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We study an ill-posed forward-backward parabolic equation with techniques of nonstandard analysis. Equations of this form arise in many applications, ranging from the description of phase transitions, to the dynamics of aggregating populations, and to image enhancing. The equation is ill-posed in the sense that it only has measure-valued solutions that in general are not unique. By using grid functions of nonstandard analysis, we derive a continuous-in-time and discrete-in-space formulation for the ill-posed problem. This nonstandard formulation is well-posed and formally equivalent to the classical PDE, and has a unique grid solution that satisfies the properties of an entropy measure-valued solution for the original problem. By exploiting the strength of the nonstandard formulation, we are able to characterize the asymptotic behaviour of the grid solutions and to prove that they satisfy a conjecture formulated by Smarrazzo for the measure-valued solutions to the ill-posed problem. [ABSTRACT FROM AUTHOR]

We will present the benefits of using methods of non-standard analysis in dynamic projective geometry. One major application will be the desingularization of geometric constructions. [ABSTRACT FROM AUTHOR]

We investigate the large time behavior of solutions to the following nonlinear parabolic equations { u t − div ( | ∇ u | p ( x ) − 2 ∇ u ) + f ( x , u ) = g in Ω × R + , u = 0 on ∂ Ω × R + , u ( x , 0 ) = u 0 ( x ) in Ω , where u 0 , g ∈ L 1 ( Ω ) . We first provide the existence and uniqueness of an entropy solution for the problem. Then through some delicate analysis, we establish some regularity results on the solution, by which we prove the existence of a global attractor for the solution semigroup. [ABSTRACT FROM AUTHOR]

We consider the obstacle problem related to the following energy with nonstandard growth ∫ Ω | D u | p ( x ) log ⁡ ( e + | D u | ) d x . We investigate the regularity properties of solutions to the obstacle problems along with a suitable assumptions on the variable exponent p ( ⋅ ) and the obstacle. [ABSTRACT FROM AUTHOR]

The aim of this paper is to derive the self-improving property of integrability for the spatial gradient of solutions to degenerate parabolic obstacle problem with irregular obstacles and p ( x , t ) -nonstandard growth. More precisely, we prove that the spatial gradient of the solution is integrable to a larger power than the natural one determined by the structural assumptions on the involved differential operator. [ABSTRACT FROM AUTHOR]

It is shown how delta shock waves which consist of Dirac delta distributions and classical shocks can be used to construct non-monotone solutions of the Buckley–Leverett equation. These solutions are interpreted using a recent variational definition of delta shock waves in which the Rankine–Hugoniot deficit is explicitly accounted for [6] . The delta shock waves are also limits of approximate solutions constructed using a recent extension of the weak asymptotic method to complex-valued approximations [15] . Finally, it is shown how these non-standard shocks can be fitted together to construct similarity and traveling-wave solutions which are non-monotone, but still admissible in the sense that characteristics either enter or are parallel to the shock trajectories. [ABSTRACT FROM AUTHOR]

In Di Nasso (2015) and Luperi Baglini (2012) it has been introduced a technique, based on nonstandard analysis, to study some problems in combinatorial number theory. In this paper we review such a technique and we present three of its applications: the first one is a new proof of a known result regarding the algebra of β N , namely that the center of the semigroup ( β N , ⊕ ) is N ; the second one is a generalization of a theorem of Bergelson and Hindman on arithmetic progressions of length three; the third one regards the study of which polynomials in several variables with integers coefficients have a monochromatic solution for every finite coloring of N . We will study this last application in more detail: we will prove some algebraical properties of the set P of such polynomials and we will present a few examples of nonlinear polynomials in P . In the first part of the paper we will recall the main results of the nonstandard technique that we want to use, which is based on a characterization of ultrafilters by means of nonstandard analysis. [ABSTRACT FROM AUTHOR]

Renling Jin proved that if A and B are two subsets of the natural numbers with positive Banach density, then A + B is piecewise syndetic. In this paper, we prove that, under various assumptions on positive lower or upper densities of A and B , there is a high density set of witnesses to the piecewise syndeticity of A + B . Most of the results are shown to hold more generally for subsets of Z d . The key technical tool is a Lebesgue density theorem for measure spaces induced by cuts in the nonstandard integers. [ABSTRACT FROM AUTHOR]

