Empty Set Research Articles

Novel Recurrence Relations for Volumes and Surfaces of n-Balls, Regular n-Simplices, and n-Orthoplices in Real Dimensions

This study examines n-balls, n-simplices, and n-orthoplices in real dimensions using novel recurrence relations that remove the indefiniteness present in known formulas. They show that in the negative, integer dimensions, the volumes of n-balls are zero if n is even, positive if n = −4k − 1, and negative if n = −4k − 3, for natural k. The volumes and surfaces of n-cubes inscribed in n-balls in negative dimensions are complex, wherein for negative, integer dimensions they are associated with integral powers of the imaginary unit. The relations are continuous for n ∈ ℝ and show that the constant of π is absent for 0 ≤ n < 2. For n < −1, self-dual n-simplices are undefined in the negative, integer dimensions, and their volumes and surfaces are imaginary in the negative, fractional ones and divergent with decreasing n. In the negative, integer dimensions, n-orthoplices reduce to the empty set, and their real volumes and imaginary surfaces are divergent in negative, fractional ones with decreasing n. Out of three regular, convex polytopes present in all natural dimensions, only n-orthoplices and n-cubes (and n-balls) are defined in the negative, integer dimensions.

Slice regular functions in several variables

In this paper, we lay the foundations of the theory of slice regular functions in several (non-commuting) variables ranging in any real alternative ^*-algebra, including quaternions, octonions and Clifford algebras. This higher dimensional function theory is an extension of the classical theory of holomorphic functions of several complex variables. It is based on the construction of a family of commuting complex structures on {mathbb {R}}^{2^n}. One of the relevant aspects of the theory is the validity of a Cauchy-type integral formula and the existence of ordered power series expansions. The theory includes all polynomials and power series with ordered variables and right coefficients in the algebra. We study the real dimension of the zero set of polynomials in the quaternionic and octonionic cases and give some results about the zero set of polynomials with Clifford coefficients. In particular, we show that a nonconstant polynomial always has a non empty zero set.

BEXCIS: Bayesian methods for estimating the degree of the skewness of X chromosome inactivation

BackgroundX chromosome inactivation (XCI) is an epigenetic phenomenon that one of two X chromosomes in females is transcriptionally silenced during early embryonic development. Skewed XCI has been reported to be associated with some X-linked diseases. There have been several methods measuring the degree of the skewness of XCI. However, these methods may still have several limitations.ResultsWe propose a Bayesian method to obtain the point estimate and the credible interval of the degree of XCI skewing by incorporating its prior information of being between 0 and 2. We consider a normal prior and a uniform prior for it (respectively denoted by BN and BU). We also propose a penalized point estimate based on the penalized Fieller’s method and derive the corresponding confidence interval. Simulation results demonstrate that the BN and BU methods can solve the problems of extreme point estimates, noninformative intervals, empty sets and discontinuous intervals. The BN method generally outperforms other methods with the lowest mean squared error in the point estimation, and well controls the coverage probability with the smallest median and the least variation of the interval width in the interval estimation. We apply all the methods to the Graves’ disease data and the Minnesota Center for Twin and Family Research data, and find that SNP rs3827440 in the Graves’ disease data may undergo skewed XCI towards the allele C.ConclusionsWe recommend the BN method for measuring the degree of the skewness of XCI in practice. The R package BEXCIS is publicly available at https://github.com/Wen-YiYu/BEXCIS.

Gene-Based Methods for Estimating the Degree of the Skewness of X Chromosome Inactivation.

