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A μ-way (v,k,t) trade T of volume m consists of μ pairwise disjoint collections Ti,...,Tμ, each of m blocks of size k such that for every t-subset of a v-set V, the number of blocks containing this t-subset is the same in each Ti for 1 ≤ i ≤ μ. If any t-subset of the v-set V occurs at most once in each Ti for 1 ≤ i ≤ μ, then T is called a μ-way (v, k, t) Steiner trade. In 2016, it was proved that there exists a 3-way (v, k, 2) Steiner trade of volume m when 12(k - 1) ≤ m for each k. Here we improve the lower bound to 8(k - 1) for even k, by using a recursive construction. [ABSTRACT FROM AUTHOR]

Suppose that a 0 , a 1 , ... is an integer sequence which satisfies a recurrence relation with constant coefficients, and let T (x) = f (x) / g (x) be its generating function, where f (x) and g (x) have no common factors in Z [ x ]. In this article, we study the problem of finding the rational values of x such that T (x) is an integer. We say that such a number is good for the sequence. Our first main result is that if g (x) has at least two different irreducible factors, or if g (x) has a single irreducible factor of degree at least 3, then the sequence has only finitely many good values. We also study sequences of the form 0 , 1 , ... for which the recurrence relation has order 2. Among other results, we show that under a mild condition on the recurrence relation, the sequence has infinitely many good values, and we give a constructive method to find all of them. [ABSTRACT FROM AUTHOR]

To perform the accurate and efficient dynamic load identification (DLI) on time-varying (TV) structural systems, this paper proposes a novel time-domain dynamic load identification method, which we call the inverse Wentzel–Kramers–Brillouin (WKB) real function recursive solution. The proposed method can identify the dynamic loads of TV structural systems with different masses, damping, and stiffness online. Based on the WKB real function recursive solution, we first derive the acceleration recursive solution for linear time-varying (LTV) dynamic systems. Then, by reversing the acceleration recursive solution, we obtain the recursive formula for the dynamic load as a function of time. At each time step, this method performs recursion using the load and response of the previous time step to derive the current load. Finally, we demonstrate through numerical simulations that this algorithm has higher identification accuracy and computational efficiency than the inverse Wilson-θ method, and reduces the computation time significantly. [ABSTRACT FROM AUTHOR]

We introduce and study the recursive divisor function, a recursive analog of the usual divisor function: κ x (n) = n x + ∑ d ⌊ n κ x (d) , where the sum is over the proper divisors of n. We give a geometrical interpretation of κ x (n) , which we use to derive a relation between κ x (n) and κ 0 (n). For x ≥ 2 , we observe that κ x (n) / n x < 1 / (2 − ζ (x)). We show that, for n ≥ 2 , κ 0 (n) is twice the number of ordered factorizations, a problem much studied in its own right. By computing those numbers that are more recursively divisible than all of their predecessors, we recover many of the numbers prevalent in design and technology, and suggest new ones which have yet to be adopted. [ABSTRACT FROM AUTHOR]

This article discusses the game Cutting Corners and the strategies involved in winning. The game board is a square with two blue sides and two red sides, and players draw L-shaped lines of their color. The goal is to have more areas of their color on the board after six moves. The article explores the rules of the game, provides observations about the implications of the rules, and describes a program written in Mathematica that analyzes the game and identifies winning strategies. The program has verified a winning strategy for the four-move variant of the game and has found that certain moves are winning for Red in different variants of the game. The text also provides strategy tips for both Red and Blue players. [Extracted from the article]

In this paper, we construct a new kind of weighted recursive iteration function system (IFS) and prove the existence of the unique attractor for the kind of IFS based on the Matkowski fixed point theorem. We confirm that the attractor is a bivariate fractal interpolation surface (FIS), which interpolates a given set of data. In addition, we also provide an upper error estimate of such FISs caused by changes of weights. Finally, we give their box dimension estimates for a specific type of the FISs. [ABSTRACT FROM AUTHOR]

In this paper, we present proofs of properties of semirecursive sets based directly on the definition of these sets and on the recursiveness of Kleene predicates. These proofs are shorter and clearer than traditional proofs of similar statements for recursively enumerable sets. [ABSTRACT FROM AUTHOR]

In [10] Langlands provides a formula for certain product of orbital integrals in G L (2 , Q). Its generalization has become an important question for the strategy of Beyond Endoscopy. Arthur in [1] predicts this formula should coincide with a product of polynomials associated to zeta functions of orders constructed in [15] by Zhiwei Yun. In this paper we compute, for a certain family of orders, explicit formulas for these zeta functions by a recursive method. We use these zeta functions in [5] to prove that Arthur's prediction is correct. [ABSTRACT FROM AUTHOR]

