General Class Of Solutions Research Articles

Measurement and Uncertainty Preserving Parametric Modeling for Continuous Latent Variables With Discrete Indicators and External Variables

Research in education and behavioral sciences often involves the use of latent variable models that are related to indicators, as well as related to covariates or outcomes. Such models are subject to interpretational confounding, which occurs when fitting the model with covariates or outcomes alters the results for the measurement model. This has received attention in models for continuous observable variables but to date has not been examined in the context of discrete variables. This work demonstrates that interpretational confounding can occur in models for discrete variables, and develops a multistage Bayesian estimation approach to deal with this problem. The key features of this approach are that it is (a) measurement preserving, in that it precludes the possibility of interpretational confounding, and (b) uncertainty preserving, in that the uncertainty from the earlier stage of estimating the measurement model is propagated to the second stage of estimating the relations between the latent variable(s) and any covariates or outcomes. Previous work on these methods had only considered models for continuous observed variables, and software was limited to models with a single latent variable and either covariates or outcomes. This work extends the approach and software to a more general class of solutions, including discrete variables, illustrating the procedures with analyses of real data. Functions for conducting the analyses in widely available software are provided.

Regularity for the two‐phase singular perturbation problems

We prove that an a priori BMO gradient estimate for the two phase singular perturbation problem implies Lipschitz regularity for the limits. This problem arises in the mathematical theory of combustion where the reaction-diffusion is modelled by the $p$-Laplacian. A key tool in our approach is the weak energy identity. Our method proves a natural and intrinsic characterization of the free boundary points and can be applied to more general classes of solutions.

On a general class of solutions for an optimal velocity model on an infinite lane

We consider a general class of car-following models. Such models were studied over the last decades both on a circular and on an infinite road. Many interesting phenomena have been studied, e.g. loss of stability and stop and go waves. Whereas there is good mathematical understanding for the circular road, for the infinite road only partial results are available. In this paper we study a general approach to obtain solutions which are valid for the different settings mentioned above. The well known solutions on the circular road can be seen as a subclass of solutions, the known approaches on the infinite lane can be identified and new solutions on the infinite lane are studied. Beside the classical tools from linear analysis we will employ nonlinear bifurcation techniques. Throughout the paper as a prominent example we will use the so-called optimal velocity model proposed in Bando, et al. [1995. “Dynamical model of traffic congestion and numerical simulation.” Physical Review E 51 (2): 1035–1042].

Ellis wormhole without a phantom scalar field.

In this paper, we present an exact solution for (3 + 1)-dimensional Einstein-scalar-Gauss-Bonnet theory (EsGB) in electrovacuum. The solution is characterized by only one parameter, Q, which in general can be associated with the electromagnetic field and the scalar field. We show that the solution corresponds to a charged wormhole with a throat at the region r = |Q| and is also supported by a real scalar field having a positive kinetic term. We show that the solution belongs to the most general class of solutions known as Ellis wormholes but without the need for "exotic matter" or a phantom scalar field.

Classes of metastable thermodynamic quantum time crystals

We found that thermodynamic quantum time crystals in fermi systems, defined as quantum orders oscillating periodically in the imaginary Matsubara time with zero mean, are metastable for two general classes of solutions. Mean-field time independent solutions proved to have lower free energy manifesting true thermodynamic equilibrium with either single or multiple (competing) charge, spin and superconducting symmetry breaking orders. The 'no-go' theorem is proven analytically for a case of long-range interactions between fermions in momentum space in electron-hole and Cooper channels.

Classical and quantum analysis of 3D electromagnetic pp-wave spacetime

The general classical solution of the 3D electromagnetic pp-wave spacetime has been obtained. The relevant line element contains an arbitrary essential function providing an infinite number of in-equivalent geometries as solutions. A classification is presented based on the symmetry group. To the best of our knowledge, the solution corresponding to only one of the Classes is known. The dynamics of some of the Classes was also derived from a minisuperspace Lagrangian which has been constructed. This Lagrangian contains a degree of freedom (the lapse) which can be considered either as dynamical or non-dynamical (indicating a singular or a regular Lagrangian correspondingly). Surprisingly enough, on the space of classical solutions, an equivalence of these two points of view can be established. The canonical quantization is then used in order to quantize the system for both the singular and regular Hamiltonian. A subsequent interpretation of quantum states is based on a Bohm-like analysis. The semi-classical trajectories deviate from the classical only for the regular Hamiltonian and in particular for a superposition of eigenstates (a Gaussian initial state has been used). Thus, the above mentioned equivalence is broken at the quantum level. It is noteworthy that the semi-classical trajectories tend to the classical ones in the limit where the initial wavepacket is widely spread. Hence, even with this simple superposition state, the classical solutions are acquired as a limit of the semi-classical.

A class of general solutions of the unsteady Oseen equations

In this paper, we present a class of general solutions of the unsteady Oseen equations in terms of two scalar functions. We also give a condition for a divergence-free vector to be a possible solution of Oseen equations.

