What can astrocytes compute?

I demonstrate theoretically that calcium waves in astrocytes can compute anything neurons can. A foundational result in neural computation was proving the firing rate model of neurons defines a universal function approximator. In this work I show a similar proof extends to a model of calcium waves in astrocytes, which I confirm in a series of computer simulations. I argue the major limit in astrocyte computation is not their ability to find approximate solutions, but their computational complexity. I suggest some initial experiments that might be used to confirm these predictions.

Evaluation of Green Alternatives for Blockchain Proof-of-Work (PoW) Approach

Following the footprints of Bitcoins, many other cryptocurrencies were developed mostly adopting the same or similar Proof-of-Work (PoW) approach. Since completing the PoW puzzle requires extremely high computing power, consuming a vast amount of electricity, PoW has been strongly criticised for its antithetic stand against the notion of green computing. Use of application-specific hardware, particularly application-specific integrated circuits (ASICs) has further fuelled the debate, as these devices are of no use once they become “legacy” and hence obsolete to compete in the mining race, thus contributing to electronics waste. Therefore, this paper surveys the currently available alternative approaches to PoW and evaluates their applicability - especially their appropriateness in terms of greenness.

An Integralgeometric Approach to Dorronsoro Estimates

A theorem of Dorronsoro from 1985 quantifies the fact that a Lipschitz function $f \colon \mathbb{R}^{n} \to \mathbb{R}$ can be approximated by affine functions almost everywhere, and at sufficiently small scales. This paper contains a new, purely geometric, proof of Dorronsoro’s theorem. In brief, it reduces the problem in $\mathbb{R}^{n}$ to a problem in $\mathbb{R}^{n - 1}$ via integralgeometric considerations. For the case $n = 1$, a short geometric proof already exists in the literature. A similar proof technique applies to parabolic Lipschitz functions $f \colon \mathbb{R}^{n - 1} \times \mathbb{R} \to \mathbb{R}$. A natural Dorronsoro estimate in this class is known, due to Hofmann. The method presented here allows one to reduce the parabolic problem to the Euclidean one and to obtain an elementary proof also in this setting. As a corollary, I deduce an analogue of Rademacher’s theorem for parabolic Lipschitz functions.

Unit Distances in Three Dimensions

We show that the number of unit distances determined bynpoints in ℝ3isO(n3/2), slightly improving the bound of Clarkson, Edelsbrunner, Guibas, Sharir and Welzl [5], established in 1990. The new proof uses the recently introduced polynomial partitioning technique of Guth and Katz [12]. While this paper was still in a draft stage, a similar proof of our main result was posted to the arXiv by Joshua Zahl [28].

ON THE SUPERMODULARITY OF HOMOGENEOUS OLIGOPOLY GAMES

The main purpose is to prove the supermodularity (convexity) property of a cooperative game arising from an economical situation. The underlying oligopoly situation is based on a linear inverse demand function as well as linear cost functions for the participating firms. The characteristic function of the so-called oligopoly game is determined by maximizing, for any cartel of firms, the net profit function over the feasible production levels of the firms in the cartel, taking into account their individual capacities of production and production technologies. The (rather effective) proof of the supermodularity of the characteristic function of the oligopoly game relies on the use of maximizers for the relevant maximization problems. A similar proof technique will be reviewed for a related cooperative oligopoly game arising from a slightly modified oligopoly situation where the production technology of the cartel is determined by the most efficient member firm.

An Improved Bound for k-Sets in Four Dimensions

We show that the number of halving sets of a set of n points in ℝ4 is O(n4−1/18), improving the previous bound of [10] with a simpler (albeit similar) proof.

A duality between proof systems for cyclic term graphs

This paper presents a proof-theoretic observation about two kinds of proof systems for bisimilarity between cyclic term graphs.First we consider proof systems for demonstrating that μ term specifications of cyclic term graphs have the same tree unwinding. We establish a close connection between adaptations for μ terms over a general first-order signature of the coinductive axiomatisation of recursive type equivalence by Brandt and Henglein (Brandt and Henglein 1998) and of a proof system by Ariola and Klop (Ariola and Klop 1995) for consistency checking. We show that there exists a simple duality by mirroring between derivations in the former system and formalised consistency checks, which are called ‘consistency unfoldings', in the latter. This result sheds additional light on the axiomatisation of Brandt and Henglein: it provides an alternative soundness proof for the adaptation considered here.We then outline an analogous duality result that holds for a pair of similar proof systems for proving that equational specifications of cyclic term graphs are bisimilar.

Existence of Equilibrium for Segmented Markets Models with Interest Rate Monetary Policies

Several studies have recently adopted the segmented markets model as a framework for monetary analysis. The characteristic assumption is that some households never participate in financial markets. This paper proves the existence of an equilibrium for segmented markets models where monetary policy is defined in terms of the short-term nominal interest rate. The model allows us to consider the important cases where monetary policy affects output, and responds to any sources of uncertainty, including output itself. The assumptions required for existence constrain the maximum value and the variability of the nominal interest rate. The period utility function is logarithmic. The proof is constructive, and shows how the model can be solved numerically. A similar proof can be used in the case that monetary policy is defined in terms of the bond supply.

If there is an exactly λ-free abelian group then there is an exactly λ-separable one in λ

We give a solution stated in the title to problem 3 of part 1 of the problems listed in the book of Eklof and Mekler [2], p. 453. There, in pp. 241-242, this is discussed and proved in some cases. The existence of strongly λ-free ones was proved earlier by the criteria in [5] and [3]. We can apply a similar proof to a large class of other varieties in particular to the variety of (non-commutative) groups.

Near-equational and equational systems of logic for partial functions. I

Equational logic for total functions is a remarkable fragment of first-order logic. Rich enough to lend itself to many uses, it is also quite austere. The only predicate symbol is one for a notion of equality, and there are no logical connectives. Proof theory for equational logic therefore is different from proof theory for other logics and, in some respects, more transparent. The question therefore arises to what extent a logic with a similar proof theory can be constructed when expressive power is increased.The increase mainly studied here allows one both to consider arbitrary partial functions and to express the condition that a function be total. A further increase taken into account is equivalent to a change to universal Horn sentences for partial and for total functions.Two ways of increasing expressive power will be considered. In both cases, the notion of equality is modified and nonlogical function symbols are interpreted as ranging over partial functions, instead of ranging only over total functions. In one case, the only further change is the addition of symbols that denote logical functions, such as the binary projection function Ae that maps each pair ‹a0, a1› of elements of a set A into the element a0. An addition of this kind results in a language, and also in a system of logic based on this language, which we call equational In the other case, instead of adding a symbol for Ae, one admits those special universal Horn sentences in which the conditions expressed by the antecedent are, in a sense, pure conditions of existence. Languages and systems of logic that result from a change of this kind will be called near-equational. According to whether the number of existence conditions that one may express in antecedents is finite or arbitrary, the resulting language and logic shall be finite or infinitary, respectively. Each of our finite near-equational languages turns out to be equivalent to one of our equational languages, and vice versa.