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In this paper, we study the asymptotic behavior of transmission system for coupled Kirchhoff plates, where one equation is conserved and the other has dissipative property, and the dissipation mechanism is given by fractional damping (- Δ) 2 θ v t with θ ∈ [ 1 2 , 1 ] . By using the semigroup method and the multiplier technique, we obtain the exact polynomial decay rates, and find that the polynomial decay rate of the system is determined by the inertia/elasticity ratios and the fractional damping order. Specifically, when the inertia/elasticity ratios are not equal and θ ∈ [ 1 2 , 3 4 ] , the polynomial decay rate of the system is t - 1 / (10 - 4 θ) . When the inertia/elasticity ratios are not equal and θ ∈ [ 3 4 , 1 ] , the polynomial decay rate of the system is t - 1 / (4 + 4 θ) . When the inertia/elasticity ratios are equal, the polynomial decay rate of the system is t - 1 / (4 + 4 θ) . Furthermore it has been proven that the obtained decay rates are all optimal. The obtained results extend the results of Oquendo and Suárez (Z Angew Math Phys 70(3):88, 2019) for the case of fractional damping exponent 2 θ from [0, 1] to [1, 2]. [ABSTRACT FROM AUTHOR]

By introducing two parameters in the splittings of the matrices A and B, this paper presents a parameterized accelerated iteration (PAI) method for solving the matrix equation A X B = C . The convergence property of the PAI method and the choices of the parameters are thoroughly investigated. Additionally, based on some special splittings of the matrices A and B, several variants of the PAI method are established. Furthermore, for some certain cases, the optimal parameters can be determined, and it is demonstrated that the PAI method is more efficient than the gradient-based iteration (GBI) method (Ding et al. Appl. Math. Comput. 197, 41–50 2008). Finally, by comparing it with several existing iteration methods, the effectiveness of the PAI method is verified through four numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the asymptotic monotonicity of positive solutions for fractional parabolic equation on the right half space. First, a narrow region principle for antisymmetric functions in unbounded domains is obtained, in which we remarkably weaken the decay condition u → 0 at infinity and only assume its growth rate does not exceed | x | γ (0 < γ < 2 s ) compared with (Adv. Math. 377:107463, 2021). Then we obtain asymptotic monotonicity of positive solutions of fractional parabolic equation on R + N × (0 , ∞) . [ABSTRACT FROM AUTHOR]

In this note, we make two revisions of the paper [2]. The first one is the asymptotic behavior of the energy functional as t → T (see [2, Theorem 1.6]), where T is the blow-up time. The second one is the equivalent conditions for the solutions blowing up in finite time or existing globally (see [2, Theorem 1.8]). [ABSTRACT FROM AUTHOR]

The paper is devoted to the introduction of a new class of generalized set-valued nonlinear variational-like inequality problems in the setting of Banach spaces. By means of the notion of P- η -proximal mapping, we prove its equivalence with a class of generalized implicit Wiener–Hopf equations and employ the obtained equivalence relationship and Nadler's technique to suggest a new iterative algorithm for finding an approximate solution of the considered problem. The existence of solution and the strong convergence of the sequences generated by our proposed iterative algorithm to the solution of our considered problem are verified. The problem of finding a common element of the set of solutions of a generalized nonlinear variational-like inequality problem and the set of fixed points of a total asymptotically nonexpansive mapping is also investigated. The final section deals with the investigation and analysis of the main results appeared in Kazmi and Bhat (Appl Math Comput 166:164–180, 2005) and some comments relating to them are given. The results presented in this article extend and improve some known results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a sliding method to investigate the monotonicity of solutions for nonlocal double phase equations. We first derive a narrow region principle in bounded domains. Then we illustrate how to utilize this new method of sliding to obtain monotonicity of solutions for nonlocal double phase equations in bounded domains and the whole space respectively. [ABSTRACT FROM AUTHOR]

The classical result of L. Székelyhidi states that (under some assumptions) every solution of a general linear equation must be a polynomial function. It is known that Székelyhidi's result may be generalized to equations where some occurrences of the unknown functions are multiplied by a linear combination of the variables. In this paper we study the equations where two such combinations appear. The simplest nontrivial example of such a case is given by the equation F (x + y) - F (x) - F (y) = y f (x) + x f (y) considered by Fechner and Gselmann (Publ Math Debrecen 80(1–2):143–154, 2012). In the present paper we prove several results concerning the systematic approach to the generalizations of this equation. [ABSTRACT FROM AUTHOR]

