Triviality of Iwasawa module associated to some abelian fields of prime conductors

Let p be an odd prime number and \(\ell \) an odd prime number dividing \(p-1\). We denote by \(F=F_{p,\ell }\) the real abelian field of conductor p and degree \(\ell \), and by \(h_F\) the class number of F. For a prime number \(r \ne p,\,\ell \), let \(F_{\infty }\) be the cyclotomic \(\mathbb {Z}_r\)-extension over F, and \(M_{\infty }/F_{\infty }\) the maximal pro-r abelian extension unramified outside r. We prove that \(M_{\infty }\) coincides with \(F_{\infty }\) and consequently \(h_F\) is not divisible by r when r is a primitive root modulo \(\ell \) and r is smaller than an explicit constant depending on p.     

Explicit factorizations of cyclotomic polynomials over finite fields

Let q be a prime power and let \({\mathbb {F}}_q\) be a finite field with q elements. This paper discusses the explicit factorizations of cyclotomic polynomials over \(\mathbb {F}_q\). Previously, it has been shown that to obtain the factorizations of the \(2^{n}r\)th cyclotomic polynomials, one only need to solve the factorizations of a finite number of cyclotomic polynomials. This paper shows that with an additional condition that \(q\equiv 1 \pmod p\), the result can be generalized to the \(p^{n}r\)th cyclotomic polynomials, where p is an arbitrary odd prime. Applying this result we discuss the factorization of cyclotomic polynomials over finite fields. As examples we give the explicit factorizations of the \(3^{n}\)th, \(3^{n}5\)th and \(3^{n}7\)th cyclotomic polynomials.     

Nonuniform Multiresolution Analysis on Local Fields of Positive Characteristic

A generalization of Mallat’s classic theory of multiresolution analysis (MRA) on local fields of positive characteristic was considered by Jiang et al. (J Math Anal Appl 294:523–532, 2004). In this paper, we present a notion of nonuniform MRA on local field \(K\) of positive characteristic. The associated subspace \(V_0\) of \(L^2(K)\) has an orthonormal basis, a collection of translates of the scaling function \(\varphi \) of the form \(\{ \varphi (x-\lambda ) \}_{ \lambda \in \Lambda }\) where \(\Lambda = \{ 0,r/N \}+ \mathcal{Z}, \,N \ge 1\) is an integer and \(r\) is an odd integer such that \(r\) and \(N\) are relatively prime and \(\mathcal{Z}=\{u(n): n\in \mathbb {N}_{0}\}\). We obtain the necessary and sufficient condition for the existence of associated wavelets and present an algorithm for the construction of nonuniform MRA on local fields starting from a low-pass filter \(m_{0}\) with appropriate conditions.     

On the family of elliptic curves $$\varvec{y^2=x^3-m^2x+p^2}$$

In this paper, we study the torsion subgroup and rank of elliptic curves for the subfamilies of \(E_{m,p} : y^2=x^3-m^2x+p^2\), where m is a positive integer and p is a prime. We prove that for any prime p, the torsion subgroup of \(E_{m,p}(\mathbb {Q})\) is trivial for both the cases {\(m\ge 1\), \(m\not \equiv 0\pmod 3\)} and {\(m\ge 1\), \(m \equiv 0 \pmod 3\), with \(gcd(m,p)=1\)}. We also show that given any odd prime p and for any positive integer m with \(m\not \equiv 0\pmod 3\) and \(m\equiv 2\pmod {32}\), the lower bound for the rank of \(E_{m,p}(\mathbb {Q})\) is 2. Finally, we find curves of rank 9 in this family.     

A finite spectral representation of the partition function

Let \(E_{/\mathbb {Q}}\!\) be an elliptic curve, \(p>3\) a good ordinary prime for E, and \(K_\infty \) a p-adic Lie extension of a number field k. Under some standard hypotheses, we study the asymptotic growth in both the Mordell–Weil rank and Shafarevich–Tate group for E over a tower of extensions \(K_n/k\) inside \(K_\infty \); we obtain lower bounds on the former, and upper bounds on the latter’s size.     

