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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 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]