In the present paper, the full range Strichartz estimates for homogeneous Schrödinger equations with non-degenerate and non-smooth coefficients are proved. For inhomogeneous equation, the non-endpoint Strichartz estimates are also obtained. [ABSTRACT FROM AUTHOR]

Following the popularity of the core-EP (c-EP) and weighted core-EP (w-c-EP) inverses, so called one-sided versions of the w-c-EP inverse are introduced recently in Behera et al. (Results Math 75:174 (2020). These extensions are termed as E-w-c-EP and F-w-d-c-EP g-inverses as well as the star E-w-c-EP and the F-w-d-c-EP star classes of g-inverses. The applicability of these g-inverses in solving certain restricted matrix equations has been verified. Several additional results on these classes of g-inverses are established in this paper. In addition, the Moore–Penrose E-w-c-EP inverse and the F-w-d-c-EP Moore–Penrose inverse are proposed using proper expressions that involve the Moore–Penrose inverse and the E-w-c-EP or F-we-d-c-EP inverse. Further, the W-weighted Moore–Penrose c-EP and the W-weighted c-EP Moore–Penrose g-inverses are considered with the aim to extend the considered w-c-EP generalized inverses to rectangular matrices. Characterizations, properties, representations and applications of these inverses are considered. [ABSTRACT FROM AUTHOR]

A challenging open problem in the 3D Hall-MHD theory is to ask whether or not the global weak solutions are smooth. In this paper, we prove that a weak solution is smooth if the diagonal part of the velocity gradient tensor and the non-diagonal part of the magnetic gradient tensor satisfy Ladyzhenskaya–Prodi–Serrin-type conditions. It is physically interesting since the diagonal part of a gradient tensor is related to the deformation while the non-diagonal part is related to the rotation. Moreover, our main theorems improve significantly a criterion in Ye (Comput Math Appl 70(8):2137–2154, 2015) where all entries of the velocity gradient tensor and the magnetic gradient tensor are needed. [ABSTRACT FROM AUTHOR]

In this paper we study the convergence of nonlinear Dirichlet problems for systems of variational elliptic PDEs defined on randomly perforated domains of Rn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {R}^n$$\end{document}. Under the assumption that the perforations are small balls whose centres and radii are generated by a stationary short-range marked point process, we obtain in the critical-scaling limit an averaged nonlinear analogue of the extra term obtained in the classical work of Cioranescu and Murat (Res Notes Math III, 1982). In analogy to the random setting recently introduced by Giunti, Höfer and Velázquez (Commun Part Differ Equ 43(9):1377–1412, 2018) to study the Poisson equation, we only require that the random radii have finite (n-q)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(n-q)$$\end{document}-moment, where 1q-capacity of the spherical holes is finite, and hence that the limit problem is well defined. On the other hand, it does not exclude the presence of balls with large radii, that can cluster up. We show however that the critical rescaling of the perforations is sufficient to ensure that no percolating-like structures appear in the limit. [ABSTRACT FROM AUTHOR]

This paper addresses an article by Felix Klein of 1886, in which he generalized his theory of polynomial equations of degree 5—comprehensively discussed in his Lectures on the Icosahedron two years earlier—to equations of degree 6 and 7. To do so, Klein used results previously established in line geometry. I review Klein's 1886 article, its diverse mathematical background, and its place within the broader history of mathematics. I argue that the program advanced by this article, although historically overlooked due to its eventual failure, offers a valuable insight into a time of crucial evolution of the subject. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying Hamilton-Jacobi-Bellman equations with distribution-valued coefficients, which are not well-defined in the classical sense and are understood by using the paracontrolled distribution method introduced in (Gubinelli et al. in Forum Math Pi 3(6):1, 2015). By a new characterization of weighted Hölder spaces and Zvonkin's transformation we prove some new a priori estimates, and therefore establish the global well-posedness for singular HJB equations. As applications, we obtain global well-posedness in polynomial weighted Hölder spaces for KPZ type equations on the real line, as well as modified KPZ equations for which the Cole–Hopf transformation is not applicable. [ABSTRACT FROM AUTHOR]

