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In this paper we introduce stochastic parameters into the network game model with production and knowledge externalities. This model was proposed by V. Matveenko and A. Korolev as a generalization of the two-period Romer model. Agents differ in their productivities which have deterministic and stochastic (Wiener) components. We study the dynamics of a single agent and the dynamics of a dyad where two agents are aggregated. We derive explicit expressions for the dynamics of a single agent and dyad dynamics in the form of Brownian random processes, and qualitatively analyze the solutions of stochastic equations and systems of stochastic equations.
Topic modeling is a popular technique for clustering large collections of text documents. A variety of different types of regularization is implemented in topic modeling. In this paper, we propose a novel approach for analyzing the influence of different regularization types on results of topic modeling. Based on Renyi entropy, this approach is inspired by the concepts from statistical physics, where an inferred topical structure of a collection can be considered an information statistical system residing in a non-equilibrium state. By testing our approach on four models—Probabilistic Latent Semantic Analysis (pLSA), Additive Regularization of Topic Models (BigARTM), Latent Dirichlet Allocation (LDA) with Gibbs sampling, LDA with variational inference (VLDA)—we, first of all, show that the minimum of Renyi entropy coincides with the “true” number of topics, as determined in two labelled collections. Simultaneously, we find that Hierarchical Dirichlet Process (HDP) model as a well-known approach for topic number optimization fails to detect such optimum. Next, we demonstrate that large values of the regularization coefficient in BigARTM significantly shift the minimum of entropy from the topic number optimum, which effect is not observed for hyper-parameters in LDA with Gibbs sampling. We conclude that regularization may introduce unpredictable distortions into topic models that need further research.
The problem of approximation by entire functions of exponential type defined on a countable set E of continua Gn, E = ⋃n∈ZGn⋃n∈ZGn is considered in this paper. It is assumed that all Gn are pairwise disjoint and are situated near the real axis. It is also assumed that all Gn are commensurable in a sense and have uniformly smooth boundaries. A function f is defined independently on each Gn and is bounded on E and f (r) has a module of continuity ω which satisfies the condition (1). An entire function Fσ of exponential type ≤σ is then constructed so that the following estimate of approximation of the function f by functions Fσ is valid
In this paper, we present a systematic overview of different endogenous optimization-based characteristic functions and discuss their properties. Furthermore, we define and analyze in detail a new, η-characteristic function. This characteristic function has a substantial advantage over other characteristic functions in that it can be obtained with a minimal computational effort and has a reasonable economic interpretation. In particular, the new characteristic function can be seen as a reduced version of the classical Neumann--Morgenstern characteristic function, where the players both from the coalition and from the complementary coalition use their previously computed strategies instead of solving respective optimization problems. Our finding are illustrated by a pollution control game with n non-identical players. For the considered game, we compute all characteristic functions and compare their properties. Quite surprisingly, it turns out that both the characteristic functions and the resulting cooperative solutions satisfy some symmetry relations.
Motivation
Imaging mass spectrometry (imaging MS) is a prominent technique for capturing distributions of molecules in tissue sections. Various computational methods for imaging MS rely on quantifying spatial correlations between ion images, referred to as co-localization. However, no comprehensive evaluation of co-localization measures has ever been performed; this leads to arbitrary choices and hinders method development.
Results
We present ColocML, a machine learning approach addressing this gap. With the help of 42 imaging MS experts from nine laboratories, we created a gold standard of 2210 pairs of ion images ranked by their co-localization. We evaluated existing co-localization measures and developed novel measures using term frequency–inverse document frequency and deep neural networks. The semi-supervised deep learning Pi model and the cosine score applied after median thresholding performed the best (Spearman 0.797 and 0.794 with expert rankings, respectively). We illustrate these measures by inferring co-localization properties of 10 273 molecules from 3685 public METASPACE datasets.
Functional classes on a curve in a plane (a partial case of a spatial curve) can be described by the approximation speed by functions that are harmonic in three-dimensional neighbourhoods of the curve. No constructive description of functional classes on rather general surfaces in R3 and R4 has been presented in literature so far. The main result of the paper is Theorem 1.
