Search

The Nakayama conjecture is one of the most important conjectures in ring theory. The Auslander-Reiten conjecture is closely related to it. The purpose of this paper is to show that if the Auslander-Reiten conjecture holds in codimension one for a commutative Gorenstein ring $R$, then it holds for $R$. [ABSTRACT FROM AUTHOR]

Let $a_0(n)$ and $b_0(n)$ be the normalized Fourier coefficients of the two holomorphic Hecke eigenforms $f(z)in S_{2k}(Gamma )$ and $varphi (z)in S_{2l}(Gamma )$ respectively. In 1999, Fomenko studied the following average sums of $a_0(n)$ and $b_0(n)$: begin {eqnarray*} sum _{n leq x}a_0(n)^3, quad sum _{n leq x}a_0(n)^2b_0(n), quad sum _{n leq x}a_0(n)^2b_0(n)^2, quad sum _{n leq x}a_0(n)^4. endeqnarray* }}} In this paper, we are able to improve on Fomenko's results. [ABSTRACT FROM AUTHOR]

When studying the least common multiple of some finite sequences of integers, the first author introduced the interesting arithmetic functions $g_k$ $(k in mathbb {N})$, defined by $g_k(n) := frac {n (n + 1) dots (n + k)} {operatorname {lcm}(n, n+1, dots , n + k)}$ $(forall n in mathbb {N} setminus {0})$. He proved that for each $k in mathbb {N}$, $g_k$ is periodic and $k!$ is a period of $g_k$. He raised the open problem of determining the smallest positive period $P_k$ of $g_k$. Very recently, S. Hong and Y. Yang improved the period $k!$ of $g_k$ to $operatorname {lcm}(1 , 2, dots , k)$. In addition, they conjectured that $P_k$ is always a multiple of the positive integer $frac {operatorname {lcm}(1 , 2 , dots , k , k + 1)}{k + 1}$. An immediate consequence of this conjecture is that if $(k + 1)$ is prime, then the exact period of $g_k$ is precisely equal to $operatorname {lcm}(1 , 2 , dots , k)$. par In this paper, we first prove the conjecture of S. Hong and Y. Yang and then we give the exact value of $P_k$ $(k in mathbb {N})$. We deduce, as a corollary, that $P_k$ is equal to the part of $operatorname {lcm}(1 , 2 , dots , k)$ not divisible by some prime. [ABSTRACT FROM AUTHOR]

Several recent papers have given criteria for certain polynomials to permute $mathbb {F}_q$, in terms of the periods of certain generalized Lucas sequences. We show that these results follow from a more general criterion which does not involve such sequences. [ABSTRACT FROM AUTHOR]

In this paper, we present a number of examples of commutative Banach algebras with various Arens irregularity properties. These examples illustrate in particular that strong Arens irregularity and extreme non-Arens regularity, the two natural concepts of ``maximal'' Arens irregularity for general Banach algebras as introduced by Dales-Lau and Granirer, respectively, are indeed distinct. Thereby, an open question raised by several authors is answered. We also link these two properties to another natural Arens irregularity property. [ABSTRACT FROM AUTHOR]

Hara, Takemura and Yoshida discussed toric ideals arising from two way subtable sum problems and showed that these toric ideals are generated by quadratic binomials if and only if the subtables are either diagonal or triangular. In the present paper, we show that if the subtables are either diagonal or triangular, then their toric ideals possess quadratic Gröbner bases. [ABSTRACT FROM AUTHOR]

In this paper we give a short and simple proof of the remarkable evaluations of multiple zeta values established by D. Bowman and D. M. Bradley. [ABSTRACT FROM AUTHOR]

In this paper, we show that the boundary $partial Sigma (W,S)$ of a right-angled Coxeter system $(W,S)$ is minimal if and only if $W_{tilde {S}}$ is irreducible, where $W_{tilde {S}}$ is the minimum parabolic subgroup of finite index in $W$. We also provide several applications and remarks. In particular, we show that for a right-angled Coxeter system $(W,S)$, the set ${w^{infty }|win W, o(w)=infty }$ is dense in the boundary $partial Sigma (W,S)$. [ABSTRACT FROM AUTHOR]

In this short paper we prove that, for $3 le N le 9$, the problem $ -Delta u = e^u$ on the entire Euclidean space $mathbb {R}^N$ does not admit any solution stable outside a compact set of $mathbb {R}^N$. This result is obtained without making any assumption about the boundedness of solutions. Furthermore, as a consequence of our analysis, we also prove the non-existence of finite Morse Index solutions for the considered problem. We then use our results to give some applications to bounded domain problems. [ABSTRACT FROM AUTHOR]

In this paper, we give an estimate from below of the bounding genera for homology $3$-spheres defined by Y. Matsumoto in terms of $w$-invariants. In particular, combining with Matsumoto's estimates we determine the values of the bounding genera for several infinite families of Brieskorn homology $3$-spheres. [ABSTRACT FROM AUTHOR]

