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We realize a suite of logical operations on a distance-two logical qubit stabilized using repeated error detection cycles. Logical operations include initialization into arbitrary states, measurement in the cardinal bases of the Bloch sphere, and a universal set of single-qubit gates. For each type of operation, we observe higher performance for fault-tolerant variants over non-fault-tolerant variants, and quantify the difference through detailed characterization. In particular, we demonstrate process tomography of logical gates, using the notion of a logical Pauli transfer matrix. This integration of high-fidelity logical operations with a scalable scheme for repeated stabilization is a milestone on the road to quantum error correction with higher-distance superconducting surface codes., Comment: 16 pages, 9 figures, 2 tables

An approximation space plays a vital role in the accuracy of approximations of a subset of the universal set. The main goal of this paper is to develop new kinds of soft rough sets models by using the concept of near open sets, where the accuracy of approximations is enhanced significantly. Firstly, the concepts of near soft rough approximations, denoted by “ $$J_{SR}$$ -approximations” for each $$J\in \{P,S,\gamma ,\alpha ,\beta \}$$ are proposed as a generalization some previously introduced notions. Then, their properties and relationships are disclosed. Comparisons among the proposed methods and the previous one are obtained. An algorithm has been given for decision-making problems. The proposed algorithm is tested on hypothetical data for the purpose of comparison with already existing methods.

In fuzzy set theory, the aggregation is the process that combines input fuzzy sets into a single output fuzzy set. In this manner, an aggregation operator is an important tool in the fuzzy set theory and its applications. The purpose of this study is to present some algebraic operators among neutrosophic enthalpy values and to provide some aggregation operators with the help of general t-norms and t-conorms which produce a new theoretical base in the fuzzy environment. An enthalpy value is the information energy expressed by the complement of the Shannon’s entropy and a neutrosophic enthalpy set is characterized with a truth, an indeterminacy and a falsity function defined on a universal set to $$[0,1]^{2}$$ . The first component of each function is the average of the truth, the indeterminacy and the falsity sequence of a neutrosophic multi-valued set, respectively, and the second component of each function is the fuzzy complement of the normalized Shannon’s entropy of the truth, the indeterminacy and the falsity of the same neutrosophic multi-valued set, respectively. Therefore, a neutrosophic enthalpy set contains both the level of the mean of the data and the degree of uncertainty of the data via enthalpy. Then, by using Algebraic and Einstein t-norms and t-conorms we give a multi-criteria decision making method based on these aggregation operators and a score function. This method is applied to a multi-criteria decision making problem with neutrosophic enthalpy set information and the comparison analysis is given with the existing methods to show the efficiency and sensitivity of the proposed method.

The trisecting-acting-outcome (TAO) model of three-way decisions includes trisecting a universal set into three separate and closely connected regions, devising, and applying efficient strategies on the three regions, furthermore evaluating the outcome. This paper introduces the proportional utility function (PUF), representing the ratio between an object’s initial and final quantity, to measure the outcomes from two different perspectives for movement-based three-way decision. The first perspective, is that if each object produces the same benefits or costs when they have the same movement (call region-independent), we sum up the three regions’ utility as the overall outcome. The second scenario, is that each object generates a different cost or benefit, even if they have the same movements (call region-dependent). Here, finding an optimal investment plan is an essential matter based on their equivalent classification according to their specific characteristics. For a single equivalence class, we design a strategy to reach the goal under a specific investment, although this investment may be conservative. For all equivalence classes, we give one-time optimal investment plans using PUF based on invested or reserved resources in the movement-based three-way decision. Based on the program, we adopt a series of strategies in action under these investment budgets. The experimental results show that our strategy choices have practical significance and are in line with expected results.

