Integer Index Research Articles

Snapshottable Stores

We say that an imperative data structure is snapshottable or supports snapshots if we can efficiently capture its current state, and restore a previously captured state to become the current state again. This is useful, for example, to implement backtracking search processes that update the data structure during search. Inspired by a data structure proposed in 1978 by Baker, we present a snapshottable store , a bag of mutable references that supports snapshots. Instead of capturing and restoring an array, we can capture an arbitrary set of references (of any type) and restore all of them at once. This snapshottable store can be used as a building block to support snapshots for arbitrary data structures, by simply replacing all mutable references in the data structure by our store references. We present use-cases of a snapshottable store when implementing type-checkers and automated theorem provers. Our implementation is designed to provide a very low overhead over normal references, in the common case where the capture/restore operations are infrequent. Read and write in store references are essentially as fast as in plain references in most situations, thanks to a key optimisation we call record elision . In comparison, the common approach of replacing references by integer indices into a persistent map incurs a logarithmic overhead on reads and writes, and sophisticated algorithms typically impose much larger constant factors. The implementation, which is inspired by Baker’s and the OCaml implementation of persistent arrays by Conchon and Filliâtre, is both fairly short and very hard to understand: it relies on shared mutable state in subtle ways. We provide a mechanized proof of correctness of its core using the Iris framework for the Coq proof assistant.

A note on repunit number sequence

In this paper, we investigate the classical identities of the repunit sequence with integer indices in light of the properties of Horadan-type sequences. We highlight particularly the Tagiuri-Vajda Identity and Gelin-Cesàro Identity. Additionally, we prove that no repunit is a perfect power, either even or odd. Finally, we address a divisibility criterion for the terms of repunit rn by a prime p and its powers.

Traveling Salesman Problem of optimal debris removal sequence using non-population gradient search

A new approach for debris removal sequence optimization with multiple spacecraft is developed and tested. The motion planning of the multiple rendezvous involves complex Traveling Salesman Problem (TSP) combinatorial decision, in which both the continuous parameters of the orbital maneuvers and integer indices of the debris are determined. Expected value and variance of trajectory conditioned on preceding visit, not the state of target itself, are propagated and analyzed. With the metric, likelihood of expected trajectory can be given, and the mixed integer problem is reformulated into continuous parameter estimation with maximum likelihood. A non-population gradient based optimizer is then used to search the target sequence. The method of TSP optimization is compared to the continuous parameter estimator of Extended Kalman Filter. Using this method, the classic static TSP-LIB benchmarks with up to 100 nodes are tested. We then extend the method to the TSP with space dynamics. Effectiveness and efficiency of the proposed method are tested by a sequential visit of near-earth debris considering J2 effects with a single and multiple spacecrafts.

On the refined Kaneko–Zagier conjecture for general integer indices

The refined Kaneko–Zagier conjecture claims that the algebras spanned by two kinds of “completed” finite multiple zeta values, called {widehat{mathcal {A}}}- and {widehat{mathcal {S}}}-MZVs, are isomorphic. Recently, Komori defined {widehat{mathcal {S}}}-MZVs of general integer (i.e., not necessarily positive) indices, extending the existing definition for positive indices. In view of the refined Kaneko–Zagier conjecture, Komori’s work suggests that these extended values are closely connected to {widehat{mathcal {A}}}-MZVs of general indices, which can be defined in an obvious way. In this paper, we show that the generalization of the refined Kaneko–Zagier conjecture for general integer indices is actually deduced from the conjecture for positive indices. The key ingredient is an inductive formula for {widehat{mathcal {A}}}-MZVs or {widehat{mathcal {S}}}-MZVs of indices which contain at least one non-positive entry.

