Low-degree tests at large distances

We define tests of boolean functions which distinguish between linear (or quadratic) polynomials, and functions which are very far, in an appropriate sense, from these polynomials. The tests have optimal or nearly optimal trade-offs between soundness and the number of queries. In particular, we show that functions with small Gowers uniformity norms behave ``randomly'' with respect to hypergraph linearity tests. A central step in our analysis of quadraticity tests is the proof of an inverse theorem for the third Gowers uniformity norm of boolean functions. The last result has also a coding theory application. It is possible to estimate efficiently the distance from the second-order Reed-Muller code on inputs lying far beyond its list-decoding radius

QuickMMCTest - Quick Multiple Monte Carlo Testing

Multiple hypothesis testing is widely used to evaluate scientific studies involving statistical tests. However, for many of these tests, p-values are not available and are thus often approximated using Monte Carlo tests such as permutation tests or bootstrap tests. This article presents a simple algorithm based on Thompson Sampling to test multiple hypotheses. It works with arbitrary multiple testing procedures, in particular with step-up and step-down procedures. Its main feature is to sequentially allocate Monte Carlo effort, generating more Monte Carlo samples for tests whose decisions are so far less certain. A simulation study demonstrates that for a low computational effort, the new approach yields a higher power and a higher degree of reproducibility of its results than previously suggested methods

SM Kaluza-Klein Excitations and Electroweak Precision Tests

We consider a minimal extension to higher dimensions of the Standard Model, having one compactified dimension, and we study its experimental tests in terms of electroweak data. We discuss tests from high-energy data at the $Z$-pole, and low-energy tests, notably from atomic parity violation data. This measurement combined with neutrino scattering data strongly restricts the allowed region of the model parameters. Furthermore this region is incompatible at 95% CL with the restrictions from high-energy experiments. Of course a global fit to all data is possible but the $\chi^2_{\rm min}$ for degree of freedom is unpleasantly large.Comment: LaTex, 14 pages, 2 figures. More refs. and one comment about the validity of our results for any number of extra dimensions adde

Failure mechanics in low-velocity impacts on thin composite plates

Eight-ply quasi-isotropic composite plates of Thornel 300 graphite in Narmco 5208 epoxy resin (T300/5208) were tested to establish the degree of equivalence between low-velocity impact and static testing. Both the deformation and failure mechanics under impact were representable by static indentation tests. Under low-velocity impacts such as tool drops, the dominant deformation mode of the plates was the first, or static, mode. Higher modes are excited on contact, but they decay significantly by the time the first-mode load reaches a maximum. The delamination patterns were observed by X-ray analysis. The areas of maximum delamination patterns were observed by X-ray analysis. The areas of maximum delamination coincided with the areas of highest peel stresses. The extent of delamination was similar for static and impact tests. Fiber failure damage was established by tensile tests on small fiber bundles obtained by deplying test specimens. The onset of fiber damage was in internal plies near the lower surface of the plates. The distribution and amount of fiber damage was similar fo impact and static tests

Viscoelastic properties of wood across the grain measured under water-saturated conditions up to 135\degree C: evidence of thermal degradation

In this paper, the viscoelastic properties of wood under water-saturated conditions are investigated from 10\degree C to 135\degree C using the WAVET* apparatus. Experiments were performed via harmonic tests at two frequencies (0.1 Hz and 1 Hz) for several hours. Four species of wood were tested in the radial and tangential material directions: oak (Quercus sessiliflora), beech (Fagus sylvatica), spruce (Picea abies) and fir (Abies pectinata). When the treatment is applied for several hours, a reduction of the wood rigidity is significant from temperature values as low as 80-90\degree C, and increases rapidly with the temperature level. The storage modulus of oak wood is divided by a factor two after three hours of exposure at 135\degree C. This marked reduction in rigidity is attributed to the hydrolysis of hemicelluloses. The softening temperature of wood is also noticeably affected by hygrothermal treatment. After three short successive treatments up to 135\degree C, the softening temperature of oak shifted from 79\degree C to 103\degree C, at a frequency of 1 Hz. This reduction in mobility of wood polymers is consistent with the condensation of lignins observed by many authors at this temperature level. In the same conditions, fir exhibited a softening temperature decreasing of about 4\degree C. In any case, the internal friction clearly raises

