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In the so-called generalized Turán problems we study the largest number of copies of H in an n-vertex F-free graph G. Here we introduce a variant, where F is not forbidden, but we restrict how copies of H and F can be placed in G. More precisely, given an integer n and graphs H and F, what is the largest number of copies of H in an n-vertex graph such that the vertex set of that copy does not contain and is not contained in the vertex set of a copy of F? We solve this problem for some instances, give bounds in other instances, and we use our results to determine the generalized Turán number for some pairs of graphs. [ABSTRACT FROM AUTHOR]

We introduce some generalisations of the Euler–Kronecker constant of a number field and study their arithmetic nature. [ABSTRACT FROM AUTHOR]

Let $({\cal{A}},{\cal{E}})$ be an exact category. We establish basic results that allow one to identify sub(bi)functors of ${\operatorname{Ext}}_{\cal{E}}(-,-)$ using additivity of numerical functions and restriction to subcategories. We also study a small number of these new functors over commutative local rings in detail and find a range of applications from detecting regularity to understanding Ulrich modules. [ABSTRACT FROM AUTHOR]

Occurrence of anastomotic biliary stricture (AS) remains an essential issue following hepatobiliary surgeries, and percutaneous transhepatic cholangioscopy (PTCS) has great therapeutic significance in handling refractory AS for patients with altered gastrointestinal anatomy after cholangio-jejunostomy. This present study aimed to investigate feasibility of PTCS procedures in AS patients for therapeutic indications. This study was a single-center, retrospective cohort study with a total number of 124 consecutive patients who received therapeutic PTCS due to AS. Clinical success rate, required number, and adverse events of therapeutic PTCS procedures as well as patients survival state were reviewed. These 124 patients previously underwent choledochojejunostomy or hepatico-jejunostomy, and there was post-surgical altered gastrointestinal anatomy. Overall, 366 therapeutic PTCS procedures were performed for these patients through applying rigid choledochoscope, and the median time of PTCS procedures was 3 (1–11). Among these patients, there were 34 cases (27.32%) accompanied by biliary strictures and 100 cases (80.65%) were also combined with biliary calculi. After therapeutic PTCS, most patients presented with relieved clinical manifestations and improved liver functions. The median time of follow-up was 26 months (2–86 months), and AS was successfully managed through PTCS procedures in 104 patients (83.87%). During the follow-up period, adverse events occurred in 81 cases (65.32%), most of which were tackled through supportive treatment. PTCS was a feasible, safe and effective therapeutic modality for refractory AS, which may be a promising alternative approach in clinical cases where the gastrointestinal anatomy was changed after cholangio-jejunostomy. • PTCS provide possibilities for patients complicated with biliary complications. • Most cases presented with changed gastrointestinal anatomy after cholangio-jejunostomy. • PTCS is a feasible, safe and effective modality for treating refractory AS. [ABSTRACT FROM AUTHOR]

We study the equivalence between the infinitesimal Torelli theorem for smooth hypersurfaces in rational homogeneous varieties with Picard number 1 and the theory of generalized Massey products. This equivalence shows that the differential of the period map vanishes on an infinitesimal deformation if and only if certain twisted differential forms are elements of the Jacobian ideal of the hypersurface. We also prove an infinitesimal Torelli theorem result for smooth hypersurfaces in log parallelizable varieties. [ABSTRACT FROM AUTHOR]

A quadratic form is universal if it represents all totally positive integers in a number field. In classical number theory over the rational integers there are some results where a quadratic form takes on all positive integer values except for a certain restricted set. Here we show that one of the quaternary quadratic forms proven universal for ℚ(√5) takes on all totally positive values in ℚ(√2) except for a certain infinite set. The exceptional integers have even norm and the highest power of 2 dividing that norm is 2 to the first power. [ABSTRACT FROM AUTHOR]

Fractal geometry plays an important role in the description of the characteristics of nature. Local fractional calculus, a new branch of mathematics, is used to handle the non-differentiable problems in mathematical physics and engineering sciences. The local fractional inequalities, local fractional ODEs and local fractional PDEs via local fractional calculus are studied. Fractional calculus is also considered to express the fractal behaviors of the functions, which have fractal dimensions. The interesting problems from fractional calculus and fractals are reported. With the scaling law, the scaling-law vector calculus via scaling-law calculus is suggested in detail. Some special functions related to the classical, fractional, and power-law calculus are also presented to express the Kohlrausch–Williams–Watts function, Mittag-Leffler function and Weierstrass–Mandelbrot function. They have a relation to the ODEs, PDEs, fractional ODEs and fractional PDEs in real-world problems. Theory of the scaling-law series via Kohlrausch–Williams–Watts function is suggested to handle real-world problems. The hypothesis for the tempered Xi function is proposed as the Fractals Challenge, which is a new challenge in the field of mathematics. The typical applications of fractal geometry are proposed in real-world problems. [ABSTRACT FROM AUTHOR]

