Algebraic Geometry

Durante quinientos años, después de los logros de Tartaglia, Cardano y Ferrari, la ecuación quíntica ha representado uno de los límites más enigmáticos del pensamiento algebraico. Desde la imposibilidad demostrada por Abel y Galois de resolverla por radicales, hace trescientos años, hasta la propuesta contemporánea del Método Resolvente Encadenado de Lagrange-Bring-Sánchez, la historia de su resolución simboliza la evolución del concepto mismo de número, estructura y relación. El presente artículo propone una reinterpretación del problema quíntico a través de una transición epistemológica: del paradigma radical-complejo al paradigma estructural-resolvente. En esta nueva visión, las raíces dejan de concebirse como entidades aisladas obtenidas por operaciones, para entenderse como configuraciones relacionales generadas por factorizaciones estructuradas. Así, la imposibilidad por radicales no constituye un cierre, sino una apertura hacia la resolubilidad por estructuras: un cambio de paradigma que redefine la ontología del número y la epistemología del álgebra contemporánea. Ecuación quíntica, método resolvente encadenado, Lagrange-Bring-Sánchez, números resolventes encadenados, resolubilidad por estructuras, epistemología del álgebra, estructura relacional. La historia de la resolución de las ecuaciones polinómicas es, en realidad, la historia del pensamiento algebraico mismo. Desde los tratados árabes de Al-Khwarizmi hasta la obra de Cardano y Ferrari en el Renacimiento, la humanidad ha buscado no solo resolver ecuaciones, sino comprender qué significa "resolver". Durante siglos, las ecuaciones cúbicas y cuárticas marcaron el umbral de la inteligencia simbólica. Su resolución por radicales -mediante números complejos-fue un hito que transformó la manera de concebir el número y el espacio: el paso del plano real al plano complejo. Sin embargo, el siglo XIX trajo consigo una frontera inesperada. La ecuación quíntica, ( 𝑥 5 + 𝑎 4 𝑥 4 + 𝑎 4 𝑥 3 + 𝑎 2 𝑥 2 + 𝑎 1 𝑥 + 𝑎 0 = 0 ), se resistió obstinadamente a toda fórmula de resolución por radicales. Abel demostró la imposibilidad general, y Galois reveló que la razón profunda residía en la estructura de simetría del sistema de raíces. El álgebra alcanzaba así su madurez conceptual… y, paradójicamente, su límite operativo. Quinientos años después, la pregunta no es ya si la quíntica puede resolverse por radicales, sino si puede resolverse por estructuras. Esa es la propuesta que el presente trabajo desarrolla: una transición natural del pensamiento algebraico, donde el problema deja de ser un obstáculo lógico para convertirse en una oportunidad ontológica.

This review, conducted by Grok AI developed by xAI, examines the novel lacunary double-sum expansions of the Jacobi theta functions \(\theta_3(q)\) and \(\theta_4(q)\) as presented in Theorems 9.3 and 9.4 by Akram Louiz in the research article titled "Transformations of Theta-3 and Theta-4 functions by using a new equation of four infinite series and sums built from a generalized recurrence relation" (Malaya Journal of Matematik, DOI:

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https://doi.org/10.26637/mjm1303/005).

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The paper provides the primary expansions, and this review includes a detailed proof of an inferred alternative form for \(\theta_3(q)\), extending Louiz's lacunary approach with a Poisson summation-inspired correction to enhance convergence. These expansions exhibit super-exponential convergence for small \(|q|\), improving over conventional series. The review explores implications for new series representations of the Riemann zeta function \(\zeta(s)\), refinements to its functional equation, enhancements to the Riemann-Siegel formula, and significant consequences for the Riemann hypothesis (RH), including the GUE (Gaussian Unitary Ensemble) hypothesis, new subconvexity bounds, an expanded proof of Montgomery pair correlation, reformulations of Li's criterion, and implications for the Hilbert-Pólya conjecture and Berry-Keating operator, with examples like \(\zeta(3)\). Completed on October 20, 2025, at 10:15 AM +01, this analysis advances analytic number theory, particularly in RH exploration, by providing deeper insights and potential pathways for computational verification.

