Perfect Set Research Articles

Study on a Strong and Weak n-Connected Total Perfect k-Dominating set in Fuzzy Graphs

In this paper, the concept of a strong n-Connected Total Perfect k-connected total perfect k-dominating set and a weak n-connected total perfect k-dominating set in fuzzy graphs is introduced. In the current work, the triple-connected total perfect dominating set is modified to an n-connected total perfect k-dominating set nctpkD(G) and number γnctpD(G). New definitions are compared with old ones. Strong and weak n-connected total perfect k-dominating set and number of fuzzy graphs are obtained. The results of those fuzzy sets are discussed with the definitions of spanning fuzzy graphs, strong and weak arcs, dominating sets, perfect dominating sets, generalization of triple-connected total perfect dominating sets of fuzzy graphs, complete, connected, bipartite, cut node, tree, bridge and some other new notions of fuzzy graphs which are analyzed with a strong and weak nctpkD(G) set of fuzzy graphs. The order and size of the strong and weak nctpkD(G) fuzzy set are studied. Additionally, a few related theorems and statements are analyzed.

Evolutionary Game Analysis of Industry-University-Research Cooperative Innovation in Digital Media Enterprise Cluster Based on GS Algorithm

In order to study all the advantages and disadvantages of digital media enterprises, technological innovation can only be completed through cooperation. A kind of industry-university-research cooperative innovation evolutionary game method based on GS algorithm is proposed for digital media enterprise clusters. This method analyzes the evolutionary game theory of innovation and puts forward the evolutionary stability strategy of cooperative innovation between enterprises and research institutions. The results show that decreasing V is beneficial for the evolutionary game to approach the equilibrium point (1,1); that is, the greater the cost of independent innovation is compared with collaborative innovation, the stronger the willingness of both sides of the game to collaborative innovation. Enterprises and scientific research institutions are two different subjects with different interests. If they want to complete innovation cooperation, they need to formulate a perfect set of rules so that both sides of the game can carry out cooperative innovation according to the principles, so as to achieve the goal of cooperation.

Equivalence of Lebesgue's Theorem and Baire Characterization Theorem

Let $X$ be a complete separable metric space and $Y$ be a separable Banach space. We provide a proof of equivalence by linking explicitly the following statements:\\ \noindent \textbf{\textit{Lebesgue's Theorem.}} For every $\epsilon>0$ there exists a countable collection of closed sets $\left\lbrace C_n\right\rbrace $ of $X$ such that $$X=\bigcup_{n=1}^{\infty}C_n\;\;\text{and}\;\; \omega_f\left( C_n\right)<\epsilon\;\; \text{for each} \;\; n.$$ \textbf{\textit{Baire Characterization Theorem.}} For every nonempty perfect set $K\subset X$, the function $f|_K$ has at least one point of continuity in $K$. In fact, $C(f|_K)$ is dense in $K$.\\ \indent Moreover, replacing ``closed'' by ``open'' in the Lebesgue's Theorem, we obtain a characterization of continuous functions on space $X$.

PathAL: An Active Learning Framework for Histopathology Image Analysis.

Deep neural networks, in particular convolutional networks, have rapidly become a popular choice for analyzing histopathology images. However, training these models relies heavily on a large number of samples manually annotated by experts, which is cumbersome and expensive. In addition, it is difficult to obtain a perfect set of labels due to the variability between expert annotations. This paper presents a novel active learning (AL) framework for histopathology image analysis, named PathAL. To reduce the required number of expert annotations, PathAL selects two groups of unlabeled data in each training iteration: one "informative" sample that requires additional expert annotation, and one "confident predictive" sample that is automatically added to the training set using the model's pseudo-labels. To reduce the impact of the noisy-labeled samples in the training set, PathAL systematically identifies noisy samples and excludes them to improve the generalization of the model. Our model advances the existing AL method for medical image analysis in two ways. First, we present a selection strategy to improve classification performance with fewer manual annotations. Unlike traditional methods focusing only on finding the most uncertain samples with low prediction confidence, we discover a large number of high confidence samples from the unlabeled set and automatically add them for training with assigned pseudo-labels. Second, we design a method to distinguish between noisy samples and hard samples using a heuristic approach. We exclude the noisy samples while preserving the hard samples to improve model performance. Extensive experiments demonstrate that our proposed PathAL framework achieves promising results on a prostate cancer Gleason grading task, obtaining similar performance with 40% fewer annotations compared to the fully supervised learning scenario. An ablation study is provided to analyze the effectiveness of each component in PathAL, and a pathologist reader study is conducted to validate our proposed algorithm.

