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Simulations and bisimulations between two fuzzy automata over a complete residuated lattice were defined by \'Ciri\'c et al. (2012) as fuzzy relations between the sets of states of the automata. However, they act as a crisp relationship between the automata. In particular, if there exists a (forward) bisimulation between two fuzzy automata, then the fuzzy languages recognized by them are crisply equal. Approximate simulations and bisimulations introduced by Stanimirovi\'c et al. (2020) aim at fuzzifying this phenomenon. However, they are defined only for fuzzy automata over a complete Heyting algebra and do not give the exact relationship between states of the automata. In this article, we introduce and study fuzzy simulations and bisimulations between fuzzy automata over a complete residuated lattice. These notions are novel and have good properties. They are defined for fuzzy automata over any complete residuated lattice. We prove that the fuzzy language recognized by a fuzzy automaton is fuzzily preserved by fuzzy simulations and fuzzily invariant under fuzzy bisimulations. We also prove that the notions of fuzzy simulation and bisimulation have the Hennessy-Milner properties, which are a logical characterization of the greatest fuzzy simulation or bisimulation between two fuzzy automata. In addition, we provide results showing that our notions of fuzzy simulation and bisimulation are more general and refined than the notions of simulation and bisimulation introduced by \'Ciri\'c et al. and the notions of approximate simulation and bisimulation introduced by Stanimirovi\'c et al.

Quipper is a functional programming language for quantum computing. Proto-Quipper is a family of languages aiming to provide a formal foundation for Quipper. In this paper, we extend Proto-Quipper-M with a construct called dynamic lifting , which is present in Quipper. By virtue of being a circuit description language, Proto-Quipper has two separate runtimes: circuit generation time and circuit execution time. Values that are known at circuit generation time are called parameters , and values that are known at circuit execution time are called states . Dynamic lifting is an operation that enables a state, such as the result of a measurement, to be lifted to a parameter, where it can influence the generation of the next portion of the circuit. As a result, dynamic lifting enables Proto-Quipper programs to interleave classical and quantum computation. We describe the syntax of a language we call Proto-Quipper-Dyn. Its type system uses a system of modalities to keep track of the use of dynamic lifting. We also provide an operational semantics, as well as an abstract categorical semantics for dynamic lifting based on enriched category theory. We prove that both the type system and the operational semantics are sound with respect to our categorical semantics. Finally, we give some examples of Proto-Quipper-Dyn programs that make essential use of dynamic lifting.

Semantic network is a network of concepts connected by semantic relations. It contains two forms of binary semantic network and multiplex semantic network. The associated fact prediction is a link prediction task that aims to infer the implicitly connected facts by mining the high-level representation of the network. Previous methods for associated fact prediction put much emphasis on the topological feature of network but not utilize the information of semantic expression. This paper proposes a Semantic Network Encoder (SemNE), which learns a feature mapping function from the binary semantic networks and can be applied to the multiplex semantic networks in a pre-training manner. SemNE is a two-stage framework that contains an embedding encoder and a prediction decoder. It jointly models the semantic information and network topology to enrich the network representation. A word self-organization method based on the factual boundary is proposed to unify the topological feature and the semantic feature representations. Experimental results on binary semantic networks show that SemNE achieves the state-of-the-art results in associated fact prediction and experimental results on multiplex semantic networks show that SemNE is scalable and can effectively improve the performance of existing models.

Based on the connection between the categorical derivation of classical programs from specifications and the category-theoretic approach to quantum physics, this paper contributes to extending the laws of classical program algebra to quantum programming. This aims at building correct-by-construction quantum circuits to be deployed on quantum devices such as those available at the IBM Q Experience. Quantum circuit reversibility is ensured by minimal complements, extended recursively. Measurements are postponed to the end of such recursive computations, termed "quantamorphisms", thus maximising the quantum effect. Quantamorphisms are classical catamorphisms which, extended to ensure quantum reversibility, implement quantum cycles (vulg. for-loops) and quantum folds on lists. By Kleisli correspondence, quantamorphisms can be written as monadic functional programs with quantum parameters. This enables the use of Haskell, a monadic functional programming language, to perform the experimental work. Such calculated quantum programs prepared in Haskell are pushed through Quipper to the Qiskit interface to IBM Q quantum devices. The generated quantum circuits - often quite large - exhibit the predicted behaviour. However, running them on real quantum devices incurs into a significant amount of errors. As quantum devices are constantly evolving, an increase in reliability is likely in the near future, allowing for our programs to run more accurately., 18 pages

