Dyson’s partition ranks and their multiplicative extensions

We study the Dyson rank function N(r, 3; n), the number of partitions of n with rank \(\equiv r \pmod 3\). We investigate the convexity of these functions. We extend N(r, 3; n) multiplicatively to the set of partitions, and we determine the maximum value when taken over all partitions of size n.     

Tensor ranks and symmetric tensor ranks are the same for points with low symmetric tensor rank

Fix integers n ≥ 1, d ≥ 2. Let V be an (n + 1)-dimensional vector space over a field with characteristic zero. Fix a symmetric tensor \({T\in S^d(V)\subset V^{\otimes d}}\). Here we prove that the tensor rank of T is equal to its symmetric tensor rank if the latter is at most (d + 1)/2.     

A family of non-isomorphism results

We calculate the local groups of germs associated with the higher dimensional R. Thompson groups nV. For a given \({n\in N\cup\left\{\omega\right\}}\) , these groups of germs are free abelian groups of rank r, for r ≤ n (there are some groups of germs associated with nV with rank precisely k for each index 1 ≤ k ≤ n). By Rubin’s theorem, any conjectured isomorphism between higher dimensional R. Thompson groups induces an isomorphism between associated groups of germs. Thus, if m ≠ n the groups mV and nV cannot be isomorphic. This answers a question of Brin.     

Maximum scattered linear sets and MRD-codes

The rank of a scattered \({\mathbb F}_q\)-linear set of \({{\mathrm{{PG}}}}(r-1,q^n)\), rn even, is at most rn / 2 as it was proved by Blokhuis and Lavrauw. Existence results and explicit constructions were given for infinitely many values of r, n, q (rn even) for scattered \({\mathbb F}_q\)-linear sets of rank rn / 2. In this paper, we prove that the bound rn / 2 is sharp also in the remaining open cases. Recently Sheekey proved that scattered \({\mathbb F}_q\)-linear sets of \({{\mathrm{{PG}}}}(1,q^n)\) of maximum rank n yield \({\mathbb F}_q\)-linear MRD-codes with dimension 2n and minimum distance \(n-1\). We generalize this result and show that scattered \({\mathbb F}_q\)-linear sets of \({{\mathrm{{PG}}}}(r-1,q^n)\) of maximum rank rn / 2 yield \({\mathbb F}_q\)-linear MRD-codes with dimension rn and minimum distance \(n-1\).     

Compressed Sensing and Matrix Completion with Constant Proportion of Corruptions

In this paper, we improve existing results in the field of compressed sensing and matrix completion when sampled data may be grossly corrupted. We introduce three new theorems. (1) In compressed sensing, we show that if the m×n sensing matrix has independent Gaussian entries, then one can recover a sparse signal x exactly by tractable ? ₁ minimization even if a positive fraction of the measurements are arbitrarily corrupted, provided the number of nonzero entries in x is O(m/(log(n/m)+1)). (2) In the very general sensing model introduced in Candès and Plan (IEEE Trans. Inf. Theory 57(11):7235–7254, 2011) and assuming a positive fraction of corrupted measurements, exact recovery still holds if the signal now has O(m/(log² n)) nonzero entries. (3) Finally, we prove that one can recover an n×n low-rank matrix from m corrupted sampled entries by tractable optimization provided the rank is on the order of O(m/(nlog² n)); again, this holds when there is a positive fraction of corrupted samples.     

A note on K<Emphasis Type="Italic">ä</Emphasis>hler manifolds with almost nonnegative bisectional curvature

In this note, we prove the following result. There is a positive constant ε(n, Λ) such that if M ⁿ is a simply connected compact Kähler manifold with sectional curvature bounded from above by Λ, diameter bounded from above by 1, and with holomorphic bisectional curvature H ≥ ?ε(n, Λ), then M ⁿ is diffeomorphic to the product M ₁ × ? × M _k , where each M _i is either a complex projective space or an irreducible Kähler–Hermitian symmetric space of rank ≥ 2. This resolves a conjecture of Fang under the additional upper bound restrictions on sectional curvature and diameter.     

A note on Jeśmanowicz’ conjecture

Je?manowicz [9] conjectured that, for positive integers m and n with m > n, gcd(m,n) = 1 and \({m\not\equiv n\pmod{2}}\), the exponential Diophantine equation \({(m^2-n^2)^x+(2mn)^y=(m^2+n^2)^z}\) has only the positive integer solution (x, y, z) = (2, 2, 2). We prove the conjecture for \({2 \| mn}\) and m + n has a prime factor p with \({p\not\equiv1\pmod{16}}\).     

The asymptotic formula in Waring’s problem for one square and seventeen fifth powers

Let R _k,s(n) denote the number of solutions of the equation \({n= x^2 + y_1^k + y_2^k + \cdots + y_s^k}\) in natural numbers x, y ₁, . . . , y _s . By a straightforward application of the circle method, an asymptotic formula for R _k,s(n) is obtained when k ≥ 3 and s ≥ 2^k–1 + 2. When k ≥ 6, work of Heath-Brown and Boklan is applied to establish the asymptotic formula under the milder constraint s ≥ 7 · 2^k–4 + 3. Although the principal conclusions provided by Heath-Brown and Boklan are not available for smaller values of k, some of the underlying ideas are still applicable for k = 5, and the main objective of this article is to establish an asymptotic formula for R _5,17(n) by this strategy.     