We consider finite two-player normal form games with random payoffs. Player A's payoffs are i.i.d. from a uniform distribution. Given p ∈ [ 0 , 1 ] , for any action profile, player B's payoff coincides with player A's payoff with probability p and is i.i.d. from the same uniform distribution with probability 1 − p. This model interpolates the model of i.i.d. random payoff used in most of the literature and the model of random potential games. First we study the number of pure Nash equilibria in the above class of games. Then we show that, for any positive p , asymptotically in the number of available actions, best response dynamics reaches a pure Nash equilibrium with high probability. [ABSTRACT FROM AUTHOR]

We fix a number field K and study statistical properties of the ring K [ γ ] ∩ K as γ varies over algebraic numbers of a fixed degree n ≥ 2. Given k ≥ 1 , we explicitly compute the density of γ for which K [ γ ] ∩ K = K [ 1 / k ] and show that this does not depend on the number field K. In particular, we show that the density of γ for which K [ γ ] ∩ K = K is ζ (n + 1) ζ (n) . In a recent paper [Singhal and Lin, Primes in denominators of algebraic numbers, Int. J. Number Theory (2023), doi:10.1142/S1793042124500167], the authors define X (K , γ) to be a certain finite subset of Spec ( K) and show that X (K , γ) determines the ring K [ γ ] ∩ K. We show that if 1 , 2 ∈ Spec ( K) satisfy 1 ∩ ℤ ≠ 2 ∩ ℤ , then the events 1 ∈ X (K , γ) and 2 ∈ X (K , γ) are independent. As t → ∞ , we study the asymptotics of the density of γ for which | X (K , γ) | = t. [ABSTRACT FROM AUTHOR]

We point out that there is an error in the proof of Theorem 1.1 in [The discriminant of compositum of algebraic number fields, Int. J. Number Theory 15 (2019) 353–360]. We also prove that the result of this theorem holds with an additional hypothesis. However, it is an open problem whether the result of the theorem is true in general or not. [ABSTRACT FROM AUTHOR]

A Bernoulli scheme with unequal harmonic success probabilities is investigated, together with some of its natural extensions. The study includes the number of successes over some time window, the times to (between) successive successes and the time to the first success. Large sample asymptotics, statistical parameter estimation, and relations to Sibuya distributions and Yule–Simon distributions are discussed. This toy model is relevant in several applications including reliability, species sampling problems, record values breaking and random walks with disasters. [ABSTRACT FROM AUTHOR]

The Langlands program is a vast mathematical projection linking number theory and geometry. In high-energy physics, a connection with mirror symmetry has been suggested in string theory, but it has been little studied in low-energy physics. In the framework of the Langlands program, we present a unified description of the integer and fractional quantum Hall effect and the duality found in the fractal nature of the energy spectrum of two-dimensional Bloch electrons, statistical physics, and quantum computation. The new unified view of existing dualism presented in this paper raises the entirely new question of how each theory of physics is connected as a piece of the Langlands program. [ABSTRACT FROM AUTHOR]

The unique organizational framework for polycyclic aromatic hydrocarbons developed by us has led to the discovery of number patterns for their number of isomers. Constant-isomer series are generated by repetitive circumscribing of a given set of isomers. Benzenoid, fluoranthenoid, indacenoid, and primitive coronoid constant isomer series possess twin isomer numbers with members having corresponding topologies. The concepts of strictly pericondensed, strain-free, Clar's aromatic sextet, and symmetry are interconnected in the topological correspondence between strictly pericondensed and total resonant sextet (TRS) benzenoid hydrocarbons. In a plot of TRS isomer numbers on the Formula Periodic Table for Total Resonant Sextet Benzenoids [Table PAH6(TRS)] which is a subset of Table PAH6 for ordinary benzenoids, structural correlations in isomer numbers, symmetry distributions, and empty rings between various strain-free TRS benzenoids are presented. These chemical graph theoretical results belong to the branch of mathematics called number theory. [ABSTRACT FROM AUTHOR]

Very recently, Alon and Frankl initiated the study of the maximum number of edges in n $n$‐vertex F $F$‐free graphs with matching number at most s $s$. For fixed F $F$ and s $s$, we determine this number apart from a constant additive term. We also obtain several exact results. [ABSTRACT FROM AUTHOR]

