A PSPACE Construction of a Hitting Set for the Closure of Small Algebraic Circuits

In this paper we study the complexity of constructing a hitting set for the closure of VP, the class of polynomials that can be infinitesimally approximated by polynomials that are computed by polynomial sized algebraic circuits, over the real or complex numbers. Specifically, we show that there is a PSPACE algorithm that given n,s,r in unary outputs a set of n-tuples over the rationals of size poly(n,s,r), with poly(n,s,r) bit complexity, that hits all n-variate polynomials of degree-r that are the limit of size-s algebraic circuits. Previously it was known that a random set of this size is a hitting set, but a construction that is certified to work was only known in EXPSPACE (or EXPH assuming the generalized Riemann hypothesis). As a corollary we get that a host of other algebraic problems such as Noether Normalization Lemma, can also be solved in PSPACE deterministically, where earlier only randomized algorithms and EXPSPACE algorithms (or EXPH assuming the generalized Riemann hypothesis) were known. The proof relies on the new notion of a robust hitting set which is a set of inputs such that any nonzero polynomial that can be computed by a polynomial size algebraic circuit, evaluates to a not too small value on at least one element of the set. Proving the existence of such a robust hitting set is the main technical difficulty in the proof. Our proof uses anti-concentration results for polynomials, basic tools from algebraic geometry and the existential theory of the reals

A study of singularities on rational curves via syzygies

Consider a rational projective curve C of degree d over an algebraically closed field k. There are n homogeneous forms g_1,...,g_n of degree d in B=k[x,y] which parameterize C in a birational, base point free, manner. We study the singularities of C by studying a Hilbert-Burch matrix phi for the row vector [g_1,...,g_n]. In the "General Lemma" we use the generalized row ideals of phi to identify the singular points on C, their multiplicities, the number of branches at each singular point, and the multiplicity of each branch. Let p be a singular point on the parameterized planar curve C which corresponds to a generalized zero of phi. In the "Triple Lemma" we give a matrix phi' whose maximal minors parameterize the closure, in projective 2-space, of the blow-up at p of C in a neighborhood of p. We apply the General Lemma to phi' in order to learn about the singularities of C in the first neighborhood of p. If C has even degree d=2c and the multiplicity of C at p is equal to c, then we apply the Triple Lemma again to learn about the singularities of C in the second neighborhood of p. Consider rational plane curves C of even degree d=2c. We classify curves according to the configuration of multiplicity c singularities on or infinitely near C. There are 7 possible configurations of such singularities. We classify the Hilbert-Burch matrix which corresponds to each configuration. The study of multiplicity c singularities on, or infinitely near, a fixed rational plane curve C of degree 2c is equivalent to the study of the scheme of generalized zeros of the fixed balanced Hilbert-Burch matrix phi for a parameterization of C.Comment: Typos corrected and minor changes made. To appear in the Memoirs of the AM

Toward Automation of the Supine Pressor Test for Preeclampsia

Preeclampsia leads to increased risk of morbidity and mortality for both mother and fetus. Most previous studies have largely neglected mechanical compression of the left renal vein by the gravid uterus as a potential mechanism. In this study, we first used a murine model to investigate the pathophysiology of left renal vein constriction. The results indicate that prolonged renal vein stenosis after 14 days can cause renal necrosis and an increase in blood pressure (BP) of roughly 30 mmHg. The second part of this study aimed to automate a diagnostic tool, known as the supine pressor test (SPT), to enable pregnant women to assess their preeclampsia development risk. A positive SPT has been previously defined as an increase of at least 20 mmHg in diastolic BP when switching between left lateral recumbent and supine positions. The results from this study established a baseline BP increase between the two body positions in nonpregnant women and demonstrated the feasibility of an autonomous SPT in pregnant women. Our results demonstrate that there is a baseline increase in BP of roughly 10-14 mmHg and that pregnant women can autonomously perform the SPT. Overall, this work in both rodents and humans suggests that (1) stenosis of the left renal vein in mice leads to elevation in BP and acute renal failure, (2) nonpregnant women experience a baseline increase in BP when they shift from left lateral recumbent to supine position, and (3) the SPT can be automated and used autonomously

Downward shortwave surface irradiance from 17 sites for the FIRE/SRB Wisconsin experiment

A field experiment was conducted in Wisconsin during Oct. to Nov. 1986 for purposes of both intensive cirrus cloud measurments and SRB algorithm validation activities. The cirrus cloud measurements were part of the FIRE. Tables are presented which show data from 17 sites in the First ISCCP (International Satellite Cloud Climatology Project) Regional Experiment/Surface Radiation Budget (FIRE/SRB) Wisconsin experiment region. A discussion of intercomparison results and calibration inconsistencies is also included

