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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]

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]

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 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]

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]

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]

The Wallis power function is defined on \left (-\min \left \{ p,q\right \},\infty \right) by \begin{equation*} W_{p,q}\left (x\right) =\left (\dfrac {\Gamma \left (x+p\right) }{\Gamma \left (x+q\right) }\right) ^{1/\left (p-q\right) }\text { if }p\neq q\text { and }W_{p,p}\left (x\right) =e^{\psi \left (x+p\right) }. \end{equation*} Let p_{i},q_{i}\in \mathbb {R} with 0<\delta _{i}=p_{i}-q_{i}\leq 1, \theta _{i}=\left (1-\delta _{i}\right) /2 for i=1,2 and p_{1}+q_{1}=p_{2}+q_{2}=2\sigma +1. We prove that, if q_{1}>q_{2} then for any integer m\in \mathbb {N}, the function \begin{equation*} x\mapsto \left (-1\right) ^{m}\left [ \ln \frac {W_{p_{1},q_{1}}\left (x\right) }{W_{p_{2},q_{2}}\left (x\right) }-\sum _{k=1}^{m}\frac {a_{2k}\left (\theta _{1},\theta _{2}\right) }{\left (2k+1\right) \left (2k\right) \left (x+\sigma \right) ^{2k}}\right ] \end{equation*} is completely monotonic on \left (-\sigma,\infty \right), where \begin{equation*} a_{2k}\left (\theta _{1},\theta _{2}\right) =\frac {B_{2k+1}\left (\theta _{2}\right) }{\theta _{2}-1/2}-\frac {B_{2k+1}\left (\theta _{1}\right) }{\theta _{1}-1/2}. \end{equation*} This extends and generalizes some known results. [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|

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]

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

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]

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]

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]

We show how to construct discretely ordered principal ideal subrings of \mathbb {Q}[x] with various types of prime behaviour. Given any set \mathcal D consisting of finite strictly increasing sequences (d_1,d_2,\dots, d_l) of positive integers such that, for each prime integer p, the set \{p\mathbb {Z}, d_1+p\mathbb {Z},\dots, d_l+p\mathbb {Z}\} does not contain all the cosets modulo p, we can stipulate to have, for each (d_1,\dots, d_l)\in \mathcal D, a cofinal set of progressions (f, f+d_1, \dots, f+d_l) of prime elements in our principal ideal domain R_\tau. Moreover, we can simultaneously guarantee that each positive prime g\in R_\tau \setminus \mathbb {N} is either in a prescribed progression as above or there is no other prime h in R_\tau such that g-h\in \mathbb {Z}. Finally, all the principal ideal domains we thus construct are non-Euclidean and isomorphic to subrings of the ring \widehat {\mathbb {Z}} of profinite integers. [ABSTRACT FROM AUTHOR]

Say that (x, y, z) is a positive primitive integral Pythagorean triple if x, y, z are positive integers without common factors satisfying x^2 + y^2 = z^2. An old theorem of Berggren gives three integral invertible linear transformations whose semi-group actions on (3, 4, 5) and (4, 3, 5) generate all positive primitive Pythagorean triples in a unique manner. We establish an analogue of Berggren's theorem in the context of a one-variable polynomial ring over a field of characteristic \neq 2. As its corollaries, we obtain some structure theorems regarding the orthogonal group with respect to the Pythagorean quadratic form over the polynomial ring. [ABSTRACT FROM AUTHOR]

Let f^{(r)}(n;s,k) be the maximum number of edges of an r-uniform hypergraph on n vertices not containing a subgraph with k edges and at most s vertices. In 1973, Brown, Erdős and Sós conjectured that the limit \begin{equation*} \lim _{n\to \infty } n^{-2} f^{(3)}(n;k+2,k) \end{equation*} exists for all positive integers k\ge 2. They proved this for k=2. In 2019, Glock proved this for k=3 and determined the limit. Quite recently, Glock, Joos, Kim, Kühn, Lichev and Pikhurko proved this for k=4 and determined the limit; we combine their work with a new reduction to fully resolve the conjecture by proving that indeed the limit exists for all positive integers k\ge 2. [ABSTRACT FROM AUTHOR]