A power series being given as the solution of a linear differential equation with appropriate initial conditions, minimization consists in finding a non-trivial linear differential equation of minimal order having this power series as a solution. This problem exists in both homogeneous and inhomogeneous variants; it is distinct from, but related to, the classical problem of factorization of differential operators. Recently, minimization has found applications in Transcendental Number Theory, more specifically in the computation of non-zero algebraic points where Siegel's E-functions take algebraic values. We present algorithms and implementations for these questions, and discuss examples and experiments. [ABSTRACT FROM AUTHOR]

We study extreme values of Dirichlet polynomials with multiplicative coefficients, namely D N (t) : = D f , N (t) = 1 N ∑ n ⩽ N f (n) n i t , where f is a completely multiplicative function with | f (n) | = 1 for all n ∈ N. We use Soundararajan's resonance method to produce large values of | D N (t) | uniformly for all such f. In particular, we improve a recent result of Benatar and Nishry, where they establish weaker lower bounds and only for almost all such f. [ABSTRACT FROM AUTHOR]

Let d ≥ 1 be an integer and let p be a rational prime. Recall that p is a torsion prime of degree d if there exists an elliptic curve E over a degree d number field K such that E has a K-rational point of order p. Derickx et al. (Algebra Number Theory 17(2):267–308, 2023) have computed the torsion primes of degrees 4, 5, 6 and 7. We verify that the techniques used in Derickx et al. (Algebra Number Theory 17(2):267–308, 2023) can be extended to determine the torsion primes of degree 8. [ABSTRACT FROM AUTHOR]

We study four-dimensional = 2 superconformal circular, cyclic symmetric quiver theories which are planar equivalent to = 4 super Yang-Mills. We use localization to compute nonplanar corrections to the free energy and the circular half-BPS Wilson loop in these theories for an arbitrary number of nodes, and examine their behaviour in the limit of long quivers. Exploiting the relationship between the localization quiver matrix integrals and an integrable Bessel operator, we find a closed-form expression for the leading nonplanar correction to both observables in the limit when the number of nodes and 't Hooft coupling become large. We demonstrate that it has different asymptotic behaviour depending on how the two parameters are compared, and interpret this behaviour in terms of properties of a lattice model defined on the quiver diagram. [ABSTRACT FROM AUTHOR]

To examine the relationship between early number sense and later symbolic arithmetic competence and how they are connected in young Chinese children. The present study utilized a longitudinal sample of 115 Chinese children aged 3 to 6. Young children completed measures to assess number sense, visual spatial skills (in the fall of their second kindergarten year, K2), arithmetic competence (in the spring of their second kindergarten year, K3), working memory, verbal comprehension, and inhibitory control (in the fall of their first kindergarten year, K1). The results showed that number sense predicted arithmetic competence six months later. Meanwhile, visual spatial skills played a mediating role in this association, but it became non-significant when working memory, verbal comprehension, and inhibitory control were added as covariates. These longitudinal findings confirm the close association between number sense and arithmetic competence in young Chinese children. [ABSTRACT FROM AUTHOR]

Modeling the number of air passengers correctly is essential for management policy in the global world. Based on seasonality (depending on the season of the year), data about the number of air passengers are heteroscedastic. Heteroscedasticity violates "Homoscedasticity" which is one of the central assumptions of linear regression analysis. In this study, a new weighting approach called "Weighting Absolute Centered External Variable" (WCEV) is applied to the Turkish total monthly air passenger's data to obtain correct statistical inference and forecasting. Besides scatter plot months vs. studentized residuals, the homoscedasticity assumption is checked with the studentized RCEV test as well. Consequently, the WCEV method is shown superior performance against multiple linear regressions and exponential weighted moving average (EWMA) methods. The study also provides insights into the seasonal patterns of air passenger demand in Turkey, with passenger mobility increasing in the last quarter of each year and the lowest demand in January and February. This information can be used to optimize airport and airplane maintenance schedules and increase capacity during peak months. [ABSTRACT FROM AUTHOR]

During the last decades, the study of discontinuous piecewise differential systems has become an interesting subject of research due to the important applications of this kind of systems to model natural phenomena. In the qualitative theory of differential equations, one of the interesting problems is the detection of the number of limit cycles and their configurations which remains open to date, except for very particular families of differential equations. Here, we are inspired to study the maximum number of limit cycles of the discontinuous piecewise differential systems separated by a nonregular line and formed by a linear center and one of the four classes of quadratic centers. The main tool used to prove our main results is based on the first integrals of such systems. All the computations of this paper are verified using the algebraic manipulator Mathematica. [ABSTRACT FROM AUTHOR]

