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Multivariate multipoint evaluation is the problem of evaluating a multivariate polynomial, given as a coefficient vector, simultaneously at multiple evaluation points. In this work, we show that there exists a deterministic algorithm for multivariate multipoint evaluation over any finite field \(\mathbb {F}\) that outputs the evaluations of an m-variate polynomial of degree less than d in each variable at N points in time, \(\begin{equation*} (d^m+N)^{1+o(1)}\cdot {{\sf poly}}(m,d,\log |\mathbb {F}|), \end{equation*}\) for all \(m\in \mathbb {N}\) and all sufficiently large \(d\in \mathbb {N}\). A previous work of Kedlaya and Umans (FOCS 2008 and SICOMP 2011) achieved the same time complexity when the number of variables m is at most \(d^{o(1)}\) and had left the problem of removing this condition as an open problem. A recent work of Bhargava, Ghosh, Kumar, and Mohapatra (STOC 2022) answered this question when the underlying field is not too large and has characteristic less than \(d^{o(1)}\). In this work, we remove this constraint on the number of variables over all finite fields, thereby answering the question of Kedlaya and Umans over all finite fields. Our algorithm relies on a non-trivial combination of ideas from three seemingly different previously known algorithms for multivariate multipoint evaluation, namely the algorithms of Kedlaya and Umans, that of Björklund, Kaski, and Williams (IPEC 2017 and Algorithmica 2019), and that of Bhargava, Ghosh, Kumar, and Mohapatra, together with a result of Bombieri and Vinogradov from analytic number theory about the distribution of primes in an arithmetic progression. We also present a second algorithm for multivariate multipoint evaluation that is completely elementary and, in particular, avoids the use of the Bombieri–Vinogradov theorem. However, it requires a mild assumption that the field size is bounded by an exponential tower in d of bounded height. More specifically, our second algorithm solves the multivariate multipoint evaluation problem over a finite field \(\mathbb {F}\) in time, \(\begin{equation*} (d^m+N)^{1+o(1)}\cdot {{\sf poly}}(m,d,\log |\mathbb {F}|), \end{equation*}\) for all \(m\in \mathbb {N}\) and all sufficiently large \(d\in \mathbb {N}\) , provided that the size of the finite field \(\mathbb {F}\) is at most \((\exp (\exp (\exp (\cdots (\exp (d)))))\) , where the height of this tower of exponentials is fixed. [ABSTRACT FROM AUTHOR]

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 apply a classical Wirsing's theorem and study k -free numbers, k-full numbers and perfect powers with prime factors in an arithmetic progression. [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]

Calculations about compound interest serve as the basis of most financial decisions; therefore, it is imperative to explore what mathematical knowledge people need to correctly calculate simple and compound interest. The aim of this study is to analyze the relationship between college students' competence in calculating simple and compound interest and their understanding of the arithmetic and geometric progressions. It is also pointed out whether the results vary according to gender. Population proportion tests are carried out, and gender proportion differences are considered for inferential analysis. The dichotomous Probit model was used for correlation analysis. Results demonstrate that 59.8% of students know how to formulate a whole-number succession, and only 30.9% in the case of fractional numbers. Less than 50% of students can calculate compound interest, but 76.7% can calculate simple interest. There is no significant difference between men and women. The results show a positive relationship between male students' competence in calculating compound interest with the possibility to correctly formulate a geometric succession. Findings can be useful for mathematics teaching strategy design and its applications in finance contexts with the purpose of training students to be better at finance decision making. [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]

In this article, we study the cyclicity problem of elliptic curves |$E/\mathbb{Q}$| modulo primes in a given arithmetic progression. We extend the recent work of Akbal and Güloğlu by proving an unconditional asymptotic for such a cyclicity problem over arithmetic progressions for elliptic curves E , which also presents a generalization of the previous works of Akbary, Cojocaru, M.R. Murty, V.K. Murty and Serre. In addition, we refine the conditional estimates of Akbal and Güloğlu, which gives log-power savings (for small moduli) and consequently improves the work of Cojocaru and M.R. Murty. Moreover, we study the average exponent of E modulo primes in a given arithmetic progression and obtain several conditional and unconditional estimates, extending the previous works of Freiberg, Kim, Kurlberg and Wu. [ABSTRACT FROM AUTHOR]

The equation |$x^2 + 1 = 0\mod p$| has solutions whenever p  = 2 or |$4n + 1$|⁠. A famous theorem of Fermat says that these primes are exactly the ones that can be described as a sum of two squares. The roots of the former equation are equidistributed is a beautiful theorem of Duke, Friedlander and Iwaniec. The angles associated to the representation of such prime as a sum of squares are equidistributed is a famous theorem of Hecke. We give a natural way to associate between roots and angles and prove that the joint equidistribution of the sequence of pairs of roots and angles is equidistributed as well. Our approach involves an automorphic interpretation, which reduces the problem to the study of certain Poincare series on an arithmetic quotient of |$SL_2(\mathbb{R})$|⁠. Since our Poincare series have a nontrivial dependence on their Iwasawa θ -coordinate, they do not factor into functions on the upper half plane, as in the case studied by Duke et al. Spectral analysis on these higher dimensional varieties involves the nonspherical spectrum, making this paper the first complete study of a nonspherical equidistribution problem, with an arithmetic application. A couple of notable challenges we had to overcome were that of obtaining pointwise bounds for nonspherical Eisenstein series and utilizing a non-spherical analogue of the Selberg inversion formula, which we believe may have further implications beyond this work. [ABSTRACT FROM AUTHOR]