Control systems are exposed to unintentional errors, deliberate intrusions, false data injection attacks, and various other disruptions. In this paper we propose, justify, and illustrate a rule of thumb for detecting, or confirming the absence of, such disruptions. To facilitate the use of the rule, we rigorously discuss underlying results that delineate the boundaries of the rule’s applicability. We also discuss ways to further widen the applicability of the proposed intrusion-detection methodology.
Forecasting and analyses of the dynamics of financial and economic processes such as deviations of macroeconomic aggregates (GDP, unemployment, and inflation) from their long-term trends, asset markets volatility, etc., are challenging because of the complexity of these processes. Important related research questions include, first, how to determine the qualitative properties of the dynamics of these processes, namely, whether the process is stable, unstable, chaotic (deterministic), or stochastic; and second, how best to estimate its quantitative indicators including dimension, entropy, and correlation characteristics.
These questions can be studied both empirically and theoretically. In the empirical approach, researchers consider real data expressed in terms of time series, identify the patterns of their dynamics, and then forecast the short- and long-term behavior of the process. The second approach is based on postulating the laws of dynamics for the process, deriving mathematical dynamical models based on these laws, and conducting subsequent analytical investigation of the dynamics generated by the models.
To implement these approaches, either numerical or analytical methods can be used. While numerical methods make it possible to study dynamical models, the possibility of obtaining reliable results using them is significantly limited due to the necessity of performing calculations only over finite time intervals, rounding-off errors in numerical methods, and the unbounded space of initial data sets. Analytical methods allow researchers to overcome these limitations and to identify the exact qualitative and quantitative characteristics of the dynamics of the process. However, effective analytical applications are often limited to low-dimensional models (in the literature, two-dimensional dynamical systems are most often studied).
In this paper, we develop analytical methods for the study of deterministic dynamical systems based on the Lyapunov stability theory and on chaos theory. These methods make it possible not only to obtain analytical stability criteria and to estimate limiting behavior (to localize self-excited and hidden attractors and identify multistability), but also to overcome difficulties related to implementing reliable numerical analysis of quantitative indicators such as Lyapunov exponents and the Lyapunov dimension. We demonstrate the effectiveness of the proposed methods using the mid-size firm model suggested by Shapovalov.
Studies of electronic transitions in the photoconverters with In0.4Ga0.6As quantum well-dots (QWD) layers have been carried out. It is shown that the quantum yield and electroluminescence spectral peaks are well described by e1-lh1 and e1-hh1 optical transitions in the quantum well with the same average composition and thickness. The energy of the optical transitions shifts toward longer wavelengths with an increase in the number of QWD layers. The calculated shifts of electron and hole levels due to the redistribution of elastic strain between In0.4Ga0.6As QWDs and GaAs spacer layers demonstrated a very good agreement with the experimental data.
We study approximation of bounded measurable functions on the segment [0, 1] by Kantorovich
type operators
B_n=∑_{j=0}^nC^j_nx^j(1−x)^{n−j}F_{n,j}
where F_{n, j} are functionals generated by various probability measures with sufficiently small supports. The error of approximation is estimated in terms of the second modulus of continuity. The estimate is sharp.
In practice, the critical step in building machine learning models of big data (BD) is costly in terms of time and the computing resources procedure of parameter tuning with a grid search. Due to the size, BD are comparable to mesoscopic physical systems. Hence, methods of statistical physics could be applied to BD. The paper shows that topic modeling demonstrates self-similar behavior under the condition of a varying number of clusters. Such behavior allows using a renormalization technique. The combination of a renormalization procedure with the Rényi entropy approach allows for fast searching of the optimal number of clusters. In this paper, the renormalization procedure is developed for the Latent Dirichlet Allocation (LDA) model with a variational Expectation-Maximization algorithm. The experiments were conducted on two document collections with a known number of clusters in two languages. The paper presents results for three versions of the renormalization procedure: (1) a renormalization with the random merging of clusters, (2) a renormalization based on minimal values of Kullback–Leibler divergence and (3) a renormalization with merging clusters with minimal values of Rényi entropy. The paper shows that the renormalization procedure allows finding the optimal number of topics 26 times faster than grid search without significant loss of quality.
