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Let M n = diag [ 3 p n , 3 q n ] with p n , q n ⩾ 1 for all n ⩾ 1 and let D = { (0 , 0) t , (1 , 0) t , (0 , 1) t } . One can generate a Borel probability measure μ { M n } , D = δ M 1 − 1 D ∗ δ (M 2 M 1) − 1 D ∗ δ (M 3 M 2 M 1) − 1 D ∗ ⋯. Such measure μ { M n } , D is called a Moran Sierpinski-type measure. It is known Deng et al (Acta Math. Sin. submitted) that the associated Hilbert space L 2 (μ { M n } , D) has an exponential orthonormal basis. In this paper, we first characterize all the maximal exponential orthogonal sets for L 2 (μ { M n } , D) . For such a maximal orthogonal set, we then give some sufficient conditions to determine whether it is an orthonormal basis of L 2 (μ { M n } , D) or not. [ABSTRACT FROM AUTHOR]

Serre showed that for any integer m , a (n) ≡ 0 (mod m) for almost all n , where a (n) is the n th Fourier coefficient of any modular form with rational coefficients. In this paper, we consider a certain class of cuspforms and study # { a (n) (mod m) } n ≤ x over the set of integers with O (1) many prime factors. Moreover, we show that any residue class a ∈ ℤ / m ℤ can be written as the sum of at most 13 Fourier coefficients, which are polynomially bounded as a function of m. [ABSTRACT FROM AUTHOR]

We consider a square expanding with constant speed seen from an observer moving away with constant acceleration and study the distribution of angles between rays from the observer towards the lattice points in the square. We prove the existence of the gap distribution as time tends to infinity and provide explicit formulas for the corresponding density function. [ABSTRACT FROM AUTHOR]

In the present paper, we have studied the two-dimensional deformation of an elastically orthotropic layer of finite thickness overlying a rough-rigid base due to surface loads. For a two-dimensional approximation, the governing equations are split into two cases: plane strain and anti-plane strain case. We consider plane strain case only. The Airy stress function approach is used to obtain the stresses and displacements in the integral form at any point of the layer. To solve the integrals analytically, the linear combination of exponential terms occurring in the denominator is expanded by binomial expression and then approximated by a finite sum of exponential terms using the least square method. Then, the integrals are evaluated using standard integral tables. The analytical expressions for the displacements are obtained for the normal line loading. Two orthotropic materials, such as Topaz and Barytes, have been considered and graphs showing the displacement field for Topaz, Barytes and the isotropic case have been plotted. [ABSTRACT FROM AUTHOR]

The antiplane deformation of a two layered model consisting of a homogeneous, elastically isotropic (HEI) stratum of finite thickness lying on a HEI substratum occurring from a buried inclined strike-slip fault (ISF) situated in the stratum has been studied. The stratum is in welded contact with the substratum having different rigidity. The integral expressions for the deformation field occurring from a long strike-slip line dislocation for the antiplane strain case have been obtained using Fourier transform approach. The integrals have been solved analytically by expanding the denominator into a finite sum of exponential terms (FSET) using least square method. Further, the deformation is obtained for a long buried ISF of finite width via integrating over the width of line dislocation. The variation of the displacement has been computed graphically with respect to the epicentral distance as well as with depth. [ABSTRACT FROM AUTHOR]

We improve the standard Weyl estimate for quartic exponential sums in which the argument is a quadratic irrational. Specifically we show that ∑ n ≤ N e (α n 4) ≪ ε , α N 5 / 6 + ε for any ε > 0 and any quadratic irrational α ∈ R − Q. Classically one would have had the exponent 7 / 8 + ε for such α. In contrast to the author's earlier work [2] on cubic Weyl sums (which was conditional on the abc -conjecture), we show that the van der Corput AB -steps are sufficient for the quartic case, rather than the B A A B -process needed for the cubic sum. [ABSTRACT FROM AUTHOR]