In this paper we study the problem of constructing non-trivial subtowers and supertowers of recursive towers of function fields over finite fields. [ABSTRACT FROM AUTHOR]

We show that, for a primitive recursive function h (x , t) , where x is a n-tuple of natural numbers and t is a natural number, there exists a feedforward artificial neural network N (x , t) , such that for any n-tuple of natural numbers z and a positive natural number m, the first m + 1 terms of the sequence { h (z , t) } are the same as the terms of the tuple (N (z , 0) , ... , N (z , m)) . [ABSTRACT FROM AUTHOR]

This paper proposes a cooperative distributed extremum seeking control (ESC) approach for solving distributed optimization problems in static and dynamical multiagent systems, where the agents' coupled dynamics are unknown and only local cost values can be measured. The approach begins with each agent estimating the local gradient information of the unknown local cost function through recursive identification usingmeasured local cost values. Subsequently, the distributed identification-gradient tracking (DIGT) algorithm is employed to enable each agent to track the global gradient information of the global cost function, utilizing the estimated local gradient information from itself and its neighbors in the network. The proposed cooperative distributed ESC method enables the multiagent system to reach the extreme point of the global cost function under model-free scenarios. Furthermore, the convergence properties of the approach are established for both static and dynamical multiagent systems. Finally, numerical examples are presented to demonstrate the effectiveness and stability of the proposed approach. [ABSTRACT FROM AUTHOR]

The authors analyze Diophantine sets and show that all recursively enumerable sets are Diophantine. Based on the classical results from the theory of recursive functions, a simple version of the theorem on the incompleteness of arithmetic is provided: there is a polynomial that has no positive integer solutions, and for which it is impossible to prove the absence of positive roots. [ABSTRACT FROM AUTHOR]

In the first half of this paper, we study the way that sets of real numbers closed under Turing equivalence sit inside the real line from the perspective of algebra, measure and orders. Afterwards, we combine the results from our study of these sets as orders with a classical construction from Avraham to obtain a restriction about how non trivial automorphism of the Turing degrees (if they exist) interact with 1-generic degrees. [ABSTRACT FROM AUTHOR]

The article discusses the standard course for learning recurrent sequences in the primary, secondary and high-school mathematics. We propose an extension of the curricula with adding recurrent linear homogeneous sequences in the upper high school course. Additionally, we present related problems in computer science and information technology in order to construct interdisciplinary links, aimed at demonstrating the applied side of recurrent sequences as well as motivating the more gifted students for the learning of some specific computer science basics. [ABSTRACT FROM AUTHOR]

When realized on computational devices with finite quantities of memory, feedforward artificial neural networks and the functions they compute cease being abstract mathematical objects and turn into executable programs generating concrete computations. To differentiate between feedforward artificial neural networks and their functions as abstract mathematical objects and the realizations of these networks and functions on finite memory devices, we introduce the categories of general and actual computabilities and show that there exist correspondences, i.e., bijections, between functions computable by trained feedforward artificial neural networks on finite memory automata and classes of primitive recursive functions. [ABSTRACT FROM AUTHOR]

It cannot be decided whether a pushdown automaton accepts using a pushdown height, which does not depend on the input length, i.e., when it accepts using constant height. Furthermore, when a pushdown automaton accepts in constant height, the height can be arbitrarily large with respect to the size of the description of the machine, namely it does not exist any recursive function in the size of the description of the machine bounding the height of the pushdown. In contrast, in the restricted case of pushdown automata over a one-letter input alphabet, i.e., unary pushdown automata, the situation is different. First, acceptance in constant height is decidable. Moreover, in the case of acceptance in constant height, the height is at most exponential with respect to the size of the description of the pushdown automaton. We also prove a matching lower bound. Finally, if a unary pushdown automaton uses nonconstant height to accept, then the height should grow at least as the logarithm of the input length. This bound is optimal. [ABSTRACT FROM AUTHOR]

This article considers the tracking control problem in nonlinear systems, and thus designs a novel high-gain-observer based adaptive tracking controller. For optimal control problems in the case of parameter perturbations in nonlinear systems, the adaptive control method which designs readily adjustable control laws and adaptive parameters is adopted in backstepping. To address the problem of containing immeasurable states in physical systems, a high gain observer is designed to obtain unknown states. The high gain parameters can better regulate the observed performance of the system. Meanwhile, by introducing the fuzzy logic system (FLS), the unknown construction in the system is estimated. In addition, an event triggering strategy with a fixed threshold is included to conserve network resources. Furthermore, the command filter is used in the controller design to solve "over-complication" and "excess parameterization" problems in the backstepping. Through the Lyapunov function and the recursive deduction of the command filter, the signals in the close loop system (CLS) are all ultimately bounded while the tracking error can converge to a small area. Ultimately, the effectiveness of the design method is demonstrated by simulations. [ABSTRACT FROM AUTHOR]

