A generalization of the Mitchell Lemma: The Ulmer Theorem and the Gabriel–Popescu Theorem revisited

We prove a generalization of the Mitchell Lemma, and we show that it is a key lemma that can be used in order to deduce in a unified easier way several important results. Thus, the Ulmer Theorem, the generalized Gabriel–Popescu Theorem and the generalized Takeuchi Lemma are all consequences of the generalized Mitchell Lemma.     

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 the Runge–Kutta iteration

Iterative solvers in combination with multi-grid have been used extensively to solve large algebraic systems. One of the best known is the Runge–Kutta iteration. We show that a generally used formulation [A. Jameson, Numerical solution of the Euler equations for compressible inviscid fluids, in: F. Angrand, A. Dervieux, J.A. Désidéri, R. Glowinski (Eds.), Numerical Methods for the Euler Equations of Fluid Dynamics, SIAM, Philadelphia, 1985, pp. 199–245] does not allow to form all possible polynomial transmittance functions and we propose a new formulation to remedy this, without using an excessive number of coefficients.     

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 generalization of the Hall–Witt identity

Let Y → X be a finite normal cover of a wedge of n ≥ 3 circles. We prove that for any nonzero v ∈ H ₁(Y; Q) there exists a lift \(\widetilde F\) to Y of a basepoint-preserving homotopy equivalence F: X → X such that the set of iterates \(\left\{ {{{\widetilde F}^d}\left( v \right)} \right\}:d \in \mathbb{Z} \subseteq {H_1}\left( {Y,\mathbb{Q}} \right)\) is infinite. The main achievement of this paper is the use of representation theory to prove the existence of a purely topological object that seems to be inaccessible via topology.     

A generalization of the Shapley–Ichiishi result

The Shapley–Ichiishi result states that a game is convex if and only if the convex hull of marginal vectors equals the core. In this paper, we generalize this result by distinguishing equivalence classes of balanced games that share the same core structure. We then associate a system of linear inequalities with each equivalence class, and we show that the system defines the class. Application of this general theorem to the class of convex games yields an alternative proof of the Shapley–Ichiishi result. Other applications range from computation of stable sets in non-cooperative game theory to determination of classes of TU games on which the core correspondence is additive (even linear). For the case of convex games we prove that the theorem provides the minimal defining system of linear inequalities. An example shows that this is not necessarily true for other equivalence classes of balanced games.     

A generalization of the Dedekind–MacNeille completion

In this paper we introduce the notion of \(Z_{\delta }\)-continuity as a generalization of precontinuity, complete continuity and \(s_{2}\)-continuity, where Z is a subset selection. And for each poset P, a closure space \(Z^{c}_{\delta }(P)\) arises naturally. For any subset system Z, we define a new type of completion, called \(Z_{\delta }\)-completion, extending each poset P to a Z-complete poset. The main results are: (1) if a subset system Z is subset-hereditary, then \(cl_{Z}(\Psi (P))\), the Z-closure of all principal ideals \(\Psi (P)\) of poset P in \(Z^{c}_{\delta }(P)\), is a \(Z_{\delta }\)-completion of P and \(Z^{c}_{\delta }(P) \cong Z^{c}_{\delta }(cl_{Z}(\Psi (P)))\); (2) let Z be an HUL-system and P a \(Z_{\delta }\)-continuous poset, then the \(Z_{\delta }\)-completion of P is also \(Z_{\delta }\)-continuous, and a Z-complete poset L is a \(Z_{\delta }\)-completion of P iff P is an embedded \(Z_{\delta }\)-basis of L; (3) the Dedekind–MacNeille completion is a special case of the \(Z_{\delta }\)-completion.     

On the generalization of Faltings’ Annihilator Theorem

Let R be a commutative Noetherian ring, and let n be a non-negative integer. In this article, by using the theory of Gorenstein dimensions, it is shown that whenever R is a homomorphic image of a Noetherian Gorenstein ring, then the invariants ${\inf\{i \in \mathbb{N}_0|\, \rm{dim\, Supp}(\mathfrak{b}^t H_{\mathfrak{a}}^i(M)) \geq n\, \rm{for\, all}\, t \in \mathbb{N}_0\}}$ and ${\inf\{\lambda_{\mathfrak{a} R_{\mathfrak{p}}}^{\mathfrak{b} R_{\mathfrak{p}}}(M_{\mathfrak{p}})|\, \mathfrak{p} \in {\rm Spec} \, R\, \rm{and\, dim}\, R/ \mathfrak{p} \geq n\}}$ are equal, for every finitely generated R-module M and for all ideals ${\mathfrak{a}, \mathfrak{b}}$ of R with ${\mathfrak{b}\subseteq \mathfrak{a}}$ . This generalizes Faltings’ Annihilator Theorem (see [6]).     

