Faisal Yousafzai, Muhammad Danish Zia, Yiu-Yin Lee
et al.
This paper proposes a novel decision-making framework that integrates algebraic structures within intuitionistic fuzzy logic to address complex real-world decision-making scenarios involving uncertainty. Two models are developed, including the Abel-Grassmann Intuitionistic Fuzzy Decision Matrix (AG-IFDM) for ranking non-associative neural connectivity patterns and the Group-theoretic Intuitionistic Fuzzy Ranking (GIFR) for evaluating agricultural treatment formulations. These models provide a reliable ranking of alternatives in uncertain environments where the presence or absence of associativity influences the outcomes. The integration of empirical intuition with algebraic reasoning enables a more realistic modelling of uncertainty, where theoretical precision is complemented by practical applications. Integrating both theoretical foundation and practical insight, the applications from medical neuroscience and agricultural optimization underscore the versatility of the proposed models in addressing order-sensitive real-world systems, with promising potential for large-scale validation. Furthermore, the practical illustrations of these models confirm their effectiveness, supported by comprehensive robustness, sensitivity and complexity evaluations. This work establishes a flexible and mathematically precise foundation for algebraic decision-making to effectively address order-sensitive problems and enhance the transparency of multi-criteria decisions under uncertain environments.
Raffaello Pellegrino, Roberto Paganelli, Stefania Bandinelli
et al.
A high polyphenol intake has been associated with higher bone-mineral density. In contrast, we recently demonstrated that the urinary levels of these micronutrients were associated with the long-term accelerated deterioration of the bone. To expand on the health consequences of these findings, we assessed the association between urinary level and dietary intake of polyphenols and the 9-year risk of hip fractures in the InCHIANTI study cohort. The InCHIANTI study enrolled representative samples from two towns in Tuscany, Italy. Baseline data were collected in 1998 and at follow-up visits in 2001, 2004, and 2007. Of the 1453 participants enrolled at baseline, we included 817 participants in this study who were 65 years or older at baseline, donated a 24 hour urine sample, and underwent a quantitative computerized tomography (pQCT) of the tibia. Fracture events were ascertained by self-report over 9 years of follow-up. Thirty-six hip fractures were reported over the 9-year follow-up. The participants who developed a hip fracture were slightly older, more frequently women, had a higher dietary intake of polyphenols, had higher 24-hour urinary polyphenols excretion, and had a lower fat area, muscle density, and cortical volumetric Bone Mineral Density (vBMD) in the pQCT of the tibia. In logistic regression analyses, the baseline urinary excretion of total polyphenols, expressed in mg as a gallic acid equivalent, was associated with a higher risk of developing a hip fracture. Dietary intake of polyphenols was not associated with a differential risk of fracture. In light of our findings, the recommendation of an increase in dietary polyphenols for osteoporosis prevention should be considered with caution.
We continue the study of Lagrangian-invariant objects (LI-objects for short) in the derived category $D^b(A)$ of coherent sheaves on an abelian variety, initiated in arXiv:1109.0527. For every element of the complexified ample cone $D_A$ we construct a natural phase function on the set of LI-objects, which in the case $\dim A=2$ gives the phases with respect to the corresponding Bridgeland stability (see math.AG/0307164). The construction is based on the relation between endofunctors of $D^b(A)$ and a certain natural central extension of groups, associated with $D_A$ viewed as a hermitian symmetric space. In the case when $A$ is a power of an elliptic curve, we show that our phase function has a natural interpretation in terms of the Fukaya category of the mirror dual abelian variety. As a byproduct of our study of LI-objects we show that the Bridgeland's component of the stability space of an abelian surface contains all full stabilities.
