Institute of Computer Science Pravdepodobnostn´ı modelován´ı

Transkript

Institute of Computer Science
Academy of Sciences of the Czech Republic
Pravděpodobnostnı́ modelovánı́
lokálnı́ch povětrnostnı́ch podmı́nek
s aplikacemi v dopravě 1
Pavel Juruš
Technical report No. 1018
January 3, 2008
Abstract:
Je navrhován systém pro lokálnı́ predikci koncentracı́ chemických látek a pole větru. Systém bude zaměřen
na modelovánı́ šı́řenı́ chemické látky po jejı́m úniku do ovzdušı́ při havárii nebezpečného nákladu. Důležitými
plánovanými vlastnostmi jsou regionálnı́ měřı́tko, využitı́ realistického a aktuálnı́ho popisu atmosféry a
pravděpodobnostnı́ predikce. Model je koncipován tak, aby mohl být spuštěn pro libovolné mı́sto havárie,
které je předem neznámé. To umožňuje použitı́ i dalšı́ch aplikacı́, řešı́cı́ch disperzi v regionálnı́m měřı́tku.
Použité metody lze navı́c zobecnit pro předpověď dalšı́ch veličin popisujı́cı́ch atmosféru, pro které je typický lokálnı́ charakter. Těžištěm práce je zkoumánı́ vztahu skutečných a modelových hodnot. K analýze a
predikci chyb modelů bude využito nejnovějšı́ch poznatků ze stochastického modelovánı́ atmosféry, asimilace dat a statistického postprocesingu.
Keywords:
ensemble models, atmospheric modelling, stochastic forecasts
1 Výzkum reportovaný v této zprávě byl podporován projektem GA AV ČR 1ET 400300414 “Matematické modelovánı́ čistoty ovzdušı́ s aplikacemi v krizovém managementu” programu “Informačnı́ společnost” a institucionálnı́m
výzkumným záměrem AV0Z10300504
Inverznı́ modelovánı́ emisı́
1
Jaroslav Resler
January 3, 2008
Abstrakt:
V této zprávě jsou zkoumány zdroje nepřesnostı́ regionálnı́ch chemických transportnı́ch modelů, přičemž
hlavnı́ pozornost je soustředěna na nepřesnosti emisnı́ch vstupů. Pro jejich zpřesněnı́ je možno vedle
tradičnı́ch postupů použı́t též metody inverznı́ho modelovánı́ korigujı́cı́ modelové vstupy pomocı́ přı́mých či
nepřı́mých pozorovánı́ atmosféry. Jsou porovnávány vlastnosti jednotlivých metod modelujı́cı́ch vztah emisekoncentrace a jako nejvýhodnějšı́ pro dalšı́ experimenty je vybrána metoda odvozená z metody 4DVar. Dále
jsou zkoumány vlastnosti a dostupnost jednotlivých typů pozorovánı́ atmosféry a vhodnost jejich použitı́
v uvedených metodách inverznı́ho modelovánı́. Zvláštnı́ pozornost je věnována ověřenı́ možného přı́nosu
pozorovánı́ ze satelitnı́ch nástrojů, které tvořı́ nový a zatı́m neověřený zdroj informace pro regionálnı́ modelovánı́ kvality ovzdušı́. Na základě provedených pilotnı́ch experimentů lze předpokládat přı́nos těchto
pozorovánı́ pro modelovánı́ atmosféry v regionálnı́m měřı́tku.
Keywords:
inverse modelling, atmospheric modelling, satelite instruments
1 Výzkum reportovaný v této zprávě byl podporován projektem GA AV ČR 1ET 400300414 “Matematické modelovánı́ čistoty ovzdušı́ s aplikacemi v krizovém managementu” programu “Informačnı́ společnost” a institucionálnı́m
výzkumným záměrem AV0Z10300504
G-variation in Lp-spaces and
integral representation
Terezie Šámalová
June 2008
Abstract:
This paper brings some improvements on known estimates on rates of approximation by neural networks.
We proceed along the line proposed by V. Krkov applying integral representations. We prove that existence
of integral representation of a function is equivalent to limit of classical neural networks. We give two proofs
for finiteness of G-variation for Lp activation functions. This enables Maurey-Jones-Barron-type estimates
to be applied in this more general setting. We show that the known estimates cannot distinguish between
sigmoidal activation functions and provide limitations of the approach as well. Applying presented results
we finally give estimates for some concrete approximation schemas.
