Predicate Calculus Research Articles

Spectrum and analytic functional calculus in real and quaternionic frameworks: An overview

<abstract><p>An approach to the elementary spectral theory for quaternionic linear operators was presented by the author in a recent paper, quoted and discussed in the Introduction, where, unlike in works by other authors, the construction of the analytic functional calculus used a Riesz-Dunford-Gelfand type kernel, and the spectra were defined in the complex plane. In fact, the present author regards the quaternionic linear operators as a special class of real linear operators, a point of view leading to a simpler and a more natural approach to them. The author's main results in this framework are summarized in the following, and other pertinent comments and remarks are also included in this text. In addition, a quaternionic joint spectrum for pairs of operators is discussed, and an analytic functional calculus which uses a Martinelli type kernel in two variables is recalled.</p></abstract>

Introducing and solving generalized Black–Scholes PDEs through the use of functional calculus

We introduce some families of generalized Black–Scholes equations which involve the Riemann–Liouville and Weyl space-fractional derivatives. We prove that these generalized Black–Scholes equations are well-posed in (L^1-L^infty )-interpolation spaces. More precisely, we show that the elliptic-type operators involved in these equations generate holomorphic semigroups. Then, we give explicit integral expressions for the associated solutions. In the way to obtain well-posedness, we prove a new connection between bisectorial-like operators and sectorial operators in an abstract setting. Such a connection extends the scaling property of sectorial operators to a wider family of both operators and the functions involved.

On the Variable Order Fractional Calculus Characterization for the Hidden Variable Fractal Interpolation Function

In this study, the variable order fractional calculus of the hidden variable fractal interpolation function is explored. It extends the constant order fractional calculus to the case of variable order. The Riemann–Liouville and the Weyl–Marchaud variable order fractional calculus are investigated for hidden variable fractal interpolation function. Moreover, the conditions for the variable fractional order μ on a specified range are also derived. It is observed that, under certain conditions, the Riemann–Liouville and the Weyl–Marchaud variable order fractional calculus of the hidden variable fractal interpolation function are again the hidden variable fractal interpolation functions interpolating the new data set.

Spectral order unit spaces and JB-algebras

Order unit spaces with comparability and spectrality properties as introduced by Foulis are studied. We define continuous functional calculus for order unit spaces with the comparability property and Borel functional calculus for spectral order unit spaces. Applying the conditions of Alfsen and Schultz, we characterize order unit spaces with comparability property that are JB-algebras. We also prove a characterization of Rickart JB-algebras as those JB-algebras for which every maximal associative subalgebra is monotone σ-complete, extending an analogous result of Saitô and Wright for C*-algebras.

Multivariate trace inequalities, p-fidelity, and universal recovery beyond tracial settings

Trace inequalities are general techniques with many applications in quantum information theory, often replacing the classical functional calculus in noncommutative settings. The physics of quantum field theory and holography, however, motivates entropy inequalities in type III von Neumann algebras that lack a semifinite trace. The Haagerup and Kosaki Lp spaces enable re-expressing trace inequalities in non-tracial von Neumann algebras. In particular, we show this for the generalized Araki–Lieb–Thirring and Golden–Thompson inequalities from the work of Sutter et al. [Commun. Math. Phys. 352(1), 37 (2017)]. Then, using the Haagerup approximation method, we prove a general von Neumann algebra version of universal recovery map corrections to the data processing inequality for relative entropy. We also show subharmonicity of a logarithmic p-fidelity of recovery. Furthermore, we prove that the non-decrease of relative entropy is equivalent to the existence of an L1-isometry implementing the channel on both input states.

Abstract fractional inverse source problem of order $$0<\alpha <1$$ in a Banach space

Given an abstract Cauchy problem with time-fractional derivative in the Caputo sense of order \(0<\alpha <1\) in a Banach space with an inhomogeneous term \(\varphi (t)\vartheta \), we investigate the inverse problem of determining \(\vartheta \) from it and an overdetermination in the form of an integral. Assume that the operator of evolution A generates an analytic \(C_0\)-semigroup, we give the uniqueness and unique existence of the solution in terms of the eigenvalues and spectrum of A. And via the approach of functional calculus, the representation of the solution is given.

