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This is an expository version of our paper [arXiv:1902.07384]. Our aim is to present recent Macaulay2 algorithms for computation of mixed multiplicities of ideals in a Noetherian ring which is either local or a standard graded algebra over a field. These algorithms are based on computation of the equations of multi-Rees algebras of ideals that generalises a result of Cox, Lin and Sosa. Using these equations we propose efficient algorithms for computation of mixed volumes of convex lattice polytopes and sectional Milnor numbers of hypersurfaces with an isolated singularity., Comment: The manuscript is prepared for the volume based on the International Conference in Honor of Prof. Ravi S. Kulkarni's 80th Birthday on Geometry, Groups and Mathematical Philosophy. arXiv admin note: text overlap with arXiv:1902.07384

This study was conducted to assess the knowledge level of dairy farmers in Punjab. A total of 250 dairy farmers were randomly selected from the farmers visiting Veterinary clinics of GADVASU, and participating in various extension programmes, various training programmes, animal welfare camps and pashu palan mela. The relevant information was collected through a structured interview schedule. It was observed that 64.8 per cent of the farmers have medium level of knowledge regarding breeding practices whereas 15.2 per cent farmers have high level of knowledge. It was observed that majority of the large farmers, 100 per cent have knowledge regarding heat detection and 86.11 per cent have knowledge regarding time of heat detection. The medium farmers and large farmers also possessed higher knowledge regarding vaccination of animals. It was observed that control of ectoparasite was known by 89.2 per cent of the respondents. N Laldinpuii | S K Kansal | H K Verma | J S Bhatti "Knowledge level of dairy farmers about improved breeding and health care practices in Punjab" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-3 , April 2018, URL: https://www.ijtsrd.com/papers/ijtsrd10784.pdf

The genetic diversity and population structure studies on 32 wild Malus species were conducted using the nine inter simple sequence repeat (ISSR) markers. The average value of diversity indices, viz. resolving power (Rp), polymorphic information content (PIC), effective multiplex ratio (EMR) and marker index (MI) of ISSR markers were 2.389, 0.388, 16.429 and 6.228, respectively, while the Jaccard’s similarity coefficient ranged from 0.46 to 0.97. The cluster analysis divided the selected Malus species into two major clusters and principal coordinate analysis (PCoA) further reconfirmed the result of the cluster analysis. The first three axes of PCoA explained 43.49% variation and analysis of molecular variance (AMOVA) explained 16% variation between the indigenous and exotic Malus populations. The studied Malus species were genetically differentiated into four distinct populations which were revealed through the model based population structure analysis. Thus, the present investigation revealed substantial genetic diversity among the studied wild Malus species and existing genetic diversity could be valuable genetic resources for future apple improvement programme.

Using vanishing of graded components of local cohomology modules of the Rees algebra of the normal filtration of an ideal, we give bounds on the normal reduction number. This helps to get necessary and sufficient conditions in Cohen-Macaulay local rings of dimension $d\geq 3$, for the vanishing of the normal Hilbert coefficients $\overline{e}_k(I)$ for $k\leq d,$ in terms of the normal reduction number., Comment: Theorem 5.5 corrected, to appear in Journal of Ramanujan Mathematical Society

In this expository paper we present proofs of Grothendieck-Serre Formula for multi-graded algebras and Rees algebras for admissible multi-graded filtrations. As applications, we derive formulas of Sally for postulation number of admissible filtrations and Hilbert coefficients. We also discuss a partial solution of Itoh's conjecture by Kummini and Masuti. We present an alternate proof of Huneke-Ooishi Theorem and a generalisation for multi-graded filtrations., Comment: 32 pages

We survey some of the major results about normal Hilbert polynomials of ideals. We discuss a formula due to Lipman for complete ideals in regular local rings of dimension two, theorems of Huneke, Itoh, Huckaba, Marley and Rees in Cohen-Macaulay analytically unramified local rings. We also discuss recent works of Goto-Hong-Mandal and Mandal-Singh-Verma concerning the positivity of the first coefficient of the normal Hilbert polynomial in unmixed analytically unramified local rings. Results of Moral{\'e}s and Villarreal linking normal Hilbert polynomial of monomial ideals with Ehrhart polynomials of polytopes are also presented.

In this expository paper we survey results that relate Hilbert coefficients of an m-primary ideal I in a Cohen-Macaulay local ring (R, m) with depth of the associated graded ring G(I). Several results in this area follow from two theorems of S. Huckaba and T. Marley. These were proved using homological techniques. We provide simple proofs using superficial sequences., Comment: 24 pages, expository paper to appear in Mathematics Student

This paper is a survey on major results on Hilbert functions of multigraded algebras and mixed multiplicities of ideals, including their applications to the computation of Milnor numbers of complex analytic hypersurfaces with isolated singularity, multiplicities of blowup algebras and mixed volumes of polytopes., Comment: 34 pages

Let (R, m) be a Cohen-Macaulay local ring and I be an m-primary ideal. We introduce ideals of almost minimal mixed multiplicty which are analogues of ideals studied by J. Sally. The main theorem describes the Hilbert series of fiber cones of these ideals., Comment: 12 pages