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We construct for every integer k⩾3$k\geqslant 3$ and every real μ∈(0,k−1k)$\mu \in (0, \frac{k-1}{k})$ a set of integers X=X(k,μ)$X=X(k, \mu)$ which, when coloured with finitely many colours, contains a monochromatic k$k$‐term arithmetic progression, whilst every finite Y⊆X$Y\subseteq X$ has a subset Z⊆Y$Z\subseteq Y$ of size |Z|⩾μ|Y|$|Z|\geqslant \mu |Y|$ that is free of arithmetic progressions of length k$k$. This answers a question of Erdős, Nešetřil and the second author. Moreover, we obtain an analogous multidimensional statement and a Hales–Jewett version of this result. [ABSTRACT FROM AUTHOR]

We prove a known 2-coloring of the integers [N] ≔\{1,2,3,\dots,N\} minimizes the number of monochromatic arithmetic 3-progressions under certain restrictions. A monochromatic arithmetic progression is a set of equally-spaced integers that are all the same color. Previous work by Parrilo, Robertson and Saracino conjectured an optimal coloring for large N that involves 12 colored blocks. Here, we prove that the conjectured coloring minimizes monochromatic arithmetic 3-progressions among anti-symmetric colorings with 12 or fewer colored blocks. We leverage a connection to the coloring of the continuous interval [0,1] used by Parrilo, Robertson, and Saracino as well as by Butler, Costello and Graham. Our proof identifies classes of colorings with permutations, then counts the permutations using mixed integer linear programming. [ABSTRACT FROM AUTHOR]

We characterize those strings whose suffix arrays are based on arithmetic progressions, in particular, arithmetically progressed permutations where all pairs of successive entries of the permutation have the same difference modulo the respective string length. We show that an arithmetically progressed permutation P coincides with the suffix array of a unary, binary, or ternary string. We further analyze the conditions of a given P under which we can find a uniquely defined string over either a binary or ternary alphabet having P as its suffix array. For the binary case, we show its connection to lower Christoffel words, balanced words, and Fibonacci words. In addition to solving the arithmetically progressed suffix array problem, we give the shape of the Burrows–Wheeler transform of those strings solving this problem. These results give rise to numerous future research directions. [ABSTRACT FROM AUTHOR]

Sparse Ramanujan Sequences (SRS) are periodic series derived from Ramanujan Sums (RS). RS is an exponential summation used to derive infinite series expansions for arithmetic functions, whose application pervades different fields of study from classical mathematics to digital signal processing. SRS has properties similar to RS such as periodicity and orthogonality. SRS transforms allow the sparse representation of signals, hence they have many potential applications, such as denoising and compression. The application of SRS transform to reduce the Peak to Average Power Ratio (PAPR) in Orthogonal Frequency Division Multiplexing (OFDM) is proposed in this paper. The Complementary Cumulative Distribution Function (CCDF) and Bit Error Rate (BER) are used to assess the PAPR reduction performance of the proposed system. The results show that the proposed SRS transform-based OFDM system significantly reduces PAPR and BER with reduced computational complexity in comparison with the existing systems. [ABSTRACT FROM AUTHOR]

Let S be a discrete semigroup and let S S denote the collection of all functions f : S → S . If (P , ∘) is a subsemigroup of S S by composition operation, then P induces a natural topological dynamical system. In fact, (β S , { T f } f ∈ P) is a topological dynamical system, where β S is the Stone–Čech compactification of S, x ↦ T f (x) = f β (x) : β S → β S and f β is a unique continuous22 extension of f. In this paper, we concentrate on the dynamical system (β S , { T f } f ∈ P) , when S is an arbitrary discrete semigroup and P is a subsemigroup of S S and obtain some relations between subsets of S and subsystems of β S with respect to P. As a consequence, we prove that if (S , +) is an infinite commutative discrete semigroup and C is a finite partition of S, then for every finite number of arbitrary homomorphisms g 1 , ⋯ , g l : N → S , there exist an infinite subset B of the natural numbers and C ∈ C such that for every finite summations n 1 , ⋯ , n k of B there exists s ∈ S such that { s + g i (n 1) , s + g i (n 2) , ⋯ , s + g i (n k) } ⊆ C , ∀ i ∈ { 1 , ⋯ , l }. [ABSTRACT FROM AUTHOR]

