Nonlocal Setting Research Articles

Mutational inclusions: Differential inclusions in metric spaces

The focus of interest is how to extend ordinary differential inclusions beyond the traditional border of vector spaces. We aim at an existence theorem for solutions whose values are in a given metric space. In the nineties, Aubin suggested how to formulate ordinary differential equations and their main existence theorems in metric spaces: mutational equations (which are quite similar to the quasidifferential equations of Panasyuk). Now the well-known Antosiewicz-Cellina Theorem is extended to so-called mutational inclusions. It provides new results about nonlocal set evolutions in R N .

D3 branes in a Melvin universe: A new realm for gravitational holography

The decoupling limit of a certain configuration of D3 branes in a Melvin universe defines a sector of string theory known as Puff Field Theory (PFT) - a theory with non-local dynamics but without gravity. In this work, we present a systematic analysis of the non-local states of strongly coupled PFT using gravitational holography. And we are led to a remarkable new holographic dictionary. We show that the theory admits states that may be viewed as brane protrusions from the D3 brane worldvolume. The footprint of a protrusion has finite size - the scale of non-locality in the PFT - and corresponds to an operator insertion in the PFT. We compute correlators of these states, and we demonstrate that only part of the holographic bulk is explored by this computation. We then show that the remaining space holographically encodes the dynamics of the D3 brane tentacles. The two sectors are coupled: in this holographic description, this is realized via quantum entanglement across a holographic screen - a throat in the geometry - that splits the bulk into the two regions in question. We then propose a description of PFT through a direct product of two Fock spaces - akin to other non-local settings that employ quantum group structures.

Non-Existence and Uniqueness Results for Supercritical Semilinear Elliptic Equations

Non-existence and uniqueness results are proved for several local and non-local supercritical bifurcation problems involving a semilinear elliptic equation depending on a parameter. The domain is star-shaped but no other symmetry assumption is required. Uniqueness holds when the bifurcation parameter is in a certain range. Our approach can be seen, in some cases, as an extension of non-existence results for non-trivial solutions. It is based on Rellich-Pohozaev type estimates. Semilinear elliptic equations naturally arise in many applications, for instance in astrophysics, hydrodynamics or thermodynamics. We simplify the proof of earlier results by K. Schmitt and R. Schaaf in the so-called local multiplicative case, extend them to the case of a non-local dependence on the bifurcation parameter and to the additive case, both in local and non-local settings.

Zinc cysteine active sites of metalloproteins: A density functional theory and x-ray absorption fine structure study

Density functional theory (DFT) and x-ray absorption fine structure (XAFS) spectroscopy are complementary tools for the biophysical study of active sites in metalloproteins. DFT is used to compute XAFS multiple scattering Debye Waller factors, which are then employed in genetic algorithm-based fitting process to obtain a global fit to the XAFS in the space of fitting parameters. Zn-Cys sites, which serve important functions as transcriptional switches in Zn finger proteins and matrix metalloproteinases, previously have proven intractable by this method; here these limitations are removed. In this work we evaluate optimal DFT nonlocal functionals and basis sets for determining optimal geometries and vibrational densities of states of mixed ligation Zn(His)(4-n)(Cys)(n) sites. Theoretical results are compared to experimental XAFS measurements and Raman spectra from the literature and tabulated for use.

Image Processing with Nonlocal Spectral Bases

This article studies regularization schemes that are defined using a lifting of the image pixels in a high-dimensional space. For some specific classes of geometric images, this discrete set of points is sampled along a low-dimensional smooth manifold. The construction of differential operators on this lifted space allows one to compute PDE flows and perform variational optimizations. All of these schemes lead to regularizations that exploit the manifold structure of the lifted image. Depending on the specific definition of the lifting, one recovers several well-known semilocal and nonlocal denoising algorithms that can be interpreted as local estimators over a semilocal or a nonlocal manifold. This framework also allows one to define thresholding operators in adapted orthogonal bases. These bases are eigenvectors of the discrete Laplacian on a manifold adapted to the geometry of the image. Numerical results compare the efficiency of PDE flows, energy minimizations, and thresholdings in the semilocal and nonlocal settings. The superiority of the nonlocal computations is studied through the performance of nonlinear approximation in orthogonal bases.