We characterize the optimal solution to the linear programming relaxation of the standard formulation for the minimum satisfiability problem. We give a O ( n m 2 ) combinatorial algorithm to solve the fractional version of the minimum satisfiability problem optimally where n ( m ) is the number of variables (clauses). As a by-product, we obtain a 2 ( 1 − 1 ∕ 2 k ) approximation algorithm for the minimum satisfiability problem where k is the maximum number of literals in any clause.

For each of (i) arbitrary stochastic reset, (ii) deterministic reset with arbitrary period, (iii) reset at arbitrary constant rate, and then in the sense of either (a) first-order stochastic dominance or (b) expectation (i.e. for each of the six possible combinations of the preceding), those laws of random times are precisely characterized that are rendered no bigger [rendered no smaller; left invariant] by all possible restart laws (within the classes (i), (ii), (iii), as the case may be). Partial results in the same vein for reset with branching are obtained. In particular it is found that deterministic and arbitrary stochastic restart lead to the same characterizations, but this equivalence fails to persist for exponential (constant-rate) reset.

Probabilistic programming languages (PPLs) allow programmers to construct statistical models and then simulate data or perform inference over them. Many PPLs restrict models to a particular instance of simulation or inference, limiting their reusability. In other PPLs, models are not readily composable. Using Haskell as the host language, we present an embedded domain specific language based on algebraic effects, where probabilistic models are modular, first-class, and reusable for both simulation and inference. We also demonstrate how simulation and inference can be expressed naturally as composable program transformations using algebraic effect handlers.

Many problem domains, including program synthesis and rewrite-based optimization, require searching astronomically large spaces of programs. Existing approaches often rely on building specialized data structures—version-space algebras, finite tree automata, or e-graphs—to compactly represent such spaces. At their core, all these data structures exploit independence of subterms; as a result, they cannot efficiently represent more complex program spaces, where the choices of subterms are entangled. We introduce equality-constrained tree automata (ECTAs), a new data structure, designed to compactly represent large spaces of programs with entangled subterms. We present efficient algorithms for extracting programs from ECTAs, implemented in a performant Haskell library, ecta. Using the ecta library, we construct Hectare, a type-driven program synthesizer for Haskell. Hectare significantly outperforms a state-of-the-art synthesizer Hoogle+—providing an average speedup of 8×—despite its implementation being an order of magnitude smaller.

Free monads (and their variants) have become a popular general-purpose tool for representing the semantics of effectful programs in proof assistants. These data structures support the compositional definition of semantics parameterized by uninterpreted events, while admitting a rich equational theory of equivalence. But monads are not the only way to structure effectful computation, why should we limit ourselves? In this paper, inspired by applicative functors, selective functors, and other structures, we define a collection of data structures and theories, which we call program adverbs, that capture a variety of computational patterns. Program adverbs are themselves composable, allowing them to be used to specify the semantics of languages with multiple computation patterns. We use program adverbs as the basis for a new class of semantic embeddings called Tl\"on embeddings. Compared with embeddings based on free monads, Tl\"on embeddings allow more flexibility in computational modeling of effects, while retaining more information about the program's syntactic structure., Comment: Accepted by ICFP 2022. 32 pages. Extended version

We propose an IR for quantum computing that directly exposes quantum and classical data dependencies for the purpose of optimization. The Quantum Intermediate Representation for Optimization (QIRO) consists of two dialects, one input dialect and one that is specifically tailored to enable quantum-classical co-optimization. While the first employs a perhaps more intuitive memory-semantics (quantum operations act on qubits via side-effects), the latter uses value-semantics (operations consume and produce states) to integrate quantum dataflow in the IR’s Static Single Assignment (SSA) graph. Crucially, this allows for a host of optimizations that leverage dataflow analysis. We discuss how to map existing quantum programming languages to the input dialect and how to lower the resulting IR to the optimization dialect. We present a prototype implementation based on MLIR that includes several quantum-specific optimization passes. Our benchmarks show that significant improvements in resource requirements are possible even through static optimization. In contrast to circuit optimization at run time, this is achieved while incurring only a small constant overhead in compilation time, making this a compelling approach for quantum program optimization at application scale.

