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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]

Recently, a lot of works have been done on the study of the failure rate (FR), the aging intensity function (AI) and their properties. In the article, we analyze the aging classes based on monotonicity of new functions viz., arithmetic mean, geometric mean, and harmonic mean of AI function, respectively. Also, the relationships between these functions and the classical AI function are discussed. Moreover, some characterizations of distributions based on these functions are established. Some characterizations of proportional hazard rate and AI family of distributions are discussed. Investigations are illustrated by some examples and applications of aging functions in data analysis. [ABSTRACT FROM AUTHOR]

In recent years, an increasing number of studies have examined the association between mathematical abilities and executive functions (EFs). However, it remains unknown via which mechanisms' mathematical performance is associated with EFs. The current study examined the associations of overall task proficiency, strategy selection, and strategy execution when solving multi-step subtraction problems with three EFs (i.e., updating, inhibition, and shifting). With a choice/no-choice design, 150 fifth-graders' (Mage = 10 years, 11 months) direct subtraction (DS; e.g., 712 − 346 = ?; 712 − 300 = 412, 412 − 40 = 372, and 372 − 6 = 366) and subtraction by addition (SBA; e.g., 712 − 346 = ?; 346 + 54 = 400, 400 + 300 = 700, 700 + 12 = 712, and 54 + 300 + 12 = 366) use to solve multi-step subtractions was examined. Participants were offered ten subtractions in one choice condition (free choice between DS and SBA), based on which we obtained data about task proficiency and strategy selection, and in two no-choice conditions (mandatory use of either DS or SBA), which were used to examine strategy execution. The results showed that task proficiency in the choice condition was associated with updating and inhibition but not with shifting. Furthermore, strategy selection was associated with updating but not with inhibition and shifting. Strategy execution was associated with updating but not with inhibition and shifting. The inclusion of separate strategy selection and execution parameters helped to unravel how arithmetic performance and EFs are associated. [ABSTRACT FROM AUTHOR]

In this paper, we prove explicit asymptotic formulae for some functions used in sieve methods and show that there exists no odd multiperfect number of abundancy four whose squared part is cubefree. [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]

In this paper, we address the problem of the divergent sums of general arithmetic functions over the set of primes. In classical analytic number theory, the sum of the logarithm of the prime numbers plays a crucial role. We consider the sums of powers of the logarithm of primes and its connection with Riemann's zeta function (z.f.). This connection is achieved through the second Chebyshev function of order n, which can be estimated by exploiting the symmetry properties of Riemann's zeta function. Finally, a heuristic approach to evaluating more general sums is also given. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Central China Normal University is the property of Huazhong Normal University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

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]

Let E be an Anderson A -module over C ∞. The period lattice of E is related to its module of special functions by means of a non-canonical isomorphism introduced by the authors in [GM21]. In this paper, we explain how a modification of the inverse map is canonical by interpreting it as a residue morphism along the characteristic graph. This phenomenon has already been observed in various situations. The main innovation of this text is that of costability (costable admissible opens, costable site, etc.) which provides a convenient framework to develop the notion of sheaves of E (C ∞) -valued meromorphic functions on the rigid analytic plane. [ABSTRACT FROM AUTHOR]

In this paper we consider two functions related to the arithmetic and geometric means of element orders of a finite group, showing that certain lower bounds on such functions strongly affect the group structure. In particular, for every prime p, we prove a sufficient condition for a finite group to be p-nilpotent, that is, a group whose elements of p ′ -order form a normal subgroup. Moreover, we characterize finite cyclic groups with prescribed number of prime divisors. [ABSTRACT FROM AUTHOR]