For arbitrary real t > 1 we examine the set { ⌊ x / n t ⌋ : n ≤ x }. Asymptotic formulas for the cardinality of this set and the number of primes in this set are given. The prime counting result uses an alternate Vaughan's decomposition for the von Mangoldt function, with triple exponential sums instead of double exponential sums. These sets are the sparsest known sets that satisfy the prime number theorem, in the sense that the number of primes is asymptotically given by the cardinality of the set divided by the natural logarithm of the cardinality of the set. [ABSTRACT FROM AUTHOR]

We construct a perverse sheaf related to the the matrix exponential sums investigated by Erdélyi and Tóth [ Matrix Kloosterman sums , 2021, arXiv:2109.00762]. As this sheaf appears as a summand of certain tensor product of Kloosterman sheaves, we can establish the exact structure of the cohomology attached to the sums by relating it to the Springer correspondence and using the recursion formula of Erdélyi and Tóth. [ABSTRACT FROM AUTHOR]

We construct an algorithm that reduces the complexity for computing generalized Dedekind sums from exponential to polynomial time. We do so by using an efficient word rewriting process in group theory. [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 study mean values for exponential sums of divisor functions. We improve previous results of [M. Pandey, Moment estimates for the exponential sum with higher divisor functions, C. R. Math. Acad. Sci. Paris 360 (2022) 419–424]. [ABSTRACT FROM AUTHOR]

Given a matrix A\in \mathrm {GL}_d(\mathbb {Z}). We study the pseudorandomness of vectors \mathbf {u}_n generated by a linear recurrence relation of the form \begin{equation*} \mathbf {u}_{n+1} \equiv A \mathbf {u}_n \pmod {p^t}, \qquad n = 0, 1, \ldots, \end{equation*} modulo p^t with a fixed prime p and sufficiently large integer t \geqslant 1. We study such sequences over very short segments of length which has not been accessible via previously used methods. Our technique is based on the method of N. M. Korobov [Mat. Sb. (N.S.) 89(131) (1972), pp. 654–670, 672] of estimating double Weyl sums and a fully explicit form of the Vinogradov mean value theorem due to K. Ford [Proc. London Math. Soc. (3) 85 (2002), pp. 565–633]. This is combined with some ideas from the work of I. E. Shparlinski [Proc. Voronezh State Pedagogical Inst., 197 (1978), 74–85 (in Russian)] which allows us to construct polynomial representations of the coordinates of \mathbf {u}_n and control the p-adic orders of their coefficients in polynomial representations. [ABSTRACT FROM AUTHOR]

In this article we prove an explicit sub-Weyl bound for the Riemann zeta function ζ (s) on the critical line s = 1 / 2 + i t. In particular, we show that | ζ (1 / 2 + i t) | ≤ 66.7 t 27 / 164 for t ≥ 3. Combined, our results form the sharpest known bounds on ζ (1 / 2 + i t) for t ≥ exp ⁡ (61). [ABSTRACT FROM AUTHOR]

Let p be a fixed odd prime and Q (x , y , z) = a x 2 + b x y + c y 2 + d x z + e y z + f z 2 be a fixed quadratic form in Z [ x , y , z ] which is non-degenerate in F p [ x , y , z ] and gcd (a (4 a c - b 2) , p) = 1. Let (x 0 , y 0 , z 0) be a fixed point in Z 3 . We study the behavior of solutions (x, y, z) of congruences of the form Q (x , y , z) ≡ 0 mod q with q = p n , where max { | x - x 0 | , | y - y 0 | , | z - z 0 | } ≤ N and gcd (z , p) = 1. In fact, we consider a smooth version of this problem and establish an asymptotic formula (thus the existence of such solutions) when n → ∞ , under the condition N ≥ q 1 2 + ε . [ABSTRACT FROM AUTHOR]

In this paper, we can show that S Λ (x) = ∑ 1 ≤ n ≤ x Λ ([ x n ]) = ∑ n = 1 ∞ Λ (n) n (n + 1) x + O (x 7 / 15 + 1 / 195 + ε) , where Λ (n) is the von Mangdolt function. Moreover, we can also give similar results related to the divisor function, which improve previous results. [ABSTRACT FROM AUTHOR]

