Related papers: On dummy variables of structure-preserving transfo…
An isomorphism between two hermitian unitals is proved, and used to treat isomorphisms of classical groups that are related to the isomorphism between certain simple real Lie algebras of types A and D (and rank 3).
Satisfiability is a classic problem in computational complexity theory, in which one wishes to determine whether an assignment of values to a collection of Boolean variables exists in which all of a collection of clauses composed of logical…
Most classical mechanical systems are based on dynamical variables whose values are real numbers. Energy conservation is then guaranteed if the dynamical equations are phrased in terms of a Hamiltonian function, which then leads to…
This paper is especially written for students and demonstrates the correct use of nominal and ordinal scaled variables in regression analysis by means of so-called dummy variables. We start out with examples of body mass index (BMI)…
In this paper we perform Lie group analysis of systems of partial differential equations which describe different cases of classical plasma equilibria, and find groups of transformations admitted by those equations in several important…
A class of variable coefficient (1+1)-dimensional nonlinear reaction-diffusion equations of the general form $f(x)u_t=(g(x)u^nu_x)_x+h(x)u^m$ is investigated. Different kinds of equivalence groups are constructed including ones with…
We discuss a general method by which a higher order difference equation on a group is transformed into an equivalent triangular system of two difference equations of lower orders. This breakdown into lower order equations is based on the…
The singularity structure of solutions of a class of Hamiltonian systems of ordinary differential equations in two dependent variables is studied. It is shown that for any solution, all movable singularities, obtained by analytic…
Using the method of equivariant moving frames, we present a procedure for constructing symmetry-preserving finite element methods for second-order ordinary differential equations. Using the method of lines, we then indicate how our…
Partial differential equations (PDEs) describing thermodynamically isolated systems typically possess conserved quantities (like mass, momentum, and energy) and dissipated quantities (like entropy). Preserving these conservation and…
By a theorem of Chevalley the image of a morphism of varieties is a constructible set. The algebraic version of this fact is usually stated as a result on "extension of specializations" or "lifting of prime ideals". We present a difference…
Group classification of a class of Benjamin-Bona-Mahony (BBM) equations with time dependent coefficients is carried out. Two equivalent lists of equations possessing Lie symmetry extensions are presented: up to point equivalence within the…
Motivated by positivity-, monotonicity-, and convexity preserving differential equations, we introduce a definition of shape preserving operator semigroups and analyze their fundamental properties. In particular, we prove that the class of…
A method to calculate the adjoint solution for a large class of partial differential equations is discussed. It differs from the known continuous and discrete adjoint, including automatic differentiation. Thus, it represents an alternative,…
We generalize the classical Lie results on a basis of differential invariants for a one-parameter group of local transformations to the case of arbitrary number of independent and dependent variables. It is proved that if universal…
We construct a derived variant of Emerton's eigenvarieties using the locally analytic representation theory of $p$-adic groups. The main innovations include comparison and exploitation of two homotopy equivalent completed complexes…
In combinatorics, the probabilistic method is a very powerful tool to prove the existence of combinatorial objects with interesting and useful properties. Explicit constructions of objects with such properties are often very difficult, or…
We briefly review two different methods of applying Lie group theory in the numerical solution of ordinary differential equations. On specific examples we show how the symmetry preserving discretization provides difference schemes for which…
Equivalence transformations play one of the important roles in continuum mechanics. These transformations reduce the original equations to simpler forms. One of the classes of nonlocal equivalence transformations is the class of reciprocal…
The $\varepsilon$-form of a system of differential equations for Feynman integrals has led to tremendeous progress in our abilities to compute Feynman integrals, as long as they fall into the class of multiple polylogarithms. It is…