We consider the three dimensional free-boundary compressible elastodynamic system under the Rayleigh–Taylor sign condition. This describes the motion of an isentropic inviscid elastic medium with moving boundary. The deformation tensor is assumed to satisfy the neo-Hookean linear elasticity. The local well-posedness was proved by Trakhinin (J Differ Eq 264(3):1661–1715, 2018) by Nash–Moser iteration. In this paper, we give a new proof of the local well-posedness by the combination of classical energy method and hyperbolic approach. In the proof, we apply the tangential smoothing method to define the approximation system. The key observation is that the structure of the wave equation of pressure together with Christodoulou–Lindblad (Commun Pure Appl Math 53(12):1536–1602, 2000) elliptic estimates reduces the energy estimates to the control of tangentially-differentiated wave equations despite a potential loss of derivative in the source term. To the best of our knowledge, we first establish the nonlinear energy estimate without loss of regularity for free-boundary compressible elastodynamics. The energy estimate is also uniform in sound speed which yields the incompressible limit, that is, the solutions of the free-boundary compressible elastodynamic equations converge to the incompressible counterpart provided the convergence of initial datum. It is worth emphasizing that our method is completely applicable to compressible Euler equations. Our observation also shows that it is not necessary to include the full time derivatives in the boundary energy and analyze higher order wave equations as in Lindblad–Luo (Commun Pure Appl Math 71(7):1273–1333, 2018) and Luo (Ann. PDE 4(2):2506–2576, 2018) even if we require the energy is uniform in sound speed. Moreover, the enhanced regularity for compressible Euler equations obtained in Lindblad–Luo (Commun Pure Appl Math 71(7):1273–1333, 2018) and Luo (Ann. PDE 4(2):2506–2576, 2018) can still be recovered for a slightly compressible elastic medium by further delicate analysis of the Alinhac good unknowns, which is completely different from Euler equations. [ABSTRACT FROM AUTHOR]

Existing research suggests that young children can develop a partial understanding of the equal sign as an operator rather than as a relational symbol of equivalence. This partial understanding can be the result of overemphasis on canonical equation syntaxes of the type a + b = c in elementary school mathematics. This paper presents an examination of context and syntax nuances of relevant sections from the grade 1 Greek series of textbooks and workbooks. Using a conceptual framework of context variation, the analysis shows qualitative differences between equations of similar syntax and provides a nuanced determination of contextual and structural aspects of 'variation' in how the equal sign is presented in elementary mathematics. The paper proposes that since equations have context-specific meanings, context variations should constitute a separate element of analysis when investigating how the equal sign is presented. The implication for practice and future research is that nuanced considerations of equation syntax within varied contexts are needed for elaborating analyses of the equal sign presentation that move beyond dichotomized categorizations of canonical/non-canonical syntaxes. [ABSTRACT FROM AUTHOR]

In this paper, an efficient (G ′ / G , 1 / G) -expansion method is adopted to resolve a famous (2 + 1)-dimensional fractional Ablowitz–Kaup–Newell–Segur (AKNS) water wave equation for the non-conservative system that plays a significant role in understanding the wave propagation. This work addresses the physical and dynamic behavior of some new exact trigonometric, hyperbolic, and rational solitary wave solutions in the form of 3D-plots and contour plots using different measures of parameters. The obtained results show the efficiency of the proposed method for the analytical treatment of nonlinear problems in mathematics, science and engineering and may be helpful in better understanding the propagating wave dynamics in diverse situations. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of Choquard equations with Hardy–Littlewood–Sobolev lower or upper critical exponent in the whole space R N . We combine an argument of L. Jeanjean and H. Tanaka (see (Proc. Am. Math. Soc. 131:2399–2408, 2003) with a concentration–compactness argument, and then we obtain the existence of ground state solutions, which extends and complements the earlier results. [ABSTRACT FROM AUTHOR]

It was proved in Forti and Schwaiger (C R Math Acad Sci Soc R Can 11(6):215–220, 1989), Schwaiger (Aequ Math 35:120–121, 1988) and with different methods in Schwaiger (Developments in functional equations and related topics. Selected papers based on the presentations at the 16th international conference on functional equations and inequalities, ICFEI, Bȩdlewo, Poland, May 17–23, 2015, Springer, Cham, pp 275–295, 2017) that under the assumption that every function defined on suitable abelian semigroups with values in a normed space such that the norm of its Cauchy difference is bounded by a constant (function) is close to some additive function, i.e., the norm of the difference between the given function and that additive function is also bounded by a constant, the normed space must necessarily be complete. By Schwaiger (Ann Math Sil 34:151–163, 2020) this is also true in the non-archimedean case. Here we discuss the situation when the bound is a suitable non-constant function. [ABSTRACT FROM AUTHOR]

