The formulation of a new analysis on a zero measure Cantor set C(⊂ I = [0, 1]) is presented. A non-Archimedean absolute value is introduced in C exploiting the concept of relative infinitesimals and a scale invariant ultrametric valuation of the form log ε −1 (ε/x) for a given scale ε > 0 and infinitesimals 0 < x < ε, x ∈ I\C. Using this new absolute value, a valued (metric) measure is defined on C and is shown to be equal to the finite Hausdorff measure of the set, if it exists. The formulation of a scale invariant real analysis is also outlined, when the singleton {0} of the real line R is replaced by a zero measure Cantor set. The Cantor function is realized as a locally constant function in this setting. The ordinary derivative dx/dt in R is replaced by the scale invariant logarithmic derivative d log x/d log t on the set of valued infinitesimals. As a result, the ordinary real valued functions are expected to enjoy some novel asymptotic properties, which might have important applications in number theory and in other areas of mathematics.

A class of ultrametric Cantor sets (C, d u ) introduced recently (Raut, S and Datta, D P (2009), Fractals, 17, 45-52) is shown to enjoy some novel properties. The ultrametric d u is defined using the concept of relative infinitesimals and an inversion rule. The associated (infinitesimal) valuation which turns out to be both scale and reparametrisation invariant, is identified with the Cantor function associated with a Cantor setC where the relative infinitesimals are supposed to live in. These ultrametrics are both metrically as well as topologically inequivalent compared to the topology induced by the usual metric. Every point of the original Cantor set C is identified with the closure of the set of gaps ofC. The increments on such an ultrametric space is accomplished by following the inversion rule. As a consequence, Cantor functions are reinterpreted as locally constant functions on these extended ultrametric spaces. An interesting phenomenon, called growth of measure, is studied on such an ultrametric space. Using the reparametrisation invariance of the valuation it is shown how the scale factors of a Lebesgue measure zero Cantor set might get deformed leading to a deformed Cantor set with a positive measure. The definition of a new valuated exponent is introduced which is shown to yield the fatness exponent in the case of a positive measure (fat) Cantor set. However, the valuated exponent can also be used to distinguish Cantor sets with identical Hausdorff dimension and thickness. A class of Cantor sets with Hausdorff dimension log 3 2 and thickness 1 are constructed explicitly.

The framework of a new scale invariant analysis on a Cantor set C ⊂ I = [0, 1] , presented originally in S. Raut and D. P. Datta, Fractals, 17, 45-52, (2009), is clarified and extended further. For an arbitrarily small ε > 0, elementsx in I\C satisfying 0

Microorganism cells movement in fluid is universal and affects many ecological and biological processes, including infection, reproduction and marine life ecosystem. There are many biological and medical applications that require understanding of transport process in nanofluids containing suspension of microorganism. The present problem deals with the bioconvection of Casson nanofluid containing suspension of motile gyrotactic microorganisms over inclined stretching sheet in the presence of thermal radiation, viscous dissipation, and chemical reaction and magnetic field. At the surface, influence of the thermo-solutal Marangoni convection and suction / injection impact are considered. The governing equations are solved numerically by using fourth order Runge-Kutta method with shooting technique. Impact of the major pertinent parameters on the velocity, temperature, nanoparticles concentration and density of the motile microorganism is illustrated graphically. Finally, correlation of various crucial parameters on skin friction, local Nusselt number, Sherwood number and local motile microorganism density number are displayed through graphs and tables.

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In this article, we have studied a memory dependent prey–predator model with logistic type prey growth subject to the Allee effect and Holling type‐III functional response. To include the memory effect, we have used here a fractional order system. The model contains at most four equilibrium points among which one is trivial, two are axial, and the last one is interior. Here we have studied behavior of these four equilibrium points using eigen analysis method. Our investigation shows that the trivial equilibrium point is stable for strong Allee effect, and unstable for weak Allee effect. Between the two axial equilibrium points, one is stable under certain condition but the other is always unstable. This unstable equilibrium point exists only for strong Allee case. The interior equilibrium point arises when the axial equilibrium point is unstable. Stability condition of this interior equilibrium point depends on the parametric relationship. The schematic diagram divides the a‐p plane into five sub‐regions in both the cases of memory dependent and memory less system. The number and nature of the equilibrium points in different sub‐regions are different. Finally, we have drawn the bifurcation diagrams considering all the equilibrium points in different sub‐regions and the results are concluded at the end.

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