In this paper, a revised version of interval ASM method is proposed for solving multi objective interval transportation problems without converting them to their crisp equivalent forms. Multi-objective transportation problems (MOITP) involving interval parameters are considered. By assigning suitable weights to each objective function, the multiple objective interval transportation problem is reduced to a single objective interval transportation problem (SOITP). Initial basic feasible solution is obtained by applying the proposed method with the help of new interval arithmetic and ranking functions and the optimality is tested using the interval version of MODI method. A numerical example is provided to illustrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new convexity notion for inter-valued functions, known as Geometrically–Arithmetically Cr-h-convex functions (abbreviated as GA-Cr-h-CFs) and explore its properties. The family of GA-Cr-h-CFs simultaneously covers the family of GA-CFs, GA-h-CFs and GA-Cr-CFs. On the other hand, we demonstrate its equivalence with convex functions (CFs), h-CFs, Cr-CFs and Cr-h-CFs via exponential function. We also investigate necessary and sufficient conditions for GA-Cr-h-CFs via two corresponding real-valued GA-h-CFs. Due to significance and applications of mathematical inequalities, we apply our findings and propose several inequalities such as Jensen–Mercer inequality (JMI), Hermite–Hadamard inequality (HHI) and weighted Hermite–Hadamard-type inequalities (wHHIs) for GA-Cr-h-CFs. Through our results, we re-capture many known results and inequalities for subclasses of GA-Cr-h-CFs studied in the recent literature. [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]

Summary: In Wireless Sensor Networks (WSN), energy‐efficient, reliable routing is the core objective for the data transmission process. Anomaly nodes in the communication environment can affect reliable routing and network efficiency. Therefore, the present research created a novel Arithmetic Optimization‐based Rumor Routing Protocol with SKINNY Crypto mechanism (AORRP‐SCrypt) for secure Routing in WSN. Primarily, the required nodes are deployed in the related WSN environment. The arithmetic function traced the nodes' higher energy consumption and malicious action to avoid security vulnerabilities during data transmission. The sensed data packets are transmitted based on the Rumor routing function of the proposed system. Furthermore, to avoid data access by unauthorized users, it is encrypted using the SKINNY Crypto mechanism. Finally, the network efficiency and security performance are validated in the NS3 platform. The model gained 98.1% network throughput, 95.6% PDR, a lower energy consumption of between 0.001 and 0.01 J, and a 99% confidentiality rate. The developed system provided increased network security and efficiency. [ABSTRACT FROM AUTHOR]

Paul Erdos and Carl Pomerance have proofs on an asymptotic upper bound on the number of preimages of Euler's totient function ϕ and the sum-of-divisors functions σ. In this paper, we will extend the upper bound to the number of preimages of iterates of ϕ and σ. Using these new asymptotic upper bounds, a conjecture in de Koninck and Kátai's paper, "On the uniform distribution of certain sequences involving the Euler totient function and the sum of divisors function" is now proven and many corollaries follow from their proven conjecture. [ABSTRACT FROM AUTHOR]

We investigate function field analogs of the distribution of primes, and prime k-tuples, in "very short intervals" of the form I(f) := {f(x)+a : a ∈ Fp} for f(x) ∈ Fp[x] and p prime, as well as cancellation in sums of function field analogs of the Möbius µ function and its correlations (similar to sums appearing in Chowla's conjecture). For generic f, i.e., for f a Morse polynomial, the error terms are roughly of size O(√p) (with typical main terms of order p). For non-generic f we prove that independence still holds for "generic" set of shifts. We can also exhibit examples for which there is no cancellation at all in Möbius/Chowla type sums (in fact, it turns out that (square root) cancellation in Möbius sums is equivalent to (square root) cancellation in Chowla type sums), as well as intervals where the heuristic "primes are independent" fails badly. The results are deduced from a general theorem on correlations of arithmetic class functions; these include characteristic functions on primes, the Möbius µ function, and divisor functions (e.g., function field analogs of the Titchmarsh divisor problem can be treated). We also prove analogous, but slightly weaker, results in the more delicate fixed characteristic setting, i.e., for f(x) ∈ Fq[x] and intervals of the form f(x)+ a for a ∈ Fq, where p is fixed and q = pl grows. [ABSTRACT FROM AUTHOR]

In this paper, we discuss spectrum and energy of Arithmetic function signed graphs with respect to an arithmetical function and present some results. [ABSTRACT FROM AUTHOR]

A necessary and sufficient condition for sum of secondary range symmetric matrices to be secondary range symmetric is given here. Also, the concept of parallel summable secondary range symmetric matrices and some of its characterizations are being explored. [ABSTRACT FROM AUTHOR]

