scholarly journals Product set phenomena for measured groups

2017 ◽  
Vol 38 (8) ◽  
pp. 2913-2941 ◽  
Author(s):  
MICHAEL BJÖRKLUND
Keyword(s):  
Free Group ◽  
Finite Rank ◽  
The Other ◽  
Independent Interest ◽  
Large Sets ◽  
Finite Set ◽  
Piecewise Syndetic ◽  
Product Sets ◽  
Special Case ◽  
Theoretical Results

We strengthen and extend in this paper some recent results by Di Nasso, Goldbring, Jin, Leth, Lupini and Mahlburg on piecewise syndeticity of product sets in countable amenable groups to general countable measured groups. We also address several fundamental differences between the behavior of products of ‘large’ sets in Liouville and non-Liouville measured groups. As a (very) special case of our main results, we show that if $G$ is a free group of finite rank, and $A$ and $B$ are ‘spherically large’ subsets of $G$, then there exists a finite set $F\subset G$ such that $AFB$ is thick. The position of the set $F$ is curious, but seems to be necessary; in fact, we can produce left thick sets $A,B\subset G$ such that $B$ is ‘spherically large’, but $AB$ is not piecewise syndetic. On the other hand, if $A$ is spherically large, then $AA^{-1}$ is always piecewise syndetic and piecewise left syndetic. However, contrary to what happens for amenable groups, $AA^{-1}$ may fail to be syndetic. The same phenomena occur for many other (even amenable, but non-Liouville) measured groups. Our proofs are based on some ergodic-theoretical results concerning stationary actions which should be of independent interest.

AUTOMORPHISMS OF RELATIVELY FREE GROUPS IN THE VARIETY N2A ∧ AN2 ∧ Nc

2001 ◽  
Vol 63 (3) ◽  
pp. 607-622 ◽  
Author(s):  
ATHANASSIOS I. PAPISTAS
Keyword(s):  
Automorphism Group ◽  
Free Group ◽  
Finite Index ◽  
Finite Rank ◽  
Free Groups ◽  
Finite Set ◽  
Tame Automorphism ◽  
Positive Integers ◽  
Relatively Free Group

For positive integers n and c, with n [ges ] 2, let Gn, c be a relatively free group of finite rank n in the variety N2A ∧ AN2 ∧ Nc. It is shown that the subgroup of the automorphism group Aut(Gn, c) of Gn, c generated by the tame automorphisms and an explicitly described finite set of IA-automorphisms of Gn, c has finite index in Aut(Gn, c). Furthermore, it is proved that there are no non-trivial elements of Gn, c fixed by every tame automorphism of Gn, c.

COMMUTATOR SUBGROUPS OF FREE NILPOTENT GROUPS

2007 ◽  
Vol 17 (05n06) ◽  
pp. 1021-1031
Author(s):  
N. GUPTA ◽  
I. B. S. PASSI
Keyword(s):  
Nilpotent Group ◽  
Free Group ◽  
Finite Rank ◽  
Free Groups ◽  
Nilpotent Groups ◽  
Finite Chain ◽  
Free Nilpotent Group ◽  
Infinite Rank ◽  
Derived Subgroup ◽  
Finite Set

For fixed m, n ≥ 2, we examine the structure of the nth lower central subgroup γn(F) of the free group F of rank m with respect to a certain finite chain F = F(0) > F(1) > ⋯ > F(l-1) > F(l) = {1} of free groups in which F(k) is of finite rank m(k) and is contained in the kth derived subgroup δk(F) of F. The derived subgroups δk(F/γn(F)) of the free nilpotent group F/γn(F) are isomorphic to the quotients F(k)/(F(k) ∩ γn(F)) and admit presentations of the form 〈xk,1,…,xk,m(k): γ(n)(F(k))〉, where γ(n)(F(k)), contained in γn(F), is a certain partial lower central subgroup of F(k). We give a complete description of γn(F) as a staggered product Π1 ≤ k ≤ l-1(γ〈n〉(F(k))*γ[n](F(k)))F(k+1), where γ〈n〉(F(k)) is a free factor of the derived subgroup [F(k),F(k)] of F(k) having countable infinite rank and generated by a certain set of reduced commutators of weight at least n, and γ[n](F(k)) is the subgroup generated by a certain finite set of products of non-reduced ordered commutators of weight at least n. There are some far-reaching consequences.