In this paper we study a class of semilinear wave-type equations with viscoelastic damping and delay feedback with time variable coefficient. By combining semigroup arguments, careful energy estimates and an iterative approach we are able to prove, under suitable assumptions, a well-posedness result and an exponential decay estimate for solutions corresponding to small initial data. This extends and concludes the analysis initiated in Nicaise and Pignotti (J Evol Equ 15:107–129, 2015) and then developed in Komornik and Pignotti (Math Nachr, to appear, 2018), Nicaise and Pignotti (Evol Equ 18:947–971, 2018). [ABSTRACT FROM AUTHOR]

Makeenko and Migdal (Phys Lett B 88(1):135–137, 1979) gave heuristic identities involving the expectation of the product of two Wilson loop functionals associated to splitting a single loop at a self-intersection point. Kazakov and Kostov (Nucl Phys B 176(1):199–215, 1980) reformulated the Makeenko–Migdal equations in the plane case into a form which made rigorous sense. Nevertheless, the first rigorous proof of these equations (and their generalizations) was not given until the fundamental paper of Lévy (2017). Subsequently Driver, Kemp, and Hall Commun. Math. Phys. 351(2), 741–774 (2017) gave a simplified proof of Lévy's result and then with Driver, Gabriel, Kemp, and Hall Commun. Math. Phys. 352(3), 967–978 (2017) we showed that these simplified proofs extend to the Yang–Mills measure over arbitrary compact surfaces. All of the proofs to date are elementary but tricky exercises in finite dimensional integration by parts. The goal of this article is to give a rigorous functional integral proof of the Makeenko–Migdal equations guided by the original heuristic machinery invented by Makeenko and Migdal. Although this stochastic proof is technically more difficult, it is conceptually clearer and explains "why" the Makeenko–Migdal equations are true. It is hoped that this paper will also serve as an introduction to some of the problems involved in making sense of quantizing Yang–Mill's fields. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the hydrodynamic limit to rarefaction waves of the compressible Euler system for the Landau equation with Coulomb potentials as the Knudsen number ε > 0 is vanishing. Precisely, whenever ε > 0 is small, for the Cauchy problem on the Landau equation with suitable initial data involving a scaling parameter a ∈ [ 2 3 , 1 ] , we construct the unique global-in-time uniform-in- ε solution around a local Maxwellian whose fluid quantities are the rarefaction wave of the corresponding Euler system. In the meantime, we establish the convergence of solutions to the Riemann rarefaction wave uniformly away from t = 0 at a rate ε 3 5 - 2 5 a | ln ε | as ε → 0 . The proof is based on the refined energy approach combining Guo (Commun Math Phys 231:391–434, 2002) and Liu et al. (Physica D 188:178–192, 2004) under the scaling transformation (t , x) → (ε - a t , ε - a x) . [ABSTRACT FROM AUTHOR]

In this paper, the author studies the local existence of strong solutions and their possible blow-up in time for a quasilinear system describing the interaction of a short wave induced by an electron field with a long wave representing an extension of the motion of the director field in a nematic liquid crystal's asymptotic model introduced in [Saxton, R. A., Dynamic instability of the liquid crystal director. In: Current Progress in Hyperbolic Systems (Lindquist, W. B., ed.), Contemp. Math., Vol.100, Amer. Math. Soc., Providence, RI, 1989, pp.325–330] and [Hunter, J. K. and Saxton, R. A., Dynamics of director fields, SIAM J. Appl. Math., 51, 1991, 1498–1521] and studied in [Hunter, J. K. and Zheng, Y., On a nonlinear hyperbolic variational equation I, Arch. Rat. Mech. Anal., 129, 1995, 305–353], [Hunter, J. K. and Zheng, Y., On a nonlinear hyperbolic variational equation II, Arch. Rat. Mech. Anal., 129, 1995, 355–383] and in [Zhang, P. and Zheng, Y., On oscillation of an asymptotic equation of a nonlinear variational wave equation, Asymptotic Anal., 18, 1998, 307–327] and, more recently, in [Bressan, A., Zhang, P. and Zheng, Y., Asymptotic variational wave equations, Arch. Rat. Mech. Anal., 183, 2007, 163–185]. [ABSTRACT FROM AUTHOR]

The isospectral deformations of the Frobenius–Stickelberger–Thiele (FST) polynomials introduced in Spiridonov et al. (Commun Math Phys 272:139–165, 2007) are studied. For a specific choice of the deformation of the spectral measure, one is led to an integrable lattice (FST lattice), which is indeed an isospectral flow connected with a generalized eigenvalue problem. In the second part of the paper the spectral problem used previously in the study of the modified Camassa–Holm (mCH) peakon lattice is interpreted in terms of the FST polynomials together with the associated FST polynomials, resulting in a map from the mCH peakon lattice to a negative flow of the finite FST lattice. Furthermore, it is pointed out that the degenerate case of the finite FST lattice unexpectedly maps to the interlacing peakon ODE system associated with the two-component mCH equation studied in Chang et al. (Adv Math 299:1–35, 2016). [ABSTRACT FROM AUTHOR]

