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About 6,115 papers for “Commutator”

Commutator,

Dashuk,P N · Defense Technical Information Center (DTIC) · 1979

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Commutator

A wooden circle with six metal knobs spaced evenly around the perimeter. Used as a switch to record data.

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Commutator and self-commutator approximants II

Philip Maher · Filomat · 2010

We minimize the quantities (i) ||T -(AX-X A) ||, (ii) ||T -(X*X-XX*)|| and (iii) ||T -(AX - X B)|| where T is isometric and where in (i) A is paranormal and commutes with T , in (ii) X*(or X ) is paranormal and commutes with T , and in (iii) A and B are paranormal and AT = T B and T A = BT. The upshot is that these quantities are minimized when 0 = AX - XA = X*X - XX*= AX - XB. To prove these results we obtain the power norm equality for paranormal operators: if A is paranormal then ||An|| = ||A||n if n?N.

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STATIC COMMUTATOR

CROSSMAN,R.E., THOMPSON,J.D., THORPE,C.R. · Defense Technical Information Center (DTIC) · 1961

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Commutator and stable commutator lengths in groups

Korkmaz, Mustafa · OpenMETU (Middle East Technical University) · 2019

Suppose that a and b are odd positive integers.In 2012, T. Miyazaki and A. Togb,[3], gave all of the solutions of the Diophantine equations (2am-1) x +(2m) y = (2am + 1) z and b x + 2 y = (b + 2) z with a > 1 and b 5 in positive integers.In this work, we propose a similar problem (which we call the shuffle variant of a Diophantine equation of Miyazaki and Togb).Here we first prove that the Diophantine equation (2am + 1) x + (2m) y = (2am -1) z has only the solution (a, m, x, y, z) = (2, 1, 2, 1, 3) with any fixed positive integer a > 1 in positive integers.Then using this result, we show that the Diophantine equation b x +2 y = (b-2) z has no solutions with any fixed odd positive integer b 7 in positive integers.On the proofs, we use elementary methods and Laurent's refinement of Baker's theorem.

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Commutator Gymnastics

Barry G. Adams · 1994

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10-COMMUTATOR AND 13-COMMUTATOR A.S. DZHUMADIL’DAEV

2006

Abstract. Skew-symmetric sum of N! compositions of N vector fields in all possible order is called N-commutator. We construct 10-commutator and 13-commutator on V ect(3) and 10-commutator on a space of divergenceless vector fields V ect0(3). We show that 2-commutator, 10-commutator and 13-commutator form final list of N-commutators on V ect(3) and under these polylinear operations V ect(3) has a structure of sh-Lie algebra. We establish that the list of 2-and 10-commutators on V ect0(3) is also final. Constructions are based on calculations of powers of odd derivations. Let (A, ◦) be an algebra with vector space A and multiplication ◦. Let C &lt; t1,..., tk&gt; be a space of non-commutative non-associative polynomials. Any f ∈ C &lt; t1,..., tk&gt; induces a k-ary map f: A × · · · × A → A, k that corresponds to any a1,..., ak ∈ A element f(a1,..., ak) calculated in terms of multiplication ◦. If this map is trivial, i.e., f(a1,..., ak) = 0, for any a1,..., ak ∈ A then f = 0 is said an identity on (A, ◦). If f is polylinear, then f induces a k-ary multiplication on A. For example, if s2 = t1t2 − t2t1 ∈ C &lt; t1, t2&gt;, then s2(a, b) = a ◦ b − b ◦ a is an ordinary commutator.

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Commutator Approximants

Philip Maher · Proceedings of the American Mathematical Society · 1992

This paper deals with minimizing $||B - (AX - XA)|{|_p}$, where $A$ and $B$ are fixed, $B \in {\mathcal {C}_p}$, and $X$ varies such that $AX - XA \in {\mathcal {C}_p}$. (Here, ${\mathcal {C}_p}$ denotes the von Neumann-Schatten class and ${\left \| \right \|_p}$ denotes its norm.) The main result (Theorem 3.2) says that if $A$ is normal and $AB = BA$ then $||B - (AX - XA)|{|_p},1 \leq p < \infty$, is minimized if and for $1 < p < \infty$ only if, $AX - XA = 0$; and that the map $X \to ||B - (AX - XA)||_p^p,1 < p < \infty$, has a critical point at $X = V$ if and only if $AV - VA = 0$.

Cited by 13

Commutator Machines

2025

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Commutator theory

Ralph Freese, Ralph McKenzie, George F. McNulty et al. · Mathematical surveys and monographs · 2022

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Transistorized Commutator

Charles W. McMullen · Review of Scientific Instruments · 1959

Views Icon Views Article contents Figures & tables Video Audio Supplementary Data Peer Review Share Icon Share Twitter Facebook Reddit LinkedIn Tools Icon Tools Reprints and Permissions Cite Icon Cite Search Site Citation Charles W. McMullen; Transistorized Commutator. Rev. Sci. Instrum. 1 July 1959; 30 (7): 595–597. https://doi.org/10.1063/1.1716695 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentAIP Publishing PortfolioReview of Scientific Instruments Search Advanced Search |Citation Search

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Commutator Approximants

Philip Maher · 2017

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Commutator equations

Stefano Fioravanti · International Journal of Algebra and Computation · 2024

In this paper, we investigate properties of varieties of algebras described by a novel concept of equation that we call commutator equation. A commutator equation is a relaxation of the standard term equality obtained substituting the equality relation with the commutator relation. Namely, an algebra [Formula: see text] satisfies the commutator equation [Formula: see text] if for each congruence [Formula: see text] in [Formula: see text] and for each substitution [Formula: see text] of elements in the same [Formula: see text]-class, we have [Formula: see text]. This notion of equation draws inspiration from the definition of a weak difference term and allows for further generalization of it. Furthermore, we present an algorithm that establishes a connection between congruence equations valid in the variety generated by the abelian algebras of the idempotent reduct of a given variety and congruence equations that hold in the entire variety. Additionally, we provide a proof that if the variety generated by the abelian algebras of the idempotent reduct of a variety satisfies a nontrivial idempotent Mal’cev condition, then also the entire variety satisfies a nontrivial idempotent Mal’cev condition, a statement that follows also from [ 12 , Theorem 3.13].

