Mirror descent algorithm on the indefinite control horizon

We consider the problem of optimal control in a random environment in a minimax setting as applied to data processing. It is assumed that the random environment provides two methods of data processing, the effectiveness of which is not known in advance. The goal of the control in this case is to find the optimal strategy for the application of processing methods and to minimize losses. To solve this problem, the mirror descent algorithm is used, including its modifications for batch processing. The use of algorithms for batch processing allows us to get a significant gain in speed due to the parallel processing of batches. In the classical statement, the search for the optimal strategy is considered on a fixed control horizon but this article considers an indefinite control horizon. With an indefinite horizon, the control algorithm cannot use information about the value of the horizon when searching for an optimal strategy. Using numerical modeling, the operation of the mirror descent algorithm and its modifications on an indefinite control horizon is studied and obtained results are presented.

On the number of holomorphic mappings between Riemann surfaces of finite analytic type

The set of non-constant holomorphic mappings between two given compact Riemann surfaces of genus greater than 1 is always finite. This classical statement was made by de Franchis. Furthermore, bounds on the cardinality of the set depending only on the genera of the surfaces have been obtained by a number of mathematicians. The analysis is carried over in this paper to the case of Riemann surfaces of finite analytic type (i.e. compact Riemann surfaces minus a finite set of points) so that the finiteness result, together with a crude but explicit bound depending only on the topological data, may be extended for the number of holomorphic mappings between such surfaces.

The lemma of the logarithmic derivative for subharmonic functions

The classical statement of the lemma in question [7], [3] is about meromorphic functions f on ℂ and says thatfor all r > 0, with the possible exception of a set of finite Lebesgue measure. Here T(r, f) is the Nevanlinna characteristic of f. The lemma plays an important role in value distribution theory.

GENERALIZED KAZAKOV-MIGDAL-KONTSEVICH MODEL: GROUP THEORY ASPECTS

The Kazakov-Migdal model, if considered as a functional of external fields, can always be represented as an expansion over characters of the GL group. The integration over “matter fields” can be interpreted as going over the model (the space of all highest weight representations) of GL. In the case of compact unitary groups the integrals should be substituted by discrete sums over the weight lattice. The D=0 version of the model is the generalized Kontsevich integral, which in the above-mentioned unitary (discrete) situation coincides with the partition function of 2D Yang-Mills theory with the target space of genus g=0 and m=0, 1, 2 holes. This particular quantity is always a bilinear combination of characters and appears to be a Toda lattice τ function. (This is a generalization of the classical statement that individual GL characters are always singular KP τ functions.) The corresponding element of the universal Grassmannian is very simple and somewhat similar to the one arising in investigations of the c=1 string models. However, in certain circumstances the formal sum over representations should be evaluated by the steepest descent method, and this procedure leads to some more-complicated elements of the Grassmannian. This “Kontsevich phase,” as opposed to the simple “character phase,” deserves further investigation.

Calvin's Institutes of the Christian Religion

On account of its influence upon many generations of Protestants, Calvin's Institutes of the Christian Religion may be regarded as the classical statement of the Protestant Christian faith. It reflects more clearly than any other book produced by the Reformation the thought which inaugurated the whole Protestant movement.

Greek Political Heritage and Totalitarianism

IN A FAMOUS dialogue between the Athenian ambassadors and the Council of the small island of Melos, Thucydides has given the classical statement of the “right” of the stronger. “The brave Milesians soon see that they cannot appeal to the Athenians' sense of justice, because the Athenians recognize no standard but their own political advantage…By making the Athenians justify the right of the stronger through the law of nature, and transform God from the guardian of justice into the pattern of all earthly authority and force, Thucydides gives the realistic policy of Athens the depth and validity of a philosophical doctrine.” The Dutch, in the days of Peter Breughel, used to say: “the big fish devour the little fish” to which Spinoza added “by natural right.” That is the doctrine of the “state,”as inherited from the Greeks. Similar situations still haunt us. Did the Russians by natural right seek to destroy Finnish independence?