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Most of the existing synthetic aperture radar (SAR) automatic target recognition (ATR) methods aim at the closed set situation, in which the classes of targets in the test set have appeared in the training set. However, in practice, the classifier is likely to encounter the targets from unseen categories and classify them incorrectly, which brings a huge challenge to current SAR ATR techniques. To overcome this problem, this letter proposes an open set recognition (OSR) method based on multitask learning, and the method is developed from generative adversarial network (GAN). Essentially, this method decomposes OSR into two tasks: classification and abnormal detection. The classification task is the same as that in the closed set situation, while the abnormal detection task is used to determine whether the targets belongs to the unseen categories. Correspondingly, the network structure of GAN is modified and the other full-connection network branch is added to the end of the discriminator, so it has the ability to accomplish the above two tasks. Finally, according to the results of two tasks, the OSR for SAR targets can be realized. The experimental results on moving and stationary target acquisition (MSTAR) dataset demonstrate that the proposed method has the better recall, precision, F1, and accuracy than other OSR methods.

Real-world applications now deal with a massive amount of data, and information about the world is inaccurate, incomplete, or uncertain. Therefore, we present in our paper a proposed model for solving problems. This model is based on the class of locally generalized closed sets, namely, locally simply* alpha generalized closed* sets and locally simply* alpha generalized closed** sets (briefly, L S-M*alpha GC*-sets and L S-M*alpha GC**-sets), based on simply* alpha open set. We also introduce various concepts of their properties and their relationship with other types, and we are studying several of their properties. Finally, we apply the concept of the simply* alpha open set to illustrate the importance of our method in decision-making for information systems about the infections of Coronavirus in humans. In fact, we were able to decide the impact factors of Coronavirus infection. The results were also programmed using the MATLAB program. Therefore, it is recommended that our proposed concept be used in future decision-making.

In this paper, the concept of soft closed groups is presented using the soft ideal pre-generalized open and soft pre-open, which are -ᶅ- - -closed sets " -closed", Which illustrating several characteristics of these groups. We also use some games and - Separation Axiom, such as (Ʈ0, Ӽ, ᶅ) that use many tables and charts to illustrate this. Also, we put some proposals to study the relationship between these games and give some examples.

In traditional semantic segmentation, knowing about all existing classes is essential to yield effective results with the majority of existing approaches. However, these methods trained in a Closed Set of classes fail when new classes are found in the test phase, not being able to recognize that an unseen class has been fed. This means that they are not suitable for Open Set scenarios, which are very common in real-world computer vision and remote sensing applications. In this paper, we discuss the limitations of Closed Set segmentation and propose two fully convolutional approaches to effectively address Open Set semantic segmentation: OpenFCN and OpenPCS. OpenFCN is based on the well-known OpenMax algorithm, configuring a new application of this approach in segmentation settings. OpenPCS is a fully novel approach based on feature-space from DNN activations that serve as features for computing PCA and multi-variate gaussian likelihood in a lower dimensional space. In addition to OpenPCS and aiming to reduce the RAM memory requirements of the methodology, we also propose a slight variation of the method (OpenIPCS) that uses an iteractive version of PCA able to be trained in small batches. Experiments were conducted on the well-known ISPRS Vaihingen/Potsdam and the 2018 IEEE GRSS Data Fusion Challenge datasets. OpenFCN showed little-to-no improvement when compared to the simpler and much more time efficient SoftMax thresholding, while being some orders of magnitude slower. OpenPCS achieved promising results in almost all experiments by overcoming both OpenFCN and SoftMax thresholding. OpenPCS is also a reasonable compromise between the runtime performances of the extremely fast SoftMax thresholding and the extremely slow OpenFCN, being able to run close to real-time. Experiments also indicate that OpenPCS is effective, robust and suitable for Open Set segmentation, being able to improve the recognition of unknown class pixels without reducing the accuracy on the known class pixels. We also tested the scenario of hiding multiple known classes to simulate multimodal unknowns, resulting in an even larger gap between OpenPCS/OpenIPCS and both SoftMax thresholding and OpenFCN, implying that gaussian modeling is more robust to settings with greater openness.

