Descriptive Complexity of Finite Structures: Saving the Quantifier Rank

Given a relational structure M on n elements, let D(M) be the minimum quantifier rank of a first order formula identifying M up to isomorphism in the class of n-element structures. The obvious upper bound is D(M)\le n. We show that if the relations in M have arity at most k, then D(M)<(1-\frac{1}{2k})n+k^2-k+2. The coefficient at n, which equals 1-\frac{1}{2k}, is probably not best possible but this is the first known bound having it strictly below 1 (for fixed k). If one is content to have the worse coefficient 1-\frac{1}{2k^2+2}, then one can choose an identifying formula of a very special form: a prenex formula with at most one quantifier alternation. A few other results in this vein are presented.Comment: 39 page

The First Order Definability of Graphs: Upper Bounds for Quantifier Rank

We say that a first order formula A distinguishes a graph G from another graph G' if A is true on G and false on G'. Provided G and G' are non-isomorphic, let D(G,G') denote the minimal quantifier rank of a such formula. We prove that, if G and G' have the same order n, then D(G,G')\le(n+3)/2, which is tight up to an additive constant of 1. The analogous questions are considered for directed graphs (more generally, for arbitrary structures with maximum relation arity 2) and for k-uniform hypergraphs. Also, we study defining formulas, where we require that A distinguishes G from any other non-isomorphic G'.Comment: 52 page

Chiral vortaic effect and neutron asymmetries at NICA

We study the possibility of testing experimentally signatures of P-odd effects related with the vorticity of the medium. The Chiral Vortaic Effect is generalized to the case of conserved charges different from the electric one. In the case of baryonic charge and chemical potential such effect should manifest itself in neutron asymmetries at the NICA accelerator complex measured by the MPD detector. The required accuracy may be achieved in a few months of accelerator running. We also discuss polarization of the hyperons and P-odd correlations of particle momenta (handedness) as probes of vorticity.Comment: LaTeX, 10 pages, 3 figures (V2: new figure and comments added

Succinct Definitions in the First Order Theory of Graphs

We say that a first order sentence A defines a graph G if A is true on G but false on any graph non-isomorphic to G. Let L(G) (resp. D(G)) denote the minimum length (resp. quantifier rank) of a such sentence. We define the succinctness function s(n) (resp. its variant q(n)) to be the minimum L(G) (resp. D(G)) over all graphs on n vertices. We prove that s(n) and q(n) may be so small that for no general recursive function f we can have f(s(n))\ge n for all n. However, for the function q^*(n)=\max_{i\le n}q(i), which is the least monotone nondecreasing function bounding q(n) from above, we have q^*(n)=(1+o(1))\log^*n, where \log^*n equals the minimum number of iterations of the binary logarithm sufficient to lower n below 1. We show an upper bound q(n)<\log^*n+5 even under the restriction of the class of graphs to trees. Under this restriction, for q(n) we also have a matching lower bound. We show a relationship D(G)\ge(1-o(1))\log^*L(G) and prove, using the upper bound for q(n), that this relationship is tight. For a non-negative integer a, let D_a(G) and q_a(n) denote the analogs of D(G) and q(n) for defining formulas in the negation normal form with at most a quantifier alternations in any sequence of nested quantifiers. We show a superrecursive gap between D_0(G) and D_3(G) and hence between D_0(G) and D(G). Despite it, for q_0(n) we still have a kind of log-star upper bound: q_0(n)\le2\log^*n+O(1) for infinitely many n.Comment: 41 pages, Version 2 (small corrections

