**632**

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**Assessment of Independent EEG Components Obtained by Different Methods for BCI Based on Motor Imagery**

Húsek, Dušan; Frolov, A. A.; Kerechanin, J. V.; Bobrov, P.D.

2021 - English
Eight methods of decomposition of a multichannel EEG signal are compared in terms of their ability to identify the most physiologically significant components. The criterion for the meaningfulness of a method is its ability to reduce mutual information between components; to create components that can be attributed to the activity of dipoles located in the cerebral cortex; find components that are provided by other methods and for this case; and at the same time, these components should most contribute to the accuracy of the BCI based on imaginary movement. Independent component analysis methods AMICA, RUNICA and FASTICA outperform others in the first three criteria and are second only to the Common Spatial Patterns method in the fourth criterion. The components created by all methods for 386 experimental sessions of 27 subjects were combined into more than 100 clusters containing more than 10 elements. Additionally, the components of the 12 largest clusters were analyzed. They have proven to be of great importance in controlling BCI, their origins can be modeled using dipoles in the brain, and they have been detected by several degradation methods. Five of the 12 selected components have been identified and described in our previous articles. Even if the physiological and functional origins of the rest of identified components’ are to be the subject of further research, we have shown that their physiological nature is at least highly probable.\n
Keywords:
*brain–computer interface; motor imagery; blind source separation; independent component analysis; common spatial patterns; cluster analysis; EEG pattern extraction; EEG analysis; ICA; CSP; BCI; motor imagery*
Available at various institutes of the ASCR
Assessment of Independent EEG Components Obtained by Different Methods for BCI Based on Motor Imagery

Eight methods of decomposition of a multichannel EEG signal are compared in terms of their ability to identify the most physiologically significant components. The criterion for the meaningfulness of ...

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**Nearly All Reals Can Be Sorted with Linear Time Complexity**

Jiřina, Marcel

2021 - English
We propose a variant of the counting sort modified for sorting reals in a linear time. It is assumed that the sorting key and pointers to the items being sorted are moved and individual items remain at the same place in the memory (in place sorting). In this case, the space complexity of the new variant of the algorithm is the same as the complexity of the quicksort. We also quantify the practical limits for possible sorting reals in a linear time. This possibility is assured under additional assumptions on the distribution of the sorting key, mainly the independence and identity of the distribution. Here we give a more general criteria easily applicable in practice. We also show that the algorithm is applicable for data that do not fulfill criteria for linear time complexity but even that the computation is faster than the system quicksort. A new, faster version of the algorithm is attached.
Keywords:
*sorting; algorithm; real sorting key; time complexity; linear complexity*
Available in digital repository of the ASCR
Nearly All Reals Can Be Sorted with Linear Time Complexity

We propose a variant of the counting sort modified for sorting reals in a linear time. It is assumed that the sorting key and pointers to the items being sorted are moved and individual items remain ...

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**Visual Images Segmentation based on Uniform Textures Extraction**

Goltsev, A.; Gritsenko, V.; Húsek, Dušan

2020 - English
A new effective procedure for partial texture segmentation of visual images is proposed. The procedure segments any input image into a number of non-overlapping homogeneous ne-grained texture areas. The main advantages of the proposed procedure are as follows. It is completely unsupervised, that is, it processes the input image without any prior knowledge of either the type of textures or the number of texture segments in the image. In addition, the procedure segments arbitrary images of all types. This means that no changes to the procedure parameters are required to switch from one image type to another. Another major advantage of the procedure is that in most cases it extracts the uniform ne-grained texture segments present in the image, just as humans do. This result is supported by series of experiments that demonstrate the ability of the procedure to delineate uniform ne-grained texture segments over a wide range of images. At a minimum, image processing according to the proposed technique leads to a signficant reduction in the uncertainty of the internal structure of the analyzed image.
Keywords:
*Texture feature; Texture window; Homogeneous ne-grained texture segment; Texture segment extraction; Texture segmentation*
Available at various institutes of the ASCR
Visual Images Segmentation based on Uniform Textures Extraction

A new effective procedure for partial texture segmentation of visual images is proposed. The procedure segments any input image into a number of non-overlapping homogeneous ne-grained texture areas. ...

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**Two limited-memory optimization methods with minimum violation of the previous quasi-Newton equations**

Vlček, Jan; Lukšan, Ladislav

2020 - English
Limited-memory variable metric methods based on the well-known BFGS update are widely used for large scale optimization. The block version of the BFGS update, derived by Schnabel (1983), Hu and Storey (1991) and Vlček and Lukšan (2019), satisfies the quasi-Newton equations with all used difference vectors and for quadratic objective functions gives the best improvement of convergence in some sense, but the corresponding direction vectors are not descent directions generally. To guarantee the descent property of direction vectors and simultaneously violate the quasi-Newton equations as little as possible in some sense, two methods based on the block BFGS update are proposed. They can be advantageously combined with methods based on vector corrections for conjugacy (Vlček and Lukšan, 2015). Global convergence of the proposed algorithm is established for convex and sufficiently smooth functions. Numerical experiments demonstrate the efficiency of the new methods.
Keywords:
*unconstrained minimization; variable metric methods; limited-memory methods; variationally derived methods; global convergence; numerical results*
Available in a digital repository NRGL
Two limited-memory optimization methods with minimum violation of the previous quasi-Newton equations

Limited-memory variable metric methods based on the well-known BFGS update are widely used for large scale optimization. The block version of the BFGS update, derived by Schnabel (1983), Hu and Storey ...