Fix an integer κ ≥ 2 . Let P ≥ 2 be a prime, and F be the symmetric-square lift of a Hecke newform f ∈ S κ * (P) . We study the exponential sum L F (α) = ∑ n ∼ N A F (n , 1) e (n α) by implementing an average over a family in such a way to investigate the best possible magnitude of the level aspect bound for L F (α) . We prove a uniform bound with respect to any α ∈ R and the level parameter P , and present that there exist certain forms with fairly strong oscillations in L F (α) , if the associated level of f is allowed to vary. As applications, we consider the shifted convolution sums for GL (3) × GL (d) , for any d ≥ 2 , in a family as well as theWaring-Goldbach problem associated to Fourier coefficients of SL (3 , Z) -Maa β forms. [ABSTRACT FROM AUTHOR]

Let \mu (n) be the Möbius function and e(\alpha)=e^{2\pi i\alpha }. In this paper, we study upper bounds of the classical sum \[ S(x,\alpha)≔\sum _{1\leq n\leq x}\mu (n)e(\alpha n). \] We can improve some classical results of Baker and Harman [J. London Math. Soc. (2) 43 (1991), pp. 193–198]. [ABSTRACT FROM AUTHOR]

Traces of singular moduli can be approximated by exponential sums of quadratic irrationals. Recently Andersen and Duke used theory of Maass forms to estimate generalized twisted traces with power-saving error bounds. We establish an asymptotic formula with effective error bounds for such traces. Our methods depend on an explicit bound for sums of Kloosterman sums on Γ 0 (4) . [ABSTRACT FROM AUTHOR]

We obtain new bounds, pointwise and on average, for Dedekind sums s (λ , p) modulo a prime p with λ of small multiplicative order d modulo p. Assuming the infinitude of Mersenne primes, the range of our results is optimal. Moreover, we relate high moments of L (1 , χ) over subgroups of characters to some correlations of Dedekind sums, and use recent results of the second and third author to study these correlations. [ABSTRACT FROM AUTHOR]

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

We prove that the sequence 1!, 2!, 3!, . . . produces at least (√ 2+o(1))√p distinct residues modulo prime p. Moreover, the factorials within an interval I⊆ {0, 1, . . ., p - 1} of length N > p7/8+ε produce at least (1 + o(1)) √p distinct residues modulo p. As a corollary, we prove that every non-zero residue class can be expressed as a product of seven factorials n1! ... n7! modulo p, where ni = O(p6/7+ε) for all i = 1, . . ., 7, which provides a polynomial improvement upon the preceding results. [ABSTRACT FROM AUTHOR]

An explicit subconvexity bound for the Riemann zeta function ζ (s) on the critical line s = 1 / 2 + i t is proved. This bound uses a new version of the Kusmin–Landau lemma and improves on current explicit bounds for | ζ (1 / 2 + i t) |. [ABSTRACT FROM AUTHOR]

