Algebraic Geometry

Khovanov homology for knots has generated a flurry of activity in the topology community. This paper studies the Khovanov type cohomology for graphs with a special attention to torsion. When the underlying algebra is Z[x]/(x 2), we determine precisely those graphs whose cohomology contains torsion. For a large class of algebras, we show that torsion often occurs. Our investigation of torsion led to other related general results. Our computation could potentially be used to predict the Khovanov-Rozansky sl(m) homology of knots (in particular (2,n) torus knot). We also predict that our work is

In the paper, the inverse multi-parametric problem is investigated in the following form: for the given sequence of eigen values {(λ 1,n , λ 2,n , ..., λ m,n)} n=1,2,... ⊂ R m with real coordinates and the sequences of appropriate given eigen elements {Φ n } n=1,2,... = { φ 1,n ⊗ φ 2,n ⊗ ... ⊗ φ m,n } n=1,2,.

This article is concerned with the existence of positive solutions of a fourthorder p-Laplacian boundary value problem. Based on a priori estimates achieved by utilizing Jensen's integral inequalities for convex and concave functions, we use fixed point index theory to establish the existence of positive solutions for the above problem.

This paper continues the work done in and is an attempt to establish a conceptual framework which generalizes the work of Manin [21] on the relation between non-linear second order ODE of type Painlevé VI and integrable systems. The principle behind everything is a strong interaction between K-theory and Picard-Fuchs type differential equations via Abel-Jacobi maps. Our main result is an extension of a theorem of Donagi and Markman .

We compute the transcendental part of the normal function corresponding to the Deligne class of a cycle in K1 of a mirror family of quartic K3 surfaces. The resulting multivalued function does not satisfy the hypergeometric differential equation of the periods and we conclude that the cycle is indecomposable for most points in the mirror family. The occuring inhomogenous Picard-Fuchs equations are related to Painlevé VI type differential equations. 1991 Mathematics Subject Classification. 14C25, 19E20 . We are grateful to DFG, UAM and Univ. Essen for supporting this project.

We develop a theory of differential equations associated to families of algebraic cycles in higher Chow groups (i.e., motivic cohomology groups). This formalism is related to inhomogenous Picard-Fuchs type differential equations. For families of K3 surfaces the corresponding non-linear ODE turns out to be similar to Chazy's equation.

We use L 2 -Higgs cohomology to determine the Hodge numbers of the parabolic cohomology H 1 ( S, j * V), where the local system V arises from the third primitive cohomology of family of Calabi-Yau threefolds over a curve S. The method gives a way to predict the presence of algebraic 2-cycles in the total space of the family and is applied to some examples.

This paper presents a systematic exploration of convex polyhedra derived from the intersection of symmetric slab regions in R 3. Building upon previous work [8] that introduced a methodology based on a set of 22 planes related to cubic, octahedral, and hexagonal symmetries and analyzed a specific family P(C), this study significantly expands the investigation to a wider range of families formed by combinations of these symmetric slab types. We explore families such as R(C cube , C hex), yielding forms like the 16-vertex orthogonal triangulated octagonal prismatoid, H(C), which primarily generates infinite prismatic forms, and RCO(C cube , Coct), which can generate known semi-regular polyhedra such as the cuboctahedron. The investigation then focuses on more general and complex combinations: the COE polyhedra, formed by the intersection of Octahedral and Hexagonal slabs with independent parameters (Coct, C hex), and the RCOE polyhedra, resulting from the intersection of Cubic, Octahedral, and Hexagonal slabs with parameters (C cube , Coct, C hex). Through a systematic computational exploration of parameter space examples for these families, particularly ROE (a subset of COE) and RCOE, we demonstrate that these methods are powerful generators of a rich and diverse set of convex polyhedra. This exploration has led to the discovery and detailed characterization of several new irregular convex polyhedra. In the ROE family, we discovered multiple distinct geometric realizations for known combinatorial types of Deltahedra (polyhedra with only triangular faces) with 8, 10, and 12 vertices. Although the combinatorial types may be cataloged, their specific geometric forms, derived from our parameterized intersection method, are novel. Analysis of the duals of these irregular Deltahedra has shown faces corresponding to the vertex figures of the primals, including irregular forms such as kites (deltoids). In the RCOE family, we identified new irregular forms, including specific types with 24 and 32 vertices distinct from previously cataloged forms. We provide a rigorous computational analysis of their precise combinatorial properties (V, E, F counts, face shapes), detailed geometric features (distinct face types, edge lengths, angles, dual properties) and classify them based on uniformity and transitivity. The identification of distinct combinatorial types arising from parameter variation across these families, including different structures for the same vertex count, reveals the complex and multifaceted structure of the parameter spaces. This work contributes novel methodologies for generating and studying convex polyhedra from structured inputs, provides detailed characterizations of likely unprecedented irregular forms and new geometric realizations, and highlights the significant combinatorial diversity accessible through these intersection methods, expanding the known atlas of polyhedral examples.