This paper is concerned with strictly k-convex large solutions to Hessian equations Sk(D2u(x)) = b(x)f(u(x)), x ∈ Ω, where Ω is a strictly (k − 1)-convex and bounded smooth domain in ℝn, b ∈ C ∞ ( Ω ¯) is positive in Ω, but may be vanishing on the boundary. Under a new structure condition on f at infinity, the author studies the refined boundary behavior of such solutions. The results are obtained in a more general setting than those in [Huang, Y., Boundary asymptotical behavior of large solutions to Hessian equations, Pacific J. Math., 244, 2010, 85–98], where f is regularly varying at infinity with index p > k. [ABSTRACT FROM AUTHOR]

In this paper, we consider a system of two viscoelastic equations with Dirichlet boundary conditions. For certain class of relaxation functions and initial data, we establish general and optimal decay results. This result extends earlier one of Liu (Nonlinear Anal. TMA 71:2257–2267, 2010), in which only the usual exponential and polynomial decay rates are considered. The conditions of the relaxation functions g 1 (t) and g 2 (t) in our work appeared first in Messaoudi and Khulaifi (Appl. Math. Lett. 66:16–22, 2017). [ABSTRACT FROM AUTHOR]

In previous work (Alves et al. in Z Angew Math Phys 69:106, 2018), by using the linear semigroup theory, Alves et al. investigated the existence and exponential stability results for a Moore–Gibson–Thompson model encompassing memory of type 1, 2 or 3 in a history space framework. In this paper, we continue to consider the similar problem with type 1 and establish explicit and general decay results of energy for system in both the subcritical and critical cases, by introducing suitable energy and perturbed Lyapunov functionals and following convex functions ideas presented in Guesmia (J Math Anal Appl 382:748–760, 2011). Our results allow a much larger class of the convolution kernels which improves the earlier related results. [ABSTRACT FROM AUTHOR]

In this paper, finite time blow-up property of solutions to a p-Kirchhoff-type parabolic equation with general nonlinearity is considered. Some sufficient conditions are given for the weak solutions to blow up in finite time. An upper bound for the blow-up time is also derived. The results partially generalize some recent ones reported by Han and Li (Comput Math Appl. 2018;75:3283–3297). [ABSTRACT FROM AUTHOR]

In this paper, we investigate the stability of the solutions of a viscoelastic plate equation with a logarithmic nonlinearity. We assume that the relaxation function g satisfies the minimal condition g ′ (t) ≤ − ξ (t) G (g (t)) , where ξ and G satisfy some properties. With this very general assumption on the behavior of g, we establish explicit and general energy decay results from which we can recover the exponential and polynomial rates when G (s) = s p and p covers the full admissible range [ 1 , 2) . Our new results substantially improve and generalize several earlier related results in the literature such as Gorka (Acta Phys. Pol. 40:59–66, 2009), Hiramatsu et al. (J. Cosmol. Astropart. Phys. 2010(06):008, 2010), Han and Wang (Acta Appl. Math. 110(1):195–207, 2010), Messaoudi and Al-Khulaifi (Appl. Math. Lett. 66:16–22, 2017), Mustafa (Math. Methods Appl. Sci. 41(1):192–204, 2018), and Al-Gharabli et al. (Commun. Pure Appl. Anal. 18(1):159–180, 2019). [ABSTRACT FROM AUTHOR]

In this paper, we prove a localised version of the bounded L 2 -curvature theorem of (Klainerman et al. Invent Math 202(1):91–216, 2015). More precisely, we consider initial data for the Einstein vacuum equations posed on a compact spacelike hypersurface Σ with boundary, and show that the time of existence of a classical solution depends only on an L 2 -bound on the Ricci curvature, an L 4 -bound on the second fundamental form of ∂ Σ ⊂ Σ , an H 1 -bound on the second fundamental form, and a lower bound on the volume radius at scale 1 of Σ . Our localisation is achieved by first proving a localised bounded L 2 -curvature theorem for small data posed on B(0, 1), and then using the scaling of the Einstein equations and a low regularity covering argument on Σ to reduce from large data on Σ to small data on B(0, 1). The proof uses the author's previous works and the bounded L 2 -curvature theorem as black boxes. [ABSTRACT FROM AUTHOR]

