Related Theorems Research Articles

Single machine scheduling with interval grey processing time

PurposeThis paper aims to investigate the grey scheduling, which is the combination of grey system theory and scheduling problems with uncertain processing time. Based on the interval grey number and its related definitions, properties, and theorems, the single machine scheduling with uncertain processing time and its general forms are studied as the research object. Then several single machine scheduling models are reconstructed, and an actual production case is developed to illustrate the rationality of the research.Design/methodology/approachIn this paper, the authors first summarize the definitions and properties related to interval grey numbers, especially the transitivity of the partial order of interval grey numbers, and give an example to illustrate that the transitivity has a positive effect on the computational time complexity of multiple interval grey number comparisons. Second, the authors redefine the general form of the single machine scheduling problem with uncertain processing time according to the definitions and theorems of interval grey numbers. The authors then reconstruct three single machine scheduling models with uncertain processing time, give the corresponding heuristic algorithms based on the interval grey numbers and prove them. Finally, the authors develop a case study based on the engine test shop of K Company, the results show that the proposed single machine scheduling models and algorithms with uncertain processing time can provide effective guidance for actual production in an uncertain environment.FindingsThe main findings of this paper are as follows: (1) summarize the definitions and theorems related to interval grey numbers and prove the transitivity of the partial order of interval grey numbers; (2) define the general form of the single machine scheduling problem with interval grey processing time; (3) reconstruct three single machine scheduling models with uncertain processing time and give the corresponding heuristic algorithms; (4) develop a case study to illustrate the rationality of the research.Research limitations/implicationsIn the further research, the authors will continue to summarize more advanced general forms of grey scheduling, improve the theory of grey scheduling and prove it, and further explore the application of grey scheduling in the real world. In general, grey scheduling needs to be further combined with grey system theory to form a complete theoretical system.Originality/valueIt is a fundamental work to define the general form of single machine scheduling with uncertain processing time used the interval grey number. However, it can be seen as an important theoretical basis for the grey scheduling, and it is also beneficial to expand the application of grey system theory in real world.

Mod-φ convergence of Stirling distributions and limit theorems for zeros of their generating functions

We study mod-φ convergence of several probability distributions on the set of positive integers that involve Stirling numbers of both kinds and, as a consequence, derive various limit theorems for these distributions. We also derive closely related limit theorems for the distribution of zeros of the corresponding generating functions. For example, we identify the asymptotic distribution of zeros for the generating polynomial of the number of occupied boxes when n balls are allocated equiprobably and independently among θ boxes in the regime when θ grows linearly with n.

Representation of relative sheaf cohomology

Abstract We study the cohomology theory of sheaf complexes for open embeddings of topological spaces and related subjects. The theory is situated in the intersection of the general Čech theory and the theory of derived categories. That is to say, on the one hand the cohomology is described as the relative cohomology of the sections of the sheaf complex, which appears naturally in the theory of Čech cohomology of sheaf complexes. On the other hand it is interpreted as the cohomology of a complex dual to the mapping cone of a certain morphism of complexes in the theory of derived categories. We prove a “relative de Rham-type theorem” from the above two viewpoints. It says that, in the case the complex is a soft or fine resolution of a certain sheaf, the cohomology is canonically isomorphic with the relative cohomology of the sheaf. Thus the former provides a handy way of representing the latter. Along the way we develop various theories and establishes canonical isomorphisms among the cohomologies that appear therein. The second viewpoint leads to a generalization of the theory to the case of cohomology of sheaf morphisms. Some special cases together with applications are also indicated. Particularly notable is the application of the Dolbeault complex case to the Sato hyperfunction theory and other problems in algebraic analysis.

A large deviation theorem for a supercritical super-Brownian motion with absorption

AbstractWe consider a one-dimensional superprocess with a supercritical local branching mechanism $\psi$ , where particles move as a Brownian motion with drift $-\rho$ and are killed when they reach the origin. It is known that the process survives with positive probability if and only if $\rho<\sqrt{2\alpha}$ , where $\alpha=-\psi'(0)$ . When $\rho<\sqrt{2 \alpha}$ , Kyprianou et al. [18] proved that $\lim_{t\to \infty}R_t/t =\sqrt{2\alpha}-\rho$ almost surely on the survival set, where $R_t$ is the rightmost position of the support at time t. Motivated by this work, we investigate its large deviation, in other words, the convergence rate of $\mathbb{P}_{\delta_x} (R_t >\gamma t+\theta)$ as $t \to \infty$ , where $\gamma >\sqrt{2 \alpha} -\rho$ , $\theta \ge 0$ . As a by-product, a related Yaglom-type conditional limit theorem is obtained. Analogous results for branching Brownian motion can be found in Harris et al. [13].