In this continuation of (L-Y3) (arXiv:0709.1515 (math.AG)), (L-L-S-Y) (arXiv:0809.2121 (math.AG)), (L-Y4) (arXiv:0901.0342 (math.AG)), (L-Y5) (arXiv:0907.0268 (math.AG)), and (L-Y6) (arXiv:0909.2291 (math.AG)), we give an overview of the posted part of the project and then take it as background to introduce Azumaya noncommutative C ∞ -manifold and the four aspects of morphisms therefrom to a projective complex manifold. This gives us then a description of supersymmetric D-branes of A-type in a Calabi-Yau manifold along the line of the Polchinski-Grothendieck Ansatz. The notion of Kahler differentials and their tensors for an Azumaya noncommutative space are introduced. Donaldson's picture of Lagrangian and special Lagrangian submanifolds as selected from the zero-locus of a moment-map on a related space of maps can be merged into the setting of morphisms from Azumaya man- ifolds with a fundamental module. As a pedagogical toy model for illustration, we study D-branes of A-type in a Calabi-Yau torus. Simple as it is, it reveals already several features of D-branes of A-type, including their assembling/disassembling. The short-vs.-long string wrapping behavior of matrix-strings in the string-theory literature can be produced in this context as well. In addition to the previous comparison with stringy works made, the 4th theme (subtitled: "Gomez-Sharpe vs. Polchinski-Grothendieck") of Sec. 2.4 is to be read with the work (G-Sh) (arXiv:hep-th/0008150), while the 2nd theme (subtitled: "Donagi- Katz-Sharpe vs. Polchinski-Grothendieck") of Sec. 4.2 is to be read with the work (D-K-S) (arXiv:hep-th/0309270). Sec. 4.3, though not yet ready to be subtitled "Denef vs. Polchinski- Grothendieck", is to be read with the work (De) (arXiv:hep-th/0107152). Some directly related string-theory remarks are added to the end of each section.
Abstract Supersymmetric heterotic string models, built from a Calabi–Yau threefold X endowed with a stable vector bundle V , usually lead to an anomaly mismatch between c 2 ( V ) and c 2 ( X ) ; this leads to the question whether the difference can be realized by a further bundle in the hidden sector. In [M.R. Douglas, R. Reinbacher, S.-T. Yau, Branes, Bundles and Attractors: Bogomolov and Beyond, math.AG/0604597], a conjecture is stated which gives sufficient conditions on cohomology classes on X to be realized as the Chern classes of a stable reflexive sheaf V ; a weak version of this conjecture predicts the existence of such a V if c 2 ( V ) is of a certain form. In this note, we prove that on elliptically fibered X infinitely many cohomology classes c ∈ H 4 ( X , Z ) exist which are of this form and for each of them a stable S U ( n ) vector bundle with c = c 2 ( V ) exists.
Given a suitable action on a complex projective variety X of a non-reductive affine algebraic group H, this paper considers how to choose a reductive group G containing H and a projective completion of G x_H X which is a reductive envelope in the sense of math.AG/0703131. In particular it studies the family of examples given by moduli spaces of hypersurfaces in the weighted projective plane P(1,1,2) obtained as quotients by linear actions of the (non-reductive) automorphism group of P(1,1,2).
This note is a follow up of math.AG/0612267v2 and it is largely inspired by a beautiful description of Baker-Norine of non-effective degree (g-1) divisors via chip-firing game. We consider the set of all theta characteristics on a tropical curve and identify the Riemann constant as a unique non-effective one among them.
This is a sequel to [Ca01]=math.AG/0110051. We define the bimeromorphic {\it category} of geometric orbifolds. These interpolate between (compact K\" ahler) manifolds and such manifolds with logarithmic structure. These geometric orbifolds are considered from the point of view of their geometry, and thus equipped with the usual invariants of varieties: morphisms and bimeromorphic maps, differential forms, fundamental groups and universal covers, fields of definition and rational points. The most elementary properties, directly adapted from the case of varieties without orbifold structure, are established here. The arguments of [Ca01] can then be directly adapted to extend the main structure results to this orbifold category. We hope to come back to deeper aspects later. The motivation is that the natural frame for the theory of classification of compact K\" ahler (and complex projective) manifolds includes at least the category of orbifolds, as shown in [Ca01] by the fonctorial decomposition of {\it special} manifolds as tower of orbifolds with either $\kappa_+=-\infty$ or $\kappa=0$, and also, seemingly, by the minimal model program, in which most proofs work only after the adjunction of a "boundary". Also, fibrations enjoy in the bimeromorphic category of geometric orbifolds extension properties not satisfied in the category of varieties without orbifold structure, permitting to express invariants of the total space from those of the generic fibre and of the base. For example, the natural sequence of fundamental groups is exact there; also the total space is special if so are the generic fibre and the base. This makes this category suitable to lift properties from orbifolds having either $\kappa_+=-\infty$ or $\kappa=0$ to those which are special.