Noise threshold for a fault-tolerant two-dimensional lattice architecture

We consider a model of quantum computation in which the set of operations is limited to nearest-neighbor interactions on a 2D lattice. We model movement of qubits with noisy \SWAP\ operations. For this architecture we design a fault-tolerant coding scheme using the concatenated $[[7,1,3]]$ Steane code. Our scheme is potentially applicable to ion-trap and solid-state quantum technologies. We calculate a lower bound on the noise threshold for our local model using a detailed failure probability analysis. We obtain a threshold of $1.85 \times 10^{-5}$ for the local setting, where memory error rates are one-tenth of the failure rates of gates, measurement, and preparation steps. For the analogous nonlocal setting, we obtain a noise threshold of $3.61 \times 10^{-5}$. Our results thus show that the additional \SWAP\ operations required to move qubits in the local model affect the noise threshold only moderately.

Generalized reconstruction model ofSiC(001)−(n×2)surfaces and Si ad-dimer strings

We report ab initio calculations on the atomic and electronic structure of the $7\ifmmode\times\else\texttimes\fi{}2$ and $8\ifmmode\times\else\texttimes\fi{}2$ surfaces of cubic $\mathrm{SiC}(001)$ carried out within local density approximation of the density functional theory. Supercells, nonlocal norm-conserving pseudopotentials and Gaussian orbital basis sets are used to describe the surface systems. Our results for the surface structure are in good accord with a lot of experimental data. These investigations complement our previous ab initio studies on $c(4\ifmmode\times\else\texttimes\fi{}2)$, $3\ifmmode\times\else\texttimes\fi{}2$ and $5\ifmmode\times\else\texttimes\fi{}2$ reconstructions of the $\mathrm{SiC}(001)$ surface. On the basis of all our results, we suggest a general two adlayer asymmetric dimer model (TAADM) for $n\ifmmode\times\else\texttimes\fi{}2$ reconstructions of $\mathrm{SiC}(001)$. The model is based on two structural building blocks. The first contains an asymmetric dimer in one partial adlayer while the second contains an asymmetric dimer configuration of six Si adatoms in two partial adlayers. These are characteristic for the $c(4\ifmmode\times\else\texttimes\fi{}2)$ and $3\ifmmode\times\else\texttimes\fi{}2$ reconstructions, respectively. Employing the model for higher $n$ values allows us to suggest optimal structures for a large host of $n\ifmmode\times\else\texttimes\fi{}2$ reconstructions. Comparing the formation energies of the $c(4\ifmmode\times\else\texttimes\fi{}2)$ and all $n\ifmmode\times\else\texttimes\fi{}2$ structures considered, we find in agreement with experiment that the $3\ifmmode\times\else\texttimes\fi{}2$ and $c(4\ifmmode\times\else\texttimes\fi{}2)$ reconstructions are most favorable for silicon-rich and silicon-poor growth conditions, respectively. The formation energies of all other reconstructions fall in between these two limits. In particular, the general TAADM allows us to rationalize the occurrence of periodic and nonperiodic Si ad-dimer-string arrangements at Si-terminated $3\mathrm{C}\text{\ensuremath{-}}\mathrm{SiC}(001)$ surfaces in very good accord with experimental data.

Metallization of the3C−SiC(001)−(3×2)surface induced by hydrogen adsorption: A first-principles investigation

