Given a fixed small graph H and a larger graph G, an H-factor is a collection of vertex-disjoint subgraphs $H'\subset G$, each isomorphic to H, that cover the vertices of G. If G is the complete graph $K_n$ equipped with independent U(0,1) edge weights, what is the lowest total weight of an H-factor? This problem has previously been considered for $H=K_2$, for example. We show that if H contains a cycle, then the minimum weight is sharply concentrated around some $L_n = \Theta(n^{1-1/d^*})$ (where $d^*$ is the maximum 1-density of any subgraph of H). Some of our results also hold for H-covers, where the copies of H are not required to be vertex-disjoint.

For an arbitrary countable discrete infinite group G, non-singular rank-one actions are introduced. It is shown that the class of non-singular rank-one actions coincides with the class of non-singular $(C,F)$-actions. Given a decreasing sequence of cofinite subgroups in G with $\bigcap _{n=1}^\infty \bigcap _{g\in G}g\Gamma _ng^{-1}=\{1_G\}$, the projective limit of the homogeneous G-spaces $G/\Gamma _n$ as $n\to \infty $ is a G-space. Endowing this G-space with an ergodic non-singular non-atomic measure, we obtain a dynamical system which is called a non-singular odometer. Necessary and sufficient conditions are found for a rank-one non-singular G-action to have a finite factor and a non-singular odometer factor in terms of the underlying $(C,F)$-parameters. Similar conditions are also found for a rank-one non-singular G-action to be isomorphic to an odometer. Minimal Radon uniquely ergodic locally compact Cantor models are constructed for the non-singular rank-one extensions of odometers. Several concrete examples are constructed and several facts are proved that illustrate a sharp difference of the non-singular non-commutative case from the classical finite measure preserving one: odometer actions which are not of rank-one and factors of rank-one systems which are not of rank one; however, each probability preserving odometer is a factor of an infinite measure preserving rank-one system, etc.

For a division ring $D$ , denote by ${{\mathcal{M}}_{D}}$ the $D$ -ring obtained as the completion of the direct limit $\underset{\to n}{\mathop \lim }\,{{M}_{{{2}^{n}}}}(D)$ with respect to themetric induced by its unique rank function. We prove that, for any ultramatricial $D$ -ring $B$ and any non-discrete extremal pseudo-rank function $N$ on $B$ , there is an isomorphism of $D$ -rings $\overline{B}\,\cong \,{{\mathcal{M}}_{D}}$ , where $\overline{B}$ stands for the completion of $B$ with respect to the pseudo-metric induced by $N$ . This generalizes a result of von Neumann. We also show a corresponding uniqueness result for $*$ -algebras over fields $\text{F}$ with positive definite involution, where the algebra ${{\mathcal{M}}_{\text{F}}}$ is endowed with its natural involution coming from the $*$ -transpose involution on each of the factors ${{M}_{{{2}^{n}}}}\,(F)$ .

The objective of this study was to determine the factors associated with selling price of animals at livestock marts around Ireland. Data consisted of four distinct maturity categories: calves (2 to 84 days of age, n = 53 838); weanlings (6 to 12 months of age, n = 19 972); post-weanlings (12 to 36 months of age, n = 93 081) and cows (>30 months to 12 years of age, n = 94 839); sold through livestock marts between 2000 and 2008. Factors associated with animal price were determined within each maturity category separately using mixed models; random effects were mart, date of sale nested within mart, and herd of origin nested within year of sale. Mean selling price was €157, €580, €655 and €592 for calves, weanlings, post-weanlings and cows, respectively. The greatest prices were paid for singleton crossbred male calves, weanlings and post-weanlings from older dams. With the exception of the Aberdeen Angus, beef breeds and their crosses consistently received higher prices than their dairy counterparts across all four maturity categories; increased proportion of Belgian Blue and Charolais was associated with greater prices compared with other beef breeds. When live-weight was included in the multiple regression models the association between price and all factors regressed toward zero but most factors remained associated with price. The highest price was recorded in the spring months for calves, post-weanlings and cows, and in the autumn months for weanlings. Results from this study may be used to help farmers make more informed management decisions, as well as provide information for bio-economic models for evaluating alternative production systems or estimating economic values.

This article sketches a subtheory of human intelligence based on the component construct. Components differ in their levels of generality and in their functions. Metacomponents are higher-order control process used for planning how a problem should be solved, for making decisions regarding alternative courses of action during problem solving, and for monitoring solution processes. Performance components are processes used in the execution of a problem-solving strategy. Acquisition components are processes used in learning new information. Retention components are processes used in retrieving previously stored knowledge. Transfer components are used in generalization, that is, in carrying over knowledge from one task or task context to another. A mechanism for the interaction among components of different kinds and multiple components of the same kind can account for certain interesting aspects of laboratory and everyday problem solving. A brief historical overview of alternative basic units for understanding intelligence is followed by a detailed description of one of these units, the component and by a differentiation among various kinds of components. Examples of each kind of component are given, and the use of each of these components in a problem-solving situation is illustrated. Then, a system of interrelations among the various kinds of components is described. Finally, the functions of components in human intelligence are assessed by considering how the proposed subtheory can account for various empirical phenomena in the literature on human intelligence.

We study ensembles of similar systemsunder load of environmental factors. The phenomenon of adaptationhas similar properties for systems of different nature. Typically,when the load increases above some threshold, then the adaptingsystems become more different (variance increases), but thecorrelation increases too. If the stress continues to increasethen the second threshold appears: the correlation achievesmaximal value, and start to decrease, but the variance continue toincrease. In many applications this second threshold is a signalof approaching of fatal outcome.
This effect is supported by many experiments and observation ofgroups of humans, mice, trees, grassy plants, and on financialtime series. A general approach to explanation of the effectthrough dynamics of adaptation is developed. H. Selye introduced“adaptation energy" for explanation of adaptation phenomena. Weformalize this approach in factors – resource models anddevelop hierarchy of models of adaptation. Different organizationof interaction between factors (Liebig's versus synergisticsystems) lead to different adaptation dynamics. This gives anexplanation to qualitatively different dynamics of correlationunder different types of load and to some deviation from thetypical reaction to stress.
In addition to the “quasistatic" optimization factor – resourcemodels, dynamical models of adaptation are developed, and asimple model (three variables) for adaptation to one factor loadis formulated explicitly.

We investigate the role of projections in norming a $C^*$-algebra by a type II1 subfactor. Applications are given for factors with property $\varGamma$ and for free-product factors.