The set of degrees of maps D(M,N), where M,N are closed oriented n-manifolds, always contains 0 and the set of degrees of self-maps D(M) always contains 0 and 1. Also, if a,b\in D(M), then ab\in D(M); a set A\subseteq \mathbb {Z} so that ab\in A for each a,b\in A is called multiplicative. On the one hand, not every infinite set of integers (containing 0) is a mapping degree set (Neofytidis, Wang, and Wang [Bull. Lond. Math. Soc. 55 (2023), pp. 1700–1717]) and, on the other hand, every finite set of integers (containing 0) is the mapping degree set of some 3-manifolds (Costoya, Muñoz and Viruel [ Finite sets containing zero are mapping degree sets , arXiv: 2301.13719 ]). We show the following: Not every multiplicative set A containing 0,1 is a self-mapping degree set. For each n\in \mathbb {N} and k\geq 3, every D(M,N) for n-manifolds M and N is D(P,Q) for some (n+k)-manifolds P and Q. As a consequence of (ii) and Costoya, Muñoz and Viruel, every finite set of integers (containing 0) is the mapping degree set of some n-manifolds for all n\neq 1,2,4,5. [ABSTRACT FROM AUTHOR]

We prove that for even integer k\geq k_0, almost all of zeros of the even period polynomial associated to a newform of weight k on \Gamma _0(N) are on the circle |z|=1/\sqrt {N}. [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]

We provide a simple criterion on a family of functions that implies a square function estimate on L^p for every even integer p \geq 2. This defines a new type of superorthogonality that is verified by checking a less restrictive criterion than any other type of superorthogonality that is currently known. [ABSTRACT FROM AUTHOR]

This note proves two theorems regarding Fermat-type equation x^r + y^r = dz^p where r \geq 5 is a prime. Our main result shows that, for infinitely many integers d, the previous equation has no non-trivial primitive solutions such that 2 \mid x+y or r \mid x+y, for a set of exponents p of positive density. We use the modular method with a symplectic argument to prove this result. [ABSTRACT FROM AUTHOR]

Let k be an even positive integer, p be a prime and m be a nonnegative integer. We find an upper bound for orders of zeros (at cusps) of a linear combination of classical Eisenstein series of weight k and level p^m. As an immediate consequence we find the set of all eta quotients that are linear combinations of these Eisenstein series and hence the set of all eta quotients of level p^m whose derivatives are also eta quotients. [ABSTRACT FROM AUTHOR]

In 2008, Schmidt and Tuller stated a conjecture concerning optimal packing and covering of integers by translates of a given three-point set. In this note, we confirm their conjecture and relate it to several other problems in combinatorics. [ABSTRACT FROM AUTHOR]

For the ideal \mathfrak {p} in k[x, y, z] defining a space monomial curve, we show that \mathfrak {p}^{(2 n - 1)} \subseteq \mathfrak {m} \mathfrak {p}^{n} for some positive integer n, where \mathfrak {m} is the maximal ideal (x, y, z). Moreover, the smallest such n is determined. It turns out that there is a counterexample to a claim due to Grifo, Huneke, and Mukundan, which states that \mathfrak {p}^{(3)} \subseteq \mathfrak {m} \mathfrak {p}^2 if k is a field of characteristic not 3; however, the stable Harbourne conjecture holds for space monomial curves as they claimed. [ABSTRACT FROM AUTHOR]

We present an infinite family of recursive formulas that count binary integer partitions satisfying natural divisibility conditions and show that these counts are interrelated via partial sums. Moreover, we interpret the partitions we study in the language of graded polynomial rings and apply this to the mod 2 Steenrod algebra to compute the free rank of certain homology modules in stable homotopy theory. [ABSTRACT FROM AUTHOR]

Let M^n be a closed hypersurface with constant mean curvature and constant scalar curvature in the unit sphere \mathbb {S}^{n+1}. Denote by S and H the squared length of the second fundamental form and the mean curvature of M, respectively. For a fixed integer k, where 1\leq k\leq n-2, we prove that there exists a positive constant \gamma (n) depending only on n, such that if |H|<\gamma (n) and \alpha _k(n,H)\leq S \leq \alpha _{k+1}(n,H), then either S=\alpha _k(n,H) and M must be the Clifford torus \mathbb {S}^{n-k}\big (\frac {1}{\sqrt {1+\lambda _k^2}}\big)\times \mathbb {S}^{k}\big (\frac {\lambda _k}{\sqrt {1+\lambda _k^2}}\big), or S=\alpha _{k+1}(n,H) and M is the Clifford torus \mathbb {S}^{n-k-1}\big (\frac {1}{\sqrt {1+\lambda _{k+1}^2}}\big)\times \mathbb {S}^{k+1}\big (\frac {\lambda _{k+1}}{\sqrt {1+\lambda _{k+1}^2}}\big). Here \alpha _k(n,H)=n+\frac {n^3}{2k(n-k)}H^2-\frac {n(n-2k)}{2k(n-k)}\sqrt {n^2H^4+4k(n-k)H^2} and \lambda _k=\frac {n|H|+\sqrt {n^2H^2+4k(n-k)}}{2(n-k)}. [ABSTRACT FROM AUTHOR]