In answer set programming, two groups of rules are considered strongly equivalent if they have the same meaning in any context. In some cases, strong equivalence of programs in the input language of the grounder gringo can be established by deriving rules of each program from rules of the other. The possibility of such proofs has been demonstrated for a subset of that language that includes comparisons, arithmetic operations, and simple choice rules, but not aggregates. This method is extended here to a class of programs in which some uses of the #count aggregate are allowed. This paper is under consideration for acceptance in TPLP., Paper presented at the 38th International Conference on Logic Programming (ICLP 2022), 16 pages

In this study we analyzed turbulent heat fluxes over a seasonal ice cover on a boreal lake located in southern Finland. Eddy covariance (EC) flux measurements of sensible (H) and latent heat (LE) from four ice-on seasons between 2014 and 2019 are compared to three different bulk transfer models: one with a constant transfer coefficient and two with stability-adjusted transfer coefficients: the Lake Heat Flux Analyzer and SEA-ICE. All three models correlate well with the EC results in general while typically underestimating the magnitude and the standard deviation of the flux in comparison to the EC observations. Differences between the models are small, with the constant transfer coefficient model performing slightly better than the stability-adjusted models. Small difference in temperature and humidity between surface and air results in low correlation between models and EC. During melting periods (surface temperature T0>0 ∘C), the model performance for LE decreases when compared to the freezing periods (T0 ∘C), while the opposite is true for H. At low wind speed, EC shows relatively high fluxes (±20 W m−2) for H and LE due to non-local effects that the bulk models are not able to reproduce. The complex topography of the lake surroundings creates local violations of the Monin–Obukhov similarity theory, which helps explain this counterintuitive result. Finally, the uncertainty in the estimation of the surface temperature and humidity affects the bulk heat fluxes, especially when the differences between surface and air values are small.

Wraparound channels in Torus Network-on-Chip (NoC) help in reducing overall hop counts traversed by traffic. However, the cyclic paths created by wraparound channel make Torus NoC deadlock prone. Turn model and Channel Dependency Graph (CDG) are two classical approaches used for detecting and avoiding deadlock in NoC. In this work, we propose an Arc model for avoiding deadlock in Torus NoC. Arc model is an extension to Turn model and is useful for deadlock avoidance in Torus. Directional Dependency Graph (DDG) is also presented in this work by combining both Turn model and CDG for detecting deadlock in Torus NoC. DDG is a simpler approach for identifying deadlock scenarios, formulating deadlock avoidance and showing deadlock freedom while using Arc model along with a set of Turns.

The lattice-valued intuitionistic fuzzy set was introduced by Gerstenkorn and Tepavcevi as a generalization of both the fuzzy set and the L -fuzzy set by incorporating membership functions, nonmembership functions from a nonempty set X to any lattice L , and lattice homomorphism from L to the interval 0,1 . In this article, lattice-valued intuitionistic fuzzy subgroup type-3 (LIFSG-3) is introduced. Lattice-valued intuitionistic fuzzy type-3 normal subgroups, cosets, and quotient groups are defined, and their group theocratic properties are compared with the concepts in classical group theory. LIFSG-3 homomorphism is established and examined in relation to group homomorphism. The research findings are supported by provided examples in each section.

Families of 1- and 2-periodic knots and weavings that have isogonal (vertex-transitive) piecewise-linear embeddings are described. In these structures there is just one thread, or multiple threads with parallel or collinear axes. The principal structures are a large family of 1-periodic knots and related multi-thread infinite links, knitting patterns and chain-link weaving. The relevance to synthetic chemistry is described in terms of targets for designed synthesis such as mechanically interlocked polymers.