A locally conformally product (LCP) structure on compact manifold M is a conformal structure c together with a closed, non-exact and non-flat Weyl connection D with reducible holonomy. Equivalently, an LCP structure on M is defined by a reducible, non-flat, incomplete Riemannian metric h D on the universal cover M ̃ of M , with respect to which the fundamental group π 1 (M) acts by similarities. It was recently proved by Kourganoff that in this case (M ̃ , h D) is isometric to the Riemannian product of the flat space ℝ q and an incomplete irreducible Riemannian manifold (N , g N). In this paper, we show that for every LCP manifold (M , c , D) , there exists a metric g ∈ c such that the Lee form of D with respect to g vanishes on vectors tangent to the distribution on M defined by the flat factor ℝ q , and use this fact in order to construct new LCP structures from a given one by taking products. We also establish links between LCP manifolds and number field theory, and use them in order to construct large classes of examples, containing all previously known examples of LCP manifolds constructed by Matveev–Nikolayevsky, Kourganoff and Oeljeklaus–Toma (OT-manifolds). [ABSTRACT FROM AUTHOR]

This paper concerns the visibility properties of lattice points in multiple random walks on N k , where k ≥ 2 is an integer. We study two aspects of the visibility: simultaneous visibility in multiple random walkers; and that only some of these walkers are visible. Combining tools from number theory and probability theory, we prove the corresponding densities of the above two parts. [ABSTRACT FROM AUTHOR]

The call for papers for this Special Issue Iidentifies contemporary humanity as experiencing a global "biogeochemical ... political, economic, technological, ethical, and therefore, biocultural" crisis and asks scholars to consider how "religion may function as an adaptive or maladaptive presence" in response. Unasked is the adaptive capacity of scholarship as a crisis response. When buildings fall in earthquakes, or cities burn in wildfires, or second stories flood, few people just keep on doing what they were doing, "with a change of focus". This is "studying Rome while it burns". It's time to put out the fire if we can and survive it if we cannot. We scholar/teachers can't go on doing the same things and expecting different results. Unprecedented circumstances call for unprecedented actions in response. What would actual crisis responses on our part look like? What steps do we need to take as human beings in response to this crisis? How will that affect us as professionals? Seeking an ecology, rather than unanimity, of action and thought, and guided by Brian Walker's resilience theory and a number of Indigenous scholars, I suggest a process of reintegration, analogous to regenerative agriculture, which is at once both socio-cultural and ecological. This process, necessarily rooted in place, progressively situates us experientially in a dynamic, creative, and relational world characterized by connection, collaboration, and relation. As scholars, we will find forms of discovery, discussion, and dissemination that share these qualities. As teachers, we will model this world to our students and embody it in our classrooms and curricula. [ABSTRACT FROM AUTHOR]

We reviewed Human Communication Research (HCR) articles that included named theories in the: (a) abstract, (b) keywords, or (c) section headers to determine theory trends, identifying 592 instances of named theories in 447 articles. We conducted a follow-up analysis of 76 articles to illustrate how HCR researchers have contributed to theory development. Our review demonstrated systematic growth in theory use; the number of named theories in our sample doubled from the first 10 years to the last. There remains the propensity for theory shopping in the discipline with over 50% of theories in each decade cited only once. We also analyzed theory development strategies. There was evidence of theorizing that: (a) extended the range of theories and (b) explored effects. There was less theory synthesis that might provide frameworks for organizing related but distinct theories. We concluded by offering recommendations for how researchers can contribute to future theoretical development. [ABSTRACT FROM AUTHOR]

Communication is a theory-driven discipline, but does it always need to be? This article raises questions related to the role of theory in communication science, with the goal of providing a thoughtful discussion about what theory is, why theory is (or is not) important, the role of exploration in theory development, what constitutes a theoretical contribution, and the current state of theory in the field. We describe communication researchers' interest with theory by assessing the number of articles in the past decade of research that mention theory (nearly 80% of papers have attended to theory in some way). This article concludes with a forward-looking view of how scholars might think about theory in their work, why exploratory research should be valued more and not considered as conflicting with theory, and how conceptual clarity related to theoretical interests and contributions are imperative for human communication research. [ABSTRACT FROM AUTHOR]

We show that completeness at higher levels of the theory of the reals is a robust notion (under changing the signature and bounding the domain of the quantifiers). This mends recognized gaps in the hierarchy, and leads to stronger completeness results for various computational problems. We exhibit several families of complete problems which can be used for future completeness results in the real hierarchy. As an application we sharpen some results by Bürgisser and Cucker on the complexity of properties of semialgebraic sets, including the Hausdorff distance problem also studied by Jungeblut, Kleist, and Miltzow. [ABSTRACT FROM AUTHOR]