Matrix-free weighted quadrature for a computationally efficient isogeometric kk-method

The $k$-method is the isogeometric method based on splines (or NURBS, etc.) with maximum regularity. When implemented following the paradigms of classical finite element methods, the computational resources required by the $k-$method are prohibitive even for moderate degree. In order to address this issue, we propose a matrix-free strategy combined with weighted quadrature, which is an ad-hoc strategy to compute the integrals of the Galerkin system. Matrix-free weighted quadrature (MF-WQ) speeds up matrix operations, and, perhaps even more important, greatly reduces memory consumption. Our strategy also requires an efficient preconditioner for the linear system iterative solver. In this work we deal with an elliptic model problem, and adopt a preconditioner based on the Fast Diagonalization method, an old idea to solve Sylvester-like equations. Our numerical tests show that the isogeometric solver based on MF-WQ is faster than standard approaches (where the main cost is the matrix formation by standard Gaussian quadrature) even for low degree. But the main achievement is that, with MF-WQ, the $k$-method gets orders of magnitude faster by increasing the degree, given a target accuracy. Therefore, we are able to show the superiority, in terms of computational efficiency, of the high-degree $k$-method with respect to low-degree isogeometric discretizations. What we present here is applicable to more complex and realistic differential problems, but its effectiveness will depend on the preconditioner stage, which is as always problem-dependent. This situation is typical of modern high-order methods: the overall performance is mainly related to the quality of the preconditioner

On Axis-Parallel Tests for Tensor Product Codes

Many low-degree tests examine the input function via its restrictions to random hyperplanes of a certain dimension. Examples include the line-vs-line (Arora, Sudan 2003), plane-vs-plane (Raz, Safra 1997), and cube-vs-cube (Bhangale, Dinur, Livni 2017) tests. In this paper we study tests that only consider restrictions along axis-parallel hyperplanes, which have been studied by Polishchuk and Spielman (1994) and Ben-Sasson and Sudan (2006). While such tests are necessarily "weaker", they work for a more general class of codes, namely tensor product codes. Moreover, axis-parallel tests play a key role in constructing LTCs with inverse polylogarithmic rate and short PCPs (Polishchuk, Spielman 1994; Ben-Sasson, Sudan 2008; Meir 2010). We present two results on axis-parallel tests. (1) Bivariate low-degree testing with low-agreement. We prove an analogue of the Bivariate Low-Degree Testing Theorem of Polishchuk and Spielman in the low-agreement regime, albeit with much larger field size. Namely, for the 2-wise tensor product of the Reed-Solomon code, we prove that for sufficiently large fields, the 2-query variant of the axis-parallel line test (row-vs-column test) works for arbitrarily small agreement. Prior analyses of axis-parallel tests assumed high agreement, and no results for such tests in the low-agreement regime were known. Our proof technique deviates significantly from that of Polishchuk and Spielman, which relies on algebraic methods such as Bezout\u27s Theorem, and instead leverages a fundamental result in extremal graph theory by Kovari, Sos, and Turan. To our knowledge, this is the first time this result is used in the context of low-degree testing. (2) Improved robustness for tensor product codes. Robustness is a strengthening of local testability that underlies many applications. We prove that the axis-parallel hyperplane test for the m-wise tensor product of a linear code with block length n and distance d is Omega(d^m/n^m)-robust. This improves on a theorem of Viderman (2012) by a factor of 1/poly(m). While the improvement is not large, we believe that our proof is a notable simplification compared to prior work