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Quantum annealing machines are next-generation computers for solving combinatorial optimization problems. Although physical simulations are one of the most promising applications of quantum annealing machines, a method how to embed the target problem into the machines has not been developed except for certain simple examples. In this study, we focus on a method of representing real numbers using binary variables, or quantum bits. One of the most important problems for conducting physical simulation by quantum annealing machines is how to represent the real number with quantum bits. The variables in physical simulations are often represented by real numbers but real numbers must be represented by a combination of binary variables in quantum annealing, such as quadratic unconstrained binary optimization (QUBO). Conventionally, real numbers have been represented by assigning each digit of their binary number representation to a binary variable. Considering the classical annealing point of view, we noticed that when real numbers are represented in binary numbers, there are numbers that can only be reached by inverting several bits simultaneously under the restriction of not increasing a given Hamiltonian, which makes the optimization very difficult. In this work, we propose three new types of real number representation and compared these representations under the problem of solving linear equations. As a result, we found experimentally that the accuracy of the solution varies significantly depending on how the real numbers are represented. We also found that the most appropriate representation depends on the size and difficulty of the problem to be solved and that these differences show a consistent trend for two annealing solvers. Finally, we explain the reasons for these differences using simple models, the minimum required number of simultaneous bit flips, one-way probabilistic bit-flip energy minimization, and simulation of ideal quantum annealing machine. [ABSTRACT FROM AUTHOR]

We study the Cantor real base numeration system which is a common generalization of two positional systems, namely the Cantor system with a sequence of integer bases and the Rényi system with one real base. We focus on the case of an alternate base B given by a purely periodic sequence (β n) n ≥ 1 of real numbers greater than 1. We answer an open question of Charlier et al. (J Number Theory 254:184–198, 2024, https://doi.org/10.1016/j.jnt.2023.07.008) on the set of numbers with eventually periodic B -expansions. We also investigate for which bases all sufficiently small rationals have a purely periodic B -expansion. We show that a necessary condition for this phenomenon is that δ = ∏ n = 1 p β n (where p is the period-length of B ) is a Pisot or a Salem unit. We also provide a sufficient condition. We thus generalize the results known for the Rényi numeration system, i.e. for the case when p = 1 . We provide a class of alternate bases in which all rational numbers in the interval [0, 1) have a purely periodic B -expansion. [ABSTRACT FROM AUTHOR]

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 establish higher dimensional versions of a recent theorem by Chen and Haynes [Int. J. Number Theory 19 (2023), 1405–1413] on the expected value of the smallest denominator of rational points in a randomly shifted interval of small length, and of the closely related 1977 Kruyswijk–Meijer conjecture recently proved by Balazard and Martin [Bull. Sci. Math. 187 (2023), Paper No. 103305]. We express the distribution of smallest denominators in terms of the void statistics of multidimensional Farey fractions and prove convergence of the distribution function and certain finite moments. The latter was previously unknown even in the one‐dimensional setting. We furthermore obtain a higher dimensional extension of Kargaev and Zhigljavsky's work on moments of the distance function for the Farey sequence [J. Number Theory 65 (1997), 130–149] as well as new results on pigeonhole statistics. [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]

There are three main types of number used in modern, industrialized societies. Cardinals count sets (e.g., people, objects) and quantify elements of conventional scales (e.g., money, distance), ordinals index positions in ordered sequences (e.g., years, pages), and nominals serve as unique identifiers (e.g., telephone numbers, player numbers). Many studies that have cited number frequencies in support of claims about numerical cognition and mathematical cognition hinge on the assumption that most numbers analyzed are cardinal. This paper is the first to investigate the relative frequencies of different number types, presenting a corpus analysis of morphologically unmarked numbers (not, e.g., "eighth" or "21st") in which we manually annotated 3,600 concordances in the Corpus of Contemporary American English. Overall, cardinals are dominant—both pure cardinals (sets) and measurements (scales)—except in the range 1,000–10,000, which is dominated by ordinal years, like 1996 and 2004. Ordinals occur less often overall, and nominals even less so. Only for cardinals do round numbers, associated with approximation, dominate overall and increase with magnitude. In comparison with other registers, academic writing contains a lower proportion of measurements as well as a higher proportion of ordinals and, to some extent, nominals. In writing, pure cardinals and measurements are usually represented as number words, but measurements—especially larger, unround ones—are more likely to be numerals. Ordinals and nominals are mostly represented as numerals. Altogether, this paper reveals how numbers are used in American English, establishing an initial baseline for any analyses of number frequencies and shedding new light on the cognitive and psychological study of number. [ABSTRACT FROM AUTHOR]

