Counter Example Research Articles

On star Rothberger spaces modulo an ideal

In this article, we introduce the ideal star-Rothberger property by coupling the notion of a star operator to that of an ideal Rothberger space, after which some of its topological characteristics are analysed. By creating relationships between a numbers of topological features with structures similar to the ideal star-Rothberger space, we reinforce the concept. In order to illustrate the differences between a number of related topological properties, we also provide several counter examples. Certain preservation-related properties under subspaces and functions are investigated. Lastly we find a way to express ideal star-Rothberger space by means of families of closed sets by bringing some modifications to the SSI^I property.

On some covering properties via a- open sets.

In this paper the notion of a-open set used as a tool to introduce certain types of covering properties which is similar to the familiar property of Hurewicz, and prove that we can use a-open sets instead of open sets in the definition of a-compact and a-Hurewicz space and investigated. Some properties and counter examples are given, also the relationship between these spaces was considered.

Rigidity on quantum symmetry for a certain class of graph C*-algebras

Quantum symmetry of graph C*-algebras has been studied, under the consideration of different formulations, in the past few years. It is already known that the compact quantum group (C(S1)∗C(S1)∗⋯∗C(S1)︸|E(Γ)|−times,Δ) [in short, *|E(Γ)|C(S1),Δ] always acts on a graph C*-algebra for a finite, connected, directed graph Γ in the category introduced by Joardar and Mandal, where |E(Γ)| ≔ number of edges in Γ. In this article, we show that for a certain class of graphs including Toeplitz algebra, quantum odd sphere, matrix algebra etc. the quantum symmetry of their associated graph C*-algebras remains *|E(Γ)|C(S1),Δ in the category as mentioned before. More precisely, if a finite, connected, directed graph Γ satisfies the following graph theoretic properties: (i) there does not exist any cycle of length ≥2 (ii) there exists a path of length (|V(Γ)| − 1) which consists all the vertices, where |V(Γ)| ≔ number of vertices in Γ (iii) given any two vertices (may not be distinct) there exists at most one edge joining them, then the universal object coincides with *|E(Γ)|C(S1),Δ. Furthermore, we have pointed out a few counter examples whenever the above assumptions are violated.

Entropy generation in mixed convection rotating flow over a heated surface

This work aims to study entropy generation in mixed convection flow of Newtonian viscous fluid. Water and kerosene oil we can take as counter examples in this case. This fluid flows past a vertical plate. This fluid near the wall is heated with isothermal heating. The plate moves with constant acceleration and the fluid flows in a rotating fame. Mixed convection is considered here due to buoyancy force and constant plate linear translation. This problem is modeled in terms of partial differential equations (PDEs) and then using suitable variables, the problem is transformed into dimensionless form. Exact solutions of the problem are attained using the Laplace transform. The equations for Bejan number and entropy generation are also developed in dimensionless form, and then by via the exact solutions of velocity and energy, the results of entropy generation and Bejan number are figured. Graphical results are provided to discuss about the behavior of embedded parameters. Results of Nusselt number and skin friction are also figured in tabular form and discussed.

Generalized Alternating Projections on Manifolds and Convex Sets

In this paper, we extend the previous convergence results for the generalized alternating projection method applied to subspaces in [arXiv:1703.10547] to hold also for smooth manifolds. We show that the algorithm locally behaves similarly in the subspace and manifold settings and that the same rates are obtained. We also present convergence rate results for when the algorithm is applied to non-empty, closed, and convex sets. The results are based on a finite identification property that implies that the algorithm after an initial identification phase solves a smooth manifold feasibility problem. Therefore, the rates in this paper hold asymptotically for problems in which this identification property is satisfied. We present a few examples where this is the case and also a counter example for when this is not.

Efficient Numerical Method for Solving a Quadratic Riccati Differential Equation

This study presents families of the fourth-order Runge–Kutta methods for solving a quadratic Riccati differential equation. From these families, the England version is more efficient than other fourth-order Runge–Kutta methods and practically well-suited for solving initial value problems in general and quadratic Riccati differential equation in particular. The stability analysis of the present method is well-established. In order to verify the accuracy, we compared the numerical solutions obtained using the England version of fourth-order Runge–Kutta method with the recently published works reported in the literature. Several counter examples are solved using the present methods to demonstrate their reliability and efficiency.

