Negative Eigenvalues Research Articles

Riemannian Trust Region Method for Minimization of the Fourth Central Moment for Localized Molecular Orbitals.

The importance of localized molecular orbitals (MOs) in correlation treatments beyond mean-field calculation and in the illustration of chemical bonding (and antibonding) can hardly be overstated. However, the generation of orthonormal localized occupied MOs is significantly more straightforward than obtaining orthonormal localized virtual MOs. Orthonormal MOs allow facile use of highly efficient group theoretical methods (e.g., graphical unitary group approach) for calculation of Hamiltonian matrix elements in multireference configuration interaction calculations (such as MRCISD) and in quasi-degenerate perturbation treatments, such as the Generalized Van Vleck Perturbation Theory. Moreover, localized MOs can elucidate qualitative understanding of bonding in molecules, in addition to high-accuracy quantitative descriptions. We adopt the powers of the fourth moment cost function introduced by Jørgensen and coworkers. Because the fourth moment cost functions are prone to having multiple negative Hessian eigenvalues when starting from easily available canonical (or near-canonical) MOs, standard optimization algorithms can fail to obtain the orbitals of the virtual or partially occupied spaces. To overcome this drawback, we applied a trust region algorithm on an orthonormal Riemannian manifold with an approximate retraction from the tangent space built into the first and second derivatives of the cost function. Moreover, the Riemannian trust region outer iterations were coupled to truncated Conjugate Gradient inner loops, which avoided any costly solutions of simultaneous linear equations or eigenvector/eigenvalue solutions. Numerical examples are provided on model systems, including the high-connectivity H10 set in 1-, 2-, and 3-dimensional arrangements, and on a chemically realistic description of cyclobutadiene (c-C4H4) and the propargyl radical (C3H3). In addition to demonstrating the algorithm on occupied and virtual blocks of orbitals, the method is also shown to work on the active space at the MCSCF level of theory.

Mixed convective-quadratic radiative MoS 2–SiO 2/H 2 O hybrid nanofluid flow over an exponentially shrinking permeable Riga surface with slip velocity and convective boundary conditions: Entropy and stability analysis

In this work, the flow of a hybrid Mo S 2 − Si O 2 /water nanofluid across a porous exponentially diminishing Riga surface with the effects of quadratic heat radiation is taken into account. We also considered the effects of mixed convection, slip velocity, and convective thermal boundary conditions. Before numerically solving the governing partial differential equations, the MATLAB function bvp4c was used to get a comprehensive description of all relevant flow characteristics. The ramifications of physical characteristics concerning fluid flow and thermal profile are investigated using graphs and tables. The primary objective of this study is to investigate the Nusselt number, skin friction coefficient, and entropy production. Using the second law of thermodynamics, the irreversibility factor is computed. For a given value of the mass suction parameter, the findings also indicate the occurrence of dual-nature solutions in the shrinking sheet area, with a stable upper solution branch and unstable bottom solution branch. In addition, the first solution produces a positive minimum eigenvalue, whereas the second solution produces a negative eigenvalue, showing the stability of the first solution. For heat transfer enhancement, the existence of quadratic thermal radiative parameter impacts is more advantageous. With a larger value for the modified Hartman number, the velocity field has expanded. Also, recent research demonstrates that raising the thermal Biot number enhances the thermal gradient. Increases in nanoparticle volume fraction and thermal radiation parameters lead to a commensurate rise in entropy production. Because of these findings, we can better regulate the temperature of various environments by controlling the flow of heat.

Indices of Some Meromorphic Functions of Degree $3$ on Tori

The index of a meromorphic function g on a compact Riemann surface is an invariant of g, which is defined as the number of negative eigenvalues of the differential operator L:=—Δ—| dG |2, where Δ is the Laplacian with respect to a conformal metric ds2 on the Riemann surface and G:M→S2 is the holomorphic map corresponding to g. We consider the meromorphic function w on the Riemann surface Ma={(z,w)∈ℂ^2|w2=z(z−a)(z+1a)}(a≥1) homeomorphic to a torus. We find a0>1 and determine the index of tw for all a in the range 1≤a≤a0 and all t>0.