For an n$n$‐bit positive integer a$a$ written in binary as a=∑j=0n−1εj(a)2j,$ {a} = \sum _{j=0}^{n-1} \varepsilon _{j}(a) \,2^j,$ where εj(a)∈{0,1}$\varepsilon _j(a) \in \lbrace 0,1\rbrace$, j∈{0,...,n−1}$j\in \lbrace 0, \ldots , n-1\rbrace$, εn−1(a)=1$\varepsilon _{n-1}(a)=1$, let us define a←=∑j=0n−1εj(a)2n−1−j,$ \overleftarrow{a} = \sum _{j=0}^{n-1} \varepsilon _j(a)\,2^{n-1-j},$ the digital reversal of a$a$. Also let Bn={2n−1⩽a<2n:aodd}${\mathcal {B}}_n= \lbrace 2^{n-1}\leqslant a<2^n:\nobreakspace a \text{ odd}\rbrace$. With a sieve argument, we obtain an upper bound of the expected order of magnitude for the number of p∈Bn$p \in {\mathcal {B}}_n$ such that p$p$ and p←$\overleftarrow{p}$ are prime. We also prove that for sufficiently large n$n$, {a∈Bn:max{Ω(a),Ω(a←)}⩽8}⩾c2nn2,$$\begin{equation*} \hspace*{30pt}{\left| \lbrace a \in {\mathcal {B}}_n:\nobreakspace \max \lbrace \Omega (a), \Omega (\overleftarrow{a})\rbrace \leqslant 8 \rbrace \right|} \geqslant c\, \frac{2^n}{n^2}, \end{equation*}$$where Ω(n)$\Omega (n)$ denotes the number of prime factors counted with multiplicity of n$n$ and c>0$c > 0$ is an absolute constant. Finally, we provide an asymptotic formula for the number of n$n$‐bit integers a$a$ such that a$a$ and a←$\overleftarrow{a}$ are both squarefree. Our method leads us to provide various estimates for the exponential sum ∑a∈Bnexp2πi(αa+ϑa←)(α,ϑ∈R).$$\begin{equation*} \hspace*{30pt}\sum _{a \in {\mathcal {B}}_n} \exp {\left(2\pi i (\alpha a + \vartheta \overleftarrow{a})\right)} \quad (\alpha,\vartheta \in \mathbb {R}). \end{equation*}$$ [ABSTRACT FROM AUTHOR]

Let 0 < ρ < 1 and let { a j , b j , n j } j = 1 ∞ be a sequence of positive integers with an upper bound. Associated with them, there exists a unique Borel probability measure μ ρ , { 0 , a j , b j } , { n j } generated by the following infinite convolution of discrete measures: μ ρ , { 0 , a j , b j } , { n j } = δ ρ n 1 ⁢ { 0 , a 1 , b 1 } ∗ δ ρ n 1 + n 2 ⁢ { 0 , a 2 , b 2 } ∗ δ ρ n 1 + n 2 + n 3 ⁢ { 0 , a 3 , b 3 } ∗ ⋯ , where gcd ⁡ (a j , b j) = 1 for all j ∈ ℕ . In this paper, we show that L 2 ⁢ (μ ρ , { 0 , a j , b j } , { n j } ) admits an exponential orthonormal basis if and only if the following two conditions are satisfied: (i) { a j , b j } ≡ { ± 1 } ⁢ (mod ⁢ 3) for all j ≥ 1 ; (ii) there exists a natural number r such that ρ - r ∈ 3 ⁢ ℕ and n j ∈ r ⁢ ℕ for all j ≥ 2 . [ABSTRACT FROM AUTHOR]

Although body fluid volume control by the kidneys may be classified as a long-term arterial pressure (AP) control system, it does not necessarily follow that the urine flow (UF) response to changes in AP is slow. We quantified the dynamic characteristics of the UF response to short-term AP changes by changing mean AP between 60 mmHg and 100 mmHg every 10 s according to a binary white noise sequence in anesthetized rats (n = 8 animals). In a baro-on trial (the carotid sinus baroreflex was enabled), the UF response represented the combined synergistic effects of pressure diuresis (PD) and neurally mediated antidiuresis (NMA). In a baro-fix trial (the carotid sinus pressure was fixed at 100 mmHg), the UF response mainly reflected the effect of PD. The UF step response was quantified using the sum of two exponential decay functions. The fast and slow components had time constants of 6.5 ± 3.6 s and 102 ± 85 s (means ± SD), respectively, in the baro-on trial. Although the gain of the fast component did not differ between the two trials (0.49 ± 0.21 vs. 0.66 ± 0.22 lL·min-1·kg-1·mmHg-1), the gain of the slow component was greater in the baro-on than in the baro-fix trial (0.51 ± 0.14 vs. 0.09 ± 0.39 lL·min-1·kg-1·mmHg-1, P = 0.023). The magnitude of NMA relative to PD was calculated to be 32.2 ± 29.8%. In conclusion, NMA contributed to the slow component, and its magnitude was approximately one-third of that of the effect of PD. [ABSTRACT FROM AUTHOR]