Definitions with no quantifier alternation

Let $D(G)$ be the minimum quantifier depth of a first order sentence $\Phi$ that defines a graph $G$ up to isomorphism. Let $D_0(G)$ be the version of $D(G)$ where we do not allow quantifier alternations in $\Phi$. Define $q_0(n)$ to be the minimum of $D_0(G)$ over all graphs $G$ of order $n$. We prove that for all $n$ we have $\log^*n-\log^*\log^*n-1\le q_0(n)\le \log^*n+22$, where $\log^*n$ is equal to the minimum number of iterations of the binary logarithm needed to bring $n$ to 1 or below. The upper bound is obtained by constructing special graphs with modular decomposition of very small depth.Comment: 24 pages, we complement the lower bound proved in the first version with a tight upper bound. The title of the paper has been change

Jordan-Wigner fermionization for spin-1/2 systems in two dimensions: A brief review

We review the papers on the Jordan-Wigner transformation in two dimensions to comment on a possibility of examining the statistical mechanics properties of two-dimensional spin-1/2 models. We discuss in some detail the two-dimensional spin-1/2 isotropic XY and Heisenberg models.Comment: 19 pages, 10 figures include

Symmetries of the hypergeometric function mF_m-1

In this paper, we show that the generalized hypergeometric function mF_m-1 has a one parameter group of local symmetries, which is a conjugation of a flow of a rational Calogero-Mozer system. We use the symmetry to construct fermionic fields on a complex torus, which have linear-algebraic properties similar to those of the local solutions of the generalized hypergeometric equation. The fields admit a non-trivial action of the quaternions based on the above symmetry. We use the similarity between the linear-algebraic structures to introduce the quaternionic action on the direct sum of the space of solutions of the generalized hypergeometric equation and its dual. As a side product, we construct a ``good'' basis for the monodromy operators of the generalized hypergeometric equation inspired by the study of multiple flag varieties with finitely many orbits of the diagonal action of the general linear group by Magyar, Weyman, and Zelevinsky. As an example of computational effectiveness of the basis, we give a proof of the existence of the monodromy invariant hermitian form on the space of solutions of the generalized hypergeometric equation (in the case of real local exponents) different from the proofs of Beukers and Heckman and of Haraoka. As another side product, we prove an elliptic generalization of Cauchy identity.Comment: This is the final version of the paper. An Appendix on the Calogero-Mozer system has been added to the previous version. The paper is to appear in TAMS without the Appendi

The Secular Aberration Drift and Future Challenges for VLBI Astrometry

The centrifugial acceleration of the Solar system, resulting from the gravitational attraction of the Galaxy centre, causes a phenomenon known as 'secular aberrration drift'. This acceleration of the Solar system barycentre has been ignored so far in the standard procedures for high-precision astrometry. It turns out that the current definition of the celestial reference frame as epochless and based on the assumption that quasars have no detectable proper motions, needs to be revised. In the future, a realization of the celestial reference system (realized either with VLBI, or GAIA) should correct source coordinates from this effect, possibly by providing source positions together with their proper motions. Alternatively, the galactocentric acceleration may be incorporated into the conventional group delay model applied for VLBI data analysis.Comment: Proceedings of the "Journees-2011" meeting hold in Vienna University of Technology, 19-21 September, 201

Metal pad instabilities in liquid metal batteries

A mechanical analogy is used to analyze the interaction between the magnetic field, electric current and deformation of interfaces in liquid metal batteries. It is found that, during charging or discharging, a sufficiently large battery is prone to instabilities of two types. One is similar to the metal pad instability known for aluminum reduction cells. Another type is new. It is related to the destabilizing effect of the Lorentz force formed by the azimuthal magnetic field induced by the base current and the current perturbations caused by the local variations of the thickness of the electrolyte layer

Some Aspects of Three-Quark Potentials

We analytically evaluate the expectation value of a baryonic Wilson loop in a holographic model of an SU(3) pure gauge theory. We then discuss three aspects of a static three-quark potential: an aspect of universality which concerns properties independent of a geometric configuration of quarks; a heavy diquark structure; and a relation between the three and two-quark potentials.Comment: 35 pages, many figures; v2: appendix and two figures added, minor improvements in presentation; v3: improved presentation, some points clarified, reference adde