On the superfield effective potential in three dimensions

We extend the periodicity of birational rowmotion for rectangular posets to the case when the base field is replaced by a noncommutative ring (under appropriate conditions). This resolves a conjecture from 2014. The proof uses a novel approach and is fully self-contained. Consider labelings of a finite poset \(P\) by \(\left\vert P\right\vert + 2\) elements of a ring \(\mathbb{K}\): one label associated with each poset element and two constant labels for the added top and bottom elements in \(\widehat{P}\). Birational rowmotion is a partial map on such labelings. It was originally defined by Einstein and Propp for \(\mathbb{K}=\mathbb{R}\) as a lifting (via detropicalization) of piecewise-linear rowmotion, a map on the order polytope \(\mathcal{O}(P) := \{\text{order-preserving } f: P \to[0,1]\}\). The latter, in turn, extends the well-studied rowmotion map on the set of order ideals (or more properly, the set of order filters) of \(P\), which correspond to the vertices of \(\mathcal{O}(P)\). Dynamical properties of these combinatorial maps sometimes (but not always) extend to the birational level, while results proven at the birational level always imply their combinatorial counterparts. Allowing \(\mathbb{K}\) to be noncommutative, we generalize the birational level even further, and some properties are in fact lost at this step.In 2014, the authors gave the first proof of periodicity for birational rowmotion on rectangular posets (when \(P\) is a product of two chains) for \(\mathbb{K}\) a field, and conjectured that it survives (in an appropriately twisted form) in the noncommutative case. In this paper, we prove this noncommutative periodicity and a concomitant antipodal reciprocity formula. We end with some conjectures about periodicity for other posets, and the question of whether our results can be extended to (noncommutative) semirings.It has been observed by Glick and Grinberg that, in the commutative case, periodicity of birational rowmotion can be used to derive Zamolodchikov periodicity in the type \(AA\) case, and vice-versa. However, for noncommutative \(\mathbb{K}\), Zamolodchikov periodicity fails even in small examples (no matter what order the factors are multiplied), while noncommutative birational rowmotion continues to exhibit periodicity. Thus, our result can be viewed as a lateral generalization of Zamolodchikov periodicity to the noncommutative setting.Mathematics Subject Classifications: 06A07, 05E99Keywords: Rowmotion, posets, noncommutative rings, semirings, Zamolodchikov periodicity, root systems, promotion, trees, graded posets, Grassmannian, tropicalization

The spin-one Duffin-Kemmer-Petiau oscillator in uniform magnetic field is studied in noncommutative formalism. The corresponding energy is obtained and thereby the corresponding thermal properties are obtained for both commutative and noncommutative cases.

Since the late sixties, various Auslander regularity conditions have been widely investigated in both commutative and noncommutative cases, e.g., [Ba2], [FGR], [Roos], [Lev], etc .... In particular, the dimension and multiplicity theory for modules, the pure (holonomic) module theory over noncommutative Auslander regular rings have been developed in the study of rings of differential operators and enveloping algebras, e.g., [Ka], [Be], [B-B], [K-T], etc ... A typical result concerning purity is the so-called Gabber-Kashiwara theorem (see [Gi] Theorem V8) which has been used to: (1). determine the existence of holonomic submodules in D-modules and the existence of a b-function; (2). determine the equidimensionality of the characteristic variety of a simple module over the enveloping algebra of a Lie algebra. In this chapter we first present a general study of Auslander regularity of filtered rings and graded rings; Then we generalize the important Gabber-Kashiwara theorem and Roos theorem to Zariski rings with commutative associated graded rings including most of the examples given in Ch.II. §2; and consequently the pure module theory over Zariski ring is developed as well.KeywordsExact SequenceSpectral SequenceNoetherian RingRegular RingProjective ResolutionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This chapter is an overview of the arguments relating symmetry, experimental data, and inference that appear in the theory, methods, and applications of dihedral analysis. The example introduced here is based on the only commutative case (n = 2) and yet it includes all the basic elements present in the non-commutative (n > 2) cases.

1. This paper contains a generalization in the commutative case of the structure theorem of W. Ambrose for H*-algebras. An immediate result of Ambrose's axioms is that the orthogonal complement of an ideal is an ideal of the same nature. This fact suggests thinking in terms of a Banach algebra whose conjugate space is a Banach algebra such that the annihilator of an ideal in one algebra is an ideal in the other algebra. This is the essential idea in the definition of a commutative, semi-simple, reflexive Banach algebra given in ?2. This section shows that such an algebra is essentially an 1,-space in which a product has been defined by means of co-ordinate multiplication. ?3 contains a few remarks and examples concerning the author's conjecture that a similar structure theorem must exist in the noncommutative case. The term annihilator will be constantly needed in both its ring and vector space sense. The phrase ring annihilator will be used to distinguish the use of the term in the ring sense from its use in the vector space sense. B' will be used to denote the conjugate space of the given algebra B. Greek letters e, 8, and p will be used to denote complex numbers. Elements in B will be denoted by letters x, y, and z, while elements in B' will be denoted by f, g, and h. The value of the linear functionalf at the point x will be denoted by (x, f). Projections in B will be denoted by means of the letter E, while those in B' will be denoted by means of E'. The following theorems from the general theory will be used.

