment on the ``θ\theta-term renormalization in the (2+1)-dimensional CPN−1CP^{N-1} model with θ\theta term''

It is found that the recently published first coefficient of nonzero $\beta$-function for the Chern-Simons term in the $1/N$ expansion of the $CP^{N-1}$ model is untrue numerically. The correct result is given. The main conclusions of Park's paper are not changed.Comment: 3 pages, LATE

Zeta Nonlocal Scalar Fields

We consider some nonlocal and nonpolynomial scalar field models originated from p-adic string theory. Infinite number of spacetime derivatives is determined by the operator valued Riemann zeta function through d'Alembertian $\Box$ in its argument. Construction of the corresponding Lagrangians L starts with the exact Lagrangian $\mathcal{L}_p$ for effective field of p-adic tachyon string, which is generalized replacing p by arbitrary natural number n and then taken a sum of $\mathcal{L}_n$ over all n. The corresponding new objects we call zeta scalar strings. Some basic classical field properties of these fields are obtained and presented in this paper. In particular, some solutions of the equations of motion and their tachyon spectra are studied. Field theory with Riemann zeta function dynamics is interesting in its own right as well.Comment: 13 pages, submitted to Theoretical and Mathematical Physic

The fundamental solution of the unidirectional pulse propagation equation

The fundamental solution of a variant of the three-dimensional wave equation known as "unidirectional pulse propagation equation" (UPPE) and its paraxial approximation is obtained. It is shown that the fundamental solution can be presented as a projection of a fundamental solution of the wave equation to some functional subspace. We discuss the degree of equivalence of the UPPE and the wave equation in this respect. In particular, we show that the UPPE, in contrast to the common belief, describes wave propagation in both longitudinal and temporal directions, and, thereby, its fundamental solution possesses a non-causal character.Comment: accepted to J. Math. Phy

Discretized rotation has infinitely many periodic orbits

For a fixed k in (-2,2), the discretized rotation on Z^2 is defined by (x,y)->(y,-[x+ky]). We prove that this dynamics has infinitely many periodic orbits.Comment: Revised after referee reports, and added a quantitative statemen