Mathematics

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The interrelationship between pathogen infection and host mobility is of great importance for successful spread of disease in spatial pest population. As spread of infection depends on horizontal transmission of pathogen, there are numerous factors like susceptibility, latent period, host movement that influence overall effectiveness of the control policy. Initiation of new infection cycle depends on density of infected inoculum in the site. So, spatial movement of infected hosts during the course of infection influence the dynamics. Also, infected individuals are more vulnerable to predators and hence production of virus particles in the site depends on predation to some extent. We derive a four dimensional delayed reaction-diffusion model in one spatial dimension and compute the minimum travelling speed of transmission of infection. We show that the minimum speed is sensitive to several parameters of the system. For example, the minimum speed decreases only with increase in delay in lysis process, but otherwise it increases with increase in force of infection, diffusivity of infectives or per capita virus production. A concluding discussion with numerical simulation is presented in the end. © 2013.

World University root node

(SoNS) School of Natural Sciences

Mathematical Biosciences

Spatiotemporal dynamics of insect pest population under viral infection

Let K be a finite extension of a characteristic zero field F. We say that a pair of n × n matrices (A,B) over F represents K if K ≅ F[A]/〈B〉, where F[A] denotes the subalgebra of Mn(F) containing A and 〈B〉 is an ideal in F[A], generated by B. In particular, A is said to represent the field K if there exists an irreducible polynomial q(x) ∈ F[x] which divides the minimal polynomial of A and K ≅ F[A]/〈q(A)〉. In this paper, we identify the smallest order circulant matrix representation for any subfield of a cyclotomic field. Furthermore, if p is a prime and K is a subfield of the p-th cyclotomic field, then we obtain a zero-one circulant matrix A of size p × p such that (A, J) represents K, where J is the matrix with all entries 1. In case, the integer n has at most two distinct prime factors, we find the smallest order 0, 1-companion matrix that represents the n-th cyclotomic field. We also find bounds on the size of such companion matrices when n has more than two prime factors. © 2013 The Indian National Science Academy.

Fulltext

Indian Journal of Pure and Applied Mathematics

Representation of cyclotomic fields and their subfields

In this paper, we discuss a one parameter family of complex Born-Infeld solitons arising from a one parameter family of minimal surfaces. The process enables us to generate a new solution of the B-I equation from a given complex solution of a special type (which are abundant). We illustrate this with many examples. We find that the action or the energy of this family of solitons remains invariant in this family and find that the well-known Lorentz symmetry of the B-I equations is responsible for it. © Indian Academy of Sciences.

Proceedings of the Indian Academy of Sciences: Mathematical Sciences

One-parameter family of solitons from minimal surfaces

Let M be a complex manifold and let PM C∞([0, 1], M) be space of smooth paths over M. We prove that the induced almost complex structure on PM is weak integrable by extending the result of Indranil Biswas and Saikat Chatterjee of [Geometric structures on path spaces, Int. J. Geom. Meth. Mod. Phys. 8(7) (2011) 1553-1569]. Further we prove that if M is smooth manifold with corner and N is any complex manifold then induced almost complex structure ℑ on Fréchet manifold C∞(M, N) is weak integrable. © 2013 World Scientific Publishing Company.

International Journal of Geometric Methods in Modern Physics

Almost complex structure on path space

For a fixed positive integer n, let Wn be the permutation matrix corresponding to permutation. In this article, it is shown that a symmetric matrix with rational entries is circulant if, and only if, it lies in the subalgebra of Q[x]/ (xn-1) generated by Wn+Wn -1. On the basis of this, the singularity of graphs on n-vertices is characterized algebraically. This characterization is then extended to characterize the singularity of directed circulant graphs. The kth power matrix Wnk+Wn -k defines a circulant graph Cnk. The results above are then applied to characterize its singularity, and that of its complement graph. The digraph Cr,s,t is defined as that whose adjacency matrix is circulant circ(a), where a is a vector with the first r-components equal to 1, and the next s, t and n - (r + s + t) components equal to zero, one, and zero respectively. The singularity of this digraph (graph), under certain conditions, is also shown to depend algebraically upon these parameters. A slight generalization of these graphs are also studied.

Electronic Journal of Linear Algebra

Non-singular circulant graphs and digraphs

The Birkhoff-James orthogonality is a generalization of Hilbert space orthogonality to Banach spaces. We investigate this notion of orthogonality when the Banach space has more structures. We start by doing so for the Banach space of square matrices moving gradually to all bounded operators on any Hilbert space, then to an arbitrary C*-algebra and finally a Hilbert C*-module. © 2013 Elsevier Ltd.