In this paper, using Q *-closed sets, we introduce a new version of normality called Q *-normality, which is a weak form of normality. Further utilizing Q * g-closed sets, we obtain some characterizations of Q *-normal and normal spaces and also obtain some preservation theorems for Q *-normal spaces. Keywords-Q *-closed, g-closed, Q * g-closed sets, Q *-continuous and almost Q *-continuous functions, Normal, Q *-normal spaces.

Let F be a local field, ψ a nontrivial unitary additive character of F , and V a finite dimensional vector space over F . Let us say that a complex function on V is elementary if it has the form where C ∈ C, Q is a rational function (the phase function), P j are polynomials, and χ j multiplicative characters of F . For generic χ j , this function canonically extends to a distribution on V (if char(F )=0). Occasionally, the Fourier transform of an elementary function is also an elementary function (the basic example is the Gaussian integral: k = 0, Q is a nondegenerate quadratic form). It is interesting to determine when exactly this happens. This question is the main subject of our study. In the first part of this paper we show that for F = R or C, if the Fourier transform of an elementary function g = 0 with phase function -Q such that det d 2 Q = 0 is another elementary function g * with phase function Q * , then Q * is the Legendre transform of Q (the "semiclassical condition"). We study properties and examples of phase functions satisfying this condition, and give a classification of phase functions such that both Q and Q * are of the form f (x)/t, where f is a homogeneous cubic polynomial and t is an additional variable (this is one of the simplest possible situations). Unexpectedly, the proof uses Zak's classification theorem for Severi varieties. 1 In the second part of the paper we give a necessary and sufficient condition for an elementary function to have an elementary Fourier transform (in an appropriate "weak" sense) and explicit formulas for such Fourier transforms in the case when Q and P j are monomials, over any local field F . We also describe a generalization of these results to the case of monomials of norms of finite extensions of F . Finally, we generalize some of the above results (including Fourier integration formulas) to the case when F = C and Q comes from a prehomogeneous vector space.

We prove the equivalence of two conjectural constructions of unramified cuspidal automorphic functions on the adelic group GLn(A) associated to an irreducible ℓ-adic local system of rank n on an algebraic curve X over a finite field. The existence of such a function is predicted by the Langlands conjecture. The first construction, which was proposed by Shalika [32] and Piatetski-Shapiro [30] following Weil and Jacquet-Langlands [15] (n = 2), is based on considering the Whittaker function. The second construction, which was proposed recently by Laumon [25] following Drinfeld (n = 2) and Deligne (n = 1), is geometric: the automorphic function is obtained via Grothendieck's "faisceaux-fonctions" correspondence from a complex of sheaves on an algebraic stack. Our proof of their equivalence is based on a local result about the spherical Hecke algebra, which we prove for an arbitrary reductive group. We also discuss a geometric interpretation of this result.

In this paper, we provide the irregularity properties of trees with strong support vertex by analyzing two prominent topological indices: the Albertson index and the Sigma index. We further establish extremal bounds for both indices across families of trees defined by given degree sequences. Let T 1 and T 2 be a star trees of order n, where T 1 ∼ = T 2 , then, we provide Albertson index of T 1 ∼ = T 2. Let T n,∆ be a class of trees with n vertices, there are a tree T ′ ∈ T n,∆ such that irr(T ′) < irr(T).