This is a survey of formal axiomatic systems for the three main varieties of constructive analysis, in a common language and with intuitionistic logic, which are as nearly as possible compatible with classical analysis and with one another. Classically sound consequences of principles of intuitionistic mathematics are emphasized. Compatibility with classical analysis is of two kinds. On the one hand, Bishop's constructive mathematics and a very substantial part of intuitionistic analysis are classically correct, sharing with constructive recursive mathematics a neutral subsystem adequate for recursive function theory and elementary real analysis. On the other hand, each constructive system considered here is separately consistent with the negative interpretation of each of its classically sound subsystems, establishing internal compatibility with the classical context it is intended to refine. A possibly new criterion for the transposability of unique existential quantifiers and a recent theorem of Vafeiadou on the internal compatibility of classical and intuitionistic analysis are included. This article is part of the theme issue 'Modern perspectives in Proof Theory'. [ABSTRACT FROM AUTHOR]

The "somewhat vague, intuitive" notion from computability theory of an effective procedure (method) or algorithm can be fairly precisely defined, even if it does not have a purely mathematical definition—and even if (as many have asserted) for that reason, the Church–Turing thesis (that the effectively calculable functions on natural numbers are exactly the general recursive functions), cannot be proved. However, it is logically provable from the notion of an effective procedure, without reliance on any (partially) mathematical thesis or conjecture concerning effective procedures, such as the Church–Turing thesis, that the class of effective procedures is undecidable, i.e., that there is no effective procedure for ascertaining whether a given procedure is effective. The proof does not even appeal to a precise definition of 'effective procedure'. Instead, it relies solely and entirely on a basic grasp of the intuitive notion of such a procedure. Though the result itself is not surprising, it is also not without significance. It has the consequence, for example, that the solution to a decision problem, if it is to be complete, must be accompanied by a separate argument that the proposed ascertainment procedure is, in fact, a decision procedure, i.e., effective—for example, that it invariably terminates with the correct verdict. [ABSTRACT FROM AUTHOR]

A novel kernel recursive second-order sine adaptive (KRSOSA) algorithm was devised for identifying non-linear systems, which was constructed using a symmetry squared sine function to develop a novel kernel loss function and recursive scheme. In the proposed KRSOSA algorithm, the squared sine function provides resistance to impulsive noise due to the sine operation, which was well-derived and investigated in the framework of kernel adaptive filtering (KAF). The behavior of the proposed KRSOSA algorithm was investigated and analyzed using computer simulations, which provided good performance for identifying non-linear systems under impulsive noises. [ABSTRACT FROM AUTHOR]

Digital notch filters are useful for removing sinusoidal signals with a single frequency or multi-frequencies. For some applications, the sinusoidal signals being processed may change frequencies, and thus the notch filter must also have a variable notch frequency. Such a digital filter with a variable notch-frequency (VNF) is called variable notch-frequency (VNF) digital filter. Generally speaking, a VNF filter has the ability to change its notch frequency continuously, and can remove sinusoidal signals with variable frequencies. To design a recursive VNF filter, guaranteeing its stability is the top-priority issue. In this paper, a recursive VNF filter is designed by utilizing the least Lp-norm criterion, and the stability is guaranteed through adopting the stability-guarantee methodology based on parameter transformations. To perform the parameter transformations, a function that meets a special condition is required, and such a function is called unity-bounded function. In this paper, various unity-bounded functions are developed for the parameter-transformation-based stability guarantee. The unity-bounded functions play a crucial role in the stability guarantee. This paper details how to design a stable VNF filter by incorporating the parameter transformations employing the unity-bounded functions into the Lp-norm minimization of the magnitude response. Theoretically, the stability of the recursive VNF filter resulting from the design is definitely guaranteed. Moreover, since the coefficients of the designed VNF are variable, changing the filter coefficients enables the user to get a VNF filter with updated notch frequency. [ABSTRACT FROM AUTHOR]

Accurate forecast of natural gas consumption is of significance for policy makers to formulate energy plans, save costs and improve energy structure. This study designs a novel three-parameter discrete direct grey model denoted as TDDGM, which overcomes existing modeling drawbacks about the accumulating generation for monotonous series. Moreover, the optimal initial condition of the proposed model is deduced by the ordinary least square method and the final recursive function is derived. Error checking method of TDDGM and five different numerical cases are introduced to verify the superiority and effectiveness of the new approach. The results show that TDDGM performs better than other several benchmark models in multiple tests. Finally, it is utilized to forecast China's natural gas consumption and the predicted results will maintain a steady upward trend, reaching 322.06 billion cubic meters in 2022. The research results have positive significance for enriching grey system theory and improving energy structure in China. [ABSTRACT FROM AUTHOR]

