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The paper lists the main open questions left in the area both theoretically and practically, mainly related with the inherent tension between communication efficiency and quality of assumptions in proof systems. This special issue of Designs, Codes and Cryptography on the topic Mathematics of Zero-Knowledge aims at encouraging mathematicians and computer scientists with a strong interest in Mathematics to contribute to the burgeoning topic of zero-knowledge proofs. [Extracted from the article]

Very recently, Pushpa and Vasuki (Arab. J. Math. 11, 355–378, 2022) have proved Eisenstein series identities of level 5 of weight 2 due to Ramanujan and some new Eisenstein identities for level 7 by the elementary way. In their paper, they introduced seven restricted color partition functions, namely P ∗ (n) , M (n) , T ∗ (n) , L (n) , K (n) , A (n) , and B(n), and proved a few congruence properties of these functions. The main aim of this paper is to obtain several new infinite families of congruences modulo 2 a · 5 ℓ for P ∗ (n) , modulo 2 3 for M(n) and T ∗ (n) , where a = 3 , 4 and ℓ ≥ 1 . For instance, we prove that for n ≥ 0 , P ∗ (5 ℓ (4 n + 3) + 5 ℓ - 1) ≡ 0 (mod 2 3 · 5 ℓ). In addition, we prove witness identities for the following congruences due to Pushpa and Vasuki: M (5 n + 4) ≡ 0 (mod 5) , T ∗ (5 n + 3) ≡ 0 (mod 5). [ABSTRACT FROM AUTHOR]

Recent rank-based attacks have reduced the security of Rainbow, which is one of the multi-layer UOV signatures, below the NIST security requirements by speeding up iterative kernel-finding operations using classical mathematics techniques. If quantum algorithms are applied to perform these iterative operations, the rank-based attacks may be more threatening to multi-layer UOV, including Rainbow. In this paper, we propose a quantum rectangular MinRank attack called the Q-rMinRank attack, the first quantum approach to key recovery attacks on multi-layer UOV signatures. Our attack is a general model applicable to multi-layer UOV signature schemes, and in this paper, we provide examples of its application to Rainbow and the Korean TTA standard, HiMQ. We design two quantum oracle circuits to find the kernel in consideration of the depth-width trade-off of quantum circuits. One is to reduce the width of the quantum circuits using qubits as a minimum, and the other is to reduce the depth using parallelization instead of using a lot of qubits. By designing quantum circuits to find kernels with fewer quantum resources and complexity by adding mathematical techniques, we achieve quadratic speedup for the MinRank attack to recover the private keys of multi-layer UOV signatures. We also estimate quantum resources for the designed quantum circuits and analyze quantum complexity based on them. The width-optimized circuit recovers the private keys of Rainbow parameter set V with only 1089 logical qubits. The depth-optimized circuit recovers the private keys of Rainbow parameter set V with a quantum complexity of 2 174 , which is lower than the complexity of 2 221 recovering the secret key of AES-192, which provides the same security level as parameter set III. [ABSTRACT FROM AUTHOR]

Uncertainty theory, founded in 2007, has become a branch of mathematics to model uncertainty rather than randomness. As an indispensable part of uncertainty theory, uncertain graph and uncertain network optimization has received the wide attention of many scholars. Naturally, a series of original research achievements have been obtained on uncertain graph and uncertain network optimization. This paper aims to present a state-of-the-art review on the recent advance in uncertain graph and uncertain network optimization. Furthermore, it hopes to predict the possible future research directions. Based on Web of Science database, this paper retrieves 144 related papers from 2011 to 2021 to analyze the features of published articles. More precisely, we analyze the annual number of publications, key topics and sub-fields, journals, and most-cited articles. In addition, the main results and models for uncertain graph and uncertain network optimization are summarized. Furthermore, the limitations of existing literature and the possible development trend are discussed. [ABSTRACT FROM AUTHOR]

Here, we systematically review research on teaching knowledge in the context of undergraduate STEM education, with particular attention to what this research reveals about knowledge that is important for evidence-based teaching. Evidence-based teaching can improve student outcomes in undergraduate STEM education. However, the enactment of promising evidence-based teaching strategies depends greatly on the instructor and potentially on the teaching knowledge they are able to deploy. The review includes an overview of prevalent teaching knowledge theory, including pedagogical content knowledge, mathematical knowledge for teaching, and pedagogical knowledge. We compare and contrast teaching knowledge theory and terminology across STEM disciplines in order to build bridges for researchers across disciplines. Our search for peer-reviewed investigations of teaching knowledge in undergraduate science, engineering and mathematics yielded 45 papers. We examined the theoretical frameworks used in each study and analyzed study approaches, comparing across disciplines. Importantly, we also synthesized findings from research conducted in the context of evidence-based teaching. Overall, teaching knowledge research is sparse and siloed by discipline, and we call for collaborative work and better bridge-building across STEM disciplines. Though disciplinary divergences are common in discipline-based education research, the effect is magnified in this research area because the theoretical frameworks are themselves siloed by discipline. Investigations of declarative knowledge were common, and we call for increased attention to knowledge used in the practice of teaching. Finally, there are not many studies examining teaching knowledge in the context of evidence-based teaching, but the existing work suggests that components of pedagogical content knowledge, pedagogical knowledge, and content knowledge influence the implementation of evidence-based teaching. We describe implications for future teaching knowledge research. We also call on those who develop and test evidence-based strategies and curriculum to consider, from the beginning, the teaching knowledge needed for effective implementation. [ABSTRACT FROM AUTHOR]

