arXiv subjects
Publications and source records attributed to Arpita Ghosh.
This short note establishes an abstract Hales--Jewett theorem for semigroups equipped with a finite family of retractions. The proof relies on the interplay between retractions and tensor products of ultrafilters.
The set of sums of two squares plays a significant role in elementary number theory. In this article, we establish the existence of several rich monochromatic configurations in the natural numbers by exploiting algebraic structures induced by the set of sums of two squares. The proofs rely largely on the algebraic properties arising from the induced structures on the Stone-Čech compactification of the natural numbers.
Motivation: Motivated by the growing interest in understanding the role of non-metricity in describing dense stellar systems, in this paper, we study compact stellar configurations within the framework of linear $f(Q)$ gravity. Methodology: By adopting a linear modification of the form $f(Q) = αQ+β$, we analyze the internal structure and physical properties of an anisotropic relativistic star within the framework of $f(Q)$ gravity. We employ the Karmarkar's condition together with the Vaidya-Tikekar metric ansatz to obtain a closed-form interior solution of the star. The interior solution is then matched to the Schwarzschild exterior solution across the boundary of the star. By varying the model parameters, we analyze physical features of the resultant stellar configuration. Results: We note distinctive features in the density, pressure, anisotropy and total mass of the star under a such modification. By enforcing the condition that the central pressure remains finite, we obtain the maximum compactness bound which is shown to depend solely on the Vaidya-Tikekar curvature parameter $K$. We recover the Buchdahl bound for the curvature parameter $K=0$, which corresponds to the solution for an isotropic and homogeneous fluid sphere. Utilizing the energy density and radial pressure profiles, we numerically integrate the modified Tolman-Oppenheimer-Volkoff equations and obtain the mass-radius ($M-R$) relationships for different values of the model parameter $α$. We note that for higher values of $α$, the maximum mass and radius decrease, shifting the stable branch towards ultra-compact configurations. An interesting observation in our analysis is that a linearly modified $f(Q)$ gravity model can support comparatively low mass stars. Utilizing the observed mass of some known pulsars, we demonstrate how our model can be used to fine-tune the radius of the star.
There are several notions of size for semigroups that have natural analogues for partial semigroups. Among these are thick, syndetic, central, piecewise syndetic, IP, J, and the more recently introduced notion of combinatorially rich, abbreviated CR. We investigate the notion of CR set for adequate partial semigroups, its relation to other notions, especially J sets, and some surprising differences among them.
The incorporation of quantum ansatz with machine learning classification models demonstrates the ability to extract patterns from data for classification tasks. However, taking advantage of the enhanced computational power of quantum machine learning necessitates dealing with various constraints. In this paper, we focus on constraints like finding suitable datasets where quantum advantage is achievable and evaluating the relevance of features chosen by classical and quantum methods. Additionally, we compare quantum and classical approaches using benchmarks and estimate the computational complexity of quantum circuits to assess real-world usability. For our experimental validation, we selected the gene expression dataset, given the critical role of genetic variations in regulating physiological behavior and disease susceptibility. Through this study, we aim to contribute to the advancement of quantum machine learning methodologies, offering valuable insights into their potential for addressing complex classification challenges in various domains.
H. Furstenberg defined Central sets in $\mathbb{N}$ by using the notions of topological dynamics, later Bergelson and Hindman characterized central sets in $\mathbb{N}$ and also in arbitrary semigroup in terms of algebra of Stone-Čech compactification of that set. We state the new notion of large sets in a partial semigroup setting and characterize the algebraic structure of the sets by using the algebra of Stone-Čech compactification. By using these notions, we introduce the \emph{Partial Semigroup Partial Dynamical System(PSPDS)} and show that topological dynamical characterization of central sets in a partial semigroup is equivalent to the usual algebraic characterization.
Anisotropy is one factor that appears to be significantly important in the studies of relativistic compact stars. In this paper, we make a generalization of the Buchdahl limit by incorporating an anisotropic effect for a selected class of exact solutions describing anisotropic stellar objects. In the isotropic case of a homogeneous distribution, we regain the Buchdahl limit $2M/R \leq 8/9$. Our investigation shows a direct link between the maximum allowed compactness and pressure anisotropy vi-a-vis geometry of the associated $3$-space.
Social computing encompasses the mechanisms through which people interact with computational systems: crowdsourcing systems, ranking and recommendation systems, online prediction markets, citizen science projects, and collaboratively edited wikis, to name a few. These systems share the common feature that humans are active participants, making choices that determine the input to, and therefore the output of, the system. The output of these systems can be viewed as a joint computation between machine and human, and can be richer than what either could produce alone. The term social computing is often used as a synonym for several related areas, such as "human computation" and subsets of "collective intelligence"; we use it in its broadest sense to encompass all of these things. Social computing is blossoming into a rich research area of its own, with contributions from diverse disciplines including computer science, economics, and other social sciences. Yet a broad mathematical foundation for social computing is yet to be established, with a plethora of under-explored opportunities for mathematical research to impact social computing. As in other fields, there is great potential for mathematical work to influence and shape the future of social computing. However, we are far from having the systematic and principled understanding of the advantages, limitations, and potentials of social computing required to match the impact on applications that has occurred in other fields. In June 2015, we brought together roughly 25 experts in related fields to discuss the promise and challenges of establishing mathematical foundations for social computing. This document captures several of the key ideas discussed.