The bilateral birth–death chain generated by the associated Jacobi polynomials

AbstractWe give a probabilistic interpretation of the associated Jacobi polynomials, which can be constructed from the three‐term recurrence relation for the classical Jacobi polynomials by shifting the integer index n by a real number t. Under certain restrictions, this will give rise to a doubly infinite tridiagonal stochastic matrix, which can be interpreted as the one‐step transition probability matrix of a discrete‐time bilateral birth–death chain with state space on . We also study the unique UL and LU stochastic factorizations of the transition probability matrix, as well as the discrete Darboux transformations and corresponding spectral matrices. Finally, we use all these results to provide an urn model on the integers for the associated Jacobi polynomials.

Theory for constructing effective electronic models of bilayer graphene systems

We present and discuss practical techniques for formulating effective models to describe the low-energy electronic properties of bilayer graphene systems. We show that such effective models are constructed from a collection of appropriate single-layer Bloch states of two graphene layers. In general, the obtained effective models allow the construction of a so-called moiré band structure. However, it is not the result of an irreducible representation of a translation symmetry group of the bilayer lattices except for the commensurate bilayer configurations. We also point out that the commensurate bilayer configurations are classified into three categories depending on the divisibility of the difference between two commensurate integer indices by 3. The electronic band structure of three lattice configurations, one for each category, is shown. Especially by combining with a real-space calculation, we validate the working ability of constructed effective models for generic bilayer graphene systems by showing that the effects of interlayer sliding are diminished by twisting. This result is consistent with the invariance of effective models under the interlayer sliding operation.

Nontrivial Quantum Cellular Automata in Higher Dimensions

We construct a three-dimensional quantum cellular automaton (QCA), an automorphism of the local operator algebra on a lattice of qubits, which disentangles the ground state of the Walker-Wang three fermion model. We show that if this QCA can be realized by a quantum circuit of constant depth, then there exists a two-dimensional commuting projector Hamiltonian which realizes the three fermion topological order which is widely believed not to be possible. We conjecture in accordance with this belief that this QCA is not a quantum circuit of constant depth, and we provide two further pieces of evidence to support the conjecture. We show that this QCA maps every local Pauli operator to a local Pauli operator, but is not a Clifford circuit of constant depth. Further, we show that if the three-dimensional QCA can be realized by a quantum circuit of constant depth, then there exists a two-dimensional QCA acting on fermionic degrees of freedom which cannot be realized by a quantum circuit of constant depth; i.e., we prove the existence of a nontrivial QCA in either three or two dimensions. The square of our three-dimensional QCA can be realized by a quantum circuit of constant depth, and this suggests the existence of a $\mathbb{Z}_2$ invariant of a QCA in higher dimensions, totally distinct from the classification by positive rationals (i.e., by one integer index for each prime) in one dimension. In an appendix, unrelated to the main body of this paper, we give a fermionic generalization of a result of Bravyi and Vyalyi on ground states of 2-local commuting Hamiltonians.

Parameter Identification of Fractional Index Viscoelastic Model for Vegetable-Fiber Reinforced Composite.

In the present work, parameters for adapting the behavior of the uniaxial three-element viscoelastic constitutive model with integer and fractional index derivatives to the mechanical evolution of an epoxy-composite material reinforced with long random henequen fibers, were determined. Cyclic loading-unloading with 0.1%, 0.2%, 0.3%, …, 1.0% controlled strain and staggered fluency experiments at 5 MPa, 10 MPa, and 15 MPa constant tension were performed in stages, and the obtained data were used to determine and validate the model's parameter values. The Inverse Method of Identification was used to calculate the parameters, and the Particle Swarm Optimization (PSO) method was employed to achieve minimization of the error function. A comparison between the simulated uniaxial results and the experimental data is demonstrated graphically. There exists a strong dependence between properties of the composite and the fiber content (0 wt%, 9 wt%, 14 wt%, 22 wt%, and 28 wt% weight percentage fiber/matrix), and therefore also of the model parameter values. Both uniaxial models follow the viscoelastic behavior of the material and the fractional index version presents the best accuracy. The latter method was noted to be adequate for determination of the aforementioned constants using non-large experimental data and procedures that are easy to implement.