Some stable and closed-shell structures of anticancer drugs by graph theoretical parameters

The eigenvalues are significant in mathematics, but they are also relevant in other domains like as chemistry, economics, and a variety of others. In terms of our research, eigenvalues are used in chemistry to represent not only the form of energy but also the various physicochemical aspects of a chemical substance. We must comprehend the connection between mathematics and chemistry. The antibonding level is related to positive eigenvalues, the bonding level is associated to negative eigenvalues, and the nonbonding level is linked to zero eigenvalues. In this work, we studied some anticancer drug structures in terms of nullity, matching number, eigenvalues of adjacency matrix, and characteristics polynomials. As a result, Carmustine, Caulibugulone-E, Aspidostomide-E anticancer drug structures are stable, closed-shell molecules since their nullity is equal to zero.

An Ahmad-Lazer-Paul-type result for indefinite mixed local-nonlocal problems

We prove the existence and multiplicity of weak solutions for a mixed local-nonlocal problem at resonance. In particular, we consider a not necessarily positive operator which appears in models describing the propagation of flames. A careful adaptation of well known variational methods is required to deal with the possible existence of negative eigenvalues.

Isomorphism between the Białynicki-Birula and the Landau-Peierls Fock space quantization of the electromagnetic field in position representation

We first present a summary of the quantization of the electromagnetic field in position space representation, using two main approaches: the Landau-Peierls approach in the Coulomb gauge and the Białynicki-Birula (BB) approach, based on the Riemann–Silberstein vector. We describe both in a framework that starts with a classical Hamiltonian structure and builds the quantum model in a bosonic Fock space by a precisely defined principle of correspondence. We show that the two approaches are completely equivalent. This is formulated by showing that there is a unitary map between the Fock spaces that makes them isomorphic. Since all the physically measurable quantities can be expressed in terms of scalar products, this implies that the two quantizations lead to exactly the same physical properties. We show furthemore that the isomorphism is preserved in the time evolutions. To show the equivalence, we use the concepts of helicity and frequency operators. The combination of these two operators provides a formulation that allows one to make the link between these two methods of quantization in a precise way. We also show that the construction in the BB quantization that avoids the presence of negative eigenvalues in the Hamiltonian, in analogy with the one for the Dirac equation for electrons and positrons, can be performed through an alternative choice of the canonical variables for Maxwell’s equations.

Chain Reaction of Fenton Autoxidation of Tartaric Acid: Critical Behavior at Low pH

Autoxidation of tartaric acid in air-saturated aqueous solutions in the presence of Fe(II) at low pH, 2.5, shows autocatalytic behavior with distinct initiation, propagation, and termination phases. With increasing pH, the initiation phase speeds up, while the propagation phase shortens and reduces to none. We show that the propagation phase is a chain reaction that occurs via activation of oxygen in the initiation stage with the production of hydrogen peroxide. The subsequent Fenton oxidation that regenerates hydrogen peroxide with a positive feedback is typical of a self-sustained chain reaction. The conditions for such a chain reaction are shown to be similar to those of a dynamical system with critical behavior; namely, the system becomes unstable when the kinetic matrix of pseudo-first-order reaction becomes negatively defined with a negative eigenvalue giving the rate of exponential (chain) growth of the reactive species.

Period-1 Motions to Twin Spiral Homoclinic Orbits in the Rössler System

Abstract In this paper, period-1 motions to twin spiral homoclinic orbits in the Rössler system are presented. The period-1 motions varying with a system parameter are predicted semi-analytically through an implicit mapping method, and the corresponding stability and bifurcations of the period-1 motions are determined through eigenvalue analysis. The approximate homoclinic orbits are obtained, which can be detected through the periodic motions with the positive and negative infinite large eigenvalues. The two limit ends of the bifurcation diagram of the period-1 motion are at twin spiral homoclinic orbits. For comparison, numerical and analytical results of stable period-1 motion are presented. The approximate spiral homoclinic orbits are demonstrated for a better understanding of complex dynamics of homoclinic orbits. Herein, only initial results on periodic motions to homoclinic orbits are presented for the Rössler system. In fact, the Rössler system has rich complex dynamics existing in other high-dimensional nonlinear systems. Thus, the further studies of bifurcation trees of periodic motions to infinite homoclinic orbits will be completed in sequel.