In this paper, we find conditions under which distribution functions of randomly stopped minimum, maximum, minimum of sums and maximum of sums belong to the class of generalized subexponential distributions. The results presented in this article complement the closure properties of randomly stopped sums considered in the authors' previous work. In this work, as in the previous one, the primary random variables are supposed to be independent and real-valued, but not necessarily identically distributed. The counting random variable describing the stopping moment of random structures is supposed to be nonnegative, integer-valued and not degenerate at zero. In addition, it is supposed that counting random variable and the sequence of the primary random variables are independent. At the end of the paper, it is demonstrated how randomly stopped structures can be applied to the construction of new generalized subexponential distributions. [ABSTRACT FROM AUTHOR]

In this paper, we study the Newton polygon of the L -function of a generalized Kloosterman polynomial with two variables over finite fields. We give the explicit form of the monomial basis of the top dimensional cohomology space of the p -adic complex associated to the L -function. One consequence of this result is a concrete method for computing the Hodge polygon. Using decomposition theorems of Wan, we determine when a generalized Kloosterman polynomial is ordinary and when its Newton polytope is generically ordinary. [ABSTRACT FROM AUTHOR]

White-light superflares from ultra-cool stars are thought to be resulted from magnetic reconnection, but the magnetic dynamics in a fully convective star is not clear yet. In this paper, we report a stellar superflare detected with the ground wide angle camera (GWAC), along with rapid follow-ups with the F60A, Xinglong 2.16-m, and LCOGT telescopes. The effective temperature of the counterpart is estimated to be 2200 ± 50 K by the BT-Settl model, corresponding to a spectral type of L0. The R -band light curve can be modelled as a sum of three exponential decay components, where the impulsive component contributes a fraction of 23 per cent of the total energy, while the gradual and the shallower decay phases emit 42 per cent and 35 per cent of the total energy, respectively. The strong and variable Balmer narrow emission lines indicate the large amplitude flare is resulted from magnetic activity. The bolometric energy released is about 6.4 × 1033 erg, equivalent to an energy release in a duration of 143.7 h at its quiescent level. The amplitude of Δ R  = −8.6 mag (or Δ V  = −11.2 mag), placing it one of the highest amplitudes of any ultra-cool star recorded with excellent temporal resolution. We argue that a stellar flare with such rapidly decaying and huge amplitude at distances greater than 1 kpc may be false positive in searching for counterparts of catastrophic events such as gravitational wave events or gamma-ray bursts, which are valuable in time-domain astronomy and should be given more attention. [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]

We consider families of exponential sums indexed by a subgroup of invertible classes modulo some prime power q. For fixed d , we restrict to moduli q so that there is a unique subgroup of invertible classes modulo q of order d. We study distribution properties of these families of sums as q grows and we establish equidistribution results in some regions of the complex plane which are described as the image of a multi-dimensional torus via an explicit Laurent polynomial. In some cases, the region of equidistribution can be interpreted as the one delimited by a hypocycloid, or as a Minkowski sum of such regions. [ABSTRACT FROM AUTHOR]

In this paper, we define Mersenne, Mersenne-Lucas hybrid quaternions. We give the Binet's formula, the generating functions, exponential generating functions and sum formula of these quaternions. We find some relations between Mersenne-Lucas hybrid quaternions, Jacobsthal hybrid quaternions, Jacobsthal-Lucas hybrid quaternions and Mersenne hybrid quaternions. [ABSTRACT FROM AUTHOR]

Let |$k_i\ (i=1,2,\ldots ,t)$| be natural numbers with |$k_1>k_2>\cdots >k_t>0$|⁠ , |$k_1\geq 2$| and |$t