Let B be an associative ring with identity, A a subring of B containing the identity of B. If B is commutative then it is customary to define an element b of B to be integral over A if it satisties an equation of the form for some a1, a2, …, an A. This definition does not generalize readily to the case when B is non-commutative. Van der Waerden ([11], p. 75) defines b ∈ B to be integral over A if all powers of b belong to a finite A-module. This definition is quite satisfactory when A satisfies the ascending chain condition for left ideals, but in the general case this type of integrity is not necessarily transitive, even when B is commutative. Krull [6] calls an element b ∈ B which satisfies the above condition almost integral over A (but he only considers the commutative case). The subset Ā of B consisting of all almost integral elements over A is called the complete integral closure of A in B. If Ā = A, A is said to be completely integrally closed in B. More recently (in [3]), Gilmer and Heinzer (see also Bourbaki, [1]) have discussed these properties in the commutative case and have shown that the complete integral closure of A in B need not be completely integrally closed in B. If B is not commutative, the set A of elements of B almost integral over A, may not even form a ring. In [5] p. 122, Jacobson uses a definition equivalent to Van der Waerden's for the non-commutative case but the definition applies only for a very restricted class of rings.

Weak amenability of a weighted group algebra, or a Beurling algebra, is a long-standing open problem. The commutative case has been extensively investigated and fully characterized. We study the non-commutative case. Given a weight function $\omega$ on a locally compact group $G$, we characterize derivations from $L^1(G,\omega)$ into its dual in terms of certain functions. Then we show that for a locally compact IN group $G$, if there is a non-zero continuous group homomorphism $\varphi$: $G\to \mathbb{C}$ such that $\varphi(x)/\omega(x)\omega(x^{-1})$ is bounded on $G$, then $L^1(G,\omega)$ is not weakly amenable. Some useful criteria that rule out weak amenability of $L^1(G,\omega)$ are established. Using them we show that for many polynomial type weights the weighted Heisenberg group algebra is not weakly amenable, neither is the weighted $\boldsymbol{ax+b}$ group algebra. We further study weighted quotient group algebra $L^1(G/H,\hat\omega)$, where $\hat\omega$ is the canonical weight on $G/H$ induced by $\omega$. We reveal that the kernel of the canonical homomorphism from $L^1(G,\omega)$ to $L^1(G/H,\hat\omega)$ is complemented. This allows us to obtain some sufficient conditions under which $L^1(G/H,\hat\omega)$ inherits weak amenability of $L^1(G,\omega)$. We study further weak amenability of Beurling algebras of subgroups. In general, weak amenability of a Beurling algebra does not pass to the Beurling algebra of a subgroup. However, in some circumstances this inheritance can happen. We also give an example to show that weak amenability of both $L^1(H,\omega|_H)$ and $L^1(G/H,\hat\omega)$ does not ensure weak amenability of $L^1(G,\omega)$.

In 1969 Lindenstrauss and Rosenthal showed that if a Banach space is isomorphic to a complemented subspace of an Lp-space, then it is either an Lp-space or isomorphic to a Hilbert space. This is the motivation of this paper where we study non- Hilbertian complemented operator subspaces of non-commutative Lp-spaces and show that this class is much richer than in the commutative case. We investigate the local properties of some new classes of operator spaces for every 2 < p < 1 which can be considered as operator space analogues of the Rosenthal sequence spaces from Banach space theory, constructed in 1970. Under the usual conditions on the defining sequence we prove that most of these spaces are operator Lp-spaces, not completely isomorphic to previously known such spaces. However, it turns out that some column and row versions of our spaces are not operator Lp-spaces and have a rather complicated local structure which implies that the Lindenstrauss-Rosenthal alternative does not carry over to the non-commutative case.

Equivariant matrices, commuting with a group of permutation matrices, are considered. Such matrices typically arise from PDEs and other computational problems where the computational domain exhibits discrete geometrical symmetries. In these cases, group representation theory provides a powerful tool for block diagonalizing the matrix via the Generalized Fourier Transform (GFT). This technique yields substantial computational savings in problems such as solving linear systems, computing eigenvalues and computing analytic matrix functions such as the matrix exponential. The paper is presenting a comprehensive self contained introduction to this field. Building upon the familiar special (finite commutative) case of circulant matrices being diagonalized with the Discrete Fourier Transform, we generalize the classical convolution theorem and diagonalization results to the noncommutative case of block diagonalizing equivariant matrices. Applications of the GFT in problems with domain symmetries have been developed by several authors in a series of papers. In this paper we elaborate upon the results in these papers by emphasizing the connection between equivariant matrices, block group algebras and noncommutative convolutions. Furthermore, we describe the algebraic structure of projections related to non-free group actions. This approach highlights the role of the underlying mathematical structures, and provides insight useful both for software construction and numerical analysis. The theory is illustrated with a selection of numerical examples.