Constructive degree bounds for group-based models

Group-based models arise in algebraic statistics while studying evolution processes. They are represented by embedded toric algebraic varieties. Both from the theoretical and applied point of view one is interested in determining the ideals defining the varieties. Conjectural bounds on the degree in which these ideals are generated were given by Sturmfels and Sullivant. We prove that for the 3-Kimura model, corresponding to the group G=Z2xZ2, the projective scheme can be defined by an ideal generated in degree 4. In particular, it is enough to consider degree 4 phylogenetic invariants to test if a given point belongs to the variety. We also investigate G-models, a generalization of abelian group-based models. For any G-model, we prove that there exists a constant $d$, such that for any tree, the associated projective scheme can be defined by an ideal generated in degree at most d.Comment: Boundedness results for equations defining the projective scheme were extended to G-models (including 2-Kimura and all JC

Toric geometry of the 3-Kimura model for any tree

In this paper we present geometric features of group based models. We focus on the 3-Kimura model. We present a precise geometric description of the variety associated to any tree on a Zariski open set. In particular this set contains all biologically meaningful points. Our motivation is a conjecture of Sturmfels and Sullivant on the degree in which the ideal associated to 3-Kimura model is generated

The Importance of Calculating the Potential Gross Domestic Product in the Context of the Taylor Rule

Taylor stated humorously that his rule was so easy that it could be written down on the back of a business card. The reality shows that the practical use of this type of rule implies accepting many assumptions about its final shape. The article mentions only the matter of influence of calculating the potential GDP and output gap on the empirical relevance of the Taylor rule. Two ways of calculating potential GDP were presented, i.e. the HP filter and linear trend of the current and the real GDP both seasonally adjusted (an additive model with seasonal dummies; TRAMO/SEATS procedure).Taylor rule, output gap.

Obstructions to combinatorial formulas for plethysm

Motivated by questions of Mulmuley and Stanley we investigate quasi-polynomials arising in formulas for plethysm. We demonstrate, on the examples of $S^3(S^k)$ and $S^k(S^3)$, that these need not be counting functions of inhomogeneous polytopes of dimension equal to the degree of the quasi-polynomial. It follows that these functions are not, in general, counting functions of lattice points in any scaled convex bodies, even when restricted to single rays. Our results also apply to special rectangular Kronecker coefficients.Comment: 7 pages; v2: Improved version with further reaching counterexamples; v3: final version as in Electronic Journal of Combinatoric

Standardization union effects: the case of EU enlargement

The analysis of trade policy shows growing interest in various types of “standards”. While technical regulations and standards are introduced to protect the interest of consumers, they can also act as technical barriers to trade (TBT), as foreign suppliers complying with national regulations might be required to bear certain costs of adjustment to the new regime. Recent literature focused on the concept of standards and concluded that shared standards promote trade. We instead set our attention to technical regulations of the European Union and concentrate on their effects on trade costs. The analysis is inspired by Gandal and Shy’s (2001) cost reducing standardization union theory. This paper summarizes results of research undertaken within a larger product assessing importance of technical barriers to trade for new EU members. The recent empirical study by Hagemejer (2005), based on detailed trade data of the EU. He has shown that in sectors where the EU technical regulations are most complicated and require costly adaptation, the trade within EU is booming. He argues that the trade between EU members is more concentrated within the high-TBT products, while the imports from outside are focused on the low-TBT or no-TBT products. Thus, EU technical regulations might in fact be trade diverting if the difference in productivity between intra and extra-EU partners is large. In this context we analyze the pattern of new members’ exports to the “old” EU. We calculate the trade coverage of various standardisation approaches and analyze the comparative advantage structure of the new EU members. We demonstrate that the structure of TBT’s affecting exports from new EU members is slowly converging with the one that characterizes intra-EU trade. Therefore, we expect that CEEC’s countries will benefit from applying common technical regulations of the EU after accession. In the last section of our paper we report the results of questionnaire-based research made among Polish companies in December of 2004, i.e. after the Eastern enlargement. It seems that the adjustment costs were moderate and the adaptation process to new technical regulations is already completed. Therefore, one can expected welfare gains for new members of the EU. We perform a CGE simulation using a GTAP model to assess these gains.EU enlargement; technical barriers to trade; international trade

Derived category of toric varieties with Picard number three

We construct a full, strongly exceptional collection of line bundles on the variety X that is the blow up of the projectivization of the vector bundle O_{P^{n-1}}\oplus O_{P^{n-1}}(b) along a linear space of dimension n-2, where b is a non-negative integer

Secant cumulants and toric geometry

We study the secant line variety of the Segre product of projective spaces using special cumulant coordinates adapted for secant varieties. We show that the secant variety is covered by open normal toric varieties. We prove that in cumulant coordinates its ideal is generated by binomial quadrics. We present new results on the local structure of the secant variety. In particular, we show that it has rational singularities and we give a description of the singular locus. We also classify all secant varieties that are Gorenstein. Moreover, generalizing (Sturmfels and Zwiernik 2012), we obtain analogous results for the tangential variety.Comment: Some improvements to previous results, with other minor changes. Updated reference