This work is an attempt to apply Lambert series in the theory of univalent functions. We first consider the Hadamard product of Rabotnov function and Lambert series with coefficients derived from the arithmetic function σ n to introduce a normalized linear operator J R α , β z . We then acquire sufficient conditions for J R α , β z to be univalent, starlike and convex, respectively. Furthermore, we discuss the inclusion results in some special classes, namely, spiral-like and convex spiral-like subclasses. In addition, we extend the findings by incorporating two Robin's inequalities, one of which is analogous to the Riemann hypothesis. [ABSTRACT FROM AUTHOR]

In 2019, Andrews and Newman ['Partitions and the minimal excludant', Ann. Comb. 23 (2) (2019), 249–254] introduced the arithmetic function $\sigma \textrm {mex}(n)$ , which denotes the sum of minimal excludants over all the partitions of n. Baruah et al. ['A refinement of a result of Andrews and Newman on the sum of minimal excludants', Ramanujan J. , to appear] showed that the sum of minimal excludants over all the partitions of n is the same as the number of partition pairs of n into distinct parts. They proved three congruences modulo $4$ and $8$ for two functions appearing in this refinement and conjectured two further congruences modulo $8$ and $16$. We confirm these two conjectures by using q -series manipulations and modular forms. [ABSTRACT FROM AUTHOR]

Let G be a connected graph on n vertices with adjacency matrix A G. Associated to G is a polynomial d G (x 1 , ... , x n) of degree n in n variables, obtained as the determinant of the matrix M G (x 1 , ... , x n) , where M G = Diag (x 1 , ... , x n) − A G. We investigate in this article the set V d G (r) of non-negative values taken by this polynomial when x 1 , ... , x n ≥ r ≥ 1. We show that V d G (1) = Z ≥ 0. We show that for a large class of graphs one also has V d G (2) = Z ≥ 0. When V d G (2) ≠ Z ≥ 0 , we show that for many graphs V d G (2) is dense in Z ≥ 0. We give numerical evidence that in many cases, the complement of V d G (2) in Z ≥ 0 might in fact be finite. As a byproduct of our results, we show that every graph can be endowed with an arithmetical structure whose associated group is trivial. [ABSTRACT FROM AUTHOR]

This paper considers some preliminary conclusions of Fejér's integral inequality relevant to strongly geometric arithmetic convex functions that is a type of the class of convex functions and also a mapping to produce a novel trapezoidal form. This mapping is used to derive new theorems and results. By utilization these, some applications were given. [ABSTRACT FROM AUTHOR]

A specially multiplicative arithmetic function is the Dirichlet convolution of two completely multiplicative arithmetic functions. The aim of this paper is to prove explicitly that two mathematical objects, namely (a, b)-Fibonacci sequences and specially multiplicative prime-independent arithmetic functions, are equivalent in the sense that each can be reconstructed from the other. Replacing one with another, the exploration space of both mathematical objects expands significantly [ABSTRACT FROM AUTHOR]

We apply the character sums method of Lenstra, Moree, and Stevenhagen to explicitly compute the constants in the Titchmarsh divisor problem for Kummer fields and division fields of Serre curves. We derive our results as special cases of a general result on the product expressions for the sums in the form ∑ n = 1 ∞ g (n) # G (n) in which g(n) is a multiplicative arithmetic function and { G (n) } is a certain family of Galois groups. Our results extend the application of the character sums method to the evaluation of constants, such as the Titchmarsh divisor constants, that are not density constants. [ABSTRACT FROM AUTHOR]

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]

String constraint solving is the core of various testing and verification approaches for scripting languages. Among algorithms for solving string constraints, flattening is a well-known approach that is particularly useful in handling satisfiable instances. As string/integer conversion is an important function appearing in almost all scripting languages, Abdulla et al. extended the flattening approach to this function recently. However, their approach supports only a special flattening pattern and leaves the support of the general flat regular constraints as an open problem. In this paper, we fill the gap by proposing a complete flattening approach for the string/integer conversion. The approach is built upon a new quantifier elimination procedure for the linear-exponential arithmetic (namely, the extension of Presburger arithmetic with exponential functions, denoted by ExpPA) improved from the one proposed by Cherlin and Point in 1986. We analyze the complexity of our quantifier elimination procedure and show that the decision problem for existential ExpPA formulas is in 3-EXPTIME. Up to our knowledge, this is the first elementary complexity upper bound for this problem. While the quantifier elimination procedure is too expensive to be implemented efficiently, we propose various optimizations and provide a prototypical implementation. We evaluate the performance of our implementation on the benchmarks that are generated from the string hash functions as well as randomly. The experimental results show that our implementation outperforms the state-of-the-art solvers. [ABSTRACT FROM AUTHOR]