Skewed X chromosome inactivation (XCI-S) has been reported to be associated with some X-linked diseases, and currently several methods have been proposed to estimate the degree of the XCI-S (denoted as ) for a single locus. However, no method has been available to estimate for genes. Therefore, in this paper, we first propose the point estimate and the penalized point estimate of for genes, and then derive its confidence intervals based on the Fieller’s and penalized Fieller’s methods, respectively. Further, we consider the constraint condition of and propose the Bayesian methods to obtain the point estimates and the credible intervals of , where a truncated normal prior and a uniform prior are respectively used (denoted as GBN and GBU). The simulation results show that the Bayesian methods can avoid the extreme point estimates (0 or 2), the empty sets, the noninformative intervals () and the discontinuous intervals to occur. GBN performs best in both the point estimation and the interval estimation. Finally, we apply the proposed methods to the Minnesota Center for Twin and Family Research data for their practical use. In summary, in practical applications, we recommend using GBN to estimate of genes.

RETRACTED ARTICLE: Sentiment analysis using treebank filtered preprocess with relevant vector boost classifier

The procedure of identifying and classifying opinions in a piece of text to find out whether customer reviews toward a particular product or service are positive, negative, or neutral is termed as sentiment analysis. Stock market prediction is one of the most attractive topics in academic and real-life business. Many data mining techniques about sentiment analysis are suffering from the inaccuracy of prediction. The low classification accuracy has a direct effect on the reliability of stock market indicators. Treebank filtering Data Preprocessing based Ochiai-Barkman Relevance Vector Linear Programming Boost Classification technique is used for stock market prediction using sentimental analysis with higher prediction accuracy and lesser classification time for enhancing accuracy of stock market based on product review. Initially, the customer reviews and feedback on services or products are collected from the large database. After that, the collected customer reviews are preprocessed by performing the process such as tokenization, stemming, filtering. In order to achieve sentimental analysis through classifying customer reviews as positive and negative, Ochiai-Barkman Relevance Vector Linear Programming Boost Classification algorithm is used. The Linear Programming Boost Classification algorithm constructs with an empty set of weak classifiers as the Ochiai-Barkman Relevance Vector machine. The customer reviews are classified based on the Ochiai-Barkman similarity coefficient. The ensemble technique combines the weak classification results into strong by minimizing the error. In this way, the classification performance gets improved and the prediction of the stock market is carried out in a more accurate manner. Experimental evaluation is carried out on factors such as prediction accuracy, sensitivity, specificity, and prediction time versus amount of customer reviews.

Prediction and outlier detection in classification problems.

We consider the multi‐class classification problem when the training data and the out‐of‐sample test data may have different distributions and propose a method called BCOPS (balanced and conformal optimized prediction sets). BCOPS constructs a prediction set C(x) as a subset of class labels, possibly empty. It tries to optimize the out‐of‐sample performance, aiming to include the correct class and to detect outliers x as often as possible. BCOPS returns no prediction (corresponding to C(x) equal to the empty set) if it infers x to be an outlier. The proposed method combines supervised learning algorithms with conformal prediction to minimize a misclassification loss averaged over the out‐of‐sample distribution. The constructed prediction sets have a finite sample coverage guarantee without distributional assumptions. We also propose a method to estimate the outlier detection rate of a given procedure. We prove asymptotic consistency and optimality of our proposals under suitable assumptions and illustrate our methods on real data examples.

Numbers with empty rational Korselt sets

Let $N$ be a positive integer, and $\alpha=\dfrac{\alpha_{1}}{\alpha_{2}}\in \mathbb{Q}\setminus \{0,N\}$ with $\gcd(\alpha_{1}, \alpha_{2})=1$. $N$ is called an $\alpha$-Korselt number, equivalently $\alpha$ is said an $N$-Korselt base, if $\alpha_{2}p-\alpha_{1}$ divides $\alpha_{2}N-\alpha_{1}$ for every prime divisor $p$ of $N$. The set of $N$-Korselt bases in $\mathbb{Q}$ is denoted by $\mathbb{Q}$-$\mathcal{KS}(N)$ and called the set of rational Korselt bases of $N$.In this paper rational Korselt bases are deeply studied, where we give in details their belonging sets and their forms in some cases. This allows us to deduce that for each integer $n\geq 3$, there exist infinitely many squarefree composite numbers $N$ with $n$ prime factors and empty rational Korselt sets.