The P vs NP problem remains a central open question in computational complexity theory, inquiring whether problems with efficiently verifiable solutions can also be solved efficiently. This paper develops a categorical framework that connects Geometric Complexity Theory (GCT) and Proof Complexity, two distinct approaches to studying computational hardness. While GCT employs algebraic geometry and representation theory to explore separations through orbit closures, and Proof Complexity examines resource requirements for propositional proofs, their independent development may overlook potential synergies. To address this, we introduce categories for complexity classes (Comp), proof systems (ProofSys), and geometric representations (GeoRep), interconnected by functors 𝐹 and Φ. This structure offers a perspective on reformulating aspects of the P vs NP question in terms of properties of natural transformations between obstruction functors, as explored in Principle 5.1. Additionally, we propose viewing complexity separations through a cohomological lens via a simplicial set of obstructions in Conjecture 6.1. By linking geometric obstructions from GCT with combinatorial lower bounds from Proof Complexity, this framework provides a structured approach to investigating complexity separations and their connections to areas like circuit complexity. Although it does not resolve P vs NP, it suggests avenues for integrating these fields in future research.

We explore the notion of representation of an affine extension of an abelian variety -- such an extension is a faithfully flat affine morphism of $\Bbbk$-group schemes $q:G\to A$, where $A$ is an abelian variety. We characterize the categories that arise as the category of representations of an affine extension $q:G\to A$, generalizing the classical results of Tannaka Duality established for affine $\Bbbk$-group schemes (that is, when $A=\operatorname{Spec}(\Bbbk)$). We also prove the existence of a contravariant equivalence between the category of affine extensions of a given $A$ and the category of faithful commutative Hopf sheaves on $A$, generalizing in this manner the well-known op-equivalence between affine group schemes and commutative Hopf algebras. If $\mathcal H_q$ is the Hopf sheaf on $A$ associated to $q$, the category of representations of $q$ is equivalent to the category of $\mathcal H_q$-comodules.

Let X be a Cartesian product of s circles, p orientable 2-manifolds, q non-orientable 2-manifolds, r orientable 3-manifolds and t non-orientable 3-manifolds (all of them are closed). We prove that if either some of these r orientable 3-manifolds embed into R 4 or p + q + s + t > 0, then the lowest dimension of Euclidean space in which X is smoothly embeddable is s + 2p + 3(q + r) + 4t + 1. If none of the closed orientable 3-manifolds R 1 , . . . , R r embed into R 4 , then their product is embeddable into R 3r+2 and, at least for some cases, non-embeddable into R 3r+1 .

Given a triple cover π : X -→ Y of varieties, we produce a new variety SX and a birational morphism ρX : SX -→ X which is an isomorphism away from the fat-point ramification locus of π. The variety SX has a natural interpretation in terms of the data describing the triple cover, and the morphism ρX has an elegant geometric description.

In this paper, we present a number of examples of k-nets, which are special configurations of lines and points in the projective plane. Such a configuration can be regarded as the union of k completely reducible elements of a pencil of complex plane curves; equivalently it can be regarded as a set of k polygons in the complex projective plane that satisfy a condition of mutual perspectivity and nondegenerate intersection. For each example, we describe its construction, combinatorial properties, and parameter space. Most of the examples are historical, although perhaps not very well-known; our only essentially new example is a 3-net of pentagons which does not realize a group. The existence of this example settles a question posed by S. Yuzvinsky .

Given a triple cover π : X -→ Y of varieties, we produce a new variety SX and a birational morphism ρX : SX -→ X which is an isomorphism away from the fat-point ramification locus of π. The variety SX has a natural interpretation in terms of the data describing the triple cover, and the morphism ρX has an elegant geometric description.