Using deep learning, the supervised hyperspectral image (HSI) classification methods are based on the ideal assumption of closed sets, in which all testing classes are defined a priori. However, it...

In this paper pgrw-locally closed set, pgrw-locally closed*-set and pgrw-locally closed**-set are introduced. A subset A of a topological space (X,t) is called pgrw-locally closed (pgrw-lc) if A=GÇF where G is a pgrw-open set and F is a pgrw-closed set in (X,t). A subset A of a topological space (X,t) is a pgrw-lc* set if there exist a pgrw-open set G and a closed set F in X such that A= GÇF. A subset A of a topological space (X,t) is a pgrw-lc**-set if there exists an open set G and a pgrw-closed set F such that A=GÇF. The results regarding pgrw-locally closed sets, pgrw-locally closed* sets, pgrw-locally closed** sets, pgrw-lc-continuous maps and pgrw-lc-irresolute maps and some of the properties of these sets and their relation with other lc-sets are established.

In this paper, it is shown that for a minimal system (X, G), if H is a normal subgroup of G with finite index n, then X can be decomposed into n components of closed sets such that each component is minimal under H-action. Meanwhile, we prove that for a residual set of points in a minimal system with finitely many commuting homeomorphisms, the set of return times to any non-empty open set contains arbitrarily long geometric progressions in multidimension, extending a previous result by Glasscock, Koutsogiannis and Richter.

The ‘the’ in the title hides a subtlety. A metric space induces not one but four topologies - by means of open sets, closed sets, closure, and interior - they just so happen to coincide. The agreement between these four structures arising from a metric function d : X × X → [ 0 , ∞ ] is due to a combination of the metric axioms and the lattice structure of [ 0 , ∞ ] . Further motivation materializes from Lawvere's observation from 1973 to the effect that a (slightly generalized) metric space is a category enriched in [ 0 , ∞ ] . Metric spaces taking values in structures other than [ 0 , ∞ ] are relevant for generalizations of metric spaces and find a natural home in Lawvere's categorical setting. In particular, in recent years quantales emerged as structures occupying an important niche in between [ 0 , ∞ ] and arbitrary monoidal categories. Since a category enriched in a quantale Q is the same thing as a metric space taking values in Q one may ask whether such a thing belongs to algebra or geometry. Further, does the quadruplet of topologies associated to a Q-valued space/category still consist of identical siblings? We propose a litmus test for the geometricity of Q-valued spaces as we investigate these issues.

The popularity of printing devices has multiplied the diffusion of printed documents, raising concerns regarding the security and integrity of their content. The same device that prints reliable contracts, newspapers, and others, can also be used for malicious purposes, such as printing fake money, forging fake contracts, and producing illegal packaging, thus calling for the development of image forensics techniques to pinpoint criminal printed materials and trace back to their origin. Despite some recent advances, previous works model such a problem as a big data-focused closed-set classification problem. In this work, we address the source linking problem of printed color documents by treating it as a verification problem. Specifically, we aim at deciding if two documents have been printed by the same printer or not. To achieve this goal, and to cope with the data scarcity deriving from the difficulty of gathering massive amounts of printed and scanned documents, we propose to use an ensemble of Siamese Neural Networks, with unique architectures expressly designed to work with a small training dataset. As a further unique feature, the proposed approach is suited to work in an open set scenario, where the printers used to produce the documents analyzed at the test time are not included in the training set. Results obtained under both open and closed set conditions, with a thorough comparison with available baseline methods, showed classification performance higher than 97% in the closed set scenario and higher than 86% in the open set case, highlighting the practicality of such approaches in real-world scenarios.