Keywords:
Neural networks, rates of approximation, G-variation, integral representation
Pruning algorithms for
one-hidden-layer feedforward neural
networks
Terezie Šámalová, Robert Šámal
July 2008
Abstract:
In the article we propose randomized algoritms based on a modification of probabilistic proof of theorem on
rates of approximation in convex closures from [DDGS93]. On high-level we derive time complexities of the
proposed algorithms and compare them to iterative algorithms in several concrete situations. Observing
properties of one type of probabilistic algorithm we propose it as a solution for pruning a too big neural
network. We derive theoretically bounds on error such introduced.
Too big networks arise for example from regularized minimization problems. We overview kernel-based
regularization and propose use of the probabilistic algorithm. Further we discuss new types of composite
kernels and their advantages. Experiments on use of probabilistic pruning on these schemas are running.
Keywords:
probabilistic algorithm, randomized pruning, error estimate, time complexity, regularization, kernel
methods
An integral upper bound for
neural-network approximation
Paul C. Kainen1 and Věra Kůrková
2
May 2008
Abstract:
Complexity of one-hidden-layer networks is studied using tools from nonlinear approximation and integration
theory. For functions with suitable integral representations in the form of networks with infinitely many
hidden units, upper bounds are derived on the speed of decrease of approximation error with an increasing
number of network units. A unifying framework for derivation of such bounds is obtained using properties
of Bochner integral. The results are applied to perceptron networks.
Keywords:
Model complexity of neural networks, integral representation in the form of network with infinitely many
hidden units, rates of variable-basis approximation, variational norm, Bochner integral, perceptron
networks.
1 Department
of Mathematics, Georgetown University, Washington, D. C. 20057-1233, USA, [email protected]
of Computer Science, Academy of Sciences of the Czech Republic, Pod Vodárenskou věžı́ 2, 182 07,
Prague 8, Czech Republic, [email protected]
2 Institute
Communicating Mobile
Nano-Machines and Their
Computational Power1
Jiřı́ Wiedermann2 and Lukáš Petrů3
May 2008
Abstract:
A computational model of molecularly communicating mobile nanomachines is defined. Nanomachines
are modelled by timed probabilistic automata augmented by a severely restricted communication mechanism. We show that for molecular communication among such machines an asynchronous stochastic
protocol originally designed for wireless communication in so-called amorphous computers with static computational units can also be used. We design an algorithm that using randomness and timing delays selects
with a high probability a leader from among sets of anonymous candidates. This enables a simulation of
counter automata proving that networks of mobile nanomachines possess universal computing power.
Keywords:
molecular communication; nanomachines; timed probabilistic automata; universal computing
1 This research was carried out within the institutional research plan AV0Z10300504 and partially supported by the
GA ČR grant No. 1ET100300419 and GD201/05/H014.
2 Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod Vodárenskou věžı́ 2, 182 07
Prague 8, Czech Republic
3 Faculty of Mathematics and Physics, Charles University, Malostranské náměstı́ 25, 118 00 Prague 1, Czech Republic,
[email protected]
Decomposition of Correlation Integral
to Local Functions
Marcel Ji ina and Marcel Ji ina, jr.
Technical Report No. V-1025
June 2008
Abstract
We show that the correlation integral can be decomposed into functions each related to a particular
point of data space. For these functions, one can use similar polynomial approximations such as the
correlation integral. The essential difference is that the value of the exponent, which would correspond
to the correlation dimension, differs in accordance to the position of the point in question. Moreover, we
show that the multiplicative constant in that polynomial approximation is proportional to the probability
density estimation at that point. This finding is used to construct a classifier.
Keywords:
multivariate data, correlation dimension, correlation integral, decomposition, probability density
estimation
2
2
Learning Weighted Metrics Method with a Nonsmooth
Learning Process
July 2008
Abstract
A new approach to the Learning Weighted Metrics method for optimized classification of data with 1-NN
rule is proposed. New approach is based on application of updating rule similar to one of Madaline
neural network, and on dynamic optimization of the step size similar to Runge’s method of half step. A
short theory is given and the classification ability is demonstrated.
Keywords:
multivariate data, pattern classification, 1-NN classifier, weighted distances, error minimization.