Lacunary Statistical Convergence for Double Sequences on $\mathscr{L}-$ Fuzzy Normed Space

On $\mathscr{L}-$ fuzzy normed spaces, which is the generalization of fuzzy spaces, the notion of lacunary statistical convergence for double sequences which is a generalization of statistical convergence, are studied and developed in this paper. In addition, the definitions of lacunary statistical Cauchy and completeness for double sequences and related theorems are given on $\mathscr{L}-$ fuzzy normed spaces. Also, the relationship of lacunary statistical Cauchyness and lacunary statistical boundedness for double sequences with respect to $\mathscr{L}-$ fuzzy norm is shown.

Existence and Uniqueness of Polyhedra with Given Values of the Conditional Curvature

The theory of polyhedra and the geometric methods associated with it are interesting not only in their own right but also have a wide outlet in the general theory of surfaces. Certainly, it is only sometimes possible to obtain the corresponding theorem on surfaces from the theorem on polyhedra by passing to the limit. Still, the theorems on polyhedra give directions for searching for the related theorems on surfaces. In the case of polyhedra, the elementary-geometric basis of more general results is revealed. In the present paper, we study polyhedra of a particular class, i.e., without edges and reference planes perpendicular to a given direction. This work is a logical continuation of the author’s work, in which an invariant of convex polyhedra isometric on sections was found. The concept of isometry of surfaces and the concept of isometry on sections of surfaces differ from each other. Examples of isometric surfaces that are not isometric on sections and examples of non-isometric surfaces that are isometric on sections. However, they have non-empty intersections, i.e., some surfaces are both isometric and isometric on sections. In this paper, we prove the positive definiteness of the found invariant. Further, conditional external curvature is introduced for “basic” sets, open faces, edges, and vertices. It is proved that the conditional curvature of the polyhedral angle considered is monotonicity and positive definiteness. At the end of the article, the problem of the existence and uniqueness of convex polyhedra with given values of conditional curvatures at the vertices is solved.

Solution of road network problem with the help of m-polar fuzzy graph using isometric and antipodal concept

The isometry in crisp graph theory is a well-known fact. But, isometry under a fuzzy environment was developed recently and studied many facts. In a m-polar fuzzy graph, we have to think m components for each node and edge. Since, in our consideration, we consider m components for each nodes as well as edges, therefore we can not handle this type of situation using fuzzy model as their is a single components for this concept. Again, we can not apply bipolar or intuitionistic fuzzy graph model as each edges or nodes have just two components. Thus, these mPFG models give more efficient fuzziness results than other fuzzy model. Also, it is very interesting to develop and analyze such types of mPFGs with examples and related theorems. Considering all those things together, we have presented isometry under a m-polar fuzzy environment. In this paper, we have discussed the isometric m-polar fuzzy graph along with many exciting facts about it. Metric space properties have also been implemented on m-polar fuzzy isometric graph. We also have initiated a generalized fuzzy graph, namely antipodal m-polar fuzzy graphs, along with several issues. The degree of it is also presented along with edge regularity properties. We also give a relation between m-polar fuzzy antipodal graphs and their underlying crisp graphs. Its properties have also been discussed on m-polar fuzzy odd as well as even cycles, complete graphs, etc. Finally, a real-life application on a road network system in a m-polar fuzzy environment using the mu-distance concept is also presented.

Data block decomposition and intelligent secure acquisition of microdata

P-sets (P stands for Packet) is a set model with dynamic characteristics, which is obtained by introducing dynamic characteristics into Cantor set and improving Cantor set. According to the fact that the characteristics of class I big data are completely consistent with the basic characteristics of P-sets, this paper gives research on theory and application on class I big data from the view of mathematics. Here we introduce Class I big data which need some new definitions of data block, microdata and data link. Based on these concepts, decomposition theorem of data block and microdata relation theorem are given, and then attribute reasoning theorem and microdata intelligent discovery and the intelligent secure acquisition algorithm of microdata are also proposed. By using these theoretical results, the applications of secure acquisition of microdata are presented. In summary, P-sets mathematical model provides a new theory and method for studying class I big data.

