• Canada
• 010102 general mathematics close

Abstract In this paper, we consider a discrete logistic equation x ( n +1)= x ( n ) exp r ( n ) 1 − x ( n ) K ( n ) where {r(n)} and {K(n)} are positive ω-periodic sequences. Sufficient conditions are obtained for the existence of a positive and globally asymptotically stable ω-periodic solution. Counterexamples are given to illustrate that the conclusions in [1] are incorrect.

In this paper, we investigate two-sided bounds for the small ball probability of a mixed fractional Brownian motion with a general deterministic trend function, in terms of respective small ball probability of a mixed fractional Brownian motion without trend. To maximize the lower bound, we consider various ways to split the trend function between the components of the mixed fractional Brownian motion for the application of Girsanov theorem, and we show that the optimal split is the solution of a Fredholm integral equation. We find that the upper bound for the probability is also a function of this optimal split. The asymptotic behaviour of the probability as the ball becomes small is analyzed for zero trend function and for the particular choice of the upper limiting function.

AbstractIn this paper, we study the expressive power and the complexity of first-order logic with arithmetic, as a query language over relational and constraint databases. We consider constraints over various domains (N, Z, Q, and R), and with various arithmetical operations (⩽, +, ×, etc.).We first consider the data complexity of first-order queries. We prove in particular that linear queries can be evaluated in AC0 over finite integer databases, and in NC1 over linear constraint databases. This improves previously known bounds. We also show that over all domains, enough arithmetic lead to arithmetical queries, therefore, showing the frontiers of constraints for database purposes.We then tackle the problem of the expressive power, with the definability of the parity and the connectivity, which are the most classical examples of queries not expressible in first-order logic over finite structures. We prove that these two queries are first-order definable in the presence of (enough) arithmetic. Nevertheless, we show that they are not definable with constraints of interest for constraint databases such as linear constraints for instance. Finally, we developed reduction techniques for queries over constraint databases, that allow us to draw conclusions with respect to their undefinability in various constraint query languages.

We find an infinite word w on four symbols with the following property: Two occurrences of any block in w must be separated by more than the length of the block. That is, in any subword of w of the form xyx, the length of y is greater than the length of x. This answers a question of C. Edmunds connected to the Burnside problem for groups. The research of the first author was supported by an NSERC Operating Grant. The second author was supported by an NSERC Undergraduate Summer Research Award. https://www.cambridge.org/core/journals/canadian-mathematical-bulletin/article/words-without-nearrepetitions/F86509D865F222F1FC63ACA8545C069E

The deep geometrical relationships holding among the NLS equation, the vortex filament equation,the Heisenberg spin model, and the Schrodinger map equation are extended to the general setting of Hermitian symmetric spaces. New results are obtained by utilizing a generalized Hasimoto variable which arises from applying the general theory of parallel moving frames. The example of complex projective space CP^N= SU(N+1)/U(N) is used to illustrate the method and results. 48 pages. Published version

AbstractDe Simone showed that prime bull- and chair-free graphs containing a co-diamond are either bipartite or an induced cycle of odd length at least five. Based on this result, we give a complete structural characterization of prime (bull,chair)-free graphs having stability number at least four as well as of (bull,chair,co-chair)-free graphs. This implies constant-bounded clique width for these graph classes which leads to linear time algorithms for some algorithmic problems. Moreover, we obtain a robust O(nm) time algorithm for the maximum weight stable set problem on bull- and chair-free graphs without testing whether the (arbitrary) input graph is bull- and chair-free. This improves previous results with respect to structural insight, robustness and time bounds.

Because surgical procedures require clinicians to develop and maintain procedural expertise and because blinding in randomized clinical trials of such therapies is often challenging, their critical appraisal raises unique issues. Risk of bias of trials of surgical procedures increases if investigators fail to rigorously conceal allocation and, where possible, to ensure blinding of those involved in the trial. Variability in surgeons' expertise can also increase bias and lead to important limitations in applicability. To address these issues, this Users' Guide to the Medical Literature reviews the use of remote randomization systems, blinding, sham-controlled trials, split-body trials, expertise-based trials, and mechanistic vs practical trials. Consideration of risk of bias and applicability issues will allow clinicians to make optimal use of trials addressing surgical procedures.

Abstract We propose a reaction–diffusion system with nonlocal delays to model the growth of plankton communities feeding on a limiting nutrient supplied at a constant rate. Two delays are incorporated into the model: one describes the delayed nutrient recycling process and the other models the delayed growth response of the plankton. It is assumed that both delays are nonlocal in the sense that there are delayed not only in time but also in space. Local and bifurcation analyses are carried out. It has been shown that Turing-type spatial patterns occur when the diffusion coefficients vary and temporal or spatial-temporal patterns occur when the delay involved in the growth response changes.

We provide a simple algorithm for finding the optimal upper bound for sums of products of matrix entries of the form S_pi(N) := sum_{j_1, ..., j_2m = 1}^N t^1_{j_1 j_2} t^2_{j_3 j_4} ... t^m_{j_2m-1 j_2m} where some of the summation indices are constrained to be equal. The upper bound is easily obtained from a graph G associated to the constraints in the sum. 20 pages

We reformulate notions from the theory of quasi-Poisson g-manifolds in terms of graded Poisson geometry and graded Poisson-Lie groups and prove that quasi-Poisson g-manifolds integrate to quasi-Hamiltonian g-groupoids. We then interpret this result within the theory of Dirac morphisms and multiplicative Manin pairs, to connect our work with more traditional approaches, and also to put it into a wider context suggesting possible generalizations. 39 pages