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We consider the Dirac operator with a periodic potential on the half-line with the Dirichlet boundary condition at zero. Its spectrum consists of an absolutely continuous part plus at most one eigenvalue in each open gap. The resolvent admits a meromorphic continuation onto a two-sheeted Riemann surface with a unique simple pole on each open gap: on the first sheet (an eigenvalue) or on the second sheet (a resonance). These poles are called levels and there are no other poles. If the potential is shifted by real parameter t, then the continuous spectrum does not change but the levels can change their positions. We prove that each level is smooth and in general, non-monotonic function of t. We prove that a level is a strictly monotone function of t for a specific potential. Using these results, we obtain formulas to recover potentials of special forms.
International news plays an important role in shaping public opinion about the foreign policy and leadership of a country. Yet research shows that the bias in favor of the current political leadership is prevalent in foreign news coverage. In this study, we explore whether these assumptions hold in the case of digital news outlets in media systems outside of established democracies. We examine the representations of Russia in digital news streams of Kazakhstan and Ukraine based on a collection of news published by about 30 top news websites in each of the countries during 2018 (n = 2,339,583 news items). To study the coverage of Russia, we follow an approach combining topic modeling for extraction of news agendas and qualitative analysis of news framing. Then, we compare Kazakhstani and Ukrainian news agendas and their framing. The results suggest that digital news media in the selected cases follow expectations based on the research of offline media despite the transformations that happened in news production with the advance of the Internet.
For n-person multicriteria game with chance moves in extensive form, we prove the existence of pure strategy subgame perfect Pareto equilibrium (SPPE). Then we provide and demonstrate an algorithm which allows to reasonably select and construct a unique SPPE.
We consider Schrödinger operators with periodic potentials on periodic discrete graphs. Their spectrum consists of a finite number of bands. We determine trace formulas for the Schrödinger operators. The proof is based on the decomposition of the Schrödinger operators into a direct integral and a specific representation of fiber operators. The traces of the fiber operators are expressed as finite Fourier series of the quasimomentum. The coefficients of the Fourier series are given in terms of the potentials and cycles in the quotient graph from some specific cycle sets. We also present the trace formulas for the heat kernel and the resolvent of the Schrödinger operators and the determinant formulas.
Stochastic parameters are introduced into a model of network games with production and knowledge externalities. The model was formulated by V. Matveenko and A. Korolev and generalizes Romer’s two-period model. The agents’ productivities have both deterministic and Wiener components. The research represents the dynamics of a single agent and the dynamics in a triangle that occurs in the process of combining the agents. Explicit expressions for the dynamics of a single agent and dyad agents are obtained in the form of Brownian random processes. Solutions of stochastic equations and systems are analyzed qualitatively.
In this paper, stochastic parameters are introduced into the network games model with production and knowledge’s externalities. This model was formulated by V. Matveenko and A. Korolev and generalized two-period Romer model. Agents’ productivities have deterministic and Wiener components. We consider the dynamics that occur when two complete networks are combined. Explicit expressions in the form of Brownian random processes are obtained. A qualitative analysis of the solution of a system of stochastic equations is carried out.
Hierarchical topic modeling is a potentially powerful instrument for determining topical structures of text collections that additionally allows constructing a hierarchy representing the levels of topic abstractness. However, parameter optimization in hierarchical models, which includes finding an appropriate number of topics at each level of hierarchy, remains a challenging task. In this paper, we propose an approach based on Renyi entropy as a partial solution to the above problem. First, we introduce a Renyi entropy-based metric of quality for hierarchical models. Second, we propose a practical approach to obtaining the “correct” number of topics in hierarchical topic models and show how model hyperparameters should be tuned for that purpose. We test this approach on the datasets with the known number of topics, as determined by the human mark-up, three of these datasets being in the English language and one in Russian. In the numerical experiments, we consider three different hierarchical models: hierarchical latent Dirichlet allocation model (hLDA), hierarchical Pachinko allocation model (hPAM), and hierarchical additive regularization of topic models (hARTM). We demonstrate that the hLDA model possesses a significant level of instability and, moreover, the derived numbers of topics are far from the true numbers for the labeled datasets. For the hPAM model, the Renyi entropy approach allows determining only one level of the data structure. For hARTM model, the proposed approach allows us to estimate the number of topics for two levels of hierarchy.