Dilation and Birkhoff-James orthogonality

For any two elements x and y in a normed space X, x is said to be Birkhoff-James orthogonal to y, denoted by x⊥By, if ‖x+λy‖≥‖x‖ for every scalar λ. Also, for any ε∈[0,1), x is said to be ε-approximate Birkhoff-James orthogonal to y, denoted by x⊥Bεy, if‖x+λy‖2≥‖x‖2−2ε‖x‖‖λy‖,for all scalarsλ. For any ρ>0, a unitary operator U acting on a Hilbert space K is said to be a unitary ρ-dilation of an operator T on a Hilbert space H if H⊆K and Tn=ρPHUn|H for every nonnegative integer n, where PH:K→H is the orthogonal projection. Also, when ρ=1 and T is a contraction, U is called a unitary dilation of T. We obtain the following main results.(1)We find necessary and sufficient conditions such that for any two contractions T,A on H, their Schäffer unitary dilations UT˜ and UA˜ on the space ⊕−∞∞H are Birkhoff-James orthogonal. Also, counter example shows that in general UT˜⊥̸BUA˜ even if T⊥BA.(2)For any ρ>0 and for two Hilbert space operators T,A with T⊥BA, we show that if ‖T‖=ρ then UT⊥BUA for any unitary ρ-dilations UT of T and UA of A acting on a common Hilbert space. Also, we show by an example that the condition that ‖T‖=ρ cannot be ignored.(3)For any ρ>0, we explicitly construct examples of Hilbert space operators T,A such that T⊥̸BA but any of their unitary ρ-dilations UT,UA acting on a common Hilbert space are Birkhoff-James orthogonal.(4)We find a characterization for the ε-approximate Birkhoff-James orthogonality of operators on complex Hilbert spaces.(5)For any ρ>0 and for any Hilbert space operators T,A, we find a sharp bound on ε such that T⊥BA implies UT⊥BεUA for any unitary ρ-dilations UT of T and UA of A acting on a common space. Also, we show by an example that in general the bound on ε cannot be improved.(6)We construct families of generalized Schäffer-type unitary dilations for a Hilbert space contraction in two different ways. Then we show that one of them preserves Birkhoff-James orthogonality while any two members UT,UA from the other family are always Birkhoff-James orthogonal irrespective of the orthogonality of T and A.(7)We show that Andô dilation of a pair of commuting contractions of the form (T,ST), where S is a unitary that commutes with T, are orthogonal. Also, we explore orthogonality of regular unitary dilation of a pair of commuting contractions. However, Birkhoff-James orthogonality is independent of commutativity of operators.

On impurity functions in decision trees

Impurity functions are crucial in decision trees. These functions help determine the impurity level of a node in a decision tree, guiding the splitting criteria. However, two primary ambiguities have surrounded impurity functions: (1) the question of their non negativity and (2) the debate over their concavity. In this paper, we address these uncertainties by delving into the characteristics of impurity functions. We establish that the non negativity of an impurity function is inconsequential. Through counter examples, we disprove the equivalence between an impurity function and a concave function. We identify an impurity function that is not concave and a concave function that is not an impurity function. Interestingly, we find an impurity function that results in a negative impurity reduction. Furthermore, we validate several significant properties of impurity functions. For example, we demonstrate that when an impurity function is concave, the impurity reduction remains nonnegative for multiway divisions. We also discuss the sufficient conditions for a concave function to be an impurity function. Our numerical results further indicate that a positive linear combination of the two most popular impurity functions, namely Gini Index and Entropy, may surpass the individual performance of each when applied to the well-known German credit dataset.