Building a consensus: A rectangle covering problem

2011 ◽  
Vol 21 (2) ◽  
pp. 119-128
Author(s):  
RICHARD S. BIRD
Keyword(s):  
Real Line ◽  
Type Inference ◽  
The Other ◽  
Covering Problem ◽  
Inference Algorithm ◽  
The Real ◽  
Crucial Component ◽  
Finite Set ◽  
Special Case

The other day, over a very pleasant lunch in the restaurant of Oxford's recently renovated Ashmolean Museum, Oege de Moor gave me a problem about rectangles. The problem is explained more fully later, but roughly speaking one is given a finite set of rectangles RS and a rectangle R completely covered by RS. The task is to construct a single rectangle covering R among the elements of a larger set of rectangles associated with RS, called the saturation of RS. The saturation of RS is the closure of RS under so-called consensus operations, a term coined in (Quine, 1959), in which two rectangles are combined in two distinct ways to form new rectangles. The rectangle problem is a simplified version of containment-checking, a crucial component in a type inference algorithm for Datalog programs (Schäfer & de Moor, 2010). 19 In the Schäfer-de Moor algorithm the problem is generalised to cubes in n-space rather than rectangles in two-space, the components of each cube are given by propositional formulae rather than by intervals on the real line, and certain equality and inhabitation constraints are taken into account. Oege felt that the central proof, Lemma 15 in (Schäfer & de Moor, 2010), deserved to be simplified so he posed the rectangle problem as a special case. This pearl was composed in response to the challenge.

Behaviour of a Cylindrical Shell Subject to Sinusoidal and Shock Couples

1965 ◽  
Vol 7 (3) ◽  
pp. 283-291 ◽  
Author(s):  
J. L. Hedges ◽  
B. Mills
Keyword(s):  
Cylindrical Shell ◽  
Experimental Verification ◽  
Horizontal Plane ◽  
The Other ◽  
Thin Cylindrical Shell ◽  
Shock Excitation ◽  
Support Points ◽  
Shell Diameter ◽  
Special Case ◽  
Theoretical Results

The response of a thin cylindrical shell subject to sinusoidal and shock excitation is predicted using the Rayleigh theory of inextensibility. In the theoretical analysitwo separate cases are considered; in the first the loads are applied at each end of the shell while in the second the loads are applied at one end only. In each case the position of the loads is perfectly general. Experimental verification is provided for the special case of the load and support points being in the horizontal plane through a shell diameter; one end of the shell being supported while the other end is subjected to sinusoidal and shock excitation. The correlation between the experimental and theoretical results indicated that the response of the shell could be predicted from the theory of inextensibility.

Unified Inverse Dynamics of Variable Topologies of a Metamorphic Parallel Mechanism Using Screw Theory

2013 ◽  
Author(s):  
Dongming Gan ◽  
Jorge Dias ◽  
Lakmal D. Seneviratne
Keyword(s):  
Inverse Kinematics ◽  
Parallel Mechanism ◽  
Inverse Dynamics ◽  
Screw Theory ◽  
Parallel Robot ◽  
The Other ◽  
Dynamics Modeling ◽  
Two Phases ◽  
Special Case ◽  
Theoretical Results

This paper presents the topology variation of a 3(rT)PS metamorphic parallel mechanism which can change its mobility from 3 to 6. The reconfiguration stems from a reconfigurable (rT)PS limb of which the two phases can be unified by taking one as a special case of the other. Based on this, unified inverse kinematics is solved and a unified dynamics modeling is built using screw theory which naturally represents the geometric constraint and actuation forces in the same manner. The obtained modeling covers all the topologies of the parallel mechanism. A numerical example demonstrates the theoretical results which provide basis for this metamorphic parallel robot with applications in reconfiguration-required environment.

scholarly journals Jordan products of quantum channels and their compatibility

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Mark Girard ◽  
Martin Plávala ◽  
Jamie Sikora
Keyword(s):  
Quantum State ◽  
Sufficient Conditions ◽  
The Other ◽  
Independent Interest ◽  
Quantum Channels ◽  
Semidefinite Programs ◽  
Jordan Product ◽  
Marginal Problem ◽  
Jordan Products ◽  
Product Of Matrices

AbstractGiven two quantum channels, we examine the task of determining whether they are compatible—meaning that one can perform both channels simultaneously but, in the future, choose exactly one channel whose output is desired (while forfeiting the output of the other channel). Here, we present several results concerning this task. First, we show it is equivalent to the quantum state marginal problem, i.e., every quantum state marginal problem can be recast as the compatibility of two channels, and vice versa. Second, we show that compatible measure-and-prepare channels (i.e., entanglement-breaking channels) do not necessarily have a measure-and-prepare compatibilizing channel. Third, we extend the notion of the Jordan product of matrices to quantum channels and present sufficient conditions for channel compatibility. These Jordan products and their generalizations might be of independent interest. Last, we formulate the different notions of compatibility as semidefinite programs and numerically test when families of partially dephasing-depolarizing channels are compatible.

