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In this paper we resolve a conjecture of Kresch and Tamvakis [Duke Math. J. 110 (2001), pp. 359–376]. Our result is the following. Theorem : For any positive integer D and any integers i,j (0\leq i,j \leq D), \; the absolute value of the following hypergeometric series is at most 1: \begin{equation*} {_4F_3} \left [ \begin {array}{c} -i, \; i+1, \; -j, \; j+1 \\ 1, \; D+2, \; -D \end{array} ; 1 \right ]. \end{equation*} To prove this theorem, we use the Biedenharn-Elliott identity, the theory of Leonard pairs, and the Perron-Frobenius theorem. [ABSTRACT FROM AUTHOR]

This paper deals with the chaotic behavior of linear operators on Banach spaces in both discrete and continuous cases. The inheritances of chaos for linear operators and C_0-semigroups are obtained. More precisely, for any positive integer n\geq 2, both Li-Yorke n-chaos and distributional n-chaos are proved to be inherited under iterations for a linear operator. One further shows that a C_0-semigroup \{T_t\}_{t\geq 0} and every single operator T_t share the same Li-Yorke n-scrambled set and distributionally n-scrambled set for any positive integer n\geq 2. In particular, the Li-Yorke n-chaos and distributional n-chaos become the Li-Yorke chaos and distributional chaos when n=2, respectively. Some equivalent criteria for dense n-chaos and generic n-chaos of linear operators and C_0-semigroups are also established for any n\geq 2. [ABSTRACT FROM AUTHOR]

In this paper, we mainly establish a congruence for a sum involving Apéry numbers, which was conjectured by Z.-W. Sun. Namely, for any prime p>3 and positive odd integer m, we prove that there is a p-adic integer c_m only depending on m such that \begin{equation*} \sum _{k=0}^{p-1}(2k+1)^{m}(-1)^kA_k\equiv c_mp\left (\frac {p}{3}\right)\pmod {p^3}, \end{equation*} where A_k=\sum _{j=0}^{k}\binom {k}{j}^2\binom {k+j}{j}^2 is the Apéry number and (\frac {.}{p}) is the Legendre symbol. [ABSTRACT FROM AUTHOR]

Let \psi :\mathbb {R}_+\to \mathbb {R}_+ be a non-increasing function. A real number x is said to be \psi-Dirichlet improvable if the system \begin{equation*} |qx-p|< \psi (t) \ \ {\text {and}} \ \ |q|

Let x_1, x_2, ..., x_n be n points on the sphere S^2. Determining the value \inf \sum _{1\leq k

Let S= \{ a_{1}, a_{2}, \dots, a_{n} \} be a finite set of non-zero integers. R. Balasubramanian, F. Luca and R. Thangadurai [Proc. Amer. Math. Soc. 138 (2010), pp. 2283–2288] gave an exact formula for the degree of the multi-quadratic field \mathbb {K}= \mathbb {Q}(\sqrt {a_{1}}, \sqrt {a_{2}}, \dots, \sqrt {a_{n}}) over \mathbb {Q}. In this paper, we calculate the explicit structure of the Galois group \operatorname {Gal}(\mathbb {K}/\mathbb {Q}) in terms of its action on \sqrt {a_{i}} for 1 \leq i \leq n. [ABSTRACT FROM AUTHOR]

Let \lambda \in (1,\sqrt {2}] be an algebraic integer with Mahler measure 2. A classical result of Garsia shows that the Bernoulli convolution \mu _\lambda is absolutely continuous with respect to the Lebesgue measure with a density function in L^\infty. In this paper, we show that the density function is continuous. [ABSTRACT FROM AUTHOR]