We study lower bounds for the Riemann zeta function ζ ⁢ ( s ) {\zeta(s)} along vertical arithmetic progressions in the right-half of the critical strip. We show that the lower bounds obtained in the discrete case coincide, up to the constants in the exponential, with the ones known for the continuous case, that is when the imaginary part of s ranges on a given interval. Our methods are based on a discretization of the resonance method for estimating extremal values of ζ ⁢ ( s ) {\zeta(s)} . [ABSTRACT FROM AUTHOR]

Let , with and , be the set of locally integrable functions with the zero integrals over all balls of radius in . We study the interpolation problem , with , for functions in with growth constraints at infinity. Under consideration is the case that is a set of points on a certain straight line in which is close in some sense to a finite union of arithmetic progressions and is a sequence of complex numbers satisfying the condition . We show that this interpolation problem is solvable in the class of those functions in which, together with their derivatives, satisfy a special decay condition at infinity. The condition is an upper bound that implies power decay in the directions orthogonal to and also cannot be significantly improved along the straight line . [ABSTRACT FROM AUTHOR]

Let f be a polynomial with integer coefficients whose degree is at least 2. We consider the problem of covering the orbit Orb f (t) = { t , f (t) , f (f (t)) , ... } , where t is an integer, using arithmetic progressions each of which contains t. Fixing an integer k ≥ 2 , we prove that it is impossible to cover Orb f (t) using k such arithmetic progressions unless Orb f (t) is contained in one of these progressions. In fact, we show that the relative density of terms covered by k such arithmetic progressions in Orb f (t) is uniformly bounded from above by a bound that depends solely on k. In addition, the latter relative density can be made as close as desired to 1 by an appropriate choice of k arithmetic progressions containing t if k is allowed to be large enough. [ABSTRACT FROM AUTHOR]

We study the sup-norm bound (both individually and on average) for Eisenstein series on certain arithmetic hyperbolic orbifolds producing sharp exponents for the modular surface and Picard 3-fold. The methods involve bounds for Epstein zeta functions, and counting restricted values of indefinite quadratic forms at integer points. [ABSTRACT FROM AUTHOR]

A covering system of the integers is a finite collection of arithmetic progressions whose union is the set of integers. A well-known problem on covering systems is the minimum modulus problem posed by Erdős in 1950, who asked whether the minimum modulus in such systems with distinct moduli can be arbitrarily large. This problem was resolved by Hough in 2015, who showed that the minimum modulus is at most 1016. In 2022, Balister, Bollobás, Morris, Sahasrabudhe and Tiba reduced Hough's bound to 616 , 000 by developing Hough's method. They call it the distortion method. In this paper, by applying this method, we mainly prove that there does not exist any covering system of multiplicity s in any global function field of genus g over F q for q ≥ (1.14 + 0.16 g) e 6.5 + 0.97 g s 2. In particular, there is no covering system of F q [ x ] with distinct moduli for q ≥ 759. [ABSTRACT FROM AUTHOR]

Let U be the set of positive odd numbers that can not be written in the form p + 2 k. Recently, by analyzing possible prime divisors of b , Chen proved b ≥ 11184810 and ω (b) ≥ 7 if an arithmetic progression a (mod b) is in U , with ω (b) = 7 if and only if b = 11184810 , where ω (n) is the number of distinct prime divisors of n. In this paper, we take a computational approach to prove b ≥ 11184810 and provide all possible values of a if a (mod 11184810) is in U. Moreover, we explicitly construct nontrivial arithmetic progressions a (mod b) in U with ω (b) = 8 , 9, 10, or 11, and provide potential nontrivial arithmetic progressions a (mod b) in U such that ω (b) = s for any fixed s ≥ 12. Furthermore, we improve the upper bound estimate of numbers of the form p + 2 k by Habsieger and Roblot in 2006 to 0.490341088858244 by enhancing their algorithm and employing GPU computation. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic distribution of coefficients of general L -functions over arithmetic progressions without the Ramanujan conjecture. As an application, we consider the high mean of Fourier coefficients of holomorphic forms or Maass forms for Γ = SL (2 , Z) over arithmetic progressions, and improve the results of Jiang and Lü [10]. Our new results remove the restriction to prime module and improve the interval length of module q. [ABSTRACT FROM AUTHOR]

We give a purely combinatorial proof for the infinitary van der Waerden's theorem. [ABSTRACT FROM AUTHOR]

