scholarly journals Non-stable K1-functors of Multiloop Groups

2016 ◽  
Vol 68 (1) ◽  
pp. 150-178 ◽  
Author(s):  
Anastasia Stavrova
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Reductive Group ◽  
Two Dimensional ◽  
Laurent Polynomials ◽  
Full Automorphism Group ◽  
The Difference ◽  
Split Torus ◽  
Whitehead Groups

AbstractLet k be a field of characteristic 0. Let G be a reductive group over the ring of Laurent polynomials . Assume that G contains amaximal R-torus, and that every semisimple normal subgroup of G contains a two-dimensional split torus G2m. We show that the natural map of non-stable K1-functors, also called Whitehead groups, KG1(R) → KG1 ( k((x1)) … ((xn))) is injective, and an isomorphism if G is semisimple. As an application, we provide a way to compute the difference between the full automorphism group of a Lie torus (in the sense of Yoshii–Neher) and the subgroup generated by exponential automorphisms.

Extended two-dimensional belief function based on divergence measurement

2020 ◽  
pp. 1-8
Author(s):  
Jianping Fan ◽  
Jing Wang ◽  
Meiqin Wu
Keyword(s):  
Belief Function ◽  
Distribution Functions ◽  
Divergence Measure ◽  
Uncertain Information ◽  
Belief Functions ◽  
Two Dimensional ◽  
Probability Distribution Functions ◽  
Basic Probability ◽  
The Difference ◽  
Supporting Degree

The two-dimensional belief function (TDBF = (mA, mB)) uses a pair of ordered basic probability distribution functions to describe and process uncertain information. Among them, mB includes support degree, non-support degree and reliability unmeasured degree of mA. So it is more abundant and reasonable than the traditional discount coefficient and expresses the evaluation value of experts. However, only considering that the expert’s assessment is single and one-sided, we also need to consider the influence between the belief function itself. The difference in belief function can measure the difference between two belief functions, based on which the supporting degree, non-supporting degree and unmeasured degree of reliability of the evidence are calculated. Based on the divergence measure of belief function, this paper proposes an extended two-dimensional belief function, which can solve some evidence conflict problems and is more objective and better solve a class of problems that TDBF cannot handle. Finally, numerical examples illustrate its effectiveness and rationality.

Free subgroups in k(x 1,... ,x n )(X;σ) and k(x,y)(k;σ)

2019 ◽  
Vol 31 (3) ◽  
pp. 769-777
Author(s):  
Jairo Z. Gonçalves
Keyword(s):  
Normal Subgroup ◽  
Polynomial Ring ◽  
Multiplicative Group ◽  
Free Subgroup ◽  
Laurent Polynomials ◽  
Skew Polynomial Ring ◽  
Ring Of Fractions ◽  
Skew Polynomial ◽  
Free Subgroups ◽  
Mild Restriction

Abstract Let k be a field, let {\mathfrak{A}_{1}} be the k-algebra {k[x_{1}^{\pm 1},\dots,x_{s}^{\pm 1}]} of Laurent polynomials in {x_{1},\dots,x_{s}} , and let {\mathfrak{A}_{2}} be the k-algebra {k[x,y]} of polynomials in the commutative indeterminates x and y. Let {\sigma_{1}} be the monomial k-automorphism of {\mathfrak{A}_{1}} given by {A=(a_{i,j})\in GL_{s}(\mathbb{Z})} and {\sigma_{1}(x_{i})=\prod_{j=1}^{s}x_{j}^{a_{i,j}}} , {1\leq i\leq s} , and let {\sigma_{2}\in{\mathrm{Aut}}_{k}(k[x,y])} . Let {D_{i}} , {1\leq i\leq 2} , be the ring of fractions of the skew polynomial ring {\mathfrak{A}_{i}[X;\sigma_{i}]} , and let {D_{i}^{\bullet}} be its multiplicative group. Under a mild restriction for {D_{1}} , and in general for {D_{2}} , we show that {D_{i}^{\bullet}} , {1\leq i\leq 2} , contains a free subgroup. If {i=1} and {s=2} , we show that a noncentral normal subgroup N of {D_{1}^{\bullet}} contains a free subgroup.

scholarly journals Colored HOMFLY-PT for hybrid weaving knot $$ {\hat{\mathrm{W}}}_3 $$(m, n)

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Vivek Kumar Singh ◽  
Rama Mishra ◽  
P. Ramadevi
Keyword(s):  
Closed Form ◽  
Topological Strings ◽  
Chebyshev Polynomials ◽  
Fibonacci Numbers ◽  
Infinite Family ◽  
Closed Form Expression ◽  
Two Dimensional ◽  
Laurent Polynomials ◽  
Form Expression ◽  
Torus Knots

Abstract Weaving knots W(p, n) of type (p, n) denote an infinite family of hyperbolic knots which have not been addressed by the knot theorists as yet. Unlike the well known (p, n) torus knots, we do not have a closed-form expression for HOMFLY-PT and the colored HOMFLY-PT for W(p, n). In this paper, we confine to a hybrid generalization of W(3, n) which we denote as $$ {\hat{W}}_3 $$ W ̂ 3 (m, n) and obtain closed form expression for HOMFLY-PT using the Reshitikhin and Turaev method involving $$ \mathrm{\mathcal{R}} $$ ℛ -matrices. Further, we also compute [r]-colored HOMFLY-PT for W(3, n). Surprisingly, we observe that trace of the product of two dimensional $$ \hat{\mathrm{\mathcal{R}}} $$ ℛ ̂ -matrices can be written in terms of infinite family of Laurent polynomials $$ {\mathcal{V}}_{n,t}\left[q\right] $$ V n , t q whose absolute coefficients has interesting relation to the Fibonacci numbers $$ {\mathrm{\mathcal{F}}}_n $$ ℱ n . We also computed reformulated invariants and the BPS integers in the context of topological strings. From our analysis, we propose that certain refined BPS integers for weaving knot W(3, n) can be explicitly derived from the coefficients of Chebyshev polynomials of first kind.

