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Score equivalency of two Computer-Based Testing (henceforth CBT) and Paper-and-Pencil-Based Testing (henceforth PBT) versions has turned into a controversial issue during the last decade in Iran. The comparability of mean scores obtained from two CBT and PBT formats of test should be investigated to see if test takers' testing performance is influenced by the effects of testing administration mode. This research was conducted to examine score equivalency across modes as well as the relationship of gender, and testing mode preference with test takers' performance on computerized testing. The information on testing mode preference and attitudes towards CBT and its features was supported by a focus group interview. Findings indicated that the scores of test takers were not different in both modes and there was no statistically significant relationship between moderator above variables and CBT performance. [ABSTRACT FROM AUTHOR]

The Randić index (R) and the geometric-arithmetic index (GA) are found to be useful tools in QSPR and QSAR studies. In the Journal of Inequalities and Applications 180, 1-7, Lokesha, Shwetha Shetty, Ranjini, Cangul, and Cevik gave "New bounds for Randić and GA indices." In the paper, we first point out that Theorems 1, 2, and 4 are incorrect and in this short note we present the correct inequalities for Randić and GA indices. In the same paper, we provide the equality cases for Theorems 3, 5, and 6. [ABSTRACT FROM AUTHOR]

In this tutorial note we consider the class of non-convex semi-infinite optimization problems which are defined by (one or) finitely many objective functions as well as infinitely many constraints in a finitedimensional space. We present an overview of recent results on the so-called p-power transformation which changes the original problem equivalently locally around a solution point. This transformation is a convexification procedure for the Lagrangian where the functions in the original problem are substituted by their p-th powers. As a consequence, the convexity of the so-transformed Lagrangian allows the application of local duality theory and corresponding solution methods locally around this solution point of the original problem. [ABSTRACT FROM AUTHOR]

The sixth paper, "On Some New Ostrowski-Mercer-Type Inequalities for Differentiable Functions", by Sial et al. [[6]], establishes a new integral identity involving differentiable functions and uses it for some new Ostrowski-Mercer-type inequalities for differentiable convex functions. The importance of inequalities in Mathematics is beautifully summarized in a citation attributed to Respected Professor Andrey Nikolaevich Kolmogorov: HT

We have studied initial value problems for Caputo fractional differential equations with singular nonlinearities involving the p-Laplacian operator. We have given a precise mathematical analysis of the equivalence of the fractional differential equations and Volterra integral equations studied in this paper. A theorem for the global existence of the solution was proven. In addition, an example was given at the end of the article as an application of the results found in this paper. [ABSTRACT FROM AUTHOR]

In this meta syntheses paper, I explore gender as a social factor that interacts with both the cognitive and affective domains of education by explaining what it means for mathematics to be gendered and how we can use mathematics education to promote gender fairness as a form of social justice. I start by discussing gender as a dichotomy and how educational research transitioned from a physiological to a socio-cultural perspective. Guided by Ferguson’s and Bjerrun-Nielsen’s Four-Perspective Gender model, and Cohen and Martin’s Four Dimensional Model of Gender. I argue that mathematics education can be designed to promote gender equity only by considering these dimensions, and when there is a deliberate effort to examine how our exercise of education widens or narrows gender gap in terms of the four dimensions of gender equality: equality to access, learning experience, educational outcomes, and most importantly, external results. I discuss mathematics as an area perceived to embody a symbolic gender and what can be done through mathematics education to break that stereotype and contribute to the furthering equality to external results. I focus this discussion into the context of the Philippines and why it may be essential for educational reforms to zoom out to include other equally important dimensions of gender other than equality to access. [ABSTRACT FROM AUTHOR]

In this note we show that the presented proof of [1, th. 12] , being based on a false statement appearing in this proof, is not viable for all of the proclaimed values of the involved parameters. We also determine necessary and sufficient conditions for those parameters, which provides that the considered statement, and therefore [1, th. 12] itself, still remains valid. [ABSTRACT FROM AUTHOR]