Let n, r and f be positive integers. Let p be a prime number and \psi be an arbitrary fixed nontrivial additive character of the finite field \mathbb F_q with q=p^f elements. Let F be a polynomial in \mathbb F_q[x_1,\dots,x_n] and V be the affine algebraic variety defined over \mathbb {F}_q by the simultaneous vanishing of the polynomials \{F_i\}_{i=1}^r\subseteq \mathbb F_q[x_1,\dots,x_n]. Let \mathbb {Z}_{\ge 0} stand for the set of all nonnegative integers and A be an arbitrary nonempty subset of \{1,\dots,n\}. For a polynomial H(X)=\sum _{{\mathbf {d}}}\alpha _{\mathbf {d}}X^{\mathbf {d}} with {\mathbf {d}}=(d_1,\dots,d_n)\in \mathbb {Z}_{\ge 0}^n, X^{\mathbf {d}}=x_1^{d_1}\dots x_n^{d_n} and \alpha _{\mathbf {d}}\in \mathbb {F}_q^*, we define \deg _A(H)=\max _{{\mathbf {d}}}\{\sum _{i\in A}d_i\} to be the A-degree of H. In this paper, for the exponential sum S(F,V,\psi)=\sum _{X\in V(\mathbb {F}_q)}\psi (F(X)) with V(\mathbb {F}_q) being the set of the \mathbb {F}_q-rational points of V, we show that \begin{equation*} \mathrm {ord}_q S(F,V,\psi)\ge \frac {|A|-\sum _{i=1}^r\deg _A(F_i)} {\max _{1\le i\le r}\{\deg _A(F),\deg _A(F_i)\}} \end{equation*} if \deg _A(F)>0 or \deg _A(F_i)>0 for some i\in \{1,\dots,r\}. This estimate improves Sperber's theorem obtained in 1986. This also leads to an improvement of the p-adic valuation of the number N(V) of \mathbb {F}_q-rational points on the variety V which strengthens the Ax-Katz theorem. Moreover, we use the A-degree and p-weight A-degree to establish p-adic estimates on multiplicative character sums and twisted exponential sums which improve Wan's results gotten in 1995. [ABSTRACT FROM AUTHOR]

For any n\in \mathbb {N}=\{0,1,2,\ldots \} and b,c\in \mathbb {Z}, the generalized central trinomial coefficient T_n(b,c) denotes the coefficient of x^n in the expansion of (x^2+bx+c)^n. Let p be an odd prime. In this paper, we determine the summations \sum _{k=0}^{p-1}T_k(b,c)^2/m^k modulo p^2 for integers m with certain restrictions. As applications, we confirm some conjectural congruences of Sun [Sci. China Math. 57 (2014), pp. 1375–1400]. [ABSTRACT FROM AUTHOR]

Let \mathcal {B} be the collection of rectangular parallelepipeds in \mathbb {R}^3 whose sides are parallel to the coordinate axes and such that \mathcal {B} consists of parallelepipeds with side lengths of the form s, 2^j s, t, where s, t > 0 and j lies in a nonempty subset S of the integers. In this paper, we prove the following: If S is a finite set, then the associated geometric maximal operator M_\mathcal {B} satisfies the weak type estimate \begin{equation*} \left |\left \{x \in \mathbb {R}^3 : M_{\mathcal {B}}f(x) > \alpha \right \}\right | \leq C \int _{\mathbb {R}^3} \frac {|f|}{\alpha }\left (1 + \log ^+ \frac {|f|}{\alpha }\right)\; \end{equation*} but does not satisfy an estimate of the form \begin{equation*} \left |\left \{x \in \mathbb {R}^3 : M_{\mathcal {B}}f(x) > \alpha \right \}\right | \leq C \int _{\mathbb {R}^3} \phi \left (\frac {|f|}{\alpha }\right) \end{equation*} for any convex increasing function \phi : \mathbb [0, \infty) \rightarrow [0, \infty) satisfying the condition \begin{equation*} \lim _{x \rightarrow \infty }\frac {\phi (x)}{x (\log (1 + x))} = 0. \end{equation*} On the other hand, if S is an infinite set, then the associated geometric maximal operator M_\mathcal {B} satisfies the weak type estimate \begin{equation*} \left |\left \{x \in \mathbb {R}^3 : M_{\mathcal {B}}f(x) > \alpha \right \}\right | \leq C \int _{\mathbb {R}^3} \frac {|f|}{\alpha } \left (1 + \log ^+ \frac {|f|}{\alpha }\right)^{2} \end{equation*} but does not satisfy an estimate of the form \begin{equation*} \left |\left \{x \in \mathbb {R}^3 : M_{\mathcal {B}}f(x) > \alpha \right \}\right | \leq C \int _{\mathbb {R}^3} \phi \left (\frac {|f|}{\alpha }\right) \end{equation*} for any convex increasing function \phi : \mathbb [0, \infty) \rightarrow [0, \infty) satisfying the condition \begin{equation*} \lim _{x \rightarrow \infty }\frac {\phi (x)}{x (\log (1 + x))^2} = 0. \end{equation*} [ABSTRACT FROM AUTHOR]

