Simple Variational Method Research Articles

HiCT: Hierarchical Comprehend of Transformer for Weakly Supervised Object Localization

The weakly supervised object localization (WSOL) has always been a very challenging research subject in the field of computer vision, which aims to predict the localization of objects in an image using only an image-level class labeling approach. The traditional convolutional neural network (CNN) based-approaches utilize the local class activation and discrimination for classification guidance, and the biggest drawback of CNN is that it cannot capture the remote feature dependencies between pixels. Recently, the transformer architecture has been deployed in the WSOL, but the transformer cannot well capture local features. To address the above problems, we propose HiCT (Hierarchical comprehend of transformer), a simple and effective visual converter variation method. Moreover, we also propose a discriminative-based attention layer (DAL), which aims to mine the local feature information by utilizing the global token attention graph mechanism. To further improve the coverage of object localization, we introduce the spatial aware digging module (SADM). In addition, a set of complementarity loss calculators to patch hierarchy (CPH) is proposed to improve the sample class aggregation capability of our model. Finally, we conducted experiments on two commonly used datasets of CUB-200-2011 and ILSVRC, so as to verify the effectiveness of our method.

Design of non-zero dispersion shifted fiber with large effective area based on variational method

The determination of design parameters of large effective area fiber with segmented-core profile is presented by using simple variational method to obtain the effective area, mode field diameter, and dispersion of the fiber. The designed fiber has a large effective area of 171.1 μm2 with mode field diameter, dispersion, and normalized cut off frequency of 9.7 μm, 2.85 ps/nm.km, 2.491, respectively. The results have shown that the fiber with a bending radius of 35 mm, has a very low bending loss of 0.0053 dB/km. The calculated of parameters values of designed non-zero dispersion-shifted fiber (NZDSF) have shown that with a segmented-core profile of radius 3.5 μm, with a ring width of 0.2 μm, the core effective area Aeff increases from 152.3 μm2 to 189 μm2, which in practical point of view, is an achievement to reduce the nonlinearity effects in the NZDSF as transmission medium.

Mass spectra and transition magnetic moments of low lying charmed baryons in a quark model

Excited state mass spectra of the low lying single charmed baryons with nonstrangeness have been calculated in a hypercentral approach. The six-dimensional hyperradial Schrödinger equation is solved by applying a simple variational method. We extend our scheme to predict the magnetic moments and the [Formula: see text] transition magnetic moments of [Formula: see text] and [Formula: see text] state baryons. A comparison of our results with the experimental data and predictions obtained in recent models is also presented.

Calculations of Bethe logarithm for hydrogen and helium using B-splines in different gauges

The efficient and simple B-splines variational method is successfully used to calculate Bethe logarithm for the hydrogen atom in the velocity and length gauges. The ground state Bethe logarithm of hydrogen with fourteen accurate figures is obtained in the velocity gauge, and in the length gauge the ground state value has eleven accurate figures. Present velocity- and length-gauge results for the ns, np, nd, and nf states up to n = 200 of hydrogen are above the 10−10 level of relative difference, which represent the successful variational attempt to calculate Bethe logarithm of hydrogen in the velocity and length gauges. In addition, the B-splines variational method is successfully extended to calculate Bethe logarithm for the helium atom combined with configuration interaction. Results of numerical calculations for n1,3S up to n = 8 states are presented in the acceleration gauge, the velocity gauge, and a hybrid of the velocity and acceleration gauges(also called pa-gauge). For the 23S state, the acceleration-gauge value of 4.364 036 7(2) a.u. and the pa-gauge value of 4.364 036 4(2) a.u. both have eight significant digits. For other triplet S states, present results in three different gauges all have five to seven accurate figures. While for the singlet S states, the best convergent values are obtained in the pa-gauge, of which the relative differences with the latest results are at the 10−6 to 10−8 orders of magnitude.