Optimization of Hyperparameters in Object Detection Models Based on Fractal Loss Function

Hyperparameters involved in neural networks (NNs) have a significant impact on the accuracy of model predictions. However, the values of the hyperparameters need to be manually preset, and finding the best hyperparameters has always puzzled researchers. In order to improve the accuracy and speed of target recognition by a neural network, an improved genetic algorithm is proposed to optimize the hyperparameters of the network by taking the loss function as the research object. Firstly, the role of all loss functions in object detection is analyzed, and a mathematical model is established according to the relationship between loss functions and hyperparameters. Secondly, an improved genetic algorithm is proposed, and the feasibility of the improved algorithm is verified by using complex fractal function and fractional calculus. Finally, the improved genetic algorithm is used to optimize the hyperparameters of the neural network, and the prediction accuracy of the model before and after the improvement is comprehensively analyzed. By comparing with state-of-the-art object detectors, our proposed method achieves the highest prediction accuracy in object detection. Based on an average accuracy rate of 95%, the detection speed is 20 frames per second, which shows the rationality and feasibility of the optimized model.

The Quantified Argument Calculus with Two- and Three-valued Truth-valuational Semantics

We introduce a two-valued and a three-valued truth-valuational substitutional semantics for the Quantified Argument Calculus (Quarc). We then prove that the 2-valid arguments are identical to the 3-valid ones with strict-to-tolerant validity. Next, we introduce a Lemmon-style Natural Deduction system and prove the completeness of Quarc on both two- and three-valued versions, adapting Lindenbaum’s Lemma to truth-valuational semantics. We proceed to investigate the relations of three-valued Quarc and the Predicate Calculus (PC). Adding a logical predicate T to Quarc, true of all singular arguments, allows us to represent PC quantification in Quarc and translate PC into Quarc, preserving validity. Introducing a weak existential quantifier into PC allows us to translate Quarc into PC, also preserving validity. However, unlike the translated systems, neither extended system can have a sound and complete proof system with Cut, supporting the claim that these are basically different calculi.

Measurable functional calculi and spectral theory in the reflexive Banach spaces

In this paper, we consider the spectral theory. More precisely, we study the spectral families and their corresponding operators on reflexive Banach spaces. Assume [Formula: see text] is a well-bounded operator on reflexive Lebesgue spaces then the operator [Formula: see text] is a scalar-type spectral operator. We establish that if a weak spectral family [Formula: see text] is concentrated on [Formula: see text] then there exists a linear well-bounded operator [Formula: see text] on the reflexive Banach space [Formula: see text], such that [Formula: see text] holds for all [Formula: see text] and [Formula: see text]. We prove that by assuming [Formula: see text] is a functional calculus on the measurable space [Formula: see text], then there are a semi-finite measure space [Formula: see text] and solitary operator [Formula: see text], and an injective pointwise continuous *-homomorphism [Formula: see text], such that [Formula: see text] holds for any operator of the multiplication [Formula: see text] that multiplies by function [Formula: see text].

Fuzzy Differential Subordination and Superordination Results Involving the q-Hypergeometric Function and Fractional Calculus Aspects

The concepts of fuzzy differential subordination and superordination were introduced in the geometric function theory as generalizations of the classical notions of differential subordination and superordination. Fractional calculus is combined in the present paper with quantum calculus aspects for obtaining new fuzzy differential subordinations and superordinations. For the investigated fuzzy differential subordinations and superordinations, fuzzy best subordinates and fuzzy best dominants were obtained, respectively. Furthermore, interesting corollaries emerge when using particular functions, frequently involved in research studies due to their geometric properties, as fuzzy best subordinates and fuzzy best dominants. The study is finalized by stating the sandwich-type results connecting the previously proven results.

Axially Harmonic Functions and the Harmonic Functional Calculus on the S-spectrum

The spectral theory on the S-spectrum was introduced to give an appropriate mathematical setting to quaternionic quantum mechanics, but it was soon realized that there were different applications of this theory, for example, to fractional heat diffusion and to the spectral theory for the Dirac operator on manifolds. In this seminal paper we introduce the harmonic functional calculus based on the S-spectrum and on an integral representation of axially harmonic functions. This calculus can be seen as a bridge between harmonic analysis and the spectral theory. The resolvent operator of the harmonic functional calculus is the commutative version of the pseudo S-resolvent operator. This new calculus also appears, in a natural way, in the product rule for the F-functional calculus.