In this paper we study the theory Q. We prove a basic result that says that, in a sense explained in the paper, Q can be split into two parts. We prove some consequences of this result. (i) Q is not a poly-pair theory. This means that, in a strong sense, pairing cannot be defined in Q. (ii) Q does not have the Pudlák Property. This means that there two interpretations of $$\mathsf{S}^1_2$$ in Q which do not have a definably isomorphic cut. (iii) Q is not sententially equivalent with $$\mathsf{PA}^-$$ . This tells us that we cannot do much better than mutual faithful interpretability as a measure of sameness of Q and $$\mathsf{PA}^-$$ . We briefly consider the idea of characterizing Q as the minimal-in-some-sense theory of some kind modulo some equivalence relation. We show that at least one possible road towards this aim is closed. [ABSTRACT FROM AUTHOR]

In this paper, we show that the hyperstability of the general linear equation recently proved by Piszczek (Aequationes Math 88:163–168, 2014) is a direct consequence of that of the Cauchy equation proved earlier by Brzdȩk (Acta Math Hung 141:58–67, 2013). [ABSTRACT FROM AUTHOR]

In this paper we prove optimal regularity for the convex envelope of supersolutions to general fully nonlinear elliptic equations with unbounded coefficients. More precisely, we deal with coefficients and right hand sides (RHS) in Lq with q ≥ n . This extends the result of Caffarelli on the C loc 1 , 1 regularity of the convex envelope of supersolutions of fully nonlinear elliptic equations with bounded RHS. Moreover, we also provide a regularity result with estimates for ω -semiconvex functions that are supersolutions to the same type of equations with unbounded RHS (i.e, RHS in L q , q ≥ n ). By a completely different method, our results here extend the recent regularity results obtained by Braga et al. (Adv Math 334:184–242, 2018) for q > n , as far as fully nonlinear PDEs are concerned. These results include, in particular, the apriori estimate obtained by Caffarelli et al. (Commun Pure Appl Math 38(2):209–252, 1985) on the modulus of continuity of the gradient of ω -semiconvex supersolutions (for linear equations and bounded RHS) that have a Hölder modulus of semiconvexity. [ABSTRACT FROM AUTHOR]

A one-sided limit order book is modeled as a noncooperative game for several players. Agents offer various quantities of an asset at different prices, competing to fulfill an incoming order, whose size is not known a priori. Players can have different payoff functions, reflecting different beliefs about the fundamental value of the asset and probability distribution of the random incoming order. In a previous paper, the existence of a Nash equilibrium was established by means of a fixed point argument. The main issue discussed in the present paper is whether this equilibrium can be obtained from the unique solution to a two-point boundary value problem, for a suitable system of discontinuous ordinary differential equations. Some additional assumptions are introduced, which yield a positive answer. In particular, this is true when there are exactly two players, or when all players assign the same exponential probability distribution to the incoming order. In both of these cases, we also prove that the Nash equilibrium is unique. A counterexample shows that these assumptions cannot be removed, in general. [ABSTRACT FROM AUTHOR]

This paper is concerned with the spreading (persistence) and vanishing (extinction) of a disease which is characterized by a diffusive SIRS model with a bilinear incidence rate and free boundary. Through discussing the dynamics of a free boundary problem of an SIRS model, the spreading of a disease is described. We get the sufficient conditions which ensure the disease spreading or vanishing. In addition, the estimate of the expanding speed is also given when the free boundaries extend to the whole $$\mathbb {R}$$ . [ABSTRACT FROM AUTHOR]

We establish a logarithmic-type rate of convergence for the homogenization of fully nonlinear uniformly elliptic second-order pde in strongly mixing media with similar, i.e., logarithmic, decorrelation rate. The proof consists of two major steps. The first, which is actually the only place in the paper where probability plays a role, establishes the rate for special (quadratic) data using the methodology developed by the authors and Wang to study the homogenization of nonlinear uniformly elliptic pde in general stationary ergodic random media. The second is a general argument, based on the new notion of δ-viscosity solutions which is introduced in this paper, that shows that rates known for quadratic can be extended to general data. As an application of this we also obtain here rates of convergence for the homogenization in periodic and almost periodic environments. The former is algebraic while the latter depends on the particular equation. [ABSTRACT FROM AUTHOR]