Motivated by applications in declarative data analysis, in this article, we study DatalogZ--an extension of Datalog with stratified negation and arithmetic functions over integers. This language is known to be undecidable, so we present the fragment of limit DatalogZ programs, which is powerful enough to naturally capture many important data analysis tasks. In limit DatalogZ, all intensional predicates with a numeric argument are limit predicates that keep maximal or minimal bounds on numeric values. We show that reasoning in limit DatalogZ is decidable if a linearity condition restricting the use of multiplication is satisfied. In particular, limit-linear DatalogZ is complete for -EXP 2 and captures -P2 over ordered datasets in the sense of descriptive complexity. We also provide a comprehensive study of several fragments of limit-linear DatalogZ. We show that semi-positive limit-linear programs (i.e., programs where negation is allowed only in front of extensional atoms) capture coNP over ordered datasets; furthermore, reasoning becomes coNEXP-complete in combined and coNP-complete in data complexity, where the lower bounds hold already for negation-free programs. In order to satisfy the requirements of data-intensive applications, we also propose an additional stability requirement, which causes the complexity of reasoning to drop to EXP in combined and to P in data complexity, thus obtaining the same bounds as for usual Datalog. Finally, we compare our formalisms with the languages underpinning existing Datalog-based approaches for data analysis and show that core fragments of these languages can be encoded as limit programs; this allows us to transfer decidability and complexity upper bounds from limit programs to other formalisms. Therefore, our article provides a unified logical framework for declarative data analysis which can be used as a basis for understanding the impact on expressive power and computational complexity of the key constructs available in existing languages. [ABSTRACT FROM AUTHOR]

In this paper, we establish a mean value result of arithmetic functions over shorter intervals with the Selberg–Delange method using the Hooley–Huxley contour. [ABSTRACT FROM AUTHOR]

Let λf (n) be the n-th normalized Fourier coefficient of f, which is a primitive holomorphic cusp form of even integral weight k ≥ 2 for the full modular group S L2(ℤ). Let also σ(n) and φ(n) be the sum-of-divisors function and the Euler totient function, respectively. In this paper, we are able to establish the asymptotic formula of the sum of the hybrid arithmetic function λfl(n)σc(n)φd(n) over the sparse sequence {n : n = a² + b²}, namely, ∑n≤x λfl(n)σc (n)φd(n)r2(n) for 1 ≤ l ≤ 8, where x is a sufficiently large real number, the function r2(n) denotes the number of representations of n as n = a² + b², a, b, l ∈ ℤ and c, d ∈ ℝ. [ABSTRACT FROM AUTHOR]

For various arithmetic functions f: N - R, the behavior of f(n!) and that of Σn≤Nf(n!) can be intriguing. For instance, for some functions f, we have f(n!) = Σk≤nf(k), for others, we have f(n!) = ΣpͰnf(p) (where the sum runs over all the primes p ≤ n). Also, for some f, their minimum order coincides with limn→∞ f(n!), for others, it is their maximum order that does so. Here, we elucidate such phenomena and more generally, we embark on a study of f(n!) and of Σn≤Nf(n!) for a wide variety of arithmetical functions f. In particular, letting d(n) and σ(n) stand respectively for the number of positive divisors of n and the sum of the positive divisors of n, we obtain new accurate asymptotic expansions for d(n!) and σ(n!). Furthermore, setting σ1(n): = max{d|n:d≤√n} and observing that no one has yet obtained an asymptotic value for Σn≤Nσ(n) as N → →, we show how one can obtain the asymptotic value of Σn≤Nσ1(n!). [ABSTRACT FROM AUTHOR]

In this paper we give upper estimates and asymptotic results for some classes of multiplicative functions. For the proofs we mainly use tools from the convolution arithmetic of number theoretical functions (see [4]). [ABSTRACT FROM AUTHOR]