Signed networks form a particular class of complex networks that has many applications in sociology, recommender and voting systems. The contribution of this paper is twofold. First, we propose an approach aimed at determining the characteristic subgraphs of the network. Second, we apply the developed approach to the analysis of the network describing the Wikipedia adminship elections. It is shown that this network agrees with the status theory if one does not consider strongly tied vertices, i.e., the vertices that are connected in both directions. At the same time, the strongly connected vertices mostly agree with the structural balance theory. This result indicates that there is a substantial difference between single and double connections, the fact that deserves a detailed analysis within a broader context of directed signed networks.
In the paper, the differential games on networks with partner sets are considered. The payoffs of a given player depend on his actions and the actions of the players from his partner set. The cooperative version of the game is proposed, and a special type of characteristic function is introduced. It is proved the constructed cooperative game is convex. Using the properties of the payoff functions and the constructed characteristic function, the Shapley Value and $\tau$-value are computed. It is also proved that in this special class of differential games the Shapley value is time-consistent.
A class of cooperative differential games on networks is considered. It is supposed that players have the possibility to cut connections with neighbors at each time instant of the game. This gives the possibility to compute the values of a characteristic function for each coalition as a joint payoff of players from this coalition without payments induced by actions of players outside the coalition. Thus the characteristic function is evaluated along the cooperative trajectory. It measures the worth of coalitions under the process of cooperation instead of under minimax confrontation or the Nash non-cooperative stance. The approach essentially simplifies the construction of the characteristic function and cooperative solutions such as the Shapley value and others. Also, it is proved that the proposed characteristic function is convex and as a result, the Shapley value belongs to the core.
Consider a difference operator H with periodic coefficients on the octant of the lattice. We show that for any integer N and any bounded interval I, there exists an operator H having N eigenvalues, counted with multiplicity on this interval, and does not exist other spectra on the interval. Also right and to the left of it are spectra and the corresponding subspaces have an infinite dimension. Moreover, we prove similar results for other domains and any dimension. The proof is based on the inverse spectral theory for periodic Jacobi operators.
We consider a Sturm–Liouville operator with an integrable potential 𝑞 on the unit interval𝐼 = [0,1].We consider a Schrödinger operator with a real compactly supported potential on the half line and on the line, where this potential coincides with 𝑞 on the unit interval and vanishes outside 𝐼. We determine the relationships between eigenvalues of such operators and obtain estimates of eigenvalues in terms of potentials.
Under minimal assumptions, we prove that an empirical estimator of the tail conditional allocation (TCA), also known as the marginal expected shortfall, is consistent. Examples are provided to confirm the minimality of the assumptions. A simulation study illustrates the performance of the estimator in the context of developing confidence intervals for the TCA. The philosophy adopted in the present paper relies on three principles: easiness of practical use, mathematical rigor, and practical justifiability and verifiability of assumptions.
Efficient representation of data aggregations is a fundamental problem in modern big data applications, where network topologies and deployed routing and transport mechanisms play a fundamental role in optimizing desired objectives such as cost, latency, and others. In traditional networking, applications use TCP and UDP transports as a primary interface for implemented applications that hide the underlying network topology from end systems. On the flip side, to exploit network infrastructure in a better way, applications restore characteristics of the underlying network. In this work, we demonstrate that both specified extreme cases can be inefficient to optimize given objectives. We study the design principles of routing and transport infrastructure and identify extra information that can be used to improve implementations of compute-aggregate tasks. We build a taxonomy of compute-aggregate services unifying aggregation design principles, propose algorithms for each class, analyze them theoretically, and support our results with an extensive experimental study.
