Related papers: The general linear equation on open connected sets

Solutions of nonlinear functional equations are generally not expressed as a finite number of combinations and compositions of elementary and known special functions. One of the approaches to study them is, firstly, to find formal solutions…

We study forms $I=(f_1,\ldots,f_r)$, $\deg f_i=d_i$, in $F$ which is the free associative algebra $k\langle x_1,\ldots,x_n\rangle$ or the polynomial ring $k[x_1,\ldots,x_n]$, where $k$ is a field and $\deg x_i=1$ for all $i$. We say that…

The long root geometry $A_{n,\{1,n\}}(\mathbb{K})$ for the special linear group $\mathrm{SL}(n+1,\mathbb{K})$ admits an embedding in the (projective space of) the vector space of the traceless square matrices of order $n+1$ with entries in…

We prove that if a level set of a degree $n$ general inverse $\sigma_k$ equation $f(\lambda_1, \cdots, \lambda_n) = \lambda_1 \cdots \lambda_n - \sum_{k = 0}^{n-1} c_k \sigma_k(\lambda) = 0$ is contained in $q + \Gamma_n$ for some $q \in…

Let $G=(V,E)$ be a connected finite graph and $C(V)$ be the set of functions defined on $V$. Let $\Delta_p$ be the discrete $p$-Laplacian on $G$ with $p>1$ and $L=\Delta_p-k$, where $k\in C(V)$ is positive everywhere. Consider the operator…

Assume $n\geq 2$. Consider the elementary symmetric polynomials $e_k(y_1,y_2,\ldots, y_n)$ and denote by $E_0,E_1,\ldots,E_{n-1}$ the elementary symmetric polynomials in reverse order \begin{align*}…

Motivated by the Cauchy-Davenport theorem for sumsets, and its interpretation in terms of Cayley graphs, we prove the following main result : There is a universal constant e > 0 such that, if G is a connected, regular graph on n vertices,…

We prove that if a linear equation, whose coefficients are continuous rational functions on a nonsingular real algebraic surface, has a continuous solution, then it also has a continuous rational solution. This is known to fail in higher…

A parametric version of Brouwer's Fixed Point Theorem, which is proven using the fixed-point index, states that for every continuous mapping $f : (X \times Y) \to Y$, where $X$ is nonempty, compact, and connected subset of a Hausdorff…

The author presents the generalized Stokes theorem for R-linear forms on Lie algebroids (which can be non-local). We apply the Stokes formula on forms to prove that two homotopic homomorphisms of Lie algebroids implies the existence of a…

Consider an ordinary differential equation which has a Lax pair representation A'(x)= [A(x),B(x)], where A(x) is a matrix polynomial with a fixed regular leading coefficient and the matrix B(x) depends only onA(x). Such an equation can be…

Two natural extensions of Jensen's functional equation on the real line are the equations $f(xy)+f(xy^{-1}) = 2f(x)$ and $f(xy)+f(y^{-1}x) = 2f(x)$, where $f$ is a map from a multiplicative group $G$ into an abelian additive group $H$. In a…

We prove a general categorical theorem that enables us to state that under certain conditions, the range of a functor is large. As an application, we prove various results of which the following is a prototype: If every diagram, indexed by…

A linear connection on a Finsler manifold is called compatible to the Finsler function if its parallel transports preserve the Finslerian length of tangent vectors. Generalized Berwald manifolds are Finsler manifolds equipped with a…

This paper is devoted to finding the general solutions of the functional equation $\sumin \sumjm h(p_iq_j)=\sumin h(p_i)+\sumjm k_j(q_j)+\lambda\sumin h(p_i)\sumjm k_j(q_j)$ valid for all complete probability distributions…

We deal with linear programming problems involving absolute values in their formulations, so that they are no more expressible as standard linear programs. The presence of absolute values causes the problems to be nonconvex and nonsmooth,…

Let $F$ be an ordered topological vector space (over $\mathbb{R}$) whose positive cone $F_+$ is weakly closed, and let $E \subseteq F$ be a subspace. We prove that the set of positive continuous linear functionals on $E$ that can be…

We derive a generalized Stokes' theorem, valid in any dimension and for arbitrary loops, even if self intersecting or knotted. The generalized theorem does not involve an auxiliary surface, but inherits a higher rank gauge symmetry from the…

In the paper we investigate the continuity properties of the mapping $\Phi$ which sends any non-empty compact connected hv-convex planar set $K$ to the associated generalized conic function $f_K$. The function $f_K$ measures the average…

We show that the Generalized Vanishing Conjecture $$\forall_{m \ge 1} [\Lam^m f^m = 0] \Longrightarrow \forall_{m \gg 0} [\Lam^m (g f^m) = 0]$$ for a fixed differential operator $\Lam \in k[\partial]$ follows from a special case of it,…