Analysis of Fractional Kundu-Eckhaus and Massive Thirring Equations Using a Hybridization Scheme

This paper deals with the study of fractional Kundu-Eckhaus equation (FKEE) and fractional massive Thirring problem (FMTP) that appear in the quantum field theory, weakly nonlinear dispersive water waves, and nonlinear optics. Since the variational iteration method involves integration, the Laplace transform involves convolution theorem in recurrence relation to derive the series solution. To avoid some assumptions and hypothesis, we apply a two-scale approach for such a nonlinear complex model. The fractional differential equation may be transformed into its partner equation using He’s fractional complex transform, and then, the nonlinear elements can be readily handled using the homotopy perturbation method (HPM). Numerical results are derived in a rapid converge series form to improve the accuracy of the scheme greatly. Graphical representations and error distribution show that the two-scale approach is a very convenient tool.

A quantitative relative index theorem and Gromov’s conjectures on positive scalar curvature (with an appendix by Jinmin Wang and Zhizhang Xie)

In this paper, we prove a quantitative relative index theorem. It provides a conceptual framework for studying some conjectures and open questions of Gromov on positive scalar curvature. More precisely, we prove a \lambda -Lipschitz rigidity theorem for (possibly incomplete) Riemannian metrics on spheres with certain types of subsets removed. This \lambda -Lipschitz rigidity theorem is asymptotically optimal. As a consequence, we obtain an asymptotically optimal \lambda -Lipschitz rigidity theorem for positive scalar curvature metrics on hemispheres. These give positive answers to the corresponding open questions raised by Gromov. As another application, we prove Gromov's cube inequality on the bound of distances between opposite faces of spin manifolds with cube-like boundaries with a suboptimal constant. As immediate consequences, this implies Gromov’s cube inequality on the bound of widths of Riemannian cubes and Gromov’s conjecture on the bound of widths of Riemannian bands with suboptimal constants. Further geometric applications will be discussed in a forthcoming paper.

Complex Analysis and Residue Theorem

Mathematics is knitted into our lives by laying the fundamental base of many things. But people still think it is unnecessary to study math deeply due to its complexity. A lot of the great work from mathematicians is not noticed by other fields. Among all the essential mathematical methods and theorems, the Residue Theorem is one of the most significant ones. Therefore, this article will interpret the Residue Theorem step by step with its related theorems. Before jumping into the Residue Theorem, the article will introduce a series of mathematical methods, including the Triangle Inequality, the Euler's Formula, the Analytic function, the Taylor Series, and the Laurent Series. The applications and graphs of these mathematical methods will help to understand them further. Consequently, this article will demonstrate the most crucial part of this article — the Residue Theorem by its definition, formula, and example. Last but not least, this article will finish with how the Residue Theorem is used in our wind power generation system to express further how math correlates to our daily life. The result of this article shows that the Residue Theorem can be used to help explain other theorems and to extend new mathematical theories. It can be used not only in mathematics but also in a lot of fields, such as computer science, physics, and engineering. This article hopes a more extensive population can acknowledge the contribution of math to our world through the Residue Theorem.

Group Role Assignment With Constraints (GRA+): A New Category of Assignment Problems

This article systematically establishes a new category of assignment problems by reviewing and extending the problems related to group role assignment (GRA) from a novel vision. After reviewing seven related GRA with Constraints (GRA+) problems, this article specifies three new major assignment problems to make GRA+ problems complete and coherent. In addition, this article proves a series of new related theorems, proposes new conditions for the specified problems to have feasible solutions, and verifies the hardness of the newly specified problems. This article finally verifies the value of the presented theoretical work and provides a generalized formalization of this category of problems, i.e., the one highly abstract optimization problem, which is specified the first time. This article contributes to the literature of assignment problems with a novel category of well-defined problems, i.e., GRA+. The presented problem category is original, consistent, but far from complete. It will initiate further innovations in assignment research along with the presented directions. This article again demonstrates the power of the methodology role-based collaboration (RBC), and the environments—classes, agents, roles, groups, and objects (E-CARGO) model.