Two curves are affinely equivalent if there exists an affine mapping transforming one of them onto the other. Thus, detecting affine equivalence comprises, as important particular cases, similarity, congruence and symmetry detection. In this paper we generalized previous results by the authors to provide an algorithm for computing the affine equivalences between two parametric curves of certain types, in any dimension. In more detail, the algorithm is valid for rational curves, and for parametric curves with nonrational but meromorphic components, it admits an also meromorphic, and in fact rational, inverse. Unlike other algorithms already known for rational curves, the algorithm completely avoids polynomial system solving, and instead uses bivariate factoring as a fundamental tool. The algorithm has been implemented in the computer algebra system Maple and can be freely downloaded and used. [ABSTRACT FROM AUTHOR]

The paper presents a critical review and new accomplishments on the equivalence of the first-order displacement-based zigzag theories for laminated composite and sandwich structures. Zigzag theories (ZZTs) have widely spread among researchers over the last few decades thanks to their accuracy in predicting the response of multilayered composite and sandwich structures while retaining the simplicity of their underlying equivalent single-layer (ESL) theory. The displacement field consists of two main contributions: the global one, able to describe the overall structural behaviour, and the local layer-wise one that considers the transverse shear continuity at the layer interfaces that describe the "zigzag" displacement pattern typical of multilayered structures. In the framework of displacement-based linear ZZTs, various assumptions have been made on the local contribution, and different theories have been deduced. This paper aims to provide a unified formulation for first-order ZZTs, highlighting some common aspects and underlying equivalencies with existing formulations. The mathematical demonstrations and the numerical examples prove the equivalence of the approaches to characterising local zigzag enrichment. Finally, it is demonstrated that the kinematic assumptions are the discriminants of the ZZTs' accuracy. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of a Quasi B-algebra, building upon the foundational work on Post algebras by G. Epstein and the investigation of semi B-algebras. We systematically explore the properties of Quasi B-algebras, establishing equivalent conditions for the transition from a semi B-algebra to a Quasi B-algebra. [ABSTRACT FROM AUTHOR]

This paper discusses the large deviations for stochastic predator–prey model driven by multiplicative Lévy noise. Using Galerkin approximation, we initially prove the existence and uniqueness of solution. Due to the equivalence between Laplace principle and large deviation principle under a Polish space, the method of weak convergence has been followed in order to establish our results for this coupled system of equations. [ABSTRACT FROM AUTHOR]

In this paper, we first prove a relation between the critical points of the skew-circulant matrix and the eigenvalues of its principal matrix. Furthermore, we reprove the inequality about the zeros of a polynomial and its critical points by using the properties of skew-circulant matrix, which is to show that we can not only find the skew-circulant matrix, but also give more structure matrices to prove this inequality. [ABSTRACT FROM AUTHOR]

The paper presents a novel approach to generalize the two-point weighted Ostrowski's formula for Riemann-Stieltjes integrals by utilizing a unique class of functions of bounded r - variation. The proposed approach yields several results that exhibit sharp and better bounds compared to existing established results by using parameters and weights. Additionally, the paper also captures many of the known results as special cases. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of weakly semiprime ideals and weakly n -systems in noncommutative rings. We establish the equivalence between an ideal P being a weakly semiprime ideal and R − P being a weakly n -system. We provide alternative definitions and explore the properties of weakly semiprime ideals. Additionally, we investigate scenarios where all ideals in a given ring are weakly semiprime and demonstrate that in Noetherian rings, where every ideal is weakly semiprime, the prime radical and the Jacobson radical coincide. [ABSTRACT FROM AUTHOR]

Running time analysis of evolutionary algorithms for continuous optimization is one research challenge in the field of evolutionary algorithms (EAs). However, the theoretical analysis results have rarely been applied to evolutionary algorithms for continuous optimization in practice, let alone their variants for evolution strategy. In this paper, we regarded the first hitting time of evolution strategy as the stopping time of the renewal process on the basis of the renewal process and in combination with Wald's inequality and stopping time theory. Afterwards, to demonstrate the application of the proposed model in the first hitting time analysis of (1 + 1) ES, we analyzed it with different mutation operators on the sphere function. First, we significantly improved the lower bound on the first hitting time of (1 + 1) ES with a uniform mutation operator, i.e., from Ω (n) to Ω (e c n) . Next, O (n 2 n) was the upper bound on the first hitting time of (1 + 1) ES with a Gaussian mutation operator from the initial distance R to half of the initial distance R /2. The numerical experimental results showed that the theoretical calculation was consistent with the actual running time, which provides a novel method for analyzing the first hitting time of EAs. [ABSTRACT FROM AUTHOR]