The Quantum Second Law
and Quantum Information1
Ladislav Andrey
September 2008
Abstract:
The big progress of novel quantum information science brings about a possibility to study the role the
quantum entanglement may play in different fields, e.g., quantum computation, quantum communication,
quantum teleportation, quantum cryptography, et cetera. It is proved rigorously here that every quantum
operation generating quantum entanglement states in the multipartite quantum system of qubits leads to
the reduction of quantum entropy of whole system. The novel quantum second law is then formulated on
this basis. As the consequence it follows the quantum entanglement can also explain rather surprisingly
many misleading and false expectations of so called challenges to the second law which have been proposed
over the last ten years, or so.
Keywords:
Quantum information, quantum entanglement, quantum second law
1 This work was partly funded by the grant 1ET300100403 of GAAV and by the Institutional Research Plan
AVOZ10300504.
A Characterization of Hitting Sets
for 1 Branching Programs of Width 3
Jiřı́ Šı́ma1 , Stanislav Žák2
October 2008
Abstract:
This report is an extension to the previous paper [1] by the authors where the necessary definitions and
references can be found. This paper presents a very preliminary technical result of ongoing work-in-progress
on a characterization of hitting sets for read-once branching programs of width 3.
Keywords:
derandomization, hitting set, branching programs of bounded width
1 {sima|stan}@cs.cas.cz
2 Research partially supported by project 1M0545 of The Ministry of Education of the Czech Republic (J. Šı́ma), the
“Information Society” project 1ET100300517 (S. Žák), and the Institutional Research Plan AV0Z10300504.
!#"%$'&)(*,+.-/0 "213(4'"%(657-/98;:<(>=@?%(2"A:CB('DFE)GIHJ1K"
L
V
M
P
NPOQRNJ
TOW
TOQOQQ^J
M
YM
Z
U6N
OQ]N
_`PO
anbrdteyg)iAjlknmUo
prqtsvuhwyx sfz{U|}qy~v|2[r thy
snvqv| hyh
Q| z]sf¡
"! #$%&#'(*)+!#,-.$/!01$%234"-65!78-.!'(*--.6
#9
&:;
-$/!=<(>@?&A$/'B9)!6C-.$/!0$/# *'D!!E"!F$/!78$%'(!@E"$=
"6#.!
#!.GHI'(-.&J#='(!-.K
:-.$%
"#L"!J"$/'(!,#'BA'B +'B#6"!L!0M!;
!!0#-N/O.'(!7"-'BP:"$/'(.!?QR!LI!0#6:+2"!LS TH
-.'(!J#K
!E
-.I'(!
#.I"UV$/#?/WX
+'B "#/
I'B#YP#%.$=
-.$/!0#'B"!Z"
#[$/&'B3$%2P!\[]^'B-'B:_ON##$P?AM!H$%.E'(-1#-68'BM#SA!7=:#Z#
.!6
+-I'(!L#I
"#:.-6`N="a
'(E#.E'(-a#I"a $%2?
b[Rq cd;|ht¡
|}Jzv|He_wyx qAqv{ qyqvw\qfu\crr{ x¯w\q\zv| qy{°{ x¯~t®x s ~v|n{ qv
The dynamics of complex logistic
equation and consequences1
Andrey L.2 , Ando H.3 Aihara K.4
October 2008
Abstract:
The new approach to study the dynamics of the complex valued deterministic logistic equation in the form
of equivalent 2D mapping is presented. Numerical simulations then indicate surprisingly, that there is not
the analogy of canonical Hnon map type deterministic chaos in this case. The important role of property
”to be complex valued” in nonlinear dynamics in general with possible consequences, e. g., to quantum
chaos, is considered.
Keywords:
Complex numbers in quantum mechanics; complex logistic equation; non-chaotic behavior; quantum
chaos; quantum entanglement; quantum communication
1 This
work was supported by the JSPS exchange fellowship program and the grant 1ET300100403.
of Computer Science, Academy of Sciences of the Czech Republic, Prague 182 07, Czech Republic
3 Institute of Industrial Science, University of Tokyo, Tokyo, 53-8505, Japan
4 Institute of Industrial Science, University of Tokyo, Tokyo, 53-8505, Japan
2 Institute
Number-free reductions in
logic-based fuzzy mathematics
Libor Běhounek∗
Technical report No. V–1032
December 2008
Abstract:
This report collects preliminary ideas on rendering classical mathematical notions in formal theories
over suitable deductive fuzzy logics in such a way that references to real numbers are eliminated
from their definitions and removed to the semantics of fuzzy logic. The conceptual simplification
of the theory (in exchange for non-classical reasoning) is demonstrated on several examples.
