1,186 research outputs found
Two Structural Results for Low Degree Polynomials and Applications
In this paper, two structural results concerning low degree polynomials over
finite fields are given. The first states that over any finite field
, for any polynomial on variables with degree , there exists a subspace of with dimension on which is constant. This result is shown to be tight.
Stated differently, a degree polynomial cannot compute an affine disperser
for dimension smaller than . Using a recursive
argument, we obtain our second structural result, showing that any degree
polynomial induces a partition of to affine subspaces of dimension
, such that is constant on each part.
We extend both structural results to more than one polynomial. We further
prove an analog of the first structural result to sparse polynomials (with no
restriction on the degree) and to functions that are close to low degree
polynomials. We also consider the algorithmic aspect of the two structural
results.
Our structural results have various applications, two of which are:
* Dvir [CC 2012] introduced the notion of extractors for varieties, and gave
explicit constructions of such extractors over large fields. We show that over
any finite field, any affine extractor is also an extractor for varieties with
related parameters. Our reduction also holds for dispersers, and we conclude
that Shaltiel's affine disperser [FOCS 2011] is a disperser for varieties over
.
* Ben-Sasson and Kopparty [SIAM J. C 2012] proved that any degree 3 affine
disperser over a prime field is also an affine extractor with related
parameters. Using our structural results, and based on the work of Kaufman and
Lovett [FOCS 2008] and Haramaty and Shpilka [STOC 2010], we generalize this
result to any constant degree
Pattern formation in a ring cavity with temporally incoherent feedback
We present a theoretical and experimental study of modulation instability and pattern formation in a passive nonlinear optical cavity that is longer than the coherence length of the light circulating in it. Pattern formation in this cavity exhibits various features of a second-order phase transition, closely resembling laser action
Spreadsheets with programmatically accessible version history
Spreadsheets enable users to store and track values of a variable. To graph changes in values of a variable over time, a user typically stores in the spreadsheet values of the variable at different time points. Retaining all values in individual, visible cells of a spreadsheet can make the spreadsheet difficult to read and/or maintain.
Modern spreadsheet programs include features to track changes made to the cells of a spreadsheet, e.g., as a version history. The version history enables a user to revert to a prior version. Techniques of this disclosure enable programmatic access to historical values stored in the cells of a spreadsheet. The techniques eliminate the need for a user to explicitly store multiple values for a variable over a time period. Spreadsheets that implement the techniques enable users to generate a graph by programmatically accessing historical values of the variable in the spreadsheet
DETECTING SEARCH QUERY LANGUAGE
A language detection system can be used for detecting language of a search query that is provided at a search engine by a user. The system receives the search query in a first language, that is detected by the system. The search engine determines a first set of search results in response to the search query. The system receives the first set of search results. The system then identifies language of the first set of search results. The identified language is concluded to be the search query language (first language)
The Intimate Relationship between Cavitation and Fracture
Nearly three decades ago, the field of mechanics was cautioned of the obscure
nature of cavitation processes in soft materials [Gent, A.N., 1990. Cavitation
in rubber: a cautionary tale. Rubber Chemistry and Technology, 63(3)]. Since
then, the debate on the mechanisms that drive this failure process is ongoing.
Using a high precision volume controlled cavity expansion procedure, this paper
reveals the intimate relationship between cavitation and fracture. Combining a
Griffith inspired formulation for crack propagation, and a Gent inspired
formulation for cavity expansion, we show that despite the apparent complexity
of the fracture patterns, the pressure-volume response follows a predictable
path. In contrast to available studies, both the model and our experiments are
able to track the entire process including the unstable branch, by controlling
the volume of the cavity. Moreover, this minimal theoretical framework is able
to explain the ambiguity in previous experiments by revealing the presence of
metastable states that can lead to first order transitions at onset of
fracture. The agreement between the simple theory and all of the experimental
results conducted in PDMS samples with shear moduli in the range of 25-246
[kPa], confirms that cavitation and fracture work together in driving the
expansion process. Through this study we also determine the fracture energy of
PDMS and show its significant dependence on strain stiffening
Morphogenesis and proportionate growth: A finite element investigation of surface growth with coupled diffusion
Modeling the spontaneous evolution of morphology in natural systems and its
preservation by proportionate growth remains a major scientific challenge. Yet,
it is conceivable that if the basic mechanisms of growth and the coupled
kinetic laws that orchestrate their function are accounted for, a minimal
theoretical model may exhibit similar growth behaviors. The ubiquity of surface
growth, a mechanism by which material is added or removed on the boundaries of
the body, has motivated the development of theoretical models, which can
capture the diffusion-coupled kinetics that govern it. However, due to their
complexity, application of these models has been limited to simplified
geometries. In this paper, we tackle these complexities by developing a finite
element framework to study the diffusion-coupled growth and morphogenesis of
finite bodies formed on uniform and flat substrates. We find that in this
simplified growth setting, the evolving body exhibits a sequence of distinct
growth stages that are reminiscent of natural systems, and appear spontaneously
without any externally imposed regulation or coordination. The computational
framework developed in this work can serve as the basis for future models that
are able to account for growth in arbitrary geometrical settings, and can shed
light on the basic physical laws that orchestrate growth and morphogenesis in
the natural world
Rate Amplification and Query-Efficient Distance Amplification for Linear LCC and LDC
The main contribution of this work is a rate amplification procedure for LCC. Our procedure converts any q-query linear LCC, having rate ? and, say, constant distance to an asymptotically good LCC with q^poly(1/?) queries.
