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The International Sirius Mathematics Center, located on the Black Sea coast, was founded in 2019 by the Talent and Success Foundation for organizing mathematical conferences and other events promoting the development of mathematics in Russia. In 2024, the Center has acquired its own Russian-language journal for publication of original research papers by participants of conferences held at the Center. The authors are invited to submit papers to the journal at https://ef.msp.org/submit/sirius [ABSTRACT FROM AUTHOR]

Example-generation tasks have been suggested as an effective way to both promote students’ learning of mathematics and assess students’ understanding of concepts. E-assessment offers the potential to use example-generation tasks with large groups of students, but there has been little research on this approach so far. Across two studies, we investigate students’ responses to a particular sequence of example-generation tasks, posed either on paper or through e-assessment. We identify a striking difference in students’ example-generation strategies and success, for e-assessment and paper-based versions of the same tasks. This suggests the use of example-generation tasks in e-assessment may not be straightforward, and we conclude by discussing the implications for teaching and research. [ABSTRACT FROM AUTHOR]

This note provides a counterexample to a theorem announced in the last part of the paper (Vicente and Custódio Math Program 133:299–325, 2012). The counterexample involves an objective function f : R → R which satisfies all the assumptions required by the theorem but contradicts some of its conclusions. A corollary of this theorem is also affected by this counterexample. The main flaw revealed by the counterexample is the possibility that a directional direct search method (dDSM) generates a sequence of trial points (x k) k ∈ N converging to a point x ∗ where f is discontinuous, lower semicontinuous and whose objective function value f (x ∗) is strictly less than lim k → ∞ f (x k) . Moreover the dDSM generates trial points in only one of the continuity sets of f near x ∗ . This note also investigates the proof of the theorem to highlight the inexact statements in the original paper. Finally this work introduces a modification of the dDSM that allows, in usual cases, to recover the properties broken by the counterexample. [ABSTRACT FROM AUTHOR]

Grasping mathematical objects as related to processes is often considered critical for mathematics understanding. Yet, the ontology of mathematical objects remains under debate. In this paper, we theoretically oppose internalist approaches that claim mental entities as the endpoints of process–object transitions and externalist approaches that stress mathematical artifacts—such as physical manipulatives and formulas—as constituting mathematical objects. We search for a view on process–object duality that overcomes the dualism of mind and body. One such approach is commognition that describes mathematical objects as discursive entities. This paper expands the nature of mathematical objects beyond discourse and highlights the role of learners' interaction with the environment by adopting ecological onto-epistemology. We develop a functional dynamic systems perspective on process–object duality in mathematics learning emphasizing embodied actions and the re-invention of artifacts' affordances. As a main result, we reconsider process–object duality as a reification of repetitive actions into a cultural artifact that consists of two steps: (1) forming a new sensory-motor coordination that brings new perception to the fore and (2) crystallizing a new artifact in a mathematical environment that captures this new perception. An empirical example from research on embodied action-based design for trigonometry illustrates our theoretical ideas. [ABSTRACT FROM AUTHOR]

In this paper, as an improvement of the paper (Ishige et al. in SIAM J Math Anal 49:2167–2190, 2017), we obtain the higher order asymptotic expansions of the large time behavior of the solution to the Cauchy problem for inhomogeneous fractional diffusion equations and nonlinear fractional diffusion equations. [ABSTRACT FROM AUTHOR]

In Kim et al. (Nagoya Math J 178: 37-53, 2005), Lee and Lee (J Geometry Phys 56(10): 2011-2023, 2006), the authors developed a nice formula to compute the Nielsen number of a self-map on an infra-nilmanifold. For the case of nilmanifolds this formula was extended to n-valued maps in Deconinck and Dekimpe (J Fixed Point Theory Appl 25(4): Paper No. 84, 29, 2023). In this paper, we extend these results further and establish the averaging formula to compute the Nielsen number of any n-valued affine map on an infra-nilmanifold. [ABSTRACT FROM AUTHOR]

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]

We study locally maximal attractors of expanding dynamical systems. Our main result is a representation of these attractors with the help of topological Markov chains corresponding to the Markov partitions of these attractors, which allows us to describe the dynamics of system on them. Ya. G. Sinai was the first who constructed and used Markov partitions for Anosov's diffeomorphisms [Funk. Anal. Prilozh., 2, No 1, 64; No 3, 70 (1968); English translation:Funct. Anal. Appl., 2, No 1, 61; No 3, 245 (1968)]. Expanding endomorphisms regarded as the simplest representatives of endomorphisms were first studied by M. Shub [Amer. J. Math., 91, No 1, 175 (1969)]. To construct Markov partitions for expanding endomorphisms, we update Sinai's approach in the proper way. A more detailed historical overview can be found in the work by O. M. Sharkovsky [Ukr. Mat. Zh., 74, No. 12, 1709 (2023); English translation:Ukr. Math. J., 74, No. 12, 1950 (2023)]. In this work, Sharkovsky indicated that the methods used to prove the main results presented in [Dokl. Akad. Nauk SSSR, 170, No. 6, 1276 (1966); English translation:Sov. Math. Dokl., 7, No. 5, 1384 (1966)] were, in fact, published in the collection of papers "Dynamical systems and the problems of stability of solutions of differential equations" (1973) issued by the Institute of Mathematics of the Academy of Sciences of Ukraine. This collection is difficultly accessible and was never translated into English. Note that, in the indicated paper, these methods were applied to somewhat different objects. To the best of our knowledge, there is no information about publications of similar results. In view of the outlined history and importance of the described approach (based on Markov partitions and topological Markov chains) for the description of construction of the attractors, it seems reasonable to publish these results anew. [ABSTRACT FROM AUTHOR]