In the present paper, we have obtained common best proximity point theorems of nonself maps in Hausdorff topological space. Further, our results extend the results due to Gerald F. Jungck, thereby proving a generalized version of Kirk's theorem (J. London Math. 1(1):107–111, 1969). [ABSTRACT FROM AUTHOR]

Reference (Ericksen in Q. J. Mech. Appl. Math. 45:545–554, 1992), a short and dense paper by J.L. Ericksen, is taken as a primary source of terms frequently used in continuum thermodynamics, in an attempt to make their significance as precise as possible. [ABSTRACT FROM AUTHOR]

In this paper, we construct a symbol calculus yielding short exact sequences for the dual Toeplitz algebra generated by all bounded dual Toeplitz operators on the Hardy space associated with the polydisk D n in the unitary space C n , that have been introduced and well studied in our earlier paper (Benaissa and Guediri in Taiwan J Math 19: 31–49, 2015), as well as for the C*-subalgebra generated by dual Toeplitz operators with symbols continuous on the associated hypertorus T n . [ABSTRACT FROM AUTHOR]

This note remedies an error in our paper tilted Conormal Spaces and Whitney Stratifications (Found. Comput. Math., 2022). [ABSTRACT FROM AUTHOR]

In this paper, we establish some reversed dynamic inequalities of Hilbert type on time scales nabla calculus by applying reversed Hölder's inequality, chain rule on time scales, and the mean inequality. As particular cases of our results (when T = N and T = R ), we get the reversed form of discrete and continuous inequalities proved by Chang-Jian, Lian-Ying and Cheung (Math. Slovaca 61(1):15–28, 2011). [ABSTRACT FROM AUTHOR]

In the paper, we investigate the uniqueness problem of a meromorphic function and its difference operator to the most general setting via two shared set problems and thus improve a recent result of Chen–Chen (Bull Malays Math Sci Soc 35(3): 765-774, 2012). [ABSTRACT FROM AUTHOR]

The notion of approximate fixed point sequence, emphasized in Chidume (Geometric properties of Banach spaces and nonlinear iterations. Lecture Notes in Mathematics, 1965. Springer-Verlag London, Ltd., London, 2009), is a very useful tool in proving convergence theorems for fixed point iterative schemes in the class of nonexpansive-type mappings. In the present paper, our aim is to present simple and unified alternative proofs of some classical fixed point theorems emerging from Banach contraction principle, by using a technique based on the concepts of approximate fixed point sequence and graphic contraction. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the Harary index and the hyper-Wiener index of the H-generalized join of graphs and the generalized corona product of graphs. As a consequence, we deduce some of the results in (Das et al. in J. Inequal. Appl. 2013:339, 2013) and (Khalifeh et al. in Comput. Math. Appl. 56:1402–1407, 2008). Moreover, we calculate the Harary index and the hyper-Wiener index of the ideal-based zero-divisor graph of a ring. [ABSTRACT FROM AUTHOR]

In this paper, by revisiting intersection rules for normal cones, we give formulas for estimating or computing the Fréchet/Mordukhovich/Moreau–Rockafellar subdifferentials of optimal value functions of constrained parametric optimization problems under metric qualification conditions. The results are then applied to derive chain rules for composite functions in both convex and nonconvex situations. Illustrative examples and comparisons to existing results, including those of Mordukhovich and Shao (Trans Amer Math Soc 348:1235–1280, 1996), Mordukhovich et al. (Math Program Ser B 116:369–396, 2009) and of An and Jourani (J Optim Theory Appl 192:82–96, 2022), are also addressed. [ABSTRACT FROM AUTHOR]

In this paper, we establish a relation-theoretic set-valued version of the fixed point result of Ćirić and Prešić (Acta Math. Univ. Comen. LXXVI(2):143–147, 2007) on metric spaces endowed with an arbitrary binary relation. The results of this paper, generalize and unify the fixed point results of Ćirić and Prešić (Acta Math. Univ. Comen. LXXVI(2):143–147, 2007), Shukla and López (Quaest. Math. 45(3):1–16, 2019), and Shukla and Radenović (An. Ştiinţ. Univ. 'Al.I. Cuza' Iaşi, Mat. 63(2):339–350, 2017) in product spaces. Some examples are provided that justify and establish the importance of our results. As applications of our main result, we have established the existence of solutions to differential inclusion problems and the weak asymptotical stability and a global attractivity of the equilibrium point of a difference inclusion problem. The use of arbitrary binary relations in our results permits us to apply the results to the differential inclusion problems and difference inclusion problems with weaker assumptions than those used in the papers mentioned above. [ABSTRACT FROM AUTHOR]