The global impact of the SARS-CoV-2 pandemic has underscored the need for a deeper understanding of viral evolution to anticipate new viruses or variants. Genetic recombination is a fundamental mechanism in viral evolution, yet it remains poorly understood. In this study, we conducted a comprehensive research on the genetic regions associated with genetic recombination features in SARS-CoV-2. With this aim, we implemented a two-phase transfer learning approach using genomic spectrograms of complete SARS-CoV-2 sequences. In the first phase, we utilized a pre-trained VGG-16 model with genomic spectrograms of HIV-1, and in the second phase, we applied HIV-1 VGG-16 model to SARS-CoV-2 spectrograms. The identification of key recombination hot zones was achieved using the Grad-CAM interpretability tool, and the results were analyzed by mathematical and image processing techniques. Our findings unequivocally identify the SARS-CoV-2 Spike protein (S protein) as the pivotal region in the genetic recombination feature. For non-recombinant sequences, the relevant frequencies clustered around 1/6 and 1/12. In recombinant sequences, the sharp prominence of the main hot zone in the Spike protein prominently indicated a frequency of 1/6. These findings suggest that in the arithmetic series, every 6 nucleotides (two triplets) in S may encode crucial information, potentially concealing essential details about viral characteristics, in this case, recombinant feature of a SARS-CoV-2 genetic sequence. This insight further underscores the potential presence of multifaceted information within the genome, including mathematical signatures that define an organism's unique attributes. [ABSTRACT FROM AUTHOR]

Recently, Mangerel extended the Matomäki–Radziwiłł theorem to a large collection of unbounded multiplicative functions in typical short intervals. In this paper, we combine Mangerel's result with Halász-type result recently established by Granville, Harper and Soundararajan to consider the distribution of a class of multiplicative functions in short intervals. First, we prove cancellation in the sum of the coefficients of the standard L-function of an automorphic irreducible cuspidal representation of GL m over Q with unitary central character in typical intervals of length h (log X) c with h = h (X) → ∞ and some constant c > 0 (under Vinogradov–Korobov zero-free region and GRC). Then we also establish a non-trivial bound for the product of divisor-bounded multiplicative functions with the Liouville function in arithmetic progressions over typical short intervals. [ABSTRACT FROM AUTHOR]

The Hurwitz zeta-function ζ (s , α) , s = σ + i t , with parameter 0 < α ⩽ 1 is a generalization of the Riemann zeta-function ζ (s) ( ζ (s , 1) = ζ (s) ) and was introduced at the end of the 19th century. The function ζ (s , α) plays an important role in investigations of the distribution of prime numbers in arithmetic progression and has applications in special function theory, algebraic number theory, dynamical system theory, other fields of mathematics, and even physics. The function ζ (s , α) is the main example of zeta-functions without Euler's product (except for the cases α = 1 , α = 1 / 2 ), and its value distribution is governed by arithmetical properties of α. For the majority of zeta-functions, ζ (s , α) for some α is universal, i.e., its shifts ζ (s + i τ , α) , τ ∈ R , approximate every analytic function defined in the strip { s : 1 / 2 < σ < 1 } . For needs of effectivization of the universality property for ζ (s , α) , the interval for τ must be as short as possible, and this can be achieved by using the mean square estimate for ζ (σ + i t , α) in short intervals. In this paper, we obtain the bound O (H) for that mean square over the interval [ T − H , T + H ] , with T 27 / 82 ⩽ H ⩽ T σ and 1 / 2 < σ ⩽ 7 / 12 . This is the first result on the mean square for ζ (s , α) in short intervals. In forthcoming papers, this estimate will be applied for proof of universality for ζ (s , α) and other zeta-functions in short intervals. [ABSTRACT FROM AUTHOR]

Finding the Frobenius number and the genus of any numerical semigroup S is a well-known open problem. Similarly, it has been studied how to express the Frobenius number and the genus of a quotient of a numerical semigroup. In this paper, by enumerating the Hilbert series of each type of numerical semigroup, we show an expression for the genus of a quotient of numerical semigroups generated by one of the following series: arithmetic progression, geometric series, and Pythagorean triple. [ABSTRACT FROM AUTHOR]