Theory of ultrasonic attenuation in one and two dimensional metals

1976 ◽  
Vol 54 (14) ◽  
pp. 1454-1460 ◽  
Author(s):  
T. Tiedje ◽  
R. R. Haering
Keyword(s):  
Ultrasonic Attenuation ◽  
Three Dimensional ◽  
Electronic Systems ◽  
Two Dimensional ◽  
Temperature Dependent ◽  
One Dimensional ◽  
The Difference

The theory of ultrasonic attenuation in metals is extended so that it applies to quasi one and two dimensional electronic systems. It is shown that the attenuation in such systems differs significantly from the well-known results for three dimensional systems. The difference is particularly marked for one dimensional systems, for which the attenuation is shown to be strongly temperature dependent.

NUMERICAL SOLUTION OF TWO-DIMENSIONAL PROBLEMS OF THERMOVISCOELASTICITY

2021 ◽  
Vol 335 (1) ◽  
pp. 60-64
Author(s):  
M. Bukenov ◽  
Ye. Mukhametov
Keyword(s):  
A Priori ◽  
Elastic Collision ◽  
Qualitative Behavior ◽  
Stress Profile ◽  
Two Dimensional ◽  
A Priori Information ◽  
Deformable Solids ◽  
Methods Of Calculation ◽  
The Difference ◽  
Priori Information

This paper considers the numerical implementation of two-dimensional thermoviscoelastic waves. The elastic collision of an aluminum cylinder with a two-layer plate of aluminum and iron is considered. In work [1] the difference schemes and algorithm of their realization are given. The most complete reviews of the main methods of calculation of transients in deformable solids can be found in [2, 3, 4], which also indicates the need and importance of generalized studies on the comparative evaluation of different methods and identification of the areas of their most rational application. In the analysis and physical interpretation of numerical results in this work it is also useful to use a priori information about the qualitative behavior of the solution and all kinds of information about the physics of the phenomena under study. Here is the stage of evolution of contact resistance of collision – plate, stress profile.

On Automorphisms and Derivations of a Lie Algebra

2006 ◽  
Vol 13 (01) ◽  
pp. 119-132 ◽  
Author(s):  
V. R. Varea ◽  
J. J. Varea
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Large Class ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Dimension ◽  
Nilpotent Lie Algebra ◽  
Identity Component ◽  
Trivial Center

We study automorphisms and derivations of a Lie algebra L of finite dimension satisfying certain centrality conditions. As a consequence, we obtain that every nilpotent normal subgroup of the automorphism group of L is unipotent for a very large class of Lie algebras. This result extends one of Leger and Luks. We show that the automorphism group of a nilpotent Lie algebra can have trivial center and have yet a unipotent identity component.

Perception of Risk as Related to Choice in a Two-Dimensional Risk Situation

1978 ◽  
Vol 43 (3_suppl) ◽  
pp. 1059-1062 ◽  
Author(s):  
John W. Dickson
Keyword(s):  
Risky Choice ◽  
Expected Value ◽  
Two Dimensions ◽  
Choice Situation ◽  
Two Dimensional ◽  
Objective Risk ◽  
Subjective Risk ◽  
Risk Situation ◽  
The Difference ◽  
Two Alternatives

A risky choice was created by manipulating two dimensions of risk for 21 managers attending a conference. The first dimension varied risk by altering the difference in expected value between two alternatives of widely differing variance. The second dimension varied the expectancy of achieving a particular outcome. Whereas choice was significantly related to both dimensions of risk, it was not significantly related to estimates of the subjective risk inherent in the choice situation. It appears that subjective risk does not mediate between objective risk and choice.

scholarly journals The paradox of constant oceanic plastic debris: evidence for evolved microbial biodegradation?

2017 ◽  
Author(s):  
Ricard Solé ◽  
Ernest Fontich ◽  
Blai Vidiella ◽  
Salva Duran-Nebreda ◽  
Raúl Montañez ◽  
...  
Keyword(s):  
North Atlantic ◽  
Nonlinear Coupling ◽  
Global Studies ◽  
Two Dimensional ◽  
Plastic Debris ◽  
The North ◽  
Microbial Biodegradation ◽  
The North Atlantic ◽  
The Difference ◽  
Microbial Model

Although the presence of vast amounts of plastic in the open ocean has generated great concern due to its potential ecological consequences, recent studies reveal that its measured abundance is much smaller than expected. Regional and global studies indicate that the difference between expected and actual estimates is enormous, suggesting that a large part of the plastic has been degraded by either physical and biotic processes. A paradoxical observation is the lack of a trend in plastic accumulation found in the North Atlantic Subtropical Gyre, despite the rapid increase in plastic production and disposal. In this paper we show, using mathematical and computer models, that this observation could be explained by the nonlinear coupling between plastic (as a resource) and an evolved set of organisms (the consumers) capable of degrading it. The result is derived using two different resource-consumer mathematical approaches as well as a spatially-dependent plastic-microbial model incorporating a minimal hydrodynamical coupling with a two-dimensional fluid. The potential consequences of the evolution of marine plastic garbage and its removal are outlined.

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