For non-zero integers n and k\geq 2, a generalized Diophantine m-tuple with property D_k(n) is a set of m positive integers \{a_1,a_2,\ldots, a_m\} such that a_ia_j + n is a k-th power for 1\leq i< j\leq m. Define M_k(n)≔\sup \{|S| : S has property D_k(n)\}. In this paper, we study upper bounds on M_k(n), as we vary n over positive integers. In particular, we show that for k\geq 3, M_k(n) is O(\log n) and further assuming the Paley graph conjecture, M_k(n) is O((\log n)^{\epsilon }). The problem for k=2 was studied by a long list of authors that goes back to Diophantus. [ABSTRACT FROM AUTHOR]

In this paper, we consider the mean field equation \begin{equation*} \Delta u+e^{u}=\sum _{i=0}^{3}4\pi n_{i}\delta _{\frac {\omega _{i}}{2}}\text { in }E_{\tau }, \end{equation*} where n_{i}\in \mathbb {Z}_{\geq 0}, E_{\tau } is the flat torus with periods \omega _{1}=1, \omega _{2}=\tau and \operatorname {Im}\tau >0. Assuming N=\sum _{i=0}^{3}n_{i} is odd, a non-critical case for the above PDE, we prove: (i) If among \{n_{i}|i=0,1,2,3\} there is only one odd integer, then there is always an even solution. Furthermore, if n_{0} = 0 and n_{3} is odd, then up to SL_{2}(\mathbb {Z}) action, except for finitely many E_{\tau }, there are exactly \frac {n_{3}+1}{2} even solutions. (ii) If there are exactly three odd integers in \{n_{i}|i=0,1,2,3\}, then the equation has no even solutions for any flat torus E_{\tau }. Our second result might suggest the symmetric solution of the above mean field equation does not hold in general. [ABSTRACT FROM AUTHOR]

In this paper we study the b-ary expansions of the square roots of the function defined by the recurrence ƒb(n) = b ƒb(n−1) + n with initial value ƒ(0) = 0 taken at odd positive integers n, of which the special case b = 10 is often referred to as the ''schizophrenic'' or ''mock-rational'' numbers. Defined by Darling in 2004 and studied in more detail by Brown in 2009, these irrational numbers have the peculiarity of containing long strings of repeating digits within their decimal expansion. The main contribution of this paper is the extension of schizophrenic numbers to all integer bases b ≥ 2 by formally defining the schizophrenic pattern present in the b-ary expansion of these numbers and the study of the lengths of the non-repeating and repeating digit sequences that appear within. [ABSTRACT FROM AUTHOR]