This paper is concerned with finite sequences of integers that may be written as sums of squares of two nonzero integers. We first find infinitely many integers n such that n , n + h and n + k are all sums of two squares where h and k are two arbitrary integers, and as an immediate corollary obtain, in parametric terms, three consecutive integers that are sums of two squares. Similarly we obtain n in parametric terms such that all the four integers n , n + 1 , n + 2 , n + 4 are sums of two squares. We also find infinitely many integers n such that all the five integers n , n + 1 , n + 2 , n + 4 , n + 5 are sums of two squares, and finally, we find infinitely many arithmetic progressions, with common difference 4, of five integers all of which are sums of two squares. [ABSTRACT FROM AUTHOR]

We investigate the relations between the rings E , G and D of values taken at algebraic points by arithmetic Gevrey series of order either −1 (E -functions), 0 (analytic continuations of G -functions) or 1 (renormalization of divergent series solutions at ∞ of E -operators) respectively. We prove in particular that any element of G can be written as multivariate polynomial with algebraic coefficients in elements of E and D , and is the limit at infinity of some E -function along some direction. This prompts to defining and studying the notion of mixed functions, which generalizes simultaneously E -functions and arithmetic Gevrey series of order 1. Using natural conjectures for arithmetic Gevrey series of order 1 and mixed functions (which are analogues of a theorem of André and Beukers for E -functions) and the conjecture D ∩ E = Q ‾ (but not necessarily all these conjectures at the same time), we deduce a number of interesting Diophantine results such as an analogue for mixed functions of Beukers' linear independence theorem for values of E -functions, the transcendence of the values of the Gamma function and its derivatives at all non-integral algebraic numbers, the transcendence of Gompertz constant as well as the fact that Euler's constant is not in E. [ABSTRACT FROM AUTHOR]

The adversary makes malicious samples capable of triggering erroneous judgments in deep learning models by introducing imperceptible perturbations to the original benign texts. These malicious samples are referred to as adversarial texts. The exploration of adversarial text generation methods not only facilitates our understanding of the robustness of mainstream deep neural networks against such adversarial attacks but also aids in developing appropriate defensive strategies. Nevertheless, the mainstream research on textual adversarial attacks has mainly focused on attack effectiveness, overlooking the associated attack cost. For real-world attacks, considerations such as the time cost, material cost, manpower cost, and various constraints are also crucial. In this paper, we propose a low-cost adversarial text generation method based on the universal strategy in the black-box attack scenario, Universal Chinese Text Tricker (UCTT), for tendency classification on Chinese texts. UCTT is both text-independent and model-independent, which markedly reduces its attack cost. Instead of crafting adversarial texts for a specific text, UCTT generates universal perturbations based on a universal word substitution list, which is applicable to any data in tendency classification datasets. Given a perturbation rate, we can use the word list to craft adversarial texts by simple substitutions without accessing the target model. In the framework of adversarial text generation based on word importance, UCTT utilizes count, arithmetic progression, linear normalization, and nonlinear normalization to calculate the scores of the important words in the dataset and then computes the candidate word frequencies, which in turn constructs the universal word substitution list. Compared with other black-box methods, the experimental results on real-world tendency classification datasets show that UCTT exhibits an effective attack capability while significantly reducing the attack cost. Compared to the powerful baseline we designed that exceeds the SOTA, UCTT improves the efficiency of adversarial text generation by up to a factor of 7 without accessing the target model. In addition to demonstrating excellent attack performance on mainstream models, UCTT is also capable of attacking the powerful ChatGPT in the physical world, which cannot be directly attacked by traditional adversarial text generation methods due to the hard labels produced by the target model. [ABSTRACT FROM AUTHOR]

Let τ denote the divisor function and ℋ = { h 1 , ... , h k } be an admissible set. We prove that there are infinitely many n for which the product ∏ i = 1 k (n + h i) is square-free and ∑ i = 1 k τ (n + h i) ≤ ⌊ ρ k ⌋ , where ρ k is asymptotic to 2 1 2 6 2 8 5 3 k 2 . It improves a previous result of Ram Murty and Vatwani, replacing 3 / 4 by 2 1 2 6 / 2 8 5 3. The main ingredients in our proof are the higher rank Selberg sieve and Irving–Wu–Xi estimate for the divisor function in arithmetic progressions to smooth moduli. [ABSTRACT FROM AUTHOR]