Generalized Mercier stability criterion for stellarators

The Mercier criterion is a well-known stability criterion for tokamaks. It is derived from a 2 × 2 matrix eigenvalue problem arising from the expansion of resonant solutions about a singular surface where m−nq=0, with m and n being the poloidal and toroidal mode numbers, respectively, and q being the safety factor. The stability criterion is that the eigenvalues must be real, otherwise, the solution oscillates, violating the Newcomb crossing criterion. Because of the non-axisymmetry of stellarators, different toroidal as well as poloidal harmonics couple to each other. It follows that each singular surface can have multiple resonant harmonics, with multiplicity M≥1. The corresponding matrix eigenvalue problem involves a 2M×2M matrix, resulting in M pairs of positive and negative eigenvalues. The generalized stability criterion is that all eigenvalues must be real. While the original Mercier criterion can be expressed in terms of quadratures of equilibrium quantities over the singular surface, which can be evaluated anywhere, the generalized Mercier criterion can only be evaluated on rational q surfaces with a given set of resonant harmonics.

Electromagnetic Steklov eigenvalues: existence and distribution in the self-adjoint case

In previous works, it was suggested to use Steklov eigenvalues for Maxwell equations as target signature for nondestructive testing, and it was recognized that this eigenvalue problem cannot be reformulated as a standard eigenvalue problem for a compact operator. Consequently, a modified eigenvalue problem with the desired properties was proposed. We report that apart for a countable set of particular frequencies, the spectrum of the original self-adjoint eigenvalue problem consists of three disjoint parts: The essential spectrum consisting of the origin, an infinite sequence of positive eigenvalues which accumulate only at infinity and an infinite sequence of negative eigenvalues which accumulate only at zero. The analysis is based on a suitable topological decomposition, a representation of the operator as block operator and Schur-factorizations. For each Schur-complement, the existence of an infinite sequence of eigenvalues is proven via an intermediate value technique. The modified eigenvalue problem arises as limit of one Schur-complement.

Flow dynamics of MHD hybrid nanofluid past a moving thin needle with a temporal stability test: A Galerkin method approach

Heat transmission processes have numerous industrial applications. A novel form of nanofluid known as the hybrid nanofluid that has a higher thermal exponent helps in improving the ability of regular fluids to transfer heat. This article aims to analyze the impact of hybrid nanofluid with effects like magnetohydrodynamics (MHD), Brownian motion and thermophoresis, heat generation/absorption, and chemical reaction past a permeable moving thin needle. Two assorted nanomaterials (MoS2 and Fe3O4 ) are added in a base fluid such as ethylene glycol (EG) to access the performance of the hybrid nanofluid. Using suitable similarity variables, the governing equations of the problem are turned into customary ordinary differential equations. The resulting equations are solved analytically using the Galerkin method (GM) and dual solutions are obtained in a certain range of moving parameter λ. Due to this, a stability test is implemented to find a stable solution. With the help of a bvp4c function in Matlab software, the smallest eigenvalues are computed, where positive eigenvalues are linked with the stable solution while negative eigenvalues are linked with the unstable solution. It is concluded that only the first solution is stable while the second solution is not. The skin friction factor and heat transfer rate are reduced by growing the needle size. The magnetic field has an inverse relation with the flow field and direct relation with the temperature field. The improvement in heat generation parameter causes the fluid temperature to rise. The fluid concentration is enhanced with the increment of the chemical reaction parameter.