Abelian covers of graphs and maps between outer automorphism groups of free groups

We explore the existence of homomorphisms between outer automorphism groups of free groups Out(F _n ) → Out(F _m ). We prove that if n > 8 is even and n ≠ m ≤ 2n, or n is odd and n ≠ m ≤ 2n ? 2, then all such homomorphisms have finite image; in fact they factor through det : \({{\rm Out}(F_n) \to \mathbb{Z}/2}\) . In contrast, if m = r ⁿ (n ? 1) + 1 with r coprime to (n ? 1), then there exists an embedding \({{\rm Out}(F_n) \hookrightarrow {\rm Out}(F_m)}\) . In order to prove this last statement, we determine when the action of Out(F _n ) by homotopy equivalences on a graph of genus n can be lifted to an action on a normal covering with abelian Galois group.     

On the minimum size of binary codes with length 2<Emphasis Type="Italic">R</Emphasis> + 4 and covering radius <Emphasis Type="Italic">R</Emphasis>

The minimum size of a binary code with length n and covering radius R is denoted by K(n, R). For arbitrary R, the value of K(n, R) is known when n ≤  2R +  3, and the corresponding optimal codes have been classified up to equivalence. By combining combinatorial and computational methods, several results for the first open case, K(2R +  4, R), are here obtained, including a proof that K(10, 3) =  12 with 11481 inequivalent optimal codes and a proof that if K(2R +  4, R) <  12 for some R then this inequality cannot be established by the existence of a corresponding self-complementary code.     

Prime and composite integers close to powers of a number

We prove that, for any real numbers ξ ≠ 0 and ν, the sequence of integer parts [ξ2 ⁿ  + ν], n = 0, 1, 2, . . . , contains infinitely many composite numbers. Moreover, if the number ξ is irrational, then the above sequence contains infinitely many elements divisible by 2 or 3. The same holds for the sequence [ξ( ? 2) ⁿ  + ν _n ], n = 0, 1, 2, . . . , where ν ₀, ν ₁, ν ₂, . . . all lie in a half open real interval of length 1/3. For this, we show that if a sequence of integers x ₁, x ₂, x ₃, . . . satisfies the recurrence relation x _n+d  = cx _n  + F(x _n+1, . . . , x _n+d-1) for each n  ≥  1, where c ≠ 0 is an integer, \({F(z_1,\dots,z_{d-1}) \in \mathbb {Z}[z_1,\dots,z_{d-1}],}\) and lim _{n→ ∞}|x _n | = ∞, then the number |x _n | is composite for infinitely many positive integers n. The proofs involve techniques from number theory, linear algebra, combinatorics on words and some kind of symbolic computation modulo 3.     

A note on a mixed routing and scheduling problem on a grid graph

We consider a particular case of the Fleet Quickest Routing Problem (FQRP) on a grid graph of m × n nodes that are placed in m levels and n columns. Starting nodes are placed at the first (bottom) level, and nodes of arrival are placed at the mth level. A feasible solution of FQRP consists in n Manhattan paths, one for each vehicle, such that capacity constraints are respected. We establish m*, i.e. the number of levels that ensures the existence of a solution to FQRP in any possible permutation of n destinations. In particular, m* is the minimum number of levels sufficient to solve any instance of FQRP involving n vehicles, when they move in the ways that the literature has until now assumed. Existing algorithms give solutions that require, for some values of n, more levels than m*. For this reason, we provide algorithm CaR, which gives a solution in a graph m* × n, as a minor contribution.     

The Frölicher spectral sequence can be arbitrarily non-degenerate

The Frölicher spectral sequence of a compact complex manifold X measures the difference between Dolbeault cohomology and de Rham cohomology. If X is Kähler then the spectral sequence collapses at the E ₁term and no example with d _n  ≠  0 for n > 3 has been described in the literature.We construct for n ≥  2 nilmanifolds with left-invariant complex structure X _n such that the n-th differential d _n does not vanish. This answers a question mentioned in the book of Griffiths and Harris.     

Most small $${\varvec{}}$$-grous have an automorhism of order 2

Let f(p, n) be the number of pairwise nonisomorphic p-groups of order \(p^n\), and let g(p, n) be the number of groups of order \(p^n\) whose automorphism group is a p-group. We prove that the limit, as p grows to infinity, of the ratio g(p, n) / f(p, n) equals 1/3 for \(n=6,7\).     