Generation of entropy on blood conveying silver nanoparticles embedded in curved surfaces

The brain is a complicated system that experiences energy and energy exchange problems. While silver nanoparticles are essential in cancer treatment, entropy analysis can be used to study the brain and its illness. In addition to the nature of the blood's thermophysical properties and the utility of silver nanoparticles over curved stretched surfaces, nothing is known about the dynamics of transient blood carrying silver nanoparticles when entropy generation and Lorentz force are important. So, a transitory irreversibility analysis problem (entropy generation/s law analysis) is studied in this work. As a counter example to Casson fluid, blood is used as the base fluid, which has a suspension of silver nanoparticles on a curved, stretched surface. Convective boundary conditions and thermal radiation are considered. By using the necessary similarity transformations, the governing equations for the current flow are converted into a nonlinear system. The finite element method is employed to obtain multiple results for the condensed system. The significance of various emerging parameters related to current study is stimulated and analyzed thoroughly trough the graphs of profiles of temperature and velocity, moreover, temperature gradient and drag force are also taken into account. The outcomes indicating that radiation and dissipation effects increase entropy As a result, entropy can play a crucial role in the artificial process used to analyze brain activity. Due to the presence of silver nanoparticles in blood flow, the velocity and temperature also increase.

Problematizing Possible -isms in Adult Second Language Classrooms

Abstract While addressing issues of prejudice and exclusion is crucial for helping adult second language (L2) learners acquire and critique socio-interactional norms of their pluralistic communities, there is still a lack of work examining how teachers can problematize prejudiced talk when it arises in the classroom. Within the detail-oriented frameworks of conversation analysis and membership categorization analysis, this study examines teacher practices for problematizing students’ hearably prejudiced, stereotypical, or exclusionary talk—that is, students’ possible -isms. Drawing on data from 55 hours of video-recorded English L2 classes in the United States, I explore three problematizing responses to possible -isms: (i) introducing complexities, (ii) offering counter examples, and (iii) treating -isms as absurd. While introducing complexities neutralizes students’ hearable -isms as somehow inadequate, the latter two practices treat the -isms as entirely inaccurate through illustrated and invited corrections. The analysis reveals these practices to be finely attuned to the institutional goals and features of the adult L2 classroom, allowing teachers to incorporate a critical perspective and delicately problematize possible -isms without threatening solidarity with their diverse adult learners.

Instabilities and Stabilities of Additive Functional Equation in Paranormed Spaces

In this paper, we solve the general solution in vector space and prove the Hyers-Ulam stability of the following additive functional equationin paranormed spaces by using the direct and fixed point methods. Also we present its pertinent counter examples for instabilities.

A Phenomenological Solution to Gettier’s Problem

In “Is Justified True Belief, Knowledge?” Gettier shows us two counter examples of analyzing Knowledge, as “Justified True Belief” or “JTB”. Lots of scholars have reconstructed similar counter examples to JTB but we can see they follow a similar algorithm. Other scholars have tried to re-analyze knowledge by adding a fourth element to JTB and reformulating knowledge in a “JTB+X” formula and some replaced justification with another alternative component (Y) and proposed a “YTB” analysis of knowledge. In this article I first overview Gettier’s problem and I show that we can construct a similar Gettier problem for each “JTB+X” or “YTB” formula. After that, I will focus on re-analyzing knowledge with a phenomenological attitude that can avoid Gettier’s problem.

Limitations of Chebyshev interpolation nodes : Counter examples and Insights

Interpolation using uniformly spaced nodes often encounters Runge’s phenomenon when applied to smooth functions. To mitigate this issue, Chebyshev roots are frequently recommended as better interpolation nodes. However, our study reveals the limitations of Chebyshev roots for interpolation by presenting a series of counterexamples. These counterexamples encompass functions that exhibit both differentiable and non-differentiable characteristics. By examining scenarios where Chebyshev interpolation falls short of producing satisfactory results, we shed light on when alternative interpolation methods should be considered.

Neutrosophic Soft Regular Semiopen Sets in Neutrosophic Soft Topological Spaces

In this paper, we introduce and study the concept of neutrosophic soft regular semiopen and neutrosophic soft regular semiclosed sets in neutrosophic soft topological spaces. By using neutrosophic soft regular semiopen set, we define a new neutrosophic soft interior operator namely neutrosophic soft regular semi interior operator and neutrosophic soft regular semi neighborhood. Also the relation between these sets and counter examples for the reverse relation are given.