THE CONSISTENCY STRENGTH OF THE PERFECT SET PROPERTY FOR UNIVERSALLY BAIRE SETS OF REALS

AbstractWe show that the statement “every universally Baire set of reals has the perfect set property” is equiconsistent modulo ZFC with the existence of a cardinal that we call virtually Shelah for supercompactness (VSS). These cardinals resemble Shelah cardinals and Shelah-for-supercompactness cardinals but are much weaker: if $0^\sharp $ exists then every Silver indiscernible is VSS in L. We also show that the statement $\operatorname {\mathrm {uB}} = {\boldsymbol {\Delta }}^1_2$ , where $\operatorname {\mathrm {uB}}$ is the pointclass of all universally Baire sets of reals, is equiconsistent modulo ZFC with the existence of a $\Sigma _2$ -reflecting VSS cardinal.

Step roots of Littlewood polynomials and the extrema of functions in the Takagi class

AbstractWe give a new approach to characterising and computing the set of global maximisers and minimisers of the functions in the Takagi class and, in particular, of the Takagi–Landsberg functions. The latter form a family of fractal functions $f_\alpha:[0,1]\to{\mathbb R}$ parameterised by $\alpha\in(-2,2)$ . We show that $f_\alpha$ has a unique maximiser in $[0,1/2]$ if and only if there does not exist a Littlewood polynomial that has $\alpha$ as a certain type of root, called step root. Our general results lead to explicit and closed-form expressions for the maxima of the Takagi–Landsberg functions with $\alpha\in(-2,1/2]\cup(1,2)$ . For $(1/2,1]$ , we show that the step roots are dense in that interval. If $\alpha\in (1/2,1]$ is a step root, then the set of maximisers of $f_\alpha$ is an explicitly given perfect set with Hausdorff dimension $1/(n+1)$ , where n is the degree of the minimal Littlewood polynomial that has $\alpha$ as its step root. In the same way, we determine explicitly the minima of all Takagi–Landsberg functions. As a corollary, we show that the closure of the set of all real roots of all Littlewood polynomials is equal to $[-2,-1/2]\cup[1/2,2]$ .

COMPLEXITY OF INDEX SETS OF DESCRIPTIVE SET-THEORETIC NOTIONS

AbstractDescriptive set theory and computability theory are closely-related fields of logic; both are oriented around a notion of descriptive complexity. However, the two fields typically consider objects of very different sizes; computability theory is principally concerned with subsets of the naturals, while descriptive set theory is interested primarily in subsets of the reals. In this paper, we apply a generalization of computability theory, admissible recursion theory, to consider the relative complexity of notions that are of interest in descriptive set theory. In particular, we examine the perfect set property, determinacy, the Baire property, and Lebesgue measurability. We demonstrate that there is a separation of descriptive complexity between the perfect set property and determinacy for analytic sets of reals; we also show that the Baire property and Lebesgue measurability are both equivalent in complexity to the property of simply being a Borel set, for $\boldsymbol {\Sigma ^{1}_{2}}$ sets of reals.

Some perfect sets in ideal nano topological spaces

We introduce the notions of nano L*-perfect, nano R*-perfect, and nano C*-perfect sets in ideal nano spaces and study their properties. We obtained a characterization for compatible ideals via nano R*-perfect sets and and investigate further their important properties

THE SET OF INCOMPLETE SUMS OF THE MODIFIED GUTHRIE-NYMANN SERIES

In this paper we study topological and metric properties of the set of incomplete sums for positive series $\sum {a_k}$, where $a_{2n-1}=3/4^n+3/4^{in}$ and $a_{2n}=2/4^n+2/4^{in}$, $n \in N$. The series depends on positive integer parameter $i \geq 2$ and it is some perturbation of the known Guthrie-Nymann series. We prove that the set of incomplete sums of this series is a Cantorval (which is a specific union of a perfect nowhere dense set of zero Lebesgue measure and an infinite union of intervals), and its Lebesgue measure is given by formula: $\lambda(X^+_i)=1+\frac{1}{4^i-3}.$ The main idea of ??proving the theorem is based on the well-known Kakey theorem, the closedness of sets of incomplete sums of the series and the density of the set everywhere in a certain segment. The work provides a full justification of the facts for the case $i=2$. To justify the main facts, the ratio between the members and the remainders of the series is used. For $i=2$ we have $r_0=\sum {a_k}=2$, $a_{2n}-r_{2n}= \frac{1}{3} \cdot \frac{1}{4^n} + \frac{5}{3} \cdot \frac{1}{16^n}$ $r_{2n-1}-a_{2n-1}= \frac{2}{3} \cdot \frac{ 1}{4^n}-\frac{2}{3} \cdot \frac{1}{16^n}$. The relevance of the study of the object is dictated by the problems of the geometry of numerical series, fractal analysis and fractal geometry of one-dimensional objects and the theory of infinite Bernoulli convolutions, one of the problems of which is the problem of the singularity of the convolution of two singular distributions.