A result of Monsky states that the Hilbert–Kunz function of a one-dimensional local ring of prime characteristic has a term \phi that is eventually periodic. For example, in the case of a power series ring in one variable over a prime-characteristic field, \phi is the zero function and is therefore immediately periodic with period 1. In additional examples produced by Kunz [Amer. J. Math. 91 (1969), pp. 772–784] and Monsky [Math. Ann. 263 (1983), pp. 43–49], \phi is immediately periodic with period 2. We show that, for every positive integer \pi, there exists a ring for which \phi is immediately periodic with period \pi. [ABSTRACT FROM AUTHOR]

Let F be the function field of a curve over a non-archimedean local field. Let m \geq 2 be an integer coprime to the characteristic of the residue field of the local field. In this article, we show that every element in H^{3}(F, \mu _{m}^{\otimes 2}) is of the form \chi \cup (f) \cup (g), where \chi is in H^{1}(F, \mathbb {Z}/m\mathbb {Z}) and (f), (g) in H^{1}(F, \mu _{m}). This extends a result of Parimala and Suresh [Ann. of Math. (2) 172 (2010), pp. 1391–1405], where they show this when m is prime and when F contains \mu _{m}. [ABSTRACT FROM AUTHOR]

We study directional mean dimension of \mathbb {Z}^k-actions (where k is a positive integer). On the one hand, we show that there is a \mathbb {Z}^2-action whose directional mean dimension (considered as a [0,+\infty ]-valued function on the torus) is not continuous. On the other hand, we prove that if a \mathbb {Z}^k-action is continuum-wise expansive, then the values of its (k-1)-dimensional directional mean dimension are bounded. This is a generalization (with a view towards Meyerovitch and Tsukamoto's theorem on mean dimension and expansive multiparameter actions) of a classical result due to Mañé: Any compact metrizable space admitting an expansive homeomorphism (with respect to a compatible metric) is finite-dimensional. [ABSTRACT FROM AUTHOR]

We give effective lower bounds on caliber numbers of the parametric family of real quadratic fields \mathbb {Q}(\sqrt {t^4-n^2}) as t varies over positive integers for a congruent number n. Furthermore, we provide lower bounds on class numbers of Richaud-Degert type real quadratic fields of the form \mathbb {Q}(\sqrt {n^2k^4-1}) for positive integers k and congruent numbers n whose elliptic curves have algebraic rank greater than 2. [ABSTRACT FROM AUTHOR]

In the classical theory, a famous by-product of the continued fraction expansion of quadratic irrational numbers \sqrt {D} is the solution to Pell's equation for D. It is well-known that, once an integer solution to Pell's equation exists, we can use it to generate all other solutions (u_n,v_n)_{n\in \mathbb {Z}}. Our object of interest is the polynomial version of Pell's equation, where the integers are replaced by polynomials with complex coefficients. We then investigate the factors of v_n(t). In particular, we show that over the complex polynomials, there are only finitely many values of n for which v_n(t) has a repeated root. Restricting our analysis to \mathbb {Q}[t], we give an upper bound on the number of "new" factors of v_n(t) of degree at most N. Furthermore, we show that all "new" linear rational factors of v_n(t) can be found when n \leq 3, and all "new" quadratic rational factors when n \leq 6. [ABSTRACT FROM AUTHOR]

We show that the Liouville function \lambda changes sign infinitely often on n^2+ d for any non-zero integer d. [ABSTRACT FROM AUTHOR]

Given a primitive collection of vectors in the integer lattice, we count the number of ways it can be extended to a basis by vectors with sup-norm bounded by T, producing an asymptotic estimate as T \to \infty. This problem can be interpreted in terms of unimodular matrices, as well as a representation problem for a class of multilinear forms. In the 2-dimensional case, this problem is also connected to the distribution of Farey fractions. As an auxiliary lemma we prove a counting estimate for the number of integer lattice points of bounded sup-norm in a hyperplane in \mathbb R^n. Our main result on counting basis extensions also generalizes to arbitrary lattices in \mathbb R^n. Finally, we establish some basic properties of sparse representations of integers by multilinear forms. [ABSTRACT FROM AUTHOR]

