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We consider the sixth-order convective-viscous Cahn-Hilliard equation, different from the standard fourth-order Cahn-Hilliard equation due to the modified expression for the thermodynamic potential. In such modified thermodynamic potential the coefficient at the square gradient term is order-parameter-dependent. It also contains the square of the Laplacian. This results in a sixth-order differential equation and additional nonlinear terms in the equation. We obtained several exact static- and traveling wave solutions and studied the dependence of solutions on the parameters of the system.

We compute the gravitational interaction of two coalescing compact objects at sixth post-Newtonian order in the static limit, employing the diagrammatic approach within the effective field theory framework of General Relativity. The calculation requires the evaluation of six-loop Feynman diagrams that are mapped onto two-point integrals with a gauge-theory-like structure, which are computed here for the first time. The resulting seventh-order contribution in Newton's constant is finite in three space dimensions. This result provides the most technically demanding missing ingredient for the determination of the conservative dynamics of the gravitational two-body system at sixth post-Newtonian order.

This work investigates the well-posedness and optimal control of a sixth-order Cahn-Hilliard equation, a higher-order variant of the celebrated and well-established Cahn-Hilliard equation. The equation is endowed with a source term, where the control variable enters as a distributed mass regulator. The inclusion of additional spatial derivatives in the sixth-order formulation enables the model to capture curvature effects, leading to a more accurate depiction of isothermal phase separation dynamics in complex materials systems. We provide a well-posedness result for the aforementioned system when the corresponding nonlinearity of double-well shape is regular and then analyze a corresponding optimal control problem. For the latter, existence of optimal controls is established, and the first-order necessary optimality conditions are characterized via a suitable variational inequality. These results aim at contributing to improve the understanding of the mathematical properties and control aspects of the sixth-order Cahn-Hilliard equation, offering potential applications in the design and optimization of materials with tailored microstructures and properties.

For each of the forty-eight exceptional algebraic solutions $u(x)$ of the sixth equation of Painlevé, we build the algebraic curve $P(u,x)=0$ of a degree conjectured to be minimal, then we give an optimal parametric representation of it. This degree is equal to the number of branches, except for fifteen solutions.

This work explores the solvability of a sixth-order Cahn--Hilliard equation with an inertial term, which serves as a relaxation of a higher-order variant of the classical Cahn--Hilliard equation. The equation includes a source term that disrupts the conservation of the mean value of the order parameter. The incorporation of additional spatial derivatives allows the model to account for curvature effects, leading to a more precise representation of isothermal phase separation dynamics. We establish the existence of a weak solution for the associated initial and boundary value problem under the assumption that the double-well-type nonlinearity is globally defined. Additionally, we derive uniform stability estimates, which enable us to demonstrate that any family of solutions satisfying these estimates converges in a suitable topology to the unique solution of the limiting problem as the relaxation parameter approaches zero. Furthermore, we provide an error estimate for specific norms of the difference between solutions in terms of the relaxation parameter.

We prove nonuniqueness results for constant sixth order $Q$-metrics on complete locally conformally flat $n$-dimensional Riemannian manifolds with $n\geqslant 7$. More precisely, assuming a positive Green function exists for the sixth order GJMS operator, our objective is two-fold. First, we use a classical bifurcation technique to prove that there exists infinitely many constant $Q$-curvature metrics on $\mathbb{S}^1\times\mathbb{S}^{n-1}$. As a by-product, we find the sixth order Yamabe invariant on this product manifold can be arbitrarily close to that of the round dimensional sphere, generalizing a result of Schoen about the classical Yamabe invariant. Second, when the underlying manifold is noncompact, we apply a bifurcation technique on Riemannian covering to construct infinitely many complete metrics with constant sixth order $Q$-curvature conformal to $\mathbb{S}^{n_1} \times \mathbb{R}^{n_2}$ or $\mathbb{S}^{n_1} \times \mathbb{H}^{n_2}$, where $n_1+n_2\geqslant 7$. Consequently, we obtain infinitely many solutions to the singular constant GJMS equation on round spheres $\mathbb{S}^n\setminus \mathbb{S}^k$ blowing up along a minimal equatorial subsphere with $0 \leqslant k

In this paper, we investigate a sixth order elliptic equation with the simply supported boundary conditions in a polygonal domain. We propose a new method that decouples the sixth order problem into a system of second order equations. Unlike the direct decomposition, which yields three Poisson problems but is restricted to polygonal domains with the largest interior angle no more than $π/{2}$, we rigorously analyze and construct extra Poisson problems to confine the solution into the same function space as that of the original sixth order problem. Consequently, the proposed method can be applied to general polygonal domains. In turn, we also present a $C^0$ finite element algorithm to discretize the new resulting system and establish optimal error estimates for the numerical solution on quasi-uniform meshes. Finally, numerical experiments are performed to validate the theoretical findings.

This EPTCS volume contains the papers from the Sixth International Workshop on Formal Methods for Autonomous Systems (FMAS 2024), which was held between the 11th and 13th of November 2024. FMAS 2024 was co-located with 19th International Conference on integrated Formal Methods (iFM'24), hosted by the University of Manchester in the United Kingdom, in the University of Manchester's Core Technology Facility.

We consider the Riemann-Hilbert correspondence associated with the $q$-difference sixth Painlevé equation in the crystal limit, i.e. $q\rightarrow 0$, and show two main results. First, the limit of this generically highly transcendental mapping is shown to exist. Second, we show that the limiting map is bi-rational and describe it explicitly.