We consider Schrödinger operators on the line with potentials that are periodic with respect to the coor-dinate variable and real analytic with respect to the energy variable. We prove that if the imaginary part of the potential is bounded in the right half-plane, then the high energy spectrum is real, and the corresponding asymptotics are determined. Moreover, the Dirichlet and Neumann problems are considered. These results are used to analyze the good Boussinesq equation.
Let p(ζ) be a positive function defined on the unit circle T and satisfying the condition|p(ζ2)−p(ζ1)|≤c0loge|ζ2−ζ1|,ζ1,ζ2∈T,p−=minζ∈Tp(ζ). Futhermore, let 0<α<1, r≥0, r∈Z, and assume that p−>1α. Define a class of analytic functions in the unit disk D as follows: f∈Hr+αp(⋅) ifsup0<ρ<1sup0<|θ|<π∫02π|f(r)(ρei(λ+θ))−f(r)(ρeiλ)|θ|α|p(eiλ)dλ<∞.The following main results are proved.
Theorem 1. Let f∈Hr+αp(⋅), and let I be an inner function, f/I∈H1. Then f/I∈Hr+αp(⋅).
Theorem 2. Let f∈Hr+αp(⋅), and let I be an inner function, f/I∈H∞. Assume that the multiplicity of every zero of f in D is at least r+1. Then fI∈Hr+αp(⋅).
We consider massless Dirac operators on the half-line with compactly supported potentials. We solve the inverse problems in terms of Jost function and scattering matrix (including characterization). We study resonances as zeros of Jost function and prove that a potential is uniquely determined by its resonances. Moreover, we prove the following: (1) resonances are free parameters and a potential continuously depends on a resonance, (2) the forbidden domain for resonances is estimated, (3) asymptotics of resonance counting function is determined, (4) these results are applied to canonical systems.
We discuss inverse resonance scattering for the Laplacian on a rotationally symmetric manifold M = (0,∞) × Y whose rotation radius is constant outside some compact interval. The Laplacian on M is unitarily equivalent to a direct sum of one-dimensional Schrödinger operators with compactly supported potentials on the half-line. We prove • Asymptotics of counting function of resonances at large radius. • The rotation radius is uniquely determined by its eigenvalues and resonances. • There exists an algorithm to recover the rotation radius from its eigenvalues and resonances. The proof is based on some non-linear real analytic isomorphism between two Hilbert spaces.
Irregular fluctuations in economy lead to unpredictable effects and disrupt its stable functioning. Various tools could be used to stabilize irregular dynamics in economic models. For example, to introduce control into the model as an external function, as well as to take into account the internal characteristics of economic agents in the economy under consideration, we consider agents that use the variables that are under their control to achieve optimum, by minimizing or maximizing cost, profit, or welfare function. However, optimal behavior in economics does not necessarily lead to simple model dynamics. It is therefore important to find the conditions for and understand the mechanism of emergence of complex dynamics. We study two New Keynesian models, including one with externalitites, in continuous-time under different monetary and fiscal policy regimes, which represent the economy where the economic agents solve constrained optimization problems. We show that in case of explosive equilibrium dynamics, limit cycles or more complicated attracting sets could appear, including chaotic attractors of various nature. In this case it is possible to control irregular dynamics, including by adjusting policy parameters that serve as bifurcation parameters, in order to alleviate the implied economic uncertainty and bring agents’ expectations in line with the intended steady state.
In this paper, a method for constructing and proving a formal mathematical model was considered that implements the security functions of information security systems based on a discretionary access control policy. An approach is shown that implements the object-subject formalization.
An example of creating a finite state machine is shown using the CPN Tools software solution.
Second session of the XXXVI International Seminar on Stability Problems for Stochastic Models