Abstract In this paper, we compute the cup product structure of the preprojective algebra Dynkin quivers of type D and E over a field of characteristic zero. This is a continuation of the work done in [P. Etingof, C. Eu, Hochschild and cyclic homology of preprojective algebras of ADE quivers, arXiv: math.AG/0609006 ] where the additive structure of the Hochschild cohomology (together with its grading) was computed. Together with the results in [K. Erdmann, N. Snashall, On Hochschild cohomology of preprojective algebras. I, J. Algebra 205 (2) (1998) 391–412, II, J. Algebra 205 (2) (1998) 413–434] (where the A -case was studied), this yields a complete description of the product in the Hochschild cohomology of ADE quivers over a field of characteristic zero.
We show that a generic principally polarized abelian variety (ppav) is uniquely determined by its theta hyperplanes. These are the non-projectivized version of those studied by Caporaso and Sernesi (see math.AG/0204164), which in a sense are a generalization to ppavs of bitangents of plane quartics, and of hyperplanes tangent to a canonical curves of genus $g$ in $g-1$ points. More precisely, we show that, generically, the set of gradients of all odd theta functions at the point zero uniquely determines a ppav with level (4,8) structure. We also show that our map is an immersion of the moduli space of ppavs.
This is a thoroughly revised version of math.AG/0502516v1 (24 Feb. 2005). Let k be a field of characteristic zero. Let Y=G/H, where G is a connected linear algebraic group over k and H is a connected closed k-subgroup of G. Let X be a smooth compactification of Y over k. We prove the conjecture set forward in the previous version : the Galois-lattice given by the geometric Picard group of X is flasque. The previous version had only partial results in this direction. They were obtained at the expense of a long d'etour via local and global fields. We owe the drastic improvement to a suggestion by O. Gabber. The previous version also assumed G semisimple simply connected. We can dispense with this. The result now covers the previously known case Y=G, with G an arbitrary connected linear algebraic group. ----- Ceci est une version enti`erement r'evis'ee de math.AG/0502516v1 (24 f'ev. 2005). Soient k un corps de caract'eristique z'ero, G un k-groupe lin'eaire connexe et H un k-sous-groupe ferm'e connexe de G. Notons Y=G/H. Soit X une k-compactification lisse de Y. Dans la pr'ec'edente version, nous avancions la conjecture : le module galoisien donn'e par le groupe de Picard g'eom'etrique de X (c'est un r'eseau) est un module flasque. Nous 'etablissions des cas particuliers de cette conjecture, sous l'hypoth`ese suppl'ementaire que G est semi-simple simplement connexe, au moyen d'une r'eduction alambiqu'ee au cas des corps p-adiques. Le pr'esent texte, qui doit beaucoup \`a une suggestion d'O. Gabber, 'etablit la conjecture dans le cas g'en'eral.
Graber, Harris and Starr proved, when n >= 2d, the irreducibility of the Hurwitz space H^0_{d,n}(Y) which parametrizes degree d coverings of a smooth, projective curve Y of positive genus, simply branched in n points, with full monodromy group S_d (math.AG/0205056). We sharpen this result and prove that H^0_{d,n}(Y) is irreducible if n >= max{2,2d-4} and in the case of elliptic Y if n >= max{2,2d-6}. We extend the result to coverings simply branched in all but one point of the discriminant. Fixing the ramification multiplicities over the special point we prove that the corresponding Hurwitz space is irreducible if the number of simply branched points is >= 2d-2. We study also simply branched coverings with monodromy group different from S_d and when n is large enough determine the corresponding connected components of H_{d,n}(Y). Our results are based on explicit calculation of the braid moves associated with the standard generators of the n-strand braid group of Y.
Abstract Let X be a smooth projective curve of genus g⩾2 defined over an algebraically closed field k of characteristic p>0. Let MX(r) be the moduli space of semi-stable rank r vector bundles with fixed trivial determinant. The relative Frobenius map F : X→X 1 induces by pull-back a rational map V : M X 1 (r)→ M X (r) . We determine the equations of V in the following two cases (1) (g,r,p)=(2,2,2) and X nonordinary with Hasse–Witt invariant equal to 1 (see math.AG/0005044 for the case X ordinary), and (2) (g,r,p)=(2,2,3). We also show the existence of base points of V, i.e., semi-stable bundles E such that F ∗ E is not semi-stable, for any triple (g,r,p).
We study a reduction of the Toda lattice in the limit of infintesimal lattice spacing. Using this reduction, we formulate a conjecture for the equivariant Gromov-Witten invariants of the sphere, which we prove in genus 0. This conjecture follows from recent work of Okounkov and Pandharipande (math.AG/0207233), combined with results from the sequel to this paper.