Exact results for duality-covariant integrated correlators in $\mathcal{N}=4$ SYM with general classical gauge groups

We present exact expressions for certain integrated correlators of four superconformal primary operators in the stress tensor multiplet of \mathcal{N}=4𝒩=4 supersymmetric Yang–Mills (SYM) theory with classical gauge group, G_NGN= SO(2N)=SO(2N), SO(2N+1)SO(2N+1), USp(2N)USp(2N). These integrated correlators are expressed as two-dimensional lattice sums by considering derivatives of the localised partition functions, generalising the expression obtained for SU(N)SU(N) {gauge group} in our previous works. These expressions are manifestly covariant under Goddard-Nuyts-Olive duality. The integrated correlators can also be formally written as infinite sums of non-holomorphic Eisenstein series with integer indices and rational coefficients. Furthermore, the action of the hyperbolic Laplace operator with respect to the complex coupling \tau=\theta/(2\pi) + 4\pi i /g^2_{_{YM}}τ=θ/(2π)+4πi/gYM2 on any integrated correlator for gauge group G_NGN relates it to a linear combination of correlators with gauge groups G_{N+1}GN+1, G_NGN and G_{N-1}GN−1. These ``Laplace-difference equations’’ determine the expressions of integrated correlators for all classical gauge groups for any value of NN in terms of the correlator for the gauge group SU(2)SU(2). The perturbation expansions of these integrated correlators for any finite value of NN agree with properties obtained from perturbative Yang–Mills quantum field theory, together with various multi-instanton calculations which are also shown to agree with those determined by supersymmetric localisation. The coefficients of terms in the large-NN expansion are sums of non-holomorphic Eisenstein series with half-integer indices, which extend recent results and make contact with low order terms in the low energy expansion of type IIB superstring theory in an AdS_5\times S^5/\mathbb{Z}_2AdS5×S5/ℤ2 background.

Hybrid Quaternions of Leonardo

In this article, we intend to investigate the Leonardo sequence presenting the hybrid Leonardo quaternions. To explore Hybrid Quaternions of Leonardo, the priori, sequence of Leonardo,quaternions and hybrid numbers were presented. Soon after, its recurrence, characteristic equation, its relation with the Fibonacci quaternions, generating function, Binet’s formula, as well as its extension to non-positive integer indices were developed. Finally, identities involving Leonardo’s hybrid quaternions are presented.

NUMERICAL SOLUTION OF THE DYNAMIC PROBLEM OF AXISYMMETRIC VIBRATIONS OF REINFORCED SHELLS

The reliability of the results obtained in the work is determined by the rigor and correctness of the statements of the initial problems; theoretical substantiation of the finite-difference schemes used; controlled accuracy of numerical calculations; conducting test calculations; compliance of the established regularities with the general properties of oscillations of thin-walled structural elements. The correctness of the formulation of the problems is achieved by using the well-known equations of the theory of shells and rods of the Tymoshenko type, which are approximations of the original equations of the three-dimensional theory of elasticity. When deriving the equations, the equations of oscillations of the multilayer shell in the smooth region and the equations of oscillations of reinforcing ribbed elements (transverse ribs) were obtained. It is not difficult to show that the indicated equations by the classification of equations in partial derivatives are equations of the hyperbolic type, which are an approximation of the oscillating equations of three-dimensional elastic bodies and sufficiently correctly reproduce wave processes in non-homogeneous shell structures, taking into account spatial gaps. Numerical algorithms for approximate solutions of the original equations are based on the use of the integro-interpolation method of constructing difference schemes. When constructing difference schemes, kinematic quantities refer to difference points with integer indices, and the values of deformations and moment forces refer to difference points with half-integer indices. This approximation of the initial kinematic and static values allows the fulfillment of the law of conservation of the total mechanical energy of the elastic structure at the difference level. The numerical algorithm is based on the use of separate finite-difference relations in the smooth domain and on the lines of spatial discontinuities with the second order of accuracy in spatial and temporal coordinates.

STARE INTO THE FUTURE OF GEODATA INTEGRATIVE ANALYSIS.

