scholarly journals Virtual Markov Chains

10.53733/147 ◽  
2021 ◽  
Vol 52 ◽  
pp. 511-559
Author(s):  
Steven Evans ◽  
Adam Jaffe
Keyword(s):  
Markov Chains ◽  
Convex Sets ◽  
Compact Convex ◽  
Initial Distribution ◽  
Projective Limit ◽  
Von Neumann ◽  
Neumann Theorem ◽  
Finite State ◽  
Virtual Setting ◽  
Von Neumann Theorem

We introduce the space of virtual Markov chains (VMCs) as a projective limit of the spaces of all finite state space Markov chains (MCs), in the same way that the space of virtual permutations is the projective limit of the spaces of all permutations of finite sets.We introduce the notions of virtual initial distribution (VID) and a virtual transition matrix (VTM), and we show that the law of any VMC is uniquely characterized by a pair of a VID and VTM which have to satisfy a certain compatibility condition.Lastly, we study various properties of compact convex sets associated to the theory of VMCs, including that the Birkhoff-von Neumann theorem fails in the virtual setting.

scholarly journals When does the Weyl-von Neumann Theorem hold?

2017 ◽  
Vol 49 (4) ◽  
pp. 742-744
Author(s):  
Hiroshi Ando ◽  
Yasumichi Matsuzawa
Keyword(s):  
Von Neumann ◽  
Neumann Theorem ◽  
Von Neumann Theorem

An extension of a convergence theorem for Markov chains arising in population genetics

2016 ◽  
Vol 53 (3) ◽  
pp. 953-956 ◽  
Author(s):  
Martin Möhle ◽  
Morihiro Notohara
Keyword(s):  
Population Genetics ◽  
Markov Chains ◽  
Convergence Theorem ◽  
Transition Matrix ◽  
Initial Distribution ◽  
Generator Matrix ◽  
Finite Dimensional ◽  
Finite State ◽  
Initial Distributions ◽  
Finite State Space

AbstractAn extension of a convergence theorem for sequences of Markov chains is derived. For every positive integer N let (XN(r))r be a Markov chain with the same finite state space S and transition matrix ΠN=I+dNBN, where I is the unit matrix, Q a generator matrix, (BN)N a sequence of matrices, limN℩∞cN= limN→∞dN=0 and limN→∞cN∕dN=0. Suppose that the limits P≔limm→∞(I+dNQ)m and G≔limN→∞PBNP exist. If the sequence of initial distributions PXN(0) converges weakly to some probability measure μ, then the finite-dimensional distributions of (XN([t∕cN))t≥0 converge to those of the Markov process (Xt)t≥0 with initial distribution μ, transition matrix PetG and limN→∞(I+dNQ+cNBN)[t∕cN]

scholarly journals Superunitary Representations of Heisenberg Supergroups

2018 ◽  
Vol 2020 (19) ◽  
pp. 5926-6006 ◽  
Author(s):  
Axel de Goursac ◽  
Jean-Philippe Michel
Keyword(s):  
Representation Theory ◽  
Heisenberg Group ◽  
Fourier Transformation ◽  
Coadjoint Orbits ◽  
New Approach ◽  
Von Neumann ◽  
Neumann Theorem ◽  
Lie Supergroups ◽  
Definition Of ◽  
Von Neumann Theorem

Abstract Numerous Lie supergroups do not admit superunitary representations (SURs) except the trivial one, for example, Heisenberg and orthosymplectic supergroups in mixed signature. To avoid this situation, we introduce in this paper a broader definition of SUR, relying on a new definition of Hilbert superspace. The latter is inspired by the notion of Krein space and was developed initially for noncommutative supergeometry. For Heisenberg supergroups, this new approach yields a smooth generalization, whatever the signature, of the unitary representation theory of the classical Heisenberg group. First, we obtain Schrödinger-like representations by quantizing generic coadjoint orbits. They satisfy the new definition of irreducible SURs and serve as ground to the main result of this paper: a generalized Stone–von Neumann theorem. Then, we obtain the superunitary dual and build a group Fourier transformation, satisfying Parseval theorem. We eventually show that metaplectic representations, which extend Schrödinger-like representations to metaplectic supergroups, also fit into this definition of SURs.

scholarly journals FUNDAMENTAL SYMMETRIES OF THE EXTENDED SPACETIME

2008 ◽  
Vol 23 (08) ◽  
pp. 1266-1269
Author(s):  
ORCHIDEA MARIA LECIAN ◽  
GIOVANNI MONTANI
Keyword(s):  
Uncertainty Relations ◽  
Fundamental Symmetries ◽  
Von Neumann ◽  
Neumann Theorem ◽  
Von Neumann Theorem

On the basis of Fourier duality and Stone-von Neumann theorem, we will examine polymer-quantization techniques and modified uncertainty relations as possible 1-extraD compactification schemes for a phenomenological truncation of the extraD tower.

The Weyl–von Neumann theorem in von Neumann factors

2017 ◽  
Vol 23 (1) ◽  
Author(s):  
Krzysztof Kamiński
Keyword(s):  
Von Neumann ◽  
Neumann Theorem ◽  
Von Neumann Theorem

AbstractIn a von Neumann factor

scholarly journals Coherent state transforms and the Mackey-Stone-Von Neumann theorem

2014 ◽  
Vol 55 (10) ◽  
pp. 102101 ◽  
Author(s):  
William D. Kirwin ◽  
José M. Mourão ◽  
João P. Nunes
Keyword(s):  
Coherent State ◽  
Von Neumann ◽  
Neumann Theorem ◽  
Von Neumann Theorem
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