A generalization of the Burkholder–Davis–Gundy inequalities

In this paper, by applying some improved inequalities, we extend the Burkholder–Davis–Gundy inequalities for α ∈ (0,1) to more general functions and submartingales. Moreover, a series of inequalities for a logarithmic function are also obtained correspondingly. Finally, we give an application to a stopped Brownian motion.     

A generalization of the Sherman–Morrison–Woodbury formula

In this paper, we develop conditions under which the Sherman–Morrison–Woodbury formula can be represented in the Moore–Penrose inverse and the generalized Drazin inverse forms. These results generalize the original Sherman–Morrison–Woodbury formula.     

A Generalization of the Gallai–Roy Theorem

 A well-known and essential result due to Roy ([4], 1967) and independently to Gallai ([3], 1968) is that if D is a digraph with chromatic number χ(D), then D contains a directed path of at least χ(D) vertices. We generalize this result by showing that if ψ(D) is the minimum value of the number of the vertices in a longest directed path starting from a vertex that is connected to every vertex of D, then χ(D) ≤ψ(D). For graphs, we give a positive answer to the following question of Fajtlowicz: if G is a graph with chromatic number χ(G), then for any proper coloring of G of χ(G) colors and for any vertex v∈V(G), there is a path P starting at v which represents all χ(G) colors. Received: May 20, 1999 Final version received: December 24, 1999     

A generalization of the Bombieri–Pila determinant method

The so-called determinant method was developed by Bombieri and Pila in 1989 for counting integral points of bounded height on affine plane curves. In this paper, we give a generalization of that method to varieties of higher dimension, yielding a proof of Heath-Brown’s “Theorem 14” by real-analytic considerations alone. Bibliography: 11 titles.     

A new generalization of the Pareto–geometric distribution

In this paper we introduce a new distribution called the beta Pareto–geometric. We provide a comprehensive treatment of the mathematical properties of the proposed distribution and derive expressions for its moment generating function and the rth generalized moment. We discuss estimation of the parameters by maximum likelihood and obtain the information matrix that is easily numerically determined. We also demonstrate its usefulness on a real data set.     

A generalization of the category O of Bernstein–Gelfand–Gelfand

In the study of simple modules over a simple complex Lie algebra, Bernstein, Gelfand and Gelfand introduced a category of modules which provides a natural setting for highest weight modules. In this note, we define a family of categories which generalizes the BGG category. We classify the simple modules for some of these categories. As a consequence we show that these categories are semisimple.     

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 Generalized Version of the Baker–Pixley Theorem

A famous theorem by Baker and Pixley states that the term functions of a finite algebra with a (d + 1)-ary near-unanimity term are precisely the functions under which all subalgebras of the algebra's dth power are closed. In this paper, we generalize the theorem to a completely abstract level. Indeed, we obtain a version of the theorem that is stated in purely category-theoretic terms, making it applicable in any concrete or abstract category. To motivate this rather abstract result, we also discuss some of its concrete applications.     

A Non-Autonomous Version of the Denjoy–Wolff Theorem

The aim of this work is to establish the celebrated Denjoy–Wolff Theorem in the context of generalized Loewner chains. In contrast with the classical situation where essentially convergence to a certain point in the closed unit disk is the unique possibility, several new dynamical phenomena appear in this framework. Indeed, ω-limits formed by suitable closed arcs of circumferences appear now as natural possibilities of asymptotic dynamical behavior.     

A counting proof of the Graham–Pollak Theorem

We give a counting based proof of the Graham–Pollak Theorem.     

A tropical proof of the Brill–Noether Theorem

We produce Brill–Noether general graphs in every genus, confirming a conjecture of Baker and giving a new proof of the Brill–Noether Theorem, due to Griffiths and Harris, over any algebraically closed field.     

A Nonlinear Extension of the Coale–Lopez Theorem

For a sequence (T _n)_nof nonlinear positive operators on a real Banach space, certain monotonicity conditions are specified under which asymptotic equality holds for the inhomogeneous iterates T _n T _n-1 T ₁ T ₀. As a consequence, weak ergodicity or even path stability holds in the norm-topology. Finally, a nonlinear extension of the Coale–Lopez theorem in demography is shown.