Symmetric monoidal theories (SMTs) generalise algebraic theories in a way that make them suitable to express resource-sensitive systems, in which variables cannot be copied or discarded at will. In SMTs, traditional tree-like terms are replaced by string diagrams, topological entities that can be intuitively thought of as diagrams of wires and boxes. Recently, string diagrams have become increasingly popular as a graphical syntax to reason about computational models across diverse fields, including programming language semantics, circuit theory, quantum mechanics, linguistics, and control theory. In applications, it is often convenient to implement the equations appearing in SMTs as rewriting rules. This poses the challenge of extending the traditional theory of term rewriting, which has been developed for algebraic theories, to string diagrams. In this paper, we develop a mathematical theory of string diagram rewriting for SMTs. Our approach exploits the correspondence between string diagram rewriting and double pushout (DPO) rewriting of certain graphs, introduced in the first paper of this series. Such a correspondence is only sound when the SMT includes a Frobenius algebra structure. In the present work, we show how an analogous correspondence may be established for arbitrary SMTs, once an appropriate notion of DPO rewriting (which we call convex) is identified. As proof of concept, we use our approach to show termination of two SMTs of interest: Frobenius semi-algebras and bialgebras.

The main objective of this article is to establish generalized fractional Hermite–Hadamard and related type integral inequalities forh-Godunova–Levin convexity andh-Godunova–Levin preinvexity with extended Wright generalized Bessel function acting as kernel. Moreover, Hermite–Hadamard-type and trapezoid-type inequalities for several known convexities including Godunova–Levin function, classical convex,s-Godunova–Levin function,P-function, ands-convex function are deduced as corollaries. These obtained results are analyzed in the form of generalization of fractional inequalities.

In this paper, we aim to develop formulas of spectral radius for an operator S in terms of generalized Aluthge transform, numerical radius, iterated generalized Aluthge transform, and asymptotic behavior of powers of S . These formulas generalize some of the formulas of spectral radius existing in literature. As an application, these formulas are used to obtain several characterizations of normaloid operators.

The notion of Δ -metric spaces has been proposed in this study as a generalization of b -metric spaces, extended b -metric spaces, and p -metric spaces. A number of topological characteristics of such spaces have been investigated in this paper. On such spaces, a noncompactness measure has been established, and some results in the framework of noncompactness measure have been achieved. We prove an analogous of the Banach contraction principle in such spaces based on this approach. In order to investigate the validity of the underlying space and our proven fixed-point theorems, supporting examples have been presented. Furthermore, the well-posedness of the fixed-point problem has been tested using our fixed-point result.

The ‘cosmological constant problem’ (CCP) has historically been understood as describing a conflict between cosmological observations in the framework of general relativity (GR) and theoret...

In Probabilistic Abductive Logic Programming we are given a probabilistic logic program, a set of abducible facts, and a set of constraints. Inference in probabilistic abductive logic programs aims to find a subset of the abducible facts that is compatible with the constraints and that maximizes the joint probability of the query and the constraints. In this paper, we extend the PITA reasoner with an algorithm to perform abduction on probabilistic abductive logic programs exploiting Binary Decision Diagrams. Tests on several synthetic datasets show the effectiveness of our approach.

Describing systems in terms of choices and their resulting costs and rewards offers the promise of freeing algorithm designers and programmers from specifying how those choices should be made; in implementations, the choices can be realized by optimization techniques and, increasingly, by machine-learning methods. We study this approach from a programming-language perspective. We define two small languages that support decision-making abstractions: one with choices and rewards, and the other additionally with probabilities. We give both operational and denotational semantics. In the case of the second language we consider three denotational semantics, with varying degrees of correlation between possible program values and expected rewards. The operational semantics combine the usual semantics of standard constructs with optimization over spaces of possible execution strategies. The denotational semantics, which are compositional, rely on the selection monad, to handle choice, augmented with an auxiliary monad to handle other effects, such as rewards or probability. We establish adequacy theorems that the two semantics coincide in all cases. We also prove full abstraction at base types, with varying notions of observation in the probabilistic case corresponding to the various degrees of correlation. We present axioms for choice combined with rewards and probability, establishing completeness at base types for the case of rewards without probability.

Partially lossy queue monoids (plq monoids) model the behavior of queues that can non-deterministically forget specified parts of their content at any time. We call the subsets of this monoid partially lossy queue languages (plq languages). While many decision problems on recognizable plq languages are decidable, most of them are undecidable if the languages are rational. In particular, in this monoid the classes of rational and recognizable languages do not coincide. This is due to the fact that the class of recognizable plq languages is not closed under multiplication and iteration. However, we can generate the recognizable plq languages using special rational expressions consisting of the Boolean operations and restricted versions of multiplication and iteration. From these special rational expressions we can also obtain an MSO logic describing the recognizable plq languages. Moreover, we provide similar results for the class of aperiodic languages in the plq monoid.

