Special Instances Research Articles

Cause and Effect: A Study on the Concept of Karma in the Buddhist Tradition

The most fundamental aspect of the Buddhist tradition is the concept of Karma. Karma is the basis of Buddhist ethics and principles that maintain that the entire universe is bound to a similar law. Our actions will decide what will happen to us, for we caused it. The current state of our existence is with our past Karma, and the present action will determine our future condition of existence. Karma is a special instance of cause and effect, according to which all our actions of body, speech, and mind are causes and all our experiences are their effects. The Law of Karma explains why each individual has a unique mental disposition, a unique physical appearance, and unique experiences. These are the various effects of the countless actions that each individual has performed in the past. We cannot find any two people who have created the same history of actions throughout their past lives. So we cannot find two people with identical states of mind, identical experiences, and identical physical appearances. Therefore, each person has different individual Karma, and each individual has to face the action of one's own Karma. Thus, the significance of this study is to determine the true concept of karma and its implication beyond our life in the Buddhist tradition.

Discrete Optimal Transport with Independent Marginals is #P-Hard

We study the computational complexity of the optimal transport problem that evaluates the Wasserstein distance between the distributions of two -dimensional discrete random vectors. The best known algorithms for this problem run in polynomial time in the maximum of the number of atoms of the two distributions. However, if the components of either random vector are independent, then this number can be exponential in even though the size of the problem description scales linearly with . We prove that the described optimal transport problem is #P-hard even if all components of the first random vector are independent uniform Bernoulli random variables, while the second random vector has merely two atoms, and even if only approximate solutions are sought. We also develop a dynamic programming–type algorithm that approximates the Wasserstein distance in pseudo-polynomial time when the components of the first random vector follow arbitrary independent discrete distributions, and we identify special problem instances that can be solved exactly in strongly polynomial time.

General ordering theorem

The problem of ordering operators has afflicted quantum mechanics since its foundation. Several orderings have been devised, but a systematic procedure to move from one ordering to another is still missing. The importance of establishing relations among different orderings is demonstrated by Wick's theorem (which relates time ordering to normal ordering), which played a crucial role in the development of quantum field theory. We prove the General Ordering Theorem (GOT), which establishes a relation among any pair of orderings, that act on operators satisfying generic (i.e. operatorial) commutation relations. We expose the working principles of the GOT by simple examples, and we demonstrate its potential by recovering two famous algebraic theorems as special instances: the Magnus expansion and the Baker-Campbell-Hausdorff formula. Remarkably, the GOT establishes a formal relation between these two theorems, and it provides compact expressions for them, unlike the notoriously complicated ones currently known.

Key theories and technologies and implementation mechanism of parallel computing for ternary optical computer.

Ternary Optical Computer (TOC) is more advanced than traditional computer systems in parallel computing, which is characterized by huge amounts of repeated computations. However, the application of the TOC is still limited because of lack of key theories and technologies. In order to make the TOC applicable and advantageous, this paper systematically elaborates the key theories and technologies of parallel computing for the TOC through a programming platform, including reconfigurability and groupable usability of optical processor bits, parallel carry-free optical adder and the TOC's application characteristics, communication file to express user's needs and data organization method of the TOC. Finally, experiments are carried out to show the effectiveness of the present theories and technologies for parallel computing, as well as the feasibility of the implementation method of the programming platform. For a special instance, it is shown that the clock cycle on the TOC is only 0.26% of on a traditional computer, and the computing resource spent on the TOC is 25% of that on a traditional computer. Based on the study of the TOC in this paper, more complex parallel computing can be realized in the future.

On efficient matching of spatiotemporal rules

Spatiotemporal rules are ubiquitous nowadays. For example, a user may need to pay for a phone call or the usage of the Internet services. The payment may depend on many factors such as the duration of the call, the locations of the caller and callee, etc. Thus, the rules are spatiotemporal in nature. Given a set of such spatiotemporal rules and the behavior of a user, the problem of spatiotemporal rule matching (STRM) is to find the activated rules for the behavior. In this paper, we consider user behaviors as special instances of spatiotemporal data and formalize the STRM problem. We show how to represent common rules as spatiotemporal rules. Then, we show how to process multiple rules in advance to generate non-conflicting rules and organize them into tree-based index structures. After that, we propose novel algorithms, taking advantage of the tree-based index structures, to solve the STRM problem. Extensive experiments show that our proposed algorithms can outperform the baseline methods by a large margin, and thus are applicable in large scale real-life scenarios.