Moderate deviations in cycle count

AbstractWe prove moderate deviations bounds for the lower tail of the number of odd cycles in a random graph. We show that the probability of decreasing triangle density by , is whenever . These complement results of Goldschmidt, Griffiths, and Scott, who showed that for , the probability is . That is, deviations of order smaller than behave like small deviations, and deviations of order larger than behave like large deviations. We conjecture that a sharp change between the two regimes occurs for deviations of size , which we associate with a single large negative eigenvalue of the adjacency matrix becoming responsible for almost all of the cycle deficit. We give analogous results for the ‐cycle density, for all odd . Our results can be interpreted as finite size effects in phase transitions in constrained random graphs.

Nondegeneracy of ground states for nonlinear scalar field equations involving the Sobolev-critical exponent at high frequencies in three and four dimensions

We consider nonlinear scalar field equations involving the Sobolev-critical exponent at high frequencies ω. Since the limiting profile of the ground states as ω→∞ is the Aubin–Talenti function and degenerate in a certain sense, from the point of view of perturbation methods, the nondegeneracy problem for the ground states at high frequencies is subtle. In addition, since the limiting profile (Aubin–Talenti function) fails to lie in L2(Rd) for d=3,4, the nondegeneracy problem for d=3,4 is more difficult than that for d≥5 and an applicable methodology is not known. In this paper, we solve the nondegeneracy problem for d=3,4 by modifying the arguments in Akahori et al. (2020) and Akahori et al. (2019). We also show that the linearized operator around the ground state has exactly one negative eigenvalue.

A Stability Analysis for Radiation Effects on Marangoni Convection Boundary Layer Over a Permeable Surface

The objectives are to derive the stability analysis theoretically and to set up the stability analysis numerically for radiation effects on Marangoni convection boundary layer over a permeable surface. The stability analysis is used to determine which branch solutions are stable and physically realisable. The stability can be tested via the smallest eigenvalue. Negative smallest eigenvalue produces an initial growth of disturbance and the flow becomes unstable. In contrast, the positive smallest eigenvalue results in an initial decay of the disturbance, thus the flow is stable. The research has an implication in order to identify which solution is stable, whether the first or the second solution.

Dual solutions of radiative Ag-MoS_2/water hybrid nanofluid flow with variable viscosity and variable thermal conductivity along an exponentially shrinking permeable Riga surface: Stability and entropy generation analysis

ABSTRACT In this analysis, the Silver-Molybdenum disulfide ()/water hybrid nanofluid flow past a permeable exponentially shrinking Riga surface with thermal radiant energy is considered. The novelty of this work is to find out the effects of variable viscosity, variable thermal conductivity with hybrid nanofluid’s slip velocity, and convective thermal boundary conditions over a shrinking Riga surface on velocity, temperature, skin-friction coefficient, Nusselt number, and entropy generation. The MATLAB programming platform with a fourth-order method to solve boundary value problem (bvp4c) was employed to get a good account of all pertinent flow parameters before solving the controlling partial differential equations numerically. Here of Silver () and of Molybdenum disulfide () nanoparticles in the hybrid model are considered in the base fluid water. The basic achievement of this analysis is to explore the Nusselt number, skin friction coefficient along with entropy formation, and Bejan number. The irreversibility factor is calculated using the thermodynamics second law. The findings also point to the presence of dual-nature solutions in the shrinking sheet region for a given value of a high mass suction parameter, with a stable upper solution branch and unstable lower solution branch. Two solutions are found for the limited range of shrinking parameter when and the solutions terminate at in the shrinking region. Additionally, the first solution generates a positive minimum eigenvalue (), whereas the second solution generates a negative eigenvalue (), indicating the solution’s stability. Nusselt number is enhanced for variable viscosity parameter but diminished by variable thermal conductivity parameter for the stable solution. The entropy generation and the Bejan number both increase significantly with the increment of the thermal radiant-energy parameter, Biot number, and Eckert number. These results are crucial in the long term because they allow us to optimize heat transmission for cooling and heating applications. Hence, this investigation is incredibly significant in the present era, particularly in the automotive industry, home industry, paper plastics, ceramics, food packaging, food colorants, cancer treatment, cosmetics, pharmaceuticals, fabrics, paints, and soaps as well.

