Probability Theory for Fuzzy Quantum Spaces with Statistical Applications

Bartková, Renáta; Riečan, Beloslav; Tirpáková, Anna

doi:10.2174/97816810853881170101

Cited by 6 publications

(7 citation statements)

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“…where H : R → [0, 1] is a continuous distribution function, increasing on an interval. Then, H has one of three distributions with parameters µ, σ, α > 0 (Figure 2): Proof of Theorem 5.1 can be found in [63]. Proof of Theorem 5 can be found in [63].…”

Section: Extreme Value Theorems For Fuzzy Quantum Spacementioning

confidence: 97%

“…Then, H has one of three distributions with parameters µ, σ, α > 0 (Figure 2): Proof of Theorem 5.1 can be found in [63]. Proof of Theorem 5 can be found in [63].…”

Section: Extreme Value Theorems For Fuzzy Quantum Spacementioning

confidence: 97%

“…Let be a sequence of independent fuzzy observables, identically distributed in fuzzy state , with mean value and variance ∈ (0, ∞). Then, for any ∈ ℝ , the following equality holds: Regarding Theorem 1.7.5 [63], the validity of the following argument is obvious:…”

Section: Limit Theorems For Fuzzy Quantum Spacementioning

confidence: 97%

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Several Limit Theorems on Fuzzy Quantum Space

Ďuriš

Bartková²,

Tirpáková

2021

Mathematics

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The probability theory using fuzzy random variables has applications in several scientific disciplines. These are mainly technical in scope, such as in the automotive industry and in consumer electronics, for example, in washing machines, televisions, and microwaves. The theory is gradually entering the domain of finance where people work with incomplete data. We often find that events in the financial markets cannot be described precisely, and this is where we can use fuzzy random variables. By proving the validity of the theorem on extreme values of fuzzy quantum space in our article, we see possible applications for estimating financial risks with incomplete data.

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Section: Extreme Value Theorems For Fuzzy Quantum Spacementioning

confidence: 97%

“…Then, H has one of three distributions with parameters µ, σ, α > 0 (Figure 2): Proof of Theorem 5.1 can be found in [63]. Proof of Theorem 5 can be found in [63].…”

Section: Extreme Value Theorems For Fuzzy Quantum Spacementioning

confidence: 97%

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Several Limit Theorems on Fuzzy Quantum Space

Ďuriš

Bartková²,

Tirpáková

2021

Mathematics

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“…Note that Egorov's theorem can also be found in the literature under the name Egoroff's theorem (see [13]) or Jegorov's theorem (see [12]). In [14], the authors studied an almost uniform convergence and the Egorov's theorem for fuzzy observables in the fuzzy quantum space. Since the intuitionistic fuzzy sets are an extension of fuzzy sets, it is interesting to study an almost uniform convergence on the family of the intuitionistic fuzzy sets.…”

Section: Introductionmentioning

confidence: 99%

On Another Type of Convergence for Intuitionistic Fuzzy Observables

Čunderlíková

2023

Mathematics

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The convergence theorems play an important role in the theory of probability and statistics and in its application. In recent times, we studied three types of convergence of intuitionistic fuzzy observables, i.e., convergence in distribution, convergence in measure and almost everywhere convergence. In connection with this, some limit theorems, such as the central limit theorem, the weak law of large numbers, the Fisher–Tippet–Gnedenko theorem, the strong law of large numbers and its modification, have been proved. In 1997, B. Riečan studied an almost uniform convergence on D-posets, and he showed the connection between almost everywhere convergence in the Kolmogorov probability space and almost uniform convergence in D-posets. In 1999, M. Jurečková followed on from his research, and she proved the Egorov’s theorem for observables in MV-algebra using results from D-posets. Later, in 2017, the authors R. Bartková, B. Riečan and A. Tirpáková studied an almost uniform convergence and the Egorov’s theorem for fuzzy observables in the fuzzy quantum space. As the intuitionistic fuzzy sets introduced by K. T. Atanassov are an extension of the fuzzy sets introduced by L. Zadeh, it is interesting to study an almost uniform convergence on the family of the intuitionistic fuzzy sets. The aim of this contribution is to define an almost uniform convergence for intuitionistic fuzzy observables. We show the connection between the almost everywhere convergence and almost uniform convergence of a sequence of intuitionistic fuzzy observables, and we formulate a version of Egorov’s theorem for the case of intuitionistic fuzzy observables. We use the embedding of the intuitionistic fuzzy space into the suitable MV-algebra introduced by B. Riečan. We formulate the connection between the almost uniform convergence of functions of several intuitionistic fuzzy observables and almost uniform convergence of random variables in the Kolmogorov probability space too.

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“…One of them is to allow two kinds of uncertainty, sometimes called randomness and vagueness/fuzziness (for a review, see, [4]), which leads to the formulation of combined probability and possibility theories [5] (see, also, [6][7][8][9]). Various interconnections between vagueness and quantum probability calculus were considered in [10][11][12][13], including the treatment of inaccuracy in measurements [14,15], non-sharp amplitude densities [16] and the related concept of partial Hilbert spaces [17].…”

Section: Introductionmentioning

confidence: 99%

Obscure Qubits and Membership Amplitudes

Duplij¹,

Vogl²

2021

Topics on Quantum Information Science

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We propose a concept of quantum computing which incorporates an additional kind of uncertainty, i.e. vagueness (fuzziness), in a natural way by introducing new entities, obscure qudits (e.g. obscure qubits), which are characterized simultaneously by a quantum probability and by a membership function. To achieve this, a membership amplitude for quantum states is introduced alongside the quantum amplitude. The Born rule is used for the quantum probability only, while the membership function can be computed from the membership amplitudes according to a chosen model. Two different versions of this approach are given here: the “product” obscure qubit, where the resulting amplitude is a product of the quantum amplitude and the membership amplitude, and the “Kronecker” obscure qubit, where quantum and vagueness computations are to be performed independently (i.e. quantum computation alongside truth evaluation). The latter is called a double obscure-quantum computation. In this case, the measurement becomes mixed in the quantum and obscure amplitudes, while the density matrix is not idempotent. The obscure-quantum gates act not in the tensor product of spaces, but in the direct product of quantum Hilbert space and so called membership space which are of different natures and properties. The concept of double (obscure-quantum) entanglement is introduced, and vector and scalar concurrences are proposed, with some examples being given.

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Probability Theory for Fuzzy Quantum Spaces with Statistical Applications

Cited by 6 publications

References 0 publications

Several Limit Theorems on Fuzzy Quantum Space

Several Limit Theorems on Fuzzy Quantum Space

On Another Type of Convergence for Intuitionistic Fuzzy Observables

Obscure Qubits and Membership Amplitudes

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