This paper gives a description of the relative Brauer group ${rm Br}(E/F)$ when $F$ has characteristic $p$, $[E:F]=p$, and the Galois group ${rm Gal}(E_1/F)$ is solvable, where $E_1$ is the Galois closure of $E$ over $F$. [ABSTRACT FROM AUTHOR]

In this paper we state a Lipschitz version of a theorem due to Cambern concerning into linear isometries between spaces of vector-valued continuous functions and deduce a Lipschitz version of a celebrated theorem due to Jerison concerning onto linear isometries between such spaces. [ABSTRACT FROM AUTHOR]

In this brief paper, we observe that basic results from rational homotopy theory provide formulas for the rational homotopy groups of gauge groups of principal bundles $K to P to B$ in terms of the rational homotopy groups of $K$ and cohomology groups of $B$ alone. [ABSTRACT FROM AUTHOR]

For a given finite-type quiver $varGamma $, we will consider scalar-removed representations $(S_{d}, R_{d}(varGamma ))$, where $S_{d}$ is a direct product of special linear algebraic groups and $R_{d}(varGamma )$ is the representation defined naturally by $varGamma $ and a dimension vector $d$. In this paper, we give a necessary and sufficient condition on $d$ that $R_{d}(varGamma )$ has only finitely many $S_{d}$-orbits. This condition can be paraphrased as a condition concerning lattices of small rank spanned by positive roots of $varGamma $. To determine such scalar-removed representations having only finitely many orbits is very fundamental to the open problem of classification of the so-called semisimple finite prehomogeneous vector spaces. We consider everything over an algebraically closed field of characteristic zero. [ABSTRACT FROM AUTHOR]

In this paper we prove a theorem more general than the following: par noindent ``If $(X,|·|)$ is an $L_1$-predual, $B$ is any boundary of $X$ and ${x_n:n in N }$ is any subset of $X$, then the closure of ${x_n:n in N }$ with respect to the topology of pointwise convergence on $B$ is separable with respect to the topology generated by the norm, whenever ${rm Ext}(B_{X^*})$ is weak$^*$ Lindelöf.'' Several applications of this result are also presented. [ABSTRACT FROM AUTHOR]

In this paper we describe all gradings by a finite abelian group $G$ on the following Lie algebras over an algebraically closed field $F$ of characteristic $pneq 2$: $mathfrak {sl}_n(F)$ ($n$ not divisible by $p$), $mathfrak {so}_n(F)$ ($ngeq 5$, $nneq 8$) and $mathfrak {sp}_n(F)$ ($ngeq 6$, $n$ even). [ABSTRACT FROM AUTHOR]

Chung, Diaconis, and Graham considered random processes of the form $X_{n+1}=a_nX_n+b_npmod p$ where $p$ is odd, $X_0=0$, $a_n=2$ always, and $b_n$ are i.i.d. for $n=0,1,2,dots $. In this paper, we show that if $P(b_n=-1)=P(b_n=0)=P(b_n=1)=1/3$, then there exists a constant $c>1$ such that $clog _2p$ steps are not enough to make $X_n$ get close to being uniformly distributed on the integers mod $p$. [ABSTRACT FROM AUTHOR]

For a positive operator $Q$ on a Banach lattice, one defines $langle Q]={Tgeq 0:~TQleq QT}$ and $[Qrangle ={Tgeq 0: TQgeq QT}$. There have been several recent results asserting that, under certain assumptions on $Q$, $[Qrangle $ has a common invariant subspace. In this paper, we use the technique of minimal vectors to establish similar results for $langle Q]$. [ABSTRACT FROM AUTHOR]

In this paper we describe an algorithm for producing infinitely many examples of set-theoretic complete intersection monomial curves in $mathbb {P}^{n+1}$, starting with a single set-theoretic complete intersection monomial curve in $mathbb {P}^{n}$. Moreover we investigate the numerical criteria to decide when these monomial curves can or cannot be obtained via semigroup gluing. [ABSTRACT FROM AUTHOR]

First we prove some elementary but useful identities in the group ring of $mathbb {Q}/mathbb {Z}$. Our identities have potential applications to several unsolved problems which involve sums of Farey fractions. In this paper we use these identities, together with some analytic number theory and results about divisors in short intervals, to estimate the cardinality of a class of sets of fundamental interest. [ABSTRACT FROM AUTHOR]

In this paper we present some results concerning bounded harmonic $1$-forms on manifolds of compact type. As a corollary we obtain a rigidity result for the first cohomology group of locally isometric Riemannian manifolds. [ABSTRACT FROM AUTHOR]

In this short paper we prove that every sufficiently large odd integer can be written as a sum of one prime, two squares of primes and $83$ powers of $2$. [ABSTRACT FROM AUTHOR]

In our paper "On solvable compact Clifford-Klein forms" [Proc. Amer. Math. Soc. 145 (2017), pp. 1819-1832] we have found a small gap in the proof of the main theorem. In this note we show that the result is correct and that the proof goes through after a slight modification. [ABSTRACT FROM AUTHOR]

In this paper we prove a weighted version of the Khintchine inequalities. [ABSTRACT FROM AUTHOR]