This paper relies on nested postulates of separate, linear and arc-continuity of functions to define analogous properties for sets that are weaker than the requirement that the set be open or closed. ...This allows three novel characterisations of open or closed sets under convexity or separate convexity postulates: the first pertains to separately convex sets, the second to convex sets and the third to arbitrary subsets of a finite-dimensional Euclidean space. By relying on these constructions, we also obtain new results on the relationship between separate and joint continuity of separately quasiconcave, or separately quasiconvex functions. We present examples to show that the sufficient conditions we offer cannot be dispensed with.
This work presents a generalization of the functions used in Fraeijs de Veubeke's quadrilateral (FVQ) element and in Hsieh, Clough and Tocher's (HCT) triangle, allowing for an arbitrary degree of the ...approximation, d$$ d $$, and for arbitrary order of continuity, r$$ r $$. These functions can be used to define finite elements that lead by direct assembly to Cr$$ {C}^r $$ approximations. Their determination for each element is procedural, generalizing the reasoning used by Zienkiewicz to explain the FVQ element. The completeness of each basis is confirmed by comparing its dimension with that of a broken basis, defined primitive element wise, upon which the continuity constraints are successively imposed. The order of continuity at the vertices that is required to enable the independent definition of the normal derivatives at the sides is confirmed numerically. The corresponding interpolation functions are symbolically derived using Mathematica, and simple matlab codes using the data thus obtained are made available, to illustrate their application.
In this paper we study the following class of fractional Kirchhoff problems: ... where > 0 is a small parameter, s ∈ (0,1), N > 2, (- )s is the fractional Laplacian, V : RN → R is a positive ...continuous function, M : 0, ∞) → R is a Kirchhoff function satisfying suitable conditions and f : R → R fulfills Berestycki-Lions type assumptions of subcritical or critical type. Using suitable variational arguments, we prove the existence of a family of positive solutions (uE) which concentrates at a local minimum of V as → 0.
Abstract
The main objective is to make a study on the basic properties and relation between Pythagorean nano
a
-interior and Pythagorean nano
a
-closure by defining them in Pythagorean nano ...topological spaces. Also certain Pythagorean nano
a
-continuous, Pythagorean nano semi and pre continuous functions which are weak forms of Pythagorean nano continuous functions are also defined.
The DiRECT trial assessed remission of type 2 diabetes during a primary care-led weight-management programme. At 1 year, 68 (46%) of 149 intervention participants were in remission and 36 (24%) had ...achieved at least 15 kg weight loss. The aim of this 2-year analysis is to assess the durability of the intervention effect.
DiRECT is an open-label, cluster-randomised, controlled trial done at primary care practices in the UK. Practices were randomly assigned (1:1) via a computer-generated list to provide an integrated structured weight-management programme (intervention) or best-practice care in accordance with guidelines (control), with stratification for study site (Tyneside or Scotland) and practice list size (>5700 or ≤5700 people). Allocation was concealed from the study statisticians; participants, carers, and study research assistants were aware of allocation. We recruited individuals aged 20-65 years, with less than 6 years' duration of type 2 diabetes, BMI 27-45 kg/m
, and not receiving insulin between July 25, 2014, and Aug 5, 2016. The intervention consisted of withdrawal of antidiabetes and antihypertensive drugs, total diet replacement (825-853 kcal per day formula diet for 12-20 weeks), stepped food reintroduction (2-8 weeks), and then structured support for weight-loss maintenance. The coprimary outcomes, analysed hierarchically in the intention-to-treat population at 24 months, were weight loss of at least 15 kg, and remission of diabetes, defined as HbA
less than 6·5% (48 mmol/mol) after withdrawal of antidiabetes drugs at baseline (remission was determined independently at 12 and 24 months). The trial is registered with the ISRCTN registry, number 03267836, and follow-up is ongoing.
The intention-to-treat population consisted of 149 participants per group. At 24 months, 17 (11%) intervention participants and three (2%) control participants had weight loss of at least 15 kg (adjusted odds ratio aOR 7·49, 95% CI 2·05 to 27·32; p=0·0023) and 53 (36%) intervention participants and five (3%) control participants had remission of diabetes (aOR 25·82, 8·25 to 80·84; p<0·0001). The adjusted mean difference between the control and intervention groups in change in bodyweight was -5·4 kg (95% CI -6·9 to -4·0; p<0·0001) and in HbA
was -4·8 mmol/mol (-8·3 to -1·4 -0·44% (-0·76 to -0·13); p=0·0063), despite only 51 (40%) of 129 patients in the intervention group using anti-diabetes medication compared with 120 (84%) of 143 in the control group. In a post-hoc analysis of the whole study population, of those participants who maintained at least 10 kg weight loss (45 of 272 with data), 29 (64%) achieved remission; 36 (24%) of 149 participants in the intervention group maintained at least 10 kg weight loss. Serious adverse events were similar to those reported at 12 months, but were fewer in the intervention group than in the control group in the second year of the study (nine vs 22).
The DiRECT programme sustained remissions at 24 months for more than a third of people with type 2 diabetes. Sustained remission was linked to the extent of sustained weight loss.
Diabetes UK.
A new class of functions called ‘almost δgβ-continuous functions’ is introduced and their basic properties are studied. Moreover, the relationships between this new class and graphs are also ...investigated.
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