This paper aims at clarifying the questions regarding the effects of the scaling method on the discrepancy function of the metric measurement invariance model. We provide examples and a formal account showing that neither the choice of the scaling method in general nor the choice of a particular referent indicator affects the value of the discrepancy function. Thus, the test statistic is not affected by the scaling method, either. The results rely on an appropriate application of the scaling restrictions, which can be phrased as a simple rule: "Apply the scaling restriction in one group only!" We develop formulas to calculate the degrees of freedom of χ²-difference tests comparing metric models to the corresponding configural model. Our findings show that it is impossible to test the invariance of the estimated loading of exactly one indicator, because metric MI models aimed at doing so are actually equivalent to the configural model. [ABSTRACT FROM AUTHOR]

In this paper, we present some further refinements of the Tan-Xie inequality for sector matrices and its applications due to Nasiri and Furuichi [J. Math. Inequal., 15 (2021), 1425-1434]. [ABSTRACT FROM AUTHOR]

Alexander Grothendieck suggested creating a new branch of topology, called "topologie modérée". In a paper by N. A'Campo, L. Ji, and A. Papadopoulos the authors conclude that no such tame topology has been developed at a purely topological level. We see our theory of sets with distinguished families of subsets, which we call smopologies, as realising Grothendieck's idea and the demands of the mentioned paper. Dropping the requirement of stability under infinite unions makes it possible to obtain several equivalences of categories of spaces with categories of lattices. We show several variations in Stone duality and Esakia duality for categories of small or locally small spaces and some subclasses of strictly continuous (or bouned continuous) mappings. Such equivalences are better than the spectral reflector functor for usual topological spaces. Some spectralifications of Kolmogorov locally small spaces can be obtained by Stone duality. Small spaces or locally small spaces seem to be generalised topological spaces. However, looking at them as topological spaces with an additional structure is better. The language of smopologies and bounded continuous mappings simplifies the language of certain Grothendieck sites and permits us to glue together infinite families of definable sets in structures with topologies, which was important when developing the o-minimal homotopy theory. [ABSTRACT FROM AUTHOR]

In this paper we obtain some refinements and reverses of Callebaut's inequality for isotonic functionals via a result of Young's inequality due to Cartwright and Field. [ABSTRACT FROM AUTHOR]

In the paper, the authors simply review recent results of inequalities, monotonicity, signs of determinants, determinantal formulas, closed-form expressions, and identities of the Bernoulli numbers and polynomials, establish an identity involving the differences between the Bernoulli polynomials and the Bernoulli numbers, present two identities among the differences between the Bernoulli polynomials and the Bernoulli numbers in terms of a determinant and a partial Bell polynomial, and derive a determinantal formula of the differences between the Bernoulli polynomials and the Bernoulli numbers. [ABSTRACT FROM AUTHOR]

Let X be a 2-connected and 6-dimensional CW-complex such that H3(X) ⊗ Z2 = 0. This paper aims to describe the group E(X) of the self-homotopy equivalences of X modulo its normal subgroup E*(X) of the elements that induce the identity on the homology groups. Making use of the Whitehead exact sequence of X, denoted by WES(X), we define the group GS(X) of G-automorphisms of WES(X) and we prove that E(X)/E*(X) ≌ ΓS(X). [ABSTRACT FROM AUTHOR]

A common feature of non-monotonic logics is that the classical notion of equivalence does not preserve the intended meaning in light of additional information. Consequently, the term strong equivalence was coined in the literature and thoroughly investigated. In the present paper, the knowledge representation formalism under consideration is claimaugmented argumentation frameworks (CAFs) which provide a formal basis to analyze conclusion-oriented problems in argumentation by adapting a claim-focused perspective. CAFs extend Dung AFs by associating a claim to each argument representing its conclusion. In this paper, we investigate both ordinary and strong equivalence in CAFs. Thereby, we take the fact into account that one might either be interested in the actual arguments or their claims only. The former point of view naturally yields an extension of strong equivalence for AFs to the claim-based setting while the latter gives rise to a novel equivalence notion which is genuine for CAFs. We tailor, examine and compare these notions and obtain a comprehensive study of this matter for CAFs. We conclude by investigating the computational complexity of naturally arising decision problems. [ABSTRACT FROM AUTHOR]

