39 research outputs found
Algebraic and algorithmic aspects of Zm × Fn : fixed subgroups and quantification of inertia
This work is based on the family of groups Z^m x F_n, namely free-abelian times free groups, direct products of finitely many copies of Z and a finitely generated free group F_n. These are a special type of right-angled Artin groups. In this work we use general combinatorics, one dimensional geometry and algebraic techniques to play with elements of Z^m x F_n to solve some algorithmic problems concerning ranks of the subgroups, automorphisms and its fixed point subgroups, subgroup intersection problem heading towards a cryptography application using the group Z^m x F_n. The core methodology of this work involves the use of Stallings graph to work with subgroups of F_n and to deal with the abelian part we use linear algebra, systems of equations, Smith normal form of integral matrices, etc. This thesis is to study algorithmic problems of Z^m x F_n, a natural extension of free groups and a first step towards further generalization into another two main directions: semi-direct products, and partially commutative groups. The three principal projects of this thesis are the following:
(1) In the lattice of subgroups of a free group, the rank function is not monotone with respect to inclusion (i.e., H<=K does not imply r(H)<=r(K)). This makes it interesting to define and study relaxed versions for this monotonic property. Based on the classical definitions of compressed and inert subgroups, we introduce the concepts of degrees of compression and inertia (dc and di) of a finitely generated subgroup H of a given group G, as an attempt to quantify how close (or far) is H from being compressed and inert and so, from satisfying the aforementioned monotonic property. In the case of Z^m x F_n, we show that the dc is algorithmically computable, we give an upper bound for the di, and relate both degrees with those of the projection H\pi to the free part. With some extra assumptions over the supremum involved in the definition of di, we define another notion called restricted degree of inertia. In the case of Z^m x F_n, beyond giving the upper bound, we are able to give an equality formula for the restricted degree of inertia.
(2) The study of the properties of fixed point subgroups by automorphisms of free groups F_n is much more complicated but they are well studied in the literature. The classical result by Dyer–Scott about fixed subgroups of finite order automorphisms of F_n being free factors of F_n is no longer true in Z^m x F_n, gives us the feeling that fixed point subgroups in Z^m x F_n have more degenerated behaviour than in the free group. Within this more general context, we prove a relaxed version in the spirit of Bestvina-Handel Theorem: the rank of fixed subgroups of finite order automorphisms is uniformly bounded in terms of n and m. For any given automorphism it is a very natural question to ask for the elements which are fixed by that automorphism. The dual problem to this is, for a given finitely generated subgroup H, to ask whether there exists a finite collection of automorphisms which fix exactly that particular subgroup H point-wise. In this text we also solve this dual problem and give an algorithm to compute auto-fixed closures of finitely generated subgroups of Z^m x F_n.
(3) In this project we develop a secret-sharing scheme taking advantage of the fact that Z^m x F_n does not satisfy the Howson property, i.e., it contains finitely generated subgroups whose intersection is not finitely generated. Concretely, the shares for the k players are going to be k finitely generated subgroups of Z^m x F_n such that every intersection of shares is not finitely generated, except the full intersection, which is taken as the secret. This way we significantly increase the difficulty for an illegal coalition of players to extract any practical additional information about the secret. We prove that for any integer k, one can effectively built such a family of subgroups of and to give an effective algorithm to compute the secret.Aquest treball versa sobre la família de grups ZM x Fn, és a dir els grups lliure-abelians per lliure, productes directes d'un nombre finit de còpies de Z i d'un grup lliure Fn. Són un tipus especial de grups right-angled Artin. En aquest treball usem eines de combinatòria, geometria 1-dimensional, i tècniques algebraiques per a jugar amb els elements de ZMm x Fn ti resoldre alguns problemes algorítmics sobre els rangs de subgrups, sobre automorfismes i els seus subgrups de punts fixos, el problema de la intersecció de subgroups i fins i tot una aplicació a la criptografia usant el grup ZM x F_n. La part metodològica principal d'aquest treball involucra l'ús dels grafs de Stallings per a treballar amb els subgrups de Fn; i per a tractar amb la part abeliana usem l'àlgebra lineal estàndard, els sistemes d'equacions i la forma normal de Smith de les matrius enteres. En aquesta tesi estudiem problemes algorísmics de ZMm x Fn, una extensió natural dels grups lliures i un primer pas cap a majors generalitzacions en dues direccions principals: productes semidirectes, i