This article proposes a new fuzzy time series model that can interpolate historical data, and forecast effectively for the future. It is combination of the improved steps from the existing models. There are problems to use the percentage variations of series between consecutive periods of time as a universal set, to divide the universal set into clusters by the automatic algorithm based on the similarity between elements, to determine the relationships between elements in the series and the divided clusters by the improved fuzzy cluster analysis algorithm, and to interpolate the historical data and to forecast for future by new principle. The proposed algorithm is performed quickly and efficiently by the established Matlab procedure. It is illustrated by an example, and tested for many other data sets, especially for 3003 series in M3-Competition data. Comparing to the existing models, the proposed model always gives the best result. We also apply the proposed model in forecasting the salty peak for a coastal province of Vietnam. Examples and application show the potential of the studied problem.

Driving discovery in the study of law and legal institutions often requires infrastructure in the form of databases and other tools. The challenge is how to build the infrastructure. For obvious reasons, transplanting coding rules and variables from one dataset to the next is perilous; specialized knowledge of local conditions is necessary before one piece of datum is collected. Also required is adherence to a universal set of principles that distinguish high‐quality infrastructure; namely, that the tool is capable of addressing real‐world problems, accessible, reproducible and reliable, sustainable and updatable, and foundational. These principles guided construction of the Israeli Supreme Court Database, new and original infrastructure encoding information from all panel cases opened between 2010 and 2018 in the Israeli Supreme Court.

Apresentation is provided of a method – the Membership Table Method (MTM) – to determine the validity of categorical syllogisms. This method makes it possible for each syllogism to be assigned a specific set. If this set is equal to the universal set U, then the categorical syllogism considered is valid, and if that set is not equal to U, then that categorical syllogism is not valid. In other words, any categorical syllogism is valid if and only if its respective set, according to the MTM, is equal to the universal set U. The conclusion of a valid categorical syllogism whose premises are true is true.

The subject of the work was the experimental verification of the negative influence of gravity sets in a pressure volumetric infusion pump with an impact on the accuracy of infusion solution dosing. The quality criterion for gravity sets was the accuracy of the flow versus the reference set by the volumetric infusion pump. The solution consisted of 14-hour measurements with two types of gravity sets, Intrafix® Primeline sets were used as universal sets and Standardline IS‑103 gravity sets. The insulin pump flow rate was set at 300 ml/h and 50 ml/h, and the actual flow rate of the infusion solution was recorded every hour using a graduated cylinder. Used gravity sets were also processed by mechanical tests, unused sets were subjected to these tests and the obtained data were compared with each other. Experiments carried out showed that at the set flow rate of 300 ml/h, the flow error with the universal set was -3% and at a set flow rate of 50 ml/h the error was +2.3%. Flow accuracy using gravity sets was worse, a flow error of -7.2% was detected for a flow rate of 50 ml/h and a flow error of -7.7% was measured for 300 ml/h. The volumetric pump used declares a tolerated inaccuracy of ± 3% when used with standard infusion sets. Based on the data, it can be concluded that the replacement of set types has an influence on the dosage of infusion solutions.

In economic science, a new scientific direction has appeared, associated with the so-called prospects theory, which has been among the leading for decision-making under conditions of uncertainty, not only for economic science. The fuzzy set theory, in which the fuzzy set (FS) is one of the types of numerous Non-factors, needs a qualitative modernization since we have a limited possibility to solve many classes of problems. The usage of fuzzy sets is almost universal for solving problems in different fields if there is uncertainty, inaccuracy. At the same time, the difficulty in choosing the membership function should be considered. The process of choosing an MF by a person (expert) is based on common sense, experience, and is not always rational. For solving tasks in conditions of uncertainty, the authors proposed the use of the tensor methodology. The authors showed the possibility to form a subset of ordered pairs based on the tensorization of the interval of a universal set with subsequent decomposition. It was suggested to use Toeplitz matrix, most effectively modeling fuzziness, as one of the ways of tensorization and taking into account the phenomenon of fuzziness. The universal set (US) on which the FS is formed contains, in the tensor format, hidden information that can be used to make a decision no less effectively than the heuristically assigned membership function, in addition, the existence of a formally calculated subset of ordered pairs (SsOP) can serve as a convincing comparative example: Having this SsOP, you can objectively refuse to assign a heuristic MF; SsOP has significantly less interval uncertainty. Alternative SsOP are formed in a uniform manner. Basic characteristics for FS and SsOP comparison are quite close or practically coincide. Examples that illustrate the higher efficiency of SsOP usage in comparison with standardly spaced FS are given.

