Local Proof Research Articles

Extensions of Noether's Second Theorem: from continuous to discrete systems

A simple local proof of Noether's Second Theorem is given. This proof immediately leads to a generalization of the theorem, yielding conservation laws and/or explicit relationships between the Euler–Lagrange equations of any variational problem whose symmetries depend on a set of free or partly constrained functions. Our approach extends further to deal with finite-difference systems. The results are easy to apply; several well-known continuous and discrete systems are used as illustrations.

Existential graphs and proofs of pragmaticism

We show how Peirce's architectonics folds on itself and finds local consequences that correspond to the major global hypotheses of the system. In particular, we study how the pragmaticist maxim (i.e., the pragmatic maxim fully modalized, support of Peirce's architectonics) can be technically represented in Peirce's existential graphs, well-suited to reveal an underlying continuity in logical operations, and can provide suggestive philosophical analogies. Further, using the existential graphs, we formalize — and prove one direction of — a “local proof of pragmaticism,” trying thus to explain the prominent place that existential graphs can play in the architectonics of pragmaticism, as Peirce persistently advocated. Finally, we present a web of “continuous iterations” of some key Peircean concepts (maxim, classification, abduction) that supports a “lattice of partial proofs” of pragmaticism.

On the (non)rigidity of the Frobenius endomorphism over Gorenstein rings

It is well-known that for a large class of local rings of positive characteristic, including complete intersection rings, the Frobenius endomorphism can be used as a test for finite projective dimension. In this paper, we exploit this property to study the structure of such rings. One of our results states that the Picard group of the punctured spectrum of such a ring $R$ cannot have $p$-torsion. When $R$ is a local complete intersection, this recovers (with a purely local algebra proof) an analogous statement for complete intersections in projective spaces first given in SGA and also a special case of a conjecture by Gabber. Our method also leads to many simply constructed examples where rigidity for the Frobenius endomorphism does not hold, even when the rings are Gorenstein with isolated singularity. This is in stark contrast to the situation for complete intersection rings. Also, a related length criterion for modules of finite length and finite projective dimension is discussed towards the end.

Low Lying Spectrum of Weak-Disorder Quantum Waveguides

We study the low-lying spectrum of the Dirichlet Laplace operator on a randomly wiggled strip. More precisely, our results are formulated in terms of the eigenvalues of finite segment approximations of the infinite waveguide. Under appropriate weak-disorder assumptions we obtain deterministic and probabilistic bounds on the position of the lowest eigenvalue. A Combes-Thomas argument allows us to obtain so-called 'initial length scale decay estimates' at they are used in the proof of spectral localization using the multiscale analysis.

Reply: On Role of Symmetries in Kelvin Wave Turbulence

In the Ref. (Lebedev and L’vov in J. Low Temp. Phys. 161, 2010, doi:10.1007/s10909-010-0215-2), this issue, two of us (VVL and VSL) considered symmetry restriction on the interaction coefficients of Kelvin waves and demonstrated that linear in small wave vector asymptotic, obtained analytically, is not forbidden, as Kosik and Svistunov (KS) expect by naive reasoning. Here we discuss this problem in additional details and show that theoretical objections by KS, presented in Ref. (Kozik and Svistunov in J. Low Temp. Phys. 161, 2010, doi:10.1007/s10909-010-0242-z), this issue, are irrelevant and their recent numerical simulation, presented in Ref. (Kozik and Svistunov in arXiv:1007.4927v1, 2010) is hardly convincing. There is neither proof of locality nor any refutation of the possibility of linear asymptotic of interaction vertices in the KS texts, Refs. (Kozik and Svistunov in J. Low Temp. Phys. 161, 2010, doi:10.1007/s10909-010-0242-z; arXiv:1006.0506v1, 2010). Therefore we can state again that we have no reason to doubt in this asymptote, that results in the L’vov–Nazarenko energy spectrum of Kelvin waves.

Good reduction of affinoids on the Lubin-Tate tower

We analyze the geometry of the tower of Lubin-Tate deformation spaces, which parametrize deformations of a one-dimensional formal module of height h together with level structure. According to the conjecture of Deligne-Carayol, these spaces realize the local Langlands correspondence in their \ell -adic cohomology. This conjecture is now a theorem, but currently there is no purely local proof. Working in the equal characteristic case, we find a family of affinoids in the Lubin-Tate tower with good reduction equal to a rather curious nonsingular hypersurface, whose equation we present explicitly. Granting a conjecture on the L -functions of this hypersurface, we find a link between the conjecture of Deligne-Carayol and the theory of Bushnell-Kutzko types, at least for certain class of wildly ramified supercuspidal representations of small conductor.

A note on fractional moments for the one-dimensional continuum Anderson model

We give a proof of dynamical localization in the form of exponential decay of spatial correlations in the time evolution for the one-dimensional continuum Anderson model via the fractional moments method. This follows via exponential decay of fractional moments of the Green function, which is shown to hold at arbitrary energy and for any single-site distribution with bounded, compactly supported density.