While quantum multifractality has been widely studied in the physics literature and is by now well understood from the point of view of physics, there is little work on this subject in the mathematical literature. I will report on the proof of multifractal scaling laws for arithmetic Seba billiards. I will explain the mathematical approach to defining the Rényi entropy associated with a sequence of eigenfunctions and sketch how arithmetic methods permit us to obtain a precise asymptotic in the semiclassical regime and how this allows us to compute the fractal exponents explicitly. Moreover, I will discuss how the symmetry relation for the fractal exponent is related to the functional equation of certain zeta functions. [ABSTRACT FROM AUTHOR]

The zeros of the nth D'Arcais polynomial, also known in combinatorics as the Nekrasov–Okounkov polynomial, dictate the vanishing properties of the nth Fourier coefficients of all complex powers x of the Dedekind η -function. In this paper, we prove that these coefficients are non-vanishing for | x | > κ (n - 1) and κ ≈ 9.7225 . Numerical computations imply that 9.72245 is a lower bound for κ . This significantly improves previous results by Kostant, Han, and Heim–Neuhauser. The polynomials studied in this paper include Chebyshev polynomials of the second kind, 1-associated Laguerre polynomials, Hermite polynomials, and polynomials associated with overpartitions and plane partitions. [ABSTRACT FROM AUTHOR]

We unify in a large class of additive functions the results obtained in the first part of this work. The proof rests on series involving the Riemann zeta function and certain sums of primes which may have their own interest. [ABSTRACT FROM AUTHOR]

In this paper, we consider sums related to the floor function. We improve previous results for some special arithmetic functions considered by O. Bordellès [On certain sums of number theory, Int. J. Number Theory, 18(9):2053– 2074, 2022], J. Stucky [The fractional sum of small arithmetic functions, J. Number Theory, 238:731–739, 2022], and J. Wu [Note on a paper by Bordellès, Dai, Heyman, Pan and Shparlinski, Period. Math. Hung., 80(1):95–102, 2020]. It is worth emphasizing that we use much simpler methods to give much better results than previous. [ABSTRACT FROM AUTHOR]

Satisfiability Modulo Theories (SMT) refers to the problem of deciding the satisfiability of a formula with respect to certain background first-order theories. In this article, we focus on Satisfiablity Modulo Integer Arithmetic, which is referred to as SMT(IA), including both linear and non-linear integer arithmetic theories. Dominant approaches to SMT rely on calling a CDCL-based SAT solver, either in a lazy or eager flavour. Local search, a competitive approach to solving combinatorial problems including SAT, however, has not been well studied for SMT. We develop the first local-search algorithm for SMT(IA) by directly operating on variables, breaking through the traditional framework. We propose a local-search framework by considering the distinctions between Boolean and integer variables. Moreover, we design a novel operator and scoring functions tailored for integer arithmetic, as well as a two-level operation selection heuristic. Putting these together, we develop a local search SMT(IA) solver called LocalSMT. Experiments are carried out to evaluate LocalSMT on benchmark sets from SMT-LIB. The results show that LocalSMT is competitive and complementary with state-of-the-art SMT solvers, and performs particularly well on those formulae with only integer variables. A simple sequential portfolio with Z3 improves the state-of-the-art on satisfiable benchmark sets from SMT-LIB. [ABSTRACT FROM AUTHOR]

Let q be a prime power and, for each positive integer n ≥ 1 , let F q n be the finite field with q n elements. Motivated by the well known concept of normal elements over finite fields, Huczynska et al. (Finite Fields Appl. 24:170-183, 2013) introduced the notion of k-normal elements. More precisely, for a given 0 ≤ k ≤ n , an element α ∈ F q n is k-normal over F q if the F q -vector space generated by the elements in the set { α , α q , ... , α q n - 1 } has dimension n - k . The case k = 0 recovers the notion of normal elements. If q and k are fixed, one may consider the number λ q , n , k of elements α ∈ F q n that are k-normal over F q and the density λ q , k (n) = λ q , n , k q n of such elements in F q n . In this paper we prove that, for arbitrary q and k, the arithmetic function λ q , k (n) has positive mean value, in the sense that the limit lim t → + ∞ 1 t ∑ 1 ≤ n ≤ t λ q , k (n) , exists and it is positive. [ABSTRACT FROM AUTHOR]