The original model of rough sets was advanced by Pawlak, which was mainly involved with the approximation of things using an equivalence relation on the universal set of his approximation space. In this paper, two kinds of approximation operators via ideals which represent extensions of Pawlak’s approximation operator have been presented. In both kinds, the definitions of upper and lower approximations based on ideals have been given. Moreover, a new type of approximation spaces via two ideals which is called bi-ideal approximation spaces was introduced for the first time. This type of approximations was analyzed by two different methods, their properties are investigated, and the relationship between these methods is proposed. The importance of these methods was its dependent on ideals which were topological tools, and the two ideals represent two opinions instead of one opinion. At the end of the paper, an applied example had been introduced in the chemistry field by applying the current methods to illustrate the definitions in a friendly way.

Three-way granular computing is a paradigm of thinking and information processing in three granules. We introduce a framework of three-way granular computing based on a hexagon induced by a trisection of a set. This enables us to unify rough-set concept analysis and formal concept analysis. For an object granule (i.e., a subset of a universal set of objects), we construct three attribute granules. Conversely, for an attribute granule (i.e., a subset of a universal set of attributes), we construct three object granules. Rough-set concept analysis uses one kind of trisections to study four types of disjunctive formal concepts and the associated concept lattices. Formal concept analysis uses another kind of trisections to study four types of conjunctive formal concepts and the associated concept lattices. The eight concept lattices can be grouped into two classes such that four lattices in each class are isomorphic.

Problem solving under uncertainty in the form of fuzzy mathematics based on methods and models of fuzzy set theory with limited possibilities for determining (assigning) membership functions is considered. A method for solving problems of this type is proposed. The method is to determine hidden knowledge in the form of subsets of ordered pairs calculated using a singular value decomposition of special (Toeplitz, Hankel, etc.) matrices formed based on the universal set. Examples that illustrate the effectiveness of the proposed method are given.

Path choice modelling is typically conducted by considering a subset of paths, not the universal set of all feasible paths as this is computationally challenging. This study proposes a two-stage modelling approach. In the first stage, it develops a new probabilistic importance sampling protocol by using fuzzy logic. In the second stage, it tested different structures of the discrete choice models, where different strategy attributes are considered along with the traditional variables. The results prove that the new sampling protocol performs better than traditional sampling protocol. Again, the inclusion of the strategy attributes proves to yield better prediction. The results of the study recommend considering different strategy attributes in the path generation process as well as in the transit assignment models. The study also discusses the effect of the choice set size on the model performance. Household travel survey data of south-east Queensland, Australia is used to develop the models.

A major challenge in developing discrete choice models when the feasible choice set is extensively large is to find an appropriate choice set that is computationally manageable for being integrated into the choice modelling specification while it also reflects the behaviour of the decision-maker. This study aims to develop a new sampling method for choice set formation which is then used to improve the knowledge about recreational destination choice behaviour. The study proposes a two-stage framework where in the first stage a rule-based fuzzy logic model is used to form a latent choice set. The second stage incorporates the set in a discrete choice model to estimate destination selection behaviour. In the travel demand modelling literature, there has been little research into modelling recreational activity destination choices. In this area, the primary focus has been on other activities, such as work, study and shopping. In this context, the sampling methods used to form the choice set using the large set of possible options in the universal set are unconvincing. The proposed two-stage approach is examined using a household travel survey from Victoria, Australia, where nine different recreational destination choice models are developed. Compared to the conventional modelling framework, these models show impressive model fit and prediction potential.