Utility-refined and budget-refined [formula omitted]-competitive equilibria

In finitely additive exchange economies the usual definition of competitive equilibrium can result in an empty equilibrium set, and therefore one has to consider notions of ɛ-competitive equilibria. In this paper we investigate the relationship between two notions of ɛ-competitive equilibria.

On the complexity of local-equitable coloring of graphs

An equitable k-partition (k≥2) of a vertex set S is a partition of S into k disjoint subsets (may be empty sets) such that the sizes of any two subsets of S differ by at most one. A local-equitable k-coloring of G is an assignment of k colors to the vertices of G such that, for every maximal clique of G, the coloring of this clique forms an equitable k-partition of itself. The local-equitable coloring of G is a stronger version of clique-coloring of graphs. Chordal graphs are 2-clique-colorable but not necessarily local-equitably 2-colorable. In this paper, we prove that it is NP-complete to decide the local-equitable 2-colorability in chordal graphs and even in split graphs. In addition, we prove that claw-free split graphs are local-equitably k-colorable when k≤4, but not necessarily local-equitably k-colorable when k≥5. A sufficient and sharp condition of local-equitably k-colorability is also given in claw-free split graphs. Secondly, we show that, given a split graph G, deciding the local-equitable k-colorability of G is solvable in polynomial time when k=ω(G)−1, where ω(G) is the clique number of G. At last, we prove that the decision problem of local-equitable 2-coloring of planar graphs is solvable in polynomial time.

Zero – an Uncommon Number: Preschoolers‘ Conceptual Understanding of Zero

The conceptual development of natural number in preschoolers is well-researched. However, less is known about the conceptual development of zero. Recent studies suggest that children develop an understanding of zero after learning to count. It remains unclear, when a conceptual understanding of “zero” as number word for an empty set emerges. This paper integrates numerical and language theories about how, where and when the concept of zero is formed and is integrated into the class of natural numbers. The counting skills of 107 preschoolers were assessed for the number range between zero and eight as well as for their ordinal understanding of zero. The results show that compared to the natural numbers, zero was substantially more difficult. Children are able to list zero in a number word sequence (0, 1, 2, 3 .... or 3, 2, 1, 0), but were unable to describe a set as having zero numbers. This latter conception contradicts findings regarding natural numbers, in that an empty set is counter intuitive. Zero could be correctly placed when consecutive order was required, but addition and subtraction by counting was more difficult. The results suggest that the conceptual development of zero differs qualitatively from the natural numbers. Based on the results, the ordinal understanding of zero as a predecessor to one, together with its matching linguistic concepts is proposed to be the key to the conceptual development of zero.

Properties of Group Neutro-Topological Space

Unlike traditional algebraic structures, where all operations are well-defined and all axioms are completely true, NeutroAlgebras and AntiAlgebras allow operations to be partially well-defined and axioms to be partially true or fully outer-defined, and axioms to be completely false. These NeutroAlgebras and AntiAlgebras represent a new research subject based on real-world examples. Since an empty set is not a subgroup of a group by observing this, the article leads to learning group neutro-topological space. We introduced the notion of a group neutro-topological space and investigated its properties.

Spectra of the Energy Operator of Two-Electron System in the Impurity Hubbard Model

We consider two-electron systems for the impurity Hubbard Model and investigate the spectrum of the system in a singlet state for the v-dimensional integer valued lattice Zv. We proved the essential spectrum of the system in the singlet state is consists of union of no more then three intervals, and the discrete spectrum of the system in the singlet state is consists of no more then five eigenvalues. We show that the discrete spectrum of the system in the triplet and singlet states differ from each other. In the singlet state the appear additional two eigenvalues. In the triplet state the discrete spectrum of the system can be empty set, or is consists of one-eigenvalue, or is consists of two eigenvalues, or is consists of three eigenvalues. For investigation the structure of essential spectra and discrete spectrum of the energy operator of two-electron systems in an impurity Hubbard model, for which the momentum representation is convenient. In addition, we used the tensor products of Hilbert spaces and tensor products of operators in Hilbert spaces and described the structure of essential spectrum and discrete spectrum of the energy operator of two-electron systems in an impurity Hubbard model.