Learning to calculate with natural numbers by structuring seems promising but how this can be taught in a sustainable manner remains an open question. An eight-month-long intervention based on the idea of using a structural approach to addition and subtraction, and particularly bridging through ten, was implemented in four Swedish first-grade classes. One goal was that by the end of first grade, students would be able to solve tasks such as subtracting 8 from 15 by using part-whole number relations. In this paper, we report on learning outcomes from task-based interviews with intervention and control groups before, immediately after, and one year after the intervention, in order to investigate long-term effects and whether students used a structural approach when solving tasks in a higher number range in the second grade. In comparison to controls, students in the intervention group showed higher increases in their learning outcomes. Moreover, the intervention group used a structural approach to a larger extent when solving tasks in higher number ranges, whereas students in the control group more commonly used single-unit counting to solve such tasks. These findings have implications both for teaching and for research on students' development of arithmetic skills. [ABSTRACT FROM AUTHOR]

In this paper, we consider how to accurately solve the singular value problem and the linear system for a class of Polynomial-Vandermonde-type (PVT) matrices, which belongs to the class of negative matrices introduced by Huang and Xue (Adv Comput Math 47:73, 2021), and these negative matrices arise in some applications such as interpolation problems. In order to parameterize PVT matrices, we present the explicit expressions of initial minors of such matrices. An algorithm is designed to accurately compute the parametrization matrix for PVT matrices. Based on the accurate parametrization algorithm, all the singular values, both large and small, of PVT matrices are computed, and the linear systems associated with PVT matrices are solved to high relative accuracy. Numerical experiments are performed to confirm the claimed high relative accuracy. [ABSTRACT FROM AUTHOR]

The multi-step inertial randomized Kaczmarz (MIRK) method is an iterative method for solving large-scale linear systems. In this paper, we enhance the MIRK method by incorporating the greedy probability criterion, coupled with the introduction of a tighter threshold parameter for this criterion. We prove that the proposed greedy MIRK (GMIRK) method enjoys an improved deterministic linear convergence compared to both the MIRK method and the greedy randomized Kaczmarz method. Furthermore, we exhibit that the multi-step inertial extrapolation approach can be geometrically interpreted as an orthogonal projection method, and establish its relationship with the sketch-and-project method in Gower and Richtárik (SIAM J Matrix Anal Appl 36(4):1660–1690, 2015) and the oblique projection technique in Li et al. (Results Appl. Math. 16:100342, 2022). Numerical experiments are provided to confirm our results. [ABSTRACT FROM AUTHOR]

The exchange of medical images and patient data over the internet has attracted considerable attention in the past decade, driven by advancements in communication and health services. However, transferring confidential data through insecure channels, such as the internet, exposes it to potential manipulations and attacks. To ensure the authenticity of medical images while concealing patient data within them, this paper introduces a high-capacity and reversible fragile watermarking model in which an authentication watermark is initially generated from the cover image and merged with the patient's information, photo, and medical report to form the global watermark. This watermark is subsequently encrypted using the chaotic Chen system technique, enhancing the model's security and ensuring patient data confidentiality. The cover image then undergoes a Discrete Fourier Transform (DFT) and the encrypted watermark is inserted into the frequency coefficients using a new embedding technique. The experimental results demonstrate that the proposed method achieves great watermarked image quality, with a PSNR exceeding 113 dB and an SSIM close to 1, while maintaining a high embedding capacity of 3 BPP (Bits Per Pixel) and offering perfect reversibility. Furthermore, the proposed model demonstrates high sensitivity to attacks, successfully detecting tampering in all 18 tested attacks, and achieves nearly perfect watermark extraction accuracy, with a Bit Error Rate (BER) of 0.0004%. This high watermark extraction accuracy is crucial in our situation where patient data need to be retrieved from the watermarked images with almost no alteration. [ABSTRACT FROM AUTHOR]