Generalized topological bulk-edge correspondence in bulk-Hermitian continuous systems with non-Hermitian boundary conditions

The bulk-edge correspondence (BEC) is the hallmark of topological systems. In continuous (nonlattice) Hermitian systems with an unbounded wave vector, it was recently shown that the BEC of Chern insulators is modified. How would it be further affected in non-Hermitian systems, experiencing loss and/or gain? In this work, we take the first step in this direction, by studying a bulk-Hermitian continuous system with non-Hermitian boundary conditions. We find in this case that edge modes emerge at the roots of the scattering matrix, as opposed to the Hermitian case, where they emerge at its poles (or, more accurately, coalescence of roots and poles). This entails a nontrivial modification to the relative Levinson's theorem. We then show that the topological structure remains the same as in the Hermitian case, and the generalized BEC holds, provided one employs appropriately modified contours in the wave-vector plane so that the scattering matrix phase winding counts the edge modes correctly. We exemplify all this using a paradigmatic model of waves in a shallow ocean or active systems in the presence of odd viscosity, as well as 2D electron gas with Hall viscosity. We use this opportunity to examine the case of large odd viscosity, where the scattering matrix becomes $2\ifmmode\times\else\texttimes\fi{}2$, which has not been discussed in previous works on the Hermitian generalized BEC.

A full-Eulerian approach for simulation of a system of fluid–rigid–elastic structure interaction based on the vorticity-stream function formulation

A monolithic mathematical framework for understanding the fluid–rigid–elastic structure interaction problem is proposed. A numerical method in a secondary formulation of the Navier–Stokes equations accompanying a technique for imposing the rigid boundaries is applied. The one-fluid formulation of the incompressible Navier–Stokes equation, containing the terms governing the elastic structure, is transformed into the vorticity-stream function formulation. The rigid structure is imposed in the flow field based on the velocity–vorticity kinematic relation and harmonic function theorem. The vorticity, level-set function, and left Cauchy–Green deformation tensor are updated utilizing three transport equations to investigate the evolution of the velocity field, elastic structure(s) configuration, and elastic stress tensor. The method is implemented to solve three challenging problems, and the results show its capabilities in proper imposing the rigid structures in the flow field and also the simultaneous modeling the rigid and elastic structure interactions with incompressible fluid flow.

On g-regularity and g-normality in Fuzzy Soft Topological Spaces

The main aim of this work is to introduce and study the notions of generalized regularity, normality, and symmetric in fuzzy soft topological spaces via fuzzy soft generalized closed sets. Investigating some of their basic properties. Many related theorems and relations of these notions are presented. Moreover, the hereditary property and some preservation theorems are discussed.

Sato hyperfunctions via relative Dolbeault cohomology

The relative Dolbeault cohomology which naturally comes up in the theory of Čech–Dolbeault cohomology turns out to be canonically isomorphic with the local (relative) cohomology of Grothendieck and Sato so that it provides a handy way of representing the latter. In this paper we use this cohomology to give simple explicit expressions of Sato hyperfunctions, some fundamental operations on them and related local duality theorems. This approach also yields a new insight into the theory of hyperfunctions and leads to a number of further results and applications. As one of such, we give an explicit embedding morphism of Schwartz distributions into the space of hyperfunctions.

An S-Original Algebraic Property Of Lie Algebras

This paper is dedicated to answer the following question:If L is a Lie algebra over a field with characteristic zero, and if S is an original property in Lie algebras class, then does S is a characteristic ideal of L?.For this goal, we have proved many related theorems, and illustrated some examples.

On Some Results About P - Lindelöf Spaces

In this work, we study p-Lindelöf spaces, a concept which is stronger than Lindelöf spaces. Every p-compact space is proven to be p-Lindelöf space, so the concept of p-Lindelöf is a generalization of the concept of p-compact. Most of the theorems which are valid for p-compact spaces will be also valid for p-Lindelöf spaces, (with a slight modification). We also study local p-Lindelöf spaces with many related theorems and examples.

On The Foundations of Fuzzy Number Theory and Fuzzy Diophantine Equations

Despite the great and rapid progress in the study of Fuzzy Logic and its applications in various scientific fields, it has not yet been used to build a consistent number theory like classical number theory. This research provides for the first time a conception of the concepts of number theory based on fuzzy logic and fuzzy membership functions, where it defines the division process, the fuzzy congruence, the greatest common divisor between integers with a fuzzy membership function. On the other hand, it presents many famous Diophantine equations formulated using fuzzy sets, in addition to many properties of fuzzy number theoretical systems, through many related theorems and accompanying illustrative examples. Also, in this research, we are raising many open research questions related to fuzzy number theory, which we believe will represent the future of progress in the study of this new mathematical branch.

On Intuitionistic Fuzzy Subgroups of (M-N) Type and Their Algebraic Properties

The objective of this paper is to define the intuitionistic fuzzy subgroup of type (M-N) and to study some of its elementary properties and substructures such as normality, direct and inverse images.Also, many related theorems and examples will be presented and illustrated.