Harbour (2016) argues for a parsimonious universal set of features for grammatical person distinctions, and suggests (ch. 7) that the same features may also form the basis for systems of deixis. We apply this proposal to an analysis of Heiltsuk, a Wakashan language with a particularly rich set of person-based deictic contrasts (Rath 1981). Heiltsuk demonstratives and third-person pronominal enclitics distinguish proximal-to-speaker, proximal-to-addressee, and distal (in addition to an orthogonal visibility contrast). There are no forms marking proximity to third persons (e.g., ‘near them’) or identifying the location of discourse participants (e.g., ‘you near me’ vs. ‘you over there’), nor does the deictic system make use of the clusivity contrast that appears in the pronoun paradigm (e.g., ‘this near you and me’ vs. ‘this near me and others’). We account for the pattern by implementing Harbour's spatial element χ as a function that yields proximity to its first- or second-person argument.

Systems of Boolean functions inducing algebras of Bernoulli distributions whose universal set has a single limit point are considered. A criterion for an algebra generated by a given distributions set to have a single limit point is proved.

Experimental realization of a universal set of quantum logic gates with high-fidelity is critical to quantum information processing, which is always challenging by inevitable interaction between the quantum system and environment. Geometric quantum computation is noise immune, and thus offers a robust way to enhance the control fidelity. Here, we experimentally implement the recently proposed extensible nonadiabatic holonomic quantum computation with solid spins in diamond at room-temperature, which maintains both flexibility and resilience against decoherence and system control errors. Compared with previous geometric method, the fidelities of a universal set of holonomic single-qubit and two-qubit quantum logic gates are improved in experiment. Therefore, this work makes an important step towards fault-tolerant scalable geometric quantum computation in realistic systems., Comment: 9 pages, 6 figures

Reliable species identification is critical for detection and monitoring of biological invasions. In this study, we propose four sets of primers for efficient amplification of several loci, including the mitochondrial cytochrome oxidase-c (COI) subunit I gene which is a basis for DNA barcoding. This set of primers gives a shorter product which can be used in high-throughput sequencing systems for metabarcoding purposes. Another mitochondrial locus encoding the large ribosomal subunit (16S) may be useful to study the population structure and as an additional source of information in the metabarcoding of communities. We propose to use a set of primers for the nuclear locus of the small ribosomal subunit (18S) as a positive control and to verify the results of the barcoding. Our proposed sets of primers demonstrate a high amplification efficiency and a high specificity both for freshwater alien and indigenous fishes. The proposed research design makes it possible to carry out extremely cheap studies on the assessment of biological diversity using genetic analysis without expensive equipment, and with the technique for conducting laboratory work and processing of the results available to any researcher. The paper also presents original data on the genetic polymorphism of all mass alien fish species in the Volga-Kama region. High efficiency of DNA identification based on our primers is shown as compared to traditional monitoring of biological invasions.

A set $I$ is said to be a universal indifferent set for $1$-genericity if for every $1$-generic $G$ and for all $X \subseteq I$, $G \Delta X$ is also $1$-generic. Miller showed that there is no infinite universal indifferent set for $1$-genericity. We introduce two variants (union and subsequence sets for $1$-genericity) of the notion of universal indifference and prove that there are no non-trivial universal sets for $1$-genericity with respect to these notions. In contrast, we show that there is a non-computable subsequence set for weak-$1$-genericity., Comment: 9 pages

In this paper, we investigate the problem of simulation of quantum-mechanical dynamical system with prescribed properties using of exactly solvable Hamiltonians. A method for solving time-dependent matrix Schrodinger equations in an explicit analytical form will be developed. It is based on exactly solvable time-independent problems, a certain choice of the initial conditions, and special time-dependent gauge transformations converting time-independent problems to time-dependent ones. An application of the exactly solvable time-dependent problems thus obtained to the construction of a universal set of gates for quantum computers is also presented.