This paper studies the Liouville type theorems for stationary the tropical climate model on the whole space R 3 . It shows that if (u , v , θ) satisfies certain anisotropic integrability conditions on the components of the u or (u , v) , also θ satisfies certain isotropic integrability conditions, there is only a trivial solution to the stationary tropical climate model. The results are a further extension of the recent work by Chae (Appl. Math. Lett. 142:108655, 2023). [ABSTRACT FROM AUTHOR]

It is known that the so-called Beck's conjecture, i.e. the equality of the finite clique and chromatic numbers of a zero-divisor graph, holds for partially ordered sets Halaš and Jukl (Discrete Math 309(13):4584–4589, 2009). In this paper we present a simple direct proof of this fact. Also, the case when the finiteness assumption of the clique number is omitted is investigated. We have shown that the conjecture fails in general and a bunch of counterexamples is presented. [ABSTRACT FROM AUTHOR]

From a physiological perspective, the delineation between steady-state and non-steady-state exercise, also referred to as the maximal metabolic steady state, holds paramount importance for evaluating athletic performance and designing and monitoring training programs. The critical power and the maximal lactate steady state are two widely used indices to estimate this threshold, yet previous studies consistently reported significant discrepancies between their associated power outputs. These findings have fueled the debate regarding the interchangeability of critical power and the maximal lactate steady state in practice. This paper reviews the methodological intricacies intrinsic to the determination of these thresholds, and elucidates how inappropriate determination methods and methodological inconsistencies between studies have contributed to the documented differences in the literature. Through a critical examination of relevant literature and by integration of our laboratory data, we demonstrate that differences between critical power and the maximal lactate steady state may be reconciled to only a few Watts when applying appropriate and strict determination criteria, so that both indices may be used to estimate the maximal metabolic steady-state threshold in practice. To this end, we have defined a set of good practice guidelines to assist scientists and coaches in obtaining the most valid critical power and maximal lactate steady state estimates. [ABSTRACT FROM AUTHOR]

In 1967, Grünbaum conjectured that any d-dimensional polytope with d + s ⩽ 2 d vertices has at least ϕ k (d + s , d) = d + 1 k + 1 + d k + 1 - d + 1 - s k + 1 k-faces. This conjecture along with the characterization of equality cases was recently proved by the author (A proof of Grünbaum's lower bound conjecture for general polytopes. Israel J. Math. 245(2), 991–1000 (2021)). In this paper, several extensions of this result are established. Specifically, it is proved that lattices with the diamond property (e.g., abstract polytopes) and d + s ⩽ 2 d atoms have at least ϕ k (d + s , d) elements of rank k + 1 . Furthermore, in the case of face lattices of strongly regular CW complexes representing normal pseudomanifolds with up to 2d vertices, a characterization of equality cases is given. Finally, sharp lower bounds on the number of k-faces of strongly regular CW complexes representing normal pseudomanifolds with 2 d + 1 vertices are obtained. These bounds are given by the face numbers of certain polytopes with 2 d + 1 vertices. [ABSTRACT FROM AUTHOR]

We consider a C 0 -semigroup on a Banach space such that the resolvent is uniformly bounded on the right half-plane. In this paper we provide a condition on the resolvent which is sufficient and necessary for the uniform exponential stability of such a semigroup. As a consequence, we give an alternative proof of Gearhart's theorem (Trans. Amer. Math. Soc. 236, 385–394 (1978)). The approach lies on a complex inversion formula and tempered distributions. [ABSTRACT FROM AUTHOR]

This paper presents a comprehensive study of a model called the (2 + 1) q -deformed tanh-Gordon model. This model is particularly useful for studying physical systems with violated symmetries, as it provides insights into their behavior. To solve the (2 + 1) q -deformed equation for specific parameter values, the (H + G ′ G 2) -expansion approach is employed. This technique generates analytical solutions that reveal valuable information about the system's dynamics and behavior. These solutions offer insights into the underlying mathematics and deepen the understanding of the system's properties. To validate the accuracy of the analytical solutions, the finite difference technique is also used to find a numerical solution to the q -deformed equation. This numerical approach ensures the correctness of the solutions and enhances the reliability of the results. Tables and graphics are presented in the publication to aid comprehension and comparison. These visuals improve the clarity and interpretability of the data, allowing readers to better understand the similarities and differences between the analytical and numerical solutions. [ABSTRACT FROM AUTHOR]

In the paper, we consider a system of thermoelasticity of type I with viscoelastic damping and nonlinear sources. By using the Galerkin method and the Banach fixed point theorem, we first prove the local existence and uniqueness of weak solution. Secondly, by extending the potential well method, we prove that the local solution exists globally if its initial position starts inside a family of "potential wells." In particular, we also establish an explicit and optimal decay rate of energy driven by the decay rate of the relaxation function which includes exponential, algebraic, and logarithmic decay rates. Finally, by using the continuation theorem and the concavity arguments due to Levine (Trans Am Math Soc 192:1–21, 1974), we show that the local solution blows up at finite time in the sense of Ball (Q J Math Oxf 28(4): 473–486, 1977) if its initial position starts outside the "potential wells." Further, an upper bound for the blow-up time is also given explicitly. Notice that our results imply a sharp result on the global existence and blow-up of the local weak solution and they also allow a relatively large class of relaxation functions that generalize the existing results in the literature. [ABSTRACT FROM AUTHOR]