Recent Progress in Fighting Ghosts in Quantum Gravity

We review some of the recent results which can be useful for better understanding of the problem of stability of vacuum and in general classical solutions in higher derivative quantum gravity. The fourth derivative terms in the purely gravitational vacuum sector are requested by renormalizability already in both semiclassical and complete quantum gravity theories. However, because of these terms, the spectrum of the theory has unphysical ghost states which jeopardize the stability of classical solutions. At the quantum level, ghosts violate unitarity, and thus ghosts look incompatible with the consistency of the theory. The “dominating” or “standard” approach is to treat higher derivative terms as small perturbations at low energies. Such an effective theory is supposed to glue with an unknown fundamental theory in the high energy limit. We argue that the perspectives for such a scenario are not clear, to say the least. On the other hand, recently, there was certain progress in understanding physical conditions which can make ghosts not offensive. We survey these results and discuss the properties of the unknown fundamental theory which can provide these conditions satisfied.

A family of solutions to the Einstein–Maxwell system of equations describing relativistic charged fluid spheres

In this paper, we present a formalism to generate a family of interior solutions to the Einstein–Maxwell system of equations for a spherically symmetric relativistic charged fluid sphere matched to the exterior Reissner–Nordström space–time. By reducing the Einstein–Maxwell system to a recurrence relation with variable rational coefficients, we show that it is possible to obtain closed-form solutions for a specific range of model parameters. A large class of solutions obtained previously are shown to be contained in our general class of solutions. We also analyse the physical viability of our new class of solutions.

Static spherically symmetric black holes of de Rham–Gabadadze–Tolley massive gravity in arbitrary dimensions

This article is devoted to static spherically symmetric black hole solutions of dRGT (de Rham–Gabadadze–Tolley) massive gravity in the presence of cosmological constant. The unitary and non-unitary gauges are used to find the solutions in three, four and five dimensions. We show that there are two general classes of solutions. In one of them, the effect of massive potential is appeared as the effective cosmological constant. By investigating these solutions in different dimensions, we find an expression for effective cosmological constant in arbitrary dimensions.

Exact solutions and critical chaos in dilaton gravity with a boundary

We consider (1 + 1)-dimensional dilaton gravity with a reflecting dynamical boundary. The boundary cuts off the region of strong coupling and makes our model causally similar to the spherically-symmetric sector of multidimensional gravity. We demonstrate that this model is exactly solvable at the classical level and possesses an on-shell SL(2, ℝ) symmetry. After introducing general classical solution of the model, we study a large subset of soliton solutions. The latter describe reflection of matter waves off the boundary at low energies and formation of black holes at energies above critical. They can be related to the eigenstates of the auxiliary integrable system, the Gaudin spin chain. We argue that despite being exactly solvable, the model in the critical regime, i.e. at the verge of black hole formation, displays dynamical instabilities specific to chaotic systems. We believe that this model will be useful for studying black holes and gravitational scattering.

Reaction enhancement by chemotaxis

An interesting problem arising in many contexts of mathematical biology is the study of the relevance of chemotaxis in reaction–diffusion processes. We approach this problem through a mathematical model representing the fertilization process of corals. As a result we obtain a system of partial differential equations describing the cell dynamics being affected by three basic phenomena: diffusion, chemotaxis and a surrounding flux. First we prove that our model has in general global classical solutions. Next, we compare the asymptotic behavior with and without chemotaxis. We show the relevance of chemotaxis after making a systematic adaptation of the well-known moments technique on bounded domains (usually used for proving blow-up in Keller–Segel systems) to analyze the behavior of the cell dynamics when the chemotactic signal increases. In the context of the proposed model, our analysis shows that for suitable initial data, the remaining fraction of unfertilized eggs at any given time τ>0 becomes arbitrarily small if the chemotaxis signal is sufficiently large and the sperm and eggs are concentrated enough at the initial time.

On global regular solutions to magnetohydrodynamics in axi-symmetric domains

We consider mhd equations in three-dimensional axially symmetric domains under the Navier boundary conditions for both velocity and magnetic fields. We prove the existence of global, regular axi-symmetric solutions and examine their stability in the class of general solutions to the mhd system. As a consequence, we show the existence of global, regular solutions to the mhd system which are close in suitable norms to axi-symmetric solutions.

ROTATING ACCRETION FLOWS: FROM INFINITY TO THE BLACK HOLE

Accretion onto a supermassive black hole of a rotating inflow is a particularly difficult problem to study because of the wide range of length scales involved. There have been broadly utilized analytic and numerical treatments of the global properties of accretion flows, but detailed numerical simulations are required to address certain critical aspects. We use the ZEUS code to run hydrodynamical simulations of rotating, axisymmetric accretion flows with Bremsstrahlung cooling, considering solutions for which the centrifugal balance radius significantly exceeds the Schwarzschild radius, with and without viscous angular momentum transport. Infalling gas is followed from well beyond the Bondi radius down to the vicinity of the black hole. We produce a continuum of solutions with respect to the single parameter Mdot_Bondi/Mdot_Edd, and there is a sharp transition between two general classes of solutions at an Eddington ratio of Mdot_Bondi/Mdot_Edd ~ few x 10^(-2). Our high inflow solutions are very similar to the standard Shakura & Sunyaev (1973) results. But our low inflow results are to zeroth order the stationary Papaloizou and Pringle (1984) solution, which has no accretion. To next order in the small, assumed viscosity they show circulation, with disk and conical wind outflows almost balancing inflow. These solutions are characterized by hot, vertically extended disks, and net accretion proceeds at an extremely low rate, only of order alpha times the inflow rate. Our simulations have converged with respect to spatial resolution and temporal duration, and they do not depend strongly on our choice of boundary conditions.