We consider the problem of finding a valid covariance matrix in the foreign exchange market given an initial non-PSD estimate of such a matrix. The common no-arbitrage assumption imposes additional linear constraints on such matrices, whereby inevitably making them singular. As a result, even the most advanced numerical techniques will predictably balk at a seemingly standard optimization task. The reason is that the problem is ill-posed while its PSD-solution is not strictly feasible. In order to deal with this issue we describe a low-dimensional face of the PSD cone that contains the feasible set. After projecting the initial problem onto this face, we come out with a reduced problem, which is both well-posed and of a smaller scale. We show that after solving the reduced problem the solution to the initial problem can be uniquely recovered in one step. We run numerous numerical experiments to compare performance of different algorithms in solving the reduced problem and to demonstrate the advantages of dealing with the reduced problem as opposed to the original one. The smaller scale of the reduced problem implies that its solution can be effectively found by application of virtually any numerical method.
In statistical classification and machine learning, as well as in social and other sciences, a number of measures of association have been proposed for assessing and comparing individual classifiers, raters, as well as their groups. In this paper, we introduce, justify, and explore several new measures of association, which we call CO-, ANTI-, and COANTI-correlation coefficients, that we demonstrate to be powerful tools for classifying confusion matrices. We illustrate the performance of these new coefficients using a number of examples, from which we also conclude that the coefficients are new objects in the sense that they differ from those already in the literature.
In this paper, we consider the following problem - what affects the Nash equilibrium amount of investment in knowledge when one of the complete graph enters another full one. The solution of this problem will allow us to understand exactly how game agents will behave when deciding whether to enter the other net, what conditions and externalities affect it and how the level of future equilibrium amount of investments in knowledge can be predicted.
Difference and differential Stackelberg games of opinion control on marketing networks are considered. The principal allocates financial resources to the firms for marketing purposes. It is supposed that the structure of a target audience described by a weighted directed graph is already determined in the stage of network analysis, and marketing control actions are applied only to the members of strong subgroups (opinion leaders). Conditions of homeostasis (phase constraints) which reflect the requirements of sustainable management are introduced additionally. The Stackelberg equilibria are found analytically. It is shown that the interests of the principal and the firms are completely compatible.
Clustering large and heterogeneous data of user-profiles from social media is problematic as the problem of finding the optimal number of clusters becomes more critical than for clustering smaller and homo- geneous data. We propose a new approach based on the deformed R ́enyi entropy for determining the optimal number of clusters in hierarchical clustering of user-profile data. Our results show that this approach allows us to estimate R ́enyi entropy for each level of a hierarchical model and find the entropy minimum (information maximum). Our approach also shows that solutions with the lowest and the highest number of clusters correspond to the entropy maxima (minima of information).
We discuss the effect of self-heating on performance of injection microdisk lasers operating in continuous-wave (CW) regime at room and elevated temperature. A model is developed that allows one to obtain analytical expressions for the peak optical power limited by the thermal rollover effect, the corresponding injection current and excess temperature of the device. The model predicts, there exists the maximum temperature of microlaser operation in CW regime and the minimum mircrodisk diameter, at which CW lasing is possible. The model allows one to determine the dependence of the device characteristics on its diameter and the inherent parameters, such as thermal resistance, electrical resistance, non-radiative recombination and characteristic temperature of the threshold current. It is found that a rapid growth of the threshold current density with decreasing the diameter (which takes place even in the absence of the self-heating effect) is the main internal reason leading to the dependence of the temperature characteristics of the mirodisk laser on its size. In the calculations, we used a set of parameters extracted from experiments with InGaAs quantum dot microdisk lasers. The simulation results (in particular, the light-current curve and the dependence of the minimum microdisk diameter on ambient temperature) comply well with the measured dependences.