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**Linear-time Algorithms for Largest Inscribed Quadrilateral**

Keikha, Vahideh

2020 - English
Let P be a convex polygon of n vertices. We present a linear-time algorithm for the problem of computing the largest-area inscribed quadrilateral of P. We also design the parallel version of the algorithm with O(log n) time and O(n) work in CREW PRAM model, which is quite work optimal. Our parallel algorithm also computes all the antipodal pairs of a convex polygon with O(log n) time and O(log2n+s) work, where s is the number of antipodal pairs, that we hope is of independent interest. We also discuss several approximation algorithms (both constant factor and approximation scheme) for computing the largest-inscribed k-gons for constant values of k, in both area and perimeter measures.
Keywords:
*Maximum-area quadrilateral; extreme area k-gon*
Available in a digital repository NRGL
Linear-time Algorithms for Largest Inscribed Quadrilateral

Let P be a convex polygon of n vertices. We present a linear-time algorithm for the problem of computing the largest-area inscribed quadrilateral of P. We also design the parallel version of the ...

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**The Equation |x| - |Ax| = b**

Rohn, Jiří

2020 - English
We formulate conditions on A and b under which the double absolute value equation |x| - |Ax| = b possesses in each orthant a unique solution which, moreover, belongs to the interior of that orthant.
Keywords:
*absolute value equation; double absolute value equation; orthantwise solvability; theorem of the alternatives*
Available in a digital repository NRGL
The Equation |x| - |Ax| = b

We formulate conditions on A and b under which the double absolute value equation |x| - |Ax| = b possesses in each orthant a unique solution which, moreover, belongs to the interior of that orthant.

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**The scalar-valued score functions of continuous probability distribution**

Fabián, Zdeněk

2019 - English
In this report we give theoretical basis of probability theory of continuous random variables based on scalar valued score functions. We maintain consistently the following point of view: It is not the observed value, which is to be used in probabilistic and statistical considerations, but its 'treated form', the value of the scalar-valued score function of distribution of the assumed model. Actually, the opinion that an observed value of random variable should be 'treated' with respect to underlying model is one of main ideas of the inference based on likelihood in classical statistics. However, a vector nature of Fisher score functions of classical statistics does not enable a consistent use of this point of view. Instead, various inference functions are suggested and used in solutions of various statistical problems. Inference function of this report is the scalar-valued score function of distribution.
Keywords:
*Shortcomings of probability theory; Scalar-valued score functions; Characteristics of continous random variables; Parametric estimation; Transformed distributions; Skew-symmetric distributions*
Available at various institutes of the ASCR
The scalar-valued score functions of continuous probability distribution

In this report we give theoretical basis of probability theory of continuous random variables based on scalar valued score functions. We maintain consistently the following point of view: It is not ...

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**A Logical Characteristic of Read-Once Branching Programs**

Žák, Stanislav

2019 - English
We present a mathematical model of the intuitive notions such as the knowledge or the information arising at different stages of computations on branching programs (b.p.). The model has two appropriate properties: i) The ”knowledge” arising at a stage of computation in question is derivable from the ”knowledge” arising at the previous stage according to the rules of the model and according to the local arrangement of the b.p. ii) The model confirms the intuitively well-known fact that the knowledge arising at a node of a computation depends not only on it but in some cases also on a ”mystery” information. (I. e. different computations reaching the same node may have different knowledge(s) arisen at it.) We prove that with respect to our model no such information exists in read-once b.p.‘s but on the other hand in b. p.‘s which are not read-once such information must be present. The read-once property forms a frontier. More concretely, we may see the instances of our models as a systems S = (U,D) where U is a universe of knowledge and D are derivation rules. We say that a b.p. P is compatible with a system S iff along each computation in P S derives F (false) or T (true) at the end correctly according to the label of the reached sink. This key notion modifies the classic paradigm which takes the computational complexity with respect to different classes of restricted b.p.‘s (e.g. read-once b.p.‘s, k-b.p.‘s, b.p.‘s computing in limited time etc.). Now, the restriction is defined by a subset of systems and only these programs are taken into account which are compatible with at least one of the chosen systems. Further we understand the sets U of knowledge(s) as a sets of admissible logical formulae. It is clear that more rich sets U‘s imply the large restrictions on b.p.‘s and consequently the smaller complexities of Boolean functions are detected. More rich logical equipment implies stronger computational effectiveness. Another question arises: given a set of Boolean functions (e.g. codes of some graphs) what logical equipment is optimal from the point of complexity?
Keywords:
*branching programs; computational complexity; logic*
Available in digital repository of the ASCR
A Logical Characteristic of Read-Once Branching Programs

We present a mathematical model of the intuitive notions such as the knowledge or the information arising at different stages of computations on branching programs (b.p.). The model has two ...

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**Absolute Value Mapping**

Rohn, Jiří

2019 - English
We prove a necessary and sufficient condition for an absolute value mapping to be bijective. This result simultaneously gives a characterization of unique solvability of an absolute value equation for each right-hand side.
Keywords:
*absolute value mapping; bijectivity; interval matrix; regularity; absolute value equation; unique solvability*
Available in a digital repository NRGL
Absolute Value Mapping

We prove a necessary and sufficient condition for an absolute value mapping to be bijective. This result simultaneously gives a characterization of unique solvability of an absolute value equation for ...

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**Overdetermined Absolute Value Equations**

Rohn, Jiří

2019 - English
We consider existence, uniqueness and computation of a solution of an absolute value equation in the overdetermined case.
Keywords:
*absolute value equations; overdetermined system*
Available in a digital repository NRGL
Overdetermined Absolute Value Equations

We consider existence, uniqueness and computation of a solution of an absolute value equation in the overdetermined case.

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