Regular models with quadratic equation of state

We provide new exact solutions to the Einstein-Maxwell system of equations which are physically reasonable. The spacetime is static and spherically symmetric with a charged matter distribution. We utilise an equation of state which is quadratic relating the radial pressure to the energy density. Earlier models, with linear and quadratic equations of state, are shown to be contained in our general class of solutions. The new solutions to the Einstein-Maxwell are found in terms of elementary functions. A physical analysis of the matter and electromagnetic variables indicates that the model is well behaved and regular. In particular there is no singularity in the proper charge density at the stellar centre unlike earlier anisotropic models in the presence of the electromagnetic field.

Splitting strings on integrable backgrounds

Using integrability we reduce the problem of constructing general classical splitting string solutions on $$ \mathbb{R} $$ × S 3 to a series of Birkhoff factorization problems. Namely, given any incoming string solution satisfying a necessary self-intersection property at some given instant in time, we use the integrability of the worldsheet σ-model to implicitly construct the pair of outgoing strings resulting from a split. The solution for each outgoing string is expressed recursively through a sequence of dressing transformations with parameters determined by the solutions to Birkhoff factorization problems in an appropriate real form of the loop group of SL 2( $$ \mathbb{C} $$ ).

Mixed potentials in radiative stellar collapse

We study the behavior of a radiating star when the interior expanding, shearing fluid particles are traveling in geodesic motion. We demonstrate that it is possible to obtain new classes of exact solutions in terms of elementary functions without assuming a separable form for the gravitational potentials or initially fixing the temporal evolution of the model unlike earlier treatments. A systematic approach enables us to write the junction condition as a Riccati equation which under particular conditions may be transformed into a separable equation. New classes of solutions are generated which allow for mixed spatial and temporal dependence in the metric functions. We regain particular models found previously from our general classes of solutions.

Brane gravitational interactions from 6D supergravity

We investigate the massive graviton contributions to 4D gravity in a 6D brane world scenario, whose bulk field content can include that of 6D chiral gauged supergravity. We consider a general class of solutions having 3-branes, 4D Poincaré symmetry and axisymmetry in the internal space. We show that these contributions, which we compute analytically, can be independent of the brane vacuum energy as a consequence of geometrical and topological properties of the above-mentioned codimension two brane world. These results support the idea that in such models the gravitational interactions may be decoupled from the brane vacuum energy.

Dressed giant magnons on ℂℙ3

A new example of AdS/CFT duality relating IIA string theory on AdS4 × 3 to = 6 superconformal Chern-Simons theory has recently been provided by ABJM. By now a number of papers have considered particular giant magnon classical string solutions in the 3 background, corresponding to excitations in the spin chain picture of the dual field theory. In this paper we apply the 3 = SU(4)/S(U(3) × U(1)) dressing method to the problem of constructing general classical string solutions describing various configurations of giant magnons. As a particular application we present a new giant magnon solution on 3. Interestingly the dressed solution carries only a single SO(6) charge, in contrast with the dyonic magnons found in previous applications of the dressing method.

Integrability in AdS/CFT correspondence: quasi-classical analysis

In this review, we consider a quasi-classical method applicable to integrable field theories which is based on a classical integrable structure—the algebraic curve. We apply it to the Green–Schwarz superstring on the AdS5 × S5 space. We show that the proposed method reproduces perfectly the earlier results obtained by expanding the string action for some simple classical solutions. The construction is explicitly covariant and is not based on a particular parameterization of the fields and as a result is free from ambiguities. On the other hand, the finite size corrections in some particularly important scaling limit are studied in this paper for a system of Bethe equations. For the general superalgebra , the result for the 1/L corrections is obtained. We find an integral equation which describes these corrections in a closed form. As an application, we consider the conjectured Beisert–Staudacher (BS) equations with the Hernandez–Lopez dressing factor where the finite size corrections should reproduce quasi-classical results around a general classical solution. Indeed, we show that our integral equation can be interpreted as a sum of all physical fluctuations and thus prove the complete one-loop consistency of the BS equations. We demonstrate that any local conserved charge (including the AdS energy) computed from the BS equations is indeed given at one loop by the sum of the charges of fluctuations with an exponential precision for large S5 angular momentum of the string. As an independent result, the BS equations in an sub-sector were derived from Zamolodchikovs's S-matrix. The paper is based on the author's PhD thesis.