Integral inequalities with generalized convexity play an important role in both applied and theoretical mathematics. The theory of integral inequalities is currently one of the most rapidly developing areas of mathematics due to its wide range of applications. In this paper, we study the concept of higher-order strongly exponentially convex functions and establish a new Hermite–Hadamard inequality for the class of strongly exponentially convex functions of higher order. Further, we derive some new integral inequalities for Riemann–Liouville fractional integrals via higher-order strongly exponentially convex functions. These findings include several well-known results and newly obtained results as special cases. We believe that the results presented in this paper are novel and will be beneficial in encouraging future research in this field. [ABSTRACT FROM AUTHOR]

The main objective of the paper is to study the final model for the Kirchhoff-type parabolic system. Such type problems have many applications in physical and biological phenomena. Under some smoothness of the final Cauchy data, we prove that the problem has a unique mild solution. The main tool is Banach's fixed point theorem. We also consider the non-well-posed problem in the Hadamard sense. Finally, we apply truncation method to regularize our problem. The paper is motivated by the work of Tuan, Nam, and Nhat [Comput. Math. Appl. 77(1):15–33, 2019]. [ABSTRACT FROM AUTHOR]

This paper proposes a new data clustering method using the advantages of metaheuristic (MH) optimization algorithms. A novel MH optimization algorithm, called arithmetic optimization algorithm (AOA), was proposed to address complex optimization tasks. Math operations inspire the AOA, and it showed significant performance in dealing with different optimization problems. However, the traditional AOA faces some limitations in its search process. Thus, we develop a new variant of the AOA, namely, Augmented AOA (AAOA), integrated with the opposition-based learning (OLB) and Lévy flight (LF) distribution. The main idea of applying OLB and LF is to improve the traditional AOA exploration and exploitation trends in order to find the best clusters. To evaluate the AAOA, we implemented extensive experiments using twenty-three well-known benchmark functions and eight data clustering datasets. We also evaluated the proposed AAOA with extensive comparisons to different optimization algorithms. The outcomes verified the superiority of the AAOA over the traditional AOA and several MH optimization algorithms. Overall, the applications of the LF and OLB have a significant impact on the performance of the conventional AOA. [ABSTRACT FROM AUTHOR]

In this short paper we tackle two subjects. First, we provide a lower bound for the first eigenvalue of the antiperiodic problem for a Hill's equation based on L p -conditions, and as a consequence, we introduce an adjusted statement of the main result about the asymptotic stability of periodic solutions for the general Duffing equation in (Torres in Mediterr. J. Math. 1(4):479–486, 2004) (Theorem 4). This appropriate version of the result arises because of one subtlety in the proof provided in (Torres in Mediterr. J. Math. 1(4):479–486, 2004). More precisely, the lower bound of the first antiperiodic eigenvalue associated with Hill's equations of potential a (t) employed there may be negative, thus the conclusion is not completely attained. Hence, the adjustments considered here provide a mathematically correct result. On the other hand, we apply this result to obtain a lateral asymptotic stable periodic oscillation in the Comb-drive finger MEMS model with a cubic nonlinear stiffness term and linear damping. This fact is not typical in Comb-drive finger devices, thus our results could provide a new possibility; a new design principle for stabilization in Comb-drive finger MEMS. [ABSTRACT FROM AUTHOR]

This paper proposes a novel matrix rank-one decomposition for quaternion Hermitian matrices, which admits a stronger property than the previous results in Ai W et al (Math Progr 128(1):253–283, 2011), Huang Y, Zhang S (Math Oper Res 32(3):758–768, 2007), Sturm JF, Zhang S (Math Oper Res 28(2):246–267 2003). The enhanced property can be used to drive some improved results in joint numerical range, S -Procedure and quadratically constrained quadratic programming (QCQP) in the quaternion domain, demonstrating the capability of our new decomposition technique. [ABSTRACT FROM AUTHOR]

Decision-making under uncertainty involves three fundamental components: acts, states of nature, and consequences, as first introduced by Savage (The foundations of statistics, Wiley, New York, 1954). An act is uniquely determined by a number of random consequences that are associated with different states of nature. If the consequences are identical across all states of nature, then the act is state-independent. Prior research on distortion risk measures (DRMs) has primarily focused on state-independent acts. In this paper, we extend the research to state-dependent acts by introducing a state-dependent DRM (SDRM) under the Anscombe–Aumann's framework (Anscombe and Aumannin in Ann Math Stat 34(1):199–205, 1963). The proposed SDRM is the weighted average of DRMs at each state, where the weights are determined by the decision maker's (DM's) subjective probabilities of the states. In situations where there is incomplete information about the DM's true distortion function and/or the true subjective probabilities of the states, we introduce a preference robust SDRM (PRSDRM) for acts. The PRSDRM is based on the worst-case state-dependent distortion function and the worst-case subjective probabilities over a dependent joint ambiguity set constructed with partially available information. To compute the PRSDRM numerically, we show that when the distortion functions are concave, it can be formulated as a biconvex program and further as a convex program by changing some variables. As a motivation and application, we use the PRSDRM for decision-making problems and propose an alternating iterative algorithm for solving it. Finally, we conduct numerical experiments to assess the performance of our proposed model and computational scheme. [ABSTRACT FROM AUTHOR]

