On the Taylor expansion of $\lambda$-terms and the groupoid structure of their rigid approximants

We show that the normal form of the Taylor expansion of a $\lambda$-term is isomorphic to its B\"ohm tree, improving Ehrhard and Regnier's original proof along three independent directions. First, we simplify the final step of the proof by following the left reduction strategy directly in the resource calculus, avoiding to introduce an abstract machine ad hoc. We also introduce a groupoid of permutations of copies of arguments in a rigid variant of the resource calculus, and relate the coefficients of Taylor expansion with this structure, while Ehrhard and Regnier worked with groups of permutations of occurrences of variables. Finally, we extend all the results to a nondeterministic setting: by contrast with previous attempts, we show that the uniformity property that was crucial in Ehrhard and Regnier's approach can be preserved in this setting.

Dynamics and bifurcations of a ratio-dependent predator-prey model

In this paper, we study a ratio-dependent predator-prey model with modied Holling-Tanner formalism, by using dynamical techniques and numerical continuation algorithms implemented in Matcont. We determine codim-1 and 2 bifurcation points and their corresponding normal form coecients. We also compute a curve of limit cycles of the system emanating from a Hopf point.

An analysis of two degenerate double-Hopf bifurcations

<abstract><p>The generic double-Hopf bifurcation is presented in detail in literature in textbooks like references. In this paper we complete the study of the double-Hopf bifurcation with two degenerate (or nongeneric) cases. In each case one of the generic conditions is not satisfied. The normal form and the corresponding bifurcation diagrams in each case are obtained. New possibilities of behavior which do not appear in the generic case were found.</p></abstract>

The spatially homogeneous Hopf bifurcation induced jointly by memory and general delays in a diffusive system

In this paper, by incorporating the general delay to the reaction term in the memory-based diffusive system, we propose a diffusive system with memory delay and general delay (e.g., gestation, hunting, migration and maturation delays, etc.). We first derive an algorithm for calculating the normal form of Hopf bifurcation in a diffusive system with memory and general delays. The developed algorithm for calculating the normal form can be used to investigate the direction and stability of Hopf bifurcation. Then, we consider a diffusive predator-prey model with ratio-dependent Holling type-3 functional response, which includes with memory and gestation delays. The Hopf bifurcation analysis without considering gestation delay is first studied, then the Hopf bifurcation analysis with memory and gestation delays is studied. Furthermore, by using the developed algorithm for calculating the normal form, the supercritical and stable spatially homogeneous periodic solutions induced jointly by memory and general delays are found theoretically. The stable spatially homogeneous periodic solutions are also found by the numerical simulations which confirms our analytic result.

Inhomogeneous spatial patterns in diffusive predator-prey system with spatial memory and predator-taxis

In this paper, we consider a diffusive predator-prey system with spatial memory and predator-taxis. Since in this system, the memory delay appears in the diffusion term, and the diffusion term is nonlinear, the classical normal form of Hopf bifurcation in the reaction-diffusion system with delay can't be applied to this system. Thus, in this paper, we first derive an algorithm for calculating the normal form of Hopf bifurcation in this system. Then in order to illustrate the effectiveness of our newly developed algorithm, we consider the diffusive Holling-Tanner model with spatial memory and predator-taxis. The stability and Hopf bifurcation analysis of this model are investigated, and the direction and stability of Hopf bifurcation periodic solution are also researched by using our newly developed algorithm for calculating the normal form of Hopf bifurcation. At last, we carry out some numerical simulations, two stable spatially inhomogeneous periodic solutions corresponding to the mode-1 and mode-2 Hopf bifurcations are found, which verifies our theoretical analysis results.

POSITION OF WOMEN WITH SPECIA REFERENCE TO MANUSMRITI

For centuries, women have played the secondary role to man. She was never held in esteem; only a mention about her pathetic status was mentioned here and there. But the advent of science and technology, growth of democratic ideas and impact of modern industrialization and general awareness and conscience among the masses has paved way for the emancipation of women. As per Encyclopedia of Women “women is as much human being as man; as she is equally entitled to develop her abilities, all her abilities, to choose the sort of life she wishes to lead and carry on all the activities and assume all the responsibilities that go to make up human dignity”. The Manusmriti (MS) of Manu is also called Manusamhita or Manavadharmasastra. It is a standard and the most authoritative work on Hindu law and present the normal form of Hindu society and civilization. It is a store house of information on the cultural, social, religious, ethical, educational, political, judicial and geographical life of the period and this is perfectly natural. Manusmriti has been upheld as the ultimate guide to lead a moral life, the digressing of which is to be treated with serious negative sanctions.

A method for counting models on grid Boolean formulas1

We present a novel algorithm based on combinatorial operations on lists for computing the number of models on two conjunctive normal form Boolean formulas whose restricted graph is represented by a grid graph Gm,n. We show that our algorithm is correct and its time complexity is O ( t · 1 . 618 t + 2 + t · 1 . 618 2 t + 4 ) , where t = n · m is the total number of vertices in the graph. For this class of formulas, we show that our proposal improves the asymptotic behavior of the time-complexity with respect of the current leader algorithm for counting models on two conjunctive form formulas of this kind.