Partially effective randomization in simulations between ARBITRARY and COMMON PRAMs

It is known that $\text{Θ(}\text{log}\text{n}\text{log}\text{log}\text{n}\text{)}$ steps are needed to simulate one step of ARBITRARY CRCW PRAMs by COMMON CRCW PRAMs, but it was open whether there is a faster simulation when randomization is allowed. This paper gives both positive and negative answers. (i) It is shown that one step of ARBITRARY can be simulated by $\text{O(}\text{log}\text{m}\text{k(}\text{log}\text{log}\text{m−}\text{log}\text{k)}\text{+}\text{log}\text{log}\text{n)}$ steps on randomized COMMON with error-rate n^−c, where m=n/k is the number of different memory cells into which at least one processor of the simulated PRAM attempts to write. The deterministic $\text{Θ(}\text{log}\text{n}\text{log}\text{log}\text{n}\text{)}$-step simulation does not become faster for smaller m, while our randomized simulation becomes $\text{O(}\text{log}\text{log}\text{n)}$ when $\text{m⩽}\text{n}\text{log}\text{log}\text{n}\text{log}\text{n}$. (ii) It is shown that when $\text{m=n,}\text{Ω(}\text{log}\text{n}\text{log}\text{log}\text{n}\text{)}$ steps are needed to simulate one step of ARBITRARY by COMMON even if randomization is allowed. This lower-bound result needs some assumption on processor communication but it strongly suggests randomization does not help when m is small.     

Three symmetric solutions of lidstone boundary value problems for difference and partial difference equations

We consider the boundary value problem $\text{δ}{}^{\mathrm{2m}}\text{y(k-m)=f(y(k),δ}{}^2\text{y(k-1),…,δ}{}^{\mathrm{2i}}\text{y(k-1),…,δ}{}^{\mathrm{2(m-1)}}\text{y(k-(m-1)))}\text{kϵ{a+1,…,b+1},}\text{δ}{}^{\mathrm{2i}}\text{y(a+1-m)=δ}{}^{\mathrm{2i}}\text{y(b+1+m-2i)=0,}\text{0≤i≤m-1,}$ where m ≥ 1 and (−1)^m f R^m → [0, ∞) is continuous. By using Amann and Leggett-Williams' fixed-point theorems, we develop growth conditions on f so that the boundary value problem has triple positive symmetric solutions. The results obtained are then applied in the investigation of radial solutions for certain partial difference equation subject to Lidstone type conditions.     

Unbounded solutions of quasi-linear difference equations

We study positive increasing solutions of the nonlinear difference equation $\text{δ(a}{}_{\mathrm{n}}\text{φ}{}_{\mathrm{p}}\text{(δχ}{}_{\mathrm{n}}\text{))=b}{}_{\mathrm{n}}\text{f(χ}{}_{\mathrm{n+1}}\text{,}\text{φ}{}_{\mathrm{p}}\text{(u)=|u|}{}^{\mathrm{p-2}}\text{u,}\text{p>1}$ where {a_n}, {b_n} are positive real sequences for n ≥ 1, fR → R is continuous with uf(u) > 0 for u ≠ 0. A full characterization of limit behavior of all these solutions in terms of a_n, b_n is established. Examples, showing the essential role of used hypotheses, are also included. The tools used are the Schauder fixed-point theorem and a comparison method based on the reciprocity principle.     

Regularity and stopping theorem for fuzzy martingales with continuous parameters

The notion of graph convergence for fuzzy-valued stochastic processes is introduced. Regularity for fuzzy (super) martingales with continuous parameters in the sense of $\text{gr}\text{·}\text{KM}$-convergence is discussed. Finally, we establish Doob's stopping theorem for continuous parameter (super, sub) martingales whose cut sets may be unbounded.