Spectral density estimates with partial symmetries and an application to Bahri–Lions-type results

We are concerned with the existence of infinitely many solutions for the problem \(-\Delta u=|u|^{p-2}u+f\) in \(\Omega \), \(u=u_0\) on \(\partial \Omega \), where \(\Omega \) is a bounded domain in \(\mathbb {R}^N\), \(N\ge 3\). This can be seen as a perturbation of the problem with \(f=0\) and \(u_0=0\), which is odd in u. If \(\Omega \) is invariant with respect to a closed strict subgroup of O(N), then we prove infinite existence for all functions f and \(u_0\) in certain spaces of invariant functions for a larger range of exponents p than known before. In order to achieve this, we prove Lieb–Cwikel–Rosenbljum-type bounds for invariant potentials on \(\Omega \), employing improved Sobolev embeddings for spaces of invariant functions.     

On generalized Ramanujan primes

Let \(\Delta = \sum _{m=0}^\infty q^{(2m+1)^2} \in \mathbf {F}_2[[q]]\) be the reduction mod 2 of the \(\Delta \) series. A modular form of level 1, \(f=\sum _{n\geqslant 0} c(n) \,q^n\), with integer coefficients, is congruent modulo \(2\) to a polynomial in \(\Delta \). Let us set \(W_f(x)=\sum _{n\leqslant x,\ c(n)\text { odd }} 1\), the number of odd Fourier coefficients of \(f\) of index \(\leqslant x\). The order of magnitude of \(W_f(x)\) (for \(x\rightarrow \infty \)) has been determined by Serre in the seventies. Here, we give an asymptotic equivalent for \(W_f(x)\). Let \(p(n)\) be the partition function and \(A_0(x)\) (resp. \(A_1(x)\)) be the number of \(n\leqslant x\) such that \(p(n)\) is even (resp. odd). In the preceding papers, the second-named author has shown that \(A_0(x)\geqslant 0.28 \sqrt{x\;\log \log x}\) for \(x\geqslant 3\) and \(A_1(x)>\frac{4.57 \sqrt{x}}{\log x}\) for \(x\geqslant 7\). Here, it is proved that \(A_0(x)\geqslant 0.069 \sqrt{x}\;\log \log x\) holds for \(x>1\) and that \(A_1(x) \geqslant \frac{0.037 \sqrt{x}}{(\log x)^{7/8}}\) holds for \(x\geqslant 2\). The main tools used to prove these results are the determination of the order of nilpotence of a modular form of level-\(1\) modulo \(2\), and of the structure of the space of those modular forms as a module over the Hecke algebra, which have been given in a recent work of Serre and the second-named author.     

New lower bounds for the Shannon capacity of odd cycles

The Shannon capacity of a graph G is defined as \(c(G)=\sup _{d\ge 1}(\alpha (G^d))^{\frac{1}{d}},\) where \(\alpha (G)\) is the independence number of G. The Shannon capacity of the cycle \(C_5\) on 5 vertices was determined by Lovász in 1979, but the Shannon capacity of a cycle \(C_p\) for general odd p remains one of the most notorious open problems in information theory. By prescribing stabilizers for the independent sets in \(C_p^d\) and using stochastic search methods, we show that \(\alpha (C_7^5)\ge 350\), \(\alpha (C_{11}^4)\ge 748\), \(\alpha (C_{13}^4)\ge 1534\), and \(\alpha (C_{15}^3)\ge 381\). This leads to improved lower bounds on the Shannon capacity of \(C_7\) and \(C_{15}\): \(c(C_7)\ge 350^{\frac{1}{5}}> 3.2271\) and \(c(C_{15})\ge 381^{\frac{1}{3}}> 7.2495\).     

Symmetry properties of resolving sets and metric bases in hypercubes

In this paper we consider some special characteristics of distances between vertices in the \(n\)-dimensional hypercube graph \(Q_n\) and, as a consequence, the corresponding symmetry properties of its resolving sets. It is illustrated how these properties can be implemented within a simple greedy heuristic in order to find efficiently an upper bound of the so called metric dimension \(\beta (Q_n)\) of \(Q_n\), i.e. the minimal cardinality of a resolving set in \(Q_n\). This heuristic was applied to generate upper bounds of \(\beta (Q_n)\) for \(n\) up to \(22\), which are for \(n\ge 19\) better than the existing ones. Starting from these new bounds, some existing upper bounds for \(23\le n\le 90\) are improved by a dynamic programming procedure.     

On the estimates of the upper and lower bounds of Ramanujan primes

For \(n\ge 1\), the nth Ramanujan prime is defined as the least positive integer \(R_{n}\) such that for all \(x\ge R_{n}\), the interval \((\frac{x}{2}, x]\) has at least n primes. Let \(p_{i}\) be the ith prime and \(R_{n}=p_{s}\). Sondow, Laishram, and other scholars gave a series of upper bounds of s. In this paper we establish several results giving estimates of upper and lower bounds of Ramanujan primes. Using these estimates, we discuss a conjecture on Ramanujan primes of Sondow–Nicholson–Noe and prove that if \(n>10^{300}\), then \(\pi (R_{mn})\le m\pi (R_{n})\) for \(m\ge 1\).     