In this paper, we consider the persistence of completely degenerate lower-dimensional invariant tori of the following reversible system {x = ω + Q(x)z + ε P1(x,z,ε), z + H(z)+ε P2(x,z,ε), where (x,z) = (x,y,u,v) ∈ Tn × Rm × R × R with m ≥ n + 2, Hz) = (0,v2p+1 + yml,uym − 1q)T with y = (y1, ⋅ ⋅ ⋅, ym−1,ym), p,q ≥ 0, l > 0 are integers, the involution G is (x,y,u,v) → (−x,y,−u,v), Q(x) is a n × m matrix function, ω is a Diophantine frequency, ε is a small positive parameter and ε P1 ε P2 are analytic perturbation terms. By the Kolmogorov-Arnold-Moser method, we prove that for sufficiently small ε the above reversible system admits lower-dimensional invariant tori with prescribed frequency ω if average of a part of Q(x) is non-singular. This should be the first persistence result of lower-dimensional invariant tori in completely degenerate reversible systems. [ABSTRACT FROM AUTHOR]

For any prime number p, and integer k ≥ 1, let Fpk be the finite field of pk elements. A famous problem in the theory of polynomials over finite fields is the characterization of all nonconstant polynomials F ∈ Fpk[x] for which the value set {F(α): α ∈ Fpk} has the minimum possible size ⌊(pk−1)/deg F⌋ + 1. For k ≤ 2, the problem was solved in the early 1960s by Carlitz, Lewis, Mills, and Straus. This paper solves the problem for k = 3. [ABSTRACT FROM AUTHOR]

In this paper we shall deduce a Bailey type formula as a consequence of the residual identity of a q-series transformation due to Gasper. Our formula leads to a variety of q-series identities which are related to the arithmetic function counting integer representations of the form n(An+B)/2 + r(Cr+D)/2 + Enr. [ABSTRACT FROM AUTHOR]

Consider the affine space consisting of pairs of matrices (A,B) of fixed size, and its closed subvariety given by the rank conditions rank A ≤ a, rank B ≤ b, and rank (A ⋅ B) ≤ c, for three non-negative integers a,b,c. These varieties are precisely the orbit closures of representations for the equioriented A3 quiver. In this paper we construct the (equivariant) minimal free resolutions of the defining ideals of such varieties. We show how this problem is equivalent to determining the cohomology groups of the tensor product of two Schur functors of tautological bundles on a 2-step flag variety. We provide several techniques for the determination of these groups, which is of independent interest. [ABSTRACT FROM AUTHOR]

Let R be a commutative noetherian ring. In this paper, we study specialization-closed subsets of Spec R. More precisely, we first characterize the specialization-closed subsets in terms of various closure properties of subcategories of modules. Then, for each nonnegative integer n we introduce the notion of n-wide subcategories of R-modules to consider the question asking when a given specialization-closed subset has cohomological dimension at most n. [ABSTRACT FROM AUTHOR]

Let m be an integer greater than three and l be an odd prime. In this paper we prove that at least one of the following groups: PΩ±2m(Fls), PSO±2m(Fls), PO2m±(Fls), or PGO±2m(Fls) is a Galois group of Q for infinitely many integers s > 0. This is achieved by making use of a slight modification of a group theory result of Khare, Larsen, and Savin, and previous results of the author on the images of the Galois representations attached to cuspidal automorphic representations of GL2m(AQ). [ABSTRACT FROM AUTHOR]

In this paper we show that any Bohr-zero non-periodic set B of traceless integer-valued matrices, denoted by Λ, intersects non-trivially the conjugacy class of any matrix from Λ. As a corollary, we obtain that the family of characteristic polynomials of B contains all characteristic polynomials of matrices from Λ. The main ingredient used in this paper is an equidistribution result for an SLd(ℤ) random walk on a finite-dimensional torus deduced from Bourgain-Furman-Lindenstrauss-Mozes work [J. Amer. Math. Soc. 24 (2011), 231-280]. [ABSTRACT FROM AUTHOR]