In this article, we investigate the image and preimage of the important combinatorial sets such as central sets, $C$-sets, and $J_δ$-sets which play an important role in the study of combinatorics under certain partial semigroup homomorphism. Using that we prove certain results which deal with the existence of $C$-set which are not central in partial semigroup framework.
The most powerful formulation of the Central Sets Theorem in an arbitrary semigroup was proved in the work of De, Hindman, and Strauss. The sets which satisfy the conclusion of the above Central Sets Theorem are called $C$-sets. The original Central Sets Theorem was extended by J. McLeod for adequate commutative partial semigroups. In this work, we will extend the Central Sets Theorem obtained by taking all possible adequate sequences in a commutative adequate partial semigroup. We shall also discuss a sufficient condition for being a set $C$-set in our context.
A notion of pi-tolerant equilibrium is defined that takes into account that players have some tolerance regarding payoffs in a game. This solution concept generalizes Nash and refines epsilon-Nash equilibrium in a natural way. We show that pi-tolerant equilibrium can explain cooperation in social dilemmas such as Prisoner's Dilemma and the Public Good game. We then examine the structure of particularly cooperative pi-tolerant equilibria, where players are as cooperative as they can be, subject to their tolerances, in Prisoner's Dilemma. To the extent that cooperation is due to tolerance, these results provide guidance to a mechanism designer who has some control over the payoffs in a game, and suggest ways in which cooperation can be increased.
The correlations and network structure amongst individuals in datasets today---whether explicitly articulated, or deduced from biological or behavioral connections---pose new issues around privacy guarantees, because of inferences that can be made about one individual from another's data. This motivates quantifying privacy in networked contexts in terms of "inferential privacy"---which measures the change in beliefs about an individual's data from the result of a computation---as originally proposed by Dalenius in the 1970's. Inferential privacy is implied by differential privacy when data are independent, but can be much worse when data are correlated; indeed, simple examples, as well as a general impossibility theorem of Dwork and Naor, preclude the possibility of achieving non-trivial inferential privacy when the adversary can have arbitrary auxiliary information. In this paper, we ask how differential privacy guarantees translate to guarantees on inferential privacy in networked contexts: specifically, under what limitations on the adversary's information about correlations, modeled as a prior distribution over datasets, can we deduce an inferential guarantee from a differential one? We prove two main results. The first result pertains to distributions that satisfy a natural positive-affiliation condition, and gives an upper bound on the inferential privacy guarantee for any differentially private mechanism. This upper bound is matched by a simple mechanism that adds Laplace noise to the sum of the data. The second result pertains to distributions that have weak correlations, defined in terms of a suitable "influence matrix". The result provides an upper bound for inferential privacy in terms of the differential privacy parameter and the spectral norm of this matrix.
Incentives are more likely to elicit desired outcomes when they are designed based on accurate models of agents' strategic behavior. A growing literature, however, suggests that people do not quite behave like standard economic agents in a variety of environments, both online and offline. What consequences might such differences have for the optimal design of mechanisms in these environments? In this paper, we explore this question in the context of optimal contest design for simple agents---agents who strategically reason about whether or not to participate in a system, but not about the input they provide to it. Specifically, consider a contest where $n$ potential contestants with types $(q_i,c_i)$ each choose between participating and producing a submission of quality $q_i$ at cost $c_i$, versus not participating at all, to maximize their utilities. How should a principal distribute a total prize $V$ amongst the $n$ ranks to maximize some increasing function of the qualities of elicited submissions in a contest with such simple agents? We first solve the optimal contest design problem for settings with homogenous participation costs $c_i = c$. Here, the optimal contest is always a simple contest, awarding equal prizes to the top $j^*$ contestants for a suitable choice of $j^*$. (In comparable models with strategic effort choices, the optimal contest is either a winner-take-all contest or awards possibly unequal prizes, depending on the curvature of agents' effort cost functions.) We next address the general case with heterogeneous costs where agents' types are inherently two-dimensional, significantly complicating equilibrium analysis. Our main result here is that the winner-take-all contest is a 3-approximation of the optimal contest when the principal's objective is to maximize the quality of the best elicited contribution.