Keating and Rudnick [The variance of the number of prime polynomials in short intervals and in residue classes, Int. Math. Res. Not.2014(1) (2014) 259–288] derived asymptotic formulas for the variances of primes in arithmetic progressions and short intervals in the function field setting. Here we consider the hybrid problem of calculating the variance of primes in intersections of arithmetic progressions and short intervals. Keating and Rudnick used an involution to translate short intervals into arithmetic progressions. We follow their approach but apply this involution, in addition, to the arithmetic progressions. This creates dual arithmetic progressions in the case when the modulus Q is a polynomial in q [ T ] such that Q (0) ≠ 0. The latter is a restriction which we keep throughout our paper. At the end, we discuss what is needed to relax this condition. [ABSTRACT FROM AUTHOR]

For an integer $b\geq 2$ , a positive integer is called a b -Niven number if it is a multiple of the sum of the digits in its base- b representation. In this article, we show that every arithmetic progression contains infinitely many b -Niven numbers. [ABSTRACT FROM AUTHOR]

For a positive integer t ≥ 2 , let b t (n) denote the number of t-regular partitions of a nonnegative integer n. Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo 2 for b 9 (n) and b 19 (n) . We prove some specific cases of two conjectures of Keith and Zanello on self-similarities of b 9 (n) and b 19 (n) modulo 2. For t ∈ { 6 , 10 , 14 , 15 , 18 , 20 , 22 , 26 , 27 , 28 } , Keith and Zanello conjectured that there are no integers A > 0 and B ≥ 0 for which b t (A n + B) ≡ 0 (mod 2) for all n ≥ 0 . We prove that, for any t ≥ 2 and prime ℓ , there are infinitely many arithmetic progressions A n + B for which ∑ n = 0 ∞ b t (A n + B) q n ≢ 0 (mod ℓ) . Next, we obtain quantitative estimates for the distributions of b 6 (n) , b 10 (n) and b 14 (n) modulo 2. We further study the odd densities of certain infinite families of eta-quotients related to the 7-regular and 13-regular partition functions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the exact delay range making input-delay systems unstabilizable. The exact range means that the systems are unstabilizable if and only if the delay is within this range. Contributions of this paper are to characterize the exact range and to present a computation method to derive this range. It is shown that the above range is related to unstable eigenvalues of the system matrix. In the discrete-time case, if none of the eigenvalues of the system matrix is a unit root, then the above range is a finite set. If there exist some eigenvalues which are unit roots, this range may be a finite set or may be composed of several arithmetic progressions. When this range contains finite elements, the number of these elements is bounded by the geometric multiplicities of eigenvalues. When this range contains arithmetic progressions, the number of such progressions is bounded by the above multiplicities. On the other hand, our results can provide an upper bound for the well-known delay margin, which is the maximal delay value achievable by a robust controller to stabilize systems. [ABSTRACT FROM AUTHOR]

We determine new values of certain Dirichlet series and related infinite series. These formulas extend results of several authors. To obtain these results we apply recent expansions of higher derivative formulas of trigonometric functions. We also investigate the transcendentality of values of these series and arithmetic relations of the values of certain related infinite series. [ABSTRACT FROM AUTHOR]

The document titled "Problems and Solutions" is an article from the American Mathematical Monthly. It contains a list of proposed mathematical problems along with their solutions. The problems cover various topics such as permutations, triangles, sets of positive integers, Fibonacci numbers, and integer matrices. The article provides detailed instructions on how to submit proposed problems and solutions. [Extracted from the article]

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.

Symmetric Lamb waves with a phase velocity exceeding the expansion wave velocity in an infinite medium are considered. It has been proven that internal waves (i.e., solutions of the wave equation, which have zero values of the strain and stress components on the surface and nonzero values in the plate bulk) may exist in this region of phase velocities. Parameters of the internal waves (phase velocity, frequency, and wavelength) are calculated; it has been proven that frequencies of the internal waves with the same phase velocity make an arithmetic progression. Several internal waves are considered, and cross sections of the corresponding strained plates are presented (as well as the distributions of the maximum tension and shear values). [ABSTRACT FROM AUTHOR]

In this paper, we study the class of affine semigroups generated by integral vectors, whose components are in generalized arithmetic progression and we observe that the defining ideal is determinantal. We also give a sufficient condition on the defining ideal of the semigroup ring for the equality of the Betti numbers of the defining ideal and those of its initial ideal. We introduce the notion of an affine semigroup generated by join of two affine semigroups and show that it preserves some nice properties, including Cohen-Macaulayness, when the constituent semigroups have those properties. [ABSTRACT FROM AUTHOR]