Algebraic Structure for (λ, μ)-Diophantine Neutrosophic Bisemiring

We introduce the notion of Diophantine neutrosophic subbisemiring (DioNSBS), level sets of DioNSBS of a bisemiring. The concept of DioNSBS is a generalization of fuzzy subbisemiring over bisemiring. We interact the theory for (λ, μ)-DioNSBS over bisemiring. Let α be the Diophantine neutrosophic subset in S , we show that α = ⟨(ΞT α , ΞI α , ΞF α ), (Γα, ∆α, Θα)⟩ is a DioNSBS of S if and only if all non empty level set α(t,s) is a subbisemiring of S for t, s ∈ [0, 1]. Let α be the DioNSBS of a bisemiring S and W be the strongest Diophantine neutrosophic relation of S , we observe that α is a DioNSBS of S if and only if W is a DioNSBS of S × S . Let α1, α2, ..., αn be the family of DioN SBSs of S1, S2, ..., Sn respectively. We show that α1× α2 × ... × αn is a DioNSBS of S1 × S2 × ... × Sn. The homomorphic image of DioNSBS is a DioNSBS. The homomorphic preimage of DioNSBS is a DioNSBS. Examples are provided to illustrate our results.

A Non-Euclidean Story or: How to Persist When Your Geometry Doesn’t

Too little mathematics has been written in prose. Thus we prove here, via a fantasy novellette, that a locally L-bilipschitz mapping f : X → Y between uniformly Ahlfors q-regular, complete and locally compact path-metric spaces X and Y is an L-bilipschitz map when Y is simply connected. The motivation for such a result arises from studying the asymptotic values of BLD-mappings with an empty branch set.

Sometimes nothing is simply nothing: Automatic processing of empty sets.

Previous work using the numerical comparison task has shown that an empty set, the nonsymbolic manifestation of zero, can be represented as the smallest quantity of the numerical magnitude system. In this study, we examined whether an empty set can be represented as such under conditions of automatic processing in which deliberate processing of stimuli magnitudes is not required by the task. In Experiment 1, participants performed physical and numerical comparisons of empty sets (i.e., empty frames) and of other numerosities presented as framed arrays of 1 to 9 dots. The physical sizes of the frames varied within pairs. Both tasks revealed a size congruity effect (SCE) for comparisons of non-empty sets. In contrast, comparisons to empty sets produced an inverted SCE in the physical comparison task, whereas no SCE was found for comparisons to empty sets in the numerical comparison task. In Experiment 2, participants performed an area comparison task using the same stimuli as Experiment 1 to examine the effect of visual cues on the automatic processing of empty sets. The results replicated the findings of the physical comparison task in Experiment 1. Taken together, our findings indicate that empty sets are not perceived as "zero," but rather as "nothing," when processed automatically. Hence, the perceptual dominance of empty sets seems to play a more important role under conditions of automatic processing, making it harder to abstract the numerical meaning of zero from empty sets.

A localization strategy combined with transfer learning for image annotation.

