On Restricted Connectivity and Extra Connectivity of Hypercubes and Folded Hypercubes

Given a graph G and a non-negative integer h, the h-restricted connectivity κh（G） of G is the minimum cardinality of a set of vertices of G, in which at least h neighbors of any vertex is not included, if any, whose deletion disconnects G and every remaining component has the minimum degree of vertex at least h; and the h-extra connectivity κh（G） of G is the minimum cardinality of a set of vertices of G, if any, whose deletion disconnects G and every remaining component has order more than h. This paper shows that for the hypercube Qn and the folded hypercube FQn, κ1（Qn）=κ（1）（Qn）=2n-2 for n≥3, κ2（Qn）=3n-5 for n≥4, κ1（FQn）=κ（1）（FQn）=2n for n≥4 and κ（2）（FQn）=4n-4 for n≥8.

Identifying sex-specific injury predictors as a key factor in maintaining optimal physical activity levels

BACKGROUND Optimal physical activity is known to reduce cardiovascular,respiratory and endocrine system diseases and,as a consequence,improve quality of life.An important risk factor for reinjuries during normal exercise is the initial connective tissue pathology.The variety of clinical dysplastic manifestations significantly complicate the timely diagnosis of this comorbidity.AIM To establish pathognomonic sex-specific dysplasia phenotypes that indicate a particular sensitivity to physical exertion.METHODS The study involved 117 participants with recurrent musculoskeletal injuries that occurred during normal exercise.There were 67 women(57.26%)and 50 men(42.74%),which made it possible to compare the presence of the identified signs between sexes.A validated questionnaire was used to screen their connective tissue status.RESULTS Ranking the most commonly revealed dysplasia signs depending on their clinical significance made it possible to establish pathognomonic sex-specific phenotypes that indicated a particular susceptibility to injuries.Individualized programs of optimal physical activity are necessary for men with chest deformities,flat-valgus feet,dolichostenomelia,arachnodactylia,hemorrhoids,abdominal muscle diastasis and recurrent hernias.In women,special sensitivity to physical exertion was associated with a combination of signs such as asthenic body,joint hypermobility,overly soft auricles,thin hyperelastic skin,atrophic striae,telangiectasias and varicose veins.Of particular importance were universal signs such as gothic palate,scoliosis,kyphosis,leg deformities,temporomandibular joint crunching,and moderate to high myopia.CONCLUSION Participants’connective tissue condition should be considered when designing optimal physical activity programs.Identifying the established sex-specific dysplasia phenotypes will allow timely optimization of training loads,thus reducing the risk of injury.

ON CONDITIONAL EDGE-CONNECTIVITY OF GRAPHS

Let k and h be two integers, 0≤h<k. Let G be a connected graph with minimum degree at least k. The conditional h-edge-connectivity of G, denoted by λ(h) (G), is defined as the minimum cardinality |S| of a set S of edges in G such that G-S is disconnected and is of minimum degree at least h. This type of edge-connectivity is a generalization of the traditional edge-connectivity and can more accurately measure the fault-tolerance of networks. In this paper, we will first show thatλ A(2)(G)≤g(k-2) for a k(≥3)-regular graph G provided G is neither K4 and K5 nor K3,a, where g is the length of a shortest cycle of G, then show that λ(h)(Qk)=(k-h)2h for a k-dimensional cube Qk.

Connecting Conditions and Constraint Equations in Applied Science

To solve the optimal solution of some issues in applied science, studying of connecting conditions, constraint conditions and constraint equations is made. This paper cites an example in point in vibration mechanics and seeks the connecting conditions and constraint equations of high speed compound rotating system. This paper points out that the selection of the boundary conditions or connection conditions can effect on the optimal solution of the issue as soon as the object function is determined.

ON THE CONDITIONAL EDGE CONNECTIVITY OF ENHANCED HYPERCUBE NETWORKS

Let G =（V, E） be a connected graph and m be a positive integer, the conditional edge connectivity λ;is the minimum cardinality of a set of edges,if it exists, whose deletion disconnects G and leaves each remaining component with minimum degree δ no less than m. This study shows that λ;(Q;) = 2 n,λ;(Q;) = 4 n-4（2 ≤ k ≤ n-1, n ≥ 3） for n-dimensional enhanced hypercube Q;. Meanwhile, another easy proof about λ;（Q;） = 4 n-8, for n ≥ 3 is proposed. The results of enhanced hypercube include the cases of folded hypercube.

Freezing of Water in a Slab with Boundary Conditions of the Third Kind

Predictions for freezing heat transfer in a slab with convective boundary conditions at the cold surfaceare obtained from heat balance integral approximations, which consider conduction as the only modeof heat transfer in both the solid and liquid and consider durations of precooling and freezing in theheat transfer process. The thermal penetration dimensionless parameter or is presented to distinguishtwO cases of freezing, and analytical results for α≥ 1 are given in this paper. An experimental investigation on freezing of water is reported for comparison with the one-dimensional conduction model toshow that experimental modeling for freezing heat transfer with convective boundary conditions usingPeltter devices is feasible. The comparison also demonstrated that the freezing rate in this case wasdecreased by natural convection in the liquid just as freezing with boundary conditions of the first kindand phase change approximately proceeds linearly with time rather than with square root of time atthe initiation of freezing.