Axiomatic characterizations of Ptolemaic and chordal graphs

The interval function and the induced path function are two well studied class of set functions of a connected graph having interesting properties and applications to convexity, metric graph theory. Both these functions can be framed as special instances of a general set function termed as a transit function defined on the Cartesian product of a non-empty set \(V\) to the power set of \(V\) satisfying the expansive, symmetric and idempotent axioms. In this paper, we propose a set of independent first order betweenness axioms on an arbitrary transit function and provide characterization of the interval function of Ptolemaic graphs and the induced path function of chordal graphs in terms of an arbitrary transit function. This in turn gives new characterizations of the Ptolemaic and chordal graphs.

Reachability Analysis for Attributes in ABAC With Group Hierarchy

Attribute-based access control (ABAC) models are widely used to provide fine-grained and adaptable authorization based on the attributes of users, resources, and other relevant entities. Hierarchical group and attribute based access control (HGABAC) model was recently proposed which introduces the novel notion of attribute inheritance through group membership. GURA <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> was subsequently proposed to provide an administrative model for user attributes in HGABAC, building upon the ARBAC97 and GURA administrative models. The GURA model uses administrative roles to manage user attributes. The reachability problem for the GURA model is to determine what attributes a particular user can acquire, given a predefined set of administrative rules. This problem has been previously analyzed in the literature. In this article, we study the user attribute reachability problem based on directly assigned attributes of the user and attributes inherited via group memberships. We first define a restricted form of GURA <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> , called rGURA <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> scheme, as a state transition system with multiple instances having different preconditions and provide reachability analysis for each of these schemes. In general, we show PSPACE-complete complexity for all rGURA <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> schemes. We further present polynomial time algorithms with empirical experimental evaluation to solve special instances of rGURA <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> schemes under restricted conditions.

A new algorithm to compute fuzzy subgroups of a finite group

&lt;abstract&gt;&lt;p&gt;The enumeration of fuzzy subgroups is a complex problem. Several authors have computed results for special instances of groups. In this paper, we present a novel algorithm that is designed to enumerate the fuzzy subgroups of a finite group. This is achieved through the computation of maximal chains of subgroups. This approach is also useful for writing a program to compute the number of fuzzy subgroups. We provide further elucidation by computing the number of fuzzy subgroups of the groups $ Q_8 $ and $ D_8 $.&lt;/p&gt;&lt;/abstract&gt;

A new family of Apostol–Genocchi polynomials associated with their certain identities

ABSTRACT In this paper, we provide a generating function for mix type Apostol–Genocchi polynomials of order η associated with Bell polynomials. We also derive certain important identities of Apostol Genocchi polynomials of order η based on Bell polynomials, such as the correlation formula, the implicit summation formula, the derivative formula, some correlation with Stirling numbers and their special instances. Moreover, we discover some symmetric identities and their related known results.

An extension of a supercongruence of Long and Ramakrishna

We prove two supercongruences for specific truncated hypergeometric series. These include a uniparametric extension of a supercongruence that was recently established by Long and Ramakrishna [Adv. Math. 290 (2016), pp. 773–808]. Our proofs involve special instances of various hypergeometric identities including Whipple’s transformation and the Karlsson–Minton summation.

Free convection Hartmann flow of a viscous fluid with damped thermal transport through a cylindrical tube

• Free convection flow of viscous fluid in the presence of transverse magnetic field. • The thermal transport equation with Caputo-Fabrizio fractional integral operator. • The analytical solutions are found by using integral transforms. The free convection flow of a viscous fluid in a cylindrical tube is investigated in this article in the presence of a transverse magnetic field. The Caputo-Fabrizio fractional integral operator is used to generalize the thermal transport equation. Integral transforms, more specifically the Laplace and finite Hankel transforms, are used to find the solutions. The ordinary model is recovered as a particular case by taking the limit, and several more special instances are explored. Graphically, the impacts of the fractional and physical factors are highlighted. For a small and large values of time, the fractional parameter has distinct effects on the temperature and velocity.

