A note on infinite transitive graphs

In [4] Jung and Watkins proved that for a connected infinite graph X either κ_∞(X) = ∞ holds or X is a strip, if Aut(X) contains a transitive abelian subgroup G. Here we prove the same result under weaker assumptions.     

A note on fragments of infinite graphs

Results involving automorphisms and fragments of infinite graphs are proved. In particular for a given fragmentC and a vertex-transitive subgroupG of the automorphism group of a connected graph there exists σ≠G such that σ[C] ⊂C. This proves the countable case of a conjecture of L. Babai and M. E. Watkins concerning graphs allowing a vertex-transitive torsion group action. Dedicated to Prof. K. Wagner on his 70^th birthday     

A note on differential operators of infinite order

A sufficient condition for entire functions f and g to be such that the series ∑_m=0^∞f^(m)(0)g^(m)/m! represents an entire function is established; and in that case, the growth of the resulting function is described.     

A note on the transcendence of infinite products

The paper deals with several criteria for the transcendence of infinite products of the form $\prod\limits_{n = 1}^\infty {[{b_n}{a^{{a_n}}}]/{b_n}{a^{{a_n}}}} $ where α > 1 is a positive algebraic number having a conjugate α* such that α ≠ |α*| > 1, {a _n } _n=1 ^∞ and {b _n } _n=1 ^∞ are two sequences of positive integers with some specific conditions. The proofs are based on the recent theorem of Corvaja and Zannier which relies on the Subspace Theorem (P.Corvaja, U.Zannier: On the rational approximation to the powers of an algebraic number: solution of two problems of Mahler and Mend`es France, Acta Math. 193, (2004), 175–191).     

A note on Dilworth's theorem in the infinite case

If is a poset and every antichain is finite, and if the length of the well-founded poset of antichains is less than ² ₁, then is the union of countably many chains. We also compute the length of the poset of antichains in the product of two ordinals, x.     

A note on D-spaces and infinite unions

It is shown that if X is a countably compact space that is the union of a countable family of D-spaces, then X is compact. This gives a positive answer to Arhangel'skii's problem [A.V. Arhangel'skii, D-spaces and finite unions, Proc. Amer. Math. Soc. 132 (7) (2004) 2163-2170]. In this note, we also obtain a result that if a regular space X is sequential and has a point-countable k-network, then X is a D-space.     

A NOTE ON BEURLING-DENY FORMULAE IN INFINITE DIMENSIONAL SPACES

1.IntroductionandMainResultsAsisIvellknown,theBeurling-DenyformulaeplayimportalrolesinthetheoryofregularDirichletformsonlocallycompactseparablemetricspaces.See[4]forarecentnicerepreselltationinthisconnection.ThepurposeofthispaperistoextendtheBeurlingDenyformulaetoquasi-regularDirichletforms,inparticulartoDirichletformsoninfinitedimensionalstatespaces.RecallthataDirichletformisquasi-regularifandonlyifitisassociatedwitharightcontinuousstrongMarkovprocesslivingesselltiallyonametriZableLusin…     

A note on groups of infinite rank with modular subgroup lattice

It is proved that if \(G\) is a (generalized) soluble group of infinite rank in which all proper subgroups of infinite rank are permodular, then the subgroup lattice of \(G\) is permodular. As a consequence of this theorem, we obtain shorter proofs for corresponding known results concerning normal or permutable subgroups of groups of infinite rank.     

A note on the regularity of solutions of infinite dimensional Riccati equations

This note is concerned with the regularity of solutions of algebraic Riccati equations arising from infinite dimensional LQR control problems. We show that distributed parameter systems described by certain parabolic partial differential equations often have a special structure that smooths solutions of the corresponding Riccati equation. This analysis is motivated by the need to find specific representations for Riccati operators that can be used in the development of computational schemes for problems where the input and output operators are not Hilbert-Schmidt. This situation occurs in many boundary control problems and in certain distributed control problems associated with optimal sensor/actuator placement.     

A note on terminal value problems for fractional differential equations on infinite interval

In this note we establish sufficient conditions for existence and uniqueness of solutions of terminal value problems for a class of fractional differential equations on infinite interval. Some illustrative examples are comprehended to demonstrate the proficiency and utility of our results.     

A note on Hydromagnetic oscillatory flow along an infinite flat plate with variable suction

The two-dimensional, incompressible flow past an infinite plate of a weakly conducting fluid in the presence of a transverse magnetic field is discussed when the suction velocity normal to the plate as well as the external flow velocity vary periodically with time. Expressions for the velocity and the skin-friction in the boundary layer have been obtained in a non-dimensional form.     

A note on an extension of extrapolative techniques for a class of infinite oscillatory integrals

Recent work on extrapolative processes for a class of infinite oscillatory integrals is extended by a procedure which involves the unknown abscissae at which the remainder integral vanishes identically. The asymptotic proximity of these abscissae to a known set is used as a basis for further extrapolation.Results show that the method can be useful for the very slowly converging integrals whose amplitudes decay only algebraically.     

A note on infinite systems of linear inequalities in Rn

A necessary and sufficient condition for infinite systems of linear inequalities to have solutions is given, using the Kuhn-Fourier Theorem, which deals with finite systems.     

A note on Buczolich's solution of the Weil gradient problem: a construction based on an infinite game

Summary Using an infinite game approach we reprove Buczolich's result that there exists a differentiable function f such that ∇f(0) = 0 and |∇ f |≧1 a.e.     

A note on the existence and uniqueness of the solution to neutral stochastic functional differential equations with infinite delay

In this note, we prove the existence and uniqueness of the solution to neutral stochastic functional differential equations with infinite delay (INSFDEs in short) in which the initial value belongs to the phase space BC((-∞,0]R^d), which denotes the family of bounded continuous R^d-value functions φ defined on (-∞,0] with norm ||φ||=sup_-∞<θ?0|φ(θ)|, under some Carathéodory-type conditions on the coefficients by means of the successive approximation. Especially, we extend the results appeared in Ren et al. [Y. Ren, S. Lu, N. Xia, Remarks on the existence and uniqueness of the solutions to stochastic functional differential equations with infinite delay, J. Comput. Appl. Math. 220 (2008) 364-372], Ren and Xia [Y. Ren, N. Xia, Existence, uniqueness and stability of the solutions to neutral stochastic functional differential equations with infinite delay, Appl. Math. Comput. 210 (2009) 72-79] and Zhou and Xue [S. Zhou, M. Xue, The existence and uniqueness of the solutions for neutral stochastic functional differential equations with infinite delay, Math. Appl. 21 (2008) 75-83].