An InAs/InGaAs quantum dot laser with a heterostructure epitaxially grown on a silicon substrate was used to fabricate injection microdisk lasers of different diameters (15–31 µm). A post-growth process includes photolithography and deep dry etching. No surface protection/passivation is applied. The microlasers are capable of operating heatsink-free in a continuous-wave regime at room and elevated temperatures. A record-low threshold current density of 0.36 kA/cm2 was achieved in 31 µm diameter microdisks operating uncooled. In microlasers with a diameter of 15 µm, the minimum threshold current density was found to be 0.68 kA/cm2. Thermal resistance of microdisk lasers monolithically grown on silicon agrees well with that of microdisks on GaAs substrates. The ageing test performed for microdisk lasers on silicon during 1000 h at a constant current revealed that the output power dropped by only ~9%. A preliminary estimate of the lifetime for quantum-dot (QD) microlasers on silicon (defined by a double drop of the power) is 83,000 h. Quantum dot microdisk lasers made of a heterostructure grown on GaAs were transferred onto a silicon wafer using indium bonding. Microlasers have a joint electrical contact over a residual n+ GaAs substrate, whereas their individual addressing is achieved by placing them down on a p-contact to separate contact pads. These microdisks hybridly integrated to silicon laser at room temperature in a continuous-wave mode. No effect of non-native substrate on device characteristics was found.
We review epitaxial formation, basic properties, and device applications of a novel type of nanostructures of mixed (0D/2D) dimensionality that we refer to as quantum well-dots (QWDs). QWDs are formed by metalorganic vapor phase epitaxial deposition of 4–16 monolayers of InxGa1−xAs of moderate indium composition (0.3 < x < 0.5) on GaAs substrates and represent dense arrays of carrier localizing indium-rich regions inside In-depleted residual quantum wells. QWDs are intermediate in properties between 2D quantum wells and 0D quantum dots and show some advantages of both of those. In particular, they offer high optical gain/absorption coefficients as well as reduced carrier diffusion in the plane of the active region. Edge-emitting QWD lasers demonstrate low internal loss of 0.7 cm−1 and high internal quantum efficiency of 87%. as well as a reasonably high level of continuous wave (CW) power at room temperature. Due to the high optical gain and suppressed non-radiative recombination at processed sidewalls, QWDs are especially advantageous for microlasers. Thirty-one μm in diameter microdisk lasers show a high record for this type of devices output power of 18 mW. The CW lasing is observed up to 110 °C. A maximum 3-dB modulation bandwidth of 6.7 GHz is measured in the 23 μm in diameter microdisks operating uncooled without a heatsink. The open eye diagram is observed up to 12.5 Gbit/s, and error-free 10 Gbit/s data transmission at 30 °C without using an external optical amplifier, and temperature stabilization is demonstrated.
A limiting curve of a stationary process in discrete time was defined by É. Janvresse, T. de la Rue, and Y. Velenik as the uniform limit of the certain renormalization of the process. We determine the limiting curves for the stationary sequence (f ∘ Tn(ω)) where T is the dyadic odometer and f is the weighted sum of digits function. Namely, we prove that for a.e. ω there exists a sequence such that the limiting curve exists and is equal to (−1) times the Tagaki–Landsberg function with parameter 1/2q. The result can be obtained as a corollary of a generalization of the Trollope–Delange formula to the q-weighted case.
Lasers based on semiconductor whispering gallery mode (WGM) resonators represent a perfect platform for active small footprint high-sensitive devices for biodetection. Biochemical samples typically require aqueous solution, and the resonator should be placed into a cuvette with water or in a microfluidic chip. The characteristics of modern semiconductor WGM lasers with an active region based on InAs/InGaAs quantum dots (QDs) make them promising for creating compact highly sensitive devices for biodetection. Deep localization of carriers in InAs/InGaAs QDs and suppressed lateral migration helps us to obtain room-temperature lasing in microdisk lasers immersed in an aqueous medium. In this work, we studied the sensitivity of the microdisk laser resonance spectral position to the refractive index of the surrounding material by changing the salinity of the water solution. We also successfully detected model proteins (secondary antibodies attached to the microdisk surface) via measurement of the lasing threshold power. The proteinprotein interaction on the microdisk surface manifests itself by an increase in the laser threshold power. Thus, in this work we demonstrated, for the first time, the possibility of using QD semiconductor microdisk lasers for detection of proteins in a microfluidic device.