Related papers: Solvability of equations in elementary functions
In the presence of a certain class of functions we show that there exists a smooth solution to Navier-Stokes equation. This solution entertains the property of being nonconvective. We introduce a definition for any possible solution to the…
Let $f$ be a real-valued function defined on $\mathbb{R}$, with $f(0) \neq 0$ and which is not constant in non empty open intervals. We prove the equations \begin{equation}\label{edif} \left\{ \begin{array}{rcll} (-\Delta )^{s}u & = & f(u),…
Over an algebraically closed field, we describe the affine varieties of solutions to the linear equations $a(xb)=c$ and $a(bx)=c$ over the split-octonions. We also determine the dimensions of the solution sets of arbitrary linear monomial…
We consider the differential equation \begin{align}\label{ab} -y'(x)+q(x)y(x)=f(x), \quad x \in \mathbb R, \end{align} where $f \in L_{p}(\mathbb R)$, $p\in [1,\infty)$, and $0\leq q \in L_{1}^{\rm loc}(\mathbb R)$,…
We prove a Green--Tao theorem for multiplicative functions.
A proof that the set of real numbers is denumerable is given.
Approximate solutions to functional evolution equations are constructed through a combination of series and conjugation methods, and relative errors are estimated. The methods are illustrated, both analytically and numerically, by…
We give complete, finite quasiequational axiomatisations for algebras of unary partial functions under the operations of composition, domain, antidomain, range and intersection. This completes the extensive programme of classifying algebras…
We study the problem of \emph{robust satisfiability} of systems of nonlinear equations, namely, whether for a given continuous function $f:\,K\to\mathbb{R}^n$ on a~finite simplicial complex $K$ and $\alpha>0$, it holds that each function…
We show that there are continuous, $W^{1,p}$ ($p<d-1$), incompressible vector fields for which uniqueness of solutions to the continuity equation fails.
We study multimodal logics over universally first-order definable classes of frames. We show that even for bimodal logics, there are universal Horn formulas that define set of frames such that the satisfiability problem is undecidable, even…
The purpose of this paper is to show that Non-Archimedean Mathematics (NAM), namely mathematics which uses infinite and infinitesimal numbers, is useful to model some Physical problems which cannot be described by the usual mathematics. The…
We give different proofs and prove new results on the non complete solvability of some systems of complex first order p.d.e.'s, especially related to the analysis on CR manifolds.
We prove that any mild solution in the Koch--Tataru space to the incompressible Navier--Stokes equation with initial data in $\mathrm{BMO}^{-1}$ is weak*-continuous in time, valued in $\mathrm{BMO}^{-1}$. We also show that the global mild…
We show that the class of unital $\mathrm{C}^*$-algebras is an elementary class in the language of operator systems. As a result, we have that there is a definable predicate in the language of operator systems that defines the…
The equation $v=v_0+\int_0^t(t-s)^{\lambda -1}v(s)ds$ is considered, $\lambda\neq 0,-1,-2...$ and $v_0$ is a smooth function rapidly decaying with all its derivatives. It is proved that the solution to this equation does exist, is unique…
In this paper, we consider the function field analogue of the Lehmer's totient problem. Let $p(x)\in\mathbb{F}_q[x]$ and $\varphi(q,p(x))$ be the Euler's totient function of $p(x)$ over $\mathbb{F}_q[x],$ where $\mathbb{F}_q$ is a finite…
We classify order $3$ linear difference operators over $\mathbb{C}(x)$ that are solvable in terms of lower order difference operators. To prove this result, we introduce the notion of absolute irreducibility for difference modules, and…
We prove the existence of non-smooth solutions to Special Lagrangian Equations in the non-convex case.
We show how to prove by means of the Lagrange inversion the limit of Arnol'd that $$ \lim_{x\to0}\frac{\sin(\tan x)-\tan(\sin x)}{\arcsin(\arctan x)-\arctan(\arcsin x)}=1\,. $$ In fact, we obtain a more general result in terms of formal…