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Commutator table.

Akhtar Hussain (83444), F. D. Zaman (18954725), Saud Owyed (18954728) et al. · Figshare · 2024

<div><p>In this research, we employ the potent technique of Lie group analysis to derive analytical solutions for the (3+1)-extended Kadomtsev-Petviashvili (3D-EKP) equation. The systematic application of this method enables the identification of Lie point symmetries associated with the equation, leading to the derivation of an optimal system of one-dimensional subalgebras relevant to the equation. This optimal system is utilized to obtain several invariant solutions. The Lie group method is subsequently applied to the reduced governing equations derived from the given equation. We complement our findings with Mathematica simulations illustrating some of the obtained solutions. Furthermore, a direct approach is used to investigate local conservation laws. Importantly, our study addresses a gap in the exploration of the 3D-EXP equation using group theoretic methods, making our findings novel in this context.</p></div>

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Commutator table.

Akhtar Hussain (83444), M. Usman (17542500), Fiazuddin Zaman (20501274) et al. · Figshare · 2025

<div><p>The Lie group method is a powerful technique for obtaining analytical solutions for various nonlinear differential equations. This study aimed to explore the behavior of nonlinear elastic wave equations and their underlying physical properties using Lie group invariants. We derived eight-dimensional symmetry algebra for the (3+1)-dimensional nonlinear elastic wave equation, which was used to obtain the optimal system. Group-invariant solutions were obtained using this optimal system. The same analysis was conducted for the damped version of this equation. For the conservation laws, we applied Noether’s theorem to the nonlinear elastic wave equations owing to the availability of a classical Lagrangian. However, for the damped version, we cannot obtain a classical Lagrangian, which makes Noether’s theorem inapplicable. Instead, we used an extended approach based on the concept of a partial Lagrangian to uncover conservation laws. Both techniques account for the conservation laws of linear momentum and energy within the model. These novel approaches add an application of variational calculus to the existing literature. This offers valuable insights and potential avenues for further exploration of the elastic wave equations.</p></div>

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commutator free

2008

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Commutator rings ∗

Zachary Mesyan · 2006

A ring is called a commutator ring if every element is a sum of additive commutators. In this note we give examples of such rings. In particular, we show that given any ring R, a right R-module N, and a nonempty set Ω, EndR ( ⊕

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Commutator construction,

Baxter, William, Jr. · The Journal of the Abraham Lincoln Association

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Commutator approximants

Philip Maher · Proceedings of the American Mathematical Society · 1992

This paper deals with minimizing <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue upper B minus left-parenthesis upper A upper X minus upper X upper A right-parenthesis StartAbsoluteValue EndAbsoluteValue Subscript p"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>B</mml:mi> <mml:mo>−</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mi>X</mml:mi> <mml:mo>−</mml:mo> <mml:mi>X</mml:mi> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">||B - (AX - XA)|{|_p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are fixed, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B element-of script upper C Subscript p"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">B \in {\mathcal {C}_p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> varies such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A upper X minus upper X upper A element-of script upper C Subscript p"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>X</mml:mi> <mml:mo>−</mml:mo> <mml:mi>X</mml:mi> <mml:mi>A</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">AX - XA \in {\mathcal {C}_p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. (Here, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C Subscript p"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathcal {C}_p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denotes the von Neumann-Schatten class and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar double-vertical-bar Subscript p"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow> <mml:mo symmetric="true">‖</mml:mo> <mml:mo symmetric="true">‖</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\left \| \right \|_p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denotes its norm.) The main result (Theorem 3.2) says that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is normal and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A upper B equals upper B upper A"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>B</mml:mi> <mml:mo>=</mml:mo> <mml:mi>B</mml:mi> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">AB = BA</mml:annotation> </mml:semantics> </mml:math> </inline-formula> then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue upper B minus left-parenthesis upper A upper X minus upper X upper A right-parenthesis StartAbsoluteValue EndAbsoluteValue Subscript p Baseline comma 1 less-than-or-equal-to p greater-than normal infinity"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>B</mml:mi> <mml:mo>−</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mi>X</mml:mi> <mml:mo>−</mml:mo> <mml:mi>X</mml:mi> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">||B - (AX - XA)|{|_p},1 \leq p > \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, is minimized if and for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 greater-than p greater-than normal infinity"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>></mml:mo> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">1 > p > \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> only if, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A upper X minus upper X upper A equals 0"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>X</mml:mi> <mml:mo>−</mml:mo> <mml:mi>X</mml:mi> <mml:mi>A</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">AX - XA = 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>; and that the map <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X right-arrow StartAbsoluteValue EndAbsoluteValue upper B minus left-parenthesis upper A upper X minus upper X upper A right-parenthesis StartAbsoluteValue EndAbsoluteValue Subscript p Superscript p Baseline comma 1 greater-than p greater-than normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>B</mml:mi> <mml:mo>−</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mi>X</mml:mi> <mml:mo>−</mml:mo> <mml:mi>X</mml:mi> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> <mml:mi>p</mml:mi> </mml:msubsup> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>></mml:mo> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">X \to ||B - (AX - XA)||_p^p,1 > p > \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, has a critical point at <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X equals up

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Commutator rings

Zachary Mesyan · 2008

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