Combination of the variational iteration method and numerical algorithms for nonlinear problems

A very simple and efficient local variational iteration method (LVIM), or variational iteration method with local property, for solving problems of nonlinear science is proposed in this paper. The analytical iteration formula of this method is derived first using a general form of first order nonlinear differential equations, followed by straightforward discretization using Chebyshev polynomials and collocation method. The resulting numerical algorithm is very concise and easy to use, only involving highly sparse matrix operations of addition and multiplication, and no inversion of the Jacobian in nonlinear problems. Apart from the simple yet efficient iteration formula, another extraordinary feature of LVIM is that in each local domain, all the collocation nodes participate in the calculation simultaneously, thus each local domain can be regarded as one “node” in calculation through GPU acceleration and parallel processing. For illustration, the proposed algorithm of LVIM is applied to various nonlinear problems including Blasius equations in fluid mechanics, buckled bar equations in solid mechanics, the Chandrasekhar equation in astrophysics, the low-Earth-orbit equation in orbital mechanics, etc. Using the built-in highly optimized ode45 function of MATLAB as a comparison, it is found that the LVIM is not only very accurate, but also much faster by an order of magnitude than ode45 in all the numerical examples, especially when the nonlinear terms are very complicated and difficult to evaluate.

Exchange frustration effect due to anisotropies on the spin-[formula omitted][formula omitted] model

The ground state magnetic properties of the J1−J1′−J2 spin-1∕2 Heisenberg antiferromagnetic model with exchange anisotropies is studied on the square lattice. Independent exchange anisotropies are included for first and second neighbors. The zero temperature phase diagram is obtained from a simple variational method based on a trial vector for the ground state. The data point out that these exchange anisotropies play an important role regarding not only the transitions between the ordered and the quantum disordered phases, but also the area of existence of the corresponding quantum disordered phase. This effect reveals the presence of an additional frustration, the so-called exchange frustration.

Study of Electromagnetic Properties of Light Baryons in the Hypercentral Approach

Light flavour baryons are studied in a non-relativistic potential model with colour Coulomb plus linear confinement potential using a simple variational method. The ground-state masses and magnetic moments of the light baryons are computed using a spin- and isospin-dependent potential. We extend our scheme to predict the transition magnetic moments of $B_{(J^P={3/2}^+)} \rightarrow B_{(J^P={1/2}^+)}\gamma$ processes. We also compute the radiative decay widths and branching ratios of light baryons. A comparison of our results with those of other works and experimental data is also presented.

The variational iteration method for characteristic problem of strong damping generalized sine-Gordon equation

A class of nonlinear strong damping sine-Gordon disturbed evolution differential equation is studied which appears widely in mathematics and mechanics. Firstly, we introduce a traveling wave transformation, and obtain the exact solution of degenerate equation. Then a functional calculating method for variational iteration is constructed, thus an iterative expansion is found. Finally, the approximate traveling wave analytic solutions for the original strong damping generalized sine-Gordon disturbed evolution equation are found. The arbitrary order approximate solutions, and the simple variational iteration method are obtained with higher accuracy. The approximate analytic solution can make up for the imperfection of the simple numerical simulation solution.

A variational theory of zero field electrical resistivity of colossal magnetoresistive manganites (Re1-xAxMnO3)