In this article, we study the distribution of index of Farey fractions by providing asymptotic formulas for moments of index of Farey fractions twisted by Dirichlet characters for Farey fractions with B -free denominators. Additionally, we reconsider the square-free case earlier done in Alkan et al. (Ramanujan J 16(2):131–161, 2008), and obtain new results for moments of indices with square-free denominators. We also obtain higher level correlation measures of the index function, generalizing earlier known results on two level correlations. [ABSTRACT FROM AUTHOR]

A conjecture of Erdős states that, for any large prime q, every reduced residue class (mod ⁡ q) can be represented as a product p 1 ⁢ p 2 of two primes p 1 , p 2 ≤ q . We establish a ternary version of this conjecture, showing that, for any sufficiently large cube-free integer q, every reduced residue class (mod ⁡ q) can be written as p 1 ⁢ p 2 ⁢ p 3 with p 1 , p 2 , p 3 ≤ q primes. We also show that, for any ε > 0 and any sufficiently large integer q, at least (2 3 - ε) ⁢ φ ⁢ (q) reduced residue classes (mod ⁡ q) can be represented as a product p 1 ⁢ p 2 of two primes p 1 , p 2 ≤ q . The problems naturally reduce to studying character sums. The main innovation in the paper is the establishment of a multiplicative dense model theorem for character sums over primes in the spirit of the transference principle. In order to deal with possible local obstructions we establish bounds for the logarithmic density of primes in certain unions of cosets of subgroups of ℤ q × of small index and study in detail the exceptional case that there exists a quadratic character ψ ⁢ (mod ⁡ q) such that ψ ⁢ (p) = - 1 for very many primes p ≤ q . [ABSTRACT FROM AUTHOR]

This article explores the Erdős-Szekeres conjecture, a mathematical problem concerning geometric patterns in sets of points. The conjecture was proposed in 1935 and has been the focus of extensive research. Various mathematicians, including Tóth, Valtr, and Suk, have made significant contributions to proving the conjecture. Suk's proof is described as a moment of brilliance in applying the right tools. The article concludes with a quote from Erdős emphasizing the importance of an open mind in mathematics. [Extracted from the article]

We provide a general framework for proving asymptotic equidistribution, convexity, and log-concavity of coefficients of generating functions on arithmetic progressions. Our central tool is a variant of Wright's Circle Method proven by two of the authors with Bringmann and Ono, following work of Ngo and Rhoades. We offer a selection of different examples of such results, proving asymptotic equidistribution results for several partition statistics, modular sums of Betti numbers of two- and three-flag Hilbert schemes, and the number of cells of dimension a (mod b) of a certain scheme central in work of Göttsche. [ABSTRACT FROM AUTHOR]

The document titled "Problems and Solutions" is an article from the American Mathematical Monthly that provides a list of proposed mathematical problems along with their solutions. The problems cover various topics and are contributed by different mathematicians from around the world. The article also includes instructions for submitting proposed problems and solutions. Additionally, it features a solution to a specific problem related to logarithmic trigonometric integrals. The given text presents a mathematical equation and its evaluation. The equation involves integrals and substitutions, and it is used to calculate a value called "I." The text provides step-by-step calculations and uses mathematical identities to simplify the equation. The final result is that I equals a specific value. The text also mentions the use of Fourier sine series in the calculations. The given text contains mathematical formulas and solutions to mathematical problems. The first part of the text presents a formula for the integral of a sine function, depending on whether the value of n is even or odd. The second part discusses the solution to a problem involving operator norms. The solution involves applying the Gram-Schmidt process to a basis in a certain space and obtaining an orthonormal basis. The text also mentions the names of individuals and groups who have solved the problems. The given text discusses the properties of a function f that belongs to a set S. It states that for any function f in S, the sum of the squares of the values of f and its derivative is equal to a constant. It also mentions that the supremum of [Extracted from the article]

We answer a number of questions of Erdős on the existence of arithmetic progressions in k -full numbers (i.e. integers with the property that every prime divisor necessarily occurs to at least the k th power). Further, we deduce a variety of arithmetic constraints upon such progressions, under the assumption of the a b c -conjecture of Masser and Oesterlé. [ABSTRACT FROM AUTHOR]