The present paper is the result of a contemplative study of a multi-time control problem (MCP) by considering its associated equivalent auxiliary control problem (MCP)ς via the exact l1 penalty method. Further study reveals that the solution set of the considered problem and the auxiliary problem exhibits an equivalence under the KT-pseudoinvexity hypothesis. Moreover, the study is extended towards the saddle point defined for (MCP) to establish the relationship between the solution set of multi-time control problem (MCP) and its associated equivalent auxiliary control problem (MCP)ς. Finally, we present an illustrative application to authenticate the results presented in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we prove that some Hiroshima's type inequalities for positive semidefinite block matrices are equivalent. These interesting results are due to Hiroshima [Phys. Rev. Lett. 91, (2003), 057902], Lin and Wolkowicz [Linear Multilinear Algebra 60, 11-12 (2012), 1365-1368], Turkmen, Paksoy and Zhang [Linear Algebra Appl. 437, 6 (2012), 1305-1316], Zhang and Xu [J. Math. Inequal. 14, 4 (2020), 1383-1388], respectively. [ABSTRACT FROM AUTHOR]

In this paper, we present two type of contributions to the study of two-phases problems. In such problems, the main focus is on optimising a diffusion function α under L∞ and L¹ constraints, this function a appearing in a diffusive term of the form -∇ · (a∇) in the model, in order to maximise a certain criterion. We provide a parabolic Talenti inequality and a partial bang-bang property in radial geometries for a general class of elliptic optimisation problems: namely, if a radial solution exists, then it must saturate, at almost every point, the L∞ constraints defining the admissible class. This is done using an oscillatory method. [ABSTRACT FROM AUTHOR]

In this paper, we consider a degenerate hyperbolic inequality in an exterior domain under three types of boundary conditions: Dirichlet-type, Neumann-type, and Robin-type boundary conditions. Using a unified approach, we show that all the considered problems have the same Fujita critical exponent. Moreover, we answer some open questions from the literature regarding the critical case. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the existence and uniqueness of the strong solution for the linear time-fractional partial differential equation with purely integral conditions. The aimed investigation is demonstrated based on the so-called energy inequality method and the density of the operator generated by the considered problem. To do so, we first set the position of the problem under consideration coupled with its corresponding equivalent problem, say problem (x). Afterward, we introduce some necessary functional spaces needed for exploring the existence and uniqueness of solution of problem (x). Finally, we investigate the existence and uniqueness of solution of the main operational equation. [ABSTRACT FROM AUTHOR]

In the paper, we establish two inequalities for differentiable bounded functions and Lipschitzian function, which are connected with Iyengar integral inequalities, and we present some new results. [ABSTRACT FROM AUTHOR]

In this paper, we establish two exponential double inequalities generalized Jordan's inequality. [ABSTRACT FROM AUTHOR]

In this paper, we establish the general solution of the following functional inequality ||2 f (x)+2 f (y)+2 f (z)- f (x+y)- f (y+z)|| || || ≤ (x+z)||, and then investigate the generalized Hyers-Ulam stability of this inequality in Banach spaces and in non-Archimedean Banach spaces by using two different approaches. [ABSTRACT FROM AUTHOR]

In this paper, by the use of weight functions and the technique of real analysis, a new Hilbert-type integral inequality with a general nonhomogeneous kernel as H(xv(y)) is given. A few equivalent statements related to the best possible constant factor and parameters are considered. As applications, the equivalent forms, the operator expressions and some corollaries are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we mainly give a generalized refinement of Young's inequality due to Yang and Wang [J. Math. Inequal., 17 (2023), 205-217]. More precisely, we show that... As applications, we obtain some inequalities for operator, Hilbert-Schmidt norm and trace class norm. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in generalized numerical radius inequalities for the off-diagonal part of a positive semidefinite block matrix. These inequalities produce a new set of inequalities for products and sums of matrices and some inequalities related to sectorial matrices. [ABSTRACT FROM AUTHOR]