This paper studies decision-making in the face of two stochastically independent sources of uncertainty. It characterizes axiomatically a Subjective Expected Utility representation of preferences where subjective beliefs consist of a product probability measure. The two key axioms in this characterization both involve some behavioral notions of stochastic independence. Our result can be understood as a purely subjective version of the Anscombe and Aumann (Ann Math Stat 34:199–205, 1963) theorem that avoids the controversial use of exogenous probabilities by appealing to stochastic independence. We also obtain an extension to Choquet Expected Utility representations. [ABSTRACT FROM AUTHOR]

The paper deals with the code length distributions for which the proportion of the codewords of a given length to all words of this length is a suitable power of 1/2. We derive a lower bound and an upper bound for the number of prefix codes with the above property. We also investigate the ratio of prefix codes to all uniquely decodable codes, which relates to the more general results of Woryna (Discret Math 340(2):51–57, 2017; Discret Appl Math 244:205–213, 2018). [ABSTRACT FROM AUTHOR]

We consider the following problem stated in 1993 by Buttazzo and Kawohl (Math Intell 15:7–12, 1993): minimize the functional ∫ ∫ Ω (1 + | ∇ u (x , y) | 2 ) - 1 d x d y in the class of concave functions u : Ω → [0,M], where Ω ⊂ R 2 is a convex domain and M > 0. It generalizes the classical minimization problem, which was initially stated by I. Newton in 1687 in the more restricted class of radial functions. The problem is not solved until now; there is even nothing known about the structure of singular points of a solution. In this paper we, first, solve a family of auxiliary 2D least resistance problems and, second, apply the obtained results to study singular points of a solution to our original problem. More precisely, we derive a necessary condition for a point being a ridge singular point of a solution and prove, in particular, that all ridge singular points with horizontal edge lie on the top level and zero level sets. [ABSTRACT FROM AUTHOR]

Boys in the Kingdom of Saudi Arabia consistently and significantly underperform compared to girls across different grades and subjects, forming one of the largest gender gaps in student achievement in the world. Saudi Arabia offers a unique setting in which boys and girls attend separate schools on a universal basis starting from grade 1. This means that boys and girls are educated only by male and female teachers, respectively, in effect inhabiting parallel education systems. In this context, this study examines the factors that are associated with student achievement in mathematics and science in grades 4 and 8 and the extent to which these associations are different for boys and girls, in an effort to gain insights into boys' underachievement in mathematics and science in Saudi Arabia. The paper employs data from two recent large-scale assessments of education: Trends in International Mathematics and Science Study (TIMSS) 2019 and Saudi Arabia's National Assessment of Learning Outcomes (NALO) 2018. A series of hierarchical two-level linear regression models showed that in grade 4, school climate was more strongly associated with boys' compared with girls' achievement in both mathematics and science, with boys attending schools of poorer school climate having a considerably lower performance compared with girls attending such schools. The findings also indicated that although greater literacy and numeracy readiness was linked with higher science achievement among boys and girls, grade 4 boys tended to benefit more from this readiness than girls. In addition, the results show that student absenteeism in grade 4 is particularly strongly associated with decreases in mathematics achievement among boys. In grade 8, significant interactions between gender and the extent to which students feel confident in science, the degree of schools' emphasis on academic success, and teachers' age are observed. The paper concludes by discussing some of the implications of these findings for educators and policy makers in Saudi Arabia. [ABSTRACT FROM AUTHOR]

Picard's theorem is among the most striking results in complex analysis and plays a decisive role in the development of the theory of entire and meromorphic functions. In this paper, we give an equivalence between Picard's theorem and entire solutions of (i) difference equation Δ η f + a (z) P (f) = 0 in C and (ii) differential equation D k f + h (z) g (f (z)) = 0 in terms of total derivatives in C n . We prove two Picard-type theorems that generalize the results of Lu (Kodai Math J 26:221–229, 2003). In addition, examples have been exhibited to show that certain assumptions in the main result cannot be improved. [ABSTRACT FROM AUTHOR]