For positive integers u and v, let Lu = [1 & 0 \\ u & 1] and Rv = [1 & v \\ 0 & 1 ]. Let Su,v be the monoid generated by Lu and Rv, and let Gu,v be the group generated by Lu and Rv. In this paper we expand on a characterization of matrices M = [a & b \\ c & d] in Sk,k and Gk,k when k ≥ 2 given by Esbelin and Gutan to Su,v when u,v ≥ 2 and Gu,v when u,v ≥ 3. We give a simple algorithmic way of determining if M is in Gu,v using a recursive function and the short continued fraction representation of b/d. [ABSTRACT FROM AUTHOR]

In 2013, S. H. Weintraub proved a generalization of the classical Eisenstein irreducibility criterion by providing a bound on the degrees of factors of a polynomial with integer coefficients (see [Proc. Amer. Math. Soc. 141(4) (2013), pp. 1159-1160]). In this paper, we extend this result with a much weaker hypothesis in a more general setup for polynomials having coefficients from the valuation ring of arbitrary valued field. Moreover, when a polynomial f(x) has coefficients from the valuation ring of a henselian valued field K, then we give more precise information about an irreducible factor of f(x) over K. [ABSTRACT FROM AUTHOR]

We deal with a sequence of integer-valued random variables {ZN}N=1∞ which is related to restricted partitions, or representations, of positive integers. We observe that ZN = X1 + ⋅⋅⋅ + XN for independent and bounded random variables Xj's, so ZN has finite mean EZN and variance VarZN. We want to find the limit distribution of {ZN = (ZN-EZN)/√VarZN} as N → ∞. While in many cases the limit distribution is normal, the main results established in this paper are that ZN →d→ Z*, where Z* is a bounded random variable. We find explicitly the range of values of Z* and derive some properties of its distribution. The main tools used are moment generating functions, cumulant generating functions, moments, and cumulants of the random variables involved. Useful related topics are also discussed. [ABSTRACT FROM AUTHOR]

The Schwarz genus g(ξ) of a fibration ξ : E → B is defined as the minimal integer n such that there exists a cover of B by n open sets that admit partial sections to ξ. Many important concepts, including the Lusternik-Schnirelmann category, Farber's topological complexity, and Smale-Vassiliev's complexity of algorithms can be naturally expressed as Schwarz genera of suitably chosen fibrations. In this paper we study Schwarz genus in relation with certain types of morphisms between fibrations. Our main result is the following: if there exists a fibrewise map ƒ : E → E' between fibrations ξ : E → B and ξ : E' → B which induces an n-equivalence between respective fibres for a sufficiently big n, then g(ξ) = g(ξ'). From this we derive several interesting results relating the topological complexity of a space with the topological complexities of its skeleta and subspaces (and similarly for the category). For example, we show that if a CW-complex has high topological complexity (with respect to its dimension and connectivity), then the topological complexity of its skeleta is an increasing function of the dimension. [ABSTRACT FROM AUTHOR]

In this paper we extend a result of Levin, who proved a lower bound on the star discrepancy of generalized Halton sequences in positive integer bases, to rational bases. [ABSTRACT FROM AUTHOR]

This paper studies vanishing of Ext modules over Cohen--Macaulay local rings. The main result of this paper implies that the Auslander--Reiten conjecture holds for maximal Cohen--Macaulay modules of rank one over Cohen--Macaulay normal local rings. It also recovers a theorem of Avramov--Buchweitz--Şega and Hanes--Huneke, which shows that the Tachikawa conjecture holds for Cohen--Macaulay generically Gorenstein local rings. [ABSTRACT FROM AUTHOR]

In this paper, we prove the following “weak bounded negativity conjecture", which says that given a complex smooth projective surface X, for any reduced curve C in X and integer g, assume that the geometric genus of each component of C is bounded from above by g; then the self-intersection number C2 is bounded from below. [ABSTRACT FROM AUTHOR]

In this paper, for r ∈ N with r ≥ 2 we consider several stronger versions of r-sensitivity and measure-theoretical r-sensitivities by analyzing subsets of non-negative integers, for which the r-sensitivity occurs. We obtain an Auslander-Yorke type dichotomy theorem: a minimal topological dynamical system is either thickly r-sensitive or an almost m-to-one extension of its maximal equicontinuous factor for some m ∈ {1, · · ·, r - 1}. [ABSTRACT FROM AUTHOR]