To evaluate the performance of a mathematical model to drive preoperative planning between RIRS and MiniPerc (MP) for the treatment of renal stones between 10 and 20 mm. Patients with a renal stone between 10 and 20 mm were enrolled. A mathematical model named Stone Management According to Size-Hardness (SMASH) score was calculated: hounsfield units (HU) χ stone maximum size (cm)/100. Patients were divided into 4 groups: RIRS with score < 15 (Group A), RIRS with score ≥ 15 (Group B), MP with score < 15 (Group C), MP with score ≥ 15 (Group D). Cyber Ho device was always used. Stone free rate (SFR) was assessed after 3 months. Complication rate and need for auxiliary procedures were evaluated. Between January 2019 and December 2021, 350 patients were enrolled (87, 88, 82 and 93 in Groups A, B, C and D). Mean stone size was 13.1 vs 13.3 mm in Group A vs B (p = 0.18) and 16.2 vs 18.1 mm in Group C vs D (p = 0.12). SFR was 82%, 61%, 75% and 85% for Groups A, B, C and D. SFR was comparable between Groups C and D (p = 0.32) and Groups A and C (p = 0.22). SFR was significantly higher in Group A over B (p = 0.03) and in Group D over B (p = 0.02). Complication rate was 2.2%, 3.4%, 12.1%, 12.9% for Groups A, B, C, D. RIRS and MP are both safe and effective. The mathematical model with the proposed cut-off allowed a proper allocation of patients between endoscopic and percutaneous approaches. Registration number of the study ISRCTN55546280. [ABSTRACT FROM AUTHOR]

This paper contributes to the conjecture of R. Scott and R. Styer which asserts that for any fixed relatively prime positive integers a, b and c all greater than 1 there is at most one solution to the equation a x + b y = c z in positive integers x, y and z, except for specific cases. The fundamental result proves the conjecture under some congruence condition modulo c on a and b. As applications the conjecture is confirmed to be true if c takes some small values including the Fermat primes found so far, and in particular this provides an analytic proof of the celebrated theorem of Scott (J Number Theory 44(2):153-165, 1993) solving the conjecture for c = 2 in a purely algebraic manner. The method can be generalized for smaller modulus cases, and it turns out that the conjecture holds true for infinitely many specific values of c not being perfect powers. The main novelty is to apply a special type of the p-adic analogue to Baker's theory on linear forms in logarithms via a certain divisibility relation arising from the existence of two hypothetical solutions to the equation. The other tools include Baker's theory in the complex case and its non-Archimedean analogue for number fields together with various elementary arguments through rational and quadratic numbers, and extensive computation. [ABSTRACT FROM AUTHOR]

Continued fractions have a long history in number theory, especially in the area of Diophantine approximation. The aim of this expository paper is to survey the main results on the theory of p-adic continued fractions, i.e., continued fractions defined over the field of p-adic numbers \mathbb {Q}_p, which in the last years has recorded a considerable increase of interest and research activity. We start from the very first definitions up to the most recent developments and open problems. [ABSTRACT FROM AUTHOR]

To strengthen property management and to solve the problem for emergency material transport under the sudden emergencies, the property management company redesigned an optimization method of rescue vehicle route choice with considering the transport time (timeliness of emergency logistics) and the time penalty cost (service satisfaction) for the residential communities it serves. The system is solved based on Genetic Algorithm. Also, the demand urgencies of each area were determined with improved Gray Relative Analysis (IGRA), while total demands of each point were calculated using combination of triangular fuzzy number theory and PERT (TFN-PERT). In addition, some parameters will be corrected to compare the running results before and after the optimization. Synchronously. And the objective function values, which were calculated with consideration of demand urgency and without considering it respectively, also are compared. The effectiveness and feasibility of this approach were verified. The result manifests that the model can contribute to both timeliness and service satisfaction. Regarding the calculation of urgency, which can lessen the blindness of dispatching and ensure the fairness of distribution to some extent. Research result combines with the specific process of transportation vehicles under the sudden emergencies. As such, the approach could furnish scientific decision support on emergency system authentically for property management companies. [ABSTRACT FROM AUTHOR]