Our second contribution is a distance amplification procedure for LDC that converts any linear LDC with distance ? and, say, constant rate to an asymptotically good LDC. The query complexity only suffers a multiplicative overhead that is roughly equal to the query complexity of a length 1/? asymptotically good LDC. This improves upon the poly(1/?) overhead obtained by the AEL distance amplification procedure [Alon and Luby, 1996; Alon et al., 1995].
Our work establishes that the construction of asymptotically good LDC and LCC is reduced, with a minor overhead in query complexity, to the problem of constructing a vanishing rate linear LCC and a (rapidly) vanishing distance linear LDC, respectively
Effect of Elasticity on Phase Separation in Heterogeneous Systems
A recent study has demonstrated that phase separation in binary liquid
mixtures is arrested in the presence of elastic networks and can lead to a
nearly uniformly-sized distribution of the dilute-phase droplets. At longer
timescales, these droplets exhibit a directional preference to migrate along
elastic property gradients to form a front of dissolving droplets [K. A.
Rosowski, T. Sai, E. Vidal-Henriquez, D. Zwicker, R. W. Style, E. R. Dufresne,
Elastic ripening and inhibition of liquid-liquid phase separation, Nature
Physics (2020) 1-4]. In this work, we develop a complete theoretical
understanding of this phenomenon in nonlinear elastic solids by employing an
energy-based approach that captures the process at both short and long
timescales to determine the constitutive sensitivities and the dynamics of the
resulting front propagation. We quantify the thermodynamic driving forces to
identify diffusion-limited and dissolution-limited regimes in front
propagation. We show that changes in elastic properties have a nonlinear effect
on the process. This strong influence can have implications in a variety of
material systems including food, metals, and aquatic sediments, and further
substantiates the hypothesis that biological systems exploit such mechanisms to
regulate important function.Comment: 20 pages, 7 figure
A large deformation theory for coupled swelling and growth with application to growing tumors and bacterial biofilms
There is significant interest in modelling the mechanics and physics of
growth of soft biological systems such as tumors and bacterial biofilms. Solid
tumors account for more than 85% of cancer mortality and bacterial biofilms
account for a significant part of all human microbial infections.These growing
biological systems are a mixture of fluid and solid components and increase
their mass by intake of diffusing species such as fluids and nutrients
(swelling) and subsequent conversion of some of the diffusing species into
solid material (growth). Experiments indicate that these systems swell by large
amounts and that the swelling and growth are intrinsically coupled. However,
existing theories for swelling coupled growth employ linear poroelasticity,
which is limited to small swelling deformations, and employ phenomenological
prescriptions for the dependence of growth rate on concentration of diffusing
species and the stress-state in the system. In particular, the termination of
growth is enforced through the prescription of a critical concentration of
diffusing species and a homeostatic stress. In contrast, by developing a fully
coupled swelling-growth theory that accounts for large swelling through
nonlinear poroelasticity, we show that the emergent driving stress for growth
automatically captures all the above phenomena. Further, we show that for the
soft growing systems considered here, the effects of the homeostatic stress and
critical concentration can be encapsulated under a single notion of a critical
swelling ratio. The applicability of the theory is shown by its ability to
capture experimental observations of growing tumors and biofilms under various
mechanical and diffusion-consumption constraints. Additionally, compared to
generalized mixture theories, our theory is amenable to relatively easy
numerical implementation with a minimal physically motivated parameter space
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