We obtain bounds on fractional parts of binary forms of the shape $$\Psi(x,y)=\alpha_k x^k+\alpha_l x^ly^{k-l}+\alpha_{l-1}x^{l-1}y^{k-l+1}+\cdots+\alpha_0 y^k$$ with |$\alpha_k,\alpha_l,\ldots,\alpha_0\in{\mathbb R}$| and |$l\leq k-2.$| By exploiting recent progress on Vinogradov's mean value theorem and earlier work on exponential sums over smooth numbers, we derive estimates superior to those obtained hitherto for the best exponent σ , depending on k and |$l,$| such that $$ \min_{\substack{0\leq x,y\leq X\\(x,y)\neq (0,0)}}\|\Psi(x,y)\|\leq X^{-\sigma+\epsilon}. $$ [ABSTRACT FROM AUTHOR]

A discharge-flow reactor combined with modulated molecular beam mass spectrometry technique was employed to determine the rate constants of H-atom reactions with hydrogen sulfide and thiirane. The rate constants for both reactions were determined at a total pressure of 2 Torr from 220 to 950 K under pseudo-first-order conditions by monitoring either consumption of H atoms in excess of H2S (C4H4S) or the molecular species in excess of atomic hydrogen. For H + H2S reaction, a suggested previously strong curvature of the Arrhenius plot was confirmed: kl = 8.7 × 10−13 × (T/298)2.87 × exp(−125/T) cm3 molecule−1 s−1 with a conservative uncertainty of 15% at all temperatures. Non-Arrhenius behavior was also observed for the reaction of H-atom with C2H4S, with the experimental rate constant data being best fitted to a sum of two exponential functions: k2 = 1.85 × 10−10 exp(−1410/T) + 4.17 × 10−12 exp(−242/T) cm3 molecule−1 s−1 with an independent of temperature uncertainty of 15%. [ABSTRACT FROM AUTHOR]

In this paper, we give a detailed proof of an asymptotic formula for averages of coefficients of a class of degree seven L -functions which can be factorized as a product of degree one and degree six L -functions. We can break the 3/4-barrier in the error term and get an explicit exponent 3 4 − 307 85172 + ε. Our proof relies on methods from the theory of exponential sums. [ABSTRACT FROM AUTHOR]

A generalized stochastic modification of the Colonel Blotto game, also known as the game of gladiators, is considered. In the original model, each of two players has a set of gladiators with given strengths. The battle of gladiator teams takes place through individual gladiator battles. In each fight, the probability of gladiator winning is proportional to its strength. Kaminsky et al. in 1984 had obtained a formula for the probability of winning in terms of weighted sums of exponential random variables. Here we provide an interpretation of this result from the Markov chains with continuous time point of view, and a more general statement of the problem is considered, for which a similar expression is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we first study exponential sums of the form ∑ x ∈ F q ψ (f (x)) χ (g (x)) called hybrid character sums, where f , g are functions over finite field F q , ψ is a nontrivial multiplicative character and χ is a nontrivial additive character of F q. The absolute values or explicit values of them are determined under certain conditions. We then present several constructions of complex codebooks which are also called frames. The proposed codebooks are constructed from Hadamard matrices with binary row selection sequences. The hybrid character sums play an important role in determining the maximum magnitude of the inner products between vectors in these codebooks. It is proved that the magnitude of the inner products between distinct code vectors of each codebook is two-valued, and the maximum magnitude of each codebook asymptotically achieves the Welch bound equality. The constructed codebooks are equivalent to nearly equiangular tight frames and have nice application in compressed sensing matrices. [ABSTRACT FROM AUTHOR]

Let and N be a sufficiently large real number. In this paper, we first prove that the Diophantine inequality is solvable in primes p1,p2, ..., p6. Moreover, we prove that for almost all R ∈ (N, 2N], the Diophantine inequality is solvable in primes p1,p2, p3. These results constitute further improvements upon previous results. [ABSTRACT FROM AUTHOR]