Do the topologies of each dimension have to be same of any space? I show that this is not necessary with amply soft topology produced by classical topologies. For example, an amply soft topology produced by classical topologies may have got any indiscrete topologies, discrete topologies or any topological spaces in each different dimension. The amply soft topology allows to write elements of different classical topologies in its each parameter sets. The classical topologies may be finite, infinite, countable or uncountable. This situation removes the boundary in soft topology and cause it to spread over larger areas. Amply soft topology produced by classical topologies is a special case of an amply soft topology. For this, I define a new soft topology it is called as an amply soft topology. I introduce amply soft open sets, amply soft closed sets, interior and closure of an amply soft set and subspace of any amply soft topological space. I gave parametric separation axioms which are different from Ti separation axioms. Ti questions the relationship between the elements of space itself while Pi questions the strength of the connection between their parameters. An amply soft topology is built on amply soft sets. Amply soft sets use any kind of universal parameter set or initial universe (such as finite or infinite, countable or uncountable). Also, subset, superset, equality, empty set, whole set on amply soft sets are defined. And operations such as union, intersection, difference of two amply soft sets and complement of an amply soft set are given. Then three different amply soft point such as amply soft whole point, amply soft point and monad point are defined. And also I give examples related taking a universal set as uncountable.

The paper presents the notion of $$(i,j)^{*}~\gamma $$ -generalized fuzzy closed set in a fuzzy bitopological space considering a parallel form of (i, j) fuzzy $$\gamma $$ -open set. We introduce the idea of $$\gamma $$ -generalized fuzzy q-neighbourhood of a fuzzy point. Moreover, we study various related properties and cite some interesting results on this set. Also, we show that generalized fuzzy closed sets and $$\gamma $$ -generalized fuzzy closed sets are completely independent of each other in the same environment. Furthermore, we introduce the notion of $$\gamma $$ -generalized fuzzy continuous function in a fuzzy bitopological space. Finally, we propose a new type of closure operator and establish important results based on this concept.

In this article, we discuss the relation of soft ω -open sets and soft ω -closed sets with other soft sets. We further define the concept of soft ω -interior and soft ω -closure and study their rel...

In this paper, we present and study some of the basic properties of the new class of sets called weakly b-closed sets and weakly b- open sets in fuzzy neutrosophic bi-topological spaces. We referred to some results related to the new definitions, which we taked the case of equal in the definition of b-sets instead of subset. Then, we discussed the relations between the new defined sets by hand and others fuzzy neutrosophic sets which were studied before us on the other hand on fuzzy neutrosophic bi-topological spaces. Then, we have studied some of characteristics and some relations are compared with necessary examples.

The rapid development of the Internet of Things has led to a series of security and privacy issues. Although existing network intrusion detection technologies can identify abnormal traffic, they are mostly focused on detecting in closed sets. For a real open network environment, when an unknown attack occurs, the existing detection system cannot recognize it correctly, which will severely threaten network security. In order to solve this problem, this paper investigates how the Extreme Value Theory (EVT) is applied to unknown network attack detection system and proposes a network intrusion detection method based on open set recognition. By fitting the known classes’ post recognition activations to a Weibull distribution, we build the Open-CNN model to recalculate each activation at the penultimate level, then the pseudo-probability of unknown classes can be estimated from the activation scores of known classes, realizing the detection purpose of unknown attacks. We perform experiments on multiple datasets with different types and feature distributions. All of them obtain high detection accuracy, which proves the effectiveness and robustness of the proposed method.

In this paper, we initiated new types of near open and closed sets called higher order sets. These sets generated by much iteration of topological interior and closure operations for a given set. These new sets are the terms of many sequences of topological near open and near closed sets. We studied many generalizations of the classical near open sets to these sequences. This paper is the starting point of a new way for researchers to study the high order topology.