Journal of Mathematical Analysis and Applications

Characterization of Birkhoff-James orthogonality

The norm of the mth derivative of the map that takes an operator to its fcth antisymmetric tensor power is evaluated. The case m = 1 has been studied earlier by Bhatia and Friedland [R. Bhatia and S. Friedland. Variation of Grassman powers and spectra. Linear Algebra and its Applications, 40:1-18, 1981]. For this purpose a multilinear version of a theorem of Russo and Dye is proved: it is shown that a positive m-linear map between C*-algebras attains its norm at the m-tuple (I,I,...,I). Expressions for derivatives of the maps that take an operator to its fcth tensor power and fcth symmetric tensor power are also obtained. The norms of these derivatives are computed. Derivatives of the map taking a matrix to its permanent are also evaluated.

Derivatives of tensor powers and their norms

Vaccination is a cornerstone of infectious disease prevention. However, individual vaccinating behaviour does not always result in population-level vaccine coverage patterns that are optimal for protecting public health. For example, vaccine coverage may fall below the elimination threshold due to nonvaccinators who free-ride on the herd immunity provided by vaccinators. Routine vaccination programs for many paediatric infectious diseases now have an almost worldwide coverage, but vaccine scares fuelled by such behaviours threaten eradication goals. This free-riding behaviour can be seen as a manifestation of policy resistance, where humans respond to an intervention in such a way that tends to undermine the intervention. However, policy resistance is only one such example of the types of dynamics that emerge from the interaction between vaccinating behaviour and disease incidence or prevalence. Here we explore four types of emergent dynamics of behaviour-incidence systems: policy resistance, policy reinforcement, outcome inelasticity, and outcome variability. We discuss examples of each of these dynamics in the behaviour-incidence modelling literature, and suggest potential implications for vaccination policy. © 2013 Springer Science+Business Media New York. All rights reserved.

Modeling the Interplay Between Human Behavior and the Spread of Infectious Diseases

Emergent dynamical features in behaviour-incidence models of vaccinating decisions

Background: MicroRNAs (miRNAs) are non-uniformly distributed in genomes and ~30\% of the miRNAs in the human genome are clustered. In this study we have focused on the imprinted miRNA cluster miR-379/miR-656 on 14q32.31 (hereafter C14) to test their coordinated function. We have analyzed expression profile of &gt;1000 human miRNAs in &gt;1400 samples representing seven different human tissue types obtained from cancer patients along with matched and unmatched controls.Results: We found 68\% of the miRNAs in this cluster to be significantly downregulated in glioblastoma multiforme (GBM), 61\% downregulated in kidney renal clear cell carcinoma (KIRC), 46\% in breast invasive carcinoma (BRCA) and 14\% in ovarian serous cystadenocarcinoma (OV). On a genome-wide scale C14 miRNAs accounted for 12-30\% of the total downregulated miRNAs in different cancers. Pathway enrichment for the predicted targets of C14 miRNA was significant for cancer pathways, especially Glioma (p&lt; 3.77x10-6, FDR&lt;0.005). The observed downregulation was confirmed in GBM patients by real-time PCR, where 79\% of C14 miRNAs (34/43) showed downregulation. In GBM samples, hypermethylation at C14 locus (p&lt;0.003) and downregulation of MEF2, a crucial transcription factor for the cluster was observed which likely contribute to the observed downregulation of the entire miRNA cluster.Conclusion: We provide compelling evidence that the entire C14 miRNA cluster is a tumor suppressor locus involved in multiple cancers, especially in GBM, and points toward a general mechanism of coordinated function for clustered miRNAs.Reviewers: Reviewed by: Prof. Gregory J Goodall and Dr. Alexander Max Burroughs. © 2013 Laddha et al.; licensee BioMed Central Ltd.

Biology Direct

Genome-wide analysis reveals downregulation of miR-379/miR-656 cluster in human cancers

We study the role of predation dynamics in oscillation of pest population in insect ecology. A two-dimensional pest control model (under the use of insecticides) with time delay in predation is considered in this paper. By the Hopf bifurcation theory, we prove the existence of the stable oscillation of the system. We also consider the economic viability of the control process. First we improve the Pontryagin maximum principle (PMP) where the delay in the system is sufficiently small and control function is linear, and then we apply the improved version of PMP to perform the optimal analysis of the pest control model as a special case. © 2013 Samit Bhattacharyya and Suma Ghosh.