Circular RNAs (circRNAs) are a novel class of non-coding RNA that, unlike linear RNAs, form a covalently closed loop without the 5′ and 3′ ends. Growing evidence shows that circular RNAs play important roles in life processes and have great potential implications in clinical and research fields. The accurate modeling of circRNAs structure and stability has far-reaching impact on our understanding of their functions and our ability to develop RNA-based therapeutics. The cRNAsp12 server offers a user-friendly web interface to predict circular RNA secondary structures and folding stabilities from the sequence. Through the helix-based landscape partitioning strategy, the server generates distinct ensembles of structures and predicts the minimal free energy structures for each ensemble with the recursive partition function calculation and backtracking algorithms. For structure predictions in the limited structural ensemble, the server also provides users with the option to set the structural constraints of forcing the base pairs and/or forcing the unpaired bases, such that only structures that meet the criteria are enumerated recursively. [ABSTRACT FROM AUTHOR]

The conventional B-splines possess the de Boor–Cox formula, which relates to a pyramid algorithm. However, for multi-degree splines, a de Boor–Cox-type evaluation algorithm only exists in some special cases. This paper considers any multi-degree spline with arbitrary degree and continuity, and provides two generalized de Boor–Cox-type relations. One uses several lower degree polynomials to build a combination to evaluate basis functions, whose form is similar to using the de Boor–Cox formula several times. The other is a linear combination of two functions out of the recursive definition, which keeps the combination coefficient polynomials of degree 1, so it is more similar to the de Boor–Cox formula and can be illustrated by several pyramids with different heights. In the process of calculating the recursions, a recursive representation using the Bernstein basis is used and numerically analyzed. [ABSTRACT FROM AUTHOR]

The equivalence of folding left and right over Peano numbers and lists makes it possible to minimalistically inter-derive (1) structurally recursive functions in direct style, (2) structurally tail-recursive functions that use an accumulator, and (3) structurally tail-recursive functions in delimited continuation-passing style, using Ohori and Sasano's lightweight fusion by fixed-point promotion. When the fold-left and the fold-right functions account for primitive iteration for Peano numbers, this equivalence is unconditional. When they account for primitive recursion for Peano numbers, this equivalence is modulo left permutativity of their induction-step parameter – a property which is more general than associativity and commutativity. And when they account for primitive iteration or for primitive recursion over lists, this equivalence is modulo left permutativity of their induction-step parameter if these two fold functions have the same type. Since the 1980s, however, the two fold functions for lists do not have the same type: the arguments for their induction-step parameter are swapped, a re-ordering that complicated Bird and Wadler's duality theorems and whose history is reviewed in an appendix. Without this re-ordering, Bird and Wadler's second duality theorem more visibly accounts for "re-bracketing," which is a key step to make recursive programs tail recursive in the general area of program development, from Cooper in the 1960s and onwards. [ABSTRACT FROM AUTHOR]

Contemporary proof assistants such as Coq require that recursive functions be terminating and corecursive functions be productive to maintain logical consistency of their type theories, and some ensure these properties using syntactic checks. However, being syntactic, they are inherently delicate and restrictive, preventing users from easily writing obviously terminating or productive functions at their whim. Meanwhile, there exist many sized type theories that perform type-based termination and productivity checking, including theories based on the Calculus of (Co)Inductive Constructions (CIC), the core calculus underlying Coq. These theories are more robust and compositional in comparison. So why haven't they been adapted to Coq? In this paper, we venture to answer this question with CIC $\widehat{\ast}$ , a sized type theory based on CIC. It extends past work on sized types in CIC with additional Coq features such as global and local definitions. We also present a corresponding size inference algorithm and implement it within Coq's kernel; for maximal backward compatibility with existing Coq developments, it requires no additional annotations from the user. In our evaluation of the implementation, we find a severe performance degradation when compiling parts of the Coq standard library, inherent to the algorithm itself. We conclude that if we wish to maintain backward compatibility, using size inference as a replacement for syntactic checking is impractical in terms of performance. [ABSTRACT FROM AUTHOR]