grups parcialment commutatius. Els tres resultats principals d'aquesta tesi són els següents: (1) En el reticle de subgrups del grup lliure la funció de rang no és monòtona respecte la inclusió (i.e., H<=K no implica r(H)<=r(K)). Això fa que sigui interessant definir i estudiar versions relaxades d'aquesta propietat de monotomia. Basades en les definicions clàssiques de compressió i inèrcia de subgrups, introduïm els conceptes de grau de compressió i grau d'inèrcia (dc i di, respectivament) d'un subgrup finitament generat H d'un grup donat G, com un intent de quantificar quan a prop o lluny és H de ser comprimit o inert, és a dir de satisfer l'esmentada propietat de monotonia. En el cas de ZM x Fn, demostrem que el dc és algorítmicament calculable, donem una fita superior per al di, i relacionem aquests dos graus amb els de la projecció H\pi a la part lliure. Amb alguna hipòtesi extra sobre el suprem involucrat en la definició de di, definim una noció alternativa anomenada grau d'inèrcia restringit. En el cas de ZM x Fn, més enllà de la fita superior donada, som capaços de donar una fórmula d'igualtat per aquest grau d'inèrcia restringit. (2) L'estudi de les propietats dels subgrups de punts fixos per automorfismes de grups lliures Fn és molt més complicat però estan ben estudiats a la literatura. El fet que el resultat clàssic de Dyer-Scott sobre els subgrups fixos d'automorfismes d'ordre finit de Fn que són factors lliures de Fn no sigui cert a ZM x F_n, ens dóna la intuïció que els subgrups de punts fixos a ZM x Fn es comportaran de manera més degenerada que en el grup lliure. En aquest context més general, demostrem una versió relaxada en l'esperit del teorema de Bestvina-Handel: el rang dels subgrups fixos d'automorfismes d'ordre finit de ZM x Fn està uniformement acotat en termes només de n i m. Per cada automorfisme donat, és natural de preguntar pels elements que queden fixes. El problema dual consisteix en, donat un subgrup finitament generat H, preguntar si existeix una col·lecció finita d'automorfismes que fixin exactament aquest subgrup en particular, element a element. En aquesta tesi resolem també aquest problema tot donant un algorisme per a calcular clausures auto-fixes de subgrups finitament generats de ZM x Fn. (3) En aquesta tesi desenvolupem un esquema de compartició de secrets traient profit del fet que ZM x Fn no satisfà la propietat de Howson, i.e., conté subgroups finitament generats amb intersecció no finitament generada. Concretament, les claus per als k jugadors seran k subgrups finitament generats de ZM x Fn de manera que qualsevol intersecció sigui no finitament generada, excepte la total, que serà presa com el secret. D'aquesta manera, augmentem significativament la dificultat per part d'una coalició il·legal de jugadors d'obtenir cap informaci
Pragmatic Approaches to Sustainable Development through Rural Women Empowerment in Bangladesh
Empowerment of rural women in Bangladesh can reduce gender inequality and improve their social status. Despite its significance, the issue cannot be done with solved easily as there are many factors that tend to debar its progress. In this regard, this paper aims to consider the multi-layered actuality of issues in the rural women empowerment concerned with gender equality in Bangladesh. This paper sought to identify the determinants of empowerment of rural women in Bangladesh. Analysis of some selected factors was performed to identify the empowerment factors. It focuses on the empowerment of rural women, including in relation to access to micro credit, Information Communication and Technology (ICT), agriculture etc. Keywords: Agriculture, ICT, Micro Credit, Poultry, women empowermen
Tourism in Bangladesh: Present Status and Future Prospects
Bangladesh is full of natural beauty. It is surrounded by river, coasts and beaches, archaeological sites, religious places, hills, forest, waterfalls, tea gardens etc. The sundarban, Historic mosque city of Bagerhat, Ruins of the Buddihist vihara at paharpur are the three world heritage sites existed in Bangladesh among 1007. To observe the beauty of nature, huge amount of domestic and foreign tourists visit the said places. In 2012, around six lakh tourist came Bangladesh to visit and enjoy it’s natural beauty. The total contribution of travel and tourism to GDP was 4.4% , to employment was 3.8% and to investment was 1.5% in 2013. Bangladesh Parjatan Corporation (BPC) look after the tourism sector in Bangladesh under the ministry of Civil Aviation and Tourism. Due to some limitations, Bangladesh has failed to introduce to the tourists as a tourist destination country. The paper tries to focus on the attractive tourist spot of Bangladesh and the contribution of tourism in the Bangladesh economy. The major concern of the writer is to highlight the factors by which Bangladesh can be one of the major tourist attractive country in the world in near future. Keywords: Tourism, Economy, Impact
Reaping Demographic Dividend in Bangladesh: Challenges and Prospects