Ramanujan-type congruences for overpartitions modulo 16

Let \(\bar{p}(n)\) denote the number of overpartitions of \(n\). Recently, Fortin–Jacob–Mathieu and Hirschhorn–Sellers independently obtained 2-, 3- and 4-dissections of the generating function for \(\bar{p}(n)\) and derived a number of congruences for \(\bar{p}(n)\) modulo 4, 8 and 64 including \(\bar{p}(8n+7)\equiv 0 \pmod {64}\) for \(n\ge 0\). In this paper, we give a 16-dissection of the generating function for \(\bar{p}(n)\) modulo 16 and show that \(\bar{p}(16n+14)\equiv 0\pmod {16}\) for \(n\ge 0\). Moreover, using the \(2\)-adic expansion of the generating function for \(\bar{p}(n)\) according to Mahlburg, we obtain that \(\bar{p}(\ell ^2n+r\ell )\equiv 0\pmod {16}\), where \(n\ge 0\), \(\ell \equiv -1\pmod {8}\) is an odd prime and \(r\) is a positive integer with \(\ell \not \mid r\). In particular, for \(\ell =7\) and \(n\ge 0\), we get \(\bar{p}(49n+7)\equiv 0\pmod {16}\) and \(\bar{p}(49n+14)\equiv 0\pmod {16}\). We also find four congruence relations: \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n) \pmod {16}\) for \(n\ge 0\), \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n)\pmod {32}\) where \(n\) is not a square of an odd positive integer, \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n)\pmod {64}\) for \(n\not \equiv 1,2,5\pmod {8}\) and \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n)\pmod {128}\) for \(n\equiv 0\pmod {4}\).     

Left annihilator of generalized derivations on Lie ideals in prime rings

Let \(R\) be a prime ring, \(L\) a noncentral Lie ideal of \(R\), \(F\) a generalized derivation with associated nonzero derivation \(d\) of \(R\). If \(a\in R\) such that \(a(d(u)^{l_1} F(u)^{l_2} d(u)^{l_3} F(u)^{l_4} \ldots F(u)^{l_k})^{n}=0\) for all \(u\in L\), where \(l_1,l_2,\ldots ,l_k\) are fixed non negative integers not all are zero and \(n\) is a fixed integer, then either \(a=0\) or \(R\) satisfies \(s_4\), the standard identity in four variables.     

On solutions of the simultaneous Pell equations $$ x^{2}-\left( a^{2}-1\right) y^{2}=1$$ and $$y^{2}-pz^{2}=1$$y2-pz2=1

Let \(a\ge 2\) be an integer and p prime number. It is well-known that the solutions of the Pell equation have recurrence relations. For the simultaneous Pell equations

$$\begin{aligned}&x^{2}-\left( a^{2}-1\right) y^{2} =1 \\&y^{2}-pz^{2} =1 \end{aligned}$$

assume that \(x=x_{m}\) and \(y=y_{m}\). In this paper, we show that if \(m\ge 3\) is an odd integer, then there is no positive solution to the system. Moreover, we find the solutions completely for \(5\le a\le 14\) in the cases when \(m\ge 2\) is even integer and \(m=1\).

     

On the maximality of a set of mutually orthogonal Sudoku Latin Squares

The maximum number of mutually orthogonal Sudoku Latin squares (MOSLS) of order \(n=m^2\) is \(n-m\). In this paper, we construct for \(n=q^2\), q a prime power, a set of \(q^2-q-1\) MOSLS of order \(q^2\) that cannot be extended to a set of \(q^2-q\) MOSLS. This contrasts to the theory of ordinary Latin squares of order n, where each set of \(n-2\) mutually orthogonal Latin Squares (MOLS) can be extended to a set of \(n-1\) MOLS (which is best possible). For this proof, we construct a particular maximal partial spread of size \(q^2-q+1\) in \(\mathrm {PG}(3,q)\) and use a connection between Sudoku Latin squares and projective geometry, established by Bailey, Cameron and Connelly.     