The normalization of the combination of generalized Lommel–Wright function J κ 1 , κ 2 κ 3 , m (z) (m ∈ N , κ 3 > 0 and κ 1 , κ 2 ∈ C ) defined by J κ 1 , κ 2 κ 3 , m (z) : = Γ m (κ 1 + 1) Γ (κ 1 + κ 2 + 1) 2 2 κ 1 + κ 2 z 1 − (κ 2 / 2) − κ 1 J κ 1 , κ 2 κ 3 , m (z) , where J κ 1 , κ 2 κ 3 , m (z) : = (1 − 2 κ 1 − κ 2) J κ 1 , κ 2 κ 3 , m (z) + z (J κ 1 , κ 2 κ 3 , m (z)) ′ and J κ 1 , κ 2 κ 3 , m (z) = (z 2) 2 κ 1 + κ 2 ∑ n = 0 ∞ (− 1) n Γ m (n + κ 1 + 1) Γ (n κ 3 + κ 1 + κ 2 + 1) (z 2) 2 n , was previously introduced and some of its geometric properties have been considered. In this paper, we report conditions for J κ 1 , κ 2 κ 3 , m (z) to be starlike and convex of order α, 0 ≤ α < 1 , inside the open unit disk using some technical manipulations of the gamma and digamma functions, as well as inequality for the digamma function that has been proved (Guo and Qi in Proc. Am. Math. Soc. 141(3):1007–1015, 2013). In addition, a method presented by Lorch (J. Approx. Theory 40(2):115–120 1984) and further developed by Laforgia (Math. Compet. 42(166):597–600 1984) is applied to establish firstly sharp inequalities for the shifted factorial that will be used to obtain the order of starlikeness and convexity. We compare then the obtained orders of starlikeness and convexity with some important consequences in the literature as well as the results proposed by all techniques to demonstrate the accuracy of our approach. Ultimately, a lemma by (Fejér in Acta Litt. Sci. 8:89–115 1936) is used to prove that the modified form of the function J κ 1 , κ 2 κ 3 , m (z) defined by I κ 1 , κ 2 κ 3 , m (z) = J κ 1 , κ 2 κ 3 , m (z) ∗ z / (1 + z) is in the class of starlike and convex functions, respectively. Further work regarding the function J κ 1 , κ 2 κ 3 , m (z) is underway and will be presented in a forthcoming paper. [ABSTRACT FROM AUTHOR]

We prove two key inequalities for metric and generalized projections in a certain Banach space. We then obtain some asymptotic behavior of a sequence generated by the shrinking projection method introduced by Takahashi et al. (J. Math. Anal. Appl. 341:276–286, 2008) where the computation allows some nonsummable errors. We follow the idea proposed by Kimura (Banach and Function Spaces IV (ISBFS 2012), pp. 303–311, 2014). The mappings studied in this paper are more general than the ones in (Ibaraki and Kimura in Linear Nonlinear Anal. 2:301–310, 2016; Ibaraki and Kajiba in Josai Math. Monogr. 11:105–120, 2018). In particular, the results in (Ibaraki and Kimura in Linear Nonlinear Anal. 2:301–310, 2016; Ibaraki and Kajiba in Josai Math. Monogr. 11:105–120, 2018) are both extended and supplemented. Finally, we discuss our results for finding a zero of maximal monotone operator and a minimizer of convex functions on a Banach space. [ABSTRACT FROM AUTHOR]

This paper deals with the blow-up phenomena connected to the following porous-medium problem with gradient terms under Robin boundary conditions: { u t = Δ u m + k 1 u p − k 2 | ∇ u | q in Ω × (0 , t ∗) , ∂ u ∂ ν + γ u = 0 on ∂ Ω × (0 , t ∗) , u (x , 0) = u 0 (x) ≥ 0 in Ω ‾ , where Ω ⊂ R n (n ≥ 3 ) is a bounded and convex domain with smooth boundary ∂Ω. The constants p, q, m are positive, and p > q > m > 1 , q > 2 . By making use of the Sobolev inequality and the differential inequality technique, we obtain a lower bound for the blow-up time of the solution. In addition, an example is given as an application of the abstract results obtained in this paper. Our results can be regarded as an answer to the open question raised by Li et al. in (Z. Angew. Math. Phys. 70:1–18, 2019). [ABSTRACT FROM AUTHOR]

Background: Large scale international assessments depend on invariance of measurement across countries. An important consideration when observing cross-national differential item functioning (DIF) is whether the DIF actually reflects a source of bias, or might instead be a methodological artifact reflecting item response theory (IRT) model misspecification. Determining the validity of the source of DIF has implications for how it is handled in practice. Method: We demonstrate a form of sensitivity analysis that can point to model misspecification induced by item complexity as a possible cause of DIF, and show how such a cause of DIF might be accommodated through attempts to generalize the IRT model for the studied item(s) in psychometrically and psychologically plausible ways. Results: In both simulated illustrations and empirical data from TIMSS 2011 and TIMSS 2019 4th and 8th Grade Math and Science, we have found that using a form of proposed IRT model generalization can substantially reduce DIF when IRT model misspecification is at least a partial cause of the observed DIF. Conclusions: By demonstrating item complexity as a possible valid source of DIF and showing the effectiveness of the proposed approach, we recommend additional attention toward model generalizations as a means of addressing and/or understanding DIF. [ABSTRACT FROM AUTHOR]

P-sets (P stands for Packet) is a set model with dynamic characteristics, which is obtained by introducing dynamic characteristics into Cantor set and improving Cantor set. According to the fact that the characteristics of class I big data are completely consistent with the basic characteristics of P-sets, this paper gives research on theory and application on class I big data from the view of mathematics. Here we introduce Class I big data which need some new definitions of data block, microdata and data link. Based on these concepts, decomposition theorem of data block and microdata relation theorem are given, and then attribute reasoning theorem and microdata intelligent discovery and the intelligent secure acquisition algorithm of microdata are also proposed. By using these theoretical results, the applications of secure acquisition of microdata are presented. In summary, P-sets mathematical model provides a new theory and method for studying class I big data. [ABSTRACT FROM AUTHOR]

