Conceptual Proof Research Articles

Separating Risk Assessment from Risk Management Poses Legal and Ethical Problems in Person-Centred Care.

Accelerating progress in screening technologies, e.g. genetic testing, means more individuals are facing the stressful decision of whether to request the test. Fully-informed and preference-based consent, as well as ethical practice, requires the full range of benefits and harms from any test or treatment to be identified and assessed from the individual's point of view. For both ethical and legal reasons, we see the decision on whether to undertake a genetic screening test being increasingly seen, in future, as calling for a personalised analysis of the full range of subsequent management options. The conventional dissociation of 'risk assessment' and 'risk management' phases is thereby ruled out. One way of addressing the resulting challenge is through personalised multi-criterial decision support tools. In this vision paper we provide conceptual proof of method of how such an interactive online tool could function. The polygenetic genetic screening decision is used, solely as illustration.

Lotus-on-chip: computer-aided design and 3D direct laser writing of bioinspired surfaces for controlling the wettability of materials and devices

In this study we present the combination of a math-based design strategy with direct laser writing as high-precision technology for promoting solid free-form fabrication of multi-scale biomimetic surfaces. Results show a remarkable control of surface topography and wettability properties. Different examples of surfaces inspired on the lotus leaf, which to our knowledge are obtained for the first time following a computer-aided design with this degree of precision, are presented. Design and manufacturing strategies towards microfluidic systems whose fluid driving capabilities are obtained just by promoting a design-controlled wettability of their surfaces, are also discussed and illustrated by means of conceptual proofs. According to our experience, the synergies between the presented computer-aided design strategy and the capabilities of direct laser writing, supported by innovative writing strategies to promote final size while maintaining high precision, constitute a relevant step forward towards materials and devices with design-controlled multi-scale and micro-structured surfaces for advanced functionalities. To our knowledge, the surface geometry of the lotus leaf, which has relevant industrial applications thanks to its hydrophobic and self-cleaning behavior, has not yet been adequately modeled and manufactured in an additive way with the degree of precision that we present here.

Involutions and stable subalgebras

Given a central simple algebra with involution over an arbitrary field, étale subalgebras contained in the space of symmetric elements are investigated. The method emphasizes the similarities between the various types of involutions and privileges a unified treatment for all characteristics whenever possible. As a consequence a conceptual proof of a theorem of Rowen is obtained, which asserts that every division algebra of exponent two and degree eight contains a maximal subfield that is a triquadratic extension of the centre.

Classification of simple cuspidal modules for solenoidal Lie algebras

We give a new conceptual proof of the classification of cuspidal modules for the solenoidal Lie algebra. This classification was originally published by Y. Su in 2001. Our proof is based on the theory of modules for the solenoidal Lie algebras that admit a compatible action of the commutative algebra of functions on a torus.

Brown’s moduli spaces of curves and the gravity operad

This paper is built on the following observation: the purity of the mixed Hodge structure on the cohomology of Brown's moduli spaces is essentially equivalent to the freeness of the dihedral operad underlying the gravity operad. We prove these two facts by relying on both the geometric and the algebraic aspects of the problem: the complete geometric description of the cohomology of Brown's moduli spaces and the coradical filtration of cofree cooperads. This gives a conceptual proof of an identity of Bergstr\"om-Brown which expresses the Betti numbers of Brown's moduli spaces via the inversion of a generating series. This also generalizes the Salvatore-Tauraso theorem on the nonsymmetric Lie operad.

Contact Isotropic Realisations of Jacobi Manifolds via Spencer Operators

Motivated by the importance of symplectic isotropic realisations in the study of Poisson manifolds, this paper investigates the local and global theory of contact isotropic realisations of Jacobi manifolds, which are those of minimal dimension. These arise naturally when considering multiplicity-free actions in contact geometry, as shown in this paper. The main results concern a classification of these realisations up to a suitable notion of isomorphism, as well as establishing a relation between the existence of symplectic and contact isotropic realisations for Poisson manifolds. The main tool is the classical Spencer operator which is related to Jacobi structures via their associated Lie algebroid, which allows to generalise previous results as well as providing more conceptual proofs for existing ones.