Localness of energy cascade in hydrodynamic turbulence. II. Sharp spectral filter

We investigate the scale-locality of subgrid-scale (SGS) energy flux and interband energy transfers defined by the sharp spectral filter. We show by rigorous bounds, physical arguments, and numerical simulations that the spectral SGS flux is dominated by local triadic interactions in an extended turbulent inertial range. Interband energy transfers are also shown to be dominated by local triads if the spectral bands have constant width on a logarithmic scale. We disprove in particular an alternative picture of “local transfer by nonlocal triads,” with the advecting wavenumber mode at the energy peak. Although such triads have the largest transfer rates of all individual wavenumber triads, we show rigorously that, due to their restricted number, they make an asymptotically negligible contribution to energy flux and log-banded energy transfers at high wavenumbers in the inertial range. We show that it is only the aggregate effect of a geometrically increasing number of local wavenumber triads which can sustain an energy cascade to small scales. Furthermore, nonlocal triads are argued to contribute even less to the space-average energy flux than is implied by our rigorous bounds, because of additional cancellations from scale-decorrelation effects. We can thus recover the −4/3 scaling of nonlocal contributions to spectral energy flux predicted by Kraichnan’s abridged Lagrangian-history direct-interaction approximation and test-field model closures. We support our results with numerical data from a 5123 pseudospectral simulation of isotropic turbulence with phase-shift dealiasing. We also discuss a rigorous counterexample of Eyink [Physica D 78, 222 (1994)], which showed that nonlocal wavenumber triads may dominate in the sharp spectral flux (but not in the SGS energy flux for graded filters). We show that this mathematical counterexample fails to satisfy reasonable physical requirements for a turbulent velocity field, which are employed in our proof of scale locality. We conclude that the sharp spectral filter has a firm theoretical basis for use in large-eddy simulation modeling of turbulent flows.

Local proofs for global safety properties

This paper explores locality in proofs of global safety properties of concurrent programs. Model checking on the full state space is often infeasible due to state explosion. A local proof, in contr...

A REVISED DUALITY PROOF OF SAMPLING LOCALIZATION IN RELAXATION SPECTRUM RECOVERY

AbstractThe duality proof of sampling localization given by Loy, Newbury, Anderssen and Davies in 2001 contains an oversight, as the classes of functions chosen do not assume the compact support. Here, it is shown how a minor change to the argument there yields a precise conclusion.

Local proofs for global safety properties

This paper explores locality in proofs of global safety properties of concurrent programs. Model checking on the full state space is often infeasible due to state explosion. A local proof, in contrast, is a collection of per-process invariants, which together imply the desired global safety property. Local proofs can be more compact than global proofs, but local reasoning is also inherently incomplete. In this paper, we present an algorithm for safety verification that combines local reasoning with gradual refinement. The algorithm gradually exposes facts about the internal state of components, until either a local proof or a real error is discovered. The refinement mechanism ensures completeness. Experiments show that local reasoning can have significantly better performance over the traditional reachability computation. Moreover, for some parameterized protocols, a local proof can be used as the basis of a correctness proof over all instances.

Divergent expansion, Borel summability and three-dimensional Navier–Stokes equation

We describe how the Borel summability of a divergent asymptotic expansion can be expanded and applied to nonlinear partial differential equations (PDEs). While Borel summation does not apply for non-analytic initial data, the present approach generates an integral equation (IE) applicable to much more general data. We apply these concepts to the three-dimensional Navier-Stokes (NS) system and show how the IE approach can give rise to local existence proofs. In this approach, the global existence problem in three-dimensional NS systems, for specific initial condition and viscosity, becomes a problem of asymptotics in the variable p (dual to 1/t or some positive power of 1/t). Furthermore, the errors in numerical computations in the associated IE can be controlled rigorously, which is very important for nonlinear PDEs such as NS when solutions are not known to exist globally.Moreover, computation of the solution of the IE over an interval [0,p0] provides sharper control of its p-->infinity behaviour. Preliminary numerical computations give encouraging results.

On Epr Paradox, Bell's Inequalities and Experiments that Prove Nothing

This article shows that the there is no paradox. Violation of Bell's inequalities should not be identified with a proof of non locality in quantum mechanics. A number of past experiments is reviewed, and it is concluded that the experimental results should be re-evaluated. The results of the experiments with atomic cascade are shown not to contradict the local realism. The article points out flaws in the experiments with down-converted photons. The experiments with neutron interferometer on measuring the "contextuality" and Bell-like inequalities are analyzed, and it is shown that the experimental results can be explained without such notions. Alternative experiment is proposed to prove the validity of local realism.

