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For linear differential and functional-differential equations of the $n$-th order criteria of equivalence with respect to the pointwise transformation are derived.

The paper describes the general form of functional-differential equations of the first order with $m(m\ge 1)$ delays which allows nontrivial global transformations consisting of a change of the independent variable and of a nonvanishing factor. A functional equation $$f(t,uv,u_1{v}_1,...,u_m{v}_m)=f(x,v,v_1,...,v_m)g(t,x,u,u_1,...,u_m)u+h(t,x,u,u_1,...,u_m)v$$ for $u\ne 0$ is solved on $ℝ$ and a method of proof by J. Aczél is applied.

The paper describes the general form of an ordinary differential equation of an order $n+1$ $(n\ge 1)$ which allows a nontrivial global transformation consisting of the change of the independent variable and of a nonvanishing factor. A result given by J. Aczél is generalized. A functional equation of the form $$f\left(s,w_{00}{v}_0,...,\sum _{j=0}^n{w}_{nj}{v}_j\right)=\sum _{j=0}^n{w}_{n+1j}{v}_j+w_{n+1n+1}f(x,v,v_1,...,v_n),$$ where $w_{n+10}=h(s,x,x_1,u,u_1,...,u_n)$, $w_{n+11}=g(s,x,x_1,...,x_n,u,u_1,...,u_n)$ and $w_{ij}=a_{ij}(x_1,...,x_{i-j+1},u,u_1,...,u_{i-j})$ for the given functions $a_{ij}$ is solved on $ℝ$, $u\ne 0.$

The paper describes the general form of an ordinary differential equation of the order $n+1$ $(n\ge 1)$ which allows a nontrivial global transformation consisting of the change of the independent variable. A result given by J. Aczél is generalized. A functional equation of the form $$f\left(s,v,w_{11}{v}_1,...,\sum _{j=1}^n{w}_{nj}{v}_j\right)=\sum _{j=1}^n{w}_{n+1j}{v}_j+w_{n+1n+1}f(x,v,v_1,...,v_n),$$ where $w_{ij}=a_{ij}(x_1,...,x_{i-j+1})$ are given functions, $w_{n+11}=g(x,x_1,...,x_n)$, is solved on $ℝ.$

The paper describes the general form of an ordinary differential equation of the second order which allows a nontrivial global transformation consisting of the change of the independent variable and of a nonvanishing factor. A result given by J. Aczél is generalized. A functional equation of the form $$f(t,vy,wy+uvz)=f(x,y,z)u^{2}v+g(t,x,u,v,w)vz+h(t,x,u,v,w)y+2uwz$$ is solved on $ℝ$ for $y\ne 0$, $v\ne 0.$

The paper deals with a theoretical model of the Crowel-Alipanahi correlator. The model describes a new possible effect of the DLTS spectra-exponential and nonexponential transient capacitance, normal or anomalous spectra.

The paper deals with the higher-order ordinary differential equations and the analogous higher-order difference equations and compares the corresponding fundamental concepts. Important dissimilarities appear for the moving frame method.

The Routh reduction of cyclic variables in the Lagrange function and the Jacobi-Maupertuis principle of constant energy systems are generalized. The article deals with one-dimensional variational integral subject to differential constraints, the Lagrange variational problem, that admits the Lie group of symmetries. Reduction to the orbit space is investigated in the absolute sense relieved of all accidental structures. In particular, the widest possible coordinate-free approach to the underdetermined...

Some open problems appearing in the primary article on the symmetry reduction are solved. A new and quite simple coordinate-free definition of Poincaré-Cartan forms and the substance of divergence symmetries (quasisymmetries) are clarified. The unbeliavable uniqueness and therefore the global existence of Poincaré-Cartan forms without any uncertain multipliers for the Lagrange variational problems are worth extra mentioning.