We prove the almost sure invariance principle (ASIP) with close to optimal error rates for nonuniformly hyperbolic maps. We do not assume exponential contraction along stable leaves, therefore our result covers in particular slowly mixing invertible dynamical systems as Bunimovich flowers, billiards with flat points as in Chernov and Zhang (Stoch Dyn 5:535–553, 2005a, Nonlinearity 18:1527–1553, 2005b) and Wojtkowski' (Commun Math Phys 126:507–533, 1990) system of two falling balls.For these examples, the ASIP is a new result, not covered by prior works for various reasons, notably because in absence of exponential contraction along stable leaves, it is challenging to employ the so-called Sinai's trick (Sinai in Russ Math Surv 27:21–70, 1972; Bowen, Lecture Notes in Math vol. 470 (1975)) of reducing a nonuniformly hyperbolic system to a nonuniformly expanding one. Our strategy follows our previous papers on the ASIP for nonuniformly expanding maps, where we build a semiconjugacy to a specific renewal Markov shift and adapt the argument of Berkes et al. (Ann Probab 42:794–817, 2014). The main difference is that now the Markov shift is two-sided, the observables depend on the full trajectory, both the future and the past. [ABSTRACT FROM AUTHOR]

By introducing two parameters in the splittings of the matrices A and B, this paper presents a parameterized accelerated iteration (PAI) method for solving the matrix equation A X B = C . The convergence property of the PAI method and the choices of the parameters are thoroughly investigated. Additionally, based on some special splittings of the matrices A and B, several variants of the PAI method are established. Furthermore, for some certain cases, the optimal parameters can be determined, and it is demonstrated that the PAI method is more efficient than the gradient-based iteration (GBI) method (Ding et al. Appl. Math. Comput. 197, 41–50 2008). Finally, by comparing it with several existing iteration methods, the effectiveness of the PAI method is verified through four numerical examples. [ABSTRACT FROM AUTHOR]

In 2015, Mao (J. Comput. Appl. Math., 290, 370–384, 2015) proposed the truncated Euler-Maruyama (EM) method for stochastic differential equations (SDEs) under the local Lipschitz condition plus the Khasminskii-type condition. Adapting the truncation idea from Mao (J. Comput. Appl. Math., 290, 370–384, 2015) and Mao (Appl. Numer. Math., 296, 362–375, 2016), lots of modified truncated EM methods are proposed (see, e.g., Guo et al. (Appl. Numer. Math., 115, 235–251, 2017,) and Lan and Xia (J. Comput. Appl. Math., 334, 1–17, 2018) and Li et al. (IMA J. Numer. Anal., 39(2), 847–892, 2019) and the references therein). These truncated-type EM methods Mao (J. Comput. Appl. Math., 290, 370–384, 2015) and Mao (Appl. Numer. Math., 296, 362–375, 2016) and Guo et al. (Appl. Numer. Math., 115, 235–251, 2017,) and Lan and Xia (J. Comput. Appl. Math., 334, 1–17, 2018) and Li et al. (IMA J. Numer. Anal., 39(2), 847–892, 2019) construct the numerical solutions by defining an appropriate truncation projection, then applying the truncation projection to the numerical solutions before substituting them into the coefficients in each iteration. In this paper, we develop a new class of explicit schemes for superlinear stochastic differential equations with piecewise continuous arguments (SDEPCAs), which are defined by directly truncating the coefficients. Our method has a more simple structure and is easier to implement. We not only show the explicit schemes converge strongly to SDEPCAs but also demonstrate the convergence rate is optimal 1/2. A numerical example is provided to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we use two approaches to define the concept of fuzzy nominal sets: classic and universal algebraic. We see that the fuzzy nominal sets obtained using the universal algebraic approach (so-called fuzzy nominal sets) are within finitely supported mathematics, whereas the fuzzy nominal sets derived using the classical approach (so-called fuzzy nominal ν supp -sets) are within ordinary mathematics and each fuzzy nominal set can be considered as a fuzzy nominal ν supp -set. We also go over the presheaf representation of fuzzy nominal sets and some other properties of these various types of fuzzy nominal sets. [ABSTRACT FROM AUTHOR]