On the kinetics of Hadamard-type fractional differential systems

2020 ◽  
Vol 23 (2) ◽  
pp. 553-570 ◽  
Author(s):  
Li Ma
Keyword(s):  
Differential Operator ◽  
Numerical Simulations ◽  
Type Inequality ◽  
The Other ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
Kinetics Of ◽  
Theoretical Results

AbstractThis paper is devoted to the investigation of the kinetics of Hadamard-type fractional differential systems (HTFDSs) in two aspects. On one hand, the nonexistence of non-trivial periodic solutions for general HTFDSs, which are considered in some functional spaces, is proved and the corresponding eigenfunction of Hadamard-type fractional differential operator is also discussed. On the other hand, by the generalized Gronwall-type inequality, we estimate the bound of the Lyapunov exponents for HTFDSs. In addition, numerical simulations are addressed to verify the obtained theoretical results.

scholarly journals The game of Double-Silver on intervals

2001 ◽  
Vol 26 (8) ◽  
pp. 485-496 ◽  
Author(s):  
Gerald A. Heuer
Keyword(s):  
Measure Zero ◽  
Optimal Strategies ◽  
The Other ◽  
Parameter Plane ◽  
Left Endpoint ◽  
Relative Placement ◽  
Special Case

Silverman's game on intervals was analyzed in a special case by Evans, and later more extensively by Heuer and Leopold-Wildburger, who found that optimal strategies exist (and gave them) quite generally when the intervals have no endpoints in common. They exist in about half the parameter plane when the intervals have a left endpoint or a right endpoint, but not both, in common, and (as Evans had earlier found) exist only on a set of measure zero in this plane if the intervals are identical. The game of Double-Silver, where each player has its own threshold and penalty, is examined. There are several combinations of conditions on relative placement of the intervals, the thresholds and penalties under which optimal strategies exist and are found. The indications are that in the other cases no optimal strategies exist.

Silent Victims Revisited: The Special Case of Domestic Violence

PEDIATRICS ◽  
1995 ◽  
Vol 96 (3) ◽  
pp. 511-513
Author(s):  
Barry Zuckerman ◽  
Marilyn Augustyn ◽  
Betsy McAlister Groves ◽  
Steven Parker
Keyword(s):  
Domestic Violence ◽  
Community Violence ◽  
Clinical Experience ◽  
Behavioral Problems ◽  
The Other ◽  
Media Attention ◽  
Witness Violence ◽  
Primary Focus ◽  
Special Case ◽  
At Home

In a commentary published previously, we communicated our concern regarding the plight of children who witness violence.1 Research suggests that children who witness violence suffer significant psychologic and behavioral problems that interfere with their ability to function in school, at home, and with peers. The primary focus of that commentary was children who witnessed community violence. Our ongoing clinical experience, heightened by media attention on domestic violence, including the O.J. Simpson case, leads us to revisit silent victims with a sole focus on those children who witness domestic violence. Domestic violence is a particularly devastating event for a child who, in the presence of danger, typically turns to a parent for protection and for whom there is no comfort or security if one parent is the perpetrator of violence, and the other is a terrified victim.

The Uniform Distribution of a Fluid Flowing Through a Perforated Pipe

1950 ◽  
Vol 17 (4) ◽  
pp. 431-438
Author(s):  
Willard M. Dow
Keyword(s):  
Fluid Flow ◽  
High Capacity ◽  
Practical Significance ◽  
Linear Rate ◽  
Gaseous Fuels ◽  
Extended Range ◽  
Flow Through ◽  
Perforated Pipe ◽  
Special Case ◽  
Theoretical Results

Abstract A theoretical analysis is made of the flow through a perforated pipe with a closed end for the special case of a constant linear rate of discharge along the length of the pipe. The results of the fluid-flow considerations are applicable to many practical manifold systems. The practical significance of the results with respect to pipe burners for gaseous fuels is emphasized as the results make possible the design of simple high-capacity and extended-range pipe burners of industrial importance. The capacity of commercially available pipe burners may be increased several hundred per cent. The validity of the theoretical results was verified by experiment.

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