The article belongs to the category of research articles. The article considers a class of ciphers - perfect ciphers. They are based on the model of K. Shannon and are considered undeciphered ciphers. Such ciphers, in particular, are ciphers of random gaming. An analysis of existing publications has shown that they substantiate the conclusion about the undeciphered character of the perfect ciphers according to K. Shannon. In this article, we introduced such concepts as: the probabilistic model of the cipher; The cipher made on attack on the open text at interception of the ciphered text; Cipher, committed by attacking a key while intercepting encrypted text; effective attack on plaintext or key; Ineffective attack on open text or key; decrypted cipher model; undeciphered cipher model. On the basis of the introduced concepts, the mathematical model of the perfect cipher is refined - the model of K. Shannon. The need for such a refinement is dictated by the fact that each cipher can have several independent models. The reasoning is presented, on the basis of which it is proved that the cipher of random gaming is a decipherable cipher.
Learning models with discrete latent variables using stochastic gradient descent remains a challenge due to the high variance of gradient estimates. Modern variance reduction techniques mostly consider categorical distributions and have limited applicability when the number of possible outcomes becomes large. In this work, we consider models with latent permutations and propose control variates for the Plackett-Luce distribution. In particular, the control variates allow us to optimize black-box functions over permutations using stochastic gradient descent. To illustrate the approach, we consider a variety of causal structure learning tasks for continuous and discrete data. We show that our method outperforms competitive relaxation-based optimization methods and is also applicable to non-differentiable score functions.
The article is concerned with the analysis of metapoetic statements of Genrikh Sapgir that were made by him in poetic form – in the shape of «programmatic» poems. In his work, the poet preferred to dispense with creative manifestos or forewords: there is extremely small number of them in Sapgir’s heritage. But at the same time he looked for ways to give a key to the meaning of his poetical quests and experiments, using his poetry, comprising of a new design concept: «poetics of semi-word» or particular method of «permutation».
We consider several procedures to number all outcomes of a permutation scheme, establish a one-to-one correspondence between the outcome and its number generated in the numbering procedure, and give some methods to simulate the outcomes.
"Designing" the complete works of his own is an important part of Aleksander Kondratov's creations, a neofuturist poet from Leningrad. One of the prospects of his future collection, "My Trinities" makes it possible to restore the poet's concept to create "The Concretions", a volume of concretist texts, which has not been fully implemented. These texts were obviously aimed at the demonstration of the whole paradigm of devices and forms of this trend of experimental poetry that Kondratov knew at the time. Keywords: Aleatory works, Aleksander Kondratov, dadaism, concretism, neofuturism, uncensored poetry, zero text, permutation, hollow text, found poetry.
A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.
Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.
Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.