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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.

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 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 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 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.

Continuing the idea of Part I, we deal with more involved pseudogroup of transformations $\overline{x}=\varphi \left(x\right)$, $\overline{y}=L\left(x\right)y$, $\overline{z}=M\left(x\right)z,...$ applied to the first order differential equations including the underdetermined case (i.e. the Monge equation $y^{\text{'}}=f(x,y,z,z^{\text{'}})$) and certain differential equations with deviation (if $z=y\left(\xi \right(x\left)\right)$ is substituted). Our aim is to determine complete families of invariants resolving the equivalence problem and to clarify the largest possible symmetries. Together with Part I, this article may be regarded as an introduction into the...

The article concerns the symmetries of differential equations with short digressions to the underdetermined case and the relevant differential equations with delay. It may be regarded as an introduction into the method of moving frames relieved of the geometrical aspects: the stress is made on the technique of calculations employing only the most fundamental properties of differential forms. The present Part I is devoted to a single ordinary differential equation subjected to the change of the independent...