New evaluation of neutron lifetime from UCN storage experiments and beam experiments

The analysis of experiments on measuring neutron lifetime has been made. The latest most accurate result of measuring neutron lifetime [Phys. Lett. B 605, 72 (2005)] 878.5 \pm 0.8 s differs from the world average value [Phys. Lett. B 667, 1 (2008)] 885.7 \pm 0.8 s by 6.5 standard deviations. In view of this both the analysis and the Monte Carlo simulation of experiments [Phys. Lett. B 483, 15 (2000)] and [Phys. Rev. Lett. 63, 593 (1989)] have been performed. Systematic errors about -6 s have been found in both experiments. The table of results of neutron lifetime measurements is given after corrections and additions have been made. A new world average value of neutron lifetime makes up 880.0 \pm 0.9 s. Here is also presented a separate analysis of experiments on measuring neutron lifetime with UCN and experiments on the beams. The average neutron lifetime for experiments with UCN is equal to 879.3(0.6) s, while for experiments on the beams it is equal to 889.1(2.9) s. The present difference of average values for both groups is (3.3 sigma) and needs consideration. The contribution of beam experiments into the world average value is not high, therefore it does not influence the above analysis. However, it is an independent problem to be solved. It seems desirable that the precision of beam experiments should be enhanced.Comment: 7 pages, 4 figures, 2 table

Comment on "Order parameter of A-like 3He phase in aerogel"

We argue that the inhomogeneous A-phase in aerogel is energetically more preferable than the "robust" phase suggested by I. A. Fomin, JETP Lett. 77, 240 (2003); cond-mat/0302117 and cond-mat/0401639.Comment: 2 page

A New World Average Value for the Neutron Lifetime

The analysis of the data on measurements of the neutron lifetime is presented. A new most accurate result of the measurement of neutron lifetime [Phys. Lett. B 605 (2005) 72] 878.5 +/- 0.8 s differs from the world average value [Phys. Lett. B 667 (2008) 1] 885.7 +/- 0.8 s by 6.5 standard deviations. In this connection the analysis and Monte Carlo simulation of experiments [Phys. Lett. B 483 (2000) 15] and [Phys. Rev. Lett. 63 (1989) 593] is carried out. Systematic errors of about -6 s are found in each of the experiments. The summary table for the neutron lifetime measurements after corrections and additions is given. A new world average value for the neutron lifetime 879.9 +/- 0.9 s is presented.Comment: 27 pages, 13 figures; Fig.13 update

Discrete integrable systems, positivity, and continued fraction rearrangements

In this review article, we present a unified approach to solving discrete, integrable, possibly non-commutative, dynamical systems, including the $Q$- and $T$-systems based on $A_r$. The initial data of the systems are seen as cluster variables in a suitable cluster algebra, and may evolve by local mutations. We show that the solutions are always expressed as Laurent polynomials of the initial data with non-negative integer coefficients. This is done by reformulating the mutations of initial data as local rearrangements of continued fractions generating some particular solutions, that preserve manifest positivity. We also show how these techniques apply as well to non-commutative settings.Comment: 24 pages, 2 figure

Applications of BGP-reflection functors: isomorphisms of cluster algebras

Given a symmetrizable generalized Cartan matrix $A$, for any index $k$, one can define an automorphism associated with $A,$ of the field $\mathbf{Q}(u_1, >..., u_n)$ of rational functions of $n$ independent indeterminates $u_1,..., u_n.$ It is an isomorphism between two cluster algebras associated to the matrix $A$ (see section 4 for precise meaning). When $A$ is of finite type, these isomorphisms behave nicely, they are compatible with the BGP-reflection functors of cluster categories defined in [Z1, Z2] if we identify the indecomposable objects in the categories with cluster variables of the corresponding cluster algebras, and they are also compatible with the "truncated simple reflections" defined in [FZ2, FZ3]. Using the construction of preprojective or preinjective modules of hereditary algebras by Dlab-Ringel [DR] and the Coxeter automorphisms (i.e., a product of these isomorphisms), we construct infinitely many cluster variables for cluster algebras of infinite type and all cluster variables for finite types.Comment: revised versio

Radiation of a relativistic electron with non-equilibrium own Coulomb field

The condition and specific features of the non-dipole regime of radiation is discussed in the context of the results of the recent CERN experiment NA63 on measurement of the radiation power spectrum of 149 GeV electrons in thin tantalum targets. The first observation of a logarithmic dependence of radiation yield on the target thickness that was done there is the conclusive evidence of the effect of radiation suppression in a thin layer of matter, which was predicted many years ago, and which is the direct manifestation of the radiation of a relativistic electron with non-equilibrium own Coulomb field. The special features of the angular distribution of the radiation and its polarization in a thin target at non-dipole regime are proposed for a new experimental study

Q-systems, Heaps, Paths and Cluster Positivity

We consider the cluster algebra associated to the $Q$-system for $A_r$ as a tool for relating $Q$-system solutions to all possible sets of initial data. We show that the conserved quantities of the $Q$-system are partition functions for hard particles on particular target graphs with weights, which are determined by the choice of initial data. This allows us to interpret the simplest solutions of the Q-system as generating functions for Viennot's heaps on these target graphs, and equivalently as generating functions of weighted paths on suitable dual target graphs. The generating functions take the form of finite continued fractions. In this setting, the cluster mutations correspond to local rearrangements of the fractions which leave their final value unchanged. Finally, the general solutions of the $Q$-system are interpreted as partition functions for strongly non-intersecting families of lattice paths on target lattices. This expresses all cluster variables as manifestly positive Laurent polynomials of any initial data, thus proving the cluster positivity conjecture for the $A_r$ $Q$-system. We also give an alternative formulation in terms of domino tilings of deformed Aztec diamonds with defects.Comment: 106 pages, 38 figure

Cluster algebras of type A2(1)A_2^{(1)}

In this paper we study cluster algebras \myAA of type $A_2^{(1)}$. We solve the recurrence relations among the cluster variables (which form a T--system of type $A_2^{(1)}$). We solve the recurrence relations among the coefficients of \myAA (which form a Y--system of type $A_2^{(1)}$). In \myAA there is a natural notion of positivity. We find linear bases \BB of \myAA such that positive linear combinations of elements of \BB coincide with the cone of positive elements. We call these bases \emph{atomic bases} of \myAA. These are the analogue of the "canonical bases" found by Sherman and Zelevinsky in type $A_{1}^{(1)}$. Every atomic basis consists of cluster monomials together with extra elements. We provide explicit expressions for the elements of such bases in every cluster. We prove that the elements of \BB are parameterized by \ZZ^3 via their $\mathbf{g}$--vectors in every cluster. We prove that the denominator vector map in every acyclic seed of \myAA restricts to a bijection between \BB and \ZZ^3. In particular this gives an explicit algorithm to determine the "virtual" canonical decomposition of every element of the root lattice of type $A_2^{(1)}$. We find explicit recurrence relations to express every element of \myAA as linear combinations of elements of \BB.Comment: Latex, 40 pages; Published online in Algebras and Representation Theory, springer, 201