Revisit Sparse Polynomial Interpolation based on Randomized Kronecker Substitution

In this paper, a new reduction based interpolation algorithm for black-box multivariate polynomials over finite fields is given. The method is based on two main ingredients. A new Monte Carlo method is given to reduce black-box multivariate polynomial interpolation to black-box univariate polynomial interpolation over any ring. The reduction algorithm leads to multivariate interpolation algorithms with better or the same complexities most cases when combining with various univariate interpolation algorithms. We also propose a modified univariate Ben-or and Tiwarri algorithm over the finite field, which has better total complexity than the Lagrange interpolation algorithm. Combining our reduction method and the modified univariate Ben-or and Tiwarri algorithm, we give a Monte Carlo multivariate interpolation algorithm, which has better total complexity in most cases for sparse interpolation of black-box polynomial over finite fields

Recent advances in real geometric reasoning

In the 1930s Tarski showed that real quantifier elimination was possible, and in 1975 Collins gave a remotely practicable method, albeit with doubly-exponential complexity, which was later shown to be inherent. We discuss some of the recent major advances in Collins method: such as an alternative approach based on passing via the complexes, and advances which come closer to "solving the question asked" rather than "solving all problems to do with these polynomials"

A Linear Algebra Approach for Detecting Binomiality of Steady State Ideals of Reversible Chemical Reaction Networks

Motivated by problems from Chemical Reaction Network Theory, we investigate whether steady state ideals of reversible reaction networks are generated by binomials. We take an algebraic approach considering, besides concentrations of species, also rate constants as indeterminates. This leads us to the concept of unconditional binomiality, meaning binomiality for all values of the rate constants. This concept is different from conditional binomiality that applies when rate constant values or relations among rate constants are given. We start by representing the generators of a steady state ideal as sums of binomials, which yields a corresponding coefficient matrix. On these grounds we propose an efficient algorithm for detecting unconditional binomiality. That algorithm uses exclusively elementary column and row operations on the coefficient matrix. We prove asymptotic worst case upper bounds on the time complexity of our algorithm. Furthermore, we experimentally compare its performance with other existing methods

The Sphaleron Rate in SU(N) Gauge Theory

The sphaleron rate is defined as the diffusion constant for topological number NCS = int g^2 F Fdual/32 pi^2. It establishes the rate of equilibration of axial light quark number in QCD and is of interest both in electroweak baryogenesis and possibly in heavy ion collisions. We calculate the weak-coupling behavior of the SU(3) sphaleron rate, as well as making the most sensible extrapolation towards intermediate coupling which we can. We also study the behavior of the sphaleron rate at weak coupling at large Nc.Comment: 18 pages with 3 figure

BicaudalD Actively Regulates Microtubule Motor Activity in Lipid Droplet Transport

A great deal of sub-cellular organelle positioning, and essentially all minus-ended organelle transport, depends on cytoplasmic dynein, but how dynein's function is regulated is not well understood. BicD is established to play a critical role in mediating dynein function-loss of BicD results in improperly localized nuclei, mRNA particles, and a dispersed Golgi apparatus-however exactly what BicD's role is remains unknown. Nonetheless, it is widely believed that BicD may act to tether dynein to cargos. Here we use a combination of biophysical and biochemical studies to investigate BicD's role in lipid droplet transport during Drosophila embryogenesis.Functional loss of BicD impairs the embryo's ability to control the net direction of droplet transport; the developmentally controlled reversal in transport is eliminated. We find that minimal BicD expression (near-BicD(null)) decreases the average run length of both plus and minus end directed microtubule (MT) based transport. A point mutation affecting the BicD N-terminus has very similar effects on transport during cellularization (phase II), but in phase III (gastrulation) motion actually appears better than in the wild-type.In contrast to a simple static tethering model of BicD function, or a role only in initial dynein recruitment to the cargo, our data uncovers a new dynamic role for BicD in actively regulating transport. Lipid droplets move bi-directionally, and our investigations demonstrate that BicD plays a critical-and temporally changing-role in balancing the relative contributions of plus-end and minus-end motors to control the net direction of transport. Our results suggest that while BicD might contribute to recruitment of dynein to the cargo it is not absolutely required for such dynein localization, and it clearly contributes to regulation, helping activation/inactivation of the motors

SDCCAG8 Interacts with RAB Effector Proteins RABEP2 and ERC1 and Is Required for Hedgehog Signaling

Recessive mutations in the SDCCAG8 gene cause a nephronophthisis-related ciliopathy with Bardet-Biedl syndrome-like features in humans. Our previous characterization of the orthologous Sdccag8gt/gt mouse model recapitulated the retinal-renal disease phenotypes and identified impaired DNA damage response signaling as an underlying disease mechanism in the kidney. However, several other phenotypic and mechanistic features of Sdccag8gt/gt mice remained unexplored. Here we show that Sdccag8gt/gt mice exhibit developmental and structural abnormalities of the skeleton and limbs, suggesting impaired Hedgehog (Hh) signaling. Indeed, cell culture studies demonstrate the requirement of SDCCAG8 for ciliogenesis and Hh signaling. Using an affinity proteomics approach, we demonstrate that SDCCAG8 interacts with proteins of the centriolar satellites (OFD1, AZI1), of the endosomal sorting complex (RABEP2, ERC1), and with non-muscle myosin motor proteins (MYH9, MYH10, MYH14) at the centrosome. Furthermore, we show that RABEP2 localization at the centrosome is regulated by SDCCAG8. siRNA mediated RABEP2 knockdown in hTERT-RPE1 cells leads to defective ciliogenesis, indicating a critical role for RABEP2 in this process. Together, this study identifies several centrosome-associated proteins as novel SDCCAG8 interaction partners, and provides new insights into the function of SDCCAG8 at this structure

Sparse Polynomial Interpolation by Variable Shift in the Presence of Noise and Outliers in the Evaluations

We compute approximate sparse polynomial models of the form ˜ f(x) = ∑t ej j=1 ˜cj(x−˜s) to a function f(x), of which an approximation of the evaluation f(ζ) at any complex argument value ζ can be obtained. We assume that several of the returned function evaluations f(ζ) are perturbed not just by approximation/noise errors but also by highly perturbed outliers. None of the ˜cj, ˜s, ej and the location of the outliers are known before-hand. We use a numerical version of an exact algorithm by [Giesbrecht, Kaltofen, and Lee 2003] together with a numerical version of the Reed-Solomon error correcting coding algorithm. We also compare with a simpler approach based on root finding of derivatives, while restricted to characteristic 0. In this preliminary report, we discuss how some of the problems of numerical instability and ill-conditioning in the arising optimization problems can be overcome. By way of experiments, we show that our techniques can recover approximate sparse shifted polynomial models, provided that there are few terms t, few outliers and that the sparse shift is relatively small