The Eastin-Knill theorem states that no quantum error correcting code can have a universal set of transversal gates. For CSS codes that can implement Clifford gates transversally it suffices to provide one additional non-Clifford gate, such as the T-gate, to achieve universality. Common methods to implement fault-tolerant T-gates like magic state distillation generate a significant hardware overhead that will likely prevent their practical usage in the near-term future. Recently methods have been developed to mitigate the effect of noise in shallow quantum circuits that are not protected by error correction. Error mitigation methods require no additional hardware resources but suffer from a bad asymptotic scaling and apply only to a restricted class of quantum algorithms. In this work, we combine both approaches and show how to implement encoded Clifford+T circuits where Clifford gates are protected from noise by error correction while errors introduced by noisy encoded T-gates are mitigated using the quasi-probability method. As a result, Clifford+T circuits with a number of T-gates inversely proportional to the physical noise rate can be implemented on small error-corrected devices without magic state distillation. We argue that such circuits can be out of reach for state-of-the-art classical simulation algorithms., Comment: v2: 11 pages, 7 figures; published version

Quantum walk has been regarded as a primitive to universal quantum computation. By using the operations required to describe the single particle discrete-time quantum walk on a position space we demonstrate the realization of the universal set of quantum gates on two- and three-qubit systems. The idea is to utilize the effective Hilbert space of the single qubit and the position space on which it evolves in order to realize multi-qubit states and universal set of quantum gates on them. Realization of many non-trivial gates and engineering arbitrary states is simpler in the proposed quantum walk model when compared to the circuit based model of computation. We will also discuss the scalability of the model and some propositions for using lesser number of qubits in realizing larger qubit systems., Comment: Updated to latest version, corrected grammar and moved appendix to main text

When someone responds to a problem, open-mindedness behavior is needed. The effort to familiarize and develop open-mindedness in pre-service mathematics teacher is to do problem solving activities. This research aims to investigate the problem solving process of pre-service mathematics teachers related to open-mindedness. The stage of the problem-solving process is to determine a problem with no specified universal set. The participants were ten open-minded Mathematics Education students at Khairun University. This type of research is a descriptive qualitative approach. The research stages consisted of, 1) give problems, observe, and record the behavior of participants during problem solving, 2) analyze the answer sheet, 3) doing triangulation through interviews, and 4) summarize the tendencies of open-mindedness pre-service mathematics teachers in problem solving. The results showed that when participants solve problems with no specified universal set, they tend to assume that the completeness of the information from the questions has been fulfilled so that they proceed to problem solving.. However, during the interview and given a room for discussion between the participants, they consciously realized that there were many solutions or even had no solutions depending on the problems given.. Therefore, the open-mindedness of pre-service mathematics teachers in problem solving is largely determined by the existence of a discussion process.

Motivated by works on extension sets in standard domains, we introduce a notion of the Carathéodory set that seems better suited for the methods used in proofs of results on characterization of extension sets. A special stress is put on a class of two-dimensional submanifolds in the tridisc that not only turns out to be Carathéodory but also provides examples of two-dimensional domains for which the celebrated Lempert Theorem holds. Additionally, a recently introduced notion of universal sets for the Carathéodory extremal problem is studied and new results on domains admitting (no) finite universal sets are given.

In this paper we discussed the prominence of Bipolar fuzzy graphs (BFG). Fuzzy set assigns a sequence of membership values to the elements of the universal set ranging from 0 to 1, whereas now our study about Bipolar fuzzy graphs whose membership degree range is [-1, 1]. The earnest efforts of the researchers are perceivable in the relevant establishment of the subject integrating coherent practicality and reality. When we assess the position of an object in space, we may have positive information expressed as a set of possible places and negative information expressed as a set of impossible places. This corresponds to the idea that the union of positive and negative information does not cover the whole space. Dominating sets have a vital function regarding the theory of fuzzy graphs. Traveling salesman problem, communication network, traffic route problem are largely discussed applications among the diverse applications dealing with the theory of dominations. In this paper we generalized Bipolar fuzzy graphs and explored various types of dominations on Bipolar fuzzy graphs such as Strong Dominations, Split and non-split dominations, Multiple dominations and some applications of Bipolar fuzzy graphs. Fuzzy graphs found an increasing number of applications in modeling real time systems where the information inherent in the system varies with different levels of precision. Bipolar fuzzy graphs can be used to model many problems in economics, operations research, etc; involving two similar, but opposite type of qualitative variables like success and failure, gain and loss etc.