On the dimension spectra of infinite conformal iterated function systems

The dimension spectrum of a conformal iterated function system (CIFS) is the set of all Hausdorff dimensions of its various subsystem limit sets. This brief note provides two constructions – (i) a compact perfect set that cannot be realized as the dimension spectrum of a CIFS; and (ii) a similarity IFS whose dimension spectrum has zero Hausdorff dimension, and thus is not uniformly perfect – which resolve questions posed by Chousionis, Leykekhman and Urbański, and goes on to provoke fresh conjectures and questions regarding the topological and metric properties of IFS dimension spectra.

Theoretical Framework for Determination of Linear Structures in Multidimensional Geodynamic Data Arrays

The article addresses the issue of clustering of multidimensional data arrays with a noise using the methods of discrete mathematical analysis (DMA clustering). The theory of DMA clustering through the logical densities calculus is detailed, and the new algorithm Linear Discrete Perfect Sets (LDPS) is described. The main objective of the LDPS algorithm is to identify linearly stretched anomalies in a multidimensional array of geo-spatial data (geophysical fields, geochemistry, satellite images, local topography, maps of recent crustal movements, seismic monitoring data, etc.). These types of anomalies are associated with tectonic structures in the upper part of the Earth’s crust and pose the biggest threat for integrity of the isolation properties of the geological environment, including in regions of high-level radioactive waste disposal. The main advantage of the LDPS algorithm as compared to other cluster analysis algorithms that may be used in arrays with a noise is that it is more focused on searching for clusters that are linear. The LDPS algorithm can apply not only in the analysis of spatial natural objects and fields but also to elongated lineament structures.

Perfect Sets of Paths in the Full Graph of SDN Switches

Perfect Sets of Paths in the Full Graph of SDN Switches

Trash Trade and Environmental Regulations: An Assessment

Following China's decision to ban trash import in 2018, some countries in Southeast Asia are at the forefront of banning and imposing stringent measures to crackdown trash trade. The new trend in trash trade regulations had further impacted weaker countries and made a haven for dumping waste unless they adopt a uniform mechanism to regulate or completely ban trash trade within the region. This study aimed to analyze the recent response of some developing countries to the trash trade and how likely this response impact other countries in the same region, taking into account the impact of the trash trade on the environment and health and waste handling capacity. This study adopted doctrinal analysis that combined descriptive and analytical approaches to analyze trash trade and regulation impacts in developing countries and arrived at a need for a holistic approach and national waste management policy to encourage the recycling industry in trash receiving states. This study found that although developing countries had a perfect set of rules and norms regarding sustainable waste management and protecting illicit trash trade, most of them had shortages due to internal and external factors. Solid waste management in developing and least developed countries was a never-ending problem due to the low technological requirements and economic investments. 
 KEYWORDS: Trash Trade, Developing Countries, Environmental Justice.

Constructing Continuous Functions

Summary This paper introduces a simple geometric construction method for graphs of continuous functions. Graphs of a nowhere differentiable continuous function, a Lipschitz function that is not differentiable on a perfect set, and a continuous function whose level sets are countable are constructed using this method. We argue that introducing students to this method early on in Real Analysis fosters proper geometric intuition about continuity and provides for simple geometric proofs of non-differentiability properties.

The difference of uranyl (UO22+) complexes with Nitrilotri–3–propanoic acid and Tris(2–carboxyethyl) phosphine: N–tricarboxylate versus P–tricarboxylate

Nitrilotri–3–propanoic Acid (NTPA) and Tris(2–carboxyethyl) phosphine (TCEP) possess similar skeleton. This pair of isostructural ligands is a perfect set to assess the differences of uranyl (UO22+) complexes with N–tricarboxylate versus P–tricarboxylate. The protonation reactions of NTPA and TCEP were investigated by potentiometry and calorimetry. Complexation of uranyl with the both ligands was studied by potentiometry and structural analysis. The results indicate that central trialkylamine in NTPA has a higher affinity for the proton than trialkylphosphine in TCEP. Crystal structures and NMR analyses indicate that central amine of NTPA and central phosphine of TCEP are not involved in the bonding. NTPA/UO22+ complex is more stable than TCEP/UO22+ complex. This difference cannot be explained by electronic effect of central N and P atoms because of nonbonding interactions. The higher stability of UO22+/NTPA versus UO22+/TCEP may be due to the fact that the central N atom of NTPA has a lower steric hindrance than the central P atom of TCEP. Another possible reason is that UO22+/TCEP complexes have different coordination modes from UO22+/NTPA complexes.

