Starting from nonlocal symmetries related to Bäcklund transformation (BT), many interesting results can be obtained. Taking the well known potential KdV (pKdV) equation as an example, a new type of nonlocal symmetry in elegant and compact form which comes from BT is presented and used to make researches in the following three subjects: two sets of negative pKdV hierarchies and their corresponding bilinear forms are constructed; the nonlocal symmetry is localized by introduction of suitable and simple auxiliary dependent variables to generate new solutions from old ones and to consider some novel group invariant solutions; some other models both in finite dimensions and infinite dimensions are generated by comprising the original BT and evolution under new nonlocal symmetry. The finitedimensional models are completely integrable in Liouville sense, which are shown equivalent to the results given through the nonlinearization method for Lax pair.

In nonlinear science, it is very difficult to find exact interaction solutions among solitons and other kinds of complicated waves such as cnoidal waves and Painlevé waves. Actually, even if for the most well-known prototypical models such as the Kortewet-de Vries (KdV) equation and the Kadomtsev-Petviashvili (KP) equation, this kind of problem has not yet been solved. In this paper, the explicit analytic interaction solutions between solitary waves and cnoidal waves are obtained through the localization procedure of nonlocal symmetries which are related to Darboux transformation for the well-known KdV equation. The same approach also yields some other types of interaction solutions among different types of solutions such as solitary waves, rational solutions, Bessel function solutions, and/or general Painlevé II solutions.

Female patients with sepsis have better clinical outcomes than male patients in terms of mortality and length of hospitalization and ICU stay.This is an open access article distributed under the terms of the Creative Commons Attribution-Non Commercial-No Derivatives License 4.0 (CCBY-NC-ND), where it is permissible to download and share the work provided it is properly cited. The work cannot be changed in any way or used commercially without permission from the journal. http://creativecommons.org/licenses/by-nc-nd/4.0.

No abstract

No abstract

Objectives: We investigated the impact of obesity (proxied as body mass index (BMI)), on short-and long-term mortality in sepsis patients. Methods: We conducted a retrospective analysis with adult sepsis ICU patients in a US medical institution from 2001 to 2012 in the MIMIC-III database. The WHO BMI categories were used. Multivariate logistic regression assessed the relationships between BMI and 30-day and 1-year mortality. Results: In total, 5563 patients were enrolled. Obese patients tended to be younger (P < 0.001), to be female (P < 0.001), to acquire worse SOFA scores (P < 0.001), and to receive more aggressive treatment compared with their normal weight counterparts. Obese patients had notably longer mechanical ventilation periods and ICU and hospital lengths of stay (LOSs). In the final model, overweight and obese patients had lower 30-day (OR 0.77, 95% CI 0.66-0.91; OR 0.65, 95% CI 0.56-0.77, respectively) and 1-year (OR 0.83, 95% CI 0.71-0.96; OR 0.70, 95% CI 0.60-0.81, respectively) mortality risks than normal weight patients. In contrast, underweight patients had worse 30-day and 1-year outcomes compared with normal weight patients (P = 0.01, P < 0.001, respectively). In morbidly obese, severe sepsis and septic shock patients, obesity remained protective. Conclusions: Obesity was correlated with short-and long-term survival advantages in sepsis patients.

A direct and systematic algorithm is proposed to find one-dimensional optimal system for the group invariant solutions, which is attributed to the classification of its corresponding one-dimensional Lie algebra. Since the method is based on different values of all the invariants, the process itself can both guarantee the comprehensiveness and demonstrate the inequivalence of the optimal system, with no further proof. To illustrate our method more clearly , we give a couple of well-known examples: the Korteweg-de Vries (KdV) equation and the heat equation.

Purpose – The purpose of this paper is to study the dynamical group consensus of heterogeneous multi-agent systems with fixed topologies. Design/methodology/approach – The tool used in this paper to model the topologies of multi-agent systems is algebraic graph theory. The matrix theory and stability theory are applied to research the group consensus of heterogeneous multi-agent systems with fixed topologies. The Laplace transform and Routh criterion are utilized to analyze the convergence properties of heterogeneous multi-agent systems. Findings – It is discovered that the dynamical group consensus for heterogeneous multi-agent systems with first-order and second-order agents can be achieved under the reasonable hypothesizes. The group consensus condition is only relied on the nonzero eigenvalues of the graph Laplacian matrix. Originality/value – The novelty of this paper is to investigate the dynamical group consensus of heterogeneous multi-agent systems with first-order and second-order agents and fixed topologies and obtain a sufficient group consensus condition.