We propose novel and differentially private versions of Count Sketch, particularly suited for dynamic, intermittent queries for observed frequencies of elements in a universal set. Our algorithms are designed for scenarios where the queries are made intermittently, that is, at different times during the course of the data stream. We explore several approaches, all based on the Laplace mechanism, and ultimately propose an algorithm that is robust and efficiently handles multiple queries at multiple times while keeping its utility at reasonable levels. We demonstrate the performance of the proposed algorithm in various scenarios with a numerical example.

Dynamical decoupling provides an effective method to protect nonadiabatic geometric quantum computation against environment-induced decoherence. Although some dynamical-decoupling-protected nonadiabatic geometric gates were proposed, they need to combine some other dynamical gates to perform universal quantum computation. In this paper we propose a scheme to realize a universal set of dynamical-decoupling-protected nonadiabatic geometric gates, which not only possesses the resilience to environment-induced decoherence but also allows us to perform universal quantum computation all by geometric operations.

Prominent examples of longitudinal phase separation in elastic systems include elastic necking, the propagation of a bulge in a cylindrical party balloon, and the beading of a gel fiber subject to surface tension. Here we demonstrate that if the parameters of such a system are tuned near a critical point (where the difference between the two phases vanishes), then the behavior of all systems is given by the minimization of a simple and universal elastic energy familiar from Ginzburg-Landau theory in an external field. We minimize this energy analytically, which yields not only the well known interfacial tanh solution, but also the complete set of stable and unstable solutions in both finite and infinite length systems, unveiling the elastic system's full shape evolution and hysteresis. Correspondingly, we also find analytic results for the the delay of onset, changes in criticality, and ultimate suppression of instability with diminishing system length, demonstrating that our simple near-critical theory captures much of the complexity and choreography of far-from-critical systems. Finally, we find critical points for the three prominent examples of phase separation given above, and demonstrate how each system then follows the universal set of solutions.

In this paper we perform the first simultaneous QCD global analysis of data from semi-inclusive deep inelastic scattering, Drell-Yan, $e^+e^-$ annihilation into hadron pairs, and proton-proton collisions. Consequently, we are able to extract a universal set of non-perturbative functions that describes the observed asymmetries in these reactions. The outcome of our analysis indicates single transverse-spin asymmetries in high-energy collisions have a common origin. Furthermore, we achieve the first phenomenological agreement with lattice QCD on the up and down quark tensor charges., Comment: 8 pages, 6 figures, matches published version

Nonadiabatic geometric phases are only dependent on the evolution path of a quantum system but independent of the evolution details, and therefore quantum computation based on nonadiabatic geometric phases is robust against control errors. To realize nonadiabatic geometric quantum computation, it is necessary to ensure that the quantum system undergoes a cyclic evolution and the dynamical phases are removed from the total phases. To satisfy these conditions, the evolution paths in previous schemes are usually restricted to some special forms, e.g, orange-slice-shaped loops, which make the paths unnecessarily long in general. In this paper, we put forward an approach to the realization of nonadiabatic geometric quantum computation by which a universal set of nonadiabatic geometric gates can be realized with any desired evolution paths. Our approach makes it possible to realize geometric quantum computation with an economical evolution time so the influence of environment noises on the quantum gates can be minimized further., Comment: 6 pages, 2 figures

New Foundations ($\mathrm{NF}$) is a set theory obtained from naive set theory by putting a stratification constraint on the comprehension schema; for example, it proves that there is a universal set $V$. $\mathrm{NFU}$ ($\mathrm{NF}$ with atoms) is known to be consistent through its close connection with models of conventional set theory that admit automorphisms. A first-order theory, $\mathrm{ML}_\mathrm{CAT}$, in the language of categories is introduced and proved to be equiconsistent to $\mathrm{NF}$ (analogous results are obtained for intuitionistic and classical $\mathrm{NF}$ with and without atoms). $\mathrm{ML}_\mathrm{CAT}$ is intended to capture the categorical content of the predicative class theory of $\mathrm{NF}$. $\mathrm{NF}$ is interpreted in $\mathrm{ML}_\mathrm{CAT}$ through the categorical semantics. Thus, the result enables application of category theoretic techniques to meta-mathematical problems about $\mathrm{NF}$ -style set theory. For example, an immediate corollary is that $\mathrm{NF}$ is equiconsistent to $\mathrm{NFU} + |V| = |\mathcal{P}(V)|$. This is already proved by Crabbe, but becomes intuitively obvious in light of the results of this paper. Just like a category of classes has a distinguished subcategory of small morphisms, a category modelling $\mathrm{ML}_\mathrm{CAT}$ has a distinguished subcategory of type-level morphisms. This corresponds to the distinction between sets and proper classes in $\mathrm{NF}$. With this in place, the axiom of power objects familiar from topos theory can be appropriately formulated for $\mathrm{NF}$. It turns out that the subcategory of type-level morphisms contains a topos as a natural subcategory.