A fundamental open problem in algebraic combinatorics is to find a positive combinatorial formula for Kronecker coefficients, which are multiplicities of the decomposition of the tensor product of two S r -irreducibles into irreducibles. Mulmuley and Sohoni attempt to solve this problem using canonical basis theory, by first constructing a nonstandard Hecke algebra B r , which, though not a Hopf algebra, is a u -analogue of the Hopf algebra C S r in some sense (where u is the Hecke algebra parameter). For r = 3 , we study this Hopf-like structure in detail. We define a nonstandard Hecke algebra H ˇ 3 ( k ) ⊆ H 3 ⊗ k , determine its irreducible representations over Q ( u ) , and show that it has a presentation with a nonstandard braid relation that involves Chebyshev polynomials evaluated at 1 u + u − 1 . We generalize this to Hecke algebras of dihedral groups. We go on to show that these nonstandard Hecke algebras have bases similar to the Kazhdan–Lusztig basis of H 3 and are cellular algebras in the sense of Graham and Lehrer. [ABSTRACT FROM AUTHOR]

Abstract: We consider the existence and non-existence of solutions for nonlinear elliptic equations whose model is where Ω is a smooth bounded domain in , μ is a bounded Radon measure. Our results are twofold: when μ is absolutely continuous with respect to the -capacity, the problem admits a unique solution; when μ is concentrated on a set of zero -capacity ( in ), then the problem does not admit a solution if is large enough. [Copyright &y& Elsevier]

A broad review is given of some areas of multivariate analysis that are not frequently emphasized. We start with situations in which underlying distributions are far from multivariate normal form, so that standard methods of multivariate analysis based on covariances are likely to be unsatisfactory. We emphasize the important distinction between internal and external analyses associated with multiple outcomes. A second broad theme relates to multiple outcomes generated by time or spatial series in which long-range dependence operates. Some implications are summarized. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider uniform asymptotics for the finite-time ruin probabilities of two kinds of nonstandard bidimensional renewal risk models with constant interest forces and diffusion generated by Brownian motions. In one of the models, two classes of claims have different arrival times, while in the another model, two classes of claims share the same arrival times. In both models, two classes of claim sizes are both upper tail asymptotically independent and their distributions belong to the intersection of the long-tailed distribution class and the dominatedly-varying-tailed distribution class, and the inter-arrival times follow a widely lower orthant dependence structure. In each model, we obtain three kinds of uniform asymptotics for the finite-time ruin probabilities, respectively. [Copyright &y& Elsevier]

Abstract: This paper proposes a nonstandard finite difference (NSFD) method for a class of delay differential equations. We prove that, for any step size (), this numerical method could intrinsically preserve the qualitative behavior of the dynamical system, including the local stability of equilibrium, the existence and the direction of Hopf bifurcation and the stability of bifurcating periodic solution. Finally, to illustrate the analytic results, the NSFD method is used for a model of the survival of red blood cells in animals. [Copyright &y& Elsevier]

Abstract: In this article we verify Dufloʼs conjecture for with respect to a Borel subgroup. It turns out that the non-standard notion weakly proper (faiblement propre) in the formulation of Dufloʼs conjecture is more reasonable than the standard notions proper and proper over image. [Copyright &y& Elsevier]

Abstract: We characterize Schwartz distributions having a value at a single point in the sense introduced by means of nonstandard analysis by A. Robinson. They appear to be continuous functions in a neighborhood of the point. This characterization improves a result by P. Loeb which assumes the everywhere existence of point values. [Copyright &y& Elsevier]

Abstract: Motivated by the problem of the existence of bounds on degrees and orders in checking primality of radical (partial) differential ideals, the nonstandard methods of van den Dries and Schmidt [L. van den Dries, K. Schmidt, Bounds in the theory of polynomial rings over fields. A nonstandard approach, Invent. Math. 76 (1984) 77–91] are here extended to differential polynomial rings over differential fields. Among the standard consequences of this work are: a partial answer to the primality problem, the equivalence of this problem with several others related to the Ritt problem, and the existence of bounds for characteristic sets of minimal prime differential ideals and for the differential Nullstellensatz. [Copyright &y& Elsevier]