Different kinds of observations feature different strengths, e.g. visible-infrared imagery for clouds and radar for precipitation, and when integrated better constrain scientific models and hypotheses. Even critical, fundamental operations such as cross-calibrations of related sensors operating on different platforms or orbits, e.g. spacecraft and aircraft, are integrative analyses. The great variety of Earth Science data types and the spatiotemporal irregularity of important low-level (ungridded) data has so far made their integration a customized, tedious process which scales in neither variety nor volume. Generic, higher-level (gridded) data products are easier to use, at the cost of being farther from the original observations and having to settle with grids, interpolation assumptions, and uncertainties that limit their applicability. The root cause of the difficulty in scalably bringing together diverse data is the current rectilinear geo-partitioning of Earth Science data into conventional arrays indexed using consecutive integer indices and then packaged into files. Such indices suffice for archival, search, and retrieval, but lack a common geospatial semantics, which is mitigated by adding on floating-point encoded longitude-latitude (lon-lat) information for registration. An alternative to floating-point lon-lat, the SpatioTemporal Adaptive Resolution Encoding (STARE) provides an encoding for geo-spatiotemporal location and neighborhood that transcends the use of files and native array indexing allowing diverse data to be organized on scalable, distributed computing and storage platforms.

Chiral Floquet Systems and Quantum Walks at Half-Period

We classify chiral symmetric periodically driven quantum systems on a one-dimensional lattice. The driving process is local, can be continuous or discrete in time, and we assume a gap condition for the corresponding Floquet operator. The analysis is in terms of the unitary operator at a half-period, the half-step operator. We give a complete classification of the connected classes of half-step operators in terms of five integer indices. On the basis of these indices it can be decided whether the half-step operator can be obtained from a continuous Hamiltonian driving, or not. The half-step operator determines two Floquet operators, obtained by starting the driving at zero or at half period, respectively. These are called timeframes and are chiral symmetric quantum walks. Conversely, we show under which conditions two chiral symmetric walks determine a common half-step operator. Moreover, we clarify the connection between the classification of half-step operators and the corresponding quantum walks. Within this theory we prove bulk-edge correspondence and show that a second timeframe allows to distinguish between symmetry protected edge states at $+1$ and $-1$ which is not possible for a single timeframe.

On scattering for the defocusing nonlinear Schrödinger equation on waveguide [formula omitted] (when m = 2,3)

In the article, we prove the large data scattering for two models, i.e. the defocusing quintic nonlinear Schrödinger equation on R2 × T and the defocusing cubic nonlinear Schrödinger equation on R3 × T. Both of the two equations are mass supercritical and energy critical. The main ingredients of the proofs contain global Stricharz estimate, profile decomposition and energy induction method. This paper is the second project of our series work (two papers, together with [31]) on large data scattering for the defocusing critical NLS with integer index nonlinearity on low dimensional waveguides. At this point, this category of problems are almost solved except for two remaining resonant system conjectures and the quintic NLS problem on R×T.

A novel algorithm for nested summation and hypergeometric expansions

We consider a class of sums over products of Z-sums whose arguments differ by a symbolic integer. Such sums appear, for instance, in the expansion of Gauss hypergeometric functions around integer indices that depend on a symbolic parameter. We present a telescopic algorithm for efficiently converting these sums into generalized polylogarithms, Z-sums, and cyclotomic harmonic sums for generic values of this parameter. This algorithm is illustrated by computing the double pentaladder integrals through ten loops, and a family of massive self-energy diagrams through mathcal{O}left({epsilon}^6right) in dimensional regularization. We also outline the general telescopic strategy of this algorithm, which we anticipate can be applied to other classes of sums.

The Dirac H-atom using the phase as an extra variable

In this paper, an algebraic solution for the bound states of the relativistic hydrogen atom is presented. The method discussed here adds an operator associated with the phase of the energy eigenstates to the set of variables of the problem. In terms of this set, appropriate ladder operators are constructed in order to express the full solution of the Dirac hydrogen equation. These ladder operators are used to form the Lie algebra of a su(1, 1) group, in the same way as that applied to angular momentum algebra and the ladder operators L±. The elements of the vector space associated with the representation of this algebra are related to a generalization of the Laguerre polynomials of the non integer index, also known as Sonine polynomials. In addition, we find that the eigenvalues of the operator constructed with the sum of the square of the three su(1, 1) generators gives precisely the relativistic energy spectrum of the hydrogen atom, including its angular momentum dependency.