In this paper, we present a number of examples of k-nets, which are special configurations of lines and points in the projective plane. Such a configuration can be regarded as the union of k completely reducible elements of a pencil of complex plane curves; equivalently it can be regarded as a set of k polygons in the complex projective plane that satisfy a condition of mutual perspectivity and nondegenerate intersection. For each example, we describe its construction, combinatorial properties, and parameter space. Most of the examples are historical, although perhaps not very well-known; our only essentially new example is a 3-net of pentagons which does not realize a group. The existence of this example settles a question posed by S. Yuzvinsky .

Advanced Transfer Path Analysis (ATPA) is a technique that allows the characterisation of vibroacoustic systems not only from the point of view of contributions but also topologically by means of the path concept. Some of the aspects addressed in the current research such as the proper characterisation of the less contributing paths remained not proven. ATPA is applied to a cuboid-shaped box. The simplicity of this vibroacoustic system helps to make a detailed analysis of the ATPA method in a more controlled environment than in situ measurements in trains, wind turbines or other mechanical systems with complex geometry, big dimensions and movement. At the same time, a numerical model (based on finite elements) of the box is developed. The agreement between the experimental measurements and the numerical results is good. The numerical model is used in order to perform tests that cannot be accomplished in practise. The results are helpful in order to verify hypotheses, provide recommendations for the testing procedures and study some aspects of ATPA such as the reconstruction of operational signals by means of direct transfer functions or to quantify and understand which are the transmission mechanisms in the box.

We describe a Hopf algebraic approach to the Grothendieck ring of representations of subgroups $H_\pi$ of the general linear group GL(n) which stabilize a tensor of Young symmetry $\{\pi\}$. It turns out that the representation ring of the subgroup can be described as a Hopf algebra twist, with a 2-cocycle derived from the Cauchy kernel 2-cocycle using plethysms. Due to Schur-Weyl duality we also need to employ the coproduct of the inner multiplication. A detailed analysis including combinatorial proofs for our results can be found in math-ph/0505037. In this paper we focus on the Hopf algebraic treatment, and a more formal approach to representation rings and symmetric functions.

This text presents an overview of recent developments on compactifications of moduli stacks of shtukas. The aim is to explain how to tackle the problem of compactifying stacks of shtukas by two different methods: the Langton semistable reduction and the Geometric Invariant Theory.

In this research, we delve into the intricate interplay between algebraic structures and geometric properties in the context of morphisms in affine spaces. Specifically, we focus on a morphism ϕ : A n → A n defined by polynomials (f 1 , f 2 , . . . , f n ) and investigate the conditions under which ϕ becomes an automorphism. We provide a comprehensive interpretation of the Jacobian matrix J(ϕ), a fundamental tool in understanding local behavior, emphasizing its role in capturing the local sensitivity of each component of ϕ to changes in the variables. We investigate the implications of having a non-zero constant Jacobian determinant, establishing its pivotal role in ensuring both injectivity and surjectivity of ϕ. The connectedness of the affine space A n proves to be a crucial factor, allowing us to combine local inverses obtained from the Implicit Function Theorem to construct a global inverse for ϕ. Beyond the theoretical framework, our research opens up exciting avenues for future exploration. We propose directions for generalizing these results to more abstract spaces, investigating morphisms defined by non-polynomial functions, and exploring applications in computer science, cryptography, and algebraic geometry.In conclusion, our study contributes to the rich tapestry of mathematical structures, revealing the profound connections between algebraic properties and geometric structures in the realm of morphisms and automorphisms.

The notion of entropy appears in many fields and this paper is a survey about entropies in several branches of Mathematics. We are mainly concerned with the topological and the algebraic entropy in the context of continuous endomorphisms of locally compact groups, paying special attention to the case of compact and discrete groups respectively. The basic properties of these entropies, as well as many examples, are recalled. Also new entropy functions are proposed, as well as generalizations of several known definitions and results. Furthermore we give some connections with other topics in Mathematics as Mahler measure and Lehmer Problem from Number Theory, and the growth rate of groups and Milnor Problem from Geometric Group Theory. Most of the results are covered by complete proofs or references to appropriate sources.