A Review of Regression Models in Machine Learning

Machine learning is one of the active fields and technologies to realize artificial intelligence (AI). The complexity of machine learning algorithms creates problems to predict the best algorithm. There are many complex algorithms in machine learning (ML) to determine the appropriate method for finding regression trends, thereby establishing the correlation association in the middle of variables is very difficult, we are going to review different types of regressions used in Machine Learning. There are mainly six types of regression model Linear, Logistic, Polynomial, Ridge, Bayesian Linear and Lasso. This paper overview the above-mentioned regression model and will try to find the comparison and suitability for Machine Learning. A data analysis prerequisite to launch an association amongst the innumerable considerations in a data set, association is essential for forecast and exploration of data. Regression Analysis is such a procedure to establish association among the datasets. The effort on this paper predominantly emphases on the diverse regression analysis model, how they binning to custom in context of different data sets in machine learning. Selection the accurate model for exploration is the most challenging assignment and hence, these models considered thoroughly in this study. In machine learning by these models in the perfect way and thru accurate data set, data exploration and forecast can provide the maximum exact outcomes.

Uniformly perfect and hereditarily non uniformly perfect analytic and conformal non-autonomous attractor sets

Conditions are given which imply that certain non-autonomous analytic iterated function systems (NIFSs) in the complex plane have uniformly perfect attractor sets, while other conditions imply the attractor is pointwise thin and thus hereditarily non uniformly perfect. Examples are given to illustrate the main theorems, as well as to indicate how they generalize other results. Examples are also given to illustrate how possible generalizations of corresponding results for autonomous IFS's do not hold in general in this more flexible setting. Further, applications to non-autonomous Julia sets are given. Lastly, since our definition of NIFS is in some ways more general than others found in the literature, a careful analysis is given to show when certain familiar relationships still hold, along with detailed examples showing when other relationships do not hold.

Advances in research on solid-state fermented feed and its utilization: The pioneer of private customization for intestinal microorganisms.

With sustainable development of biotechnology, increasing attention has been placed on utilization of solid-state fermented feed (SFF). Solid-state fermented feed has been a candidate strategy to alleviate the contradiction between supply and demand of feed resources, ensure food hygiene safety, promoting energy conservation, and emission reduction. In production of SFF, a variety of organic acids, enzymes, vitamins, peptides, and other unknown growth factors are produced, which could affect performance of animals. Solid-state fermented feed produced by different fermentation techniques has great instability on different physiological stages of different animals, which hinders the application and standardized production of SFF. Herein, we summarize the current advances in the role of the characteristics of SFF prepared by different manufacturing technique and its research progress in animal experiments on growth performance, gastrointestinal ecology, and immune system, so as to provide references for further acquiring a relatively perfect set of SFF production and evaluation systems.

On generalized perfect difference sets constructed from Sidon sets

Let N be the set of positive integers. For a nonempty set A of integers and every integer u, denote by dA(u) the number of (a,a′) with a,a′∈A such that u=a−a′. For a sequence S of positive integers, let S(x) be the counting function of S. The set A⊆N is called a perfect difference set if dA(u)=1 for every positive integer u. In 2008, Cilleruelo and Nathanson (2008) [4] constructed dense perfect difference sets from dense Sidon sets. In this paper, as a main result, we prove that: let f:N→N be an increasing function satisfying f(n)≥2 for any positive integer n, then for every Sidon set B and every function ω(x)→∞, there exists a set A⊆N such that dA(u)=f(u) for every positive integer u and B(x/3)−ω(x)≤A(x)≤B(x/3)+ω(x) for all x≥Cf,B,ω.

Continuity of logarithmic capacity

We prove the continuity of logarithmic capacity under Hausdorff convergence of uniformly perfect planar sets. The continuity holds when the Hausdorff distance to the limit set tends to zero at sufficiently rapid rate, compared to the decay of the parameters involved in the uniformly perfect condition. The continuity may fail otherwise.