Gaussian Bounds for the Heat Kernel Associated to Prolate Spheroidal Wave Functions with Applications

Gaussian upper and lower bounds and Hölder continuity are established for the heat kernel associated to the prolate spheroidal wave functions (PSWFs) of order zero. These results are obtained by application of a general perturbation principle using the fact that the PSWF operator is a perturbation of the Legendre operator. Consequently, the Gaussian bounds and Hölder inequality for the PSWF heat kernel follow from the ones in the Legendre case. As an application of the general perturbation principle, we also establish Gaussian bounds for the heat kernels associated to generalized univariate PSWFs and PSWFs on the unit ball in $${\mathbb R}^d$$ . Further, we develop the related to the PSWFs of order zero smooth functional calculus, which provides the necessary groundwork in developing the theory of Besov and Triebel–Lizorkin spaces associated to the PSWFs. One of our main results on Besov and Triebel–Lizorkin spaces associated to the PSWFs asserts that they are the same as the Besov and Triebel–Lizorkin spaces generated by the Legendre operator.

Manifestation of interval uncertainties for fractional differential equations under conformable derivative

We propose a generalization of conformable calculus for Type-2 interval-valued functions. We investigated the differentiability and integrability properties of such functions. The conformable generalized Hukuhara (gH) differentiability of fractional order is introduced in this study. We prove a number of essential theorems on the conformable differentiability of the sum, gH difference, and product in a Type 2 interval setting. Furthermore, we define conformable Laplace transformation of Type-2 interval-valued functions. We interpret uncertain linear differential equations by using proposed theories. Several examples are given in detail to illustrate and clarify these rules and theorems. Applications to solving Type-2 interval differential equations with conformable derivatives are shown. Type-2 interval generalizes the interval uncertainty. On the other hand, conformable calculus extends the notion of integer calculus. This paper contributes a generalized theory that includes several existing results of classical integral and differential calculus and their conformable extensions in crisp and interval environments.

Fractional powers of higher-order vector operators on bounded and unbounded domains

AbstractUsing the $H^{\infty }$-functional calculus for quaternionic operators, we show how to generate the fractional powers of some densely defined differential quaternionic operators of order $m\geq 1$, acting on the right linear quaternionic Hilbert space $L^{2}(\Omega,\mathbb {C}\otimes \mathbb {H})$. The operators that we consider are of the type \begin{align*} T=i^{m-1}\left(a_1(x) e_1\partial_{x_1}^{m}+a_2(x) e_2\partial_{x_2}^{m}+a_3(x) e_3\partial_{x_3}^{m}\right), \quad x=(x_1,\, x_2,\, x_3)\in \overline{\Omega}, \end{align*}where $\overline {\Omega }$ is the closure of either a bounded domain $\Omega$ with $C^{1}$ boundary, or an unbounded domain $\Omega$ in $\mathbb {R}^{3}$ with a sufficiently regular boundary, which satisfy the so-called property $(R)$ (see Definition 1.3), $e_1,\, e_2,\, e_3\in \mathbb {H}$ which are pairwise anticommuting imaginary units, $a_1,\,a_2,\, a_3: \overline {\Omega } \subset \mathbb {R}^{3}\to \mathbb {R}$ are the coefficients of $T$. In particular, it will be given sufficient conditions on the coefficients of $T$ in order to generate the fractional powers of $T$, denoted by $P_{\alpha }(T)$ for $\alpha \in (0,1)$, when the components of $T$, i.e. the operators $T_l:=a_l\partial _{x_l}^{m}$, do not commute among themselves. This kind of result is to be understood in the more general setting of the fractional diffusion problems. The method used to construct the fractional power of a quaternionic linear operator is a generalization of the method developed by Balakrishnan.

Universality property of the S-functional calculus, noncommuting matrix variables and Clifford operators

Spectral theory on the S-spectrum was born out of the need to give quaternionic quantum mechanics a precise mathematical foundation (Birkhoff and von Neumann [8] showed that a general set theoretic formulation of quantum mechanics can be realized on real, complex or quaternionic Hilbert spaces). Then it turned out that spectral theory on S-spectrum has important applications in several fields such as fractional diffusion problems and, moreover, it allows one to define several functional calculi for n-tuples of noncommuting operators. With this paper we show that the spectral theory on the S-spectrum is much more general and it contains, just as particular cases, the complex, the quaternionic and the Clifford settings. In fact, the S-spectrum is well defined for objects in an algebra that has a complex structure and for operators in general Banach modules. We show that the abstract formulation of the S-functional calculus goes beyond quaternionic and Clifford analysis, indeed the S-functional calculus has a certain universality property. This fact makes the spectral theory on the S-spectrum applicable to several fields of operator theory and allows one to define functions of noncommuting matrix variables, and operator variables, as a particular case.