Abstract: This paper cares about blowup solutions for a system of n-componential heat equations coupled via localized reactions and with variable exponents. The criteria for non-simultaneous and simultaneous blowup are established for radial solutions with or without assumptions on initial data, including the existence of non-simultaneous blowup for n components; any blowup must be simultaneous or non-simultaneous. [Copyright &y& Elsevier]

Abstract: We show that principles from nonstandard analysis hold to some extent for nonlinear generalized functions. The generalized functions under consideration are constructed as families of functions modulo a free filter, as it is usually done in applied analysis. In contrast with models of nonstandard analysis, we do not require the filter to be an ultrafilter. The principles are intended to be used as a tool for proving theorems, which we illustrate by means of an automatic continuity result that was not suspected by experts in the field. [Copyright &y& Elsevier]

Abstract: We prove sharp pointwise estimates for functions in the Sobolev spaces of radial functions defined in a ball. As a consequence, we obtain some imbeddings of such Sobolev spaces in weighted -spaces. We also prove similar imbeddings for Sobolev spaces of functions with partial symmetry. Our techniques lead to new Hardy type inequalities. It is important to observe that we do not require any vanishing condition on the boundary to obtain all our estimates. We apply these imbeddings to obtain radial solutions and partially symmetric solutions for a biharmonic equation of the Hénon type under both Dirichlet and Navier boundary conditions. The delicate question of the regularity of these solutions is also established. [Copyright &y& Elsevier]

Abstract: We prove -regularity for local minimizers of functionals where φ is a Young function. In order to generalize the results of Fuchs (2011) and Diening et al. (2009) , we assume for the function φ only the -condition and for a positive constant (and of course a Hölder-condition for the second derivatives). [Copyright &y& Elsevier]

Abstract: We prove that the minimal base size for the permutation action of the sporadic simple Baby monster group B on the cosets of its 7th and 8th maximal subgroup (in decreasing order of size) is 3 and 2 respectively. Motivated by the large sizes of these permutation actions, we develop new computational methods to prove that an orbit is regular and to show that two orbits are disjoint. [Copyright &y& Elsevier]

Abstract: A new kind of finite difference scheme is presented for special second order nonlinear two point boundary value problems. An artificial parameter is introduced in the scheme. Symbolic computation is proposed for the construction of the scheme. Local truncation error of the method is discussed. Numerical examples are illustrated. Numerical results show that the method is very effective. [Copyright &y& Elsevier]

Abstract: We derive a discretized SIRS epidemic model with time delay by applying a nonstandard finite difference scheme. Sufficient conditions for the global dynamics of the solution are obtained by improvements in discretization and applying proofs for continuous epidemic models. These conditions for our discretized model are the same as for the original continuous model. [Copyright &y& Elsevier]

Abstract: In the present paper we consider a prismatic cylinder occupied by an anisotropic and homogeneous compressible linear thermoelastic material within the framework of the linear theory of thermoelasticity without energy dissipation. The cylinder is subject to zero body force and heat supply and zero lateral specific boundary conditions and the motion is induced by a time-dependent displacement and thermal displacement specified pointwise over the base. Further, the motion is constrained such that the displacement, thermal displacement, velocity and temperature variation at points in the cylinder and at a prescribed time are in given proportions to, but not identical with, their respective initial values. It is shown that certain integrals of the solution spatially evolve with respect to the axial variable. Conditions are derived that show the integrals exhibit alternative behavior and in particular for the semi-infinite cylinder that there is either at least exponential growth or at most exponential decay, provided the elasticity tensor is positive definite or strongly elliptic. [Copyright &y& Elsevier]

Abstract: In previous papers, see, for example, [O.J. Brison, J.E. Nogueira, Second order linear sequence subgroups in finite fields, Finite Fields Appl. 14 (2008) 277–290], the authors studied f-sequence subgroups when was a polynomial of degree 2. Here we study non-standard groups when has degree k. The non-standard configurations found when extend to the general case, while new configurations when are described. [Copyright &y& Elsevier]