Let p be an odd prime and Zp the residue class ring modulo p. In this paper, we study representations of any element of Zp as the sum of a Lehmer number and a primitive root in Zp, and give an explicit inequality better than asymptotic formula for the number of representations. From this inequality, we obtained that each element of Zp can be represented as the sum of a Lehmer number and a primitive root for p > 2.5×1014. Moreover, using the algorithm we provided, we examined all the cases when p < 106 by computer. We also analyzed the time complexity of the algorithm and illustrated that it is extremely difficult to verify all the cases up to the bound 2.5×1014, and conjectured that any given element n ∈ Zp can be represented as the sum of a Lehmer number and a primitive root in Zp for all primes p except 2, 3, 5, 7, 11, 19, 31. [ABSTRACT FROM AUTHOR]

Mathematical concepts have been used in the last decades to predict the behavior of the spread of infectious diseases. Among them, the reproductive number concept has been used in several published papers to study the stability of the mathematical model used to predict the spread patterns. Some conditions were suggested to conclude if there would be either stability or instability. An analysis was also meant to determine conditions under which infectious classes will increase or die out. Some authors pointed out limitations of the reproductive number, as they presented its inability to help predict the spread patterns. The concept of strength number and analysis of second derivatives of the mathematical models were suggested as additional tools to help detect waves. This paper aims to apply these additional analyses in a simple model to predict the future. [ABSTRACT FROM AUTHOR]

A triangular tile latching system consists of a set Σ of equilateral triangular tiles with at least one latchable side and an attachment rule which permits two tiles to get latched along a latchable side. In this paper we determine the language generated by a triangular tile latching system in terms of planar graphs. [ABSTRACT FROM AUTHOR]

The Editors-in-Chief have retracted this paper [1] because its results are invalid. The paper also shows significant overlap with a paper by Yang et al. [2], which was simultaneously under consideration with another journal. Additionally, this paper showed evidence of peer review and authorship manipulation. The authors have not responded to any correspondence with regards to this retraction. [ABSTRACT FROM AUTHOR]

A list of manuscripts for future publication is presented on topics including equilibrium default, debt maturity and numerical equivalence.

If μ is a positive Borel measure on the interval [0, 1), we let 퓗μ be the Hankel matrix 퓗μ = ( μn,k)n,k≥0 with entries μn,k = μn+k and μn = ∫[0,1) tnd μ(t). Using 퓗μ, Ye and Zhou first defined the Derivative-Hilbert operator as D퓗μ(f)(z) = ... (n+1)zn, z ∈ 픻, where f (z)= ... is an analytic function in 픻. In this paper, we characterize the measure μ for which D퓗μ is a bounded (resp., compact) operator from Besov space Bp into Bloch space B with 1 < p < ∞. [ABSTRACT FROM AUTHOR]

In this paper, the authors establish a two-weight boundedness criterion of discrete Hardy operator and its dual operator in the scale of discrete weighted variable Lebesgue spaces. Moreover, we study the problem of compactness of the discrete Hardy operator in discrete weighted variable Lebesgue spaces. We also study a similar problem for the dual operator of discrete Hardy operator. [ABSTRACT FROM AUTHOR]

In this paper, we improve and generalize some existing inequalities for unitarily invariant norms by using the convexity of the function f (v) = ||AvXB1--v+ A 1--vXBv|| on the interval [0,1] . [ABSTRACT FROM AUTHOR]

In this paper, we obtain the reverse of Minkowski inequality for matrices, and from this we give reverse forms of the Oppenheim inequality for n-simplices and the Neuberg-Pedoe inequality for triangles. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of a strongly convex function with respect to two non-negative auxiliary functions on time scales. We establish several new dynamic inequalities for these classes of strongly convex functions. The results obtained in this paper are the generalization of the results of Rashid et al. (Mathematics, 7 (10), 956, 2019). Further, we discuss some special cases which may be deduced from our main results. Moreover, some examples of our main results are mentioned. [ABSTRACT FROM AUTHOR]

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In this paper, firstly, we introduce the concepts of continuity and boundedness of linear operators between two fuzzy quasi-normed spaces with general continuous t-norms, prove the equivalence of them, and point out that the set of all continuous linear operators forms a convex cone. Secondly, we establish the family of star quasi-seminorms on the cone of continuous linear operators, and construct a fuzzy quasi-norm of a continuous linear operator. [ABSTRACT FROM AUTHOR]