• A particular output error model is proposed via polynomial transformation, in which an extra filter is not required [18,36]. • A loss function is introduced using initial parameter and output error items, which gives a perspective for scheme design. • A recursive identification framework is derived with variable modified gain, which facilitates online operation [14,31]. In this paper, an adaptive recursive estimation scheme based on a novel recursive framework is proposed for a controlled autoregressive autoregressive moving average (CARARMA) system. A common loss function is established using the prediction error scheme, which has two shortcomings in case of interference, namely, biased estimation and minima problems. To overcome the two shortcomings, a recursive estimation scheme is proposed by using output error data with a discount factor and initial error data with a penalty operator. The former data do not involve the noise information of system data, so the biased estimation issue can be improved. The latter data include initial value information, such that the minima problem can be resolved. To achieve the target, polynomial transformation is applied to transform the CARARMA system into a particular model, then the loss function is introduced. Based on the loss function and recursive structure, a recursive estimator is developed. Moreover, the convergence of the proposed identification scheme is strictly analyzed. The advantage and practicality of the proposed estimator are evaluated by using a numerical example and real-world process. [ABSTRACT FROM AUTHOR]

In this paper we show that several classes of languages from computational complexity theory, such as EXPTIME , can be characterized in a continuous manner by using only polynomial differential equations. This characterization applies not only to languages, but also to classes of functions, such as the classes defining the Grzegorczyk hierarchy, which implies an analog characterization of the class of elementary computable functions and the class of primitive recursive functions. [ABSTRACT FROM AUTHOR]

Why are natural theories pre-well-ordered by consistency strength? In previous work, an approach to this question was proposed. This approach was inspired by Martin's Conjecture, one of the most prominent conjectures in recursion theory. Fixing a reasonable subsystem T of arithmetic, the goal was to classify the recursive functions that are monotone with respect to the Lindenbaum algebra of T. According to an optimistic conjecture, roughly, every such function must be equivalent to an iterate Con T α of the consistency operator "in the limit" within the ultrafilter of sentences that are true in the standard model. In previous work the author established the first case of this optimistic conjecture; roughly, every recursive monotone function is either as weak as the identity operator in the limit or as strong as Con T in the limit. Yet in this note we prove that this optimistic conjecture fails already at the next step; there are recursive monotone functions that are neither as weak as Con T in the limit nor as strong as Con T 2 in the limit. In fact, for every α, we produce a function that is cofinally equivalent to Con T α yet cofinally equivalent to Con T . [ABSTRACT FROM AUTHOR]

We investigate multiple testing and variable selection using the Least Angle Regression (LARS) algorithm in high dimensions under the assumption of Gaussian noise. LARS is known to produce a piecewise affine solution path with change points referred to as the knots of the LARS path. The key to our results is an expression in closed form of the exact joint law of a |$K$| -tuple of knots conditional on the variables selected by LARS, the so-called post-selection joint law of the LARS knots. Numerical experiments demonstrate the perfect fit of our findings. This paper makes three main contributions. First, we build testing procedures on variables entering the model along the LARS path in the general design case when the noise level can be unknown. These testing procedures are referred to as the Generalized |$t$| -Spacing tests and we prove that they have an exact non-asymptotic level (i.e. the Type I error is exactly controlled). This extends work of [ 31 ] where the spacing test works for consecutive knots and known variance. Second, we introduce a new exact multiple testing procedure after model selection in the general design case when the noise level may be unknown. We prove that this testing procedure has exact non-asymptotic level for general design and unknown noise level. Third, we prove exact control of the false discovery rate under orthogonal design assumption. Monte-Carlo simulations and a real data experiment are provided to illustrate our results in this case. Of independent interest, we introduce an equivalent formulation of the LARS algorithm based on a recursive function. [ABSTRACT FROM AUTHOR]

Z is a new type of non-finitist inference, i.e., an inference that involves treating some infinite collection as completed, designed for contraction free logic with unrestricted abstraction. It has been introduced in Petersen (Studia Logica 64:365–403, 2000) and shown to be consistent within a system L i D λ of contraction free logic with unrestricted abstraction. In Petersen (Arch Math Log 42(7):665–694, 2003) it was established that adding Z to L i D λ is sufficient to prove the totality of primitive recursive functions but it was also indicated that this would not extend to 2-recursive functions such as the Ackermann–Péter function, for instance. The purpose of the present paper is to expand the underlying idea in the construction of Z to gain a stronger notion, conveniently labeled Z 2 , which is sufficient to prove a form of nested double induction and thereby the totality of 2-recursive functions. [ABSTRACT FROM AUTHOR]

We give recursion-theoretic characterizations of the counting class # P , the class of those functions which count the number of accepting computations of non-deterministic Turing machines working in polynomial time. Moreover, we characterize in a recursion-theoretic manner all the levels { # P k } k ∈ N of the counting hierarchy of functions FCH , which result from allowing queries to functions of the previous level, and FCH itself as a whole. This is done in the style of Bellantoni and Cook's safe recursion, and it places # P in the context of implicit computational complexity. Namely, it relates # P with the implicit characterizations of FPTIME (Bellantoni and Cook, Comput Complex 2:97–110, 1992) and FPSPACE (Oitavem, Math Log Q 54(3):317–323, 2008), by exploiting the features of the tree-recursion scheme of FPSPACE . [ABSTRACT FROM AUTHOR]