Reliable and rapid identification of timber species is a very relevant issue for many countries in South America and especially for Peru, which is the second country with the largest extent of tropical forest, and that is because this issue is a necessity in order to develop an effective management of the forest resources, such as inspection and control of the timber commerce. Since current methods of identification are based on a closed set recognition approach, they are not reliable enough to be used in a practical application because scenarios of identification of timber species are by nature an open set recognition problem. For that reason, in this work we propose a convolutional neural network that has two main characteristics, being able to run in a real-time embedded system and being able to handle the open set recognition problem, that is, this model can discriminate between known and unknown species. In order to evaluate it, tests are performed in two timber species datasets and some experiments are developed in the embedded system Raspberry Pi3B+ to measure energy consumption. The results present high metrics, which means that it manages to discriminate the unknown species with accuracy and F1 score above 91% for two sets of images used. In addition to this, our proposed model obtain lower maximum power value (10-12%) and computational resource usage (5-13%) than a classical convolutional model and MobileNetsV2 measured on the Raspberry Pi3B+.

Anomaly classification can provide a discrimination of the type of anomaly at hand, that can help the user to narrow down the search space during troubleshooting. A limitation of most machine learning (ML) classification models is being under the closed set assumption and thereby failing to recognize unknown data. Literature refers to this issue as the Opens Set Recognition (OSR) problem. In this paper, the given use case is the detection of technical faults in HVAC systems in commercial buildings. Here, the OSR problem is of high relevance, because it is very unrealistic that all possible faults are already known beforehand. This paper proposes an ML pipeline for open set anomaly classification. To address the OSR problem, the pipeline makes use of three common neural network architecture models that successively build on each other, namely two autoencoders and one ANN classifier. The advantage of using common well-known models is the ease of implementation for solution developers. The validation is made against an HVAC dataset published by LBNL that contains 7 fault cases, where we assume 3 of them as unknown. Our validation results show that the proposed pipeline can significantly reduce the risk of false class predictions in the presence of unknown data.

For the existing Closed Set Recognition (CSR) methods mistakenly identify unknown jamming signals as a known class, a Conditional Gaussian Encoder (CG-Encoder) for 1-dimensional signal Open Set Recognition (OSR) is designed. The network retains the original form of the signal as much as possible and deep neural network is used to extract useful information. CG-Encoder adopts residual network structure and a new Kullback-Leibler (KL) divergence is defined. In the training phase, the known classes are approximated to different Gaussian distributions in the latent space and the discrimination between classes is increased to improve the recognition performance of the known classes. In the testing phase, a specific and effective OSR algorithm flow is designed. Simulation experiments are carried out on 9 jamming types. The results show that the CSR and OSR performance of CG-Encoder is better than that of the other three kinds of network structures. When the openness is the maximum, the open set average accuracy of CG-Encoder is more than 70%, which is about 30% higher than the worst algorithm, and about 20% higher than the better one. When the openness is the minimum, the average accuracy of OSR is more than 95%.

The aim of this paper is to introduce the concepts of IFS \(_{\alpha}\) -open sets. Also we discussed the relationship between this type of Open set and other existing Open sets in Intuitionistic fuzzy topological spaces. Also we introduce new class of closed sets namely IFS \(_{\alpha}\) -closed sets and its properties are studied.

The goal of domain adaptation (DA) is to train a good model for a target domain, with a large amount of labeled data in a source domain but only limited labeled data in the target domain. Conventional closed set domain adaptation (CSDA) assumes source and target label spaces are the same. However, this is not quite practical in real-world applications. In this work, we study the problem of open set domain adaptation (OSDA), which only requires the target label space to partially overlap with the source label space. Consequently, the solution to OSDA requires unknown classes detection and separation, which is normally achieved by introducing a threshold for the prediction of target unknown classes; however, the performance can be quite sensitive to that threshold. In this article, we tackle the above issues by proposing a novel OSDA method to perform soft rejection of unknown target classes and simultaneously match the source and target domains. Extensive experiments on three standard datasets validate the effectiveness of the proposed method over the state-of-the-art competitors.