Ecumenical Attributability and the Structural Ownership Condition on Moral Responsibility

This paper discusses the non-historicist structural ownership condition on moral responsibility forwarded by Benjamin Matheson. The structural ownership condition requires that a morally relevant action be grounded or partly grounded in psychological states that are generally coherent. While Matheson does not mean to settle the debate on historicism vs. non-historicism, he does mean to secure the position of the ownership condition against the problems that structuralist theories have faced in the past. This paper will focus on how the ownership condition handles cases of ambivalent agents. Intuitively, ambivalent agents should be responsible for what they do as long as what they do is expressive of their cares or commitments, or their authentic character. At a first glance it appears that the ownership condition follows intuitions about ambivalence, but with a closer look we can see that Matheson has provided a potential counter example to this position.

NEUTROSOPHIC WEAKLY SEMI CONTINUOUS FUNCTIONS

: In this paper, we introduce the concept of neutrosophic weakly semi continuous, neutrosophic regular semi q-neighbourhood in neutrosophic topological spaces. Moreover, we investigate the relationship among neutrosophic weakly semi continuous and other existing continuous functions and some counter examples to show that these types of mappings are not equivalent.

THE STUDY OF ECCENTRICITY SPECTRUM AND ENERGY IN PATH AND CYCLE GRAPHS

The eccentricity matrix is one of matrices to represent graphs. The eccentricity matrix is used as a basis for calculating the eccentricity spectrum and energy. This article aims to study the concepts of eccentricity spectrum and energy in simple graphs. For special cases, we also discuss eccentricity spectrum and energy of paths and cycles. All studies in this article focus on providing some examples to facilitate the reader's understanding of the concepts studied. In addition, this article also corrects the mistakes in the lemma about eccentricity spectrum of paths and theorem about eccentricity energy of odd-order cycles from reference articles. Corrections are made by indicating where the errors are in the referenced articles, providing counter examples, correcting inaccurate lemmas and theorems, and giving short proofs. At the end of the article, an open problem is also included to provide an overview of research ideas that can be developed from the concepts of eccentricity spectrum and energy.

Why It (Also) Matters What Infectious Disease Epidemiologists Call “Disease”

Infectious diseases figure prominently as (counter)examples in debates on how to conceptualize “disease.” But crucial epidemiological distinctions are often not heeded in the debate, and pathological and clinical perspectives focusing on individual patients are favored at the expense of perspectives from epidemiology focusing on populations. In clarifying epidemiological concepts, this paper highlights the distinct contributions infectious disease epidemiology can make to the conception of “disease,” and the fact that this is at least tacitly recognized by medical personnel and philosophers. Crucially, infectious disease epidemiology can help elucidate how carrying and transmitting infectious, communicable entities is a disease, even if the carriers themselves are not directly affected by symptoms detrimental to them.

Characterizations of generalized Fuzzy γ∗-closed sets

Generalized fuzzy open sets are playing a vital role in the study of fuzzy topological space as well as that of fuzzy bitopological space since its inception. More often, it is reported that fuzzy closed sets are always included in the family of generalized fuzzy closed sets. Very recently, it has appeared that fuzzy γ∗ -open sets are incomparable with fuzzy open sets. This paper aims to present three different kinds of fuzzy generalized closed sets in the light of fuzzy γ∗ -open set and associated closure operators with the terminologies- generalized fuzzy γ∗ -closed set, γ∗ -generalized fuzzy closed set and γ∗ -generalized fuzzy γ∗-closed set and it is found that the relation between any two concepts is not necessarily linear. Also, the interrelationships among them are established along with suitable counter examples which are properly placed to make the paper self-sufficient.

Optimal stabilizing rates of switched linear control systems under arbitrary known switchings

The problem of stabilizing discrete-time switched linear control systems using continuous control input under arbitrary mode switchings is studied. It is assumed that at each time instance the switching mode can be arbitrarily chosen but is always known by the controller designing the continuous control input; thus the continuous controller is of the general form of an ensemble of mode-dependent state feedback controllers. Under this setting, the fastest (worst-case) stabilizing rate is proposed as a quantitative metric of the systems’ stabilizability. Conditions are derived on when this stabilizing rate can be exactly achieved by an admissible control policy and a counter example is given to show that the stabilizing rate may not always be attained by a mode-dependent linear state feedback control policy. Bounds on the stabilizing rate are derived using (semi)norms. When such bounds are tight, the corresponding extremal norms are characterized geometrically. Numerical algorithms based on ellipsoid and polytope norms are developed for computing bounds of the stabilizing rate and illustrated through examples.