As a quantum affinization, the quantum toroidal algebra U q , q ¯ (gl ¨ n) is defined in terms of its "left" and "right" halves, which both admit shuffle algebra presentations (Enriquez in Transform Groups 5(2):111–120, 2000; Feigin and Odesskii in Am Math Soc Transl Ser 2:185, 1998). In the present paper, we take an orthogonal viewpoint, and give shuffle algebra presentations for the "top" and "bottom" halves instead, starting from the evaluation representation U q (gl ˙ n) ↷ C n (z) and its usual R-matrix R (z) ∈ End (C n ⊗ C n) (z) (see Faddeev et al. in Leningrad Math J 1:193–226, 1990). An upshot of this construction is a new topological coproduct on U q , q ¯ (gl ¨ n) which extends the Drinfeld–Jimbo coproduct on the horizontal subalgebra U q (gl ˙ n) ⊂ U q , q ¯ (gl ¨ n) . [ABSTRACT FROM AUTHOR]

In this paper, we use semidefinite programming and representation theory to compute new lower bounds on the crossing number of the complete bipartite graph K m , n , extending a method from de Klerk et al. (SIAM J Discrete Math 20:189–202, 2006) and the subsequent reduction by De Klerk, Pasechnik and Schrijver (Math Prog Ser A and B 109:613–624, 2007). We exploit the full symmetry of the problem using a novel decomposition technique. This results in a full block-diagonalization of the underlying matrix algebra, which we use to improve bounds on several concrete instances. Our results imply that cr (K 10 , n) ≥ 4.87057 n 2 - 10 n , cr (K 11 , n) ≥ 5.99939 n 2 - 12.5 n , cr (K 12 , n) ≥ 7.25579 n 2 - 15 n , cr (K 13 , n) ≥ 8.65675 n 2 - 18 n for all n. The latter three bounds are computed using a new and well-performing relaxation of the original semidefinite programming bound. This new relaxation is obtained by only requiring one small matrix block to be positive semidefinite. [ABSTRACT FROM AUTHOR]

This paper is concerned with the spreading behavior of a two-species strong-weak competition system with two free boundaries. The model may describe how a strong competing species invades into the habitat of a native weak competing species. The asymptotic spreading speed of invading fronts has been determined by making use of semi-wave systems in Du et al. (J Math Pures Appl 107:253–287, 2017). Here we give a sharp estimate for the asymptotic spreading speed of invading fronts. Moreover, we prove that the solution of the free boundary problem evolves eventually into a semi-wave solution when the spreading happens, while the solution of the free boundary problem exponentially converges to a semi-trivial solution of such system when the vanishing happens. [ABSTRACT FROM AUTHOR]

In the present paper, we establish an infinite dimensional Kolmogorov–Arnold–Moser (KAM) theorem for reversible systems with double normal frequencies. Applying it, we prove the existence of quasi-periodic solutions for one dimensional coupled nonlinear quantum harmonic oscillators (QHO) with a natural reversible structure. To compensate the lack of smoothing effect of perturbation, we introduce a class of vector fields with polynomial decay which extends the works of Grébert and Thomann (Commun Math Phys 307(2):383–427, 2011) for Hamiltonian QHO. To deal with the reversible, coupled perturbations in the equations, we also introduce a new class of generating vector fields during the KAM iteration. Moreover, the quasi-periodic solutions we obtain may not be linearly stable. This is obviously different from the result in Grébert and Thomann (2011) for Hamiltonian QHO. [ABSTRACT FROM AUTHOR]

In this paper, we explore the version of Hairer's regularity structures based on a greedier index set than trees, as introduced in (Otto et al. in A priori bounds for quasi-linear SPDEs in the full sub-critical regime, 2021, arXiv:2103.11039) and algebraically characterized in (Linares et al. in Comm. Am. Math. Soc. 3:1–64, 2023). More precisely, we construct and stochastically estimate the renormalized model postulated in (Otto et al. in A priori bounds for quasi-linear SPDEs in the full sub-critical regime, 2021, arXiv:2103.11039), avoiding the use of Feynman diagrams but still in a fully automated, i. e. inductive way. This is carried out for a class of quasi-linear parabolic PDEs driven by noise in the full singular but renormalizable range. We assume a spectral gap inequality on the (not necessarily Gaussian) noise ensemble. The resulting control on the variance of the model naturally complements its vanishing expectation arising from the BPHZ-choice of renormalization. We capture the gain in regularity on the level of the Malliavin derivative of the model by describing it as a modelled distribution. Symmetry is an important guiding principle and built-in on the level of the renormalization Ansatz. Our approach is analytic and top-down rather than combinatorial and bottom-up. [ABSTRACT FROM AUTHOR]

This note tries to answer issues raised in Bhardwaj and Kumar (J Optim Theory Appl 163(2): 685-696, 2014). The research summarizes that the results obtained in Khan et al. (J Optim Theory Appl 159: 536-546, 2013) are sound and correct and it fulfills all the necessary requirements of its scope and objectives. [ABSTRACT FROM AUTHOR]

