Related papers: Symmetry nonintegrability for extended $K(m,n,p)$ …
Consider the KPP-type equation of the form $\Delta u+f(u)=0$, where $f:[0,1] \to \mathbb R_{+}$ is a concave function. We prove for arbitrary dimensions that there is no solution bounded in $(0,1)$. The significance of this result from the…
We give a complete point-symmetry classification of all third-order evolution equations of the form $u_t=F(t,x,u,u_x, u_{xx})u_{xxx}+G(t,x,u,u_x, u_{xx})$ which admit semi-simple symmetry algebras and extensions of these semi-simple Lie…
An eighth-order equation in (3+1)-dimension is studied for its integrability. Its symmetry group is shown to be infinite-dimensional and is checked for Virasoro like structure. The equation is shown not to have Painlev$\acute{\rm e}$…
We present an extension of the Hardy--Littlewood inequality for multilinear forms. More precisely, let $\mathbb{K}$ be the real or complex scalar field and $m,k$ be positive integers with $m\geq k\,$ and $n_{1},\dots ,n_{k}$ be positive…
Nonclassical symmetries of a class of generalized Huxley equations of form $u_t=u_{xx}+k(x)u^2(1-u)$ are found. More precisely, for the class under consideration we completely classify reduction operators with $\tau=1$ and give a wide…
We apply the technique of integrable extensions to the symmetry pseudo-group of the r-th mdKP equation. This gives another look on deriving known coverings and allows us to find new coverings for this equation.
We complete the Lie symmetry classification of scalar nth order, $n \geq 4$, ordinary differential equations by means of the symmetry Lie algebras they admit. It is known that there are three types of such equations depending upon the…
We study the non-invertible symmetries of class $\mathcal{S}$ theories obtained by compactifying the type $\mathfrak{a}_{p-1}$ 6d (2,0) theory on a genus $g$ Riemann surface with no punctures. After setting up the general framework, we…
If q : P -> M is a principal K-bundle over the compact manifold M, then any invariant symmetric V-valued bilinear form on the Lie algebra k of K defines a Lie algebra extension of the gauge algebra by a space of bundle-valued 1-forms modulo…
It is shown that the symmetry algebra of quantum superintegrable system can be always chosen to be u(N),N being the number of degrees of freedom.
Integrable dispersionless Kadomtsev-Petviashvili (KP) hierarchy of B type is considered. Addition formula for the $\tau$-function and conformally invariant equations for the dispersionless BKP (dBKP) hierarchy are derived. Symmetry…
We consider certain generalized binomial sums $\mathcal{S}_{(r,n)}(\ell)$ and discuss the nonintegrality of their values for integral parameters $n,r \geq 1$ and $\ell \in \mathbb{Z}$ in several cases using $p$-adic methods. In particular,…
We show an example of a non-symmetric integro-differential equation of order $\alpha$, for $\alpha \in (0,1)$, for which H\"older estimates do not hold even though the kernels are comparable to the fractional Laplacian.
It is well-known that six-dimensional superconformal field theories can be exploited to unravel interesting features of lower-dimensional theories obtained via compactifications. In this short note we discuss a new application of 6d (2,0)…
In prior work, the author has characterized the real numbers $a,b,c$ and $1\leq p,q,r<\infty $ such that the weighted Sobolev space $W_{\{a,b\}}^{(q,p)}(R^{N}\backslash \{0}):=\{u\in L_{loc}^{1}(R^{N}\backslash \{0}):|x|^{\frac{a}{q}}u\in…
Determining if an (1+1)-differential-difference equation is integrable or not (in the sense of possessing an infinite number of symmetries) can be reduced to the study of the dependence of the equation on the lattice points, according to…
We compute symmetry algebras of a system of two equations y^(k)=z^(l)=0, where 2<=k<l. It appears that there are many ways to convert such system of ODEs to an exterior differential system. They lead to different series of…
A connected graph $G$ with at least $2m+2n+2$ vertices is said to have property $E(m,n)$ if, for any two disjoint matchings $M$ and $N$ of size $m$ and $n$ respectively, $G$ has a perfect matching $F$ such that $M\subseteq F$ and $N\cap…
Generalized symmetries of the Einstein equations are infinitesimal transformations of the spacetime metric that formally map solutions of the Einstein equations to other solutions. The infinitesimal generators of these symmetries are…
We discuss Lie algebras of the Lie symmetry groups of two generically non-integrable equations in one temporal and two space dimensions arising in different contexts. The first is a generalization of the KP equation and contains 9 arbitrary…