We consider the problem of designing a survey to aggregate non-verifiable information from a privacy-sensitive population: an analyst wants to compute some aggregate statistic from the private bits held by each member of a population, but cannot verify the correctness of the bits reported by participants in his survey. Individuals in the population are strategic agents with a cost for privacy, \ie, they not only account for the payments they expect to receive from the mechanism, but also their privacy costs from any information revealed about them by the mechanism's outcome---the computed statistic as well as the payments---to determine their utilities. How can the analyst design payments to obtain an accurate estimate of the population statistic when individuals strategically decide both whether to participate and whether to truthfully report their sensitive information? We design a differentially private peer-prediction mechanism that supports accurate estimation of the population statistic as a Bayes-Nash equilibrium in settings where agents have explicit preferences for privacy. The mechanism requires knowledge of the marginal prior distribution on bits $b_i$, but does not need full knowledge of the marginal distribution on the costs $c_i$, instead requiring only an approximate upper bound. Our mechanism guarantees $ε$-differential privacy to each agent $i$ against any adversary who can observe the statistical estimate output by the mechanism, as well as the payments made to the $n-1$ other agents $j\neq i$. Finally, we show that with slightly more structured assumptions on the privacy cost functions of each agent, the cost of running the survey goes to $0$ as the number of agents diverges.
We study trade networks with a tree structure, where a seller with a single indivisible good is connected to buyers, each with some value for the good, via a unique path of intermediaries. Agents in the tree make multiplicative revenue share offers to their parent nodes, who choose the best offer and offer part of it to their parent, and so on; the winning path is determined by who finally makes the highest offer to the seller. In this paper, we investigate how these revenue shares might be set via a natural bargaining process between agents on the tree, specifically, egalitarian bargaining between endpoints of each edge in the tree. We investigate the fixed point of this system of bargaining equations and prove various desirable for this solution concept, including (i) existence, (ii) uniqueness, (iii) efficiency, (iv) membership in the core, (v) strict monotonicity, (vi) polynomial-time computability to any given accuracy. Finally, we present numerical evidence that asynchronous dynamics with randomly ordered updates always converges to the fixed point, indicating that the fixed point shares might arise from decentralized bargaining amongst agents on the trade network.
Crowdsourcing is now widely used to replace judgement by an expert authority with an aggregate evaluation from a number of non-experts, in applications ranging from rating and categorizing online content to evaluation of student assignments in massively open online courses via peer grading. A key issue in these settings, where direct monitoring is infeasible, is incentivizing agents in the `crowd' to put in effort to make good evaluations, as well as to truthfully report their evaluations. This leads to a new family of information elicitation problems with unobservable ground truth, where an agent's proficiency- the probability with which she correctly evaluates the underlying ground truth- is endogenously determined by her strategic choice of how much effort to put into the task. Our main contribution is a simple, new, mechanism for binary information elicitation for multiple tasks when agents have endogenous proficiencies, with the following properties: (i) Exerting maximum effort followed by truthful reporting of observations is a Nash equilibrium. (ii) This is the equilibrium with maximum payoff to all agents, even when agents have different maximum proficiencies, can use mixed strategies, and can choose a different strategy for each of their tasks. Our information elicitation mechanism requires only minimal bounds on the priors, asks agents to only report their own evaluations, and does not require any conditions on a diverging number of agent reports per task to achieve its incentive properties. The main idea behind our mechanism is to use the presence of multiple tasks and ratings to identify and penalize low-effort agreement: the mechanism rewards agents for agreeing with a `reference' rater on a task but also penalizes for blind agreement by subtracting out a statistic term designed so that agents obtain reward only when they put effort into their observations.
We know that large amount of data and information should be transmitted through internet during transactions in E-Governance. Smart E-Governance system should deliver speedy, space efficient, cost effective and secure services among other governments and its citizens utilizing benefits of Information and Communication Technologies (ICT). This paper proposes to develop a space efficient and secured data transmission scheme using Modified Huffman algorithm for compression, which will also yield better bandwidth utilization and inner encryption technique with one way hash function SHA (Secured Hash Algorithm) to ensure Message integrity.
Formalisms for both non-relativistic as well as relativistic versions of field emission of electrons in presence of strong quantizing magnetic field, relevant for strongly magnetized neutron stars or magnetars are developed. In the non-relativistic scenario, where electrons obey Schr{$\ddot{\rm{o}}$}dinger equation, we have noticed that when Landau levels are populated for electrons in presence of strong quantizing magnetic field the transmission probability exactly vanishes unless the electrons are spin polarized in the opposite direction to the external magnetic field. On the other hand, the cold electron emission under the influence of strong electrostatic field at the poles is totally forbidden from the surface of those compact objects for which the surface magnetic field strength is $\gg 10^{15}$G (in the eventuality that they may exist). Whereas in the relativistic case, where the electrons obey Dirac equation, the presence of strong quantizing magnetic field completely forbids the emission of electrons from the surface of compact objects if $B >10^{13}$G.