In this paper we define and study a filtration (Gs)s≥0 on the algebra of values at algebraic points of analytic continuations of G-functions: Gs is the set of values at algebraic points in the disk of convergence of all Gfunctions Σ∞n=0 anzn for which there exist some positive integers b and c such that dsbncn+1an is an algebraic integer for any n, where dn = lcm(1, 2, . . ., n). We study the situation at the boundary of the disk of convergence, and using transfer results from analysis of singularities we deduce that constants in Gs appear in the asymptotic expansion of such a sequence (an). [ABSTRACT FROM AUTHOR]

In this paper, we complete the classification of quasi-alternating Montesinos links. We show that the quasi-alternating Montesinos links are precisely those identified independently by Qazaqzeh-Chbili-Qublan and Champanerkar-Ording. A consequence of our proof is that a Montesinos link L is quasi-alternating if and only if its double branched cover is an L-space, and bounds both a positive definite and a negative definite 4-manifold with vanishing first homology. [ABSTRACT FROM AUTHOR]

In this paper, we establish a general discrete Fourier restriction theorem. As an application, we make some progress on the discrete Fourier restriction problem associated with KdV equations. [ABSTRACT FROM AUTHOR]

A charge (finitely additive measure) defined on a Boolean algebra of sets taking values in a group G is called a strictly nonzero (SNZ) charge if it takes the identity value in G only for the zero element of the Boolean algebra. A study of such charges was initiated by Rüdiger Göbel and K. P. S. Bhaskara Rao in 2002. Our main result is a solution to one of the questions posed in that paper: we show that for every cardinal א, the Boolean algebra of clopen sets of {0, 1}א has a strictly nonzero integer-valued charge. The key lemma that we prove is that there exists a strictly nonzero integer-valued permutation-invariant charge on the Boolean algebra of clopen sets of {0, 1}א0. Our proof is based on linear-algebraic arguments, as well as certain kinds of polynomial approximations of binomial coefficients. We also show that there is no integer-valued SNZ charge on Ρ(ℕ). Finally, we raise some interesting problems on integer-valued SNZ charges. [ABSTRACT FROM AUTHOR]

In this paper we consider Artin compressed local algebras, that is, local algebras with maximal length in the class of those with given embedding dimension and socle type. They have been widely studied by several authors, including Boij, Iarrobino, Froberg and Laksov. In this class the Gorenstein algebras play an important role. The authors proved that a compressed Gorenstein X-algebra of socle degree 3 is canonically graded, i.e. analytically isomorphic to its associated graded ring. This unexpected result has been extended to compressed level X-algebras of socle degree 3 in a paper by De Stefani. This paper somehow concludes the investigation proving that Artin compressed Gorenstein X-algebras of socle degree s ≤ 4 are always canonically graded, but explicit examples prove that the result does not extend to socle degree 5 or to compressed level X-algebras of socle degree 4 and type > 1. As a consequence of this approach we present classes of Artin compressed X-algebras which are canonically graded. [ABSTRACT FROM AUTHOR]

Let K be a complete discrete valuation field of characteristic (0, p) with perfect residue field, and let T be an integral Zp-representation of Gal(K/K). A theorem of T. Liu says that if T/pnT is torsion semistable (resp., crystalline) of uniformly bounded Hodge-Tate weights for all n ≥ 1, then T is also semistable (resp., crystalline). In this paper, we show that we can relax the condition of "uniformly bounded Hodge-Tate weights" to an unbounded (log-)growth condition. [ABSTRACT FROM AUTHOR]

