Fatou's example

Examples due to P. Fatou and L. Biberbach show that Picard's little theorem cannot be extended to apply to holomorphic mappings of the complex space C_{n, n>1}. These examples are used to construct a pair of functionally independent entire functions of two complex variables such that the image of C² by the mapping realized by this pair does not contain any sphere. Here we give an example of the same type which implies a stronger assertion concerning the set of values not taken.Translated from Matematicheskie Zametki, Vol. 6, No. 4, pp. 437–441, October, 1969.We thank B. V. Shabat for suggesting this subject and for his valuable help in the work.     

On Krein's example

In his 1953 paper [Matem. Sbornik 33 (1953), 597-626] Mark Krein presented an example of a symmetric rank one perturbation of a self-adjoint operator such that for all values of the spectral parameter in the interior of the spectrum, the difference of the corresponding spectral projections is not trace class. In the present note it is shown that in the case in question this difference has simple Lebesgue spectrum filling in the interval and, therefore, the pair of the spectral projections is generic in the sense of Halmos but not Fredholm.

     

A generalization of Gruenhage's example

We construct, assuming the continuum hypothesis, an example of nonmetrizable n-dimensional Cantor manifold Xn(n∈N) with the following properties: 1) is hereditarily separable for all k∈N; 2) is perfectly normal for every k∈N; 3) the space F(Xn) is hereditarily normal for every seminormal functor F that preserves weights and one-to-one points and such that sp(F)={1,k}; in particular, and λ₃Xn are hereditarily normal. This example is a generalization of famous Gruenhage's example given in Gruenhage and Nyikos (1993) [4].     

Quitting games – An example

Quitting games are multi-player sequential games in which, at every stage, each player has the choice between continuing and quitting. The game ends as soon as at least one player chooses to quit; each player i then receives a payoff r _S ⁱ, which depends on the set S of players that did choose to quit. If the game never ends, the payoff to each player is zero.? We exhibit a four-player quitting game, where the “simplest” equilibrium is periodic with period two. We argue that this implies that all known methods to prove existence of an equilibrium payoff in multi-player stochastic games are therefore bound to fail in general, and provide some geometric intuition for this phenomenon. Received: October 2001     

An example of a rightq-ring

We show that Ivanov’s classification of indecomposable non-local rightq-rings is incomplete and provide a complete classification. Next, we correct and sharpen Byrd’s classification of rightq-rings. Partially supported by a grant from the National Science Council of Taiwan.     

An example in affine pi-rings

An example is given of a prime p.i. ringR finitely generated over a field, with a prime idealQ which is not finitely generated as a two-sided ideal ofR. Supported in part by the National Science Foundation.     

An example concerning fixed points

An example is given of a contractionT defined on a bounded closed convex subset of Hilbert space for which ((I+T)/2) ⁿ does not converge.     

A remark on Kulesza's example

In this note we simplify the proof of some properties of Kulesza's metric space with ind and Ind.

     

On Brunovský's example

The example due to Brunovský [Eq. (1)] is analyzed, and further paradoxical properties are exhibited.     

Sur un example de categorie presque-abelienne

Résumé  Si Λ est la catégorie des espaces localement convexes séparés et complets, nous montrons que la catégorie duale de Λ est une catégorie presque-abélienne et par suite, la catégorie Λ l’est aussi.

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An example of bad convex function

An example of a convex function having the gradient at each point (x, 0, ..., 0),x>0, which does not converge, whenx tends to zero, is given.The author would like to thank Professor Giannessi for his help, which led to an improved version of this paper.     

An example of L-fuzzy join space

On a generalized deMorgan lattice (X, ≤, ∨, ∧,′) we introduce a family of join hyperoperations * _p , parametrized by a parameterp εX. As a result we obtain a family of join spaces (X, * _p ). We show that: for everya,b εX the family {a*_pb} _pεX can be considered as thep-cuts of aL-fuzzy seta*b; in this manner we synthesize aL-fuzzy hyperoperation * which takes pairs fromX toL-fuzzy subsets ofX. We then show that (X, * _p ) is aL-fuzzy hypergroup (in the sense of Corsini) and can be considered as aL-fuzzy join space. Furthermore,a*b is aL-fuzzy interval for alla,b εX.     

A counter example for nilpotent groups

Let R be a ring with 1, I be a nilpotent subring of R (there exists a natural number n, such that In = 0), and I be generated by {xj |j ∈ J} as ring. Write U = 1 + I, and it is a nilpotent group with class ≤ n - 1. Let G be the subgroup of U which is generated by {1 + xj|j ∈ J}. The group constructed in this paper indicates that the nilpotency class of G can be less than that of U.     

An example of a nonregular semimonotoneQ-matrix

In their paper, Aganagic and Cottle gave necessary and sufficient conditions for aP ₀-matrix to be aQ-matrix. In his paper, Pang showed that the same characterization holds for anL-matrix.In this paper, we show that a similar characterization for anE ₀-matrix (semimonotone orL ₁-matrix) is not possible. This is done by providing a counterexample to the inclusionE ₀ Q R ₀, thus answering in the negative a question first posed by Pang.     

An N-dimensional version of Scarf's example

This paper presents a unification of the contributions of Scarf [Some examples of global instability of competitive equilibrium, Int. Econ. Rev. 1 (1960) 157–172] and Gale [A note on global instability of competitive equilibrium, Nav. Res. Logist. Q. 10 (1963) 81–87] by means of a single class of utility functions. The unification extends Scarf's three-commodity, three-consumer economy to an n-commodity, n-consumer economy. We find that the stability of the extended economy depends on whether the law of demand for exchange economies, i.e., downward sloping excess demand, holds or not. Moreover, it is seen that there is a transition from instability to stability as substitution effect increases or as the desire of each consumer for his own commodity decreases.     

A strange example on Property Np

 We exhibit an example of a line bundle M on a smooth complex projective variety Y s.t. M satisfies Property N ₁₀ , the 10-module of a minimal resolution of the ideal of the embedding of Y by M is nonzero and M ² does not satisfy Property N ₁₀ . Thus this is a completely convincing example showing that surprisingly it is not true that if a line bundle M satisfies Property N _p then any power of M satisfies Property N _p . We recall that in [Ru] we proved the following statement: if M is a line bundle on a smooth complex projective variety and M satisfies Property N _p then M ^s satisfies Property N _p if s≥p. Received: 5 March 2001