1,539 research outputs found
Finite element modelling of the inertia friction welding of a CrMoV alloy steel including the effects of solid-state phase transformations
Finite element (FE) process modelling of the inertia friction welding (IFW) between two tubular CrMoV components has been carried out using the DEFORM-2D (v10.2) software. This model has been validated against experimental test welds of the material; this included process data such as upset and rotational velocity as well as thermal data collected during the process using embedded thermocouples. The as-welded residual stress from the FE model has been compared to experimental measurements taken on the welded component using the hole drilling technique. The effects of the solid-state phase transformations which occur in the steel are considered and the trends in the residual stress measurements were well replicated when compared to the experimental data
On sharp bilinear Strichartz estimates of Ozawa-Tsutsumi type
We provide a comprehensive analysis of sharp bilinear estimates of
Ozawa-Tsutsumi type for solutions u of the free Schr\"odinger equation, which
give sharp control on in classical Sobolev spaces. In particular, we
provide a generalization of their estimates in such a way that provides a
unification with some sharp bilinear estimates proved by Carneiro and
Planchon-Vega, via entirely different methods, by seeing them all as special
cases of a one parameter family of sharp estimates. We show that the extremal
functions are solutions of the Maxwell-Boltzmann functional equation and
provide a new proof that this equation admits only Gaussian solutions. We also
make a connection to certain sharp estimates on involving certain
dispersive Sobolev norms.Comment: 17 pages, references update
Prisoner voting for the final general election before release is a solution that balances concerns about democratic rights
Democratic Audit has recently featured analysis of prisoner voting rights from several leading experts. In the second of two new contributions to this debate – following Peter Ramsay’s earlier post – Chris Bennett and Daniel Viehoff argue that both sides of the debate can make strong claims to democratic principles. They make a new proposal that aims to balance these competing concerns
Hardness of the (Approximate) Shortest Vector Problem: A Simple Proof via Reed-Solomon Codes
We give a
simple proof that the (approximate, decisional) Shortest Vector Problem is
\NP-hard under a randomized reduction. Specifically, we show that for any and any constant , the -approximate problem
in the norm (-\GapSVP_p) is not in unless \NP
\subseteq \mathsf{RP}. Our proof follows an approach pioneered by Ajtai (STOC
1998), and strengthened by Micciancio (FOCS 1998 and SICOMP 2000), for showing
hardness of -\GapSVP_p using locally dense lattices. We construct
such lattices simply by applying "Construction A" to Reed-Solomon codes with
suitable parameters, and prove their local density via an elementary argument
originally used in the context of Craig lattices.
As in all known \NP-hardness results for \GapSVP_p with , our
reduction uses randomness. Indeed, it is a notorious open problem to prove
\NP-hardness via a deterministic reduction. To this end, we additionally
discuss potential directions and associated challenges for derandomizing our
reduction. In particular, we show that a close deterministic analogue of our
local density construction would improve on the state-of-the-art explicit
Reed-Solomon list-decoding lower bounds of Guruswami and Rudra (STOC 2005 and
IEEE Trans. Inf. Theory 2006).
As a related contribution of independent interest, we also give a
polynomial-time algorithm for decoding -dimensional "Construction A
Reed-Solomon lattices" (with different parameters than those used in our
hardness proof) to a distance within an factor of
Minkowski's bound. This asymptotically matches the best known distance for
decoding near Minkowski's bound, due to Mook and Peikert (IEEE Trans. Inf.
Theory 2022), whose work we build on with a somewhat simpler construction and
analysis
Hardness of Bounded Distance Decoding on Lattices in ?_p Norms
Bounded Distance Decoding BDD_{p,?} is the problem of decoding a lattice when the target point is promised to be within an ? factor of the minimum distance of the lattice, in the ?_p norm. We prove that BDD_{p, ?} is NP-hard under randomized reductions where ? ? 1/2 as p ? ? (and for ? = 1/2 when p = ?), thereby showing the hardness of decoding for distances approaching the unique-decoding radius for large p. We also show fine-grained hardness for BDD_{p,?}. For example, we prove that for all p ? [1,?) ? 2? and constants C > 1, ? > 0, there is no 2^((1-?)n/C)-time algorithm for BDD_{p,?} for some constant ? (which approaches 1/2 as p ? ?), assuming the randomized Strong Exponential Time Hypothesis (SETH). Moreover, essentially all of our results also hold (under analogous non-uniform assumptions) for BDD with preprocessing, in which unbounded precomputation can be applied to the lattice before the target is available.
Compared to prior work on the hardness of BDD_{p,?} by Liu, Lyubashevsky, and Micciancio (APPROX-RANDOM 2008), our results improve the values of ? for which the problem is known to be NP-hard for all p > p? ? 4.2773, and give the very first fine-grained hardness for BDD (in any norm). Our reductions rely on a special family of "locally dense" lattices in ?_p norms, which we construct by modifying the integer-lattice sparsification technique of Aggarwal and Stephens-Davidowitz (STOC 2018)
Positroid cluster structures from relabeled plabic graphs
The Grassmannian is a disjoint union of open positroid varieties ,
certain smooth irreducible subvarieties whose definition is motivated by total
positivity. The coordinate ring of is a cluster algebra, and each reduced
plabic graph for determines a cluster. We study the effect of
relabeling the boundary vertices of by a permutation . Under suitable
hypotheses on the permutation, we show that the relabeled graph
determines a cluster for a different open positroid variety . As a key
step of the proof, we show that and are isomorphic by a nontrivial
twist isomorphism. Our constructions yield many cluster structures on each open
positroid variety , given by plabic graphs with appropriately relabeled
boundary. We conjecture that the seeds in all of these cluster structures are
related by a combination of mutations and Laurent monomial transformations
involving frozen variables, and establish this conjecture for (open) Schubert
and opposite Schubert varieties. As an application, we also show that for
certain reduced plabic graphs , the "source" cluster and the "target"
cluster are related by mutation and Laurent monomial rescalings.Comment: 45 pages, comments welcome! v2: minor change
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