Using a simple variational method, we have studied the zero field electrical resistivity ρ(T) of rare earth manganites doped with alkaline earths namely Re1-xAxMnO3which exhibit colossal magnetoresistance (CMR), metal–insulator transition and many other poorly understood phenomena. We take the two band model Hamiltonian for manganites in the strong electron-lattice Jahn–Teller (JT) coupling regime. This model is constructed for the doped manganites which exhibit colossal magnetoresistance (CMR) involving a broad spin-majority (eg-spins) conduction band (b-band) as well as nearly localized spin-minority (t2g-spins) electron states (ℓ-band). Two band models involving itinerant and localized states were also suggested earlier by both experimentalists and theorists. We have also studied the temperature dependence of electrical resistivity ρ(T) at H = 0 of these materials and observed the role of the model parameters e.g. local Coulomb repulsion U, strong ferromagnetic Hund's rule coupling JHbetween egand t2gspins and hybridization V between ℓ-polarons and b-electrons of the same spins on ρ(T). We find from the resistivity results that as the temperature is lowered below a critical temperature Tc(~ 200 K), there is a sudden drop in electrical resistivity ρ(T) at H = 0 resembling with the key feature of many CMR compounds like La2/3( Pb , Ca )1/3MnO3and ( Sm1-yGdy)0.55Sr0.45MnO3at y = 0.5. This anomaly in ρ(T) arises from the onset of magnetic ordering at 200 K and vanishes on increasing V or JHvalue. T-dependence of ρ(T) is metallic-like below Tc(~ 200 K), above which it shows insulator/semiconducting-like behavior.

Image Completion by Diffusion Maps and Spectral Relaxation

We present a framework for image inpainting that utilizes the diffusion framework approach to spectral dimensionality reduction. We show that on formulating the inpainting problem in the embedding domain, the domain to be inpainted is smoother in general, particularly for the textured images. Thus, the textured images can be inpainted through simple exemplar-based and variational methods. We discuss the properties of the induced smoothness and relate it to the underlying assumptions used in contemporary inpainting schemes. As the diffusion embedding is nonlinear and noninvertible, we propose a novel computational approach to approximate the inverse mapping from the inpainted embedding space to the image domain. We formulate the mapping as a discrete optimization problem, solved through spectral relaxation. The effectiveness of the presented method is exemplified by inpainting real images, where it is shown to compare favorably with contemporary state-of-the-art schemes.

Simple variational method for calculating energy and quantum capacitance of an electron gas with screened interactions

We describe a variational procedure for calculating the energy of an electron gas in which the long-range Coulomb interaction is truncated by the screening effect of a nearby metallic gate. We use this procedure to compute the quantum capacitance of the system as a function of electron density and spin polarization. The accuracy of the method is verified against published Monte-Carlo data. The results compare favorably with a recent experiment.

Kauffman knot invariant fromSO(N)orSp(N)Chern-Simons theory and the Potts model

The expectation value of Wilson loop operators in three-dimensional $\mathrm{SO}(N)$ Chern-Simons gauge theory gives a known knot invariant: the Kauffman polynomial. Here this result is derived, at the first order, via a simple variational method. With the same procedure the skein relation for Sp(N) are also obtained. Jones polynomial arises as special cases: Sp(2), $\mathrm{SO}(\ensuremath{-}2)$, and $\mathrm{SL}(2,\mathbb{R})$. These results are confirmed and extended up to the second order, by means of perturbation theory, which moreover let us establish a duality relation between $\mathrm{SO}(\ifmmode\pm\else\textpm\fi{}N)$ and $\mathrm{Sp}(\ensuremath{\mp}N)$ invariants. A correspondence between the first orders in perturbation theory of $\mathrm{SO}(\ensuremath{-}2)$, Sp(2) or SU(2) Chern-Simons quantum holonomy's traces and the partition function of the $Q=4$ Potts model is built.

Dynamics of filaments and membranes in a viscous fluid

Motivated by the motion of biopolymers and membranes in solution, this article presents a formulation of the equations of motion for curves and surfaces in a viscous fluid. We focus on geometrical aspects and simple variational methods for calculating internal stresses and forces, and we derive the full nonlinear equations of motion. In the case of membranes, we pay particular attention to the formulation of the equations of hydrodynamics on a curved, deforming surface. The formalism is illustrated by two simple case studies: (1) the twirling instability of straight elastic rod rotating in a viscous fluid, and (2) the pearling and buckling instabilities of a tubular liposome or polymersome.

Maximal persistent current in a type-II superconductor with an artificial pinning array at the matching magnetic field

The current carrying steady state of the pinned flux line lattice created by magnetic field is described. We calculate analytically the critical current for the case of the matching field (when the number of vortices is equal to that of the pinning centers) using a simple variational method in the framework of Ginzburg-Landau equations. The vortex cores are deformed and displaced in the current carrying state. Displacement of the centers of the vortices with respect to pinning centers and structure of these states are determined.

