A short proof of the Buchstaber-Rees theorem

We give a short proof of the Buchstaber-Rees theorem concerning symmetric powers. The proof is based on the notion of a formal characteristic function of a linear map of algebras.Comment: 11 pages. LaTeX2

On generalized symmetric powers and a generalization of Kolmogorov-Gelfand-Buchstaber-Rees theory

The classical Kolmogorov-Gelfand theorem gives an embedding of a (compact Hausdorff) topological space X into the linear space of all linear functionals C(X)^* on the algebra of continuous functions C(X). The image is specified by algebraic equations: f(ab)=f(a)f(b) for all functions a, b on X; that is, the image consists of all algebra homomorphisms of C(X) to numbers. Buchstaber and Rees have found that not only X, but all symmetric powers of X can be embedded into the space C(X)^*. The embedding is again given by algebraic equations, but more complicated. Algebra homomorphisms are replaced by the so-called "n-homomorphisms", the notion that can be traced back to Frobenius, but which explicitly appeared in Buchstaber and Rees's works on multivalued groups. We give a further natural generalization of Kolmogorov-Gelfand-Buchstaber-Rees theory. Symmetric powers of a space X or of an algebra A are replaced by certain "generalized symmetric powers" Sym^{p|q}(X) and S^{p|q}A, which we introduce, and n-homomorphisms, by the new notion of "p|q-homomorphisms". Important tool of our study is a certain "characteristic function" R(f,a,z), which we introduce for an arbitrary linear map of algebras f, and whose functional properties with respect to the variable z reflect algebraic properties of the map f.Comment: LaTeX, 7 pages (3+4). In this new version we slightly edited the main text, and added to it an Appendix giving details of some constructions and a short direct proof of Buchstaber--Rees's main theore

Quantum Cluster Theories

Quantum cluster approaches offer new perspectives to study the complexities of macroscopic correlated fermion systems. These approaches can be understood as generalized mean-field theories. Quantum cluster approaches are non-perturbative and are always in the thermodynamic limit. Their quality can be systematically improved, and they provide complementary information to finite size simulations. They have been studied intensively in recent years and are now well established. After a brief historical review, this article comparatively discusses the nature and advantages of these cluster techniques. Applications to common models of correlated electron systems are reviewed.Comment: submitted to Reviews of Modern Physic

Vector fields on mapping spaces and a converse to the AKSZ construction

The well-known AKSZ construction (for Alexandrov--Kontsevich--Schwarz--Zaboronsky) gives an odd symplectic structure on a space of maps together with a functional $S$ that is automatically a solution for the classical master equation $(S,S)=0$. The input data required for the AKSZ construction consist of a volume element on the source space and a symplectic structure of suitable parity on the target space, both invariant under given homological vector fields on the source and target. In this note, we show that the AKSZ setup and their main construction can be naturally recovered from the single requirement that the `difference' vector field arising on the mapping space be gradient (or Hamiltonian). This can be seen as a converse statement for that of AKSZ. We include a discussion of properties of vector fields on mapping spaces.Comment: Minor editing in version

The Effect of Board Structure on Bidder-Shareholders' Wealth: Further Evidence from the UK Bidding Firms

In this study we provide evidence of cross-sectional dependence of bidder-shareholder wealth and target’s board characteristics. More specifically we provide evidence that the percentage of non-executives, the board size, the stock holdings of executives, and the other directorships held by non-executives serving the target board are important in assessing the announcement of the bid, whereas in the bidder’s board only the percentage of non-executive directors is important for bidder-shareholders. In addition to that we provide evidence that some of these relationships are not monotonic in nature. Finally, in this study it is documented that bidder-shareholder wealth is favoured in acquisitions where bidders have marginally more executive than non-executive directors in their boards and therefore the question arises as to whether “dependent” boards are more efficient than “independent” ones.Takeovers, Board Characteristics, Corporate Governance, Shareholder-wealth

EPR of photochromic Mo3+ in SrTiO3

In single crystals of SrTiO_3, a paramagnetic center, characterized by S = 3/2 and hyperfine interaction with an I = 5/2 nuclear spin has been observed in the temperature range 4.2K-77K by means of EPR. The impurity center is attributed to Mo3+. No additional line splitting in the EPR spectrum due to the 105K phase transition has been observed. At 4.2K the following spin Hamiltonian parameters for this impurity ion were obtained: g = 1.9546\pm0.0010 and A = (32.0\pm0.05)\times10^-4 cm^-1.Comment: 5 pages, 2 figure

The Marriage Problem and the Fate of Bachelors

In the marriage problem, a variant of the bi-parted matching problem, each member has a `wish-list' expressing his/her preference for all possible partners; this list consists of random, positive real numbers drawn from a certain distribution. One searches the lowest cost for the society, at the risk of breaking up pairs in the course of time. Minimization of a global cost function (Hamiltonian) is performed with statistical mechanics techniques at a finite fictitious temperature. The problem is generalized to include bachelors, needed in particular when the groups have different size, and polygamy. Exact solutions are found for the optimal solution (T=0). The entropy is found to vanish quadratically in $T$. Also other evidence is found that the replica symmetric solution is exact, implying at most a polynomial degeneracy of the optimal solution. Whether bachelors occur or not, depends not only on their intrinsic qualities, or lack thereof, but also on global aspects of the chance for pair formation in society.Comment: 14 pages revtex, submitted to Physica