This study aims to solve the overfitting problem caused by insufficient labeled images in the automatic image annotation field. We propose a transfer learning model called CNN-2L that incorporates the label localization strategy described in this study. The model consists of an InceptionV3 network pretrained on the ImageNet dataset and a label localization algorithm. First, the pretrained InceptionV3 network extracts features from the target dataset that are used to train a specific classifier and fine-tune the entire network to obtain an optimal model. Then, the obtained model is used to derive the probabilities of the predicted labels. For this purpose, we introduce a squeeze and excitation (SE) module into the network architecture that augments the useful feature information, inhibits useless feature information, and conducts feature reweighting. Next, we perform label localization to obtain the label probabilities and determine the final label set for each image. During this process, the number of labels must be determined. The optimal K value is obtained experimentally and used to determine the number of predicted labels, thereby solving the empty label set problem that occurs when the predicted label values of images are below a fixed threshold. Experiments on the Corel5k multilabel image dataset verify that CNN-2L improves the labeling precision by 18% and 15% compared with the traditional multiple-Bernoulli relevance model (MBRM) and joint equal contribution (JEC) algorithms, respectively, and it improves the recall by 6% compared with JEC. Additionally, it improves the precision by 20% and 11% compared with the deep learning methods Weight-KNN and adaptive hypergraph learning (AHL), respectively. Although CNN-2L fails to improve the recall compared with the semantic extension model (SEM), it improves the comprehensive index of the F1 value by 1%. The experimental results reveal that the proposed transfer learning model based on a label localization strategy is effective for automatic image annotation and substantially boosts the multilabel image annotation performance.

DEVELOPMENT OF LEARNING TRAJECTORY ON THE SET TOPIC FOR 7TH GRADE IN THE CONTEXT OF SEDEKAH LAUT TRADITION

Set is an important topic to be mastered by students because it influences the development of mathematics in daily life. However, many students still have difficulty learning the topic. Therefore, it is necessary to design a learning trajectory using the appropriate approach, context, and media. This research resulted in the learning development using Sedekah Laut context to create meaningful learning and increase students' understanding of sets. The method used in this study was design research proposed by Gravemeijer & Cobb with three stages: preliminary design, experimental design (pilot experiments and teaching experiments), and retrospective analysis. However, this article only presented the results from the Preliminary design stage. The participants involved in this study were 7th-grade students of SMP Negeri 6 Semarang. The resulting hypothetical learning trajectory consists of a series of learning processes: observing context videos to find the concepts of sets, non-sets, empty sets, universal sets, and Venn diagrams; explaining the properties of the set; defining set operations; and solving problems related to sets.

The profinite topology of free groups and weakly generic tuples of automorphisms

AbstractLet be a countable first order structure and endow the universe of with the discrete topology. Then the automorphism group of becomes a topological group. A tuple of automorphisms is defined to be weakly generic iff its diagonal conjugacy class (in the algebraic sense) is dense (in the topological sense) and the ‐orbit of each is finite. Existence of tuples of weakly generic automorphisms are interesting from the point of view of model theory as well as from the point of view of finite combinatorics.The main results of the present work are as follows. In Theorem 2.6 we characterize the existence of tuples of weakly generic automorphisms with the aid of the profinite topology of free groups. In Corollary 2.12 we will show that if has finite topological rank r (and satisfies a further, mild technical condition) then the existence of a weakly generic tuple in implies the existence of weakly generic tuples in for all natural number . Finally, in Theorem 3.2 we show that if is a countable model of an ℵ0‐categorical, simple theory in which all types over the empty set are stationary and has a pair of weakly generic automorphisms then it has tuples of weakly generic automorphisms of arbitrary finite length.At the technical level we will combine elementary investigations about the profinite topology of free groups with the results of [11] about topological ranks of the automorphism groups of some structures.

Combined effects of singular and superlinear nonlinearities in singular double phase problems in [formula omitted

This paper is concerned with multiplicity results for parametric singular double phase problems in RN via the Nehari manifold approach. It is shown that the problem under consideration has at least two nontrivial weak solutions provided the parameter is sufficiently small. The idea is to split the Nehari manifold into three disjoint parts minimizing the energy functional on two of them. The third set turns out to be the empty set for small values of the parameter.

مجموعات من التحولات غير الثنائية رسم الخرائط على مجموعة محدودة

Abstract: The aim of this study is to find new groups that consisting of non-bijective transformations cannot subset of symmetric groups Sn on finite set A called G(Non-bT)groups. In addition, we give the necessary and sufficient condition for the elements of these groups on anon empty set A by using the quotient set of A, Moreover, we conclude that G(Non-bT)  Sn on the quotient set of A. Key words: Transformation, Symmetric Group, Equivalence Class, Quotient Set, Partition