The framework employed for the number-free formalization of mathematical concepts is that of
higher-order fuzzy logic, also known as Fuzzy Class Theory.
Keywords:
Fuzzy mathematics, fuzzy logic, Fuzzy Class Theory, distribution function, continuity, limit,
metric, fuzzy Dedekind cut, similarity.
∗
Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod Vodárenskou věžı́ 2,
182 07 Prague 8, Czech Republic. Email: [email protected]. Supported by grant No. A100300503 of
GA AV ČR and Institutional Research Plan AV0Z10300504.
Classifier Based on Inverted Indexes of Neighbors
November 2008
Abstract
A new method for the classification of data into classes is presented. The method is based on the sum
of reciprocals of neighbors’ indexes. We show that neighbors’ indexes are in close relation to the
polynomial transform of the neighbors’ distances. The sum of the reciprocals of indexes for all
neighbors forms truncated harmonic series due to a finite number of its elements. For the neighbors of
one class there is a sum of the selected elements of this truncated series. It is proved that the ratio of
these sums gives just the probability that the point to be classified – the query point – is of that class.
The classification ability is demonstrated on real-life data from the Machine Learning Repository and the
results are compared with published results obtained through other methods.
Keywords:
estimation, harmonic series, classification.
1
1
Analysis of Decay Processes Separation
Marcel Ji ina and František Hakl
November 2008
Abstract
This report summarizes results obtained on simulated data from ATLAS detector for
Higgs boson search.
Keywords:
multivariate data, Higgs boson, data classification, neural network, genetic
optimization.
Limited-memory projective variable metric
methods for unconstrained minimization
J. Vlček, L. Lukšan1
Technical report No. V 1036
December 2008
Abstract:
A new family of limited-memory variable metric or quasi-Newton methods for unconstrained
minimization is given. The methods are based on a positive definite inverse Hessian approximation in the form of the sum of identity matrix and two low rank matrices, obtained by the
standard scaled Broyden class update. To reduce the rank of matrices, various projections
are used. Numerical experience is encouraging.
Keywords:
Unconstrained minimization, variable metric methods, limited-memory methods, Broyden
class updates, projection matrix, numerical results.
1
This work was supported by the Grant Agency of the Czech Academy of Sciences, project No.
IAA1030405, and the Institutional research plan No. AV0Z10300504, L. Lukšan is also from Technical
University of Liberec, Hálkova 6, 461 17 Liberec.
Transformations enabling to construct
limited-memory Broyden class methods
J. Vlček, L. Lukšan1
Technical report No. V 1037
December 2008
Abstract:
The Broyden class of quasi-Newton updates for inverse Hessian approximation are transformed to the formal BFGS update, which makes possible to generalize the well-known
Nocedal method based on the Strang recurrences to the scaled limited-memory Broyden
family, using the same number of stored vectors as for the limited-memory BFGS method.
Two variants are given, the simpler of them does not require any additional matrix by vector
multiplications. Numerical results indicate that this approach can save computational time.
Keywords:
Unconstrained minimization, variable metric methods, limited-memory methods, Broyden
class updates, numerical results.
1
IAA1030405, and the Institutional research plan No. AV0Z10300504, L. Lukšan is also from Technical
University of Liberec, Hálkova 6, 461 17 Liberec.
Computational experience with modified
conjugate gradient methods for
unconstrained optimization
L.Lukšan, C.Matonoha, J.Vlček
1
December 2008
Abstract:
In this report, several modifications of the nonlinear conjugate gradient method are described
and investigated. Theoretical properties of these modifications are proved and their practical
performance is demonstrated using extensive numerical experiments.
Keywords:
Numerical optimization, conjugate direction methods, conjugate gradient methods, global
convergence, numerical experiments.
