Determinantal equations for secant varieties and the Eisenbud-Koh-Stillman conjecture

We address special cases of a question of Eisenbud on the ideals of secant varieties of Veronese re-embeddings of arbitrary varieties. Eisenbud's question generalizes a conjecture of Eisenbud, Koh and Stillman (EKS) for curves. We prove that set-theoretic equations of small secant varieties to a high degree Veronese re-embedding of a smooth variety are determined by equations of the ambient Veronese variety and linear equations. However this is false for singular varieties, and we give explicit counter-examples to the EKS conjecture for singular curves. The techniques we use also allow us to prove a gap and uniqueness theorem for symmetric tensor rank. We put Eisenbud's question in a more general context about the behaviour of border rank under specialisation to a linear subspace, and provide an overview of conjectures coming from signal processing and complexity theory in this context.Comment: 21 pages; presentation improved as suggested by the referees; To appear in Journal of London Mathematical Societ

Wavelet versus Detrended Fluctuation Analysis of multifractal structures

We perform a comparative study of applicability of the Multifractal Detrended Fluctuation Analysis (MFDFA) and the Wavelet Transform Modulus Maxima (WTMM) method in proper detecting of mono- and multifractal character of data. We quantify the performance of both methods by using different sorts of artificial signals generated according to a few well-known exactly soluble mathematical models: monofractal fractional Brownian motion, bifractal Levy flights, and different sorts of multifractal binomial cascades. Our results show that in majority of situations in which one does not know a priori the fractal properties of a process, choosing MFDFA should be recommended. In particular, WTMM gives biased outcomes for the fractional Brownian motion with different values of Hurst exponent, indicating spurious multifractality. In some cases WTMM can also give different results if one applies different wavelets. We do not exclude using WTMM in real data analysis, but it occurs that while one may apply MFDFA in a more automatic fashion, WTMM has to be applied with care. In the second part of our work, we perform an analogous analysis on empirical data coming from the American and from the German stock market. For this data both methods detect rich multifractality in terms of broad f(alpha), but MFDFA suggests that this multifractality is poorer than in the case of WTMM.Comment: substantially extended version, to appear in Phys.Rev.

Secants of Lagrangian Grassmannians

We study the dimensions of secant varieties of the Grassmannian of Lagrangian subspaces in a symplectic vector space. We calculate these dimensions for third and fourth secant varieties. Our result is obtained by providing a normal form for four general points on such a Grassmannian and by explicitly calculating the tangent spaces at these four points

Contact Moishezon threefolds with second Betti number one

We prove that the only contact Moishezon threefold having second Betti number equal to one is the projective space.Comment: 5 pages. v2: exposition improved as suggested by the referee. To appear in Archiv der Mat

Search for universality in one-dimensional ballistic annihilation kinetics

We study the kinetics of ballistic annihilation for a one-dimensional ideal gas with continuous velocity distribution. A dynamical scaling theory for the long time behavior of the system is derived. Its validity is supported by extensive numerical simulations for several velocity distributions. This leads us to the conjecture that all the continuous velocity distributions \phi(v) which are symmetric, regular and such that \phi(0) does not vanish, are attracted in the long time regime towards the same Gaussian distribution and thus belong to the same universality class. Moreover, it is found that the particle density decays as n(t)~t^{-\alpha}, with \alpha=0.785 +/- 0.005.Comment: 8 pages, needs multicol, epsf and revtex. 8 postscript figures included. Submitted to Phys. Rev. E. Also avaiable at http://mykonos.unige.ch/~rey/publi.html#Secon

Rzadki przypadek wznowy PEComa (Perivascular Epithelioid Cell Tumor). Przypadek kliniczny i przegląd piśmiennictwa

Perivascular epithelioid cell tumor (PEC-oma) is a rare mesenchymal neoplasm. Literature reports more than 100 cases of PEC-oma, a third of which is of uterine or uterine retroperitoneum origin. The case of a 59-year-old woman presented here is, to the best of our knowledge, the first described fast uterine PEC-oma recurrence of the tumor of the gastrointestinal tract origin. In this text the authors also present literature review concerning this rare female tumor.Perivascular epithelioid cell tumor (PEComa), są bardzo rzadkimi guzami pochodzenia mezenchymalnego. Do dnia dzisiejszego odnotowano w piśmiennictwie ponad 100 przypadków PEComa, z czego mniej więcej 1/3 występowała w macicy lub pokrywającej ją otrzewnej (retroperitoneum). Prezentowany przypadek 59 letniej pacjentki jest według nas pierwszym przedstawiającym szybką wznowę zmiany o typie PEComa zlokalizowaną w obrębie macicy o pierwotnym punkcie wyjścia z przewodu pokarmowego. W poniższym tekście przedstawiamy także przegląd piśmiennictwa dotyczący tego rzadkiego guza kobiecego narządu rodnego

A path integral approach to the dynamics of a random chain with rigid constraints

In this work the dynamics of a freely jointed random chain which fluctuates at constant temperature in some viscous medium is studied. The chain is regarded as a system of small particles which perform a brownian motion and are subjected to rigid constraints which forbid the breaking of the chain. For simplicity, all interactions among the particles have been switched off and the number of dimensions has been limited to two. The problem of describing the fluctuations of the chain in the limit in which it becomes a continuous system is solved using a path integral approach, in which the constraints are imposed with the insertion in the path integral of suitable Dirac delta functions. It is shown that the probability distribution of the possible conformations in which the fluctuating chain can be found during its evolution in time coincides with the partition function of a field theory which is a generalization of the nonlinear sigma model in two dimensions. Both the probability distribution and the generating functional of the correlation functions of the positions of the beads are computed explicitly in a semiclassical approximation for a ring-shaped chain.Comment: 36 pages, 2 figures, LaTeX + REVTeX4 + graphicx, minor changes in the text, reference adde

Convergence to equilibrium under a random Hamiltonian

We analyze equilibration times of subsystems of a larger system under a random total Hamiltonian, in which the basis of the Hamiltonian is drawn from the Haar measure. We obtain that the time of equilibration is of the order of the inverse of the arithmetic average of the Bohr frequencies. To compute the average over a random basis, we compute the inverse of a matrix of overlaps of operators which permute four systems. We first obtain results on such a matrix for a representation of an arbitrary finite group and then apply it to the particular representation of the permutation group under consideration.Comment: 11 pages, 1 figure, v1-v3: some minor errors and typos corrected and new references added; v4: results for the degenerated spectrum added; v5: reorganized and rewritten version; to appear in PR

Risk factors for cesarean section after using the Foley catheter for labor induction

Objective: The aim of the study was to investigate the value of the Bishop score and ultrasound examination of the cervix in predicting the success of labor induction with the use of the Foley catheter determined by the mode of delivery. Material and methods: Foley catheter induction of labor was performed in 135 pregnancies between 38 to 42 weeks gestation. The study group was divided into two groups, depending of the mode of delivery: vaginal vs. cesarean. Results: The Bishop score was significantly higher in the vaginal delivery group when compared to the caesarean section group (5.2; 95%CI: 4.4 – 6.2 vs. 3.9; 95%CI: 2.8-4.9). Cervical length was not statistically significantly different between the two groups. Multivariate logistic regression showed that patient-specific risk for caesarean section decreases with increasing maternal age and the Bishop score (Detection Rate [DR] of 52% at fixed False Positive Rate [FPR] of 10%). Conclusions: Failure of labor induction with the use of the Foley catheter can be predicted by maternal age and pre-induction Bishop score