Stability analysis of radiative-magnetic hybrid nanofluid slip flow due to an exponentially stretching/shrinking permeable sheet with heat generation

The present article aims to find the duality of solutions and investigate the magneto-radiative hybrid nanofluid flow over an exponentially stretching/shrinking permeable sheet with heat generation and multiple slip boundary conditions. The governing nonlinear partial differential equations are transformed into dimensionless nonlinear ordinary differential equations which are computed by bvp4c in Matlab platform. The hybridisation of alumina and copper nanoparticles enhances the system's heat transfer performance. The dual nature solution is obtained only if the suction is applied to the stretching/shrinking surface. From stability analysis, we can conclude that the first solution is stable, but the other is unstable. It is realistic to have the solution stable with the smallest positive eigenvalues, while the lower solution branch is unstable with the smallest negative eigenvalues. It is found that the upsurges of solid volume fraction of the copper nanoparticle enhanced the skin friction coefficient and the local Nusselt number.

An upper bound for difference of energies of a graph and its complement

The A-energy of a graph G, denoted by EA(G), is defined as sum of the absolute values of eigenvalues of adjacency matrix of G. Nikiforov in Nikiforov (2016) proved that EA(G¯)−EA(G)≤2μ¯1 and EA(G)−EA(G¯)≤2μ1 for any graph G and posed a problem to find best possible upper bound for EA(G)−EA(G¯), where μ1 and μ1¯ are the largest adjacency eigenvalues of G and its complement G¯ respectively. We attempt to provide an answer by giving an improved upper bound on a class of graphs where regular graphs become particular case. As a consequence, it is proved that there is no strongly regular graph with negative eigenvalues greater than −1. The obtained results also improves some of the other existing results.

On the detailed structure of quantum control landscape for fast single qubit phase-shift gate generation

In this work, we study the detailed structure of quantum control landscape for the problem of single-qubit phase shift gate generation on the fast time scale. In previous works, the absence of traps for this problem was proved on various time scales. A special critical point which was known to exist in quantum control landscapes was shown to be either a saddle or a global extremum, depending on the parameters of the control system. However, in case of a saddle, the numbers of negative and positive eigenvalues of the Hessian at this point and their magnitudes have not been studied. At the same time, these numbers and magnitudes determine the relative ease or difficulty for practical optimization in a vicinity of the critical point. In this work, we compute the numbers of negative and positive eigenvalues of the Hessian at this saddle point and, moreover, give estimates on magnitude of these eigenvalues. We also significantly simplify our previous proof of the theorem about this saddle point of the Hessian (Theorem 3 in [22]).

On the detailed structure of quantum control landscape for fast single qubit phase-shift gate generation

In this work, we study the detailed structure of quantum control landscape for the problem of single-qubit phase shift gate generation on the fast time scale. In previous works, the absence of traps for this problem was proven on various time scales. A special critical point which was known to exist in quantum control landscapes was shown to be either a saddle or a global extremum, depending on the parameters of the control system. However, in the case of a saddle the numbers of negative and positive eigenvalues of the Hessian at this point and their magnitudes have not been studied. At the same time, these numbers and magnitudes determine the relative ease or difficulty for practical optimization in a vicinity of the critical point. In this work, we compute the numbers of negative and positive eigenvalues of the Hessian at this saddle point and, moreover, give estimates on magnitude of these eigenvalues. We also significantly simplify our previous proof of the theorem about this saddle point of the Hessian [Theorem 3 in B. O. Volkov, O. V. Morzhin, A. N. Pechen, J. Phys. A: Math. Theor. 54, 215303 (2021)]. Bibliography: 42 titles.

The Rate of Accumulation of Negative Eigenvalues to Zero and the Absolutely Continuous Spectrum

For a bounded real-valued function V on ℝd, we consider two Schrödinger operators H+ = −Δ+V and H− = −Δ − V . We prove that if the negative spectra H+ and H−are discrete and the negative eigenvalues of H+ and H− tend to zero sufficiently fast, then the absolutely continuous spectra cover the positive half-line [0,∞).