Introduction: The ability to reason about equations in a robust and fluent way requires both instrumental knowledge of symbolic forms, syntax, and operations, as well as relational knowledge of how such formalisms map to meaningful relationships captured withinmentalmodels. A recent systematic reviewof studies contrasting the Cuisenaire-Gattegno (Cui) curriculum approach vs. traditional rote schooling on equational reasoning has demonstrated the positive efficacy of pedagogies that focus on integrating these two forms of knowledge. Methods: Here we seek to replicate and extend the most efficacious of these studies (Brownell) by implementing the curriculum to a high degree of fidelity, as well as capturing longitudinal changes within learners via a novel tablet-based assessment of accuracy and fluency with equational reasoning. We examined arithmetic fluency as a function of relational reasoning to equate initial performance across diverse groups and to track changes over four growth assessment points. Results: Results showed that the intervention condition that stressed relational reasoning leads to advances in fluency for addition and subtraction with small numbers. We also showed that this intervention leads to changes in problem solving dispositions toward complex challenges, wherein students in the CUI intervention were more inclined to solve challenging problems relative to those in the control who gave up significantly earlier on multi-step problems. This shift in disposition was associated with higher accuracy on complex equational reasoning problems. A treatment by aptitude interaction emerged for both arithmetic equation reasoning and complex multi-step equational reasoning problems, both of which showed that the intervention had greatest impact for children with lower initial mathematical aptitude. Two years of intervention contrast revealed a large effect (d = 1) for improvements in equational reasoning for the experimental (CUI) group relative to control. Discussion: The strong replication and extension findings substantiate the importance of embedding these teaching aides within the theory grounded curricula that gave rise to them. [ABSTRACT FROM AUTHOR]

The authors analyze Diophantine sets and show that all recursively enumerable sets are Diophantine. Based on the classical results from the theory of recursive functions, a simple version of the theorem on the incompleteness of arithmetic is provided: there is a polynomial that has no positive integer solutions, and for which it is impossible to prove the absence of positive roots. [ABSTRACT FROM AUTHOR]

Quantum-dot cellular automata (QCA) are a promising nanoscale computing technology that exploits the quantum mechanical tunneling of electrons between quantum dots in a cell and electrostatic interaction between dots in neighboring cells. QCA can achieve higher speed, lower power, and smaller areas than conventional, complementary metal-oxide semiconductor (CMOS) technology. Developing QCA circuits in a logically and physically reversible manner can provide exceptional reductions in energy dissipation. The main challenge is to maintain reversibility down to the physical level. A crucial component of a computer's central processing unit (CPU) is the arithmetic logic unit (ALU), which executes multiple logical and arithmetic functions on the data processed by the CPU. Current QCA ALU designs are either irreversible or logically reversible; however, they lack physical reversibility, a crucial requirement to increase energy efficiency. This paper shows a new multilayer design for a QCA ALU that can carry out 16 different operations and is both logically and physically reversible. The design is based on reversible majority gates, which are the key building blocks. We use QCADesigner-E software to simulate and evaluate energy dissipation. The proposed logically and physically reversible QCA ALU offers an improvement of 88.8% in energy efficiency. Compared to the next most efficient 16-operation QCA ALU, this ALU uses 51% fewer QCA cells and 47% less area. [ABSTRACT FROM AUTHOR]

In this paper, we introduce type 2 degenerate poly-Euler polynomials and numbers, briefly called degenerate poly-Euler polynomials and numbers, by using the modified degenerate polyexponential function and derive several properties on these polynomials and numbers. In the last section, we also consider the type 2 degenerate unipoly-Euler polynomials attached to an arithmetic function, by using the modified degenerate polyexponential function and investigate some identities of these polynomials. In particular, we give some new explicit expressions and identities of degenerate unipoly polynomials related to special numbers and polynomials. [ABSTRACT FROM AUTHOR]