Summary: In this work we elaborate on the idea of a formal theory for a limited but important part of structural chemistry, that described by Lewis' methods and VSEPR (Valence Shell Electron Pair Repulsion). For this purpose, recursive functions and propositional functions are defined, that apply to formal expressions of the structure, based on a finite set of symbols. This approach allows for the expression of numerous questions of chemical interest. The formalization of basic structural chemistry based on Lewis/VSEPR method is potentially useful for the automation of the related procedures, but possibly also of some use as complementary material in teaching and as a heuristic tool in structural chemistry. [ABSTRACT FROM AUTHOR]

Automation of C program verification is an important problem in modern software development. To solve this problem, the solution of the following problems must be automated: loop invariants, proof of verification conditions, and localization of errors in the case of invalid verification conditions. To this end, the C-lightVer system is under development in the Ershov Institute of Informatics Systems of the Siberian Branch of the Russian Academy of Sciences. This system uses an integrated approach to the automated deductive verification of C programs. This approach includes a symbolic method of verification of definite iterations for eliminating loop invariants, strategies for proving verification conditions, and a method for error localization. The symbolic method of verification of definite iterations is based on replacing the action of certain loops by the application of a special recursive function rep. The error localization method is based on matching the verification conditions to the source code and on generation of a report about the correspondence between the verification conditions and program fragments. Thus, the problem of automation of verification of C programs containing nested loops naturally arises. The application of the symbolic method of verification of definite iterations for such programs leads to a composition of the functions rep for outer and inner loops. A novel result obtained in this paper is a strategy of automation of proof of verification conditions for such programs. This strategy is based on induction on the index of iteration in the outer loop. To prove the induction step, another result obtained in this paper is used. This is a strategy for programs the specification of which contains functions with the concatenation property. The paper also describes strategies of error localization and modifications of the error localization method for the case of nested loops. These strategies are also used for verifying the loop properties that may indicate the presence of errors. As an example of applying the results obtained in this paper, automatic verification of insertion sort without loop invariants is considered. [ABSTRACT FROM AUTHOR]

Reliability evaluation has developed a trend of embedding unit commitment to replace power flow for more accurate operation strategies, while its potential reverse, embedding reliability evaluation in unit commitment, has yet to see equal exploration. This unrecognized, yet meaningful, technical attempt will encounter modeling and computational burden, largely caused by the recursive function and non-convex structure of numerical reliability evaluation equipped with the necessary repair mechanism. To tackle this challenge, this study proposes a linear routine that overcomes the non-convexity of reliability evaluation and accommodates it in unit commitment, which is expected to bring optimal operation and reliability evaluation to a new era by enabling them both conducted at the same pace. [ABSTRACT FROM AUTHOR]

A notion of strictly primitive recursive realizability is introduced by Damnjanovic in 1994. It is a kind of constructive semantics of the arithmetical sentences using primitive recursive functions. It is of interest to study the corresponding predicate logic. It was argued by Park in 2003 that the predicate logic of strictly primitive recursive realizability is not arithmetical. Park's argument is essentially based on a claim of Damnjanovic that intuitionistic logic is sound with respect to strictly primitive recursive realizability, but that claim was disproved by the author of this article in 2006. The aim of this paper is to present a correct proof of the result of Park. [ABSTRACT FROM AUTHOR]

Motivated by algorithmic information theory, the problem of program discovery can help find candidates of underlying generative mechanisms of natural and artificial phenomena. The uncomputability of such inverse problem, however, significantly restricts a wider application of exhaustive methods. Here we present a proof of concept of an approach based on IMP, a high-level imperative programming language. Its main advantage is that conceptually complex computational routines are more succinctly expressed, unlike lower-level models such as Turing machines or cellular automata. We investigate if a more expressive higher-level programming language can be more efficient at generating approximations to algorithmic complexity of recursive functions. • We explore an inverse problem for model discovery based on principles of algorithmic information. • The method can help find candidates of underlying generative mechanisms and first principles. • A proof of concept based on IMP, a high-level imperative programming language, is provided. • IMP expressions are succinct, unlike lower-level models such as formal automata. • We investigate if a more expressive higher-level programming language can be more efficient. [ABSTRACT FROM AUTHOR]

