Definition of stochastic process. Cylinder σ-algebra, finite-dimensional distributions, the Kolmogorov theorem

We show first the definition of variational convergence of unifunctions and their basic variational properties. In the next section, we extend this variational convergence definition in case the functions which are defined on product two sets (bifunctions or bicomponent functions). We present the definition of variational convergence of bifunctions, icluding epi/hypo convergence, minsuplop convergnece and maxinf-lop convergence, defined on metric spaces. Its variational properties are also considered. In this paper, we concern on the properties of epi/hypo convergence to apply these results on optimization proplems in two last sections. Next we move on to the main results that are approximations of typical and important optimization related problems on metric space in terms of the types of variational convergence are equilibrium problems, and multiobjective optimization. When we applied to the finite dimensional case, some of our results improve known one.

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Weyl modules and Weyl functors for hyper-map algebras

Journal of Algebra and Its Applications ◽

10.1142/s0219498821400077 ◽

2020 ◽

pp. 2140007

Author(s):

Angelo Bianchi ◽

Samuel Chamberlin

Keyword(s):

Lie Algebra ◽

Finitely Generated ◽

Weyl Modules ◽

Simple Lie Algebra ◽

Finite Dimensional ◽

Universal Properties ◽

Definition Of ◽

Unitary Algebra

We investigate the representations of the hyperalgebras associated to the map algebras [Formula: see text], where [Formula: see text] is any finite-dimensional complex simple Lie algebra and [Formula: see text] is any associative commutative unitary algebra with a multiplicatively closed basis. We consider the natural definition of the local and global Weyl modules, and the Weyl functor for these algebras. Under certain conditions, we prove that these modules satisfy certain universal properties, and we also give conditions for the local or global Weyl modules to be finite-dimensional or finitely generated, respectively.

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Security of Dependable Systems

Dependability and Computer Engineering - Advances in Computer and Electrical Engineering ◽

10.4018/978-1-60960-747-0.ch011 ◽

2012 ◽

pp. 230-264 ◽

Cited By ~ 1

Author(s):

Naveed Ahmed ◽

Christian Damsgaard Jensen

Keyword(s):

Stochastic Process ◽

Open Problem ◽

Development Process ◽

Traditional View ◽

Clear Definition ◽

Security Attacks ◽

Dependable Systems ◽

Broad Concept ◽

Common Framework ◽

Definition Of

Security and dependability are crucial for designing trustworthy systems. The approach “security as an add-on” is not satisfactory, yet the integration of security in the development process is still an open problem. Especially, a common framework for specifying dependability and security is very much needed. There are many pressing challenges however; here, we address some of them. Firstly, security for dependable systems is a broad concept and traditional view of security, e.g., in terms of confidentiality, integrity and availability, does not suffice. Secondly, a clear definition of security in the dependability context is not agreed upon. Thirdly, security attacks cannot be modeled as a stochastic process, because the adversary’s strategy is often carefully planned. In this chapter, we explore these challenges and provide some directions toward their solutions.

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Fuzzy anti-norm and fuzzy α-anti-convergence

Demonstratio Mathematica ◽

10.1515/dema-2013-0405 ◽

2012 ◽

Vol 45 (4) ◽

Cited By ~ 1

Author(s):

Bivas Dinda ◽

T. K. Samanta ◽

Iqbal H. Jebril

Keyword(s):

Linear Space ◽

Finite Dimensional ◽

Definition Of

AbstractIn this paper the definition of fuzzy antinorm is modified. Some properties of finite dimensional fuzzy antinormed linear space are studied. Fuzzy

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Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic Type

Symmetry ◽

10.3390/sym12101737 ◽

2020 ◽

Vol 12 (10) ◽

pp. 1737

Author(s):

Mariia Myronova ◽

Jiří Patera ◽

Marzena Szajewska

Keyword(s):