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

Perfect error-correcting codes allow for an optimal transmission of information while guaranteeing error correction. For this reason, proving their existence has been a classical problem in both pure mathematics and information theory. Indeed, the classification of the parameters of e-error correcting perfect codes over q-ary alphabets was a very active topic of research in the late 20th century. Consequently, all parameters of perfect e-error-correcting codes were found if e ≥ 3 , and it was conjectured that no perfect 2-error-correcting codes exist over any q-ary alphabet, where q > 3 . In the 1970s, this was proved for q a prime power, for q = 2 r 3 s and for only seven other values of q. Almost 50 years later, it is surprising to note that there have been no new results in this regard and the classification of 2-error-correcting codes over non-prime power alphabets remains an open problem. In this paper, we use techniques from the resolution of the generalised Ramanujan–Nagell equation and from modern computational number theory to show that perfect 2-error-correcting codes do not exist for 172 new values of q which are not prime powers, substantially increasing the values of q which are now classified. In addition, we prove that, for any fixed value of q, there can be at most finitely many perfect 2-error-correcting codes over an alphabet of size q. [ABSTRACT FROM AUTHOR]

Denote by h = h (− p) the class number of the imaginary quadratic field ℚ (− p) with p prime. It is well known that h = 1 for p ∈ { 3 , 7 , 1 1 , 1 9 , 4 3 , 6 7 , 1 6 3 }. Recently, all the solutions of the Diophantine equation x 2 + p s = 4 y n with h = 1 were given by Chakraborty et al. in [Complete solutions of certain Lebesgue–Ramanujan–Nagell type equations, Publ. Math. Debrecen 97(3–4) (2020) 339–352]. In this paper, we study the Diophantine equation x 2 + p s = 2 r y n in unknown integers (x , y , s , r , n) , where s ≥ 0 , r ≥ 3 , n ≥ 3 , h ∈ { 1 , 2 , 3 } and gcd (x , y) = 1. To do this, we use the known results from the modularity of Galois representations associated with Frey–Hellegoaurch elliptic curves, the symplectic method and elementary methods of classical algebraic number theory. The aim of this paper is to extend the above results of Chakraborty et al. [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]

Several known factors affect outcomes of Mohs facial defect reconstruction; however, the effect of repair timing on outcomes is ill-defined. The aim of this study was to determine postoperative complication rates between immediate and delayed repair of Mohs facial defects. Preferred Reporting Items of Systematic Reviews and Meta-Analyses guidelines were used. Articles were selected using PICO format—population: Mohs facial defect patients, intervention: defect repair, comparator: immediate (<24 hours), or delayed (>24 hours) repair, outcome: complication rate. PubMed/Medline (1946-2020), EMBASE (1947-2020), Scopus (1823-2020), Web of Science (1900-2020), Cochrane Library, and Clinicaltrials.gov were searched. Two independent reviewers screened abstracts; those in English with human subjects reporting repair timing and complication rates were included. Search criteria yielded 6,649 abstracts; 233 qualified for review. Data were gathered from six studies; they alone contained comparative data meeting inclusion criteria. While many well-written studies were encountered, reported results varied widely. A statistically sound meta-analysis could not be completed due to large heterogeneity between studies, biasing the analysis towards the largest weighted study. Clinically important differences may exist between immediate and delayed Mohs reconstruction, but small study numbers, large heterogeneity, and lack of standardized outcome measures limit definitive conclusions. More studies are needed to perform appropriate meta-analyses, including studies using standardized methods of reporting Mohs outcome data. [ABSTRACT FROM AUTHOR]

In this paper, we study the number of limit cycles bifurcated from the periodic orbits of a cubic uniform isochronous center with continuous and discontinuous quartic polynomial perturbations. Using the averaging theory of first order for continuous and discontinuous differential systems and comparing the obtained results, we show that the discontinuous systems have at least 6 more limit cycles than the continuous ones. This study needs some computations that have been verified using Maple. [ABSTRACT FROM AUTHOR]