Let k ≥ 2 . A generalization of the well-known Pell sequence is the k-Pell sequence. For this sequence, the first k terms are 0 , ... , 0 , 1 and each term afterwards is given by the linear recurrence P n (k) = 2 P n - 1 (k) + P n - 2 (k) + ⋯ + P n - k (k). In this paper, we extend the previous work (Rihane and Togbé in Ann Math Inform 54:57–71, 2021) and investigate the Padovan and Perrin numbers in the k-Pell sequence. [ABSTRACT FROM AUTHOR]

In the paper (J Math Soc Jpn 72(1):303–331, 2020), Tse-Chung Yang and the first two current authors computed explicitly the number | SSp 2 (F q) | of isomorphism classes of superspecial abelian surfaces over an arbitrary finite field F q of even degree over the prime field F p . There it was assumed that certain commutative Z p -orders satisfy an étale condition that excludes the primes p = 2 , 3 , 5 . We treat these remaining primes in the present paper, where the computations are more involved because of the ramification. This completes the calculation of | SSp 2 (F q) | in the even degree case. The odd degree case was previous treated by Tse-Chung Yang and the first two current authors in (Doc Math 21:1607–1643, 2016). To complete the proof of our main theorem, we give a classification of lattices over local quaternion Bass orders, which is a new input to our previous works. [ABSTRACT FROM AUTHOR]

While the diffuse prior has been widely used in applied economic theory for its technical convenience and as a way of modeling complete lack of knowledge, it is not formally defined, nor are ex ante payoffs in games under this prior. In this paper, we provide a formal treatment of the diffuse prior which can validate its application in games. We consider stationary games, in which players' signals are translation invariant in the true state and players' payoffs are translation invariant in actions together with the state. We show that strategies which admit well-defined expected payoffs under the diffuse prior are essentially stationary, being almost translation invariant in signals. Our analysis builds on two formal definitions. We define the diffuse prior through a limit construction, using sequences of well-defined priors that become increasingly dispersed. A class of strategy profiles is admissible if for any strategy profile, each player's ex ante payoff along these sequences converges to a limit that does not depend on the particular sequence. A secondary contribution of the paper is an extension of the concept of distributional strategies (Milgrom and Weber in Math Oper Res 10:619–632, 1985) to a class of multistage games. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the 3D incompressible Hall-magnetohydrodynamics with partial dissipation. Based on the results in (Du in Bound. Value Probl. 2022:6, 2022; Du and Liu in Acta Math. Sci. 42A:5, 2022; Fei and Xiang in J. Math. Phys. 56:051504, 2015), we establish an improved blow-up criterion for classical solutions. Furthermore, using the blow-up criterion, we also obtain the existence of the classical solutions only under the condition that the initial data ∥ V 0 ∥ H 1 + ∥ B 0 ∥ H 2 are sufficiently small. [ABSTRACT FROM AUTHOR]

Background: Blended mathematical sensemaking in science ("Math-Sci sensemaking") involves deep conceptual understanding of quantitative relationships describing scientific phenomena and has been studied in various disciplines. However, no unified characterization of blended Math-Sci sensemaking exists. Results: We developed a theoretical cognitive model for blended Math-Sci sensemaking grounded in prior work. The model contains three broad levels representing increasingly sophisticated ways of engaging in blended Math-Sci sensemaking: (1) developing qualitative relationships among relevant variables in mathematical equations describing a phenomenon ("qualitative level"); (2) developing mathematical relationships among these variables ("quantitative level"); and (3) explaining how the mathematical operations used in the formula relate to the phenomenon ("conceptual level"). Each level contains three sublevels. We used PhET simulations to design dynamic assessment scenarios in various disciplines to test the model. We used these assessments to interview undergraduate students with a wide range of Math skills. Interview analysis provided validity evidence for the categories and preliminary evidence for the ordering of the categories comprising the cognitive model. It also revealed that students tend to perform at the same level across different disciplinary contexts, suggesting that blended Math-Sci sensemaking is a distinct cognitive construct, independent of specific disciplinary context. Conclusion: This paper presents a first-ever published validated cognitive model describing proficiency in blended Math-Sci sensemaking which can guide instruction, curriculum, and assessment development. [ABSTRACT FROM AUTHOR]

Howe and Tan (Bull Am Math Soc 28:1–74, 1993) investigated a degenerate principal series representation of indefinite orthogonal groups O (b + , b -) and explicitly described its composition series. In particular it contains a unique unitarizable irreducible submodule Π , which is isomorphic to a cohomological representation. In this paper we construct orthogonal automorphic forms locally corresponding to Π as theta liftings of holomorphic Mp 2 (R) cusp forms by using the Borcherds' method (Invent Math 132:491–562, 1998). We propose a special choice of Schwartz functions to define the liftings, which yields precise descriptions of their Fourier expansions. [ABSTRACT FROM AUTHOR]