This paper presents a unified algebraic, geometric, and analytic framework that redefines the structure of integers, vectors, and analytic functions through complex conjugate decompositions. Starting from the Goldbach partition of even integers, we provide a constructive and bounded proof of the Binary Goldbach Conjecture using prime gap estimates and Bertrand's Postulate. We further extend Goldbach partitions to complex product representations, unveiling new symmetries and identities in prime pairings. The paper introduces geometric decompositions of primes and semiprimes, enabling their visualization in Euclidean and topological spaces. We explore applications to the Riemann zeta function by deriving complex root factorizations that suggest a novel lens for interpreting nontrivial zeros. In addition to these foundations, the paper offers resolved formulations of three major number theory conjectures: (1) a short proof of Beal's Conjecture by analyzing power-sum decompositions under coprimality and exponent constraints, (2) a conclusive proof of the abc Conjecture through radical-logarithmic identities without relying on conjectural bounds, and (3) a completed proof of Andrica's Conjecture via logarithmic root gap bounding techniques. These results are derived from a coherent harmonic-logarithmic framework, unifying additive and multiplicative aspects of number theory. Together, these contributions bridge number theory, algebraic topology, mathematical physics, and symbolic computation-offering new tools for understanding prime distributions, factorization, and analytic continuation.

Maryam Mirzakhani, a native of Iran, is currently a professor of mathematics at Stanford. She completed her Ph.D. at Harvard in 2004 under the direction of Curtis T. McMullen. In her thesis she showed how to compute the Weil-Petersson volume of the moduli space of bordered Riemann surfaces. Her research interests include Teichmüller theory, hyperbolic geometry, ergodic theory, and symplectic geometry.

We show that the rational homotopy type of the complement of a toric arrangement is completely determined by two sets of combinatorial data. This is obtained by introducing a differential graded algebra over Q whose minimal model is equivalent to the Sullivan minimal model of A.

The main result of this work is a computation of the Bergmann tau-function on Hurwitz spaces in any genus. This allows to get an explicit formula for the G-function of Frobenius manifolds associated to arbitrary Hurwitz spaces, get a new expression for determinant of Laplace operator in Poincaré metric on Riemann surfaces of arbitrary genus, and compute Jimbo-Miwa tau-function of an arbitrary Riemann-Hilbert problem with quasi-permutation monodromies.

Recently it has been shown that the methods of algebraic geometry first used for finding periodic and almost periodic solutions of KdV, HSh, may be successfully applied to study the solutions of nonlinear equations with a variable spectral parameter in associated zero-curvature representation. In this work following this treatment is extended to the case of the self-duality equation. It seems to be the first example of a four-dimensional non-linear equation solvable by the method of finite-gap integration. Two broad classes of finite-gap solutions for each -SU(2) and 5(7(1,1) gauge groups are constructed in terms of multidimensional theta-functions. The dynamics of the solutions is given by the movement of the hyperelliptic curve with moving branch points and a divisor of the poles in the moduli space of algebraic curves. In the general case our solutions have no periodicity property. We show how oneinstanton solution and 5JV-parametric tΉooft family of instantons may be obtained by the degeneration of general formulae.