1
IAA1030405, the Grant Agency of the Czech Republic, project No. 201/06/P397, and the institutional
research plan No. AV0Z10300504. L. Lukšan is also from Technical University of Liberec, Hálkova 6, 461
17 Liberec.
Interior-point method for nonlinear
programming with complementarity
constraints
L.Lukšan, C.Matonoha, J.Vlček
1
December 2008
Abstract:
In this report, we propose an algorithm for solving nonlinear programming problems with complementarity constraints, which is based on the interior-point approach. Main theoretical results
concern direction determination and step-length selection. We use an exact penalty function
to remove complementarity constraints. Thus a new indefinite linear system is defined with
a tridiagonal low-right submatrix. Inexact solution of this system is obtained iteratively using
indefinitely preconditioned conjugate gradient method. Furthermore, new merit function is defined, which includes barrier, exact penalty, and augmented Lagrangian terms. The algorithm
was implemented in the interactive system for universal functional optimization UFO. Results
of extensive numerical experiments are reported.
Keywords:
Nonlinear programming, complementarity constraints, interior-point methods, indefinite
systems, indefinite preconditioners, preconditioned conjugate gradient method, merit
functions, algorithms, computational experiments.
1
IAA1030405, the Grant Agency of the Czech Republic, project No. 201/06/P397, and the institutional
research plan No. AV0Z10300504 L.Lukšan is also from Technical University of Liberec, Hálkova 6, 461 17
Liberec.
UFO 2008
Interactive System for Universal Functional
Optimization
L.Lukšan, M.Tůma, J.Vlček, N.Ramešová,
M.Šiška, J.Hartman, C.Matonoha 1
December 2008
Abstract:
This report contains a description of the interactive system for universal functional optimization UFO, version
2008.
Keywords:
Numerical optimization, nonlinear programming, nonlinear approximation, algorithms, software systems.
1 This work was supported by the Grant Agency of the Czech Academy of Sciences, project code IAA1030405, L.Lukšan,
and M.Tůma are also from Technical University of Liberec, Hálkova 6, 461 17 Liberec.
Classifier Based on Inverted Indexes of Neighbors II.
- Theory and Appendix
November 2008
Abstract
A theory of a new method for the classification of data into classes is presented. The method is based
on the sum of reciprocals of neighbors’ indexes. We show that neighbors’ indexes are in close relation
to the approximate polynomial transform of the neighbors’ distances. The sum of the reciprocals of
indexes for all neighbors forms truncated harmonic series due to a finite number of its elements. For the
neighbors of one class there is a sum of the selected elements of this truncated series. It is proved that
the ratio of these sums gives just the probability that the point to be classified – the query point – is of
that class.
Keywords:
estimation, harmonic series, classification.
1
Graded properties of binary fuzzy
connectives
Libor Běhounek, Ulrich Bodenhofer, Petr Cintula,
Peter Sarkoci, and Susanne Saminger-Platz1
Technical report No. V–1042
December 2008
Abstract:
Graded properties of binary and unary connectives valued in MTL4 algebras are studied employing the
apparatus of Fuzzy Class Theory (or higher-order fuzzy logic) as a tool for easy derivation of graded
theorems on the connectives. The paper focuses on graded properties of conjunctive connectives such as tnorms, uninorms, aggregation operators, or quasicopulas. The properties studied include graded monotony,
Lipschitz property, commutativity, associativity, unit elements, and dominance.
Keywords:
Connectives, Fuzzy Class Theory, Dominance
1 Libor Běhounek ([email protected]) and Petr Cintula ([email protected]): Institute of Computer Science,
Academy of Sciences of the Czech Republic, Pod Vodárenskou věžı́ 2, 182 07 Prague 8, Czech Republic. Ulrich
Bodenhofer ([email protected]): Institute of Bioinformatics, Johannes Kepler University Linz, 4040 Linz,
Austria. Peter Sarkoci ([email protected]) and Susanne Saminger-Platz ([email protected]): Institute
of Bioinformatics, Johannes Kepler University Linz, 4040 Linz, Austria.
Matematické programovánı́
Ladislav Lukšan
1
Prosinec 2008
Abstrakt:
Tato zpráva obsahuje učebnı́ text pro předmět matematické programovánı́ na fakultě mechatroniky Technické university v Liberci.
Keywords:
Matematické programovánı́, numerická optimalizace, nelineárnı́ aproximace, systémy
nelineárnı́ch rovnic, algoritmy.
1
This work was supported by the Grant Agency of the Czech Academy of Sciences, project code
IAA1030405. L.Lukšan is also from Technical University of Liberec, Hálkova 6, 461 17 Liberec.