For rational \alpha, the fractional partition functions p_\alpha (n) are given by the coefficients of the generating function (q; q)_\infty ^\alpha. When \alpha = -1, one obtains the usual partition function. Congruences of the form p(\ell n + c)\equiv 0 \pmod {\ell } for a prime \ell and integer c were studied by Ramanujan. Such congruences exist only for \ell \in \{5, 7, 11\}. Chan and Wang recently studied congruences for the fractional partition functions and gave several infinite families of congruences using identities of the Dedekind eta-function. Following their work, we use the theory of non-ordinary primes to find a general framework that characterizes congruences modulo any integer. This allows us to prove new congruences such as p_\frac {57}{61}(17^2 n - 3) \equiv 0 \pmod {17^2}. [ABSTRACT FROM AUTHOR]

In this short note, we prove a uniqueness result for small entropy self-expanders asymptotic to a fixed cone. This is a direct consequence of the mountain-pass theorem and the integer degree argument proved by J. Bernstein and L. Wang. [ABSTRACT FROM AUTHOR]

We introduce the Schur class of functions, discrete analytic on the integer lattice in the complex plane. As a special case, we derive the explicit form of discrete analytic Blaschke factors and solve the related basic interpolation problem. [ABSTRACT FROM AUTHOR]

We prove that a sequence h of non-negative integers is the Hilbert function of some Artinian Gorenstein algebra with the strong Lefschetz property (SLP) if and only if it is an Stanley-Iarrobino-sequence. This generalizes the result by T. Harima which characterizes the Hilbert functions of Artinian Gorenstein algebras with the weak Lefschetz property. We also provide classes of Artinian Gorenstein algebras obtained from the ideal of points in \mathbb {P}^n such that some of their higher Hessians have non-vanishing determinants. Consequently, we provide families of such algebras satisfying the SLP. [ABSTRACT FROM AUTHOR]

We express the quotients of the Riemann zeta function at consecutive positive integers in terms of the limit of an average sum of quotients of special values of Dirichlet L-functions. [ABSTRACT FROM AUTHOR]

Fundamental to the theory of continued fractions is the fact that every infinite continued fraction with positive integer coefficients converges; however, it is unknown precisely which continued fractions with integer coefficients (not necessarily positive) converge. Here we present a simple test that determines whether an integer continued fraction converges or diverges. In addition, for convergent continued fractions the test specifies whether the limit is rational or irrational. An attractive way to visualise integer continued fractions is to model them as paths on the Farey graph, which is a graph embedded in the hyperbolic plane that induces a tessellation of the hyperbolic plane by ideal triangles. With this geometric representation of continued fractions our test for convergence can be interpreted in a particularly elegant manner, giving deeper insight into the nature of continued fraction convergence. [ABSTRACT FROM AUTHOR]

This note is concerned with the set of integral solutions of the equation x2 + 3y2 = 12n + 4, where n is a positive integer. We will describe a parametrization of this set using the 3-core partitions of n. In particular we construct a crank using the action of a suitable subgroup of the isometric group of the plane that we connect with the unit group of the ring of Eisenstein integers. We also show that the process goes in the reverse direction: from the solutions of the equation and the crank, we can describe the 3-core partitions of n. As a consequence we describe an explicit bijection between 3-core partitions and ideals of the ring of Eisenstein integers, explaining a result of G. Han and K. Ono obtained using modular forms. [ABSTRACT FROM AUTHOR]

We consider boundary conditions of self-adjoint banded Toeplitz matrices. We ask if boundary conditions exist for banded self-adjoint Toeplitz matrices which satisfy operator inequalities of Dirichlet-Neumann bracketing type. For a special class of banded Toeplitz matrices including integer powers of the discrete Laplacian we find such boundary conditions. Moreover, for this class we give a lower bound on the spectral gap above the lowest eigenvalue. [ABSTRACT FROM AUTHOR]

Let {sj}j=1n be positive integers. We show that for any 1 ≤ L ≤ n, ||∏j=1n 1 − zj)||L∞(|z|=1 ≥ exp (1/2e L/(s1s2 . . . sL)1/L). In particular, this gives geometric growth if a positive proportion of the {sj} are bounded. We also show that when the sj grow regularly and faster than j(log j)2+ε, some ε > 0, then the norms grow faster than exp((log n)1+δ) for some δ > 0. [ABSTRACT FROM AUTHOR]

For any graph G, assume that J(G) is the cover ideal of G. Let J(G)(k) denote the kth symbolic power of J(G). We characterize all graphs G with the property that J(G)(k) has a linear resolution for some (equivalently, for all) integer k ≥ 2. Moreover, it is shown that for any graph G, the sequence (reg(J(G)(k)))k=1∞ is nondecreasing. Furthermore, we compute the largest degree of minimal generators of J(G)(k) when G is either an unmixed of a claw-free graph. [ABSTRACT FROM AUTHOR]