Keçeci Numbers (first defined: July 27, 2022) constitute a unique numerical system that generates sequences based on specific initial conditions and a set of iterative rules. This system relies on a complex interplay of fundamental arithmetic operations (addition), special divisibility rules (periodically dividing by 2 or 3), and the primality properties of numbers. One of the most notable aspects of Keçeci Numbers is their definability across various number sets, such as integers, rational numbers, negative numbers, complex numbers, and quaternions, exhibiting distinct behaviours in each. This offers a rich ground for investigating how fundamental number theoretic principles manifest within different algebraic structures. The "Keçeci Prime Number" refers to the number (or its positive integer representative, based on the absolute value of its real or scalar part, according to the number type) that is prime and appears most frequently within the generated Keçeci Number sequence. This value can be interpreted as a kind of "characteristic signature" or "condensation point" regarding the sequence's structure. The potential significance of Keçeci Numbers lies not in being an established sequence, but in providing a "computational laboratory" for exploring how fundamental concepts in number theory (divisibility, primality, arithmetic progression) can generate new and unpredictable patterns under specific constraints and conditions. The application of such rules, particularly in higher-dimensional number systems like complex numbers and quaternions, might produce sequences exhibiting fractal-like structures or chaotic behaviour. Such sequences could potentially inspire models for certain dynamic systems in theoretical physics, or new designs for pseudo-random number generators in signal processing engineering or cryptography. As Keçeci Numbers are still in an exploratory phase, their true value will emerge with deeper mathematical analysis in the future (e.g., concerning the long-term behaviour of the sequences, their limits, periodicity, or the distribution of Keçeci Prime Numbers). Such original numerical systems demonstrate that mathematics is not only about solving existing problems but is also a self-enriching discipline that generates new structures and questions. This can serve as a source of motivation, reminding students and researchers of the vast, unexplored territories within the infinite world of numbers.

Keywords:

Number Sequences, Iterative Processes, Complex Numbers, Rational Numbers, Quaternions, Algorithmic Number Generation, Keçeci Numbers, KececiNumbers, Keçeci Prime Number

An important "stability" theorem in shape theory, due to D.A. Edwards and R. Geoghegan, characterizes those compacta having the same shape as a finite CW complex. In this note we present straightforward and self-contained proof of that theorem.

There is a canonical homomorphism ψ : π1(bdyX)→ π∞ 1 (X) from the fundamental group of the visual boundary, here denoted by bdy X, of any non-positively curved geodesic space X into its fundamental group at infinity. In this setting, the latter group coincides with the first shape homotopy group of the visual boundary: π∞ 1 (X) ≡ π̌1(bdy X). The induced homomorphism φ : π1(bdy X)→ π̌1(bdy X) provides a way to study the relationship between these groups. We present a class Z of compacta, so-called trees of manifolds, for which we can show that the homomorphisms φ : π1(Z)→ π̌1(Z) (Z ∈ Z) are injective. This class Z includes the visual boundaries Z = bdy X which arise from right-angled Coxeter groups whose nerves are closed PL-manifolds. In particular, it includes the visual boundaries of those Coxeter groups which act on Davis’ exotic open contractible manifolds [2]. 1. The first shape homotopy group of a metric compactum We recall the definition of the first shape homotopy group of a...

Mathématiques et sciences humaines, tome 34 (1971), p. 43-59 <http © Centre d'analyse et de mathématiques sociales de l'EHESS, 1971, tous droits réservés. L'accès aux archives de la revue « Mathématiques et sciences humaines » () implique l'accord avec les conditions générales d'utilisation (). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques J. P. DESCLÉS RÉSUMÉ Dans cet article, on présente un cadre d'étude de la syntaxe générative différent du cas classique : en raisonnant directement sur des arbres (i.e., les mots d'un système non associatif libre à plusieurs opérations) on« approche» les opérations non-associatives par des opérations « relativement associatives» ou « associatives dans leur ensemble ». On entreprend ici, dans un premier temps, la classification des « associations cohérentes bijectives à ensemble fini d'opérations ».

