We consider boundary-value problems for differential equations of second order containing a Brownian motion (random perturbation) and a small parameter and prove a special existence and uniqueness theorem for random solutions. We study the asymptotic behaviour of these solutions as the small parameter goes to zero and show the stochastic averaging theorem for such equations. We find the explicit limits for the solutions as the small parameter goes to zero.

Conjectures are given for Hilbert series related to polynomial invariants of finite general linear groups: one for invariants mod Frobenius powers of the irrelevant ideal and one for cofixed spaces of polynomials.

Game-theoretic p-Laplacian or normalized p-Laplacian operator is a version of classical variational p-Laplacian which was introduced recently in connection with stochastic games called Tug-of-War with noise (Peres et al. 2008, Tug-of-war with noise: A game-theoretic view of the p-laplacian. Duke Mathematical Journal145(1), 91–120). In this paper, we propose an adaptation and generalization of this operator on weighted graphs for 1 ≤ p ≤ ∞. This adaptation leads to a partial difference operator which is a combination between 1-Laplace, infinity-Laplace and 2-Laplace operators on graphs. Then we consider the Dirichlet problem associated to this operator and we prove the uniqueness and existence of the solution. We show that the solution leads to an iterative non-local average operator on graphs. Finally, we propose to use this operator as a unified framework for interpolation problems in signal processing on graphs, such as image processing and machine learning.

The Buckley–Leverett partial differential equation has long been used to model two-phase flow in porous media. In recent years, the PDE has been modified to include a rate-dependent capillary pressure constitutive equation, known as dynamic capillary pressure. Previous traveling wave analysis of the modified Buckley–Leverett equation uncovered non-classical solutions involving undercompressive shocks. More recently, thermodynamically constrained averaging theory (TCAT) has generalized the capillary pressure equation by including additional dependence on fluid properties. In this paper, the model and traveling wave analysis are updated to incorporate TCAT capillary pressure as a generalization of dynamic capillary pressure. Solutions of the corresponding Riemann problem are similar to previous results except in the physically relevant situation in which both phases are pure fluids. The results presented here shed new light on the nature of the interface between one pure fluid displacing another pure fluid, in accordance with TCAT.

Ill-posedness for the compressible Navier–Stokes equations has been proved by Chen et al. [On the ill-posedness of the compressible Navier–Stokes equations in the critical Besov spaces, Revista Mat. Iberoam.31 (2015), 1375–1402] in critical Besov space $L^{p}$$(p>6)$ framework. In this paper, we prove ill-posedness with the initial data satisfying

$$\begin{eqnarray}\displaystyle \Vert \unicode[STIX]{x1D70C}_{0}-\bar{\unicode[STIX]{x1D70C}}\Vert _{{\dot{B}}_{p,1}^{\frac{3}{p}}}\leqslant \unicode[STIX]{x1D6FF},\quad \Vert u_{0}\Vert _{{\dot{B}}_{6,1}^{-\frac{1}{2}}}\leqslant \unicode[STIX]{x1D6FF}. & & \displaystyle \nonumber\end{eqnarray}$$

$L(a)\unicode[STIX]{x1D6E5}u$

$u\cdot \unicode[STIX]{x1D6FB}u$

The maximal finite abelian subgroups, up to conjugation, of the simple algebraic group of type E 8 over an algebraically closed field of characteristic 0 are computed. This is equivalent to the determination of the fine gradings on the simple Lie algebra of type E 8 with trivial neutral homogeneous component. The Brauer invariant of the irreducible modules for graded semisimple Lie algebras plays a key role.

Given a non-negative integer n and a complete hereditary cotorsion triple , the notion of subcategories in an abelian category is introduced. It is proved that a virtually Gorenstein ring R is n-Gorenstein if and only if the subcategory of Gorenstein injective R-modules is with respect to the cotorsion triple , where stands for the subcategory of Gorenstein projectives. In the case when a subcategory of is closed under direct summands such that each object in admits a right -approximation, a Bazzoni characterization is given for to be . Finally, an Auslander–Reiten correspondence is established between the class of subcategories and that of certain subcategories of which are -coresolving covariantly finite and closed under direct summands.

Positive solutions of a Kirchhoff-type nonlinear elliptic equation with a non-local integral term on a bounded domain in ℝN , N ⩾ 1, are studied by using bifurcation theory. The parameter regions of existence, non-existence and uniqueness of positive solutions are characterized by the eigenvalues of a linear eigenvalue problem and a nonlinear eigenvalue problem. Local and global bifurcation diagrams of positive solutions for various parameter regions are obtained.

In this note semi-bounded self-adjoint extensions of symmetric operators are investigated with the help of the abstract notion of quasi boundary triples and their Weyl functions. The main purpose is to provide new sufficient conditions on the parameters in the boundary space to induce self-adjoint realizations, and to relate the decay of the Weyl function to estimates on the lower bound of the spectrum. The abstract results are illustrated with uniformly elliptic second-order partial differential equations on domains with non-compact boundaries.

We investigate the stability properties of positive steady-state solutions of semilinear initial–boundary-value problems with nonlinear boundary conditions. In particular, we employ a principle of linearized stability for this class of problems to prove sufficient conditions for the stability and instability of such solutions. These results shed some light on the combined effects of the reaction term and the boundary nonlinearity on stability properties. We also discuss various examples satisfying our hypotheses for stability results in dimension 1. In particular, we provide complete bifurcation curves for positive solutions for these examples.

