Novel critical phenomena in optimization driven processes on networks

The work presented in this Ph.D thesis was developed in the context of complex network theory, from a statistical physics standpoint. We examine two distinct problems in this research field, taking a special interest in their respective critical properties. In both cases, the emergence of criticality is driven by a local optimization dynamics. Firstly, a recently introduced class of percolation problems that attracted a significant amount of attention from the scientific community, and was quickly followed up by an abundance of other works. Percolation transitions were believed to be continuous, until, recently, an 'explosive' percolation problem was reported to undergo a discontinuous transition, in [93]. The system's evolution is driven by a metropolis-like algorithm, apparently producing a discontinuous jump on the giant component's size at the percolation threshold. This finding was subsequently supported by number of other experimental studies [96, 97, 98, 99, 100, 101]. However, in [1] we have proved that the explosive percolation transition is actually continuous. The discontinuity which was observed in the evolution of the giant component's relative size is explained by the unusual smallness of the corresponding critical exponent, combined with the finiteness of the systems considered in experiments. Therefore, the size of the jump vanishes as the system's size goes to infinity. Additionally, we provide the complete theoretical description of the critical properties for a generalized version of the explosive percolation model [2], as well as a method [3] for a precise calculation of percolation's critical properties from numerical data (useful when exact results are not available). Secondly, we study a network flow optimization model, where the dynamics consists of consecutive mergings and splittings of currents flowing in the network. The current conservation constraint does not impose any particular criterion for the split of current among channels outgoing nodes, allowing us to introduce an asymmetrical rule, observed in several real systems. We solved analytically the dynamic equations describing this model in the high and low current regimes. The solutions found are compared with numerical results, for the two regimes, showing an excellent agreement. Surprisingly, in the low current regime, this model exhibits some features usually associated with continuous phase transitions.

Calculus of variations on time scales and applications to economics

We consider some problems of the calculus of variations on time scales. On the beginning our attention is paid on two inverse extremal problems on arbitrary time scales. Firstly, using the Euler-Lagrange equation and the strengthened Legendre condition, we derive a general form for a variation functional that attains a local minimum at a given point of the vector space. Furthermore, we prove a necessary condition for a dynamic integro-differential equation to be an Euler-Lagrange equation. New and interesting results for the discrete and quantum calculus are obtained as particular cases. Afterwards, we prove Euler-Lagrange type equations and transversality conditions for generalized infinite horizon problems. Next we investigate the composition of a certain scalar function with delta and nabla integrals of a vector valued field. Euler-Lagrange equations in integral form, transversality conditions, and necessary optimality conditions for isoperimetric problems, on an arbitrary time scale, are proved. In the end, two main issues of application of time scales in economic, with interesting results, are presented. In the former case we consider a firm that wants to program its production and investment policies to reach a given production rate and to maximize its future market competitiveness. The model which describes firm activities is studied in two different ways: using classical discretizations; and applying discrete versions of our result on time scales. In the end we compare the cost functional values obtained from those two approaches. The latter problem is more complex and relates to rate of inflation, p, and rate of unemployment, u, which inflict a social loss. Using known relations between p, u, and the expected rate of inflation π, we rewrite the social loss function as a function of π. We present this model in the time scale framework and find an optimal path π that minimizes the total social loss over a given time interval.

Antioxidant and detoxify genes polymorphisms in colorectal cancer predisposition

Colorectal cancer (CRC) results from histologic and gene alterations can lead to a massive cellular proliferation. Most of the authors assume multifactorial causes to CRC genesis. Low physical activity, a fat diet poor in fibers and smoking habits seems to have an important role in CRC. However, there are also genetic causes associated with CRC risk. It has been described that oxidative stress levels could influence CRC development. Thus, cellular balance reactive species and defense enzymes involved in oxidative stress are crucial to maintain a good tissue function and avoid neoplasic process. Therefore, genome variations on these defense enzymes, such as MNSOD, SOD3, GSTP1, GSTT1 and GSTM1, could be important biomarkers to colorectal adenocarcinomas. We intend to determine frequencies distribution of most common polymorphisms involved on oxidative stress regulation (MNSOD, SOD3, GSTP1, GSTT1 and GSTM1) in patients with sporadic colorectal adenocarcinoma (SCA) and in healthy controls, evaluation their possible correlation with SCA risk. Samples common polymorphisms of antioxidant and detoxify genes (MNSOD T175C, SOD3 R213G, GSTP1 A105G, GSTP1 C114T, GSTT1del and GSTM1del) analysis was done by PCR-SSP techniques. In this study we found a higher prevalence of MNSOD 175CC (55% vs 2%; p<0.0001; OR: 58.5; CI 13.3 to 256.7), SOD3 213GG (31% vs 2%; p<0.0001; OR: 21.89; CI 4.93 to 97.29), GSTP1 105GG (46% vs 12%; p<0.0001; OR: 6.14; CI 2.85 to 13.26), GSTP1 114TT (38% vs 0%; p<0.0001; OR: Infinity) and GSTT1 null (75% vs 28%; p<0.0001; OR: 7.71; CI 3.83 to 15.56) mutated genotypes among SCA patients, while the normal genotypes were associated with SCA absence. Furthermore, we found GSTP1 114TT mutated genotype (52% vs 27%; p=0.003; OR: 2.88; CI: 1.41 to 5.89) and GSTT1 null genotype (87% vs 65%; p=0.003; OR: 3.66; CI 1.51 to 8.84) associated with colon samples. These findings suggest a positive association between most of common polymorphisms involved on oxidative stress regulation and SCA prevalence. Dysregulation of MNSOD, SOD3, GSTP1, GSTT1 and GSTM1 genes could be associated with an increase of ROS in colon and rectum tissue and p53 pathway deregulation, induced by oxidative stress on colonic and rectal cells. The present study also provides preliminary evidence that MNSOD 175C, SOD3 213G, GSTP1 105G, GSTP1 114T and GSTT1 null polymorphisms, may be involved in SCA risk and could be useful to clarify this multifactorial disorder.