A generalization of Menger’s Theorem

This paper generalizes one of the celebrated results in Graph Theory due to Karl. A. Menger (1927), which plays a crucial role in many areas of flow and network theory. This paper also introduces and characterizes strength reducing sets of nodes and arcs in weighted graphs.     

A generalization of Chaplygin’s Reducibility Theorem

In this paper we study Chaplygin’s Reducibility Theorem and extend its applicability to nonholonomic systems with symmetry described by the Hamilton-Poincaré-d’Alembert equations in arbitrary degrees of freedom. As special cases we extract the extension of the Theorem to nonholonomic Chaplygin systems with nonabelian symmetry groups as well as Euler-Poincaré-Suslov systems in arbitrary degrees of freedom. In the latter case, we also extend the Hamiltonization Theorem to nonholonomic systems which do not possess an invariant measure. Lastly, we extend previous work on conditionally variational systems using the results above. We illustrate the results through various examples of well-known nonholonomic systems.     

A note on q-difference equations for Ramanujan’s integrals

The Ramanujan Journal - This short paper derives the relationship between solutions of q-difference equations and generating functions for q-orthogonal polynomials. The key of the method is to...     

A note on Peano’s Theorem on time scales

In this paper we investigate the dynamic Cauchy problem in Banach spaces. We check how dense a time scale must be in such a way that Peano’s Theorem holds and we present a counterexample to Peano’s Theorem on a time scale with only one right dense point.     

A generalization of Lehmann’s Theorem on the comparison of uniform location experiments

Summary We generalize a result of Lehmann on the comparison of location experiments with uniform distributions on intervals. We compare a location experiment consisting of uniform distributions on parallelepipeds with a location experiment consisting of uniform distributions on convex bodies. We show that the first experiment can only be more informative than the second one if the convex bodies in the second experiment are themselves parallelepipeds. Further we show that the length of the edges of these parallelepipeds must fulfill a condition similar to the condition on the length of the intervals in Lehmanns result.     

A generalization of Schmidt’s Theorem on groups with all subgroups nilpotent

According to Schmidt’s Theorem a finite group whose proper subgroups are all nilpotent (or a finite group without non-nilpotent proper subgroups) is solvable. In this paper we prove that every finite group with less than 22 non-nilpotent subgroups is solvable and that this estimate is sharp.     

A note on Rotfel’d Theorem in a sector

The main purpose of this work is to present a better bound than the recent result given by Zhang [Linear Multilinear Algebra. 2015;63:2511–2517].     

Jacobi’s bound for systems of algebraic differential equations

This review paper is devoted to the Jacobi bound for systems of partial differential polynomials. We prove the conjecture for the system of n partial differential equations in n differential variables which are independent over a prime differential ideal \mathfrakp\mathfrak{p}. On the one hand, this generalizes our result about the Jacobi bound for ordinary differential polynomials independent over a prime differential ideal \mathfrakp\mathfrak{p} and, on the other hand, the result by Tomasovic, who proved the Jacobi bound for linear partial differential polynomials.     

A generalization of Knopp’s Observation on Ramanujan’s tau-function

The Ramanujan Journal - We establish a vast generalization of an observation made by Marvin Knopp half a century ago concerning the nonvanishing of Ramanujan’s tau-function.     

A note on Chudnovsky?s Fuchsian equations

We show that four exceptional Fuchsian equations, each determined by the four parabolic singularities, known as the Chudnovsky equations, are transformed into each other by algebraic transformations. We describe equivalence of these equations and their counterparts on tori. The latters are the Fuchsian equations on elliptic curves and their equivalence is characterized by transcendental transformations which are represented explicitly in terms of elliptic and theta functions.     

A note on Jensen’s inequality for BSDEs

Under the Lipschitz assumption and square integrable assumption on g, Jiang proved that Jensen＇s inequality for BSDEs with generator g holds in general if and only if g is independent of y, g is super homogenous in z and g（t, 0） = 0, a.s., a.e.. In this paper, based on Jiang＇s results, under the same assumptions as Jiang＇s, we investigate the necessary and sufficient condition on g under which Jensen＇s inequality for BSDEs with generator g holds for some specific convex functions, which generalizes some known results on Jensen＇s inequality for BSDEs.     

On 2D generalization of Higuchi’s fractal dimension

We propose a new numerical method for calculating 2D fractal dimension (DF) of a surface. This method represents a generalization of Higuchi’s method for calculating fractal dimension of a planar curve. Using a family of Weierstrass–Mandelbrot functions, we construct Weierstrass–Mandelbrot surfaces in order to test exactness of our new numerical method. The 2D fractal analysis method was applied to the set of histological images collected during direct shoot organogenesis from leaf explants. The efficiency of the proposed method in differentiating phases of organogenesis is proved.     

A note on exact finite difference schemes for modified Lotka–Volterra differential equations

We construct a class of modified Lotka–Volterra ordinary differential equations (ODE’s) and show that a nonlinear change of the dependent variables transform them into a set of coupled, linear ODE’s. Using the latter equations, we calculate the corresponding exact finite difference schemes using a technique given by Mickens. Next, we show how to reconfigure these relations to obtain the exact finite difference representation of the original modified, nonlinear Lotka–Volterra ODE’s.     

A generalization of Bonnet’s theorem on Darboux surfaces

The well-known Bonnet theorem claims that, on a Darboux surface in three-dimensional Euclidean space, along each line of curvature, the corresponding principal curvature is proportional to the cube of another principal curvature. In the present paper, this theorem is generalized (with respect to dimension) to n-dimensional hypersurfaces of Euclidean spaces.     

A generalization of Fulton’s conjecture for arbitrary groups

We prove a generalization of Fulton??s conjecture which relates intersection theory on an arbitrary flag variety to invariant theory.