Demographic change in Bangladesh is opening up new economic opportunities As in many countries declining infant and child mortality helped to spark lower fertility effectively resulting in a temporary baby boom As this cohort moves into working ages Bangladesh finds itself with a potentially higher share of workers as compared with dependents If working-age people can be productively employed Bangladesh s economic growth stands to accelerate Theoretical and empirical literature on the effect of demographics on labor supply savings and economic growth underpin this effort to understand and forecast economic growth in Bangladesh Policy choices can potentiate Bangladesh s realization of economic benefits stemming from demographic change Failure to take advantage of the opportunities inherent in demographic change can lead to economic stagnatio
Reaping Demographic Dividend in Bangladesh: Challenges and Prospects
Demographic change in Bangladesh is opening up new economic opportunities As in many countries declining infant and child mortality helped to spark lower fertility effectively resulting in a temporary baby boom As this cohort moves into working ages Bangladesh finds itself with a potentially higher share of workers as compared with dependents If working-age people can be productively employed Bangladesh s economic growth stands to accelerate Theoretical and empirical literature on the effect of demographics on labor supply savings and economic growth underpin this effort to understand and forecast economic growth in Bangladesh Policy choices can potentiate Bangladesh s realization of economic benefits stemming from demographic change Failure to take advantage of the opportunities inherent in demographic change can lead to economic stagnatio
Degrees of compression and inertia for free-abelian times free groups
© 2020. ElsevierWe introduce the concepts of degree of inertia, diG(H), and degree of compression,dcG(H), of a finitely generated subgroupHof a given groupG. For the case of direct productsof free-abelian and free groups, we compute the degree of compression and give an upper boundfor the degree of inertia. Imposing some technical assumptions to the supremum involved in thedefinition of degree of inertia, we introduce the notion called restricted degree of inertia, di'G(H),and, again for the caseZm×Fn, we provide an explicit formula relating it to the restricted degreeof inertia of its projection to the free part, di'Fn(Hp).Both authors are partially supported by the Spanish Agencia Estatalde Investigación, through grant MTM2017-82740-P (AEI/ FEDER, UE), and also by the “María de Maeztu” Programme for Units of Excellence in R&D (MDM-2014-0445)Peer ReviewedPostprint (author's final draft
Intersection configurations in free and free times free-abelian groups
In this paper we delve into the behaviour of multiple intersections of
subgroups of free and free times free-abelian (FTFA) groups with respect to
finite generability.
It is well known that free groups satisfy the Howson property (the
intersection of any two finitely generated subgroups is again finitely
generated). We show that Howson property is indeed the only obstruction for
multiple intersection configurations to be realizable within .
On the contrary, FTFA groups are well known to be non-Howson. We study
multiple intersections within FTFA groups, and we give an algorithm to decide,
given finitely generated subgroups, whether their intersection
is again finitely generated and, in the affirmative case, compute a basis for
it. We also show that any finite intersection configuration is realizable in
, for and big enough. As a
consequence, we exhibit a finitely presented group where every intersection
configuration is realizable.Comment: 19 pages, 8 figure
GREEN SYNTHESIS, CHARACTERIZATION AND ANTIBACTERIAL ACTIVITY OF SILVER NANOPARTICLES
Metallic nanoparticles have gained the interest of researchers worldwide due to their unique antibacterial, antimicrobial and anti-inflammatory properties. There is a constant need for the sustainable green synthesis of the metallic nanoparticles with less involvement of the toxic chemicals. In this background, our group has synthesised the silver nanoparticles from the aqueous extracts of clove and cinnamon through green method. The aqueous spice extracts were used for the reduction of silver nitrate solution. The synthesised silver nanoparticles were characterised by the UV-Visible spectroscopy, dynamic light scattering (DLS) and transmission electron microscopy (TEM). Antibacterial properties of the nanoparticles were evaluated on the Escherichia coli and Staphylococcus aureus strains using the Kirby-Bauer antibiotic testing method. UV-Vis spectroscopy confirms the size of the nanoparticles to be around 30-60 nm which is further confirmed by the DLS and TEM techniques. Further, the antibacterial activity analysis showed that the bacterial samples (S. aureus and E. coli) treated with the synthesised silver nanoparticles showed minimum inhibitory concentration in the range of 25-30 μM. The study presents an environment friendly method to synthesise metallic nanoparticles showing good antibacterial activity. This work would help other research groups working in the field of biological application of green synthesis mediated metallic nanoparticles