Diagonal Quadratic Approximation for Parallelization of Analytical Target Cascading

Analytical target cascading (ATC) is an effective decomposition approach used for engineering design optimization problems that have hierarchical structures. With ATC, the overall system is split into subsystems, which are solved separately and coordinated via target/response consistency constraints. As parallel computing becomes more common, it is desirable to have separable subproblems in ATC so that each subproblem can be solved concurrently to increase computational throughput. In this paper, we first examine existing ATC methods, providing an alternative to existing nested coordination schemes by using the block coordinate descent method (BCD). Then we apply diagonal quadratic approximation (DQA) by linearizing the cross term of the augmented Lagrangian function to create separable subproblems. Local and global convergence proofs are described for this method. To further reduce overall computational cost, we introduce the truncated DQA (TDQA) method, which limits the number of inner loop iterations of DQA. These two new methods are empirically compared to existing methods using test problems from the literature. Results show that computational cost of nested loop methods is reduced by using BCD, and generally the computational cost of the truncated methods is superior to the nested loop methods with lower overall computational cost than the best previously reported results.

Local strength evaluation and proof test of glass components via spherical indentation

Hertzian contact crack formation combined with acoustic emission was used to evaluate the local strength of glass in situ. The relationship between the critical load and the strength of the glass sample was established. Based on this model, a strength proof test was designed to examine the reliability of glass components in service. Moreover, a new testing device was designed and made for accomplishing this test. It has been demonstrated that this simple testing method is convenient and available for measuring or estimating the surface strength of a glass component in service, without using a standard specimen and a rupture test in laboratory.

Injectivity theorems and algebraic closures of groups with coefficients

ABSTRACTRecently, Cochran and Harvey defined torsion-free derived series of groups and proved an injectivity theorem on the associated torsion-free quotients. We show that there is a universal construction which extends such an injectivity theorem to an isomorphism theorem. Our result relates injectivity theorems to a certain homology localization of groups. In order to give a concrete combinatorial description and existence proof of the necessary homology localization, we introduce a new version of algebraic closures of groups with coefficients by considering certain types of equations.

Erratum: A local proof of Petri's conjecture at the general curve

Erratum: A local proof of Petri's conjecture at the general curve

Relating invariant linear form and local epsilon factors via global methods

We use the recent proof of Jacquet's conjecture due to Harris and Kudla [HK] and the Burger-Sarnak principle (see [BS]) to give a proof of the relationship between the existence of trilinear forms on representations of GL2(ku) for a non-Archimedean local field ku and local epsilon factors which was earlier proved only in the odd residue characteristic by this author in [P1, Theorem 1.4]. The method used is very flexible and gives a global proof of a theorem of Saito and Tunnell about characters of GL2 using a theorem of Waldspurger [W, Theorem 2] about period integrals for GL2 and also an extension of the theorem of Saito and Tunnell by this author in [P3, Theorem 1.2] which was earlier proved only in odd residue characteristic. In the appendix to this article, H. Saito gives a local proof of Lemma 4 which plays an important role in the article

GPS-DENIED SOURCE SEEKING FOR A NONHOLONOMIC UNICYCLE VIA HEADING CONTROL

We consider the problem of seeking the source of a scalar signal using an autonomous vehicle modeled as the nonholonomic unicycle. This vehicle does not have the capability of sensing its position or the position of the source, it is only capable of sensing the scalar signal at the position of the vehicle sensor, which is offset from the vehicle center. We assume that the signal field is the strongest at the position of the source and decays away from this position. The vehicle does not know the functional form of this field, only that there is a maximum at the source location. We employ extremum seeking to estimate the gradient in real time and steer the vehicle to the point where the gradient is zero (the maximum, i.e. the location of the source). We have developed two control strategies – one which keeps the angular velocity at a constant nonzero value and tunes the forward velocity, and the other which keeps the forward velocity at a constant positive value and tunes the angular velocity. In this paper we present the results for the more practical scheme, tuning the angular velocy. This is more suitable for most autonomous vehicles, including aerial ones. In addition to the local stability proof of the scheme via averaging, we present simulation results for both static and moving sources.

Source seeking with non-holonomic unicycle without position measurement and with tuning of forward velocity

We consider the problem of seeking the source of a scalar signal using an autonomous vehicle modeled as the non-holonomic unicycle and equipped with a sensor of that scalar signal but not possessing the capability to sense either the position of the source nor its own position. We assume that the signal field is the strongest at the source and decays away from it. The functional form of the field is not available to our vehicle. We employ extremum seeking to estimate the gradient of the field in real time and steer the vehicle towards the point where the gradient is zero (the maximum of the field, i.e., the location of the source). We employ periodic forward–backward movement of the unicycle (implementable with mobile robots and some underwater vehicles but not with aircraft), where the forward velocity has a tunable bias term, which is appropriately combined with extremum seeking to produce a net effect of “drifting” towards the source. In addition to simulation results we present a local convergence proof via averaging, which exhibits a delicate periodic structure with two sinusoids of different frequencies—one related to the angular velocity of the unicycle and the other related to the probing frequency of extremum seeking.