Polar Varieties and Efficient Real Elimination

Let $S_0$ be a smooth and compact real variety given by a reduced regular sequence of polynomials $f_1, ..., f_p$. This paper is devoted to the algorithmic problem of finding {\em efficiently} a representative point for each connected component of $S_0$ . For this purpose we exhibit explicit polynomial equations that describe the generic polar varieties of $S_0$. This leads to a procedure which solves our algorithmic problem in time that is polynomial in the (extrinsic) description length of the input equations $f_1, >..., f_p$ and in a suitably introduced, intrinsic geometric parameter, called the {\em degree} of the real interpretation of the given equation system $f_1, >..., f_p$.Comment: 32 page

Polar Varieties, Real Equation Solving and Data-Structures: The hypersurface case

In this paper we apply for the first time a new method for multivariate equation solving which was developed in \cite{gh1}, \cite{gh2}, \cite{gh3} for complex root determination to the {\em real} case. Our main result concerns the problem of finding at least one representative point for each connected component of a real compact and smooth hypersurface. The basic algorithm of \cite{gh1}, \cite{gh2}, \cite{gh3} yields a new method for symbolically solving zero-dimensional polynomial equation systems over the complex numbers. One feature of central importance of this algorithm is the use of a problem--adapted data type represented by the data structures arithmetic network and straight-line program (arithmetic circuit). The algorithm finds the complex solutions of any affine zero-dimensional equation system in non-uniform sequential time that is {\em polynomial} in the length of the input (given in straight--line program representation) and an adequately defined {\em geometric degree of the equation system}. Replacing the notion of geometric degree of the given polynomial equation system by a suitably defined {\em real (or complex) degree} of certain polar varieties associated to the input equation of the real hypersurface under consideration, we are able to find for each connected component of the hypersurface a representative point (this point will be given in a suitable encoding). The input equation is supposed to be given by a straight-line program and the (sequential time) complexity of the algorithm is polynomial in the input length and the degree of the polar varieties mentioned above.Comment: Late

Polar Varieties and Efficient Real Equation Solving: The Hypersurface Case

The objective of this paper is to show how the recently proposed method by Giusti, Heintz, Morais, Morgenstern, Pardo \cite{gihemorpar} can be applied to a case of real polynomial equation solving. Our main result concerns the problem of finding one representative point for each connected component of a real bounded smooth hypersurface. The algorithm in \cite{gihemorpar} yields a method for symbolically solving a zero-dimensional polynomial equation system in the affine (and toric) case. Its main feature is the use of adapted data structure: Arithmetical networks and straight-line programs. The algorithm solves any affine zero-dimensional equation system in non-uniform sequential time that is polynomial in the length of the input description and an adequately defined {\em affine degree} of the equation system. Replacing the affine degree of the equation system by a suitably defined {\em real degree} of certain polar varieties associated to the input equation, which describes the hypersurface under consideration, and using straight-line program codification of the input and intermediate results, we obtain a method for the problem introduced above that is polynomial in the input length and the real degree.Comment: Late

The new resilience of emerging and developing countries: systemic interlocking, currency swaps and geoeconomics

The vulnerability/resilience nexus that defined the interaction between advanced and developing economies in the post-WWII era is undergoing a fundamental transformation. Yet, most of the debate in the current literature is focusing on the structural constraints faced by the Emerging and Developing Countries (EDCs) and the lack of changes in the formal structures of global economic governance. This paper challenges this literature and its conclusions by focusing on the new conditions of systemic interlocking between advanced and emerging economies, and by analysing how large EDCs have built and are strengthening their economic resilience. We find that a significant redistribution of ‘policy space’ between advanced and emerging economies have taken place in the global economy. We also find that a number of seemingly technical currency swap agreements among EDCs have set in motion changes in the very structure of global trade and finance. These developments do not signify the end of EDCs’ vulnerability towards advanced economies. They signify however that the economic and geoeconomic implications of this vulnerability have changed in ways that constrain the options available to advanced economies and pose new challenges for the post-WWII economic order

Real root finding for equivariant semi-algebraic systems

Let $R$ be a real closed field. We consider basic semi-algebraic sets defined by $n$-variate equations/inequalities of $s$ symmetric polynomials and an equivariant family of polynomials, all of them of degree bounded by $2d < n$. Such a semi-algebraic set is invariant by the action of the symmetric group. We show that such a set is either empty or it contains a point with at most $2d-1$ distinct coordinates. Combining this geometric result with efficient algorithms for real root finding (based on the critical point method), one can decide the emptiness of basic semi-algebraic sets defined by $s$ polynomials of degree $d$ in time $(sn)^{O(d)}$. This improves the state-of-the-art which is exponential in $n$. When the variables $x_1, \ldots, x_n$ are quantified and the coefficients of the input system depend on parameters $y_1, \ldots, y_t$, one also demonstrates that the corresponding one-block quantifier elimination problem can be solved in time $(sn)^{O(dt)}$

The effect of a boots preparation and pure natural secretin and pancreozymin on pancreatic and gastric function in man

Secretin was found to be a powerful inhibitor of basal gastric acid secretion in man. Pure natural secretin was more effective and more rapid in its action on gastric secretion than Boots secretin.Boots pancreozymin and Swedish CCK (cholecystoktnin) had a variable effect on basal gastric acid secretion and all the changes were modest and unimpressive

Progress in small-bowel physiology and disease

There have been many interesting and exciting developments in the field  of gastro-enterology in the past few years. Many of these advances have been due to technical skills in physiology, biochemistry and radiology,  but newer diagnostic and therapeutic measures have also been introduced. The complex functions of the small bowel and especially the mucosal  lining of the intestine, provide particular scope for detailed  multidisciplinary research, clinical, paraclinical and scientific. The present communication reflects our own brief selection of recent advances in the field of  small-bowel physiology and disease and does not represent a comprehensive coverage of all the advances in the past decade

The effect of a boots preparation and pure natural secretin and pancreozymin on pancreatic and 'gastric function in man

Secretin was found to be a powerful inhibitor of basal gastric acid secretion  in man. Pure natural secretin was more effective and more rapid in its action  on gastric secretion than Boots secretin. Boots pancreozymin and Swedish CCK (cholecystoktnin) had a variable effect on basal gastric acid secretion and all the changes were modest and unimpressive

Progress in small-bowel physiology and disease

There have been many interesting and exciting developments in the field of gastro-enterology in the past few years. Many of these advances have been due to technical skills in physiology, biochemistry and radiology, but newer diagnostic and therapeutic measures have also been introduced. The complex functions of the small bowel andespecially the mucosal lining of the intestine, provide particular scope for detailed multidisciplinary research, clinical, paraclinical and scientific.The present communication reflects our own brief selection of recent advances in the field of small-bowel physiology and disease and does not represent a comprehensive coverage of all the advances in the past decade