We introduce a class of Falconer distance problems, which we call of restricted type, lying between the classical version and its pinned variant. Prototypical restricted distance sets are the diagonal distance sets, k-point configuration sets given by

$$ \begin{align*}\Delta^{\mathrm{diag}}(E)= \{ \,|(x,x,\dots,x)-(y_1,y_2,\dots,y_{k-1})| : x, y_1, \dots,y_{k-1} \in E\, \}\end{align*} $$

$E\subset \mathbb {R}^d$

$k\ge 3$

$\Delta ^{\mathrm{diag}}(E)$

(0.1)

$$ \begin{align} \dim(E)> \begin{cases} \frac{2d+1}3, & k=3, \\ \frac{(k-1)d}k,& k\ge 4. \end{cases} \end{align} $$

$C^\omega $

In obese subjects, the loss of fat mass during energy restriction is often accompanied by a loss of muscle mass. The hypothesis that n-3 PUFA, which modulate protein homoeostasis via effects on insulin sensitivity, could contribute to maintain muscle mass during energy restriction was tested in rats fed a high-fat diet (4 weeks) rich in 18 : 1 n-9 (oleic acid, OLE-R), 18 : 3 n-3 (α-linolenic acid, ALA-R) or n-3 long-chain (LC-R) fatty acid and then energy restricted (8 weeks). A control group (OLE-ad libitum (AL)) was maintained with AL diet throughout the study. Rats were killed 10 min after an i.v. insulin injection. All energy-restricted rats lost weight and fat mass, but only the OLE-R group showed a significant muscle loss. The Gastrocnemius muscle was enriched with ALA in the ALA-R group and with LC-PUFA in the ALA-R and LC-R groups. The proteolytic ubiquitin–proteasome system was differentially affected by energy restriction, with MAFbx and muscle ring finger-1 mRNA levels being decreased in the LC-R group (−30 and −20 %, respectively). RAC-α serine/threonine-protein kinase and insulin receptor substrate 1 phosphorylation levels increased in the LC-R group (+70 %), together with insulin receptor mRNA (+50 %). The ALA-R group showed the same overall activation pattern as the LC-R group, although to a lesser extent. In conclusion, dietary n-3 PUFA prevent the loss of muscle mass associated with energy restriction, probably by an improvement in the insulin-signalling pathway activation, in relation to enrichment of plasma membranes in n-3 LC-PUFA.

In this paper, compositions of a natural number are studied. The number of restricted compositions is given in a closed form, and some applications are presented.

The traditional feeding programmes and nutrient requirements of broilers, as established by the NRC, are usually based upon age and fixed growing duration. This is despite the fact that broiler growth pattern characteristics have changed enormously over last decade. Such classical approaches have been questioned as to their effectiveness, and it has been argued the NRC recommendations are no longer applicable to the present methods of production, and need to be re-evaluated. The short comings of traditional or classical fixed duration of feeding phases have led many investigators to suggest and introduce alternative feeding programmes in replacement, including alternative, phase, sequential, choice, dual and subsequent feeding. The effectiveness and shortcomings of such methods are analysed in this review.