In this paper, we study α-cosymplectic manifoldMadmitting∗-Ricci tensor. First, it is shown that a∗-Ricci semisymmetric manifoldMis∗-Ricci flat and aϕ-conformally flat manifoldMis anη-Einstein manifold. Furthermore, the∗-Weyl curvature tensorW∗onMhas been considered. Particularly, we show that a manifoldMwith vanishing∗-Weyl curvature tensor is a weakϕ-Einstein and a manifoldMfulfilling the conditionRE1,E2⋅W∗=0isη-Einstein manifold. Finally, we give a characterization for α-cosymplectic manifoldMadmitting∗-Ricci soliton given as to be nearly quasi-Einstein. Also, some consequences for three-dimensional cosymplectic manifolds admitting∗-Ricci soliton and almost∗-Ricci soliton are drawn.

Based on the Navier–Stokes equation for compressible media, this work studies the acoustic properties of a human cochlear model, in which the scala vestibuli and scala tympani are filled with compressible perilymph. Since the sound waves propagate as a compression wave in perilymph, this model can precisely handle the wave–based phenomena. Time domain analysis showed that a sound wave (fast wave) first propagates in the scala vestibuli and scala tympani, and then, a traveling wave (slow wave) is generated by the sound wave with some delay. Detailed studies based on even and odd mode analysis indicate that an odd mode sound wave, that is, the difference in the sound pressures between the scala vestibuli and scala tympani, excites the Békésy’s traveling wave, while an even mode sound determines the input impedance of the cochlea.

This paper scrutinizes the consequences of radiation and heat consumption of MHD convective flow of nanofluid on a heated stretchy plate with injection/suction and convective heating/cooling conditions. The nanofluid encompasses with C u and A g nanoparticles. We enforce the suited transformation to remodel the governing mathematical models to ODE models. The HAM (homotopy analysis method) idea is applied to derive the series solutions. The divergence of fluid velocity, temperature, skin friction coefficient, local Nusselt number, entropy generation, and Bejan number on disparate governing parameters is exhibited via graphs and tables. It is seen that the fluid velocity in both directions is subsided when elevating the magnetic field and Forchheimer number. Also, the C u nanoparticles possess hefty speed compared to A g nanoparticles because the density of A g nanoparticles is high compared to that of C u nanoparticles. The fluid temperature upturns when enlarging the heat generation and radiation parameters. The skin friction coefficients and local Nusselt number are high in A g nanoparticles than in C u nanoparticles.

We calculate the centrality dependence for coherent photoproduction of very low- p T J / ψ at Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) energies within the impact parameter-dependent saturated color dipole model. By using the large equivalent photon fluxes, we present the differential cross-section of very low- p T J / ψ produced by coherent photonuclear in peripheral heavy ion collisions. The numerical results demonstrate that our calculation agrees with J / ψ data in peripheral heavy ion collisions at Relativistic Heavy Ion Collider (RHIC) energies.

A directed graph (digraph) D of order n satisfies directed cut condition D C C if there are at least S arcs from any set S ⊆ V D , S ≤ n / 2 to its complement S ¯ = V D / S . We show that a digraph D of even order n has at least 2 n − 3 arcs if D satisfies D C C .

In order to study the investment portfolio problem, this paper propose a class of stochastic programming model with rough feasible region, where randomness and roughness coexist. Based on the covering rough set, the concept of the discrete degree covering is defined to divide the rough feasible region. Furthermore, the discrete degree covering stochastic rough programming model (DDC-SRP) is constructed depending on a synthesis effect function that considers discrete degree and the expectation and variance for random objective function. Properties of the DDC-SRP model are discussed. In addition, the convexity of the DDC-SRP model is obtained in some certain conditions. Considering the random rough simulation, a genetic algorithm is introduced. Finally, a numerical example is given to show the validity of the DDC-SRP model.

The role of Fuzzy topology in logic programming and algorithm has been recognized and applied on various programs to and more accurate result. In particular, topological methods are employed in order to obtain fixed point semantics for logic programs. In this paper, we prove some fixed point theorems in fuzzy metric spaces. As an application, some consequence theorems are given in support of our result.