We report ab initio calculations on the atomic and electronic structure, formation energies and surface vibrational modes of a variety of hydrogenated $3C\text{\ensuremath{-}}\mathrm{Si}\mathrm{C}(001)\text{\ensuremath{-}}(3\ifmmode\times\else\texttimes\fi{}2)$ surfaces to shed more light on the recently discovered hydrogen-induced metallization of this semiconductor surface. The calculations are carried out employing the local density approximation of density functional theory. The adsorption systems are described using supercells, nonlocal norm-conserving pseudopotentials and Gaussian orbital basis sets. First, we show in agreement with other most recent ab initio calculations that Si dangling bonds at the third layer, originally suggested for explaining the experimentally observed surface metallization, are not stable. Instead, for low H exposure, we find H atoms to form monohydrides at the top layer and to become adsorbed in angular Si-H-Si bonds on the third layer. Due to these bonds the surface becomes metallic. We scrutinize further possibilities of surface metallization investigating numerous conceivable other H configurations on the first three layers of the $\mathrm{Si}\mathrm{C}(001)\text{\ensuremath{-}}(3\ifmmode\times\else\texttimes\fi{}2)$ surface. It turns out that H atoms can also occupy bridge positions in angular Si-H-Si bonds on the second layer yielding a locally stable metallic surface. For larger H exposure, we find H atoms to form dihydrides at the top layer and to become adsorbed on the second and/or third layer in angular Si-H-Si bridge bonds. These structures are metallic, as well, and they are energetically even more favorable. In all cases considered, the angular Si-H-Si bonds are the origin of the surface metallization giving rise to partially occupied surface state bands which close the gap and overlap in parts with the lower projected conduction bands. Finally we address surface phonons. Some of the calculated frequencies are in very good accord with experiment. Others, particularly characteristic for the Si-H-Si bridge bonds, have not been resolved in experiment, to date.

Computational structural determination and energy landscape analysis of the hepatic carcinogen 2-(acetylamino)fluorene

2-(Acetylamino)fluorene (AAF), a potent mutagen and a prototypical example of the mutagenic aromatic amines, forms covalent adducts to DNA after metabolic activation in the liver. A benchmark study of AAF is presented using a number of the most widely used molecular mechanics and semiempirical computational methods and models. The results are compared to higher-level quantum calculations and to experimentally obtained crystal structures. Hydrogen bonding between AAF molecules in the crystal phase complicates the direct comparison of gas-phase theoretical calculations with experiment, so Hartree–Fock (HF) and Becke–Perdew (BP) density functional theory (DFT) calculations are used as benchmarks for the semiempirical and molecular mechanics results. Systematic conformer searches and dihedral energy landscapes were carried out for AAF using the SYBYL and MMFF94 molecular mechanics force fields; the AM1, PM3 and MNDO semiempirical quantum mechanics methods; HF using the 3-21G*and 6-31G* basis sets; and DFT using the nonlocal BP functional and double numerical polarization basis sets. MMFF94, AM1, HF and DFT calculations all predict the same planar structures, whereas SYBYL, MNDO and PM3 all predict various nonplanar geometries. The AM1 energy landscape is in substantial agreement with HF and DFT predictions; MMFF94 is qualitatively similar to HF and DFT; and the MNDO, PM3 and SYBYL results are qualitatively different from the HF and DFT results and from each other. These results are attributed to deficiencies in MNDO, PM3 and SYBYL. The MNDO, PM3 and SYBYL models may be unreliable for compounds in which an amide group is immediately adjacent to an aromatic ring.

Understanding the renormalization programme in a mathematically rigorous framework and an intrinsic mass scale

We show there exists a mathematically consistent framework within which the renormalization programme can be understood in a natural manner. The framework does not require any violations of mathematical rigour usually associated with the renormalization programme. We use the framework of the non-local field theories (these carry a finite mass scale Λ) and set up a finite perturbative programme. We show how this programme leads to the perturbation series of the usual renormalization programme (except for one difference) if the series is restructured. We also show that the comparison becomes possible if there exists a finite mass scale Λ, with certain properties, in the quantum field theory (which we take to be the scale present in the non-local theory). We give a way to estimate the scale Λ. We also show that the finite perturbation programme differs from the usual renormalization programme by a term, which we propose can also be used to set a bound on Λ.

Automated complexity analysis based on ordered resolution

We define order locality to be a property of clauses relative to a term ordering. This property generalizes the subformula property for proofs where the terms appearing in proofs can be bounded, under the given ordering, by terms appearing in the goal clause. We show that when a clause set is order local, then the complexity of its ground entailment problem is a function of its structure (e.g., full versus Horn clauses), and the ordering used. We prove that, in many cases, order locality is equivalent to a clause set being saturated under ordered resolution. This provides a means of using standard resolution theorem provers for testing order locality and transforming non-local clause sets into local ones. We have used the Saturate system to automatically establish complexity bounds for a number of nontrival entailment problems relative to complexity classes which include polynomial and exponential time and co-NP.