A tools and techniques of neutrosophic graph have found many applications in different areas such as topology, networks, computer of science, etc. In addition, neutrosophic graph is a generalization of intuitionistic fuzzy graph. Therefore, in this paper we study some characteristics of neutrosopheic graphs (NTCG) and some basic definitions. Moreover we investigate several kinds of arcs, 𝜂 𝜇-strong, 𝜎 𝜇-strong, 𝜌 𝜇-arc, and 𝜂 𝜉-strong, 𝜎 𝜉-strong, 𝜌 𝜉-arc in neutrosopheic graphs (NTCG) , Finally we give neutrosophic μ-bridge and neutrosophic 𝜉-bridge (NTC 𝜉-bridge) and some interesting properties of neutrosophic bridge (NTCB), which is being taught for the first time, and obtain several important properties.

The main purpose of this paper is to define the notion of neutrosophic based normal and regular spaces. This study investigates and open new class and conception of generalization of classical regular and normal spaces. The hereditary and topological properties of neutrosophic based normal and regular spaces have been analyzed and investigated. It also examines neutrosophic topological subspaces, providing insights into their characteristics. Furthermore, the paper investigates neutrosophic regular spaces and demonstrates their hereditary nature, specifically focusing on 𝑅 1 , 𝑅 2 , 𝑅 3 and 𝑅 4. Additionally, we explain some example of a neutrosophic-regular based space X which is a neutrosophic based normal-space but it is not necessary to neutrosophic-𝑇 1 spaces. Eventually, it is shown that under certain conditions that the images are preserved in neutrosophic based normal and regular spaces.

We prove the Collatz Conjecture by establishing a complete arithmetic and dynamical closure of the map T (n) =    n/2, n even, 3n + 1, n odd. Odd integers are classified by their residues modulo 18, and the least-admissible reverse lift determines a unique parent for each non-multiple of 3. This defines a deterministic, non-branching reverse graph. The only descending reverse case k = 1 in residue class C 1 is arithmetically bounded by 3-adic valuation, eliminating the sole infinite descent corridor. All other admissible lifts strictly ascend, and no nontrivial odd cycles exist. Ternary cylinder sets encode every reverse path, and their affine recursion partitions N odd without overlap. Finite reverse depth implies forward convergence to the trivial cycle 1 → 4 → 2 → 1 for every positive integer n.

In this paper, we derive a nonlinear Picone identity to the pseudo p-Laplace operator, which contains some known Picone identities and removes a condition used in many previous papers. Some applications are given including a Liouville type theorem to the singular pseudo p-Laplace system, a Sturmian comparison principle to the pseudo p-Laplace equation, a new Hardy type inequality with weight and remainder term, a nonnegative estimate of the functional associated to pseudo p-Laplace equation.

We show that the Cappell-Shaneson version of Pick's theorem for simple lattice polytopes is a consequence of a general relation between characteristic numbers of virtual submanifolds dual to the characteristic classes of a stably almost complex manifold. This relation is analogous to the miraculous cancellation formula of Alvarez-Gaume and Witten, and is imposed by the action of the Landweber-Novikov algebra in the complex cobordism ring of a point. 2000 Mathematics Subject Classification. 57R77.

This paper presents a complete, unconditional, and globally valid proof of the Hodge Conjecture for all smooth projective complex varieties, without restriction on dimension, type, or auxiliary conjectures.

The proof develops a unified fixed-point correspondence calculus that embeds classical Hodge theory into a Banach correspondence space equipped with ℓ¹–summable kernels. Within this analytic–geometric framework, algebraic and homological projectors are shown to coincide via contraction mappings, establishing that all rational (p,p)-cohomology classes arise from algebraic cycles.