It may seem a funny notion to write about theorems as old and rehashed as Descartes' rule of signs, De Gua's rule or Budan's. Admittedly, these theorems were proved numerous times over the centuries. However, despite the popularity of these results, it seems that no thorough and up-to-date historical account of their proofs has ever been given, nor has an effort been made to reformulate the oldest demonstrations in modern terms. The motivation of this paper is to put these strongly related theorems back in their historical perspective. More importantly, we suggest a way to understand Descartes' original statement, which yet remains somewhat of an enigma. We found that this question is related to a certain way of counting the alternations and permanences of signs of the polynomial coefficients, and may have been the convention used by Descartes. Remarkably, this convention not only provides a ultra-simple proof of Descartes' rule, but it can also be used to simplify the proofs of other related theorems, as the four theorems from the title of the article. Without claiming to be exhaustive, we shall present in this paper an historical account of these theorems and their proofs, and clarify their mutual relation. We will explain how a suitable convention can help understand the original statement of Descartes and greatly simplify its proof, as well as the proofs of the above-mentioned theorems. With the exception of the proof of Fourier's theorem and its generalizations, which run on rudiments of infinitesimal calculus (Taylor's theorem), the proposed demonstrations are so short and elementary that they could be taught at the undergraduate level. Of the theorems listed in the title of this paper, the oldest and by far most famous theorem is Descartes' rule of sign. It was first formulated by Descartes in 1637 in his Geometry ([10]). At the beginning of his exposition, the author gives numerical examples of products of polynomials by X -α. Next, having claimed that a polynomial P has a root α if and only if it is divisible by X -α, he added

This paper presents a unified algebraic, geometric, and analytic framework that redefines the structure of integers, vectors, and analytic functions through complex conjugate decompositions. Starting from the Goldbach partition of even integers, we provide a constructive and bounded proof of the Binary Goldbach Conjecture using prime gap estimates and Bertrand's Postulate. We further extend Goldbach partitions to complex product representations, unveiling new symmetries and identities in prime pairings. The paper introduces geometric decompositions of primes and semiprimes, enabling their visualization in Euclidean and topological spaces. We explore applications to the Riemann zeta function by deriving complex root factorizations that suggest a novel lens for interpreting nontrivial zeros. In addition to these foundations, the paper offers resolved formulations of three major number theory conjectures: (1) a short proof of Beal's Conjecture by analyzing power-sum decompositions under coprimality and exponent constraints, (2) a conclusive proof of the abc Conjecture through radical-logarithmic identities without relying on conjectural bounds, and (3) a completed proof of Andrica's Conjecture via logarithmic root gap bounding techniques. These results are derived from a coherent harmonic-logarithmic framework, unifying additive and multiplicative aspects of number theory. Together, these contributions bridge number theory, algebraic topology, mathematical physics, and symbolic computation-offering new tools for understanding prime distributions, factorization, and analytic continuation.

Group and Representation Exercises Question 1 1(a) The symmetric group S n is generated by the simple transpositions s 1 ,. .. , s n-1 i. Relations among the generators The symmetric group S n is generated by simple transpositions s i = (i, i + 1), with the following relations:

We propose a modified lattice Landau gauge based on stereographically projecting the link variables on the circle S 1 → R for compact U(1) or the 3-sphere S 3 → R 3 for SU(2) before imposing the Landau gauge condition. This can reduce the number of Gribov copies exponentially and solves the Gribov problem in compact U(1) where it is a lattice artifact. Applied to the maximal Abelian subgroup this might be just enough to avoid the perfect cancellation amongst the Gribov copies in a lattice BRST formulation for SU(N), and thus to avoid the Neuberger 0/0 problem. The continuum limit of the Landau gauge remains unchanged.

Part 2. Higher rank and higher dimensional valuations 37 8. Higher rank valuations 37 9. Higher dimensional valuations 37 Part 3. Globalization 38 References 40 Proof. By Hironaka's reduction of the singularities (see ) of M 0 , we get a nonsingular projective model M ′ of K jointly with a birational morphism M ′ → M 0 that is the composition of a finite sequence of blow-ups with non-singular centers. Consider the local ring O M ′ ,P ′ of M ′ at the center P ′ of ν and chose elements f 1 , f 2 , . . . , f r ∈ O M,P such that ν(f 1 ), ν(f 2 ), . . . , ν(f r ) are Z-linearly independent. Another application of Hironaka's theorem gives a birational morphism M → M ′ , that is also a composition of a finite sequence of blow-ups with non-singular centers, such thatf = r i=1 f i , is a monomial (times a unit) in a suitable regular system