Let l be an odd prime with l ≡ 1 (mod 4) , N = l m a positive integer, ord N (2) = f , and q = 2 f , where f = ϕ (N) / t and ϕ (·) is the Euler's function. Let { u i } = (Tr q / 2 (ω i)) i = 0 q - 2 be a binary sequence of period q - 1 , where ω is a primitive element of a finite field F q . In this paper, we obtain two-valued cross correlation distributions between two sequences { u i } and their q - 1 N -decimation sequences in two cases: t = 2 and t = 4 . [ABSTRACT FROM AUTHOR]

The authors of the paper (Vitali et al 2022 J. Phys. A: Math. Theor. 55 224012) analyzed a simple CTRW model with a waiting time distribution defined as the weighted sum of two exponential distributions. They showed that their model meets many paradigmatic features that belong to the anomalous diffusion as it is observed in living systems. This comment point out the previous paper that considers a similar model and improves on the authors' result regarding the time dependence of the mean-square displacement. [ABSTRACT FROM AUTHOR]

Let A be a finite dimensional associative algebra with derivations over a field of characteristic zero, i.e., an algebra whose structure is enriched by the action of a Lie algebra L by derivations, and let c n L (A) , n ≥ 1 , be its differential codimension sequence. Such sequence is exponentially bounded and exp L ⁡ (A) = lim n → ∞ ⁡ c n L (A) n is an integer that can be computed, called differential PI-exponent of A. In this paper we prove that for any Lie algebra L , exp L ⁡ (A) coincides with exp ⁡ (A) , the ordinary PI-exponent of A. Furthermore, in case L is a solvable Lie algebra, we apply such result to classify varieties of L -algebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety has polynomial growth. [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]

In the paper, series of exponential monomials are considered. We study the problem of the distribution of singular points of the sum of a series on the boundary of its domain of convergence. We study the conditions under which, for any sequence of coefficients of the series with a chosen domain of convergence, the domain of existence of the sum of this series coincides with the given domain of convergence. We consider sequences of exponents having an angular density (measurable) and the zero condensation index. Various criteria related to the distribution of singular points of the sum of a series of exponential monomials on the boundary of its convergence domain are obtained. In particular, in the class of the indicated sequences, a criterion is obtained that all boundary points of a chosen convex domain are special for any sum of a series with a given domain of convergence. The criteria are formulated using simple geometric characteristics of the sequence of exponents and a convex domain (the angular density and the length of the boundary arc). It is also shown that the condition that the condensation index is equal to zero is essential. [ABSTRACT FROM AUTHOR]

Let $[t]$ be the integral part of the real number t and let $\mathbb {1}_{{\mathbb P}}$ be the characteristic function of the primes. Denote by $\pi _{\mathcal {S}}(x)$ the number of primes in the floor function set $\mathcal {S}(x) := \{[{x}/{n}] : 1\leqslant n\leqslant x\}$ and by $S_{\mathbb {1}_{{\mathbb P}}}(x)$ the number of primes in the sequence $\{[{x}/{n}]\}_{n\geqslant 1}$. Improving a result of Heyman ['Primes in floor function sets', Integers 22 (2022), Article no. A59], we show $$ \begin{align*} \pi_{\mathcal{S}}(x) = \int_2^{\sqrt{x}} \frac{d t}{\log t} + \int_2^{\sqrt{x}} \frac{d t}{\log(x/t)} + O(\sqrt{x}\,\mathrm{e}^{-c(\log x)^{3/5}(\log\log x)^{-1/5}}) \quad\mbox{and}\quad S_{\mathbb{1}_{{\mathbb P}}}(x) = C_{\mathbb{1}_{{\mathbb P}}} x + O_{\varepsilon}(x^{9/19+\varepsilon}) \end{align*} $$ for $x\to \infty $ , where $C_{\mathbb {1}_{{\mathbb P}}} := \sum _{p} {1}/{p(p+1)}$ , $c>0$ is a positive constant and $\varepsilon $ is an arbitrarily small positive number. [ABSTRACT FROM AUTHOR]

This paper uses exponential sum methods to detect the presence of k-chains of points whose endpoints belong to proper subsets of finite p-adic rings. [ABSTRACT FROM AUTHOR]

