Rigidity through a Projective Lens

In this paper, we offer an overview of a number of results on the static rigidity and infinitesimal rigidity of discrete structures which are embedded in projective geometric reasoning, representations, and transformations. Part I considers the fundamental case of a bar–joint framework in projective d-space and places particular emphasis on the projective invariance of infinitesimal rigidity, coning between dimensions, transfer to the spherical metric, slide joints and pure conditions for singular configurations. Part II extends the results, tools and concepts from Part I to additional types of rigid structures including body-bar, body–hinge and rod-bar frameworks, all drawing on projective representations, transformations and insights. Part III widens the lens to include the closely related cofactor matroids arising from multivariate splines, which also exhibit the projective invariance. These are another fundamental example of abstract rigidity matroids with deep analogies to rigidity. We conclude in Part IV with commentary on some nearby areas.

Modularity of PGL2(𝔽p)-representations over totally real fields

We study an analog of Serre’s modularity conjecture for projective representations ρ¯:Gal(K¯/K)→PGL2(k), where K is a totally real number field. We prove cases of this conjecture when k=F5.

Strongly Continuous Representations in the Hilbert Space: A Far-Reaching Concept

We revisit the fundamental notion of continuity in representation theory, with special attention to the study of quantum physics. After studying the main theorem in the context of representation theory, we draw attention to the significant aspect of continuity in the analytic foundations of Wigner’s work. We conclude the paper by reviewing the connection between continuity, the possibility of defining certain local groups, and their relation to projective representations.

Optimal quantum tomography with constrained measurements arising from unitary bases

The purpose of this paper is to introduce techniques of obtaining optimal ways to determine a [Formula: see text]-level quantum state or distinguish such states. It entails designing constrained elementary measurements extracted from maximal abelian subsets of a unitary basis [Formula: see text] for the operator algebra [Formula: see text] of a Hilbert space [Formula: see text] of finite dimension [Formula: see text] or, after choosing an orthonormal basis for [Formula: see text], for the ⋆-algebra [Formula: see text] of complex matrices of order [Formula: see text]. Illustrations are given for the techniques. It is shown that the Schwinger basis [Formula: see text] of unitary operators can give for [Formula: see text], a product of primes [Formula: see text] and [Formula: see text], the ideal number [Formula: see text] of rank one projectors that have a few quantum mechanical overlaps (or, for that matter, a few angles between the corresponding unit vectors). Finally, we give a combination of the tensor product and constrained elementary measurement techniques to deal with all [Formula: see text], though with more overlaps or angles depending on the factorization of [Formula: see text] as a product of primes or their powers like [Formula: see text] with [Formula: see text], all primes, [Formula: see text] for [Formula: see text], or other types. A comparison is drawn for different forms of unitary bases for the Hilbert space factors of the tensor product like [Formula: see text] or [Formula: see text], where [Formula: see text] is the Galois field of size [Formula: see text] and [Formula: see text] is the ring of integers modulo [Formula: see text]. Even though as Hilbert spaces they are isomorphic, but quantum mechanical system-wise, these tensor products are different. In the process, we also study the equivalence relation on unitary bases defined by R. F. Werner [J. Phys. A: Math. Gen. 34 (2001) 7081–7094], connect it to local operations on maximally entangled vectors bases, find an invariant for equivalence classes in terms of certain commuting systems, called fan representations, and, relate it to mutually unbiased bases and Hadamard matrices. Illustrations are given in the context of Latin squares and projective representations as well.

A pair of homotopy-theoretic version of TQFT’s induced by a Brown functor

The purpose of this paper is to study some obstruction classes induced by a construction of a homotopy-theoretic version of projective TQFT (projective HTQFT for short). A projective HTQFT is given by a symmetric monoidal projective functor whose domain is the cospan category of pointed finite CW-spaces instead of a cobordism category. We construct a pair of projective HTQFT’s starting from a [Formula: see text]-valued Brown functor where [Formula: see text] is the category of bicommutative Hopf algebras over a field [Formula: see text] : the cospanical path-integral and the spanical path-integral of the Brown functor. They induce obstruction classes by an analogue of the second cohomology class associated with projective representations. In this paper, we derive some formulae of those obstruction classes. We apply the formulae to prove that the dimension reduction of the cospanical and spanical path-integrals are lifted to HTQFT’s. In another application, we reproduce the Dijkgraaf–Witten TQFT and the Turaev–Viro TQFT from an ordinary [Formula: see text]-valued homology theory.

Diagnostic Value of the "Table Drawing" Technique during Neuropsychological Assessment of Children Aged 4–17 Years

The method of "Table Drawing" is used in child neuropsychology to evaluate projective representations. At the same time, a unified system for assessing the technique performance, age standards, and information about the validity are absent. The present study aimed to investigate the diagnostic value of the technique during the neuropsychological examination of children aged 4–17 years. A survey of 636 persons was conducted, 411 of them boys and 225 girls. The overall technique performance was evaluated in the diagnostic process. Regardless of the diagnostician, the drawings were evaluated by the level of projective representations and the geometric properties. Age standards of technique execution were obtained. Based on the material of 597 diagnostic protocols, the contribution of the state of 14 higher mental functions and psychological characteristics to the success of the test performance was studied. Constructive-spatial functions have the greatest weight; however, the efficiency also depends on visual gnosis, dynamic praxis, and thinking. The diagnostic value of the method is different at different ages, so neuropsychological interpretation of its execution results should depend on the age of the subject.

Fine Spin-Dependent Splitting of Electronic Excitations and Their Dispersion in Single-Layer Graphene and Graphite

The dispersion dependences of electronic excitations in single-layer graphene and crystalline graphite have been studied taking the electron spin into consideration. Compatibility conditions for two-valued irreducible projective representations characterizing the symmetry of spinor excitations in the above structures and the distributions of spinor quantum states over projective classes and irreducible projective representations at all high-symmetry points in the corresponding Brillouin zones are determined for the first time. The principal existence of the spin-dependent splitting (or merging) of the electronic energy states, in particular, the electronic п-bands at the Dirac points, is established. The magnitude of spin-dependent splitting can be significant, e.g., for the transition-metal chalcogenides belonging to the same spatial symmetry group as crystalline graphite. However, because of the weak spin-orbit interaction for carbon atoms, it turns out small for all carbon structures including single-layer graphene and crystalline graphite.