The containment problem and a rational simplicial arrangement

Since Dumnicki, Szemberg, and Tutaj-Gasinska gave in 2013 in [ 11 ] the first example of a set of points in the complex projective plane such that for its homogeneous ideal I the containment of the third symbolic power in the second ordinary power fails, there has been considerable interest in searching for further examples with this property and investigating the nature of such examples. Many examples, defined over various fields, have been found but so far there has been essentially just one example found of 19 points defined over the rationals, see [ 18 , Theorem A, Problem 1]. In [ 14 , Problem 5.1] the authors asked if there are other rational examples. This has motivated our research. The purpose of this note is to flag the existence of a new example of a set of 49 rational points with the same non-containment property for powers of its homogeneous ideal. Here we establish the existence and justify it computationally. A more conceptual proof, based on Seceleanu's criterion [ 22 ] will be published elsewhere [ 19 ].

Braided systems: a unified treatment of algebraic structures with several operations

Bialgebras and Hopf (bi)modules are typical algebraic structures with several interacting operations. Their structural and homological study is therefore quite involved. We develop the machinery of braided systems, tailored for handling such multi-operation situations. Our construction covers the above examples (as well as Poisson algebras, Yetter--Drinfel$'$d modules, and several other structures, treated in separate publications). In spite of this generality, graphical tools allow an efficient study of braided systems, in particular of their representation and homology theories. These latter naturally recover, generalize, and unify standard homology theories for bialgebras and Hopf (bi)modules (due to Gerstenhaber--Schack, Panaite--{\c{S}}tefan, Ospel, Taillefer); and the algebras encoding their representation theories (Heisenberg double, algebras~$\mathscr X$, $\mathscr Y$, $\mathscr Z$ of Cibils--Rosso and Panaite). Our approach yields simplified and conceptual proofs of the properties of these objects.

Weingarten calculus via orthogonality relations: new applications

Weingarten calculus is a completely general and explicit method to compute the moments of the Haar measure on compact subgroups of matrix algebras. Particular cases of this calculus were initiated by theoretical physicists -- including Weingarten, after whom this calculus was coined by the first author, after investigating it systematically. Substantial progress was achieved subsequently by the second author and coworkers, based on representation theoretic and combinatorial techniques. All formulas of `Weingarten calculus' are in the spirit of Weingarten's seminal paper [W78]. However, modern proofs are very different from Weingarten's initial ideas. In this paper, we revisit Weingarten's initial proof and we illustrate its power by uncovering two new important applications: (i) a uniform bound on the Weingarten function, that subsumes existing uniform bounds, and is optimal up to a polynomial factor, and (ii) an extension of Weingarten calculus to symmetric spaces and conceptual proofs of identities established by the second author.

A Short Conceptual Proof of Narayana's Path-Counting Formula

We deduce Narayana's formula for the number of lattice paths that fit in a Young diagram as a direct consequence of the Gessel-Viennot theorem on non-intersecting lattice paths.

Conjugate and conformally conjugate parallelisms on Finsler manifolds

In this paper we study conjugate parallelisms and their conformal changes on Finsler manifolds. We provide sufficient conditions for a Finsler manifold endowed with two conjugate (resp. conformally conjugate) covering parallelisms to become a Berwald (resp. Wagner) manifold. As an application for Lie groups, we give a new proof for a theorem of Latifi and Razavi about bi-invariant Finsler functions being Berwald. By introducing the concept of a conformal change of a parallelism, we also obtain a conceptual proof of a theorem of Hashiguchi and Ichijyō: the class of generalized Berwald manifolds is closed under conformal change.

Pentagrams, inscribed polygons, and Prym varieties

The pentagram map is a discrete integrable system on the moduli space of planar polygons. The corresponding first integrals are so-called monodromy invariants $E_1, O_1, E_2, O_2,\dots$ By analyzing the combinatorics of these invariants, R. Schwartz and S. Tabachnikov have recently proved that for polygons inscribed in a conic section one has $E_k = O_k$ for all $k$. In this paper we give a simple conceptual proof of the Schwartz-Tabachnikov theorem. Our main observation is that for inscribed polygons the corresponding monodromy satisfies a certain self-duality relation. From this we also deduce that the space of inscribed polygons with fixed values of the monodromy invariants is an open dense subset in the Prym variety (i.e., a half-dimensional torus in the Jacobian) of the spectral curve. As a byproduct, we also prove another conjecture of Schwartz and Tabachnikov on positivity of monodromy invariants for convex polygons.

Modules which are coinvariant under automorphisms of their projective covers

In this paper we study modules coinvariant under automorphisms of their projective covers. We first provide an alternative, and in fact, a more succinct and conceptual proof for the result that a module M is invariant under automorphisms of its injective envelope if and only if given any submodule N of M, any monomorphism f:N→M can be extended to an endomorphism of M and then, as a dual of it, we show that over a right perfect ring, a module M is coinvariant under automorphisms of its projective cover if and only if for every submodule N of M, any epimorphism φ:M→M/N can be lifted to an endomorphism of M.