Cholak, Groszek and Slaman proved in J Symb Log 66:881–901, 2001 that there is a nonzero computably enumerable (c.e.) degree cupping every low c.e. degree to a low c.e. degree. In the same paper, they pointed out that every nonzero c.e. degree can cup a low2 c.e. degree to a nonlow2 degree. In Jockusch et al. (Trans Am Math Soc 356:2557–2568, 2004) improved the latter result by showing that every nonzero c.e. degree c is cuppable to a high c.e. degree by a low2 c.e. degree b. It is natural to ask in which subclass of low2 c.e. degrees can b in Jockusch et al. (Trans Am Math Soc 356:2557–2568, 2004) be located. Wu proved in Math Log Quart 50:189–201, 2004 that b can be cappable. We prove in this paper that b in Jockusch, Li and Yang’s result can be noncuppable, improving both Jockusch, Li and Yang, and Wu’s results. [ABSTRACT FROM AUTHOR]

We found a minor error in the proof of paper “Universal Alignment Probability Revisited” by S.Y. Lin and Y.C. Ho (J. Optim. Theory Appl. 113(2):399–407, ). In this note, we give a counterexample and explain the reason. We also show that the conclusion of that paper is still correct despite this minor error. A new proof of the conclusion is given. [ABSTRACT FROM AUTHOR]

A major source of complexity in the mathematical modelling of an angiogenic process derives from the strong coupling of the kinetic parameters of the relevant stochastic branching-and-growth of the capillary network with a family of interacting underlying fields. The aim of this paper is to propose a novel mathematical approach for reducing complexity by (locally) averaging the stochastic cell, or vessel densities in the evolution equations of the underlying fields, at the mesoscale, while keeping stochasticity at lower scales, possibly at the level of individual cells or vessels. This method leads to models which are known as hybrid models. In this paper, as a working example, we apply our method to a simplified stochastic geometric model, inspired by the relevant literature, for a spatially distributed angiogenic process. The branching mechanism of blood vessels is modelled as a stochastic marked counting process describing the branching of new tips, while the network of vessels is modelled as the union of the trajectories developed by tips, according to a system of stochastic differential equations à la Langevin. [ABSTRACT FROM AUTHOR]

In this paper we are interested in the global existence and large time behavior of solutions to the initial- boundary value problem for sub critical nonlinear dissipative equations (Multiple line equation(s) cannot be represented in ASCII text) where the nonlinear term N(u, ux) depends on the unknown function u and its derivative ux and satisfy the estimate (Multiple line equation(s) cannot be represented in ASCII text)The linear operator IK(u) is defined as follows (Multiple line equation(s) cannot be represented in ASCII text) where the constants an, am ϵ R, n, m are integers, m > n. The aim of this paper is to prove the global existence of solutions to the initial-boundary value Problem (1). We find the main term of the asymptotic representation of solutions in sub critical case, when the nonlinear term of equation has the time decay rate less then that of the linear terms. Also we give some general approach to obtain global existence of solution of initial-boundary value problem in sub critical case and elaborate general sufficient conditions to obtain asymptotic expansion of solution. [ABSTRACT FROM AUTHOR]

The paper is a continuation of the paper where the so-called coordinate and spectral equations were derived for finding the edge Green's functions on a branched surface having branch points of order two. The coefficients of those equations contain unknown constants. To find these constants, it is necessary to state restrictions for the solutions of the equations. After that, finding the unknown constants becomes possible, for example, by a numerical procedure of determining zeros of discrepancies. The paper is devoted to the statement of the problem of finding the unknown constants. As an example, the problem of scattering by two perpendicular half-lines is considered. As the result of using a rather subtle property of the spectral equation (symmetry associated with the reciprocity theorem), one can give a set of restrictions, in which the number of unknowns is equal to the number of restrictions. Bibliography: 2 titles. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of time-periodic weak solutions for the wave equation with homogeneous boundary conditions. This paper deals with the cases where a nonlinear term has a superlinear and sublinear growth. [ABSTRACT FROM AUTHOR]

Recently two papers [K. Jacobs, Phys. Rev. A 67, 030301(R) (2003); H. M. Wiseman and J. F. Ralph, New J. Physics 8, 90 (2006)] have derived a number of control strategies for rapid purification of qubits, optimized with respect to various goals. In the former paper the proof of optimality was not mathematically rigorous, while the latter gave only heuristic arguments for optimality. In this paper we provide rigorous proofs of optimality in all cases, by applying simple concepts from optimal control theory, including Bellman equations and verification theorems. [ABSTRACT FROM AUTHOR]