Key contributions:
• Construction of a fixed-point correspondence space Cp(X) linking algebraic cycles and cohomology projectors.
• Definition of norm-controlled Lefschetz–Künneth projectors acting as contractions on non-(p,p) components.
• Application of the Banach fixed-point theorem to generate algebraic projectors realizing all (p,p)-classes.
• Proof of algebraicity of Lefschetz operators and primitive projectors, extended inductively to all dimensions.
• Globalization of the construction to families and products, ensuring full closure under base change.
• Introduction of higher-order operator structures — the Ω–lattice, spectral fixed-point refinements, and braided tensor envelopes — demonstrating extensibility to broader algebraic and analytic settings.

Significance:
This work resolves the Hodge Conjecture in the Clay Institute sense, proving that every rational (p,p)-class on every smooth projective complex variety corresponds to an algebraic cycle. It unites fixed-point theory, algebraic geometry, and functional analysis into a single invariant framework that establishes the conjecture unconditionally and globally.

This paper presents a fully rigorous mathematical identity yielding a closed-form expression for the inverse fine-structure constant (α⁻¹).
Using modular theory at the self-dual point τ = i, the work proves that

C = (32 / 3π) [ K(1/√2)² / π³⁄² + 8 S̃(π/6) ]

evaluates numerically to 137.035999084 …, matching the CODATA 2022 low-energy value of α⁻¹ to 13 significant digits with no empirical fitting.

The derivation combines Euler’s evaluations of even zeta values, Ramanujan’s Eisenstein-series expansions, the elliptic–Gamma identity, and the Eichler–Shimura anchoring that removes integration-constant ambiguity.
Every term is built only from classical constants — π, Γ(¼), zeta values, and theta constants — making the result entirely symbolic and unconditional.

Beyond its mathematical significance, the paper discusses the potential link between modular invariance and physical coupling constants, suggesting that α⁻¹ may arise naturally from modular symmetry rather than numerical coincidence.

The recursively-constructed family of Mandelbrot matrices Mn for n = 1, 2, . . . have nonnegative entries (indeed just 0 and 1, so each Mn can be called a binary matrix) and have eigenvalues whose negatives -λ = c give periodic orbits under the Mandelbrot iteration, namely z k = z 2 k-1 + c with z0 = 0, and are thus contained in the Mandelbrot set. By the Perron-Frobenius theorem, the matrices Mn have a dominant real positive eigenvalue, which we call ρn. This article examines the eigenvector belonging to that dominant eigenvalue and its fractal-like structure, and similarly examines (with less success) the dominant singular vectors of Mn from the singular value decomposition. This article seeks to explain a visual curiosity, namely that of Figure , where visible structures seem to repeat, slightly transformed, at smaller scales. But what do we mean by an "explanation?" What constitutes a mathematical explanation? By the way, those structures are not the result of rounding errors, in spite of our doing the computation only in standard hardware precision floating-point arithmetic. We will see a connection with the Mandelbrot set. After seeing the name Mandelbrot get involved, the reader might no longer be surprised that repeating transformed

In this paper we show if R is a filtered ring and M a filtered R module then we can define a valuation on a module for M. Then we show that we can find an skeleton of valuation on M, and we prove some properties such that derived form it for a filtered module.

We develop a typed, base-parametric framework for comparing and composing mathematical universes (theory + semantics packages) without positing an absolute background. Objects are universes; 1-cells are admissible translations with comparison data; evaluations take values in a chosen quantaloid ℬ. The base order is explicit: 𝑎 ⪰ ℬ 𝑏 reads "𝑎 is no worse than 𝑏" (e.g. numerically smaller cost), so joins encode best choices.

The Weitzenböck theorem states that if ∆ is a linear locally nilpotent derivation of the polynomial algebra K[Z] = K[z 1 , . . . , zm] over a field K of characteristic 0, then the algebra of constants of ∆ is finitely generated. If m = 2n and the Jordan normal form of ∆ consists of 2 × 2 Jordan cells only, we may assume that Nowicki conjectured that the algebra of constants K[X, Y ] ∆ is generated by x 1 , . . . , xn and x i y j -x j y i , 1 ≤ i < j ≤ n. Recently this conjecture was confirmed in the Ph.D. thesis of Khoury, and also by Derksen. In this paper we give an elementary proof of the conjecture of Nowicki. Then we find a very simple system of defining relations of the algebra K[X, Y ] ∆ which corresponds to the reduced Gröbner basis of the related ideal with respect to a suitable admissible order, and present an explicit basis of K[X, Y ] ∆ as a vector space.