Traditional blood flow mapping algorithms are not effective enough when the region of interest contains the sources of artifacts. This paper examines the signatures of blood flow signals, as well as noise and artifact signals. Mathematical rules are proposed on the basis of these features to discriminate the blood flow signal from other signals, including the Doppler twinkling artifact. The diagnostic efficiency of the proposed decision rules is assessed using the previously collected database of raw ultrasound Doppler signals. The study results showed that taking account of the spatial stability of blood flow velocity can increase diagnostic efficiency to 91.6%. [ABSTRACT FROM AUTHOR]

Convexity is a deeply studied concept since it is very useful in many fields of mathematics, like optimization. When we deal with imprecision, the convexity is required as well and some important applications can be found fuzzy optimization, in particular convexity of fuzzy sets. In this paper we have extended the notion of convexity for interval-valued fuzzy sets in order to be able to cover some wider area of imprecision. We show some of its interesting properties, and study the preservation under the intersection and the cutworthy property. Finally, we applied convexity to decision-making problems. [ABSTRACT FROM AUTHOR]

Due to the prior work by Köetter and Kschischang (IEEE Trans Inf Theory 54(8):3579–3591, 2008), subspace codes (especially constant-dimension codes, CDC for short) have attracted significant attention in recent years. The sub-code construction is currently one of the most popular constructions, which can be used to further improve the lower bounds for CDCs' construction. In this paper, we show how to improve the construction of subspace codes from two parallel versions of the sub-code construction. This construction allows us to obtain larger size codes for a given minimum distance, which exceeds the latest improvements on the sub-code construction (Cossidente et al. in Adv Math Commun, 2021, https://doi.org/10.3934/amc.2021007; Lao et al. in Cryptogr Commun, 2021) in the cases including A q (10 , 4 , 5) , A q (14 , 6 , 7) , and A q (16 , 8 , 8) , etc. [ABSTRACT FROM AUTHOR]

Recently, the optimization methods for computing higher-order critical points of nonconvex problems attract growing research interest (Anandkumar Conference on Learning Theory 81-102, 2016), (Cartis Found Comput Math 18:1073-1107, 2018), (Cartis SIAM J Optim 30:513-541, 2020), (Chen Math Program 187:47-78, 2021) , as they are able to exclude the so-called degenerate saddle points and reach a solution with better quality. Despite theoretical developments in (Anandkumar Conference on Learning Theory 81-102, 2016), (Cartis Found Comput Math 18:1073-1107, 2018), (Cartis SIAM J Optim 30:513-541, 2020), (Chen Math Program 187:47-78, 2021) , the corresponding numerical experiments are missing. This paper proposes an implementable higher-order method, named adaptive high order method (AHOM), to find the third-order critical points. AHOM is achieved by solving an "easier" subproblem and incorporating the adaptive strategy of parameter-tuning in each iteration of the algorithm. The iteration complexity of the proposed method is established. Some preliminary numerical results are provided to show that AHOM can escape from the degenerate saddle points, where the second-order method could possibly get stuck. [ABSTRACT FROM AUTHOR]

Research question: This paper examines the overarching question of to what extent different analytic choices may influence the inference about country-specific cross-sectional and trend estimates in international large-scale assessments. We take data from the assessment of PISA mathematics proficiency from the four rounds from 2003 to 2012 as a case study. Methods: In particular, four key methodological factors are considered as analytical choices in the rescaling and analysis of the data: (1) The selection of country sub-samples for item calibration differing at three factor levels. (2) The item sample refering to two sets of mathematics items used within PISA. (3) The estimation method used for item calibration: marginal maximum likelihood estimation method as implemented in R package TAM or an pairwise row averaging approach as implemented in the R package pairwise. (4) The type of linking method: concurrent calibration or separate calibration with successive chain linking. Findings: It turned out that analytical decisions for scaling did affect the PISA outcomes. The factors of choosing different calibration samples, estimation method and linking method tend to show only small effects on the country-specific cross-sectional and trend estimates. However, the selection of different link items seems to have a decisive influence on country ranking and development trends between and within countries. [ABSTRACT FROM AUTHOR]

For rational functions , where is a polynomial of degree at the most and , with we use simple but elegant techniques to strengthen generalizations of certain results which extend some widely known polynomial inequalities of Erdős-Lax and Turán to rational functions . In return these reinforced results, in the limiting case, lead to the corresponding refinements of the said polynomial inequalities. As an illustration and as an application of our results, we obtain some new improvements of the Erdős-Lax and Turán type inequalities for polynomials. These improved results take into account the size of the constant term and the leading coefficient of the given polynomial. As a further factor of consideration, during the course of this paper we will demonstrate how some recently obtained results could have been proved without invoking the results of Dubinin [Distortion theorems for polynomials on the circle, Sb. Math. 191(12) (2000) 1797–1807]. [ABSTRACT FROM AUTHOR]

The concept of Tribonacci sequence spaces by the domain of a regular Tribonacci matrix was introduced by Yaying and Hazarika (Math. Slovaca 70(3):697–706, 2000). In this paper, by using the domain of regular Tribonacci matrix T = (t i k) and the concept of neutrosophic convergence, we introduce some neutrosophic normed space in Tribonacci convergent spaces and prove some topological and algebraic properties based results with respect to these spaces. [ABSTRACT FROM AUTHOR]