In the first part of the paper, we propose a different viewpoint on the theory of higher Airy structures (or Airy ideals), which may shed light on its origin. We define Airy ideals in the ħ -adic completion of the Rees Weyl algebra and show that Airy ideals are defined exactly such that they are always related to the canonical left ideal generated by derivatives by automorphisms of the Rees Weyl algebra of a simple type, which we call transvections. The standard existence and uniqueness result in the theory of Airy structures then follow immediately. In the second part of the paper, we construct Airy ideals generated by the nonnegative modes of the strong generators of the principal W -algebra of sp 2 N at level - N - 1 / 2 , following the approach developed in Borot et al. (Mem Am Math Soc, 2021). This provides an example of an Airy ideal in the Heisenberg algebra that requires realizing the zero modes as derivatives instead of variables, which leads to an interesting interpretation for the resulting partition function. [ABSTRACT FROM AUTHOR]

In this paper, we study the Z / 2 action on complex Grassmann manifolds G n (C 2 n) given by taking orthogonal complement. We completely compute the associated Z / 2 Fadell–Husseini index. Our study is parallel to the study of the index of real Grassmann manifolds G n (R 2 n) by Baralić et al. [Forum Math., 30 (2018), pp. 1539–1572]. [ABSTRACT FROM AUTHOR]

The mechanism of cytoplasmic incompatibility (CI) is important in the study of Wolbachia invasion in wild mosquitoes. Su et al. (Bull Math Biol 84(9):95, 2022) proposed a delay differential equation model by relating the CI effect to maturation delay. In this paper, we investigate the dynamics of this model by allowing the same density-dependent death rate and distinct density-independent death rates. Through analyzing the existence and stability of equilibria, we obtain the parameter conditions for Wolbachia successful invasion if the maternal transmission is perfect. While if the maternal transmission is imperfect, we give the ranges of parameters to ensure failure invasion, successful invasion and partially suppressing, respectively. Meanwhile, numerical simulations indicate that the system may exhibit monostable and bistable dynamics when parameters vary. Particularly, in the bistable situation an unstable separatrix, like a line, exists when choosing constant functions as initial values; and the maturation delay affects this separatrix in an interesting way. [ABSTRACT FROM AUTHOR]

Successful early mathematical development is vital to children’s later education, employment, and wellbeing outcomes. However, established measurement tools are infrequently used to (i) assess children’s mathematical skills and (ii) identify children with or at-risk of mathematical learning difficulties. In response, this pre-registered systematic review aimed to provide an overview of measurement tools that have been evaluated for their psychometric properties for measuring the mathematical skills of children aged 0–8 years. The reliability and validity evidence reported for the identified measurement tools were then synthesised, including in relation to common acceptability thresholds. Overall, 41 mathematical assessments and 25 screeners were identified. Our study revealed five main findings. Firstly, most measurement tools were categorised as child-direct measures delivered individually with a trained assessor in a paper-based format. Secondly, the majority of the identified measurement tools have not been evaluated for aspects of reliability and validity most relevant to education measures, and only 15 measurement tools met the common acceptability thresholds for more than two areas of psychometric evidence. Thirdly, only four screeners demonstrated an acceptable ability to distinguish between typically developing children and those with or at-risk of mathematical learning difficulties. Fourthly, only one mathematical assessment and one screener met the common acceptability threshold for predictive validity. Finally, only 11 mathematical assessments and one screener were found to concurrently align with other validated measurement tools. Building on this current evidence and improving measurement quality is vital for raising methodological standards in mathematical learning and development research. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the boundedness of solutions of the Ginzburg–Landau system where and is the subelliptic operator In the stationary case, where the solutions are independent of the time variable, our result can be seen as an extension of some results in [A. Farina, B. Sciunzi, and N. Soave, Commun. Contemp. Math. 22 (5), Article no. 1950044 (2020)] from the Laplace operator to the subelliptic operator . [ABSTRACT FROM AUTHOR]

This paper is concerned with the study of a nonlinear non-local equation that has a commutator structure. The equation reads ∂ t u − F (u) (− Δ) s / 2 u + (− Δ) s / 2 (u F (u)) = 0 , x ∈ T d , with s ∈ (0 , 1 ] . We are interested in solutions stemming from periodic positive bounded initial data. The given function F ∈ C ∞ (R +) must satisfy F ′ > 0 a.e. on (0 , + ∞) . For instance, all the functions F (u) = u n with n ∈ N ∗ are admissible non-linearities. The local theory can also be developed on the whole space, however the most complete well-posedness result requires the periodic setting. We construct global classical solutions starting from smooth positive data, and global weak solutions starting from positive data in L ∞ . We show that any weak solution is instantaneously regularized into C ∞ . We also describe the long-time asymptotics of all solutions. Our methods follow several recent advances in the regularity theory of parabolic integro-differential equations, in particular (Ann. Fac. Sci. Toulouse, Math. 25(4):723–758, 2016; Ann. Fac. Sci. Toulouse, Math. 27(4):667–677, 2018). [ABSTRACT FROM AUTHOR]

