**1092**

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**A limited-memory optimization method using the infinitely many times repeated BNS update and conjugate directions**

Vlček, Jan; Lukšan, Ladislav

2018 - English
Keywords:
*Unconstrained minimization; variable metric methods; limited-memory methods; the repeated BFGS update; global convergence; numerical results*
Available in digital repository of the ASCR
A limited-memory optimization method using the infinitely many times repeated BNS update and conjugate directions

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**Numerical solution of generalized minimax problems**

Lukšan, Ladislav; Matonoha, Ctirad; Vlček, Jan

2018 - English
Keywords:
*Numerical optimization; nonlinear approximation; nonsmooth optimization; generalized minimax problems; recursive quadratic programming methods; interior point methods; smoothing methods; algorithms; numerical experiments*
Available in digital repository of the ASCR
Numerical solution of generalized minimax problems

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**Semigroup Structure of Sets of Solutions to Equation X^s = X^m**

Porubský, Štefan

2017 - English
Using an idempotent semigroup approach we describe the semigroup and group structure of the set of solutions to equation X^m = X^s in successive steps over a periodic commutative semigroup, over multiplicative semigroups of factor rings of residually finite commutative rings and finally over multiplicative semigroups of factor rings of residually finite commutative principal ideal domains. The analysis is done through the use of the maximal subsemigroups and groups corresponding to an idempotent of the corresponding semigroup and in the case of residually finite PID’s employing the available analysis of the Euler-Fermat Theorem as given in [11]. In particular the case when this set of solutions is a union of groups is handled. As a simple application we show a not yet noticed group structure of the set of solutions to x^n = x connected with the message space of RSA cryptosystems and Fermat pseudoprimes.
Keywords:
*set of solutions; idempotent; maximal semigroup corresponding to an idempotent; maximal group corresponding to an idempotent; equation X^s = X^m; finite commutative ring with identity element; residually finite commutative principal ideal domains*
Available on request at various institutes of the ASCR
Semigroup Structure of Sets of Solutions to Equation X^s = X^m

Using an idempotent semigroup approach we describe the semigroup and group structure of the set of solutions to equation X^m = X^s in successive steps over a periodic commutative semigroup, over ...

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**An adaptive recursive multilevel approximate inverse preconditioning: Computation of the Schur complement**

Kopal, Jiří; Rozložník, Miroslav; Tůma, Miroslav

2017 - English
Available in digital repository of the ASCR
An adaptive recursive multilevel approximate inverse preconditioning: Computation of the Schur complement

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**The Computational Power of Neural Networks and Representations of Numbers in Non-Integer Bases.**

Šíma, Jiří

2017 - English
We briefly survey the basic concepts and results concerning the computational power of neural networks which basically depends on the information content of weight parameters. In particular, recurrent neural networks with integer, rational, and arbitrary real weights are classified within the Chomsky and finer complexity hierarchies. Then we refine the analysis between integer and rational weights by investigating an intermediate model of integer-weight neural networks with an extra analog rational-weight neuron (1ANN). We show a representation theorem which characterizes the classification problems solvable by 1ANNs, by using so-called cut languages. Our analysis reveals an interesting link to an active research field on non-standard positional numeral systems with non-integer bases. Within this framework, we introduce a new concept of quasi-periodic numbers which is used to classify the computational power of 1ANNs within the Chomsky hierarchy.
Keywords:
*neural network; Chomsky hierarchy; beta-expansion; cut language*
Available at various institutes of the ASCR
The Computational Power of Neural Networks and Representations of Numbers in Non-Integer Bases.

We briefly survey the basic concepts and results concerning the computational power of neural networks which basically depends on the information content of weight parameters. In particular, recurrent ...

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**Idempotents, Group Membership and their Applications**

Porubský, Štefan

2017 - English
S.Schwarz in his paper [165] proved the existence of maximal subgroups in periodic semigroups and a decade later he brought [167] into play the maximal subsemigroups and thus he embodied the idempotents in the structural description of semigroups. Later in his papers he showed that a proper description of these structural elements can be used to (re)prove many useful and important results in algebra and number theory. The present paper gives a survey of selected results scattered throughout the literature where an semigroup approach based on tools like idempotent, maximal subgroup or maximal subsemigroup either led to a new insight into the substance of the known results or helped to discover new approach to solve problems. Special attention will be given to some disregarded historical connections between semigroup and ring theory.
Keywords:
*multiplicative semigroup; finite semigroups; power semigroups; idempotent elements; finite commutative rings; principal ideal domain; Euler-Fermat theorem; Wilson theorem; matrices over fields; maximal groups contained in a semigroup; periodic sequence; multiplicative semigroup of Zm; semigroup of circulant Boolean matrices*
Available on request at various institutes of the ASCR
Idempotents, Group Membership and their Applications

S.Schwarz in his paper [165] proved the existence of maximal subgroups in periodic semigroups and a decade later he brought [167] into play the maximal subsemigroups and thus he embodied the ...