Possibilistic Distributions Taking
Values in an Incomplete Lattice
Ivan Kramosil1
December 2008
Abstract:
Since the first presentation of fuzzy sets with non-numerical membership degrees by J. A. Goguen in 1969
(cf, [5]) in the greatest part of works dealing with this subject complete lattices have been considered as
the structures in which these non-numerical fuzzy sets, called also possibility or possibilistic distributions in
[8] and later on, take their values. The reasons for such a choice are to simplify mathematical procedures
when processing fuzzy or possibility degrees, nevertheless, in what follows, we will investigate possibilistic
distributions taking their values in incomplete lattices defined over the systems of all finite subsets of an
infinite basic set. This structure is completed in a nonstandard (non-boolean) way in order to minimize the
resulting ontologically independent inputs into the obtained complete lattice. Some results are introduced
and proved, dealing with complete maxitivity of the corresponding lattice-valued possibilistic measures and
with the convergence of sequences of values of such measures given a converging sequence of subsets for
which these measures are defined (continuity of the measures in question).
Keywords:
Complete lattice, possibilistic distribution
1 Institute of Computer Science AS CR, v.v.i., Pod Vodárenskou věžı́ 2, 182 07 Prague, Czech Republic,
[email protected]
Zpráva o rešení projektu
ME 949 MŠMT ČR v roce 2008
David Coufal1
Prosinec 2008
Abstrakt:
Zpráva referuje výsledky dosažené v roce 2008 v rámci řešení projektu ME 949 MŠMT ČR ”Analýza
negativních vlivů na pozornost řidičů”
Keywords:
Detekce mikrospánků, EEG data, data-mining, klasifikační stromy
1 Ústav
informatiky AV ČR v.v.i., Pod Vodárenskou věží 2, 182 07, Praha 8, E-mail: [email protected]
Zpráva o rešení projektu
1F84B/042/520 MD ČR
v roce 2008
David Coufal1
Prosinec 2008
Abstrakt:
Zpráva referuje výsledky dosažené v roce 2008 v rámci řešení projektu 1F84B/042/520 MD ČR ”Prostředky
pro detekci a prevenci poklesu pozornosti řidičů” jako součást společného projektu MESPIN ”Metody
zvýšení spolehlivosti interakce řidiče s vozidlem”
Keywords:
Detekce mikrospánků, EEG data, data-mining, klasifikační stromy
1 Ústav
informatiky AV ČR v.v.i., Pod Vodárenskou věží 2, 182 07, Praha 8, E-mail: [email protected]
Towards a Process Mediation
Framework for Semantic Web
Services1
Roman Vaculı́n2 , Roman Neruda, Katia Sycara3
Technical report No. 1048, November 2008
Abstract:
The ability to deal with incompatibilities of service requesters and providers is a critical factor for achieving
interoperability in dynamic open environments. We focus on the problem of process mediation of semantically annotated process models of the service requester and the service provider. We propose an Abstract
Process Mediation Framework identifying the key functional areas that need to be addressed by process
mediation components. Next, we present algorithms for solving the process mediation problem in two scenarios: (a) when the mediation process has complete visibility of the process model of the service provider
and the service requester (complete visibility scenario), and (b) when the mediation process has visibility
only of the process model of the service provider but not the service requester (asymmetric scenario). The
algorithms combine the AI planing and semantic reasoning with recovery techniques and the discovery
of appropriate external services such as data mediators. Finally, the Process Mediation Agent (PMA) is
introduced, which realizes an execution infrastructure for runtime mediation.
Keywords:
process mediation, semantic web services, agents and web services, automated planning
1 This research was partially supported by the the Czech Ministry of Education project ME08095 and “Information
Society” project 1ET100300517. This research was supported in part by Darpa contract FA865006C7606 and in part
by funding from France Telecom.
2 Roman Vaculı́n, Roman Neruda – Institute of Computer Science, Academy of Sciences of the Czech Republic
({vaculin,roman}@cs.cas.cz)
3 Katia Sycara – The Robotics Institute, Carnegie Mellon University ([email protected])

Institute of Computer Science Pravdepodobnostn´ı modelován´ı

Transkript

Podobné dokumenty

Dear business partners, We would like to inform

Rehabilitation programme of cerebral palsy patients in RC [jen pro

rh incompatability

czech technical university in prague basics of fuzzy logic and its