The notion of better quasi order (\mathsf {BQO}), due to Nash-Williams, is very fruitful mathematically and intriguing from the standpoint of logic, due to several long-standing open problems. In the present paper, we make a significant step towards one of these: Let \mathbf 3 be the antichain with three elements. We show that arithmetical recursion along the natural numbers (\mathsf {ACA}_0^+) follows from \mathbf 3 being \mathsf {BQO}, over the base theory \mathsf {RCA_0} from reverse mathematics. Also over the latter, we deduce arithmetical transfinite recursion (\mathsf {ATR}_0) from the assumption that \mathbf 3 is \Delta ^0_2\text {-}\mathsf {BQO}, which plays a role in work of Montalbán. [ABSTRACT FROM AUTHOR]

This paper presents the first adaptively simulation secure functional encryption (FE) schemes for attribute-weighted sums. In the proposed FE schemes, attributes are viewed as vectors and weight functions are arithmetic branching programs (ABP). We present two schemes with varying parameters and levels of adaptive simulation security. (a) We first present a one-slot scheme supporting a bounded number of ciphertext queries and an arbitrary polynomial number of secret key queries both before and after the ciphertext queries. This is the best possible level of security one can achieve in the adaptive simulation-based framework. The scheme also achieves indistinguishability-based adaptive security against an unbounded number of ciphertext and secret key queries. (b) Next, bootstrapping from the one-slot scheme, we present an unbounded-slot scheme that can support a bounded number of ciphertext and pre-ciphertext secret key queries while supporting an a-priori unbounded number of post-ciphertext secret key queries. Both schemes enjoy ciphertexts that do not grow with the number of appearances of the attributes within the weight functions. The schemes are built upon prime-order asymmetric bilinear groups and the security is derived under the standard (bilateral) k-Linear (k-Lin) assumption. Our work resolves an open problem posed by Abdalla et al (In: CRYPTO, Springer, New York, 2020), where they presented an unbounded-slot FE scheme for attribute-weighted sum achieving only semi-adaptive simulation security. Technically, we extend the recent adaptive security framework of Lin and Luo (In: EUROCRYPT, Springer, New York, 2020), devised to achieve compact ciphertexts in the context of indistinguishability-based payload-hiding security, to the setting of simulation-based adaptive attribute-hiding security. [ABSTRACT FROM AUTHOR]

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Let \digamma (n) denote a multiplicative function with range \{-1,0,1\}, and let F(x) = \sum _{n=1}^{\left \lfloor x\right \rfloor } \digamma (n). Then F(x)/\sqrt {x} = a\sqrt {x} + b + E(x), where a and b are constants and E(x) is an error term that either tends to 0 in the limit or is expected to oscillate about 0 in a roughly balanced manner. We say F(x) has persistent bias b (at the scale of \sqrt {x}) in the first case, and apparent bias b in the latter. For example, if \digamma (n)=\mu (n), the Möbius function, then F(x) = \sum _{n=1}^{\left \lfloor x\right \rfloor } \mu (n) has apparent bias 0, while if \digamma (n)=\lambda (n), the Liouville function, then F(x) = \sum _{n=1}^{\left \lfloor x\right \rfloor } \lambda (n) has apparent bias 1/\zeta (1/2). We study the bias when \digamma (p^k) is independent of the prime p, and call such functions fake \mu 's. We investigate the conditions required for such a function to exhibit a persistent or apparent bias, determine the functions in this family with maximal and minimal bias of each type, and characterize the functions with no bias. For such a function F(x) with apparent bias b, we also show that F(x)/\sqrt {x}-a\sqrt {x}-b changes sign infinitely often. [ABSTRACT FROM AUTHOR]

We prove new bounds on bilinear forms with Kloosterman sums, complementing and improving a series of results by É. Fouvry, E. Kowalski and Ph. Michel (2014), V. Blomer, É. Fouvry, E. Kowalski, Ph. Michel and D. Milićević (2017), E. Kowalski, Ph. Michel and W. Sawin (2019, 2020) and I. E. Shparlinski (2019). These improvements rely on new estimates for Type II bilinear forms with incomplete Kloosterman sums. We also establish new estimates for bilinear forms with one variable from an arbitrary set by introducing techniques from additive combinatorics over prime fields. Some of these bounds have found a crucial application in the recent work of Wu (2020) on asymptotic formulas for the fourth moments of Dirichlet L$L$‐functions. As new applications, an estimate for higher moments of averages of Kloosterman sums and the distribution of divisor function in a family of arithmetic progressions are also given. [ABSTRACT FROM AUTHOR]