**1557**

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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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**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 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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**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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**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 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*
Available at various institutes of the ASCR
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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**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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**On the optimal initial conditions for an inverse problem of model parameter estimation**

Matonoha, Ctirad; Papáček, Š.

2017 - English
Available in digital repository of the ASCR
On the optimal initial conditions for an inverse problem of model parameter estimation

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**The use of sequential quadratic programming for solving reachability problems**

Kuřátko, Jan

2017 - English
Available in digital repository of the ASCR
The use of sequential quadratic programming for solving reachability problems

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**A Generalized Limited-Memory BNS Method Based on the Block BFGS Update**

Vlček, Jan; Lukšan, Ladislav

2017 - English
A block version of the BFGS variable metric update formula is investigated. It satisfies the quasi-Newton conditions with all used difference vectors and gives the best improvement of convergence in some sense for quadratic objective functions, but it does not guarantee that the direction vectors are descent for general functions. To overcome this difficulty and utilize the advantageous properties of the block BFGS update, a block version of the limited-memory BNS method for large scale unconstrained optimization is proposed. The algorithm is globally convergent for convex sufficiently smooth functions and our numerical experiments indicate its efficiency.
Keywords:
*unconstrained minimization; block variable metric methods; limited-memory methodserical results; the BFGS update; global convergence; numerical results*
Available in digital repository of the ASCR
A Generalized Limited-Memory BNS Method Based on the Block BFGS Update

A block version of the BFGS variable metric update formula is investigated. It satisfies the quasi-Newton conditions with all used difference vectors and gives the best improvement of convergence in ...

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**Robust Regression Estimators: A Comparison of Prediction Performance**

Kalina, Jan; Peštová, Barbora

2017 - English
Regression represents an important methodology for solving numerous tasks of applied econometrics. This paper is devoted to robust estimators of parameters of a linear regression model, which are preferable whenever the data contain or are believed to contain outlying measurements (outliers). While various robust regression estimators are nowadays available in standard statistical packages, the question remains how to choose the most suitable regression method for a particular data set. This paper aims at comparing various regression methods on various data sets. First, the prediction performance of common robust regression estimators are compared on a set of 24 real data sets from public repositories. Further, the results are used as input for a metalearning study over 9 selected features of individual data sets. On the whole, the least trimmed squares turns out to be superior to the least squares or M-estimators in the majority of the data sets,\nwhile the process of metalearning does not succeed in a reliable prediction of the most suitable estimator for a given data set.
Keywords:
*robust estimation; linear regression; prediction; outliers; metalearning*
Available at various institutes of the ASCR
Robust Regression Estimators: A Comparison of Prediction Performance

Regression represents an important methodology for solving numerous tasks of applied econometrics. This paper is devoted to robust estimators of parameters of a linear regression model, which are ...

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