In earlier papers, for “large” (but otherwise unspecified) subsets A, B of Z p and for h( x) ∈ Z p [ x], Gyarmati studied the solvability of the equations a + b = h( x), resp. ab = h( x) with a ∈ A, b ∈ B, x ∈ Z p , and for large subsets A, B, C, D of Z p Sárközy showed the solvability of the equations a + b = cd, resp. ab + 1 = cd with a ∈ A, b ∈ B, c ∈ C, d ∈ D. In this series of papers equations of this type will be studied in finite fields. In particular, in Part I of the series we will prove the necessary character sum estimates of independent interest some of which generalize earlier results. [ABSTRACT FROM AUTHOR]

In a previous paper, we have described all reproductive general solutions of a Post equation, supposing that a general solution is known. In this paper we describe all general solutions of Post equation, supposing that a general solution of this equation is known (Theorem 6). As a special case we get the previous characterization of reproductive solutions and a similar result for Boolean equations (Theorem 9). [ABSTRACT FROM AUTHOR]

The paper consists of two parts. In the first part, we propose a procedure to estimate local errors of low order methods applied to solve initial value problems in ordinary differential equations (ODEs) and index 1 differential-algebraic equations (DAEs). Based on the idea of defect correction we develop local error estimates for the case when the problem data is only moderately smooth. Numerical experiments illustrate the performance of the mesh adaptation based on the error estimation developed in this paper. In the second part of the paper, we will consider the estimation of local errors in context of stochastic differential equations with small noise. [ABSTRACT FROM AUTHOR]

This paper introduces a “spectral observability condition” for a negative self-adjoint operator which is the key to proving the null-controllability of the semigroup that it generates, and to estimating the controllability cost over short times. It applies to the interior controllability of diffusions generated by powers greater than 1/2 of the Dirichlet Laplacian on manifolds, generalizing the heat flow. The critical fractional order 1/2 is optimal for a similar boundary controllability problem in dimension one. This is deduced from a subsidiary result of this paper, which draws consequences on the lack of controllability of some one-dimensional output systems from Müntz–Szász theorem on the closed span of sets of power functions. [ABSTRACT FROM AUTHOR]

In this paper we obtain sufficient conditions on the regularity of the weak solutions to guarantee conservation of the energy and the helicity for the incompressible Euler equations. The regularity of the weak solutions are measured in terms of the Triebel-Lizorkin type of norms, ...p,qs and the Besov norms ...p,qs. In particular, in the Besov space case, our results refine the previous ones due to Constantin-E-Titi (energy) and the author of this paper (helicity), where the regularity is measured by a special class of the Besov space norm ...p,∞s = ...ps, which is the Nikolskii space. We also obtain a sufficient regularity condition for the conservation of the Lp-norm of the temperature function in the weak solutions of the quasi-geostrophic equation. [ABSTRACT FROM AUTHOR]

We will show in this paper that if λ is very close to 1, then can be attained, where M is a compact–manifold with boundary. This result gives a counter–example to the conjecture of de Figueiredo and Ruf in their paper titled "On an inequality by Trudinger and Moser and related elliptic equations" ( Comm. Pure. Appl. Math., 55, 135–152, 2002). [ABSTRACT FROM AUTHOR]

In this paper, finite element methods for the problems of wave propagation in a fluid-saturated porous medium are discussed. The medium is composed of a porous elastic solid (soil, rock, etc.) saturated by a compressible viscous fluid (oil, water, etc.), and the fluid may flow relatively to the solid. Biot's lowfrequency dynamic equations are chosen to describe the problems mentioned above, with stress-given boundary conditions, ABGs (Absorbing Boundary Conditions) on artificial boundaries and conditions on interfaces between the fluid-saturated porous medium and elastic solids. In the paper, a new kind of discrete ABCs is presented, and a discrete-time Galerkin method are utilized for obtaining approximate solutions. The numerical results show that they both are effective. Two dilatational waves (fast wave P1 and slow wave P2) and one rotational wave (S wave) are clearly visible in the figures of computational results, which coincide with theoretical analysis very well. [ABSTRACT FROM AUTHOR]

In this paper, we study the behavior of solutions of a semilinear elliptic equation in the exterior of a compact set as $|x|\backslash to\; \backslash nomathbreak\; \backslash infty$. Such equations were considered by many authors (for example, Kondrat'ev, Landis, Oleinik, Veron, etc.). In the present paper, we study the case in which in the equation contains lower terms. The coefficients of the lower terms are arbitrary bounded measurable functions. It is shown that the solutions of the equation tend to zero as $|x|\backslash to\; \backslash nomathbreak\; \backslash infty$. [ABSTRACT FROM AUTHOR]