Purpose: To determine the association between retinal blood vessel flow and geometric parameters and the risk of diabetic retinopathy (DR) progression through a 2-year prospective cohort study. Methods: Patients with type 2 diabetes mellitus (T2DM) were recruited from a diabetic registry between November 2017 and March 2019. All participants underwent standardized examinations at the baseline and 2-year follow-up visit, and the presence and severity of DR were assessed based on standard seven-field color fundus photographs. They also underwent swept-source optical coherence tomography angiography (OCTA) imaging to obtain measurements of foveal avascular zone area, blood vessel density (VD), fractal dimension (FD), blood vessel tortuosity (BVT) in the superficial capillary plexus (SCP) and deep capillary plexus (DCP). Results: A total of 233 eyes of 125 patients were included, and 40 eyes (17.17%) experienced DR progression within 2 years. DR progression was significantly associated with lower baseline VD (odds ratio [OR] 2.323 per SD decrease; 95% confidence interval [CI] 1.456–3.708; P < 0.001), lower FD (OR, 2.484 per SD decrease; 95% CI 1.268–4.867; P = 0.008), and higher BVT (OR, 2.076 per SD increase; 95% CI 1.382–3.121; P < 0.001) of the DCP after adjusting for confounding factors. The addition of OCTA metrics improved the predictive ability of the original model for DR progression (area under the curve [AUC] from 0.725 to 0.805; P = 0.022). Conclusions: OCTA-derived VD, FD and BVT in the DCP were independent predictors of DR progression and showed additive value when added to established risk models predicting DR progression. [ABSTRACT FROM AUTHOR]

This document provides a summary of a special issue of the journal Mathematics, which features research papers in the field of algebraic, analytic, and computational number theory. The papers cover a wide range of topics, including generalized Lucas pseudoprimality, Fibonacci numbers, piecewise linear difference equations, properties of Euler's function, regularities in ordered semihypergroups, zero-density results for automorphic functions, properties of Fibonacci finite operator quaternions, and more. The authors of the special issue believe that the techniques used in these papers have the potential to yield new results in these areas of mathematics. [Extracted from the article]

Let F q n be a finite field with qn elements. For a positive divisor r of q n − 1 , the element α ∈ F q n * is called r-primitive if its multiplicative order is (q n − 1) / r . Also, for a non-negative integer k, the element α ∈ F q n is k-normal over F q if gcd (α x n − 1 + α q x n − 2 + ⋯ + α q n − 2 x + α q n − 1 , x n − 1) in F q n [ x ] has degree k. In this paper we discuss the existence of elements in arithmetic progressions { α , α + β , α + 2 β , ... α + (m − 1) β } ⊂ F q n with α + (i − 1) β being ri-primitive and at least one of the elements in the arithmetic progression being k-normal over F q . We obtain asymptotic results for general k , r 1 , ... , r m and concrete results when k = r i = 2 for i ∈ { 1 , ... , m } . [ABSTRACT FROM AUTHOR]

We prove Bombieri–Vinogradov type theorems for primes with a missing digit in their b$b$‐adic expansion for some large positive integer b$b$. The proof is based on the circle method, which relies on the Fourier structure of the integers with a missing digit and the exponential sums over primes in arithmetic progressions. Combining our results with the semi‐linear sieve, we obtain an upper bound and a lower bound of the correct order of magnitude for the number of primes of the form p=1+m2+n2$p=1+m^2+n^2$ with a missing digit in a large odd base b$b$. [ABSTRACT FROM AUTHOR]

Let ki∈N$k_i\in {\mathbb {N}}$(i⩾1)$(i\geqslant 1)$ satisfy 2⩽k1⩽k2⩽⋯$2\leqslant k_1\leqslant k_2\leqslant \cdots$. Freĭman's theorem shows that when j∈N$j\in {\mathbb {N}}$, there exists s=s(j)∈N$s=s(j)\in {\mathbb {N}}$ such that all large integers n$n$ are represented in the form n=x1kj+x2kj+1+⋯+xskj+s−1$n=x_1^{k_j}+x_2^{k_{j+1}}+\cdots +x_s^{k_{j+s-1}}$, with xi∈N$x_i\in {\mathbb {N}}$, if and only if ∑ki−1$\sum k_i^{-1}$ diverges. We make this theorem effective by showing that, for each fixed j$j$, it suffices to impose the condition ∑i=j∞ki−1⩾2logkj+4.71.$$\begin{equation*} \sum _{i=j}^\infty k_i^{-1}\geqslant 2\log k_j +4.71. \end{equation*}$$More is established when the sequence of exponents forms an arithmetic progression. Thus, for example, when k∈N$k\in {\mathbb {N}}$ and s⩾100(k+1)2$s\geqslant 100(k+1)^2$, all large integers n$n$ are represented in the form n=x1k+x2k+1+⋯+xsk+s−1$n=x_1^k+x_2^{k+1}+\cdots +x_s^{k+s-1}$, with xi∈N$x_i\in {\mathbb {N}}$. [ABSTRACT FROM AUTHOR]