An extension of the rayleigh-ritz method for finding upper and lower bounds of eigenvalues

A simple variational method is described for the determination of upper and lower bounds for the eigenvalues of an Hermitian operator M. For a selected vector subspace of n dimensions it involves the diagonalization of a 2n × 2n matrix, the elements of which depend on the matrix elements of M and M2 as well as on upper and lower bounds of M in the complementary subspace. The method is equivalent to one given by H. F. Weinberger; it includes the Rayleigh-Ritz procedure and a modified form of Temple's formula as limiting cases.

About the Stability of the Dodecatoplet

A new investigation is done of the possibility of binding the “dodecatoplet”, a system of six top quarks and six top antiquarks, (t6t6), using the Yukawa potential mediated by Higgs exchange. A simple variational method gives a upper bound close to that recently estimated in a mean-field calculation. It is supplemented by a lower bound provided by identities among the Hamiltonians describing the system and its subsystems.

Single heavy flavour baryons using Coulomb plus a power law interquark potential

Properties of single heavy flavor baryons in a non-relativistic potential model with colour Coulomb plus a power law confinement potential have been studied using a simple variational method. The ground-state masses of single heavy baryons and the mass difference between the J P = $ {\frac{{3}}{{2}}}$ + and J P = $ {\frac{{1}}{{2}}}$ + states are computed using a spin-dependent two-body potential. Using the spin-flavour structure of the constituting quarks and by defining an effective confined mass of the constituent quarks within the baryons, the magnetic moments are computed. The masses and magnetic moments of the single heavy baryons are found to be in accordance with the existing experimental values and with other theoretical predictions. It is found that an additional attractive interaction of the order of -200 MeV is required for the antisymmetric states of $ \Lambda_{{Q}}^{}$ (Q $ \in$ c, b) . It is also found that the spin-hyperfine interaction parameters play a decisive role in hadron spectroscopy.

Rigorous sufficient conditions for index-guided modes in microstructured dielectric waveguides

We derive a sufficient condition for the existence of index-guided modes in a very general class of dielectric waveguides, including photonic-crystal fibers (arbitrary periodic claddings, such as "holey fibers"), anisotropic materials, and waveguides with periodicity along the propagation direction. This condition provides a rigorous guarantee of cutoff-free index-guided modes in any such structure where the core is formed by increasing the index of refraction (e.g. removing a hole). It also provides a weaker guarantee of guidance in cases where the refractive index is increased "on average" (precisely defined). The proof is based on a simple variational method, inspired by analogous proofs of localization for two-dimensional attractive potentials in quantum mechanics.

On variational methods for periodic discrete problems

We prove existence results for a general 2n-th order discrete boundary value problem subjected to periodic conditions. In contrast to published work, we exploit the finite dimension of function spaces and some simple results from matrix theory to construct a nonlinear equation in which is eventually studied with the degree theory and simple variational methods. Existence and uniqueness theorems, which are obtained in this way, improve the existing results (even in the second-order case) by including non-continuous right-hand sides, higher order problems and non-invertible difference operators.

Method for obtaining exact solutions of the nonlinear Schrödinger equation for a double-square-well potential

Both symmetric and symmetry breaking analytic solutions to the one-dimensional nonlinear Schr\odinger equation with a double square well potential are known, but not straightforward to obtain numerically. The former generalize solutions to the linear equations, the latter owe their very existence to the nonlinearity. These include, for example, solutions corresponding to the wave function localized almost entirely in one of the wells. Here we propose a systematic method for generating these solutions starting from the linear limit. In particular we find a simple exact formula giving the bifurcation point in terms of the parameters of the symmetric solution. This bifurcation point is then reproduced to a surprising degree of accuracy by a simple variational method.