A new theorem on the prime-counting function

For \(x>0\), let \(\pi (x)\) denote the number of primes not exceeding x. For integers a and \(m>0\), we determine when there is an integer \(n>1\) with \(\pi (n)=(n+a)/m\). In particular, we show that, for any integers \(m>2\) and \(a\leqslant \lceil e^{m-1}/(m-1)\rceil \), there is an integer \(n>1\) with \(\pi (n)=(n+a)/m\). Consequently, for any integer \(m>4\), there is a positive integer n with \(\pi (mn)=m+n\). We also pose several conjectures for further research; for example, we conjecture that, for each \(m=1,2,3,\ldots \), there is a positive integer n such that \(m+n\) divides \(p_m+p_n\), where \(p_k\) denotes the k-th prime.     

On Locally Gabriel Geometric Graphs

Let \(P\) be a set of \(n\) points in the plane. A geometric graph \(G\) on \(P\) is said to be locally Gabriel if for every edge \((u,v)\) in \(G\), the Euclidean disk with the segment joining \(u\) and \(v\) as diameter does not contain any points of \(P\) that are neighbors of \(u\) or \(v\) in \(G\). A locally Gabriel graph(LGG) is a generalization of Gabriel graph and is motivated by applications in wireless networks. Unlike a Gabriel graph, there is no unique LGG on a given point set since no edge in a LGG is necessarily included or excluded. Thus the edge set of the graph can be customized to optimize certain network parameters depending on the application. The unit distance graph(UDG), introduced by Erdos, is also a LGG. In this paper, we show the following combinatorial bounds on edge complexity and independent sets of LGG: (i) For any \(n\), there exists LGG with \(\Omega (n^{5/4})\) edges. This improves upon the previous best bound of \(\Omega (n^{1+\frac{1}{\log \log n}})\). (ii) For various subclasses of convex point sets, we show tight linear bounds on the maximum edge complexity of LGG. (iii) For any LGG on any \(n\) point set, there exists an independent set of size \(\Omega (\sqrt{n}\log n)\).     

On a conjecture of differentially 8-uniform power functions

Let \(m \ge 5\) be an odd integer. For \(d=2^m+2^{(m+1)/2}+1\) or \(d=2^{m+1}+3\), Blondeau et al. conjectured that the power function \(F_d=x^d\) over \(\mathrm {GF}(2^{2m})\) is differentially 8-uniform in which all values \(0, \, 2, \, 4,\, 6,\, 8\) appear. In this paper, we confirm this conjecture and compute the differential spectrum of \(F_d\) for both values of d.     

Some results on generalized strong external difference families

A generalized strong external difference family (briefly \((v, m; k_1,\dots ,k_m; \lambda _1,\dots ,\lambda _m)\)-GSEDF) was introduced by Paterson and Stinson in 2016. In this paper, we give some nonexistence results for GSEDFs. In particular, we prove that a \((v, 3;k_1,k_2,k_3; \lambda _1,\lambda _2,\lambda _3)\)-GSEDF does not exist when \(k_1+k_2+k_3< v\). We also give a first recursive construction for GSEDFs and prove that if there is a \((v,2;2\lambda ,\frac{v-1}{2};\lambda ,\lambda )\)-GSEDF, then there is a \((vt,2;4\lambda ,\frac{vt-1}{2};2\lambda ,2\lambda )\)-GSEDF with \(v>1\), \(t>1\) and \(v\equiv t\equiv 1\pmod 2\). Then we use it to obtain some new GSEDFs for \(m=2\). In particular, for any prime power q with \(q\equiv 1\pmod 4\), we show that there exists a \((qt, 2;(q-1)2^{n-1},\frac{qt-1}{2};(q-1)2^{n-2},(q-1)2^{n-2})\)-GSEDF, where \(t=p_1p_2\dots p_n\), \(p_i>1\), \(1\le i\le n\), \(p_1, p_2,\dots ,p_n\) are odd integers.     

On Weighted Average Interpolation with Cardinal Splines

Given a sequence of data \(\{ y_{n} \} _{n \in \mathbb{Z}}\) with polynomial growth and an odd number \(d\), Schoenberg proved that there exists a unique cardinal spline \(f\) of degree \(d\) with polynomial growth such that \(f ( n ) =y_{n}\) for all \(n\in \mathbb{Z}\). In this work, we show that this result also holds if we consider weighted average data \(f\ast h ( n ) =y_{n}\), whenever the average function \(h\) satisfies some light conditions. In particular, the interpolation result is valid if we consider cell-average data \(\int_{n-a}^{n+a}f ( x ) dx=y_{n}\) with \(0< a\leq 1/2\). The case of even degree \(d\) is also studied.