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**Exact Inference In Robust Econometrics under Heteroscedasticity**

Kalina, Jan; Peštová, Barbora

2017 - English
The paper is devoted to the least weighted squares estimator, which is one of highly robust estimators for the linear regression model. Novel permutation tests of heteroscedasticity are proposed. Also the asymptotic behavior of the permutation test statistics of the Goldfeld-Quandt and Breusch-Pagan tests is investigated. A numerical experiment on real economic data is presented, which also shows how to perform a robust prediction model under heteroscedasticity. Theoretical results may be simply extended to the context of multivariate quantiles
Keywords:
*heteroscedasticity; robust statistics; regression; diagnostic tools; economic data*
Fulltext is available at external website.
Exact Inference In Robust Econometrics under Heteroscedasticity

The paper is devoted to the least weighted squares estimator, which is one of highly robust estimators for the linear regression model. Novel permutation tests of heteroscedasticity are proposed. Also ...

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**On the Optimization of Initial Conditions for a Model Parameter Estimation**

Matonoha, Ctirad; Papáček, Š.; Kindermann, S.

2017 - English
The design of an experiment, e.g., the setting of initial conditions, strongly influences the accuracy of the process of determining model parameters from data. The key concept relies on the analysis of the sensitivity of the measured output with respect to the model parameters. Based on this approach we optimize an experimental design factor, the initial condition for an inverse problem of a model parameter estimation. Our approach, although case independent, is illustrated at the FRAP (Fluorescence Recovery After Photobleaching) experimental technique. The core idea resides in the maximization of a sensitivity measure, which depends on the initial condition. Numerical experiments show that the discretized optimal initial condition attains only two values. The number of jumps between these values is inversely proportional to the value of a diffusion coefficient D (characterizing the biophysical and numerical process). The smaller value of D is, the larger number of jumps occurs.
Keywords:
*FRAP; sensitivity analysis; optimal experimental design; parameter estimation; finite differences*
Available in digital repository of the ASCR
On the Optimization of Initial Conditions for a Model Parameter Estimation

The design of an experiment, e.g., the setting of initial conditions, strongly influences the accuracy of the process of determining model parameters from data. The key concept relies on the analysis ...

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**The Computational Power of Neural Networks and Representations of Numbers in Non-Integer Bases**

Šíma, Jiří

2017 - English
We briefly survey the basic concepts and results concerning the computational power of neural networks which basically depends on the information content of weight parameters. In particular, recurrent neural networks with integer, rational, and arbitrary real weights are classified within the Chomsky and finer complexity hierarchies. Then we refine the analysis between integer and rational weights by investigating an intermediate model of integer-weight neural networks with an extra analog rational-weight neuron (1ANN). We show a representation theorem which characterizes the classification problems solvable by 1ANNs, by using so-called cut languages. Our analysis reveals an interesting link to an active research field on non-standard positional numeral systems with non-integer bases. Within this framework, we introduce a new concept of quasi-periodic numbers which is used to classify the computational power of 1ANNs within the Chomsky hierarchy.
Keywords:
*neural network; Chomsky hierarchy; beta-expansion; cut language*
Available at various institutes of the ASCR
The Computational Power of Neural Networks and Representations of Numbers in Non-Integer Bases

We briefly survey the basic concepts and results concerning the computational power of neural networks which basically depends on the information content of weight parameters. In particular, recurrent ...

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**UFO 2017. Interactive System for Universal Functional Optimization**

Lukšan, Ladislav; Tůma, Miroslav; Matonoha, Ctirad; Vlček, Jan; Ramešová, Nina; Šiška, M.; Hartman, J.

2017 - English
This report contains a description of the interactive system for universal functional optimization UFO, version 2017. This version contains interfaces to the MATLAB and SCILAB graphics environments.
Keywords:
*numerical optimization; nonlinear programming; nonlinear approximation; algorithms; software systems*
Available in digital repository of the ASCR
UFO 2017. Interactive System for Universal Functional Optimization

This report contains a description of the interactive system for universal functional optimization UFO, version 2017. This version contains interfaces to the MATLAB and SCILAB graphics environments.

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