Ramsey properties of semilinear graphs

A graph G is semilinear of complexity t if the vertices of G are elements of ℝd for some d ∈ ℤ+, and the edges of G are defined by the sign patterns of t linear functions f1,…, ft: ℝd × ℝd → ℝ. We show that semilinear graphs of constant complexity have very tame Ramsey properties. More precisely, we prove that if G is a semilinear graph of complexity t which contains no clique of size s and no independent set of size n, then G has at most \({O_{s,t}}\left(n \right) \cdot {\left({\log \,n} \right)^{{O_t}\left(1 \right)}}\) vertices. We also show that the logarithmic term cannot be omitted. In particular, this implies that if G is a semilinear graph of constant complexity on n vertices, and G contains no clique of size s, then G can be properly colored with polylog(n) colors. In the past 60 years, this coloring question was extensively studied for several special instances of semilinear graphs, e.g. shift graphs, intersection and disjointness graphs of certain geometric objects, and overlap graphs. Our main result provides a general upper bound on the chromatic number of all such, seemingly unrelated, graphs.Furthermore, we consider the symmetric Ramsey problem for semilinear graphs as well. It is known that if there exists an intersection graph of N boxes in ℝd (such graphs are semilinear of complexity 2d) that contains no clique or independent set of size n, then N = Od(n2(log n)d−1). That is, the exponent of n does not grow with the dimension. We prove a result about the symmetric Ramsey properties of semilinear graphs, which puts this phenomenon in a more general context.

Optimal regularity of solutions to no-sign obstacle-type problems for the sub-Laplacian

We establish the optimal $C_{H}^{1,1}$ interior regularity of solutions to \[ \Delta_{H}u=f\chi_{\{u\ne0\}}, \] where $\Delta_{H}$ denotes the sub-Laplacian operator in a stratified group. We assume the weakest regularity condition on $f$, namely $f*\Gamma$ is $C_{H}^{1,1}$, where $\Gamma$ is the fundamental solution of $\Delta_{H}$. The $C_{H}^{1,1}$ regularity is understood in the sense of Folland and Stein. In the classical Euclidean setting, the first seeds of the above problem are already present in the 1991 paper of Sakai and are also related to quadrature domains. As a special instance of our results, when $u$ is nonnegative and satisfies the above equation we recover the $C_{H}^{1,1}$ regularity of solutions to the obstacle problem in stratified groups, that was previously established by Danielli, Garofalo and Salsa. Our regularity result is sharp: it can be seen as the subelliptic counterpart of the $C^{1,1}$ regularity result due to Andersson, Lindgren and Shahgholian.

Modern applications of cross-classified random effects models in social and behavioral research: Illustration with R package PLmixed.

Cross-classified random effects models (CCREMs) have been developed for appropriately analyzing data with a cross-classified structure. Despite its flexibility and the prevalence of cross-classified data in social and behavioral research, CCREMs have been under-utilized in applied research. In this article, we present CCREMs as a general and flexible modeling framework, and present a wide range of existing models designed for different purposes as special instances of CCREMs. We also introduce several less well-known applications of CCREMs. The flexibility of CCREMs allows these models to be easily extended to address substantive questions. We use the free R package PLmixed to illustrate the estimation of these models, and show how the general language of the CCREM framework can be translated into specific modeling contexts.

Some New Bennett–Leindler Type Inequalities via Conformable Fractional Nabla Calculus

In this article, we prove several new fractional nabla Bennett–Leindler dynamic inequalities with the help of a simple consequence of Keller’s chain rule, integration by parts, mean inequalities and Hölder’s inequality for the nabla fractional derivative on time scales. As a result of this, some new classical inequalities are obtained as special cases, and we extended our inequalities to discrete and continuous calculus. In addition, when α=1, we shall obtain some well-known dynamic inequalities as special instances from our results. Symmetrical properties are critical in determining the best ways to solve inequalities.

Generic Techniques for Building Top- k Structures

A reporting query returns the objects satisfying a predicate q from an input set. In prioritized reporting , each object carries a real-valued weight (which can be query dependent), and a query returns the objects that satisfy q and have weights at least a threshold τ. A top- k query finds, among all the objects satisfying q , the k ones of the largest weights; a max query is a special instance with k = 1. We want to design data structures of small space to support queries (and possibly updates) efficiently. Previous work has shown that a top- k structure can also support max and prioritized queries with no performance deterioration. This article explores the opposite direction: do prioritized queries, possibly combined with max queries, imply top- k search? Subject to mild conditions, we provide affirmative answers with two reduction techniques. The first converts a prioritized structure into a static top- k structure with the same space complexity and only a logarithmic blowup in query time. If a max structure is available in addition, our second reduction yields a top- k structure with no degradation in expected performance (this holds for the space, query, and update complexities). Our techniques significantly simplify the design of top- k structures because structures for max and prioritized queries are often easier to obtain. We demonstrate this by developing top- k structures for interval stabbing, 3D dominance, halfspace reporting, linear ranking, and L ∞ nearest neighbor search in the RAM and the external memory computation models.