This paper builds upon the research of Corwin and Knizel (2021) who proved the existence of stationary measures for the KPZ equation on an interval and characterized them through a Laplace transform formula. Bryc et al. (2022) found a probabilistic description of the stationary measures in terms of a Doob transform of some Markov kernels; essentially at the same time, another description connecting the stationary measures to the exponential functionals of the Brownian motion appeared in Barraquand and Le Doussal (2022). Our first main result clarifies and proves the equivalence of the two probabilistic description of these stationary measures. We then use the Markovian description to give rigorous proofs of some of the results claimed in Barraquand and Le Doussal (2022). We analyze how the stationary measures of the KPZ equation on [0,τ] behave at large scale, as τ goes to infinity. We also analyze the behaviour of the stationary measures of the KPZ equation on [0,τ] without rescaling, when τ goes to infinity. Finally, we analyze the measures on [0,∞) at large scale, which according to Barraquand and Le Doussal (2022) should correspond to stationary measures of a hypothetical KPZ fixed point on [0,∞). [ABSTRACT FROM AUTHOR]

In practical applications of regression models, we may meet with the situation where a true model is misspecified in some other forms due to certain unforeseeable reasons, so that estimation and statistical inference results obtained under the true and misspecified regression models are not necessarily the same, and therefore, it is necessary to compare these results and to establish certain links between them for the purpose of reasonably explaining and utilizing the misspecified regression models. In this paper, we propose and investigate some comparison and equivalence analysis problems on estimations and predictions under true and misspecified multivariate general linear models. We first give the derivations and presentations of the best linear unbiased predictors (BLUPs) and the best linear unbiased estimators (BLUEs) of unknown parametric matrices under a true multivariate general linear model and its misspecified form. We then derive a variety of necessary and suffcient conditions for the BLUPs/BLUEs under the two competing models to be equal using a series of highly-selective formulas and facts associated with ranks, ranges and generalized inverses of matrices, as well as block matrix operations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following mixed order conformally invariant system with Hartree-type nonlinearity: ... where 0 < s =: m + 2α < +∞, m is a integer, 0 < α ≤ 2, n ≥ 3, σ ∈ (0, n). We first prove the equivalence of the PDEs system and the IEs system. Then we give the classification of the nonnegative solutions to the system (0.1) by using the method of moving spheres. Finally, we prove Liouville-type theorems results for system (0.1) in the critical and supercritical-order cases (i.e. n 2 < s =: m+ 2/α < +∞), respectively. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of weak i-hyperfilters of ordered n-ary semihypergroups, where a positive integer 1 ≤ i ≤ n and n ≥ 2, and discuss their related properties. We define Green's relations Mi, J, H and K on ordered n-ary semihypergroups and investigate the relationships between Green's relations and the equivalence relation Wi, which is generated by the weak i-hyperfilters. Also, we give the characterizations of intra-regular ordered n-ary semihypergroups via the properties of weak ihyperfilters. Finally, we introduce the concepts of (i-, Λ-)duo ordered n-ary semihypergroups and establish some interesting properties. [ABSTRACT FROM AUTHOR]

Directing shallow-water waves and their energy is highly desired in many ocean engineering applications. Coastal infrastructures can be protected by reflecting shallow-water waves to deep water. Wave energy harvesting efficiency can be improved by focusing shallow-water waves on wave energy converters. Changing water depth can effectively affect wave celerity and therefore the propagation of shallow-water waves. However, determining spatially varying bathymetry that can direct shallow-water waves to a designed location is not trivial. In this paper, we propose a novel machine learning method to design and optimize spatially varying bathymetry for directing shallow-water waves, in which the bathymetry is assumed fixed in time without considering morphodynamics. Shallow-water wave theory was applied to establish the mapping between water wave mechanics and recurrent neural networks (RNNs). Two wave-equivalent RNNs were developed to model shallow-water waves over fixed varying bathymetry. The resulting RNNs were trained to optimize bathymetry for wave energy focusing. We demonstrate that the bathymetry optimized by the wave-equivalent RNNs can effectively reflect and refract wave energy to various designed locations. We also foresee the potential that new engineering tools can be similarly developed based on the mathematical equivalence between wave mechanics and recurrent neural networks. [ABSTRACT FROM AUTHOR]