Primate social organizations, or grouping patterns, vary significantly across species. Behavioral strategies that allow for flexibility in grouping patterns offer a means to reduce the costs of group living. Chimpanzees (Pan troglodytes) have a fission‐fusion social system in which temporary subgroups ("parties") change in composition because of local socio‐ecological conditions. Notably, western chimpanzees (P. t. verus) are described as showing a higher degree of bisexual bonding and association than eastern chimpanzees, and eastern female chimpanzees (P. t. schweinfurthii) are thought to be more solitary than western female chimpanzees. However, reported comparisons in sociality currently depend on a small number of study groups, particularly in western chimpanzees, and variation in methods. The inclusion of additional communities and direct comparison using the same methods are essential to assess whether reported subspecies differences in sociality hold in this behaviorally heterogeneous species. We explored whether sociality differs between two communities of chimpanzees using the same motion‐triggered camera technology and definitions of social measures. We compare party size and composition (party type, sex ratio) between the western Gahtoy community in the Nimba Mountains (Guinea) and the eastern Waibira community in the Budongo Forest (Uganda). Once potential competition for resources such as food and mating opportunities were controlled for, subspecies did not substantially influence the number of individuals in a party. We found a higher sex‐ratio, indicating more males in a party, in Waibira; this pattern was driven by a greater likelihood in Gahtoy to be in all‐female parties. This finding is the opposite of what was expected for eastern chimpanzees, where female‐only parties are predicted to be more common. Our results highlight the flexibility in chimpanzee sociality, and caution against subspecies level generalizations. Research highlights: Party size in western and eastern chimpanzees did not differ between our two study communities suggesting that the number of individuals in a party is shaped by similar socio‐ecological pressures.In our sample, eastern chimpanzees were found to have more males in a party compared to western chimpanzees, contrary to the expectation of more female‐only parties in eastern compared to western chimpanzees.Our findings highlight the flexibility in chimpanzee grouping patterns and caution against subspecies level generalizations from limited data points and/or heterogeneous methods. [ABSTRACT FROM AUTHOR]

Let $k\ge 2$ be an integer and let A be a set of nonnegative integers. The representation function $R_{A,k}(n)$ for the set A is the number of representations of a nonnegative integer n as the sum of k terms from A. Let $A(n)$ denote the counting function of A. Bell and Shallit ['Counterexamples to a conjecture of Dombi in additive number theory', Acta Math. Hung. , to appear] recently gave a counterexample for a conjecture of Dombi and proved that if $A(n)=o(n^{{(k-2)}/{k}-\epsilon })$ for some $\epsilon>0$ , then $R_{\mathbb {N}\setminus A,k}(n)$ is eventually strictly increasing. We improve this result to $A(n)=O(n^{{(k-2)}/{(k-1)}})$. We also give an example to show that this bound is best possible. [ABSTRACT FROM AUTHOR]

The article informs about new effective results in the theory of the Riemann zeta-function, focusing on four groups providing estimates for the zeta-function and associated functions under the assumption of the Riemann hypothesis. Topic include explicit and RH estimates for various functions related to the zeta-function, including their applications to the distribution of prime numbers and other arithmetic properties, emphasizing the importance of these findings in mathematical research.

Let \mathbb {F}_q denote the finite field of characteristic p and order q. Let \mathbb {Z}_q denote the unramified extension of the p-adic rational integers \mathbb {Z}_p with residue field \mathbb {F}_q. Given two positive integers m,n, define a box \mathcal B_m to be a subset of \mathbb {Z}_q^n with q^{nm} elements such that \mathcal B_m modulo p^m is equal to (\mathbb {Z}_q/p^m \mathbb {Z}_q)^n. For a collection of nonconstant polynomials f_1,\dots,f_s\in \mathbb {Z}_q[x_1,\ldots,x_n] and positive integers m_1,\dots,m_s, define the set of common zeros inside the box \mathcal B_m to be \begin{equation*} V=\{X\in \mathcal B_m:\; f_i(X)\equiv 0\mod {p^{m_i}}\text { for all } 1\leq i\leq s\}. \end{equation*} It is an interesting problem to give the sharp estimates for the p-divisibility of |V|. This problem has been partially solved for the three cases: (i) m=m_1=\cdots =m_s=1, which is just the Ax-Katz theorem, (ii) m=m_1=\cdots =m_s>1, which was solved by Katz [Proc. Amer. Math. Soc. 137 (2009), pp. 4065–4075; Amer. J. Math. 93 (1971), pp. 485–499], Marshal and Ramage [Proc. Amer. Math. Soc. 49 (1975), pp. 35–38], and (iii) m=1, and m_1,\dots,m_s\geq 1, which was recently solved by Cao, Wan [Finite Fields Appl. 91 (2023), p. 25] and Grynkiewicz [ A generalization of the Chevalley-Warning and Ax-Katz theorems with a view towards combinatorial number theory , Preprint, arXiv: 2208.12895 , 2022]. Based on the multi-fold addition and multiplication of the finite Witt rings over \mathbb {F}_q, we investigate the remaining unconsidered case of m>1 and m\neq m_j for some 1\leq j\leq s, and finally provide a complete answer to this problem. [ABSTRACT FROM AUTHOR]