In this paper, we define λ-approximate simulations and bisimulations for fuzzy automata over complete Heyting algebras. The value λ presents the degree of language similarity or equality between observed fuzzy automata. Algorithms for computing the greatest λ-approximate simulations and bisimulations are given. We show that λ-approximate simulations and bisimulations on a fuzzy automaton can be effectively used for factorization of fuzzy automata. We present the algorithm that splits the interval of the degrees of language similarity or equality into subintervals with same minimal corresponding factor fuzzy automata.

We develop a constructive theory of finite multisets in Homotopy Type Theory, defining them as free commutative monoids. After recalling basic structural properties of the free commutative-monoid construction, we formalise and establish the categorical universal property of two, necessarily equivalent, algebraic presentations of free commutative monoids using 1-HITs. These presentations correspond to two different equational theories invariably including commutation axioms. In this setting, we prove important structural combinatorial properties of finite multisets. These properties are established in full generality without assuming decidable equality on the carrier set. As an application, we present a constructive formalisation of the relational model of classical linear logic and its differential structure. This leads to constructively establishing that free commutative monoids are conical refinement monoids. Thereon we obtain a characterisation of the equality type of finite multisets and a new presentation of the free commutative-monoid construction as a set-quotient of the list construction. These developments crucially rely on the commutation relation of creation/annihilation operators associated with the free commutative-monoid construction seen as a combinatorial Fock space., Comment: Appeared in MFPS'22

An F-system is a computational model that performs a folding operation on words of a given language, following directions coded on words of another given language. This paper considers the case in which both given languages are regular, and it shows that the class of languages generated by such F-systems is a proper subset of the class of linear context-free languages., Comment: Nine pages. This is the accepted version of a manuscript submitted for publication in Information Processing Letters

Knaster-Tarski’s theorem, characterising the greatest fix- point of a monotone function over a complete lattice as the largest post-fixpoint, naturally leads to the so-called coinduction proof principle for showing that some element is below the greatest fixpoint (e.g., for providing bisimilarity witnesses). The dual principle, used for showing that an element is above the least fixpoint, is related to inductive invariants. In this paper we provide proof rules which are similar in spirit but for showing that an element is above the greatest fixpoint or, dually, below the least fixpoint. The theory is developed for non-expansive monotone functions on suitable lattices of the form$$\mathbb {M}^Y$$MY, whereYis a finite set and$$\mathbb {M}$$Man MV-algebra, and it is based on the construction of (finitary) approximations of the original functions. We show that our theory applies to a wide range of examples, including termination probabilities, behavioural distances for probabilistic automata and bisimilarity. Moreover it allows us to determine original algorithms for solving simple stochastic games.

This chapter discusses systems of ordinary differential equations. It presents a review of matrix algebra and calculus. It defines the characteristic equation and characteristic polynomial and shows how to calculate characteristic polynomial and eigenvalues of matrices through examples. Mathematica can compute both the characteristic polynomial and eigenvalues directly with the commands CharacteristicPolynomial and Eigenvalues. The chapter illustrates the homogeneous system of first-order linear differential equations. It provides solution of homogeneous linear systems with constant coefficients through some examples. It also discusses some theorems and terminology used in establishing the fundamentals of solving systems of differential equations. To apply variation of parameters, one first calculates a fundamental matrix for the associated homogeneous equation. In many cases, Laplace transforms can be used to solve a system of linear differential equations. The chapter illustrates the techniques needed to accomplish this with Mathematica through examples. It further explains nonlinear systems, linearization, and equilibrium points.

In this paper we try to find a computational interpretation for a strong form of extensionality, which we call "converse extensionality". Converse extensionality principles, which arise as the Dialectica interpretation of the axiom of extensionality, were first studied by Howard. In order to give a computational interpretation to these principles, we reconsider Brouwer's apartness relation, a strong constructive form of inequality. Formally, we provide a categorical construction to endow every typed combinatory algebra with an apartness relation. We then exploit that functions reflect apartness, in addition to preserving equality, to prove that the resulting categories of assemblies model a converse extensionality principle.

In this paper, we develop some Hermite–Hadamard–Fejér type inequalities for n-times differentiable functions whose absolute values of n-th derivatives are (α,m)-convex function. The results obtained in this paper are extensions and generalizations of the existing ones. As a special case, the generalization of the remainder term of the midpoint and trapezoidal quadrature formulas are obtained.