Exploiting temporal nonlocality to remove scaling bottlenecks in nonadiabatic quantum dynamics

An extension of the full multiple spawning (FMS) method for quantum non-adiabatic dynamics that capitalizes on the global nature of quantum mechanics and on the deterministic nature of the FMS method is discussed. The FMS method uses a classically motivated time-dependent basis set for the wave function and here we demonstrate that the choice of a temporally nonlocal basis set can reduce the scaling of the dominant effort in ab initio multiple spawning from O(N2) to O(N), where N is the number of basis functions describing the nuclear degrees of freedom. The procedure is applied to a two-dimensional two electronic state model problem and we show that the temporally nonlocal basis set provides accurate expectation values and branching ratios over a broad range of energies.

Numerical study of pattern formation in weakly damped parametric surface waves

We present a numerical study of parametrically driven surface waves in fluids of low viscosity. The equations governing fluid motion in the bulk and the appropriate boundary conditions at the free surface are approximated by a nonlocal set of equations involving surface coordinates alone, thus allowing the numerical analysis of the large aspect ratio limit (the aspect ratio being the ratio between the size of the system and the wavelength of the waves). The numerical calculation supports the predictions of a previously given multi-scale asymptotic analysis, yielding stable patterns of square symmetry above onset in the capillary dominated regime. In the case of mixed capillary-gravity waves, patterns of eight-fold symmetry are obtained, again in agreement with the predictions of the asymptotic expansion. These calculations provide indirect evidence about the stability of such patterns, analysis that is too involved to be carried out analytically.

On the three-dimensional Boussinesq problem for an elastic nonlocal medium

After determining the nonlocal elastic moduli and the constitutive equations used, a brief review of the Kelvin problem in nonlocal setting is given. The Westergaard procedure of transition from the classical Kelvin problem to the classical Boussinesq problem is discussed, and applied to the nonlocal case using Fourier's exponential transformation. An example illustrating the application of the method to calculate the stress system in a nonlocal half-space is given.

Nonlocal Qualitative and Quantitative Analysis of a Simple Shallow Shell Problem

AbstractA nonlocal qualitative and quantitative analysis of a nonlinear problem is illustratively carried out for a shallow shell problem. The shell model is of “Marguerre” type and the problem is governed by a system of nonlinear partial differential equations. The complete nonlocal solution set is obtained equivalently by means of the boundary value problem and independently by means of each of the two genuinely dual variational problems, too. An identical morphological behaviour of the geometrically and of the statically approached complete solution set is proved. This implies a diffeomorphism between these sets and their identical stability behaviour. Thus the paradox connected with the different stability behaviour of the common solution points described by means of the two classical complementary variational problems is resolved for the considered problem. The solution set contains solution continua which are neither minimizers nor maximizers. Singular phenomena occur at any point of the involved complete degenerate solution set. This set is a natural extension of the notion of eigenvalues from linear to nonlocal nonlinear problems. The complete degenerate solution set introduced is indispensable in a nonlocal nonlinear morphology analysis. An exact numerical analysis is carried out in a space spanned by the first four active modes.

Geometric theory of local and non-local conservation laws for the sine-Gordon equation

In previous work one of the authors gave a geometric theory of those nonlinear evolution equations (n. e. es) that can be solved by the Zakharov & Shabat (1972) inverse scattering scheme as generalized by Ablowitz, Kaup, Newell & Segur (1973 b , 1974). In this paper we extend the geo­metric theory to include the Hamiltonian structure of those n. e. es solvable by the method, and we indicate the connection between the geometric theory and the theory of prolongation structures and pseudopotentials due to Wahlquist & Estabrook (1975, 1976). We exploit a ‘gauge’ in­variance of the geometric theory to derive both the well known polynomial conserved densities of the sine-Gordon equation and a non-local set of conserved densities. These act as Hamiltonian densities for a hierarchy of sine-Gordon equations which is analogous to that found by Lax (1968) for the Korteweg-de Vries equation and appears to be new. In an Appendix we derive an expression for the equation of motion for an arbitrary member of the sine-Gordon hierarchy by methods which can be applied in larger context. The results for the sine-Gordon equation lead to the conclusion that a complete set of conserved densities for an arbitrary n. e. e. solvable by the A. K. N. S. –Z. S. scheme must include non-local conserved densities.