Let $\ell $ and p be (not necessarily distinct) prime numbers and F be a global function field of characteristic $\ell $ with field of constants $\kappa $. Assume that there exists a prime $P_\infty $ of F which has degree $1$ and let $\mathcal {O}_F$ be the subring of F consisting of functions with no poles away from $P_\infty $. Let $f(X)$ be a polynomial in X with coefficients in $\kappa $. We study solutions to Diophantine equations of the form $Y^{n}=f(X)$ which lie in $\mathcal {O}_F$ and, in particular, show that if m and $f(X)$ satisfy additional conditions, then there are no nonconstant solutions. The results apply to the study of solutions to $Y^{n}=f(X)$ in certain rings of integers in $\mathbb {Z}_{p}$ -extensions of F known as constant $\mathbb {Z}_p$ -extensions. We prove similar results for solutions in the polynomial ring $K[T_1, \ldots , T_r]$ , where K is any field of characteristic $\ell $ , showing that the only solutions must lie in K. We apply our methods to study solutions of Diophantine equations of the form $Y^n=\sum _{i=1}^d (X+ir)^m$ , where $m,n, d\geq 2$ are integers. [ABSTRACT FROM AUTHOR]

We show that the Freiman–Ruzsa theorem, characterising finite sets with bounded doubling, leads to an alternative proof of a characterisation of Meyer sets, that is, relatively dense subsets of Euclidean spaces whose difference sets are uniformly discrete. [ABSTRACT FROM AUTHOR]

Numbers of the form k · p n + 1 with the restriction k < p n are called generalized Proth numbers. For a fixed prime p we denote them by T p . The underlying structure of T 2 (Proth numbers) was investigated in [2]. In this paper the authors extend their results to all primes. An efficiently computable upper bound for the reciprocal sum of primes in T p is presented. All formulae were implemented and verified by the PARI/GP computer algebra system. We show that the asymptotic density of ⋃ p ∈ P T p is log 2 . [ABSTRACT FROM AUTHOR]

An (a, d)-edge-antimagic graceful labeling is a bijection g from V (G) ∪ E(G) into f{1, 2, ..., |V (G)| + |E(G)|} such that for each edge xy ∈ E(G), |g(x) + g(y) - g(xy)| form an arithmetic progression starting from a and having a common difference d: An (a; d)- edge-antimagic graceful labeling is called super (a, d)-edge-antimagic graceful if g(V (G)) = {1, 2, ..., |V (G)|}: A graph that admits an super (a, d)-edge-antimagic graceful labeling is called a super (a, d)-edge-antimagic graceful graph. In this paper, we prove the super (a, d) edge antimagic gracefulness of regular graphs. Later, we study the non-regular graph is super (a, 1)-edge-antimagic graceful graph. Finally, we find super edge-antimagic graceful labeling of some classes of graphs. [ABSTRACT FROM AUTHOR]

The aggregation operators are important tools in the process of information aggregation. The aim of this paper is to investigate arithmetic aggregation operators under the type2 picture fuzzy environment with the help of new operations on type-2 picture fuzzy numbers. The type-2 picture fuzzy set is an extended version of Cuong’s picture fuzzy set, which not only considers the degree of acceptance or rejection, but also takes into account the neutral degree and invalid degree during the analysis. Under these environments, by using the new operations on type-2 picture fuzzy numbers, several series of arithmetic aggregation operators for type-2 picture fuzzy sets are proposed, namely type-2 picture fuzzy weighted average operators, type-2 picture fuzzy ordered weighted average operators, and type-2 picture fuzzy hybrid average operators, along with their desirable properties. Furthermore, a decision-making approach based on these operators is also presented to make the evaluation result more objective. Finally, an illustrative example is provided to demonstrate the practicality and reasonableness of the proposed method. [ABSTRACT FROM AUTHOR]