Connecting (Anti)Symmetric Trigonometric Transforms to Dual-Root Lattice Fourier–Weyl Transforms

Explicit links of the multivariate discrete (anti)symmetric cosine and sine transforms with the generalized dual-root lattice Fourier–Weyl transforms are constructed. Exact identities between the (anti)symmetric trigonometric functions and Weyl orbit functions of the crystallographic root systems A1 and Cn are utilized to connect the kernels of the discrete transforms. The point and label sets of the 32 discrete (anti)symmetric trigonometric transforms are expressed as fragments of the rescaled dual root and weight lattices inside the closures of Weyl alcoves. A case-by-case analysis of the inherent extended Coxeter–Dynkin diagrams specifically relates the weight and normalization functions of the discrete transforms. The resulting unique coupling of the transforms is achieved by detailing a common form of the associated unitary transform matrices. The direct evaluation of the corresponding unitary transform matrices is exemplified for several cases of the bivariate transforms.

Coherent state transform for Landau levels on quasi-tori

The spectrum of the Laplacian operator on the positive theta line bundle over the quasi-torus reduces to eigenvalues \pi\ell, \ell=0,1,\ldots{}, which are called Landau levels. This paper discusses the coherent state transform for each eigenspace associated with a Landau level. We construct a unitary transform valid for each eigenspace. A concrete form of the inverse formula for the proposed transform is also obtained.

Central Splitting of A2 Discrete Fourier–Weyl Transforms

Two types of bivariate discrete weight lattice Fourier–Weyl transforms are related by the central splitting decomposition. The two-variable symmetric and antisymmetric Weyl orbit functions of the crystallographic reflection group A2 constitute the kernels of the considered transforms. The central splitting of any function carrying the data into a sum of components governed by the number of elements of the center of A2 is employed to reduce the original weight lattice Fourier–Weyl transform into the corresponding weight lattice splitting transforms. The weight lattice elements intersecting with one-third of the fundamental region of the affine Weyl group determine the point set of the splitting transforms. The unitary matrix decompositions of the normalized weight lattice Fourier–Weyl transforms are presented. The interpolating behavior and the unitary transform matrices of the weight lattice splitting Fourier–Weyl transforms are exemplified.

Generalized Dual-Root Lattice Transforms of Affine Weyl Groups

Discrete transforms of Weyl orbit functions on finite fragments of shifted dual root lattices are established. The congruence classes of the dual weight lattices intersected with the fundamental domains of the affine Weyl groups constitute the point sets of the transforms. The shifted weight lattices intersected with the fundamental domains of the extended dual affine Weyl groups form the sets of labels of Weyl orbit functions. The coinciding cardinality of the point and label sets and corresponding discrete orthogonality relations of Weyl orbit functions are demonstrated. The explicit counting formulas for the numbers of elements contained in the point and label sets are calculated. The forward and backward discrete Fourier-Weyl transforms, together with the associated interpolation and Plancherel formulas, are presented. The unitary transform matrices of the discrete transforms are exemplified for the case A 2 .

REKAYASA BAHASA DAN KONSTRUKSI POLITIK “PERSATUAN-KESATUAN” DALAM WACANA “NEGARA KESATUAN REPUBLIK INDONESIA”

The formation of the Indonesian nation-state is inseparable from linguistic engineering. This includes phrases that transformed their lexical meaning to become a binding political concession produced by Indonesian political leaders in the 1940s and 1950s. The official name of the Indonesian state “The Unitary State of the Republic of Indonesia” is the result of this political concession in the statecraft of Indonesia. This article aims to examine the meanings of the “unity and unitary” phrases in the imagined form of Indonesian nation-state. Why was “unity and unitary” an effective political tool for the shaping of the imagination of “Indonesia”? Linguists and historians would these choices of words as a reflection of the power of language in the creation of facts. Language can transform a lexical fact into a material one. It is therefore essential to understand how did the phrase “unity and unitary” transform from a lexical to a political meaning in the context of Indonesian history? This article is based on literary analyses of official relevant documents, including the assembly proceeding of the Council for the Investigation of the Preparation for Indonesian Independence (Badan Penyelidik Usaha-usaha Persiapan Kemerdekaan Indonesia, BPUPKI) of August 1945, and the so-called Principal Guidelines for State Development (Garis-Garis Besar Haluan Negara, GBHN) of the New Order. It argues that the “unity and unitary” phrase represents a negotiation of diverse political elements which then shaped the crafting of Indonesian nation-state. The changes in the contexts in which the phrase was used show a changing association between the lexical and ideological meanings of the phrase. While these changes worked towards institutionalizing the Indonesian state, this article concludes that they also submerged the people’s discontents. The phrase “unity and unitary” reflected the making of people’s uniformity to a large extent.

A New Mechanism of Open System Evolution and Its Entropy Using Unitary Transformations in Noncomposite Qudit Systems

The evolution of an open system is usually associated with the interaction of the system with an environment. A new method to study the open-type system evolution of a qubit (two-level atom) state is established. This evolution is determined by a unitary transformation applied to the qutrit (three-level atom) state, which defines the qubit subsystems. This procedure can be used to obtain different qubit quantum channels employing unitary transformations into the qutrit system. In particular, we study the phase damping and spontaneous-emission quantum channels. In addition, we mention a proposal for quasiunitary transforms of qubits, in view of the unitary transform of the total qutrit system. The experimental realization is also addressed. The probability representation of the evolution and its information-entropic characteristics are considered.