Don Zagier, former director at the Bonn Max Planck Institute for Mathematics, has been awarded the Gumin Prize for Mathematics of the Year 2024. The Carl Friedrich von Siemens Foundation honors his groundbreaking research in number theory and the theory of modular forms. The Gumin Prize is awarded every three to four years to outstanding mathematicians in Germany, Austria, or Switzerland. Zagier speaks in an interview about his work, his collaboration with other mathematicians, and his former doctoral student Maryna Viazovska, who received the Fields Medal. He also comments on current developments in mathematics and mentions the Taniyama-Weil conjecture as a problem he would like to see solved. Zagier also mentions his personal story as a prodigy. [Extracted from the article]

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]

Probabilistic computing can solve complex combinatorial optimization problems more efficiently than conventional deterministic computing. A probabilistic bit (p‐bit) with an n‐p‐n bistable resistor (biristor) is demonstrated for probabilistic computing. It is fabricated on an 8‐inch wafer with complementary metal–oxide–semiconductor (CMOS) compatible technologies. Its stochastic behavior of threshold switching, which is based on the phenomenon of a single transistor latch, provides output with a Boltzmann distribution. The p‐bit is composed of a biristor, a serial resistor, and a comparator. The output probability of the biristor‐based p‐bits shows a sigmoidal relationship with the input voltage, showing typical p‐bit characteristics. Invertible Boolean logic operations with p‐bits are demonstrated, and weighted maximum Boolean satisfiability problems are solved with high energy efficiency and accuracy. The biristor‐based p‐bits with perfect CMOS compatibility show sufficient device stability, demonstrating the possibility of large‐scale integration with a p‐bit array for complex optimization solvers. [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]

This paper explores the properties of multipliers associated with discrete analogues of fractional integrals, revealing intriguing connections with Dirichlet characters, Euler's identity, and Dedekind zeta functions of quadratic imaginary fields. Employing Fourier transform techniques, the Hardy–Littlewood circle method, and a discrete analogue of the Stein–Weiss inequality on product space through implication methods, we establish ℓ p → ℓ q bounds for these operators. Our results contribute to a deeper understanding of the intricate relationship between number theory and harmonic analysis in discrete domains, offering insights into the convergence behavior of these operators. [ABSTRACT FROM AUTHOR]

We study in this article three birational invariants of projective hyper-Kähler manifolds: the degree of irrationality, the fibering gonality, and the fibering genus. We first improve the lower bound in a recent result of Voisin saying that the fibering genus of a Mumford–Tate very general projective hyper-Kähler manifold is bounded from below by a constant depending on its dimension and the second Betti number. We also study the relations between these birational invariants for projective |$K3$| surfaces of Picard number |$1$| and study the asymptotic behaviors of their degree of irrationality and fibering genus. [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]

The book "Number Theory Through the Eyes of Sophie Germain: An Inquiry Course" by David Pengelley explores the correspondence between Sophie Germain and renowned mathematicians Joseph-Louis Lagrange, Adrien-Marie Legendre, and Carl Friedrich Gauss as they attempted to find a proof for Fermat's Last Theorem. Although Germain's plan was ultimately unsuccessful, it represented significant progress on the problem and provided insights into the theorem. The book takes an inquiry-based approach, using primary source materials to teach undergraduate students elementary number theory. It offers a unique and engaging perspective on the subject, allowing students to discover and prove results themselves. [Extracted from the article]

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]

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]

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]

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]

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]

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]

In this paper, we propose a new program for introducing the sign of the functional equation to present the entire functions of order one in number theory. We suggest some open problems for the zeros of these entire functions related to the completed Dedekind zeta function, completed quadratic Dirichlet L-functions, completed Ramanujan zeta function and completed automorphic L-function. These lead to the contribution to giving the deep understanding for diffusion equations associated with the entire functions of order one in number theory. [ABSTRACT FROM AUTHOR]

In this paper, we address the entire Fourier sine and cosine integrals related to the Mittag-Leffler function. We guess that the entire functions have the real zeros in the entire complex plane. They can be connected with the well-known conjectures in analytic number theory. They are considered as the special solutions for the time-fractional diffusion equation within the Caputo fractional derivative. [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]

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]

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]

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 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]

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]

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]

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]

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]

Bell polynomials have already been used in many different fields, ranging from number theory to operator theory. Here another application in Laplace Transform (LT) theory is shown, describing a method for computing the LT of nested functions. A table containing the first few values of complete Bell polynomials is shown, and a program for deriving the next ones is given. A program for approximating the Laplace Transform of composed analytic functions is also presented. [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]

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 $k\ge 2$ 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]

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]