The purpose of this study is to introduce the new class of Hermite–Hadamard inequality for LR-convex interval-valued functions known as LR-interval Hermite–Hadamard inequality, by means of pseudo-order relation ( ≤p ). This order relation is defined on interval space. We have proved that if the interval-valued function is LR-convex then the inclusion relation “ ⊆ ” coincident to pseudo-order relation “ ≤p ” under some suitable conditions. Moreover, the interval Hermite–Hadamard–Fejér inequality is also derived for LR-convex interval-valued functions. These inequalities also generalize some new and known results. Useful examples that verify the applicability of the theory developed in this study are presented. The concepts and techniques of this paper may be a starting point for further research in this area.

In this paper, algebraic relations were established that determined the invariance of a transformed number after several transformations. The restrictions that determine the group structure of these relationships were analyzed, as was the case of the Klein group. Parametric Kr functions associated with the existence of cycles were presented, as well as the role of the number of their links in the grouping of numbers in higher-order equivalence classes. For this, we developed a methodology based on binary equivalence relations and the complete parameterization of the Kaprekar routine using Ki functions of parametric transformation.

This paper addresses an H2 optimal control problem for a class of discrete-time stochastic systems with Markov jump parameter and multiplicative noises. The involved Markov jump parameter is a uniform ergodic Markov chain taking values in a Borel-measurable set. In the presence of exogenous white noise disturbance, Gramian characterization is derived for the H2 norm, which quantifies the stationary variance of output response for the considered systems. Moreover, under the condition that full information of the system state is accessible to measurement, an H2 dynamic optimal control problem is shown to be solved by a zero-order stabilizing feedback controller, which can be represented in terms of the stabilizing solution to a set of coupled stochastic algebraic Riccati equations. Finally, an iterative algorithm is provided to get the approximate solution of the obtained Riccati equations, and a numerical example illustrates the effectiveness of the proposed algorithm.

The family of cumulative paired ϕ-entropies offers a wide variety of ordinal dispersion measures, covering many well-known dispersion measures as a special case. After a comprehensive analysis of this family of entropies, we consider the corresponding sample versions and derive their asymptotic distributions for stationary ordinal time series data. Based on an investigation of their asymptotic bias, we propose a family of signed serial dependence measures, which can be understood as weighted types of Cohen’s κ, with the weights being related to the actual choice of ϕ. Again, the asymptotic distribution of the corresponding sample κϕ is derived and applied to test for serial dependence in ordinal time series. Using numerical computations and simulations, the practical relevance of the dispersion and dependence measures is investigated. We conclude with an environmental data example, where the novel ϕ-entropy-related measures are applied to an ordinal time series on the daily level of air quality.

In this paper, the author considers twisted q-analogues of Catalan-Daehee numbers and polynomials by using p-adic q-integral on Zp. We derive some explicit identities for those twisted numbers and polynomials related to various special numbers and polynomials.

For random walks on a complex network, the configuration of a network that provides optimal or suboptimal navigation efficiency is meaningful research. It has been proven that a complete graph has the exact minimal mean hitting time, which grows linearly with the network order. In this paper, we present a class of sparse networks G(t) in view of a graphic operation, which have a similar dynamic process with the complete graph; however, their topological properties are different. We capture that G(t) has a remarkable scale-free nature that exists in most real networks and give the recursive relations of several related matrices for the studied network. According to the connections between random walks and electrical networks, three types of graph invariants are calculated, including regular Kirchhoff index, M-Kirchhoff index and A-Kirchhoff index. We derive the closed-form solutions for the mean hitting time of G(t), and our results show that the dominant scaling of which exhibits the same behavior as that of a complete graph. The result could be considered when designing networks with high navigation efficiency.

Among various modifications of the permutation entropy defined as the Shannon entropy of the ordinal pattern distribution underlying a system, a variant based on Rényi entropies was considered in a few papers. This paper discusses the relatively new concept of Rényi permutation entropies in dependence of non-negative real number q parameterizing the family of Rényi entropies and providing the Shannon entropy for q=1. Its relationship to Kolmogorov–Sinai entropy and, for q=2, to the recently introduced symbolic correlation integral are touched.