The nonlinear time-series analysis method, based on the recurrence plot theory, has received great attention from researchers and has been successfully used in multiple fields. However, traditional recurrence plots that use Heaviside step functions to determine the recursive behavior of a point in the phase space have two problems: (1) Heaviside step functions produce a rigid boundary, resulting in information loss; and (2) the selection of the critical distance, ε, is crucial; if the selection is inappropriate, it will result in a low-dimensional dynamics error, and as of now, there exists no unified method for selecting this parameter. With regard to the problems described above, the novelty of this article lies in the following: (1) when determining the state-phase point recursiveness, a Gaussian function is used to replace the Heaviside function, thereby solving the rigidity and binary value problems of the recursive analysis results caused by the Heaviside step function; and (2) texture analysis is performed on a recurrence plot, new ways of studying complex system dynamics features are proposed, and a system of complex system dynamic-like measurement methods is built. [ABSTRACT FROM AUTHOR]

Accurate survival prediction is crucial for doctors to choose appropriate treatment options and make prognoses for gastric cancer patients. While machine learning algorithms have shown promise for improving prediction accuracy, the lack of doctor involvement and guidance can reduce the credibility of prediction results in practical clinical applications. To address this issue, we propose a survival prediction expert system integrating deep forest and active preference learning to incorporate doctors' empirical knowledge into the prediction process. Our expert system first reduces feature dimensionality through feature selection, with doctor input to ensure clinical relevance. We then train estimators to simulate clinical prediction, and doctors compare these estimators in pairs and give preferences based on their experience. A recursive function based on the radial basis function is used to learn the preferences and assign weights to all the estimators. We apply our approach to two publicly available datasets and one private dataset. Our system achieves an AUC of 0.87 and a C-index of 0.75 for predictive classification and regression analysis tasks. Importantly, our system outperforms the clinical gold standard method (Nomogram) and other state-of-the-art machine learning methods with a maximum improvement of 12% and 10%, respectively, demonstrating superior performance. In conclusion, our expert system effectively incorporates doctors' empirical knowledge into the survival prediction process, enhancing the credibility of prediction results in practical clinical applications. Our approach has significant potential for improving treatment decisions and prognoses for gastric cancer patients. • A novel expert system for OS prediction with physicians in the training loop. • Dynamically corrects the feature set with both the physician and the machine. • Simulate physician preferences of clinical pathway by computing the intra-tree distances. • Validated on both private and public datasets and yields state-of-the-art performance. [ABSTRACT FROM AUTHOR]

The complexity of equivalence relations has received much attention in the recent literature. The main tool for such endeavour is the following reducibility: given equivalence relations R and S on natural numbers, R is computably reducible to S if there is a computable function f : ω → ω that induces an injective map from R-equivalence classes to S-equivalence classes. In order to compare the complexity of equivalence relations which are computable, researchers considered also feasible variants of computable reducibility, such as the polynomial-time reducibility. In this work, we explore Peq, the degree structure generated by primitive recursive reducibility on primitive recursive equivalence relations with infinitely many equivalence classes. In contrast with all other known degree structures on equivalence relations, we show that Peq has much more structure: e.g., we show that it is a dense distributive lattice. On the other hand, we also offer evidence of the intricacy of Peq, proving, e.g., that the structure is neither rigid nor homogeneous. [ABSTRACT FROM AUTHOR]

Some hydropower reservoirs are operated under different constraint levels. For these reservoirs, a multi-level (ML) dependent-chance (DC) model is established. In the model, only when the higher-level constraints are satisfied are the lower-level constraints or system benefits considered. The multi-level dependent-chance (MLDC) model is specified by two models. One is based on existing reliability-constrained (RC) dynamic programming (DP), in which the soft constraints are addressed using reliability constraints of 1, and the priorities are reflected using the order of magnitudes of Lagrange multipliers. The other is the explicit dependent-chance reasoning in the DP recursive function, in which each soft constraint is represented as an objective function of negative expected failure time and the optimum is the solution with a larger value for all higher-level objective functions. The proposed models are applied to derive long-term operation rules for the hydropower system on the middle-lower Lancang River. The results show the feasibility and performances of the explicit graded constraint control of the proposed model and the solution methods. [ABSTRACT FROM AUTHOR]

Neasu (Tibeto-Burman: China) exhibits a prefix that derives new coordinators from existing ones by elaborately changing their subcategorial properties. Prefixed and unprefixed coordinators are distinguished by the complement they take (±verbal, ±CoP) and the possibility of being stacked up at least twice (±stackable). A prefixed coordinator has two of these three features switched from "−" to "+", when compared with its unprefixed counterpart and thus see its ability to occur as the head of recursive coordination structures increased. The prefix ao- is an operator of recursion. [ABSTRACT FROM AUTHOR]