The SW26010 many-core processor is based on the Sunway architecture that is composed of management and computing processing elements (MPE and CPE, respectively), each of which is equipped with a stand-alone math library. The issue is that each Sunway Math Library (SML) version is written in assembly which is outside the power of compilers that take high-level languages as input; existing optimization approaches thus mainly rely on manual strategies, which are considered inefficient. In this paper, we leverage the concept of superblock scheduling, a well-known compilation technique, and present a tool named SMPOT to optimize the SML. SMPOT first builds a superblock using a novel tail duplication algorithm, and then uses code motion restrictions to avoid code compensation, followed by matching the machine model. Finally, it reorders instructions on the main path by an activation algorithm based on available computing resources. The experimental results show that SMPOT can effectively improve the performance of the SML. The main path performance of MPE functions is improved by 10.61% on average and overall performance by 5.40%. The main path performance of CPE functions is improved by 5.72% on average and overall performance by 2.98%. [ABSTRACT FROM AUTHOR]

Recently, Horlemann-Trautmann and Weger (Adv Math Commun, https://doi.org/10.3934/amc.2020089, 2020) proposed a general framework for a Quaternary McEliece cryptosystem over the ring Z 4 , and generalized the construction over the ring Z p m where p is a prime integer. By considering the Lee metric, the public key size for the McEliece cryptosystems can be substantially smaller than their counterparts in the Hamming metric. Furthermore, the hardness of the McEliece cryptosystems over Z p m is based on the Lee Syndrome Decoding problem, which was shown to be NP-complete. This paper aims to analyze the design and security of the Lee metric McEliece cryptosystem over Z p m in the Lee metric. We derive some necessary conditions for the quaternary codes used in the Lee metric McEliece cryptosystem, and show that the theoretical quaternary codes proposed in (Adv Math Commun, https://doi.org/10.3934/amc.2020089, 2020) do not exist. Furthermore, we propose a plaintext recovery attack on the Lee metric McEliece cryptosystem over Z p m , when the minimum Lee distance of the underlying codes with certain structures is greater than twice the Lee weight of the error. Our plaintext recovery attack is applicable to the cryptosystems over Z p m when m > 1 . We apply our plaintext recovery attack on the Quaternary McEliece cryptosystem, and manage to recover partial plaintext of length k 1 within O 2 min { k 1 , 0.0473 n } operations, where n is the length of ciphertext. Hence, the proposed theoretical parameters for the Quaternary McEliece over Z 4 in (Adv Math Commun, https://doi.org/10.3934/amc.2020089, 2020) do not achieve 128-bit security, as we can recover the partial plaintext within 0.591 s. This also implies that the use of quaternary codes in the McEliece cryptosystem may not reduce the public key size significantly. Furthermore, for some parameters of the Lee metric McEliece cryptosystems over Z 4 , our plaintext recovery attack can be more efficient than the Lee-Brickell's and Stern's Information Set Decoding Algorithm over Z 4 . [ABSTRACT FROM AUTHOR]

The paper presents results of CFD parametric study of centrifugal compressor stage vane diffusers in the Ansys CFX. The results obtained will be used to build a mathematical model of vane diffuser in optimization design system. Objects of research are vane diffusers with external relative diameter (relative to the diameter of the impeller) equal to 1.5; vane inlet angle of 20°; relative vane heights of 0.025, 0.034, 0.045, 0.06, and 0.08; and vane profile curvature angles of 10, 15, and 20 °. The characteristics of polytrophic efficiency, loss coefficient, recovery coefficient, ratio of inlet and outlet velocities, and flow deviation angle versus incidence angle are set. The analysis of the flow structure in the vane diffuser channels is presented. Unlike with a straight vane cascade, the deviation angle in the circular rows of vane diffusers tends to increase with increasing row density. This may be due to the complex nature of the interaction of the active part of the flow with separation zones. In rows with almost straight vanes at a lower density, the separation zone on the pressure side decreases and even shifts to the very end of the suction side. [ABSTRACT FROM AUTHOR]

This paper has two aspects. Mathematically, in the context of global optimization, it provides the existence of an optimum of a perturbed optimization problem that generalizes the celebrated Ekeland variational principle and equivalent formulations (Caristi, Takahashi), whenever the perturbations need not satisfy the triangle inequality. Behaviorally, it is a continuation of the recent variational rationality approach of stay (stop) and change (go) human dynamics. It gives sufficient conditions for the existence of traps in a changing environment. In this way it emphasizes even more the striking correspondence between variational analysis in mathematics and variational rationality in psychology and behavioral sciences. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of Choquard equations with Hardy–Littlewood–Sobolev lower or upper critical exponent in the whole space R N . We combine an argument of L. Jeanjean and H. Tanaka (see (Proc. Am. Math. Soc. 131:2399–2408, 2003) with a concentration–compactness argument, and then we obtain the existence of ground state solutions, which extends and complements the earlier results. [ABSTRACT FROM AUTHOR]