Graves and Murty proved that for any integer a ≥ 2 and any fixed integer k ≥ 2, there are ≫ log x/log log x primes p ≤ x such that ap-1 ≢ 1 (mod p2) and p ≡ 1 (mod k), under the assumption of the abc conjecture. In this paper, for any fixed M, the bound log x/log log x is improved to (log x/log log x)(log log log x)M. [ABSTRACT FROM AUTHOR]

We classify all possible h-vectors of graded artinian Gorenstein algebras in socle degree 4 and codimension ⩽17 and in socle degree 5 and codimension ⩽25. We obtain as a consequence that the least number of variables allowing the existence of a nonunimodal Gorenstein h-vector is 13 for socle degree 4 and 17 for socle degree 5. In particular, the smallest nonunimodal Gorenstein h-vector is (1, 13, 12, 13, 1), which was constructed by Stanley in his 1978 seminal paper on level algebras. This solves a longstanding open question in this area. All of our results are characteristic free. [ABSTRACT FROM AUTHOR]

The deviations of a graded algebra are a sequence of integers that determine the Poincaré series of its residue field and arise as the number of generators of certain DG algebras. In a sense, deviations measure how far a ring is from being a complete intersection. In this paper, we study extremal deviations among those of algebras with a fixed Hilbert series. In this setting, we prove that, like the Betti numbers, deviations do not increase when passing to an initial ideal and are maximized by the lex-segment ideal. We also prove that deviations grow exponentially for Golod rings and for certain quadratic monomial algebras. [ABSTRACT FROM AUTHOR]

In this paper we provide a complete answer to a question by Heyman and Shparlinski concerning the natural density of polynomials which are irreducible by Eisenstein's criterion after applying some shift. The main tool we use is a local to global principle for density computations over a free Zmodule of finite rank. [ABSTRACT FROM AUTHOR]

In this paper, we study cut sets of attractors of iterated function systems (IFS) in ℝd. Under natural conditions, we show that all irreducible cut sets of these attractors are perfect sets or single points. This leads to a criterion for the existence of cut points of IFS attractors. If the IFS attractors are self-affine tiles, our results become algorithmically checkable and can be used to exhibit cut points with the help of Hata graphs. This enables us to construct cut points of some self-affine tiles studied in the literature. [ABSTRACT FROM AUTHOR]

In the present paper we obtain several new results related to the problem of upper bound estimates for the number of solutions of the congruence.....where p is a large prime number and λ is an integer coprime to p. Our arguments are based on recent estimates of trigonometric sums over subgroups due to Shkredov and Shteinikov. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the simultaneous Pell equations x² - (m² - 1)y² = 1, z² - (n² - 1)y² = 1 have only a positive integer solution (x, y, z) = (m, 1, n) if m < n = m + me, 0 < e < 1 and m = 202304 1/1-e . Using a computational reduction method we can omit the lower bound for m when m < n = m 1/5 . Moreover, we apply our main result to a family of Thue equations in two parameters studied by Jadrijević. [ABSTRACT FROM AUTHOR]

Cameron and Erdős raised the question of how many maximal sum-free sets there are in {1, . . ., n}, giving a lower bound of 2⎣n/4⎦. In this paper we prove that there are in fact at most 2(1/4+o(1))n maximal sum-free sets in {1, . . ., n}. Our proof makes use of container and removal lemmas of Green as well as a result of Deshouillers, Freiman, Sós and Temkin on the structure of sum-free sets. [ABSTRACT FROM AUTHOR]

Let k be the rational function field over the finite field of q elements and ... its fixed algebraic closure. In this paper, we study algebraic relations over ... among the fundamental period ... of the Carlitz module and the positive characteristic multizeta values ζ(n) and ζ(n, n) for an "odd" integer n, where we say that n is "odd" if q - 1 does not divide n. We prove that these three elements are either algebraically independent over ... or satisfy some simple relation over k. We also prove that if 2n is "odd", then these three elements are algebraically independent over .... [ABSTRACT FROM AUTHOR]