Finer Granularity DFT Bins With Moving Window for Capacitance Sensing

In this article, the moving window discrete Fourier transform (MWDFT) with finer granularity bin indices is proposed for frequency estimation of periodic signals. The MWDFT structure is a cascade connection of comb filter and a complex resonator. The sign of comb filter is toggled between positive and negative using a multiplexer to select the integer and fractional bin indices. The sign of comb filter is chosen as negative for setting-up integer bin indices with certain resonator angle. The positive sign of comb filter rotates the zeros of the MWDFT to a specified angle in the anticlockwise direction in the z -plane. By selecting the positive feed forward coefficient, all odd harmonics of one-half of the fundamental frequency can be extracted from the MWDFT. This unique property of MWDFT with finer granularity bin indices is exploited to extract frequencies of finer resolution. A frequency locked loop is designed with MWDFT of finer granularity bins to estimate the frequency change of finer resolution. The proposed method is implemented on field programmable gate array (FPGA) as a capacitance measurement system. The method is able to provide better accuracy, wider frequency estimation range, and wider capacitance measurement range.

Sparse obstructions for minor-covering parameters

Given a finite set of graphs H and a non-negative integer k, we define Ak(H) as the set containing every graph G that has k vertices whose removal provides a graph without any of the graphs in H as a minor. It is known that if H contains at least one planar graph then each obstruction in Ak(H) has at most kcH vertices, for some cH depending only on the choice of H. In this paper, we investigate the size of the graphs in Ak(H) that belong to certain classes of sparse graphs. In particular, we prove that for every graph F, if H contains at least one planar graph and only connected graphs, all graphs in Ak(H) that are F-topological minor-free have at most cF,H⋅k vertices, where cF,H depends exclusively on the choice of H and F. Our result is a consequence of two more general conditions on graph parameters, namely the Finite Integer Index Property and Protrusion Decomposability, that can serve as a general framework for proving linear bounds for obstructions.

R\xe9nyi entanglement entropies for the compactified massless boson with open boundary conditions

We investigate the Rényi entanglement entropies for the one-dimensional massless free boson compactified on a circle, which describes the low energy sector of several interacting many-body 1d systems (Luttinger Liquid). We focus on systems on a finite segment with open boundary conditions and possible inhomogeneities in the couplings. We provide expressions for the Rényi entropies of integer indices in terms of Fredholm determinant-like expressions. Within the homogeneous case, we reduce the problem to the solution of linear integral equations and the computation of Riemann Theta functions. We mainly focus on a single interval in the middle of the system, but results for generic bipartitions are given as well.

Fractional two-parameter parabolic diffraction-free beams

We consider two-parameter real-valued parabolic diffraction-free beams. An analytical expression for the complex amplitude of the beam is obtained using the superposition of the Bessel functions. Symmetry properties of beams are analyzed and numerically demonstrated for different cases. The variety of the formed pictures increases significantly in comparison with earlier regarded parabolic beams. Particularly interesting properties exhibit beams with half-integer indices, they have symmetry properties different from the symmetry of beams with integer indices. For an arbitrary fractional index, the beams have a more complex distribution and do not have any symmetry. This fact facilitates the solution of the inverse problem when it is necessary to form similar structures, since just small changes in two parameters of the analytical expression are needed. This is much simpler than the implementation of iterative procedures, which, as a rule, provide essentially non-smooth solutions that are complex in optical implementation. Experimental generation of the fractional two-parameter parabolic beams is implemented for the first time by a spatial light modulator using phase coding of the calculated complex amplitudes. The different types of experimentally generated parabolic beams are in a good agreement with the modeling results.