Orientation-dependent pinning and homoclinic snaking on a planar lattice

We study homoclinic snaking of one-dimensional, localized states on two-dimensional, bistable lattices via the method of exponential asymptotics. Within a narrow region of parameter space, fronts connecting the two stable states are pinned to the underlying lattice. Localized solutions are formed by matching two such stationary fronts back-to-back; depending on the orientation relative to the lattice, the solution branch may “snake” back and forth within the pinning region via successive saddle-node bifurcations. Standard continuum approximations in the weakly nonlinear limit (equivalently, the limit of small mesh size) do not exhibit this behavior, due to the resultant leading-order reaction-diffusion equation lacking a periodic spatial structure. By including exponentially small effects hidden beyond all algebraic orders in the asymptotic expansion, we find that exponentially small but exponentially growing terms are switched on via error function smoothing near Stokes lines. Eliminating these otherwise unbounded beyond-all-orders terms selects the origin (modulo the mesh size) of the front, and matching two fronts together yields a set of equations describing the snaking bifurcation diagram. This is possible only within an exponentially small region of parameter space—the pinning region. Moreover, by considering fronts orientated at an arbitrary angle ψ to the x-axis, we show that the width of the pinning region is nonzero only if tan ψ is rational or infinite. The asymptotic results are compared with numerical calculations, with good agreement

Orbifold Resolution by D-Branes

We study topological properties of the D-brane resolution of three-dimensional orbifold singularities, C^3/Gamma, for finite abelian groups Gamma. The D-brane vacuum moduli space is shown to fill out the background spacetime with Fayet--Iliopoulos parameters controlling the size of the blow-ups. This D-brane vacuum moduli space can be classically described by a gauged linear sigma model, which is shown to be non-generic in a manner that projects out non-geometric regions in its phase diagram, as anticipated from a number of perspectives.Comment: 26 pages, 2 figures (TeX, harvmac big, epsf

Space Suit Portable Life Support System (PLSS) 2.0 Pre-Installation Acceptance (PIA) Testing

Following successful completion of the space suit Portable Life Support System (PLSS) 1.0 development and testing in 2011, the second system-level prototype, PLSS 2.0, was developed in 2012 to continue the maturation of the advanced PLSS design. This advanced PLSS is intended to reduce consumables, improve reliability and robustness, and incorporate additional sensing and functional capabilities over the current Space Shuttle/International Space Station Extravehicular Mobility Unit (EMU) PLSS. PLSS 2.0 represents the first attempt at a packaged design comprising first generation or later component prototypes and medium fidelity interfaces within a flight-like representative volume. Pre-Installation Acceptance (PIA) is carryover terminology from the Space Shuttle Program referring to the series of test sequences used to verify functionality of the EMU PLSS prior to installation into the Space Shuttle airlock for launch. As applied to the PLSS 2.0 development and testing effort, PIA testing designated the series of 27 independent test sequences devised to verify component and subsystem functionality, perform in situ instrument calibrations, generate mapping data, define set-points, evaluate control algorithms, evaluate hardware performance against advanced PLSS design requirements, and provide quantitative and qualitative feedback on evolving design requirements and performance specifications. PLSS 2.0 PIA testing was carried out in 2013 and 2014 using a variety of test configurations to perform test sequences that ranged from stand-alone component testing to system-level testing, with evaluations becoming increasingly integrated as the test series progressed. Each of the 27 test sequences was vetted independently, with verification of basic functionality required before completion. Because PLSS 2.0 design requirements were evolving concurrently with PLSS 2.0 PIA testing, the requirements were used as guidelines to assess performance during the tests; after the completion of PIA testing, test data served to improve the fidelity and maturity of design requirements as well as plans for future advanced PLSS functional testing

Self-Calibration of Cameras with Euclidean Image Plane in Case of Two Views and Known Relative Rotation Angle

The internal calibration of a pinhole camera is given by five parameters that are combined into an upper-triangular $3\times 3$ calibration matrix. If the skew parameter is zero and the aspect ratio is equal to one, then the camera is said to have Euclidean image plane. In this paper, we propose a non-iterative self-calibration algorithm for a camera with Euclidean image plane in case the remaining three internal parameters --- the focal length and the principal point coordinates --- are fixed but unknown. The algorithm requires a set of $N \geq 7$ point correspondences in two views and also the measured relative rotation angle between the views. We show that the problem generically has six solutions (including complex ones). The algorithm has been implemented and tested both on synthetic data and on publicly available real dataset. The experiments demonstrate that the method is correct, numerically stable and robust.Comment: 13 pages, 7 eps-figure