We present two sets of 12 integers that have the same sets of 4-sums. The proof of the fact that a set of 12 numbers is uniquely determined by the set of its 4-sums published 50 years ago is wrong, and we demonstrate an incorrect calculation in it.
In this note, we extend the definition of multiple harmonic sums and apply their stuffle relations to obtain explicit evaluations of the sums $R_n(p,t)=sum olimits_{m=0}^n m^p H_m^t$, where $H_m$ are harmonic numbers. When $tle 4$ these sums were fir
st studied by Spiess around 1990 and, more recently, by Jin and Sun. Our key step first is to find an explicit formula of a special type of the extended multiple harmonic sums. This also enables us to provide a general structural result of the sums $R_n(p,t)$ for all $tge 0$.
A $Gamma$-magic rectangle set $MRS_{Gamma}(a, b; c)$ of order $abc$ is a collection of $c$ arrays $(atimes b)$ whose entries are elements of group $Gamma$, each appearing once, with all row sums in every rectangle equal to a constant $omegain Gamma$
and all column sums in every rectangle equal to a constant $delta in Gamma$. In this paper we prove that for ${a,b} eq{2^{alpha},2k+1}$ where $alpha$ and $k$ are some natural numbers, a $Gamma$-magic rectangle set MRS$_{Gamma}(a, b;c)$ exists if and only if $a$ and $b$ are both even or and $|Gamma|$ is odd or $Gamma$ has more than one involution. Moreover we obtain sufficient and necessary conditions for existence a $Gamma$-magic rectangle MRS$_{Gamma}(a, b)$=MRS$_{Gamma}(a, b;1)$.
We construct a linear system non-local game which can be played perfectly using a limit of finite-dimensional quantum strategies, but which cannot be played perfectly on any finite-dimensional Hilbert space, or even with any tensor-product strategy.
In particular, this shows that the set of (tensor-product) quantum correlations is not closed. The constructed non-local game provides another counterexample to the middle Tsirelson problem, with a shorter proof than our previous paper (though at the loss of the universal embedding theorem). We also show that it is undecidable to determine if a linear system game can be played perfectly with a finite-dimensional strategy, or a limit of finite-dimensional quantum strategies.
We obtain an estimate on the average cardinality of the value set of any family of monic polynomials of Fq[T] of degree d for which s consecutive coefficients a_{d-1},..., a_{d-s} are fixed. Our estimate holds without restrictions on the characterist
ic of Fq and asserts that V(d,s,bfs{a})=mu_d.q+mathcal{O}(1), where V(d,s,bfs{a}) is such an average cardinality, mu_d:=sum_{r=1}^d{(-1)^{r-1}}/{r!} and bfs{a}:=(a_{d-1},.., d_{d-s}). We provide an explicit upper bound for the constant underlying the mathcal{O}--notation in terms of d and s with good behavior. Our approach reduces the question to estimate the number of Fq--rational points with pairwise--distinct coordinates of a certain family of complete intersections defined over Fq. We show that the polynomials defining such complete intersections are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning the singular locus of the varieties under consideration, from which a suitable estimate on the number of Fq--rational points is established.
We show that the set of real numbers of Lagrange value 3 has Hausdorff dimension zero by showing the appropriate generalization for each element of the Teichmueller space of the appropriate subgroup of the classical modular group.