A note on fractional covers of a graph

A fractional colouring of a graph $G$ is a function that assigns a non-negative real value to all possible colour-classes of $G$ containing any vertex of $G$, such that the sum of these values is at least one for each vertex. The fractional chromatic number is the minimum sum of the values assigned by a fractional colouring over all possible such colourings of $G$. Introduced by Bosica and Tardif, fractional covers are an extension of fractional colourings whereby the real-valued function acts on all possible subgraphs of $G$ belonging to a given class of graphs. The fractional chromatic number turns out to be a special instance of the fractional cover number. In this work we investigate fractional covers acting on $(k+1)$-clique-free subgraphs of $G$ which, although sharing some similarities with fractional covers acting on $k$-colourable subgraphs of $G$, they exhibit some peculiarities. We first show that if a simple graph $G_2$ is a homomorphic image of a simple graph $G_1$, then the fractional cover number defined on the $(k+1)$-clique-free subgraphs of $G_1$ is bounded above by the corresponding number of $G_2$. We make use of this result to obtain bounds for the associated fractional cover number of graphs that are either $n$-colourable or $a\!\!:\!\!b$-colourable.

Online Bootstrap Inference For Policy Evaluation In Reinforcement Learning

The recent emergence of reinforcement learning (RL) has created a demand for robust statistical inference methods for the parameter estimates computed using these algorithms. Existing methods for inference in online learning are restricted to settings involving independently sampled observations, while inference methods in RL have so far been limited to the batch setting. The bootstrap is a flexible and efficient approach for statistical inference in online learning algorithms, but its efficacy in settings involving Markov noise, such as RL, has yet to be explored. In this article, we study the use of the online bootstrap method for inference in RL policy evaluation. In particular, we focus on the temporal difference (TD) learning and Gradient TD (GTD) learning algorithms, which are themselves special instances of linear stochastic approximation under Markov noise. The method is shown to be distributionally consistent for statistical inference in policy evaluation, and numerical experiments are included to demonstrate the effectiveness of this algorithm across a range of real RL environments. Supplementary materials for this article are available online.

Classifying decomposition and wavelet coorbit spaces using coarse geometry

This paper is concerned with the study of Besov-type decomposition spaces, which are scales of spaces associated to suitably defined coverings of the euclidean space Rd, or suitable open subsets thereof. A fundamental problem in this domain, that is currently not fully understood, is deciding when two different coverings give rise to the same scale of decomposition spaces.In this paper, we establish a coarse geometric approach to this problem, and show how it specializes for the case of wavelet coorbit spaces associated to a particular class of matrix groups H<GL(Rd) acting via dilations. These spaces can be understood as special instances of decomposition spaces, and it turns out that the question whether two different dilation groups H1,H2 have the same coorbit spaces can be decided by investigating whether a suitably defined map ϕ:H1→H2 is a quasi-isometry with respect to suitably defined word metrics. We then proceed to apply this criterion to a large class of dilation groups called shearlet dilation groups, where this quasi-isometry condition can be characterized algebraically. We close with the discussion of selected examples.

A Real-Time Iterative Projection Scheme for Solving the Common Fixed Point Problem and its Applications

In this paper we are concerned with the Common Fixed Point Problem (CFPP) with demi-contractive operators and its special instance, the Convex Feasibility Problem (CFP) in real Hilbert spaces. Motivated by the recent result of Ordoñez et al. [35] and in general, the field of online/real-time algorithms, for example [19, 20, 30], in which the entire input is not available from the beginning and given piece-by-piece, we propose an online/real-time iterative scheme for solving CFPPs and CFPs in which the involved operators/sets emerge along time. This scheme is capable of operating on any block, for any finite number of iterations, before moving, in a serial way, to the next block.The scheme is based on the recent novel result of Reich and Zalas [37] known as the Modular String Averaging (MSA) procedure. The convergence of the scheme follows [37] and other classical results in the fields of fixed point theory and variational inequalities, such as [34].Numerical experiments for linear and nonlinear feasibility problems with applications to image recovery are presented and demonstrate the validity and potential applicability of our scheme for example to online/real-time scenarios.