We use the Bateman--Horn Conjecture from number theory to give strong evidence of a positive answer to Peter Neumann's question, whether there are infinitely many simple groups of order a product of six primes. (Those with fewer than six were classified by Burnside, Frobenius and Hölder in the 1890s.) The groups satisfying this condition are PSL2(8), PSL2(9) and PSL2(p) for primes p such that p² -- 1 is a product of six primes. The conjecture suggests that there are infinitely many such primes p, by providing heuristic estimates for their distribution which agree closely with evidence from computer searches. We also briefly discuss the applications of this conjecture to other problems in group theory, such as the classifications of permutation groups and of linear groups of prime degree, the structure of the power graph of a finite simple group, the construction of highly symmetric block designs, and the possible existence of infinitely many Kn groups for each n ≥ 5. [ABSTRACT FROM AUTHOR]

Llibre et al. in [ J. Nonlinear Sci. , 2019] studied the number of limit cycles of piecewise linear Liénard system with one anti-saddle at the central zone in some parameter regions. However, the exact number and hyperbolicity of limit cycles were still unclear as in these cases of parameter. The goal of this paper is to give the exact number of limit cycles and the hyperbolicity of limit cycles in these parameter cases, where the maximum number of limit cycles is two. Moreover, all corresponding global phase portraits in the Poincaré disc are obtained. [ABSTRACT FROM AUTHOR]

We study the minimum number of maximum matchings in a bipartite multigraph G $G$ with parts X $X$ and Y $Y$ under various conditions, refining the well‐known lower bound due to M. Hall. When ∣X∣=n $| X| =n$, every vertex in X $X$ has degree at least k $k$, and every vertex in X $X$ has at least r $r$ distinct neighbors, the minimum is r!(k−r+1) $r!(k-r+1)$ when n≥r $n\ge r$ and is [r+n(k−r)]∏i=1n−1(r−i) $[r+n(k-r)]{\prod }_{i=1}^{n-1}(r-i)$ when n≤r $n\le r$. When every vertex has at least two neighbors and ∣Y∣−∣X∣=t≥0 $| Y| -| X| =t\ge 0$, the minimum is [(n−1)t+2+b](t+1) $[(n-1)t+2+b](t+1)$, where b=∣E(G)∣−2(n+t) $b=| E(G)| -2(n+t)$. We also determine the minimum number of maximum matchings in several other situations. We provide a variety of sharpness constructions. [ABSTRACT FROM AUTHOR]

OBJECTIVES: To compare grades, National Board of Medical Examiners (NBME) Shelf Exam scores, and student satisfaction with the Obstetrics and Gynecology (OB/GYN) clerkship after transitioning from normative to criterion-based grading. METHODS: Between July 2021 and July 2022, the Icahn School of Medicine at Mount Sinai (ISMMS) adhered to a normative grading scheme in which ∼60% of students achieved a grade of Honors, 30% achieved a grade of High Pass and 10% achieved a grade of Pass for the OB/GYN clerkship. In July 2022, ISMMS transitioned to a criterion-based scheme. In this scheme, 6 competencies were created. Criteria were determined for each competency, delineating achieving a score of "Pass" versus "Honors" for the specific objective. Students needed to meet the criteria for Honors for 4 out of 6 of the competencies in order to ultimately receive a grade of Honors for the clerkship. The number of students achieving Honors, NBME shelf exam scores, and student clerkship satisfaction ratings between the normative and criterion-based schemes were compared. RESULTS: The number of students studying in academic year (AY) 2021–2022 and AY 2022–2023 were 134 and 137, respectively. A significantly lower percentage of students received Honors in AY 2021–2022 than in AY 2022–2023 (66% vs. 96%, P <.01). Mean exam scores were significantly higher for those receiving Honors in AY 2021–2022 than in AY 2022–2023 (P <.05); scores for AY 2021–2022 and AY 2022–2023 were 78.9, 95% CI [77.6, 80.1] and 76.7, 95% CI [75.6, 77.8], respectively. Mean exam scores for all students were not significantly different between the 2 academic cohorts (77.8 vs. 76.2, P =.06). Clerkship satisfaction rating was significantly higher in AY 2022–2023 than in AY 2021–2022 (4.1 vs. 3.7, P <.05). CONCLUSIONS: These findings support a paradigm that compares learner performance to predefined measures as opposed to peer performance. [ABSTRACT FROM AUTHOR]

The paper aims to conduct an analysis of pricing strategies in a dual channel supply chain under external uncertainty, utilizing Interval numbers theory and Game theory as the theoretical basis. The focus is on maximizing the expected profits of manufacturers and retailers. Four models are considered: centralized decision-making, manufacturer's Stackelberg, retailer's Stackelberg strategy, and vertical Nash model, with the decision variable being the product price. By solving the game model, the paper compares the optimal decisions under the four models and conducts sensitivity analysis to reflect the influence of key parameters and analyze their relationships. The ultimate goal is to optimize profits under various circumstances by adjusting market potential and price parameters to determine the best price level. The findings suggest that decision-maker's risk indicators have a greater impact on decision results when market demand is less sensitive to price, and that the size of the market has a negative correlation with the impact of decision-maker's risk indicators on decision results. [ABSTRACT FROM AUTHOR]