Abstract: Consider a model of location choice by two sorts of agents, called “buyers” and “sellers”: In the first period agents simultaneously choose between two identical possible locations; following this, the agents at each location play some sort of game with the other agents there. Buyers prefer locations with fewer other buyers and more sellers, and sellers have the reverse preferences. We study the set of possible equilibrium sizes for the two markets, and show that two markets of very different sizes can co-exist even if larger markets are more efficient. This extends the analysis of Ellison and Fudenberg [2003. Quart. J. Econ. 118, 1249–1278], who ignored the constraint that the number of agents of each type in each market should be an integer, and instead analyzed the “quasi-equilibria” where agents are treated as infinitely divisible. [Copyright &y& Elsevier]

Abstract: Under nonstandard growth conditions, following Moser''s iteration technique, we prove boundedness of solutions for fourth order nonlinear elliptic equation in divergence form. [Copyright &y& Elsevier]

Abstract: The relationship between Popper spaces (conditional probability spaces that satisfy some regularity conditions), lexicographic probability systems (LPS''s), and nonstandard probability spaces (NPS''s) is considered. If countable additivity is assumed, Popper spaces and a subclass of LPS''s are equivalent; without the assumption of countable additivity, the equivalence no longer holds. If the state space is finite, LPS''s are equivalent to NPS''s. However, if the state space is infinite, NPS''s are shown to be more general than LPS''s. [Copyright &y& Elsevier]

Non-standard finite difference methods (NSFDM) introduced by Mickens [Non-standard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994] are interesting alternatives to the traditional finite difference and finite volume methods. When applied to linear hyperbolic conservation laws, these methods reproduce exact solutions. In this paper, the NSFDM is first extended to hyperbolic systems of conservation laws, by a novel utilization of the decoupled equations using characteristic variables. In the second part of this paper, the NSFDM is studied for its efficacy in application to nonlinear scalar hyperbolic conservation laws. The original NSFDMs introduced by Mickens (1994) were not in conservation form, which is an important feature in capturing discontinuities at the right locations. Mickens [Construction and analysis of a non-standard finite difference scheme for the Burgers–Fisher equations, Journal of Sound and Vibration 257 (4) (2002) 791–797] recently introduced a NSFDM in conservative form. This method captures the shock waves exactly, without any numerical dissipation. In this paper, this algorithm is tested for the case of expansion waves with sonic points and is found to generate unphysical expansion shocks. As a remedy to this defect, we use the strategy of composite schemes [R. Liska, B. Wendroff, Composite schemes for conservation laws, SIAM Journal of Numerical Analysis 35 (6) (1998) 2250–2271] in which the accurate NSFDM is used as the basic scheme and localized relaxation NSFDM is used as the supporting scheme which acts like a filter. Relaxation schemes introduced by Jin and Xin [The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Communications in Pure and Applied Mathematics 48 (1995) 235–276] are based on relaxation systems which replace the nonlinear hyperbolic conservation laws by a semi-linear system with a stiff relaxation term. The relaxation parameter () is chosen locally on the three point stencil of grid which makes the proposed method more efficient. This composite scheme overcomes the problem of unphysical expansion shocks and captures the shock waves with an accuracy better than the upwind relaxation scheme, as demonstrated by the test cases, together with comparisons with popular numerical methods like Roe scheme and ENO schemes. [Copyright &y& Elsevier]

The notions of “labelled set” and “numerosity” are introduced to generalize the counting process of finite sets. The resulting numbers, called numerosities, are then used to develop nonstandard analysis. The existence of a numerosity function is equivalent to the existence of a selective ultrafilter, hence it is independent of the axioms of ZFC. [Copyright &y& Elsevier]

We consider problems of the form d2u/dt2+Au=F,αu(0)+u(T)=g,βdu/dt(0)+du/dt+Au=F, αu(0)+u(T)=g, βdu/dt(0)+du/dt(0)+du/dt(T)=h, for t∈(0,T), where A is a densely defined, linear, time independent, positive definite symmetric operator and α and β are constants. Although most of our results would hold for more general operators A, we restrict attention to the case in which A is a differential operator and determine ranges of values of α and β for which it is possible to obtain energy bounds, uniqueness results, and, in a special case, pointwise bounds. Some extensions which include a damping term or a term which arises in a generalization of the Kirchhoff string model are also discussed. [Copyright &y& Elsevier]