Tensor Product ◽

Lie Algebras ◽

Coxeter Groups ◽

Reflection Groups ◽

Geometrical Structures ◽

Simple Lie Algebras ◽

Finite Dimensional ◽

Branching Rules ◽

Definition Of ◽

Representation Orbit

The invariants of finite-dimensional representations of simple Lie algebras, such as even-degree indices and anomaly numbers, are considered in the context of the non-crystallographic finite reflection groups H2, H3 and H4. Using a representation-orbit replacement, the definitions and properties of the indices are formulated for individual orbits of the examined groups. The indices of orders two and four of the tensor product of k orbits are determined. Using the branching rules for the non-crystallographic Coxeter groups, the embedding index is defined similarly to the Dynkin index of a representation. Moreover, since the definition of the indices can be applied to any orbit of non-crystallographic type, the algorithm allowing to search for the orbits of smaller radii contained within any considered one is presented for the Coxeter groups H2 and H3. The geometrical structures of nested polytopes are exemplified.

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Isomorphism of Some Simple 2-Graded Lie Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-1977-031-4 ◽

1977 ◽

Vol 29 (2) ◽

pp. 289-294

Author(s):

Dragomir Ž. Djoković

Keyword(s):

Lie Algebras ◽

Characteristic Zero ◽

Graded Lie Algebras ◽

Finite Dimensional ◽

Definition Of

The grading is by integers modulo 2 and we refer to it as 2-grading. For the definition of 2-graded Lie algebras L and their properties we refer the reader to the papers [1; 2; 3]. All algebras considered here are finite-dimensional over a field F of characteristic zero.

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The definition of a multi-dimensional generalization of shot noise

Journal of Applied Probability ◽

10.1017/s0021900200110988 ◽

1971 ◽

Vol 8 (01) ◽

pp. 128-135 ◽

Cited By ~ 9

Author(s):

D. J. Daley

Keyword(s):

Stochastic Process ◽

Probability Distribution ◽

Absolute Convergence ◽

List Type ◽

Type Number ◽

Homogeneous Poisson Process ◽

Cauchy Principal ◽

Principal Value ◽

Definition Of ◽

Image Position

The paper studies the formally defined stochastic process where {tj } is a homogeneous Poisson process in Euclidean n-space En and the a.e. finite Em -valued function f(·) satisfies |f(t)| = g(t) (all |t | = t), g(t) ↓ 0 for all sufficiently large t → ∞, and with either m = 1, or m = n and f(t)/g(t) =t/t. The convergence of the sum at (*) is shown to depend on (i) (ii) (iii) . Specifically, finiteness of (i) for sufficiently large X implies absolute convergence of (*) almost surely (a.s.); finiteness of (ii) and (iii) implies a.s. convergence of the Cauchy principal value of (*) with the limit of this principal value having a probability distribution independent of t when the limit in (iii) is zero; the finiteness of (ii) alone suffices for the existence of this limiting principal value at t = 0.

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Process Dimension of Classical and Non-Commutative Processes

Open Systems & Information Dynamics ◽

10.1142/s1230161212500072 ◽

2012 ◽

Vol 19 (01) ◽

pp. 1250007

Author(s):

Wolfgang Löhr ◽

Arleta Szkoła ◽

Nihat Ay

Keyword(s):

Stochastic Process ◽

Ergodic Decomposition ◽

Statistical Complexity ◽

Lower Semi ◽

Decomposition Formula ◽

Close Relationship ◽

Natural Characteristic ◽

Causal States ◽

Definition Of ◽

Process Dimension

We treat observable operator models (OOM) and their non-commutative generalisation, which we call NC-OOMs. A natural characteristic of a stochastic process in the context of classical OOM theory is the process dimension. We investigate its properties within the more general formulation, which allows one to consider process dimension as a measure of complexity of non-commutative processes: We prove lower semi-continuity, and derive an ergodic decomposition formula. Further, we obtain results on the close relationship between the canonical OOM and the concept of causal states which underlies the definition of statistical complexity. In particular, the topological statistical complexity, i.e. the logarithm of the number of causal states, turns out to be an upper bound to the logarithm of process dimension.

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