This paper is aimed at using a lookup table method to improve the scalar multiplication performance of elliptic curve cryptography. The lookup table must be divided into two polynomials and requires two iterations of point doubling operation, for which negation operations are needed. It is well known that an inversion operation requires a lot of multiplication time. The advantage of this paper is that we are able to reduce one inverse element calculation for this problem and also improve the basic operations of finite fields through segmentation methods. If the normal basis method is used in the design of the inverse element operation, it must be converted to the normal basis through the standard basis. However, the conversion process requires a lot of matrix operations. Though the anti-element operation has good speed performance, it also increases the computational complexity. Using number theory and grouping methods will greatly improve the performance of inverse element operations. With application of the two-time point doubling operation in the hardware implementation, the developed approach reduces the computing time by 48% as compared with the conventional approach. The computational time of the scalar multiplication using the presented method is further improved by 67% over the traditional algorithm with only an area increase of 12%. Finally, the proposed lookup table-based technique can be utilized for software and hardware implementation, as the developed arithmetic operations are simple and are consistent in their execution. [ABSTRACT FROM AUTHOR]

In this article, we study the maximum number of limit cycles of discontinuous piecewise differential systems, formed by two Hamiltonians systems separated by a straight line. We consider three cases, when both Hamiltonians systems in each side of the discontinuity line have simultaneously degree one, two or three. We obtain that in these three cases, this maximum number is zero, one and three, respectively. Moreover, we prove that there are discontinuous piecewise differential systems realizing these maximum number of limit cycles. Note that we have solved the extension of the 16th Hilbert problem about the maximum number of limit cycles that these three classes of discontinuous piecewise differential systems separated by one straight line and formed by two Hamiltonian systems with a degree either one, two, or three, which such systems can exhibit. [ABSTRACT FROM AUTHOR]

The article extends the theoretical and applicative analysis of Zipf's law. We are concerned with a set of properties of Zipf's law that derive directly from the power law expression and from the discrete nature of the objects to which the law is applied, when the objects are words, lemmas, and the like. We also search for variations of Zipf's law that can help explain the noisy results empirically reported in the literature and the departures of the empirically obtained nonlinear graph from the theoretical linear one, with the variants analyzed differing from Mandelbrot and lognormal distributions. A problem of interest that we deal with is that of mixtures of populations obeying Zipf's law. The last problem has relevance in the analysis of texts with words with various etymologies. Computational aspects are also addressed. [ABSTRACT FROM AUTHOR]

We survey the history of the arithmetic derivative and more recent advances in research on this topic. Among other things, we discuss a few generalizations of the original arithmetic derivative and some arithmetic differential equations that are related to Goldbach's conjecture and the twin prime conjecture. Our primary purpose is to give an overview of this field, but we also aim at providing for an introductory course on discrete mathematics or number theory. Therefore, our survey contains ten exercises. [ABSTRACT FROM AUTHOR]

"An Introduction to the Circle Method" by M. Ram Murty and Kaneenika Sinha is a book that introduces the circle method in number theory. The authors assume no prior background in number theory but do expect some familiarity with complex analysis. The book provides a detailed explanation of the circle method, including the use of exponential sums and contour integration. It also covers the steps that are often omitted in other texts, making it a valuable resource for both beginners and experts in the field. The book focuses on building a narrow path towards solving Waring's Problem and Goldbach's Conjecture in additive number theory. The circle method has been successful in proving the Ternary Goldbach Theorem, but has not been able to solve Goldbach's Conjecture. The authors argue that the obstruction to solving Goldbach's Conjecture is more serious than previously thought. The book acknowledges the uncertainty of the future of the circle method due to ongoing developments in analytic number theory. Overall, the book is praised for making the circle method accessible to a wider audience. [Extracted from the article]

A (k, m)-Furstenberg set is a subset S ⊂ F q n with the property that each k-dimensional subspace of F q n can be translated so that it intersects S in at least m points. Ellenberg and Erman (Algebra Number Theory 10(7), 1415–1436 (2016)) proved that (k, m)-Furstenberg sets must have size at least C n , k m n / k , where C n , k is a constant depending only n and k. In this paper, we adopt the same proof strategy as Ellenberg and Erman, but use more elementary techniques than their scheme-theoretic method. By modifying certain parts of the argument, we obtain an improved bound on C n , k , and our improved bound is nearly optimal for an algebraic generalization the main combinatorial result. We also extend our analysis to give lower bounds for sets that have large intersection with shifts of a specific family of higher-degree co-dimension n - k varieties, instead of just co-dimension n - k subspaces. [ABSTRACT FROM AUTHOR]