Positive scalar curvature and product formulas for secondary index invariants

We introduce partial secondary invariants associated to complete Riemannian metrics which have uniformly positive scalar curvature outside a prescribed subset on a spin manifold. These can be used to distinguish such Riemannian metrics up to concordance relative to the prescribed subset. We exhibit a general external product formula for partial secondary invariants, from which we deduce product formulas for the higher rho-invariant of a metric with uniformly positive scalar curvature as well as for the higher relative index of two metrics with uniformly positive scalar curvature. Our methods yield a new conceptual proof of the secondary partitioned manifold index theorem and a refined version of the delocalized APS-index theorem of Piazza-Schick for the spinor Dirac operator in all dimensions. We establish a partitioned manifold index theorem for the higher relative index. We also show that secondary invariants are stable with respect to direct products with aspherical manifolds that have fundamental groups of finite asymptotic dimension. Moreover, we construct examples of complete metrics with uniformly positive scalar curvature on non-compact spin manifolds which can be distinguished up to concordance relative to subsets which are coarsely negligible in a certain sense. A technical novelty in this paper is that we use Yu's localization algebras in combination with the description of K-theory for graded C*-algebras due to Trout. This formalism allows direct definitions of all the invariants we consider in terms of the functional calculus of the Dirac operator and enables us to give concise proofs of the product formulas.

The double power monad is the composite power monad

In the context of quantaloid-enriched categories, we rely essentially on the classifying property of presheaf categories to give a conceptual proof of a theorem due to Höhle: the double power monad and the composite power monad, on the category of quantaloid-enriched categories, are the same. Via the theory of distributive laws, we identify the algebras of this monad to be the completely codistributive complete categories, and the homomorphisms between such algebras are the bicontinuous functors. With these results we hope to contribute to the further development of a theory of Q-valued preorders (in the sense of Pu and Zhang).

Hochster duality in derived categories and point-free reconstruction of schemes

For a commutative ring R, we exploit localization techniques and point-free topology to give an explicit realization of both the Zariski frame of R (the frame of radical ideals in R) and its Hochster dual frame, as lattices in the poset of localizing subcategories of the unbounded derived category D(R). This yields new conceptual proofs of the classical theorems of HopkinsNeeman and Thomason. Next we revisit and simplify Balmer’s theory of spectra and supports for tensor triangulated categories from the viewpoint of frames and Hochster duality. Finally we exploit our results to show how a coherent scheme (X,OX) can be reconstructed from the tensor triangulated structure of its derived category of perfect complexes.

Pre-Lie Deformation Theory

In this paper, we develop the deformation theory controlled by pre-Lie algebras; the main tool is a new integration theory for pre-Lie algebras. The main field of application lies in homotopy algebra structures over a Koszul operad; in this case, we provide a homotopical description of the associated Deligne groupoid. This permits us to give a conceptual proof, with complete formulae, of the Homotopy Transfer Theorem by means of gauge action. We provide a clear explanation of this latter ubiquitous result: there are two gauge elements whose action on the original structure restrict its inputs and respectively its output to the homotopy equivalent space. This implies that a homotopy algebra structure transfers uniformly to a trivial structure on its underlying homology if and only if it is gauge trivial; this is the ultimate generalization of the d-dbar lemma.

Drinfeld Realization of Quantum Twisted Affine Algebras via Braid Group

The Drinfeld realization of quantum affine algebras has been tremendously useful since its discovery. Combining techniques of Beck and Nakajima with our previous approach, we give a complete and conceptual proof of the Drinfeld realization for the twisted quantum affine algebras using Lusztig’s braid group action.

Fourier duality of quantum curves

There are two different ways to deform a quantum curve along the flows of the KP hierarchy. We clarify the relation between the two KP orbits: In the framework of suitable connections attached to the quantum curve they are related by a local Fourier duality. As an application we give a conceptual proof of duality results in 2D quantum gravity.

The Exponential Map Is Chaotic: An Invitation to Transcendental Dynamics

We present an elementary and conceptual proof that the complex exponential map is chaotic when considered as a dynamical system on the complex plane. (This was conjectured by Fatou in 1926 and first proved by Misiurewicz 55 years later.) The only background required is a first undergraduate course in complex analysis.