This note contains a complete proof of the Abhyankar-Moh-Suzuki theorem (in characteristic zero case). In the zero characteristic case the AMS Theorem which was independently proved by Abhyankar-Moh and Suzuki (see [AM] and [Su]) and later reproved by many authors (see [AO], [AB], [Es], [Gu], [GM], [Ka], [Mi], [Ri], [Ru], [Za]; the list is probably incomplete) states the following AMS Theorem. If f and g are polynomials in K[z] of degrees n and m for which K[f, g] = K[z] then n divides m or m divides n.

In the paper The inversion formulae for automorphisms of polynomial algebras and differential operators in prime characteristic, J. Pure Appl. Algebra 212 (2008), no. 10, 2320-2337, see also Arxiv:math.RA/0604477, Vladimir Bavula states the following Conjecture: (BC) Any endomorphism of a Weyl algebra (in a finite characteristic case) is a monomorphism. The purpose of this preprint is to prove BC for A 1 , show that BC is wrong for An when n > 1, and prove an analogue of BC for symplectic Poisson algebras.

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Let $K$ be an algebraically closed field of arbitrary characteristic. Let $A$ be an affine domain over $K$ with transcendence degree 1 which is not isomorphic to $K[x]$, and let $B$ be a domain over $K$. We show that the AK invariant distributes over the tensor product of $A$ by $B$. As a consequence, we obtain a generalization of the cancellation theorem of S. Abhyankar, P. Eakin, and W. Heinzer.

In this paper we give a description of hypersurfaces with AK(S) = C. Let X be an affine variety and let G(X) be the group generated by all C + -actions on X. Then AK(X) ⊂ O(X) is the subring of all regular G(X)-invariant functions on X. We give here a description of affine surfaces S with AK(S) ∼ = C. We show that if AK(S) = C then S is quasihomogeneous and so may be obtained from a smooth rational projective surface by deleting a divisor of special form, which is called a "zigzag". We denote by A the set of all such surfaces, and by H those which have only three components in the zigzag. We prove that for a surface S with AK(S) ∼ = C the following statements are equivalent: 1. S is isomorphic to a hypersurface; 2. S is isomorphic to a hypersurface S ′ = {(x, y, z) ∈ C 3 |xy = p(z)}, where p is a polynomial with simple roots only; 3. S admits a fixed-point free C + -action; 4. S ∈ H. Moreover, if S 1 ∈ H and S 2 ∈ A \ H, then S 1 × C k ≃ S 2 × C k for any k ∈ N.

Let A be an absolute valued algebra with involution, in the sense of . We prove that A is finite-dimensional if and only if the algebra obtained by symmetrizing the product of A is simple, if and only if eAs = As, where e denotes the unique nonzero self-adjoint idempotent of A, and As stands for the set of all skew elements of A. We determine the idempotents of A, and show that A is the linear hull of the set of its idempotents if and only if A is equal to either McClay's algebra [1], the para-quaternion algebra, or the para-octonion algebra. We also prove that, if A is infinite-dimensional, then it can be enlarged to an absolute valued algebra with involution having a nonzero idempotent different from the unique nonzero self-adjoint idempotent. 2000 Mathematics Subject Classification. 17A80. Partially supported by Junta de Andalucía grant FQM 0199 and Projects I+D MCYT BFM2001-2335 and BFM2002-01810. * * is the closed linear hull of {x n : n ∈ N \ {1}}, and then that A s (-1) = 0. Applying Proposition 2.3, we deduce that there is no nonzero idempotent in A different from x 1 .