On a sufficiently soft substrate, a resting fluid droplet will cause significant deformation of the substrate. This deformation is driven by a combination of capillary forces at the contact line and the fluid pressure at the solid surface. These forces are balanced at the surface by the solid traction stress induced by the substrate deformation. Young's Law, which predicts the equilibrium contact angle of the droplet, also indicates an a priori radial force balance for rigid substrates, but not necessarily for soft substrates that deform under loading. It remains an open question whether the contact line transmits a non-zero force tangent to the substrate surface in addition to the conventional normal (vertical) force. We present an analytic Fourier transform solution technique that includes general interfacial energy conditions, which govern the contact angle of a 2D droplet. This includes evaluating the effect of gravity on the droplet shape in order to determine the correct fluid pressure at the substrate surface for larger droplets. Importantly, we find that in order to avoid a strain singularity at the contact line under a non-zero tangential contact line force, it is necessary to include a previously neglected horizontal traction boundary condition. To quantify the effects of the contact line and identify key quantities that will be experimentally accessible for testing the model, we evaluate solutions for the substrate surface displacement field as a function of Poisson's ratio and zero/non-zero tangential contact line forces.

In this paper, we consider how to express an Iwahori–Whittaker function through Demazure characters. Under some interesting combinatorial conditions, we obtain an explicit formula and thereby a generalization of the Casselman–Shalika formula. Under the same conditions, we compute the transition matrix between two natural bases for the space of Iwahori fixed vectors of an induced representation of a $p$-adic group; this corrects a result of Bump–Nakasuji.

For a fat sub-Riemannian structure, we introduce three canonical Ricci curvatures in the sense of Agrachev–Zelenko–Li. Under appropriate bounds we prove comparison theorems for conjugate lengths, Bonnet–Myers type results and Laplacian comparison theorems for the intrinsic sub-Laplacian. As an application, we consider the sub-Riemannian structure of 3-Sasakian manifolds, for which we provide explicit curvature formulas. We prove that any complete 3-Sasakian structure of dimension $4d+3$, with $d>1$, has sub-Riemannian diameter bounded by $\unicode[STIX]{x1D70B}$. When $d=1$, a similar statement holds under additional Ricci bounds. These results are sharp for the natural sub-Riemannian structure on $\mathbb{S}^{4d+3}$ of the quaternionic Hopf fibrations:

$$\begin{eqnarray}\mathbb{S}^{3}{\hookrightarrow}\mathbb{S}^{4d+3}\rightarrow \mathbb{HP}^{d},\end{eqnarray}$$

$\unicode[STIX]{x1D70B}$

$d\geqslant 1$

This paper proposes a neural network architecture for solving systems of non-linear equations. A back propagation algorithm is applied to solve the problem, using an adaptive learning rate procedure, based on the minimization of the mean squared error function defined by the system, as well as the network activation function, which can be linear or non-linear. The results obtained are compared with some of the standard global optimization techniques that are used for solving non-linear equations systems. The method was tested with some well-known and difficult applications (such as Gauss–Legendre 2-point formula for numerical integration, chemical equilibrium application, kinematic application, neuropsychology application, combustion application and interval arithmetic benchmark) in order to evaluate the performance of the new approach. Empirical results reveal that the proposed method is characterized by fast convergence and is able to deal with high-dimensional equations systems.

We give a minimum degree condition sufficient to ensure the existence of a fractional Kr -decomposition in a balanced r-partite graph (subject to some further simple necessary conditions). This generalizes the non-partite problem studied recently by Barber, Lo, Kühn, Osthus and the author, and the 3-partite fractional K 3-decomposition problem studied recently by Bowditch and Dukes. Combining our result with recent work by Barber, Kühn, Lo, Osthus and Taylor, this gives a minimum degree condition sufficient to ensure the existence of a (non-fractional) Kr -decomposition in a balanced r-partite graph (subject to the same simple necessary conditions).

Let G be a finite D-quasirandom group and A ⊂ G k a δ-dense subset. Then the density of the set of side lengths g of corners

$$\{(a_{1},\dotsc,a_{k}),(ga_{1},a_{2},\dotsc,a_{k}),\dotsc,(ga_{1},\dotsc,ga_{k})\} \subset A$$

For an orientation H with n vertices, let T(H) denote the maximum possible number of labelled copies of H in an n-vertex tournament. It is easily seen that T(H) ≥ n!/2e(H), as the latter is the expected number of such copies in a random tournament. For n odd, let R(H) denote the maximum possible number of labelled copies of H in an n-vertex regular tournament. In fact, Adler, Alon and Ross proved that for H=C n , the directed Hamilton cycle, T(C n ) ≥ (e−o(1))n!/2n , and it was observed by Alon that already R(C n ) ≥ (e−o(1))n!/2n . Similar results hold for the directed Hamilton path P n . In other words, for the Hamilton path and cycle, the lower bound derived from the expectation argument can be improved by a constant factor. In this paper we significantly extend these results, and prove that they hold for a larger family of orientations H which includes all bounded-degree Eulerian orientations and all bounded-degree balanced orientations, as well as many others. One corollary of our method is that for any fixed k, every k-regular orientation H with n vertices satisfies T(H) ≥ (e k −o(1))n!/2e(H), and in fact, for n odd, R(H) ≥ (e k −o(1))n!/2e(H).

We analyse the decay properties of the solution semigroup S(t) generated by the linear integrodifferential equation

where the operator A is strictly positive self-adjoint with A –1 not necessarily compact. The asymptotic stability of S(t) is investigated in terms of the dependence of the parameter γ ∈ ℝ. In particular, we show that S(t) is not exponentially stable when γ ≠ 1.

We consider a possibly anisotropic integrodifferential semilinear equation, driven by a non-decreasing nonlinearity. We prove that if the solution grows less than the order of the operator at infinity, then it must be affine (possibly constant).