Quantum compiling, a process that decomposes the quantum algorithm into a series of hardware-compatible commands or elementary gates, is of fundamental importance for quantum computing. We introduce an efficient algorithm based on deep reinforcement learning that compiles an arbitrary single-qubit gate into a sequence of elementary gates from a finite universal set. It generates near-optimal gate sequences with given accuracy and is generally applicable to various scenarios, independent of the hardware-feasible universal set and free from using ancillary qubits. For concreteness, we apply this algorithm to the case of topological compiling of Fibonacci anyons and obtain near-optimal braiding sequences for arbitrary single-qubit unitaries. Our algorithm may carry over to other challenging quantum discrete problems, thus opening up a new avenue for intriguing applications of deep learning in quantum physics., 6 pages, 5 figures; Supplementary Material: 4 pages, 7 figures

The purpose of this study is to reveal the concept definitions and examples of pre-service elementary mathematics teachers regarding sets. In this context, students’ formal definitions, personal definitions and examples in sets were investigated. Concept of equal sets, subsets, union, intersection, compliment, and difference were considered as basic concepts of sets in this study. The participants of the study consisted of 79 freshmen studying in the department of elementary mathematics teacher training at a state university of Turkey. The study adopted qualitative research approach and is a sample of a case study. The data of the study was collected from Definition Knowledge Form for Sets and unstructured interviews. The results of the study indicated that most of the pre-service teachers had difficulty in giving notational definitions and daily-life examples while they gave successfully verbal definitions of the basic concepts in sets. It was revealed that verbal definitions, notational definitions, prototype examples with schema, symbols and operational properties were parts of some pre-service teachers’ concept images. Furthermore, some pre-service teachers confused equal sets with equivalent sets and difference with symmetric difference. They also did not consider the universal set in defining the compliment. It was found that most of the preservice teachers had difficulty in expressing their thoughts using mathematical language throughout the study.

The object of the paper is to formulate Quantum (Schrodinger) dynamics of spectrally bounded wavefunction. The Nyquist theorem is used to replace the wavefunction with a discrete series of numbers. Consequently, in this case, Schrodinger dynamics can be formalized as a universal set of ordinary differential Equations, with universal coupling between them, which are related to Euler’s formula. It is shown that the coefficient (m, n) is inversely proportional to the distance between the points n and m. As far as we know, this is the first time that this inverse square law was formulated.

Using author provided tags to predict tags for a new document often results in the overgeneration of tags. In the case where the author doesn't provide any tags, our documents face the severe under-tagging issue. In this paper, we present a method to generate a universal set of tags that can be applied widely to a large document corpus. Using IBM Watson's NLU service, first, we collect keywords/phrases that we call "complex document tags" from 8,854 popular reports in the corpus. We apply LDA model over these complex document tags to generate a set of 765 unique "simple tags". In applying the tags to a corpus of documents, we run each document through the IBM Watson NLU and apply appropriate simple tags. Using only 765 simple tags, our method allows us to tag 87,397 out of 88,583 total documents in the corpus with at least one tag. About 92.1% of the total 87,397 documents are also determined to be sufficiently-tagged. In the end, we discuss the performance of our method and its limitations., Work done by Maharshi R. Pandya and Jessica Reyes as IBM interns under leadership of Bob Vanderheyden. Article to be published