Accurate estimation of design floods is necessary for developing effective flood‐management strategies. Climate change (CC) studies on floods generally consider alterations in mean runoff using ensembles compared to a base period. In this study, we examined the plausibility and implications of applying individual climate model‐generated flows versus their ensembles to estimate peak floods (magnitude and timing of occurrence), using Budhigandaki River Basin of Nepal as a case study. Annual maximum one‐day floods were derived for four future climate scenario projections (cold‐dry, cold‐wet, warm‐wet, and warm‐dry) from simulated daily flow series. Future floods of six return periods estimated for the individual climate scenarios were compared with their "Ensemble" (combiner for the ensemble series is the arithmetic mean of daily floods), "Average," and 'Baseline." Results showed that magnitudes of the flood peaks are such that those estimated using "Ensemble" < "Average" < individual series. We conclude that ensemble series should not be used for flood estimation because of the averaging effect. Designers should consider at the least the "Average" instead of the "Ensemble" series while designing climate‐resilient flood structures. Furthermore, the occurrences of flood peaks are likely to be confined within the monsoon season for the "Ensemble" but spread out in the other months for the individual climate scenarios. This could have direct implications on the availability and mobilization of resources as well as the need for a year‐round operational early warning system for flood risk management. [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 paper, we investigate arithmetic progressions in integer-valued quadratic polynomials. We prove two results for integer-valued quadratic polynomials. First, we prove that there is no three-term arithmetic progression with a perfect square as a common difference for a class of integer-valued quadratic polynomials. We then find a class of numbers depending on integer-valued quadratic polynomials, which cannot work as a common difference for three-term arithmetic progression. Further, we deduce some results for arithmetic progressions in the sequence of polygonal and centered polygonal numbers. [ABSTRACT FROM AUTHOR]

Communication among hard-of-hearing individuals presents challenges, and to facilitate communication, sign language is preferred. Many people in the deaf and hard-of-hearing communities struggle to understand sign language due to their lack of sign-mode knowledge. Contemporary researchers utilize glove and vision-based approaches to capture hand movement and analyze communication; most researchers use vision-based techniques to identify disabled people's communication because the glove-based approach causes individuals to feel uncomfortable. However, the glove solution successfully identifies motion and hand dexterity, even though it only recognizes the numbers, words, and letters being communicated, failing to identify sentences. Therefore, artificial intelligence (AI) is integrated with the sign language prediction system to identify disabled people's sentence-based communication. Here, wearable glove-related sign language information is utilized to analyze the recognition system's efficiency. The collected inputs are processed using progression learning deep convolutional neural networks (PLD-CNNs). The technique known as progression learning processes sentences by dividing them into words, creating a training dataset. The model assists in efforts to understand sign language sentences. A memetic optimization algorithm is used to calibrate network performance, minimizing recognition optimization problems. This process maximizes convergence speed and reduces translation difficulties, enhancing the overall learning process. The created system is developed using the MATLAB (R2021b) tool, and its proficiency is evaluated using performance metrics. The experimental findings illustrate that the proposed system works by recognizing sign language movements with excellent precision, recall, accuracy, and F1 scores, rendering it a powerful tool in the detection of gestures in general and sign-based sentences in particular. [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]

We study the existence of products of primes in arithmetic progressions, building on the work of Ramaré and Walker. One of our main results is that if q$q$ is a large modulus, then any invertible residue class mod q$q$ contains a product of three primes where each prime is at most q6/5+ε$q^{6/5+\epsilon }$. Our arguments use results from a wide range of areas, such as sieve theory or additive combinatorics, and one of our key ingredients, which has not been used in this setting before, is a result by Heath‐Brown on character sums over primes from his paper on Linnik's theorem. [ABSTRACT FROM AUTHOR]

We make progress on a question posed by Vemuri on the optimal Gaussian decay of harmonic oscillators, proving the original conjecture up to an arithmetic progression of times. The techniques used are a suitable translation of the problem at hand in terms of the free Schrödinger equation, the machinery developed in the work of Cowling, Escauriaza, Kenig, Ponce and Vega (2010), and a lemma which relates decay on average to pointwise decay. Such a lemma produces many more consequences in terms of equivalences of uncertainty principles. Complementing such results, we provide endpoint results in particular classes induced by certain Laplace transforms, both to the decay lemma and to the remaining cases of Vemuri's conjecture, shedding light on the full endpoint question. [ABSTRACT FROM AUTHOR]