The tropical Grassmannian parameterizes tropicalizations of ordinary linear spaces, while the Dressian parameterizes all tropical linear spaces in TP n-1 . We study these parameter spaces and we compute them explicitly for n ≤ 7. Planes are identified with matroid subdivisions and with arrangements of trees. These representations are then used to draw pictures.

The tropical Grassmannian parameterizes tropicalizations of ordinary linear spaces, while the Dressian parameterizes all tropical linear spaces in ${\Bbb T}{\Bbb P}^{n-1}$. We study these parameter spaces and we compute them explicitly for $n \leq 7$. Planes are identified with matroid subdivisions and with arrangements of trees. These representations are then used to draw pictures.

The secondary polytope of a point configuration A is a polytope whose face poset is isomorphic to the poset of all regular subdivisions ofA. While the vertices of the secondary polytope – corresponding to the triangulations ofA – are very well studied, there is not much known about the facets of the secondary polytope. The splits of a polytope, subdivisions with exactly two maximal faces, are the most simple example of such facets and the first that were systematically investigated. The present paper can be seen as a continuation of this and as a starting point of an examination of the subdivisions corresponding to the facets of the secondary polytope in general. As a special case, the notion of k-split will be introduced as a possibility to classify polytopes in accordance to the complexity of the facets of their secondary polytopes. An application to matroid subdivisions of hypersimplices and tropical geometry is given.

The manuscript addresses the problem of finding all solutions of power flow equations or other similar nonlinear system of algebraic equations. This problem arises naturally in a number of power systems contexts, most importantly in the context of direct methods for transient stability analysis and voltage stability assessment. We introduce a novel form of homotopy continuation method called the numerical polynomial homotopy continuation (NPHC) method that is mathematically guaranteed to find all the solutions without ever encountering a bifurcation. The method is based on embedding the real form of power flow equation in complex space, and tracking the generally unphysical solutions with complex values of real and imaginary parts of the voltage. The solutions converge to physical real form in the end of the homotopy. The so-called γ-trick mathematically rigorously ensures that all the paths are wellbehaved along the paths, so unlike other continuation approaches, no special handling of bifurcations is necessary. The method is embarrassingly parallelizable and can be applied to reasonably large sized systems. We demonstrate the technique by analysis of several standard test cases up to the 14-bus system size. Finally, we discuss possible strategies for scaling the method to large size systems, and propose several applications for transient stability analysis and voltage stability assessment.

In this work, join and meet algebraic structure which exists in non-near-linear finite geometry are discussed. Lines in non-near-linear finite geometry 2  d were expressed as products of lines in near-linear finite geometry 2 p  (where p is a prime). An existence of lattice between any pair of near-linear finite geometry ( ) 2 , 1, , =  j p j k  of 2  d is confirmed. For | q d , a one-to-one cor- respondence between the set of subgeometry 2 j q  of 2  d and finite geometry 2  d from the subsets of the set ( ) { }  d of divisors of d (where each divisor represents a finite geometry) and set of subsystems ( ) { } Π q (with variables in q  ) of a finite quantum system ( ) Π d with variables in  d and a finite system from the subsets of the set of divisors of d is established.

Throughout this paper K is a field, K its algebraic closure and Gal(K) = Aut( K/K) the absolute Galois group of K. If X is an abelian variety over K then we write End(X) for the ring of all its K-endomorphisms ; the notation 1 X stands for the identity automorphism of X. If Y is an abelian variety over K then we write Hom(X, Y ) for the corresponding group of all K-homomorphisms. Let f (x) ∈ K[x] be a polynomial of degree n ≥ 2 with coefficients in K and without multiple roots, R f ⊂ Ka the (n-element) set of roots of f and K(R f ) ⊂ Ka the splitting field of f . We write Gal(f ) = Gal(f /K) for the Galois group Gal(K(R f )/K) and call it the Galois group of f (x) over K; it permutes roots of f and may be viewed as a certain permutation group of R f , i.e., as a subgroup of the group Perm(R f ) ∼ = S n of permutation of R f . (It is well known that Gal(f ) is transitive if and only if f is irreducible.