AI tasks based on deep neural networks have been widely applied in industrial applications, such as process control, quality inspection or predictive maintenance. Deep neural network classifiers are particularly successful, as they provide powerful and reliable algorithms for many applications such as object recognition and fault diagnosis. However, most deep classifier applications are not able to recognize class samples that are beyond the scope of their training data. Samples of unknown classes (denoted as open set data) lead to significant drops in performance, as the output of deep classifiers is limited to the known classes of the training data (denoted as closed set data). This paper presents a method to recognize open set samples without changing the neural network architecture, the training process, nor the trained models. In our method, we firstly train a neural network for normal closed set fault diagnosis. Then we compare the feature maps of testing samples and known class samples during inference using local outlier factor to recognize open set samples. We evaluate our method with two public datasets and show that our method can increase the overall accuracy by 40% when classifying open set data. Besides, we also compared our method to the state-of-the-art open set recognition approach for fault diagnosis applications and the results show that our method leads to better F1-scores.

We study Universal Domain Adaptation (UniDA) problem, which is recently proposed. Different from existing domain adaptation (DA) methods, e.g., Closed set, Open set and Partial DA, UniDA does not need any prior knowledge about the overlap across the source and target label sets. We have two challenges in UniDA problem: i) Domain shift. ii) Category shift. Towards tackling above challenges, we formulate a universal adaptation network called Dual Contrastive Network (DCN), where a contrastive module and a transferability rule are included. The experimental results reveal that DCN can work stably on different UniDA settings and exceeds the state-of-the-art performance across five benchmarks against existing DA methods.

In this work, analogous to Molodtsov Soft sets, We define a new class of Soft sets, namely Soft ARS closed sets and its complement Soft ARS open sets. Further we compare the properties of Soft ARS closed sets with Soft closed set, Soft semi closed set, Soft \(\alpha\) closed set, Soft JP closed set, Soft gsp closed set and we prove the concept of Soft ARS Closed sets is independent of the concept of Soft \(\omega\) closed sets, Soft \(\alpha\)g closed set, Soft gs closed set, Soft gp closed set, Soft pre closed set, Soft Strongly g closed set, Soft g closed, Soft \(\beta\) closed set.We show that the concept of Soft ARS closed sets is stronger than Soft semi closed set and weaker than the concept of Soft generalized semi pre closed sets. The same is applicable for Soft ARS open sets as well. We find the characterization of Soft ARS closed sets and Soft ARS open sets. We investigate fundamental properties of Soft ARS closed sets discuss their properties.

Existing video domain adaptation methods focus on addressing closed set problems. However, it is nearly impossible to guarantee different domains share exactly the same set of categories in realistic scenarios. Hence, open set video domain adaptation (OSVDA) problem, which involves unknown categories, has achieved increasingly close attention. In this paper, we propose a seminal framework, which involves spatial and temporal information to address OSVDA problem. Besides, we design a novel discrimination module, i.e., Dual Metric Discriminator (DMD), to separate known and unknown categories based on implicit and explicit similarity metrics. We conduct comprehensive experiments on several benchmarks and achieve state-of-the-art performance with 40.4%, 33.7%, and 79.2% accuracy on UCF to HMDB, HMDB to UCF, and Kinetics to UCF scenarios respectively.

Keystroke dynamics is a powerful behavioral biometric capable of user authentication based on typing patterns. As larger keystroke datasets become available, machine learning and deep learning algorithms are becoming popular. Knowledge of every possible impostor is not known during training which means that keystroke dynamics is an open set recognition problem. Treating open set recognition problems as closed set (assuming samples from all impostors are present) can cause models to incur data leakage, which can provide unrealistic overestimates of performance. It is a common problem in machine learning and can cause models to report higher accuracies than would be expected in the real world. In this paper, we outline open set recognition and discuss how, if not handled properly, it can lead to data leakage. The performance of common machine learning methods, such as SVM and MLP are investigated with and without leakage to clearly demonstrate the differences in performance. A synthetic dataset and a publicly available keystroke dynamics fixed-text dataset are used for research transparency and reproducibility.

