In [1] we presented a model for transactions when goods are given away in the expectation of a later settlement. In settings where people keep track of their social accounts we were able to redefine concepts like account balance, yield curve and the law of diminishing returns. In this paper we establish a general equilibrium theorem, conjectured in [1], by developing sufficient conditions for any instance of the standard model (or Gift Economy Model) to have a unique equilibrium. The convergence to that equilibrium is exponential and for each pair of entities P and Q the total sum of yields from all mutual transactions is equal to zero. [1] W.P. Weijland, Mathematical Foundations for the Economy of Giving, ArXiv Categories: q-fin.GN, Report 1401.4664, 2014.

In this article we are willing to give some first steps to quantum mechanics and a motivation of quantum mechanics and its interpretation for undergraduate students not from physics. After a short historical review in the development we discuss philosophical, physical and mathematical interpretation. We define local realism, locality and hidden variable theory which ends up in the EPR paradox, a place where questions on completeness and reality comes into play. The fundamental result of the last century was maybe Bell's that states that local realism is false if quantum mechanics is true. From this fact we can obtain the so called Bell inequalities. After a didactic example of the fact what these inequalities means we describe the key concept of quantum entanglement motivated here by quantum information theory. Also classical entropy and von Neuman entropy is discussed.

This paper presents an extension and an elaboration of the theory of differential similarity, which was originally proposed in arXiv:1401.2411 [cs.LG]. The goal is to develop an algorithm for clustering and coding that combines a geometric model with a probabilistic model in a principled way. For simplicity, the geometric model in the earlier paper was restricted to the three-dimensional case. The present paper removes this restriction, and considers the full $n$-dimensional case. Although the mathematical model is the same, the strategies for computing solutions in the $n$-dimensional case are different, and one of the main purposes of this paper is to develop and analyze these strategies. Another main purpose is to devise techniques for estimating the parameters of the model from sample data, again in $n$ dimensions. We evaluate the solution strategies and the estimation techniques by applying them to two familiar real-world examples: the classical MNIST dataset and the CIFAR-10 dataset.

Recent innovations in reconstructing large scale, full-precision, neuron-synapse-scale connectomes demand subsequent improvements to graph analysis methods to keep up with the growing complexity and size of the data. One such tool is the recently introduced directed $q$-analysis. We present numerous improvements, theoretical and applied, to this technique: on the theoretical side, we introduce modified definitions for key elements of directed $q$-analysis, which remedy a well-hidden and previously undetected bias. This also leads to new, beneficial perspectives to the associated computational challenges. Most importantly, we present a high-speed, publicly available, low-level implementation that provides speed-ups of several orders of magnitude on C. Elegans. Furthermore, the speed gains grow with the size of the considered graph. This is made possible due to the mathematical and algorithmic improvements as well as a carefully crafted implementation. These speed-ups enable, for the first time, the analysis of full-sized connectomes such as those obtained by recent reconstructive methods. Additionally, the speed-ups allow comparative analysis to corresponding null models, appropriately designed randomly structured artificial graphs that do not correspond to actual brains. This, in turn, allows for assessing the efficacy and usefulness of directed $q$-analysis for studying the brain. We report on the results in this paper.

In this paper we develop further the formalism of fibrations of configuration spaces as a tool for modelling motion of autonomous systems in variable environments. We analyse the situations when the external conditions may change during the motion of the system and analyse two possibilities: (a) when the behaviour of the external conditions is known in advance; and (b) when the future changes of the external conditions are unknown but we can measure the current state and the current velocity of the external conditions, at every moment of time. We prove that in the case (a) the complexity of the motion algorithm is the same as in the case of constant external conditions; this generalises the result of \cite{FGY}. In case (b) we introduce a new concept of a reaction mechanism which allows to take into account unexpected and unpredictable changes in the environment. A reaction mechanism is mathematically an infinitesimal lifting function on a fibre bundle, a nonlinear generalisation of the classical concept of an Ehresmann connection. We illustrate these notions by examples which show that nonlinear infinitesimal lifting function (reaction mechanisms) appear naturally, are inevitable and ubiquitous.

For the eight-dimensional Riemannian manifold comprised by the three-level quantum systems endowed with the Bures metric, we numerically approximate the integrals over the manifold of several functions of the curvature and of its (anti-)self-dual parts. The motivation for pursuing this research is to elaborate upon the findings of Dittmann in his paper, "Yang-Mills equation and Bures metric" (quant-ph/9806018).

Water autoionization plays a critical role in determining pH and properties of various chemical and biological processes occurring in the water mediated environment. The strikingly unsymmetrical potential energy surface of the dissociation process poses a great challenge to the mechanistic study. Here, we demonstrate that reliable sampling of the ionization path is accessible through nanosecond timescale metadynamics simulation enhanced by machine learning of the neural network potentials with ab initio precision, which is proved by quantitatively reproduced water equilibrium constant (p$K_\mathrm{w}$=14.14) and ionization rate constant (1.566$\times10^{-3}$ s$^{-1}$). Statistical analysis unveils the asynchronous character of the concerted triple proton transfer process. Based on conditional ensemble average calculations, we propose a dual-presolvation mechanism, which suggests that a pair of hypercoordinated and undercoordinated waters bridged by one \ce{H2O} cooperatively constitutes the initiation environment for autoionization, and contributes majorly to the local electric field fluctuation to promote water dissociation.

This short note contains an explicit proof of the Jacobi identity for variational Schouten bracket in $Z_2$-graded commutative setup. For the reasoning to be rigorous, it refers to the product bundle geometry of iterated variations (see arXiv:1312.1262 [math-ph]); no ad hoc regularizations occur anywhere in this theory.

In a recent Letter, Avron et. al (math-ph/0105011) introduced a notion of optimal quantum pumps. These are adiabatic quantum pumps which work without dissipation. In particular, they produce neither entropy nor noise. In the present Comment we show that in the absence of magnetic field optimal quantum pumps always have a vanishing transmission coefficient. Such `quantum pumps' do not make use of Quantum Mechanics since all tunneling or interference effects are banned by vanishing of the transmission coefficient. We leave it as an outstanding question whether genuine optimal quantum pumps with nonvanishing transmission coefficient can be constructed by making use of the magnetic field.

A mathematical model describing a steady pH-gradient in the solution of ampholytes in water has been studied with the use of analytical, asymptotic, and numerical methods. We show that at the large values of an electric current a concentration distribution takes the form of a piecewise constant function that is drastically different from a classical Gaussian form. The correspondent pH-gradient takes a stepwise form, instead of being a linear function. A discovered anomalous pH-gradient can crucially affect the understanding of an isoelectric focusing process.