On a Special Case of the Herbert Stahl Theorem

The BMV conjecture states that for n ×  n Hermitian matrices A and B the function f_A,B(t) = trace e^tA+B is exponentially convex. Recently the BMV conjecture was proved by Herbert Stahl. The proof of Herbert Stahl is based on ingenious considerations related to Riemann surfaces of algebraic functions. In the present paper we give a purely “matrix” proof of the BMV conjecture for the special case rank A = 1. This proof is based on the Lie product formula for the exponential of the sum of two matrices and does not require complex analysis.     

Multiset combinatorial batch codes

Batch codes, first introduced by Ishai, Kushilevitz, Ostrovsky, and Sahai, mimic a distributed storage of a set of n data items on m servers, in such a way that any batch of k data items can be retrieved by reading at most some t symbols from each server. Combinatorial batch codes, are replication-based batch codes in which each server stores a subset of the data items. In this paper, we propose a generalization of combinatorial batch codes, called multiset combinatorial batch codes (MCBC), in which n data items are stored in m servers, such that any multiset request of k items, where any item is requested at most r times, can be retrieved by reading at most t items from each server. The setup of this new family of codes is motivated by recent work on codes which enable high availability and parallel reads in distributed storage systems. The main problem under this paradigm is to minimize the number of items stored in the servers, given the values of n, m, k, r, t, which is denoted by N(n, k, m, t; r). We first give a necessary and sufficient condition for the existence of MCBCs. Then, we present several bounds on N(n, k, m, t; r) and constructions of MCBCs. In particular, we determine the value of N(n, k, m, 1; r) for any \(n\ge \left\lfloor \frac{k-1}{r}\right\rfloor {m\atopwithdelims ()k-1}-(m-k+1)A(m,4,k-2)\), where \(A(m,4,k-2)\) is the maximum size of a binary constant weight code of length m, distance four and weight \(k-2\). We also determine the exact value of N(n, k, m, 1; r) when \(r\in \{k,k-1\}\) or \(k=m\).     

Complexities of normal bases constructed from Gauss periods

Let q be a power of a prime p, and let \(r=nk+1\) be a prime such that \(r\not \mid q\), where n and k are positive integers. Under a simple condition on q, r and k, a Gauss period of type (n, k) is a normal element of \({\mathbb {F}}_{q}^{n}\) over \({\mathbb {F}}_q\); the complexity of the resulting normal basis of \({\mathbb {F}}_{q}^{n}\) over \({\mathbb {F}}_q\) is denoted by C(n, k; p). Recent works determined C(n, k; p) for \(k\le 7\) and all qualified n and q. In this paper, we show that for any given \(k>0\), C(n, k; p) is given by an explicit formula except for finitely many primes \(r=nk+1\) and the exceptional primes are easily determined. Moreover, we describe an algorithm that allows one to compute C(n, k; p) for the exceptional primes \(r=nk+1\). Our numerical results cover C(n, k; p) for \(k\le 20\) and all qualified n and q.     

An arithmetical equation with respect to regular convolutions

It is well known that Euler’s totient function \(\phi \) satisfies the arithmetical equation \( \phi (mn)\phi ((m, n))=\phi (m)\phi (n)(m, n) \) for all positive integers m and n, where (m, n) denotes the greatest common divisor of m and n. In this paper we consider this equation in a more general setting by characterizing the arithmetical functions f with \(f(1)\ne 0\) which satisfy the arithmetical equation \( f(mn)f((m,n)) = f(m)f(n)g((m, n)) \) for all positive integers m, n with \(m,n \in A(mn)\), where A is a regular convolution and g is an A-multiplicative function. Euler’s totient function \(\phi _A\) with respect to A is an example satisfying this equation.     

Possible isolation number of a matrix over nonnegative integers

Let ?₊ be the semiring of all nonnegative integers and A an m × n matrix over ?₊. The rank of A is the smallest k such that A can be factored as an m × k matrix times a k×n matrix. The isolation number of A is the maximum number of nonzero entries in A such that no two are in any row or any column, and no two are in a 2 × 2 submatrix of all nonzero entries. We have that the isolation number of A is a lower bound of the rank of A. For A with isolation number k, we investigate the possible values of the rank of A and the Boolean rank of the support of A. So we obtain that the isolation number and the Boolean rank of the support of a given matrix are the same if and only if the isolation number is 1 or 2 only. We also determine a special type of m×n matrices whose isolation number is m. That is, those matrices are permutationally equivalent to a matrix A whose support contains a submatrix of a sum of the identity matrix and a tournament matrix.     

A modified large-scale structure-preserving doubling algorithm for a large-scale Riccati equation from transport theory

A large scale nonsymmetric algebraic Riccati equation XCX ? XE ? AX + B = 0 arising in transport theory is considered, where the n × n coefficient matrices B,C are symmetric and low-ranked and A, E are rank one updates of nonsingular diagonal matrices. By introducing a balancing strategy and setting appropriate initial matrices carefully, we can simplify the large-scale structure-preserving doubling algorithm (SDA_ls) for this special equation. We give modified large-scale structure-preserving doubling algorithm, which can reduce the flop count of original SDA_ls by half. Numerical experiments illustrate the effectiveness of our method.