Causal functional calculus

We construct a new topology on the space of stopped paths and introduce a calculus for causal functionals on generic domains of this space. We propose a generic approach to pathwise integration without any assumption on the variation index of a path and obtain functional change of variable formulae which extend the results of Föllmer [Séminaire de probabilités 15 (1981), 143–150] and Cont and Fournié [J. Funct. Anal. 259 (2010), no. 4, 1043–1072] to a larger class of functionals, including Föllmer's pathwise integrals. We show that a class of smooth functionals possess a pathwise analogue of the martingale property. For paths that possess finite quadratic variation, our approach extends the Föllmer–Ito calculus and removes previous restriction on the time partition sequence. We introduce a foliation structure on this path space and show that harmonic functionals may be represented as pathwise integrals of closed 1-forms.

Two Points and n -th Derivatives Norm Inequalities for Analytic Functions in Banach Algebras

UDC 517.5 Let ℬ be a unital Banach algebra, let a ∈ ℬ , G be a convex domain of ℂ with σ ( a ) ⊂ G , let α , β ∈ G , and let f : G → ℂ be analytic on G . By using the analytic functional calculus, we obtain among others the following result:Some examples for the exponential function on Banach algebras are also given.

On finitely summable Fredholm modules from Smale spaces

We prove that all K K -homology classes of the stable (and unstable) Ruelle algebra of a Smale space have explicit Fredholm module representatives that are finitely summable on the same smooth subalgebra and with the same degree of summability. The smooth subalgebra is induced by a metric on the underlying Smale space groupoid and fine transversality relations between stable and unstable sets. The degree of summability is related to the fractal dimension of the Smale space. Further, the Fredholm modules are obtained by taking Kasparov products with a fundamental class of the Spanier-Whitehead K K -duality between the Ruelle algebras. Finally, we obtain general results on stability under holomorphic functional calculus and construct Lipschitz algebras on étale groupoids.

Catalan Generating Functions for Generators of Uni-parametric Families of Operators

In this paper we study solutions of the quadratic equation AY^2-Y+I=0 where A is the generator of a one parameter family of operator (C_0-semigroup or cosine functions) on a Banach space X with growth bound w_0 le frac{1}{4}. In the case of C_0-semigroups, we show that a solution, which we call Catalan generating function of A, C(A), is given by the following Bochner integral, C(A)x:=∫0∞c(t)T(t)xdt,x∈X,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} C(A)x := \\int _{0}^\\infty c(t) T(t)x \\; \\mathrm{d}t, \\quad x\\in X, \\end{aligned}$$\\end{document}where c is the Catalan kernel, c(t):=12π∫14∞e-λt4λ-1λdλ,t>0.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} c(t) := \\frac{1}{2\\pi } \\int _{\\frac{1}{4}}^\\infty e^{-\\lambda t} \\frac{\\sqrt{4\\lambda -1}}{\\lambda } \\; \\mathrm{d}\\lambda , \\quad t>0. \\end{aligned}$$\\end{document}Similar (and more complicated) results hold for cosine functions. We study algebraic properties of the Catalan kernel c as an element in Banach algebras L^1_{omega }(mathbb R^+), endowed with the usual convolution product, * and with the cosine convolution product, *_c. The Hille–Phillips functional calculus allows to transfer these properties to C_0-semigroups and cosine functions. In particular, we obtain a spectral mapping theorem for C(A). Finally, we present some examples, applications and conjectures to illustrate our results.

Asynchronous Functional Sessions: Cyclic and Concurrent

We present Concurrent GV (CGV), a functional calculus with message-passing concurrency governed by session types. With respect to prior calculi, CGV has increased support for concurrent evaluation and for cyclic network topologies. The design of CGV draws on APCP, a session-typed asynchronous pi-calculus developed in prior work. Technical contributions are (i) the syntax, semantics, and type system of CGV; (ii) a correct translation of CGV into APCP; (iii) a technique for establishing deadlock-free CGV programs, by resorting to APCP's priority-based type system.