A Characterization of Cesàro Convergence

We show that a real bounded sequence (Formula presented.) is Cesàro convergent to (Formula presented.) if and only if the sequence of averages with indices in (Formula presented.) converges to (Formula presented.) for all (Formula presented.). If, in addition, the sequence (Formula presented.) is nonnegative, then it is Cesàro convergent to 0 if and only if the same condition holds for some (Formula presented.)

A characterization of Sophie Germain primes

Let n ≥ 5 be an odd integer. It is shown that {1σ(1),...,nσ(n)} is a complete residue system modulo n for some permutation σ of {1,...,n} if and only if 1 2(n - 1) is a Sophie Germain prime. Partial results are obtained also for the case n even

A characterization of (I,J)-regular matrices

Let I, J be two ideals on N which contain the family Fin of finite sets. We provide necessary and sufficient conditions on the entries of an infinite real matrix A = (a(n, k)) which maps I-convergent bounded sequences into J-convergent bounded sequences and preserves the corresponding ideal limits. The well-known characterization of regular matrices due to Silverman-Toeplitz corresponds to the case I = J = Fin. Lastly, we provide some applications to permutation and diagonal matrices, which extend several known results in the literature

Density-like and Generalized Density Ideals

We show that there exist uncountably many (tall and nontall) pairwise nonisomorphic density-like ideals on which are not generalized density ideals. In addition, they are nonpathological. This answers a question posed by Borodulin-Nadzieja et al. in [this Journal, vol. 80 (2015), pp. 1268-1289]. Lastly, we provide sufficient conditions for a density-like ideal to be necessarily a generalized density ideal

Sums of Multivariate Polynomials in Finite Subgroups

Let R be a commutative ring, f ∈ R[X1,⋯,Xk] a multivariate polynomial, and G a finite subgroup of the group of units of R satisfying a certain constraint, which always holds if R is a field. Then, we evaluate Σ f(x1,⋯,xk), where the summation is taken over all pairwise distinct x1,⋯,xk G. In particular, let ps be a power of an odd prime, n a positive integer coprime with p - 1, and a1,⋯,ak integers such that φ(ps) divides a1 + ⋯ + ak and p - 1 does not divide Σi∈Iai for all non-empty proper subsets I ⊆ {1,⋯,k}; then ∑x1a1⋯xkak ≡ φ(ps)/gcd(n,φ(ps))(-1)k-1(k - 1)!modps, where the summation is taken over all pairwise distinct nth residues x1,⋯,xk modulo ps coprime with p

On the number of distinct prime factors of a sum of super-powers

Given k,ℓ∈N+, let xi,j be, for 1≤i≤k and 0≤j≤ℓ some fixed integers, and define, for every n∈N+, sn:=∑i=1 k∏j=0 ℓxi,j n. We prove that the following are equivalent: (a) There are a real θ>1 and infinitely many indices n for which the number of distinct prime factors of sn is greater than the super-logarithm of n to base θ.(b) There do not exist non-zero integers a0,b0,…,aℓ,bℓ such that s2n=∏i=0 ℓai (2n) and s2n−1=∏i=0 ℓbi (2n−1) for all n.We will give two different proofs of this result, one based on a theorem of Evertse (yielding, for a fixed finite set of primes S, an effective bound on the number of non-degenerate solutions of an S-unit equation in k variables over the rationals) and the other using only elementary methods. As a corollary, we find that, for fixed c1,x1,…,ck,xk∈N+, the number of distinct prime factors of c1x1 n+⋯+ckxk n is bounded, as n ranges over N+, if and only if x1=⋯=xk

Convergent subseries of divergent series

Let X be the set of positive real sequences x= (xn) such that the series ∑ nxn is divergent. For each x∈ X, let Ix be the collection of all A⊆ N such that the subseries ∑ n∈Axn is convergent. Moreover, let A be the set of sequences x∈ X such that lim nxn= 0 and Ix≠ Iy for all sequences y= (yn) ∈ X with lim inf nyn+1/ yn> 0. We show that A is comeager and that contains uncountably many sequences x which generate pairwise nonisomorphic ideals Ix. This answers, in particular, an open question recently posed by M. Filipczak and G. Horbaczewska

Combining Local and Global Direct Derivative-free Optimization for Reinforcement Learning

We consider the problem of optimization in policy space for reinforcement learning. While a plethora of methods have been applied to this problem, only a narrow category of them proved feasible in robotics. We consider the peculiar characteristics of reinforcement learning in robotics, and devise a combination of two algorithms from the literature of derivative-free optimization. The proposed combination is well suited for robotics, as it involves both off-line learning in simulation and on-line learning in the real environment. We demonstrate our approach on a real-world task, where an Autonomous Underwater Vehicle has to survey a target area under potentially unknown environment conditions. We start from a given controller, which can perform the task under foreseeable conditions, and make it adaptive to the actual environment

Planning in answer set programming while learning action costs for mobile robots

For mobile robots to perform complex missions, it may be necessary for them to plan with incomplete information and reason about the indirect effects of their actions. Answer Set Programming (ASP) provides an elegant way of formalizing domains which involve indirect effects of an action and recursively defined fluents. In this paper, we present an approach that uses ASP for robotic task planning, and demonstrate how ASP can be used to generate plans that acquire missing information necessary to achieve the goal. Action costs are also incorporated with planning to produce optimal plans, and we show how these costs can be estimated from experience making planning adaptive. We evaluate our approach using a realistic simulation of an indoor environment where a robot learns to complete its objective in the shortest time

Co-transport-induced instability of membrane voltage in tip-growing cells

A salient feature of stationary patterns in tip-growing cells is the key role played by the symports and antiports, membrane proteins that translocate two ionic species at the same time. It is shown that these co-transporters destabilize generically the membrane voltage if the two translocated ions diffuse differently and carry a charge of opposite (same) sign for symports (antiports). Orders of magnitude obtained for the time and lengthscale are in agreement with experiments. A weakly nonlinear analysis characterizes the bifurcation