The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this paper, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski's observation that the regular open sets of any topological space form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras., Postprint with minor updates described in footnote on page 1

This paper aims to present the notion of (i, j)*-fuzzy γ-open set in a fuzzy bitopological space as a parallel form of (i, j)-fuzzy γ-open set due to Tripathy and Debnath (2013) [17] and show that both of them are independent concepts. Then we extend our study to (i, j)*-generalized fuzzy γ-closed set and (i, j)*-γ-generalized fuzzy closed set. We show that (i, j)*-γ-generalized fuzzy closed set and (i, j)*-generalized fuzzy γ-closed set are also independent of each other in nature. Though every (i, j)*-fuzzy γ-closed set is a (i, j)*-generalized fuzzy γ-closed set but with (i, j)*-γ-generalized fuzzy closed set, the same relation is not linear. Similarly though every (i, j)*-fuzzy closed set is (i, j)*-γ-generalized fuzzy closed set but it is independent to (i, j)*-generalized fuzzy γ-closed set. Various properties related to (i, j)*-generalized fuzzy γ-closed set are also studied. Finally, (i, j)*-generalized fuzzy γ-continuous function and (i, j)*-generalized fuzzy γ-irresolute functions are introduced and interrelationships among them are established. We characterized these functions in different directions for different applications.

We study the asymptotic behavior of the Navier–Stokes system with Dirichlet boundary conditions posed in a domain Ω e = Ω ∖ T e . Here Ω ⊂ R 3 is a bounded open set and T e is the union of many disjoint closed sets of size e 3 and density of order 1 ∕ e 3 , with e a small positive parameter. Similarly to the periodic case we get a limit system corresponding to a Brinkman flow. The difference is that now the Brinkman matrix is not homogeneous, it depends on the density of the closed sets composing T e . The result is obtained through an adaptation of the two-scale convergence method.

A function f is LC-continuous if the inverse image of any open set is a locally closed set, i.e., an intersection of an open set and a closed set. It is well known that each LC-continuous function f is countably continuous. We prove that f is finitely continuous.

Purpose The aim of the study was to examine the precision of forced-choice (closed-set) and open-ended (open-set) word recognition (WR) tasks for identifying a change in hearing. Method WR performance for closed-set (4 and 6 choices) and open-set tasks was obtained from 70 listeners with normal hearing. Speech recognition was degraded by presenting monosyllabic words in noise (−8, −4, 0, and 4 signal-to-noise ratios) or processed by a sine wave vocoder (2, 4, 6, and 8 channels). Results The 2 degraded speech understanding conditions yielded similarly shaped, monotonically increasing psychometric functions with the closed-set tasks having shallower slopes and higher scores than the open-set task for the same listening condition. Fitted psychometric functions to the average data were the input to a computer simulation conducted to assess the ability of each task to identify a change in hearing. Individual data were also analyzed using 95% confidence intervals for significant changes in scores for words and phonemes. These analyses found the following for the most to least efficient condition: open-set (phoneme), open-set (word), closed-set (6 choices), and closed-set (4 choices). Conclusions Closed-set WR testing has distinct advantages for implementation, but its poorer precision for identifying a change than open-set WR testing must be considered.

Determining how the brain stores information is one of the most pressing problems in neuroscience. In many instances, the collection of stimuli for a given neuron can be modeled by a convex set in $\mathbb{R}^d$. Combinatorial objects known as \emph{neural codes} can then be used to extract features of the space covered by these convex regions. We apply results from convex geometry to determine which neural codes can be realized by arrangements of open convex sets. We restrict our attention primarily to sparse codes in low dimensions. We find that intersection-completeness characterizes realizable $2$-sparse codes, and show that any realizable $2$-sparse code has embedding dimension at most $3$. Furthermore, we prove that in $\mathbb{R}^2$ and $\mathbb{R}^3$, realizations of $2$-sparse codes using closed sets are equivalent to those with open sets, and this allows us to provide some preliminary results on distinguishing which $2$-sparse codes have embedding dimension at most $2$., 13 pages, 10 figures

In this paper, we defined -open set by using s-operation and -closed set, then by using -open set, we define -closed set. In addition we define -closure of subset of ( ) and -interior of subset of by using -closed set and -open set respectively. Furthermore we introduce and discuss minimal -open sets in topological spaces. We establish some basic properties of minimal -open. We obtain an application of a theory of minimal -open sets and define a -locally finite space then we prove, Let be a -locally finite space and a nonempty -open set. Then there exists at least one (finite) minimal -open set such that where is semi-regular.