Swirling has a significant effect on the main characteristics of flow and can lead to its fundamental restructuring. On the flow axis, a stagnation point with zero velocity is possible, behind which a return flow zone is formed. The apparent instability leads to the formation of secondary vortex motions and can also be the cause of vortex breakdown. In the paper, a swirling flow with a velocity profile of the Batchelor vortex type has been studied on the basis of the linear hydrodynamic stability theory. An effective numerical method for solving the spectral problem has been developed. This method includes the asymptotic solutions at artificial and irregular singular points. The stability of flows was considered for the values of the Reynolds number in the range 10≤Re≤5×106. The calculations were carried out for the value of the azimuthal wavenumber parameter n=−1. As a result of the analysis of the solutions, the existence of up to eight simultaneously occurring unstable modes has been shown. The paper presents a classification of the detected modes. The critical parameters are calculated for each mode. For fixed values of the Reynolds numbers 60≤Re≤5000, the curves of neutral stability are plotted. Branching points of unstable modes are found. The maximum growth rates for each mode are determined. A new viscous instability mode is found. The performed calculations reveal the instability of the Batchelor vortex at large values of the swirl parameter for long-wave disturbances.

We extend the formalisation of confluence results in Kleene algebras to a formalisation of coherent confluence proofs. For this, we introduce the structure of higher globular Kleene algebra, a higher-dimensional generalisation of modal and concurrent Kleene algebra. We calculate a coherent Church-Rosser theorem and a coherent Newman's lemma in higher Kleene algebras by equational reasoning. We instantiate these results in the context of higher rewriting systems modelled by polygraphs.

Applicative bisimilarity is a coinductive characterisation of observational equivalence in call-by-name lambda-calculus, introduced by Abramsky in 1990. Howe (1989) gave a direct proof that it is a congruence. In previous work with Borthelle (2020), we abstract over this result by proposing a categorical framework for specifying operational semantics, in which we prove that (an abstract analogue of) applicative bisimilarity is automatically a congruence. However, the framework presents a few infelicities: (1) it requires a non-trivial refinement of the standard approach of Fiore, Plotkin, and Turi (1999) based on monoid algebras for specifying syntax with variable binding; (2) it relies on so-called prebisimulations instead of the more standard notion of bisimulation by lifting; (3) one of the axioms, called weak compositionality, feels ad hoc; (4) the proofs involve directed unions of relations, leading to quite a few painful inductions. In this paper, we rectify all of these deficiencies. In particular, a notable novelty is that the so-called Howe closure is defined as an initial monoid algebra in a category of spans. Finally, the familiality/cellularity axiom of the previous framework is now viewed as a mere sufficient condition for the main hypothesis, preservation of functional bisimulations.

We propose to simulate chemical reaction networks with the deterministic semantics abstractly, without any precise knowledge on the initial concentrations. For this, the concentrations of species are abstracted to booleans stating whether the species is present or absent, and the derivations of the concentrations are abstracted to signs saying whether the concentration is increasing, decreasing, or unchanged. We use abstract interpretation over the structure of signs for mapping the ODEs of a reaction network to a boolean network with nondeterministic updates. The abstract state transition graph of such boolean networks can be computed by finite domain constraint programming over the finite structure of signs. Constraints on the abstraction of the initial concentrations can be added naturally, leading to an abstract simulation algorithm that produces only the part of the abstract state transition graph that is reachable from the abstraction of the initial state. We proof the soundness of our abstract simulation algorithm, discuss how we implemented it, illustrate it usefulness for exact reasoning, and show its applicability to reaction networks in SBML format from the BioModels database.

Modern quantum programming languages integrate quantum resources and classical control. They must, on the one hand, be linearly typed to reflect the no-cloning property of quantum resources. On the other hand, high-level and practical languages should also support quantum circuits as first-class citizens, as well as families of circuits that are indexed by some classical parameters. Quantum programming languages thus need linear dependent type theory. This paper defines a general semantic structure for such a type theory via certain fibrations of monoidal categories. The categorical model of the quantum circuit description language Proto-Quipper-M by Rios and Selinger (2017) constitutes an example of such a fibration, which means that the language can readily be integrated with dependent types. We then devise both a general linear dependent type system and a dependently typed extension of Proto-Quipper-M, and provide them with operational semantics as well as a prototype implementation.

Our earlier version of this study had a programming error. This fixes that error.