We present a simple proof of the approximate Eastin-Knill theorem, which connects the quality of a quantum error-correcting code (QECC) with its ability to achieve a universal set of transversal logical gates. Our derivation employs powerful bounds on the quantum Fisher information in generic quantum metrological protocols to characterize the QECC performance measured in terms of the worst-case entanglement fidelity. The theorem is applicable to a large class of decoherence models, including independent erasure and depolarizing noise. Our approach is unorthodox, as instead of following the established path of utilizing QECCs to mitigate noise in quantum metrological protocols, we apply methods of quantum metrology to explore the limitations of QECCs., 4+5 pages, 2 figures

The Toffoli gate and the Hadamard gate give rise to an approximately universal set of quantum computational gates. We study the algebraic properties of extensions of this system by introducing the notion of the extended Shi–Aharonov structure. The quotient of this structure is isomorphic to a structure based on a particular set of real numbers (the Bloch hypersphere). The aim of this paper is to bring together researchers working in different areas (quantum circuits, algebra, geometry) opening up new perspectives. The algebraic approach may be useful in circuit analysis.

In the paper, the implication and tensor operations of strong L-nested systems are found, and further we show that the family of all strong L-nested systems on a universal set forms a complete Girard residuated lattice provided the underlying complete residuated lattice is Girard. Finally, an isomorphism of complete Girard residuated lattices between the family of all strong L-nested systems and the L-power set for any fixed universal set is established, and thus a new representation theorem of L-subsets is obtained.

A multiparty quantum key agreement protocol based on three-photon entangled states is proposed. In this scheme, the quantum channel between all parties is that of a closed loop, in which the qubit transmission is one-way. Each party can obtain the sum of the other parties’ secret key values through the coding rules instead of extracting their private keys. The shared secret key cannot be determined by any subset of all the participants except the universal set and each party makes an equal contribution to the final key. Moreover, the security analysis shows our protocol can resist both outside attacks and inside attacks.

A trisecting-and-acting model of three-way decisions includes a step of dividing a universal set into three regions and another step of developing strategies and actions to process objects in the three regions. The effectiveness of three-way decisions depends on both trisecting and acting. We consider a special type of three-way decisions in which strategies facilitate movements of objects from unfavorable regions to favorable regions. We study effectiveness measures of strategies and actions based on object movements. These measures are composed of two parts. A desirability matrix represents preferred and non-preferred directions of movements. A pair of a benefit matrix and a cost matrix represents, respectively, the benefit and cost associated with a movement. An effectiveness measure of actions and strategies is the summation of benefits and costs of movements for all objects.

It is important to predict the hidden behavior of a complex system. In the existing models for predicting the hidden behavior, the hidden belief rule base (HBRB) is an effective model which can use qualitative knowledge and quantitative data. However, the frame of discernment (FoD) of HBRB which is composed of some states or propositions and the universal set including all states or propositions is not complete. The global ignorance and local ignorance cannot be considered at the same time, which may lead to the inaccurate forecasting results. To solve the problems, a new HBRB model named as PHBRB in which the hidden behavior is described on the FoD of the power set is proposed in this correspondence paper. Furthermore, by using the evidential reasoning rule as the inference tool of PHBRB, a new projection covariance matrix adaption evolution strategy is developed to optimize the parameters of PHBRB so that more accurate prediction results can be obtained. A case study of network security situation prediction is conducted to demonstrate the effectiveness of the newly proposed method.

According to the iterative conception, each set is formed out of sets that are, in some sense, prior to it. Because priority plays an essential role in explanations of why contradiction-inducing sets, such as the universal set, do not exist, the success of these explanations depends on our ability to make sense of the relevant priority relation. I argue that attempts to do this have fallen short: understanding priority in a straightforwardly constructivist sense threatens the coherence of the empty set and raises serious epistemological concerns; but the leading realist interpretations–ontological and modal interpretations of priority—are deeply problematic as well. I conclude that the purported explanatory virtues of the iterative conception are, at present, unfounded.