In recent years, “mathematical orientations on real-life problems”, which continue to increase, began to make a significant impact. Information systems for many decision-making problems consist of uncertain, incomplete, indeterminate and indiscernible structures and components. Classical set theory and interpretation methods fail to represent, express and solve the problems of these types or cause to make wrong decisions. For this reason, in this study, we provide definitions and methods to present information and problem representations in more detail and precision. This paper introduces three new topologies called covering-based rough fuzzy, covering-based rough intuitionistic fuzzy and covering-based rough neutrosophic nano topology. Some fundamental definitions such as open set, closed set, interior, closure and basis are given. Neutrosophic definitions and properties are mainly investigated. We give some real life applications of covering-based rough neutrosophic nano topology in the final part of the paper and an explanatory example of decision making application by defining core point.

In this paper we present an approach to pointless topology that is based on the study of the membership or is-element-of relation within a typed or categorical version of the component-free calculus of relations. This allows us to study several other approaches to topology, including point-set topology, on an abstract and component-free level. In particular, we will show that topologies defined by open sets, closed sets, a family of neighbourhood systems, a topological kernel-mapping, a Kuratowski closure-mapping, or a topological Aumann contact relation are cryptomorphic concepts, i.e., each concept can bijectively be transformed into any other of these concepts. All transformations are specified via relation-algebraic expressions which, in case of set-theoretic relations (that is, in case of point-set topology) can immediately be executed by the specific purpose computer algebra system RelView .

In this paper, we have discussed the definitions and the basic properties of open sets, closed sets, accumulation points or limit points and sequence. Sets may be neither open nor closed. The reader should not confuse the concept “limit point of a set” with the different, though related, concept “limit of a sequence”. Some of the solved and supplementary problems will show the relationship between these two concepts. Observe that(an ∶n ∈ N)denotes a sequence and is a function. On the other hand, {an ∶n ∈ N}denotes the range of the sequence and is a set. We have given several characterizations of these sets. We discuss the definitions of bounded sequence, convergent sequence, Cauchy sequence and their relations. Then we can study every bounded sequence of real numbers contains a convergent subsequence and every Cauchy sequence of real numbers converges to a real number. Let A be a bounded, infinite set of real numbers. Then A has at least one accumulation point. We express the definition of topology, usual topology and topological space. The creation of topology the science of spaces and figures that remains unchanged under continuous deformations represents a phenomenon of this kind, but of a distinctly modern variety. Then we begin our study of some properties topological spaces by making the idea of being connected that is being in one piece. We observe that for the usual topology on the line R and in the planeR2.Finally, we express the characterizations of the discrete topological space and indiscrete topological space.

Generalized closed sets play an important role in the study of topological spaces. In the literature we have seen the relation between closed set and generalized closed set is linear in nature, both in crisp and fuzzy environment. In particular, every closed set is a generalized closed set. In this present treatise, a new type of generalized closed set via $$(i,j)^{*}$$ -fuzzy $$\gamma $$ -open set, namely $$(i,j)^{*}$$ - $$\gamma gf\gamma $$ -closed set have been introduced in a fuzzy bitopological space and interrelationships of it with the existing ones have been made. It has been proved that this newly defined generalized fuzzy closed set is totally independent with $$(i,j)^{*}$$ -fuzzy closed set. Various characterizations of this set have been established and some interesting results are cited. As applications of this set various types of continuities are studied in the same environment. More applications are proposed by introducing a new kind of closure operator namely $$(i,j)^{*}$$ - $$\gamma $$ - $$cl_{\gamma }^{*}$$ -operator and important results have been reported based on this concept.