Solving large-scale systems of nonlinear equations (SoNE) is a central task in mathematics that traverses different areas of applications. There are several derivative-free methods for finding SoNE solutions. However, most of the methods contributed to find SoNE solutions involve a monotone cost function. Methods dealing with pseudomonotone cost function remain rare. In this paper, we introduce two classes of derivative-free spectral three-term methods to solve large-scale continuous pseudomonotone SoNE. We combine the projection method of Solodov and Svaiter with the structure of the recently developed spectral three-term conjugate gradient method for unconstrained optimization by Amini and Faramarzi. We prove that the proposed methods possess sufficient descent property, trust region property, and global convergence without relying on Lipschitz continuity. Numerical experiments show that the proposed methods are efficient and competitive with existing methods. Finally, the proposed methods have been successfully applied to recover a sparse signal from incomplete and contaminated sampling measurements, yielding promising results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some versions of relative connectedness of subspaces of a topological space and we give some facts and relations among them. We prove that these relative versions satisfy some of the classical properties of connectedness. Additionally, we apply our results to the theory of hyperspaces, aiming to address a general problem posed by Arhangel'skii (Comment Math Univ Carolin 36:305–325, 1995, Problem 3). [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of transmission system for coupled Kirchhoff plates, where one equation is conserved and the other has dissipative property, and the dissipation mechanism is given by fractional damping (- Δ) 2 θ v t with θ ∈ [ 1 2 , 1 ] . By using the semigroup method and the multiplier technique, we obtain the exact polynomial decay rates, and find that the polynomial decay rate of the system is determined by the inertia/elasticity ratios and the fractional damping order. Specifically, when the inertia/elasticity ratios are not equal and θ ∈ [ 1 2 , 3 4 ] , the polynomial decay rate of the system is t - 1 / (10 - 4 θ) . When the inertia/elasticity ratios are not equal and θ ∈ [ 3 4 , 1 ] , the polynomial decay rate of the system is t - 1 / (4 + 4 θ) . When the inertia/elasticity ratios are equal, the polynomial decay rate of the system is t - 1 / (4 + 4 θ) . Furthermore it has been proven that the obtained decay rates are all optimal. The obtained results extend the results of Oquendo and Suárez (Z Angew Math Phys 70(3):88, 2019) for the case of fractional damping exponent 2 θ from [0, 1] to [1, 2]. [ABSTRACT FROM AUTHOR]

This paper is concerned with the study of a nonlinear non-local equation that has a commutator structure. The equation reads ∂ t u − F (u) (− Δ) s / 2 u + (− Δ) s / 2 (u F (u)) = 0 , x ∈ T d , with s ∈ (0 , 1 ] . We are interested in solutions stemming from periodic positive bounded initial data. The given function F ∈ C ∞ (R +) must satisfy F ′ > 0 a.e. on (0 , + ∞) . For instance, all the functions F (u) = u n with n ∈ N ∗ are admissible non-linearities. The local theory can also be developed on the whole space, however the most complete well-posedness result requires the periodic setting. We construct global classical solutions starting from smooth positive data, and global weak solutions starting from positive data in L ∞ . We show that any weak solution is instantaneously regularized into C ∞ . We also describe the long-time asymptotics of all solutions. Our methods follow several recent advances in the regularity theory of parabolic integro-differential equations, in particular (Ann. Fac. Sci. Toulouse, Math. 25(4):723–758, 2016; Ann. Fac. Sci. Toulouse, Math. 27(4):667–677, 2018). [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]

This paper proposes a framework for constructing high-order linearly implicit exponential integrators that conserve a quadratic invariant. This is then applied to the scalar auxiliary variable (SAV) approach. Quadratic invariants are significant objects that are present in various physical equations and also in computationally efficient conservative schemes for general invariants. For instance, the SAV approach converts the invariant into a quadratic form by introducing scalar auxiliary variables, which have been intensively studied in recent years. In this vein, Sato et al. (Appl. Numer. Math. 187, 71-88 2023) proposed high-order linearly implicit schemes that conserve a quadratic invariant. In this study, it is shown that their method can be effectively merged with the Lawson transformation, a technique commonly utilized in the construction of exponential integrators. It is also demonstrated that combining the constructed exponential integrators and the SAV approach yields schemes that are computationally less expensive. Specifically, the main part of the computational cost is the product of several matrix exponentials and vectors, which are parallelizable. Moreover, we conduct some mathematical analyses on the proposed schemes. [ABSTRACT FROM AUTHOR]