With the rapid development of modern information technology, students' education and computer technology begin to blend, and the modern teaching mode is quite different from the traditional education mode known in the past. In view of the current college English teaching in the information age, this study puts forward the way of integrating computer information technology with college English teaching, improves MLP algorithm, puts forward a new artificial intelligence algorithm, improves its calculation efficiency, and uses the optimized GA-MLP-NN (Genetic Neural Network Algorithm for the Multilayer Perceptron) algorithm in college students' oral correction program. Firstly, GA-MLP-NN algorithm is used to optimize college English teaching so that more complex structures can be learned and dealt with. Incremental hidden layer unit neural network is added, which makes the operation more accurate based on S-type recursive function. Then, the oral English system is established, using the GA-MLP-NN neural network model. Finally, we evaluate the parameters of the model, design a comparative experiment and a questionnaire survey to verify the rationality and feasibility of the guess, which proves that this method can deal with more complex programs, and make students learn English more handy and close to students' needs by using computer technology. [ABSTRACT FROM AUTHOR]

The amount by which the artificial neural network weights are updated during the training process is called the learning rate. More precisely, the learning rate is an adjustable parameter used in training neural networks in which small values, often in the interval [0, 1], are handled. The learning rate determines how quickly the model updates its weights to adapt to the problem. Smaller learning rates require more training periods due to small changes to the weights per refresh cycle, while larger learning rates lead to faster changes and require fewer training periods. In this paper, the effect of changing the learning rate value in the artificial neural network designed to solve the inverse problem of fractals was studied. Some results were obtained showing the impact of this change, whether when using large values of the learning rate or small values based on the type of fractal shape required to identify the recursive functions that generate it. [ABSTRACT FROM AUTHOR]

Despite his unreserved appreciation of Turing's analysis for being a "precise and unquestionably adequate definition" of formal system or mechanical computability, Gödel nevertheless published a short note in 1972 claiming to have found a "philosophical error" in Turing's argument with regard to the finite nature of mental states and memory. A natural question arises: how could Gödel enjoy the generality conferred on his results by Turing's work, despite the error of its ways? Previous interpretative strategies by Feferman, Shagrir and others have mainly tried to resolve the disparity by distinguishing different types of arguments in Turing and taking Gödel to approve only some of them. By a more integral examination of their ideas, especially Turing's response to the "mathematical objection" based on Gödel's incompleteness theorem and Gödel's own conception of finite yet non-mechanical procedures, and taking some of the main ideas of current developments in machine learning into consideration, I will try to present a new explanation for the apparent disparity, arguing that there is no "error" on Turing's side and the seemingly conflicting views held by Turing and Gödel should best be seen as complementary, keeping intuition and ingenuity together. [ABSTRACT FROM AUTHOR]

This article presents a theory of recursion in thinking and language. In the logic of computability, a function maps one or more sets to another, and it can have a recursive definition that is semi-circular, i.e., referring in part to the function itself. Any function that is computable – and many are not – can be computed in an infinite number of distinct programs. Some of these programs are semi-circular too, but they needn't be, because repeated loops of instructions can compute any recursive function. Our theory aims to explain how naive individuals devise informal programs in natural language, and is itself implemented in a computer program that creates programs. Participants in our experiments spontaneously simulate loops of instructions in kinematic mental models. They rely on such loops to compute recursive functions for rearranging the order of cars in trains on a track with a siding. Kolmogorov complexity predicts the relative difficulty of abducing such programs – for easy rearrangements, such as reversing the order of the cars, to difficult ones, such as splitting a train in two and interleaving the two resulting halves (equivalent to a faro shuffle). This rearrangement uses both the siding and part of the track as working memories, shuffling cars between them, and so it relies on the power of a linear-bounded computer. Linguistic evidence implies that this power is more than necessary to compose the meanings of sentences in natural language from those of their grammatical constituents. [ABSTRACT FROM AUTHOR]

Several characteristics of chess are investigated with methods of computability and number theories. It is shown that for an unfinished game it is primitive recursively decidable whether the game is winnable, drawable, or absolutely losable within a specified number of future moves for the player whose turn it is to play on the last board of the game. It is also shown that there exist primitive recursive procedures to compute optimal continuations of unfinished games within specified numbers of future moves and that the set of chess games is recursive. [ABSTRACT FROM AUTHOR]

For any I l i , 1 I l i I n i , we denote by HT ht the function I g SB abl sb i if I b SB l sb i ( I m i + 1) I p i - 1 and the constant function I b SB l sb i if I b SB l sb i ( I m i + 1) I p i . Now by substituting function (010) into function (100) we get function (101), and by substituting function (112) into function (101) we get function (001). The function v z has the vector of values HT ht Since d i /= d j, then for each z D j the tuple of functions (v z (x), f 1 (x)) is correct. Keywords: enumeration closure operator; functions of three-valued logic EN enumeration closure operator functions of three-valued logic 105 114 10 04/15/22 20220401 NES 220401 Originally published in Diskretnaya Matematika (2021) 33, No2, 86-99 (in Russian). [Extracted from the article]