In my efforts to move towards an improved understanding of biculturalism, I have been particularly influenced by three of Jaan’s ideas. First, deriving from the notion that cultural means semiotically mediated, we are not “members of a culture,” but culture in terms of semiosis is part of our psyche. Second, the initation to thinking in terms of abductive class membership determination. This implies the permeability of borders, leading to the third element: co-genetic logic – the power of the triad. Borders co-define the relationship between the closed set (culture A) and the open set (non-A) which allows for the emergence of new meaning, while the bounded region in between expands and constricts over time. Especially in the field of biculturalism, we need to overcome the binary logic and recognize the power of the triad. Its components dynamically co-define each other. Jaan’s concept of inclusive separation succinctly describes bicultural negotiation processes.

In this paper, we discuss how role of interior, exterior and boundary in M-topology is different from that of similar concepts in general topology. In general topology, we have numerous results relating interior, closure, and boundary of a subset and we can characterize open sets, closed sets and clopen sets in terms of these concepts. It is shown that similar results need not be true in M- topology, and further analyses the situations for which the results are true and establishes some.

Most modern neural networks for classification fail to take into account the concept of the unknown. Trained neural networks are usually tested in an unrealistic scenario with only examples from a closed set of known classes. In an attempt to develop a more realistic model, the concept of working in an open set environment has been introduced. This in turn leads to the concept of incremental learning where a model with its own architecture and initial trained set of data can identify unknown classes during the testing phase and autonomously update itself if evidence of a new class is detected. Some problems that arise in incremental learning are inefficient use of resources to retrain the classifier repeatedly and the decrease of classification accuracy as multiple classes are added over time. This process of instantiating new classes is repeated as many times as necessary, accruing errors. To address these problems, this paper proposes the Classification Confidence Threshold approach to prime neural networks for incremental learning to keep accuracies high by limiting forgetting. A lean method is also used to reduce resources used in the retraining of the neural network. The proposed method is based on the idea that a network is able to incrementally learn a new class even when exposed to a limited number samples associated with the new class. This method can be applied to most existing neural networks with minimal changes to network architecture., Comment: Accepted to IEEE TNNLS

In this article we treat two closely related problems: 1) the upper semi-continuity property for Almgren minimal sets in regions with regular boundary; and 2) the uniqueness property for all the $2$-dimensional minimal cones in $\mathbb R^3$. ¶ Given an open set $\Omega\subset\mathbb R^n$, a closed set $E\subset \Omega$ is said to be Almgren minimal of dimension $d$ in $\Omega$ if it minimizes the $d$-Hausdorff measure among all its Lipschitz deformations in $\Omega$. We say that a $d$-dimensional minimal set $E$ in an open set $\Omega$ admits upper semi-continuity if, whenever $\{f_n(E)\}_n$ is a sequence of deformations of $E$ in $\Omega$ that converges to a set $F$, then we have ${\mathcal H}^d(F)\ge \limsup_n {\mathcal H}^d(f_n(E))$. This guarantees in particular that $E$ minimizes the $d$-Hausdorff measure, not only among all its deformations, but also among limits of its deformations. ¶ As proved in [19], when several $2$-dimensional minimal cones are all translational and sliding stable, and admit the uniqueness property, then their almost orthogonal union stays minimal. As a consequence, the uniqueness property obtained in the present paper, together with the translational and sliding stability properties proved in [18] and [20] permit us to use all known $2$-dimensional minimal cones in $\mathbb R^n$ to generate new families of minimal cones by taking their almost orthogonal unions. ¶ The upper semi-continuity property is also helpful in various circumstances: when we have to carry on arguments using Hausdorff limits and some properties do not pass to the limit, the upper semi-continuity can serve as a link. As an example, it plays a very important role throughout [19].