Telescopers for a function are linear differential (resp. difference) operators annihilating the definite integral (resp. definite sum) of this function. They play a key role in Wilf–Zeilberger theory and algorithms for computing them have been extensively studied in the past 30 years. In this paper, we introduce the notion of telescopers for differential forms with D-finite function coefficients. These telescopers appear in several areas of mathematics, for instance parametrized differential Galois theory and mirror symmetry. We give a sufficient and necessary condition for the existence of telescopers for a differential form and describe a method to compute them if they exist. Algorithms for verifying this condition are also given. [ABSTRACT FROM AUTHOR]

Semidefinite programming (SDP) problems typically utilize a constraint of the form X ⪰ x x T to obtain a convex relaxation of the condition X = x x T , where x ∈ R n . In this paper, we consider a new hyperplane branching method for SDP based on using an eigenvector of X - x x T . This branching technique is related to previous work of Saxeena et al. (Math Prog Ser B 124:383–411, 2010, https://doi.org/10.1007/s10107-010-0371-9) who used such an eigenvector to derive a disjunctive cut. We obtain excellent computational results applying the new branching technique to difficult instances of the two-trust-region subproblem. [ABSTRACT FROM AUTHOR]

Between 1873 and 1876, Felix Klein published a series of papers that he later placed under the rubric anschauliche Geometrie in the second volume of his collected works (1922). The present study attempts not only to follow the course of this work, but also to place it in a larger historical context. Methodologically, Klein's approach had roots in Poncelet's principle of continuity, though the more immediate influences on him came from his teachers, Plücker and Clebsch. In the 1860s, Clebsch reworked some of the central ideas in Riemann's theory of Abelian functions to obtain complicated results for systems of algebraic curves, most published earlier by Hesse and Steiner. These findings played a major role in enumerative geometry, whereas Plücker's work had a strongly qualitative character that imbued Klein's early studies. A leitmotif in these works can be seen in the interplay between real curves and surfaces as reflected by their transformational properties. During the early 1870s, Klein and Zeuthen began to explore the possibility of deriving all possible forms for real cubic surfaces as well as quartic curves. They did so using continuity methods reminiscent of Poncelet's earlier approach. Both authors also relied on visual arguments, which Klein would later advance under the banner of intuitive geometry (anschauliche Geometrie). [ABSTRACT FROM AUTHOR]

In the classical bargaining problem, we propose a very mild axiom of individual rationality, which we call possibility of utility gain. This requires that for at least one bargaining problem, there exists at least one player who reaches a higher utility level than their disagreement utility. This paper shows that the Nash solution (Nash in Econometrica 18(2):155–162, 1950) is characterized by possibility of utility gain and continuity with respect to feasible sets together with Nash's axioms except weak Pareto optimality. We also show that in Nash's theorem, weak Pareto optimality can be replaced by conflict-freeness (introduced by Rachmilevitch in Math Soc Sci 76(C):107–109, 2015). This demands that when the agreement most preferred by all players is feasible, this should be chosen. Furthermore, we provide alternative and unified proofs for other efficiency-free characterizations of the Nash solution. This clarifies the role of each axiom in the related results. [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 construct center stable manifolds of unstable line solitary waves for the Zakharov–Kuznetsov equation on R × T L and show the orbital stability of the unstable line solitary waves on the center stable manifolds, which yields the asymptotic stability of unstable solitary waves on the center stable manifolds near by stable line solitary waves. The construction is based on the graph transform approach by Nakanishi and Schlag (SIAM J Math Anal 44:1175–1210, 2012). Applying the bilinear estimate on Fourier restriction spaces by Molinet and Pilod (Ann Inst H Poincaré Anal Non Lineaire 32:347–371, 2015) and modifying the mobile distance in Nakanishi and Schlag (2012), we construct a contraction map on the graph space. [ABSTRACT FROM AUTHOR]

This note corrects an error in our paper Anosov flows, growth rates on covers and group extensions of subshifts, Invent. Math. 223:445–483, 2021. This leaves our main results, Theorem 1.1, Corollary 1.2, Theorem 1.3 and Theorem 5.1, unchanged. We also fill a gap in the arguments presented in Sect. 9; this requires a small modification to the results in this section. [ABSTRACT FROM AUTHOR]

Based on the expectancy-value perspective on identity and identity formation, this paper explores the relationship between math identity (MI) and the dimensions of motivation (i.e. intrinsic value, attainment value, utility value and perceived competence) and math achievement in primary school. An additional aim of our research was to explore these relationships in different cultural contexts and investigate potential gender and grade differences concerning MI. The participants were 11,782 primary school students from Norway, Sweden, Estonia, Finland, Portugal and Serbia. All predictors from the motivation spectrum were significant for students' MI across the examined countries and had a stronger association with MI than math achievement. Among the motivational dimensions, intrinsic value had the strongest association with students' MI. Boys had significantly more positive math identities than girls in Estonia, Finland, Norway and Portugal. The results showed that the grade 4 students perceived themselves less as "math persons" than their grade 3 peers in all countries. [ABSTRACT FROM AUTHOR]