Let R be a real closed field. The Pierce-Birkhoff conjecture says that any piecewise polynomial function f on R n can be obtained from the polynomial ring R[x 1 , . . . , x n ] by iterating the operations of maximum and minimum. The purpose of this paper is threefold. First, we state a new conjecture, called the Connectedness conjecture, which asserts, for every pair of points α, β ∈ Sper R[x 1 , . . . , x n ], the existence of connected sets in the real spectrum of R[x 1 , . . . , x n ], satisfying certain conditions. We prove that the Connectedness conjecture implies the Pierce-Birkhoff conjecture. Secondly, we construct a class of connected sets in the real spectrum which, though not in itself enough for the proof of the Pierce-Birkhoff conjecture, is the first and simplest example of the sort of connected sets we really need, and which constitutes the first step in our program for a proof of the Pierce-Birkhoff conjecture in dimension greater than 2. Thirdly, we apply these ideas to give two proofs of the Connectedness conjecture (and hence also of the Pierce-Birkhoff conjecture in the abstract formulation) in the special case when one of the two points α, β ∈ Sper R[x 1 , . . . , x n ] is monomial. One of the proofs is elementary while the other consists in deducing the (monomial) Connectedness conjecture as an immediate corollary of the main connectedness theorem of this paper.

In this paper, we study the structure of the graded ring associated to a limit key polynomial Qn in terms of the key polynomials that define Qn. In order to do that, we use direct limits. In general, we describe the direct limit of a family of graded rings associated to a totally ordered set of valuations. As an example, we describe the graded ring associated to a valuation-algebraic valuation as a direct limit of graded rings associated to residue-transcendental valuations.

Let R be a real closed field and A=R[x_1,...,x_n]. Let sper A denote the real spectrum of A. There are two kinds of points in sper A : finite points (those for which all of |x_1|,...,|x_n| are bounded above by some constant in R) and points at infinity. In this paper we study the structure of the set of points at infinity of sper A and their associated valuations. Let T be a subset of {1,...,n}. For j in {1,...,n}, let y_j=x_j if j is not in T and y_j=1/x_j if j is in T. Let B_T=R[y_1,...,y_n]. We express sper A as a disjoint union of sets of the form U_T and construct a homeomorphism of each of the sets U_T with a subspace of the space of finite points of sper B_T. For each point d at infinity in U_T, we describe the associated valuation v_{d*} of its image d* in sper B_T in terms of the valuation v_d associated to d. Among other things we show that the valuation v_{d*} is composed with v_d (in other words, the valuation ring R_d is a localization of R_{d*} at a suitable prime ideal).

The main result of this paper is that in order to prove the local uniformization theorem for local rings it is enough to prove it for rank one valuations. Our proof does not depend on the nature of the class of local rings for which we want to prove local uniformization. We prove also the reductions for different versions of the local uniformization theorem.

The main object of study in this paper is the module Ω of Kähler differentials of an extension of valuation rings. We show that in the case of pure extensions Ω has a very good description. Namely, it is isomorphic to the quotient of two fractional ideals of the value group of the valuation. This allows us to describe the annihilator of Ω and to give criteria for Ω to be finitely generated and to be finitely presented. We also discuss how to relate the module of Kähler differentials of pure extensions to minimal key polynomials.

In the early 1990s, a family of combinatorial CW-complexes named permutoassociahedra was introduced by Kapranov, and it was realised by Reiner and Ziegler as a family of convex polytopes. The polytopes in this family are "hybrids" of permutohedra and associahedra. Since permutohedra and associahedra are simple, it is natural to search for a family of simple permutoassociahedra, which is still adequate for a topological proof of Mac Lane's coherence. This paper presents such a family.

A skeleton of the category with finite coproducts ${\cal D}$ freely generated by a single object has a subcategory isomorphic to a skeleton of the category with finite products ${\cal C}$ freely generated by a countable set of objects. As a consequence, we obtain that ${\cal D}$ has a subcategory equivalent with ${\cal C}$ . From a proof-theoretical point of view, this means that up to some identifications of formulae the deductions of pure conjunctive logic with a countable set of propositional letters can be represented by deductions in pure disjunctive logic with just one propositional letter. By taking opposite categories, one can replace coproduct by product, i.e., disjunction by conjunction, and the other way round, to obtain the dual results.