In the structural theory of cluster algebras, a crucial role is played by a family of integer vectors, called c-vectors, which parametrize the coefficients. It has recently been shown that each c-vector with respect to an acyclic initial seed is a real root of the corresponding root system. In this paper, we obtain an interpretation of this result in terms of symmetric matrices. We show that for skew-symmetric cluster algebras, the c-vectors associated with any seed defines a quasi-Cartan companion for the corresponding exchange matrix (i.e. they form a companion basis), and we establish some basic combinatorial properties. In particular, we show that these vectors define an admissible cut of edges in the associated quivers. [ABSTRACT FROM AUTHOR]

Let p be a prime number greater than three. In the p-component of stable homotopy groups of spheres, Oka constructed a beta family from a v2-periodic map on a four cell complex. In this paper, we construct another beta family in the groups at a prime p greater than five from a v2-periodic map on an eight cell complex. [ABSTRACT FROM AUTHOR]

A classical theorem of Paley asserts the existence of an infinite family of quadratic characters whose character sums become exceptionally large. In this paper, we establish an analogous result for characters of any fixed even order. Previously our bounds were only known under the assumption of the Generalized Riemann Hypothesis. [ABSTRACT FROM AUTHOR]

It is well-known that there exist integer linear recurrence sequences {xn} such that xp = x1 (mod p) for all primes p. It is less well-known, but still classical, that there exist such sequences satisfying the stronger condition xpn = xpn-1 (mod pn) for all primes p and n ≥ 1, or even m | Σd|m μ(m/d)xd for all m ≥ 1. These congruence conditions generalize Fermat's little theorem, Euler's theorem, and Gauss's congruence, respectively. In this paper we classify sequences of these three types. Our classification for the first type is in terms of linear dependencies of the characteristic zeros; for the second, it involves recurrence sequences vanishing on arithmetic progressions; and for the last type we give an explicit classification in terms of traces of powers. [ABSTRACT FROM AUTHOR]

Let (A,m) be a noetherian local ring with maximal ideal m and infinite residue field k = A/m. Let J be an m-primary ideal, I1, . . ., I8 ideals of A, and M a finitely generated A-module. In this paper, we interpret mixed multiplicities of (I1, . . ., I8, J) with respect to M as multiplicities of joint reductions of them. This generalizes Rees's theorem on mixed multiplicity. As an application we show that mixed multiplicities are also multiplicities of Rees superficial sequences. [ABSTRACT FROM AUTHOR]

Sendov's conjecture says that if all zeros of a complex polynomial P lie in the closed unit disk and a denotes one of them, then the closed disk of center a and radius 1 contains a critical point of P (i.e. a zero of its derivative P'). The main result of this paper is to prove that, for each a, there exists an integer N such that the disk |ζ -a| ≤ 1 contains a critical point of P when the degree of P is larger than N. We obtain this by studying the geometry of the zeros and critical points of a polynomial which would eventually contradict Sendov's conjecture. [ABSTRACT FROM AUTHOR]

Since Zwegers found a connection between mock theta functions and harmonic weak Maass forms, this subject has been of vast research interest. In this paper, we obtain isomorphisms among the space H>k+1/2+(Γ0 (4m)) of (scalar valued) harmonic weak Maass forms of half integral weight whose Fourier coefficients are supported on suitable progressions, the space Hk+1/2x1,... of vector valued ones, and the space ... of certain harmonic Maass-Jacobi forms of integral weight: ... for k odd and m = 1 or a prime. This is an extension of a result developed by Eichler and Zagier, which shows that ... Here ... and Jk+1,m are the Kohnen plus space of (scalar valued) modular forms of half integral weight, the space of vector valued ones, and the space of Jacobi forms of integral weight, respectively. To extend the result, another approach is necessary because the argument by Eichler and Zagier depends on the dimension formulas for the spaces of holomorphic modular forms, but the dimensions for the spaces of harmonic weak Maass forms are not finite. Our proof relies on some nontrivial properties of the Weil representation. [ABSTRACT FROM AUTHOR]