This paper presents new six solutions for sixth degree polynomial equation in general forms basing on new theorems, where the possibility to calculate the six roots of any sixth degree equation nearly simultaneously. The proposed roots for sixth degree polynomials in this paper are structured basing on new proposed solutions for quartic polynomial equations, which we developed in order to reduce the expression of any sixth degree polynomial to an expression of fourth degree polynomial.

The high-order cumulants of conserved charges are suggested to be sensitive observables to search for the critical point of Quantum Chromodynamics (QCD). This has been calculated to the sixth order in experiments. Corresponding theoretical studies on the sixth order cumulant are necessary. Based on the universality of the critical behavior, we study the temperature dependence of the sixth order cumulant of the order parameter using the parametric representation of the three-dimensional Ising model, which is expected to be in the same universality class as QCD. The density plot of the sign of the sixth order cumulant is shown on the temperature and external magnetic field plane. We found that at non-zero external magnetic field, when the critical point is approached from the crossover side, the sixth order cumulant has a negative valley. The width of the negative valley narrows with decreasing external field. Qualitatively, the trend is similar to the result of Monte Carlo simulation on a finite-size system. Quantitatively, the temperature of the sign change is different. Through Monte Carlo simulation of the Ising model, we calculated the sixth order cumulant of different sizes of

The sixth Painlevé equation is a basic equation among the non-linear differential equations with three fixed singularities, corresponding to Gauss's hypergeometric differential equation among the linear differential equations. It is known that 2nd order Fuchsian differential equations with three singular points are reduced to the hypergeometric differential equations. Similarly, for nonlinear differential equations, we would like to determine the equation from the local behavior around the three singularities. In this paper, the sixth Painlevé equation is derived by imposing the condition that it is of type (H) at each three singular points for the homogeneous quadratic 4th-order differential equation.

We classify entire positive singular solutions to a family of critical sixth order equations in the punctured space with a non-removable singularity at the origin. More precisely, we show that when the origin is a non-removable singularity, solutions are given by a singular radial factor times a periodic solution to a sixth order IVP with constant coefficients. On the technical level, we combine integral sliding methods and qualitative analysis of ODEs, based on a conservation of energy result, to perform a topological two-parameter shooting technique. We first use the integral representation of our equation to run a moving spheres technique, which proves that solutions are radially symmetric with respect to the origin. Thus, in Emden--Fowler coordinates, we can reduce our problem to the study of an sixth order autonomous ODE with constant coefficients. The main heuristics behind our arguments is that since all the indicial roots of the ODE operator are positive, it can be decomposed into the composition of three second order operators satisfying a comparison principle. This allows us to define a Hamiltonian energy which is conserved along solutions, from which we extract their qua

This is the proceedings of the Sixth International Workshop on Languages for Modelling Variability (MODEVAR 2024) which was held at Bern, Switzerland, February 06th 2024.

Introduction to the special issue of Phil. Trans. R. Soc. A 376, 2018, `Hilbert's Sixth Problem'. The essence of the Sixth Problem is discussed and the content of this issue is introduced. In 1900, David Hilbert presented 23 problems for the advancement of mathematical science. Hilbert's Sixth Problem proposed the expansion of the axiomatic method outside of mathematics, in physics and beyond. Its title was shocking: "Mathematical Treatment of the Axioms of Physics." Axioms of physics did not exist and were not expected. During further explanation, Hilbert specified this problem with special focus on probability and "the limiting processes, ... which lead from the atomistic view to the laws of motion of continua". The programmatic call was formulated "to treat, by means of axioms, those physical sciences in which already today mathematics plays an important part." This issue presents a modern slice of the work on the Sixth Problem, from quantum probability to fluid dynamics and machine learning, and from review of solid mathematical and physical results to opinion pieces with new ambitious ideas. Some expectations were broken: The continuum limit of atomistic kinetics may differ fr

In this paper, we determine the sixth moment of the determinant of an asymmetric $n \times n$ random matrix where the entries are drawn independently from an arbitrary distribution $Ω$ with mean $0$. Furthermore, we derive the asymptotic behavior of the sixth moment of the determinant as the size of the matrix tends to infinity.

We study dynamics of solutions in the initial value space of the sixth Painlevé equation as the independent variable approaches zero. Our main results describe the repeller set, show that the number of poles and zeroes of general solutions is unbounded, and that the complex limit set of each solution exists and is compact and connected.

Hardy and Littlewood initiated the study of the $2k$-th moments of the Riemann zeta function on the critical line. In 1918 Hardy and Littlewood established an asymptotic formula for the second moment and in 1926 Ingham established an asymptotic formula for the fourth moment. Since then no other moments have been asymptotically evaluated. In this article we study the sixth moment of the zeta function on the critical line. We show that a conjectural formula for a certain family of ternary additive divisor sums implies an asymptotic formula with power savings error term for the sixth moment of the Riemann zeta function on the critical line. This provides a rigorous proof for a heuristic argument of Conrey and Gonek. Furthermore, this gives some evidence towards a conjecture of Conrey, Keating, Farmer, Rubinstein, and Snaith on shifted moments of the Riemann zeta function. In addition, this improves on a theorem of Ivic, who obtained an upper bound for the the sixth moment of the zeta function, based on the assumption of a conjectural formula for correlation sums of the triple divisor function.

Using the method of asymptotic sum rules we estimated the size of $O(m_s p_π^4)$ and $O(m_s^2 p_π^2)$ corrections to $πK$ scattering amplitude in large $N_c$ limit. These corrections arise from the sixth order effective chiral lagrangian (EChL). Our method enables us to estimate the corresponding terms of the sixth order EChL in leading order of $1/N_c$ expansion in model independent way. We found that the corrections numerically are suppressed in spite of naive expectation of 30--35\%. Our estimation gives the value of these corrections about 5--10\% .