In this paper we give a short interval version of the Balog–Ruzsa theorem concerning bounds for the L 1 norm of the exponential sum over r -free numbers. In particular, when r  = 2, for |$H \geq N^{\frac{9}{17}+\epsilon}$|⁠ , we have the lower bound result $$ \int_{\mathbb T}\left|\sum_{|n-N| \lt H} \mu^2(n)e(n \alpha)\right| d \alpha \gg H^{\frac{1}{3}}, $$ and for |$H \geq N^{\frac{18}{29}+\epsilon}$|⁠ , we have the upper bound result $$ \int_{\mathbb T}\left|\sum_{|n-N| \lt H} \mu^2(n)e(n \alpha)\right| d \alpha \ll H^{\frac{1}{3}}.$$ As an application, we show that the L 1 norm of the exponential sum |$\sum_{|n-N| \lt H} \mu(n)e(n \alpha)$| has the lower bound |$\gg H^{\frac{1}{6}}$| when |$H \geq N^{\frac{9}{17} + \varepsilon}$|⁠. [ABSTRACT FROM AUTHOR]

Zilber's Exponential-Algebraic Closedness Conjecture states that algebraic varieties in C n × (C ×) n intersect the graph of complex exponentiation, unless that contradicts the algebraic and transcendence properties of exp . We establish a case of the conjecture, showing that it holds for varieties which split as the product of a linear subspace of the additive group and an algebraic subvariety of the multiplicative group. This amounts to solving certain systems of exponential sums equations, and it generalizes old results of Zilber, which required the linear subspace to either be defined over a generic subfield of the real numbers, or it to be any subspace defined over the reals assuming unproved conjectures from Diophantine geometry and transcendence theory. The proofs use the theory of amoebas and tropical geometry. [ABSTRACT FROM AUTHOR]

In this article, we derive exact expressions for the probability density function and cumulative distribution function of the sum of independent and non-identical exponential-geometric random variables. Then we discuss the corresponding result for independent and identically distributed exponentialgeometric random variables. A saddlepoint approximation is also utilized to approximate the derived distribution. Finally, numerical simulations are used to investigate the precision of the saddlepoint approximation. [ABSTRACT FROM AUTHOR]

Let f be a Hecke–Maass cusp form of weight zero for S L 2 (ℤ) and λ f (n) be the nth Fourier coefficient. For almost all , we have ∑ n ≤ x λ f 2 (n) e (n) ≪ x 0. 8 1 2 5 + , which improves the result of Acharya [Exponential sums of squares of Fourier coefficients of cusp forms, Proc. Indian Acad. Sci. Math. Sci. 130 (2020) 24], who showed an upper bound larger than x 0. 8 2 9 7. For all α > 1 of type τ < ∞ , we also show that ∑ 1 ≤ n ≤ x λ f 2 ([ α n + β ]) − α − 1 ∑ 1 ≤ n ≤ [ α x + β ] λ f 2 (n) ≪ x 1 3 / 1 6 + + x 1 − 3 / 4 (τ + 1) + , where ℬ α , β : = { [ α n + β ] } n = 1 ∞. This result relies heavily on a generalized double sum (see Theorem 1.3). [ABSTRACT FROM AUTHOR]

In this paper, we investigate the average ∑ f ∈ κ ∗ (P) ω f − 1 ℒ f (α , β ; N) , where ℒ f (α , β ; N) = ∑ n ∼ N λ f (n) e (n 2 α + n β) for any α , β ∈ ℝ and N ≥ 2 , with f ∈ κ ∗ (P) running over primitive holomorphic cusp forms of weight κ and prime level P. As a result, we prove pointwise uniform bounds with respect to α , β for the frequency of the P-values, and present that there exist strong oscillations in the exponential sum ℒ f (α , β ; N) , so long as the level of f varies. Concerning the applications of the main result, we consider the quadratic Waring problem associated to the coefficients of cusp forms exhibiting considerable cancellations. [ABSTRACT FROM AUTHOR]