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Let X denote a quasi-projective variety over a field on which a connected linear algebraic group G acts with finitely many orbits. Then, the G-orbits define a stratification of X. We establish several key properties of the category of equivariant perverse sheaves on X, which have locally constant cohomology sheaves on each of the orbits. Under the above assumptions, we show that this category comes close to being a highest weight category in the sense of Cline, Parshall and Scott and defines a quasi-hereditary algebra. We observe that the above hypotheses are satisfied by all toric varieties and by all spherical varieties associated to connected reductive groups over any algebraically closed field. Next we show that the odd dimensional intersection cohomology sheaves vanish on all spherical varieties defined over algebraically closed fields of positive characteristics, extending similar results for spherical varieties defined over the field of complex numbers by Michel Brion and the author in prior work. Assuming that the linear algebraic group G and the action of G on X are defined over a finite field F q , and where the odd dimensional intersection cohomology sheaves on the orbit closures vanish, we also establish several basic properties of the mixed category of mixed equivariant perverse sheaves so that the associated terms in the weight filtration are finite sums of the shifted equivariant intersection cohomology complexes on the orbit closures.

We evaluate definite integrals of the form given by $\int_{0}^{\infty}R(a, x)\log (\cos (\alpha) \text{sech}(x)+1)dx$. The function $R(a, x)$ is a rational function with general complex number parameters. Definite integrals of this form yield closed forms for famous integrals in the books of Bierens de Haan [4] and Gradshteyn and Ryzhik [5].

In this article I propose a new criterion to extend the Standard Model of particle physics from a straightforward algebraic conjecture: the symmetries of physical microscopic forces originate from the automorphism groups of main Cayley–Dickson algebras, from complex numbers to octonions and sedenions. This correspondence leads to a natural enlargement of the Standard Model color sector, from a SU(3) gauge group to an exceptional Higgs-broken G(2) group, following the octonionic automorphism relation guideline. In this picture, an additional ensemble of massive G(2)-gluons emerges, which is separated from the particle dynamics of the Standard Model.

To each variety $X$ and a nonnegative integer $m$, there is a space $X_m$ over $X$, called the jet scheme of $X$ of order $m$, parametrizing $m$-th jets on $X$. Its fiber over a singular point of $X$ is called a singular fiber. For a surface with a rational double point, Mourtada gave a one-to-one correspondence between the irreducible components of the singular fiber of $X_m$ and the exceptional curves of the minimal resolution of $X$ for $m \gg 0$. In this paper, for a surface $X$ over complex number with a singularity of $A_n$ or $D_4$-type, we study the intersections of irreducible components of the singular fiber and construct a graph using this information. The vertices of the graph correspond to irreducible components of the singular fiber and two vertices are connected when the intersection of the corresponding components is maximal for the inclusion relation. In the case of $A_n$ or $D_4$-type singularity, we show that this graph is isomorphic to the resolution graph for $m \gg 0$., 18 pages

We prove that there exists a functorial correspondence between MV-algebras and partially cyclically ordered groups which are wound round of lattice-ordered groups. It follows that some results about cyclically ordered groups can be stated in terms of MV-algebras. For example, the study of groups together with a cyclic order allows to get a first-order characterization of groups of unimodular complex numbers and of finite cyclic groups. We deduce a characterization of pseudofinite MV-chains and of pseudo-simple MV-chains (i.e. which share the same first-order properties as some simple ones). We can generalize these results to some non-lineraly ordered MV-algebras, for example hyper-archimedean MV-algebras.

Given two different complex numbers a and b, a bounded linear operator T acting on an infinite-dimensional complex Hilbert space H is said to be {a, b}-quadratic if (T − a)(T − b) = 0. We provide in this paper a complete description of all surjective maps Φ (not necessarily additive) on the algebra $${\cal B}(H)$$ of all bounded linear operators on H that satisfy S − λT is {a, b}-quadratic if and only if Φ(S) − λΦ(T) is {a, b}-quadratic for every $$S,T \in {\cal B}(H)$$ and λ ∈ ℂ.

“Real Normed Algebras Revisited,” the last paper of the late Gadi Moran, attempts to reconstruct the discovery of the complex numbers, the quaternions, and the octonions, as well as proofs of their...

UDC 517.54 The aim of this work is to weaken the conditions of monogeneity for functions that take values in a given three-dimensional commutative algebra over the field of complex numbers. The monogeneity of the function is understood as a combination of its continuity and the existence of the Gateaux derivative.

Let A and B be two closed linear relations acting between two Banach spaces X and Y , and let λ be a complex number. We study the stability of the nullity and deficiency of A when it is perturbed by λ B . In particular, we show the existence of a constant ρ > 0 for which both the nullity and deficiency of A remain stable under perturbation by λ B for all λ inside the disk λ < ρ .

Upon reading the famous book on integral transforms volume II by Erdeyli et al., we encounter a formula which we use to derive a Mellin transform given by ∫ 0 ∞ x m − 1 log k a x / β 2 + x 2 γ + x d x , where the parameters a , k , β , and γ are general complex numbers. This Mellin transform will be derived in terms of the Lerch function and is not listed in current literature to the best of our knowledge. We will use this transform to create a table of definite integrals which can be used to extend similar tables in current books featuring such formulae.

We state and prove properties of the solution set of the extension of one of Wilker’s inequalities to the field of complex numbers. Wilker’s inequalities are inequalities for real numbers which involve trigonometric or hyperbolic functions. We will consider one of those inequalities on the set of complex numbers, and some important analytical properties of the solution set will be proved. Some auxiliary results will be established along the way.

We show that any two planar cubic embeddings of a pseudoplanar curve of genus 1 (over complex numbers) are linearly equivalent. We describe the ‘abstract moduli’ of nonsingular affine plane cubic curves with 3 places at infinity. We note some examples of inequivalent planar embeddings of a nonsingular affine curve of genus 0 having 2 places at infinity.

Involving a symmetric Hochschild 1-cocycle condition, we equip the space of decorated planar rooted forests with a coproduct which turns the space into a dendriform-Nijenhuis bialgebra. We combine dendriform-Nijenhuis bialgebras with operated algebras and introduce the notation of an Ω-operated DN-bialgebra. Applying the universal property of the underlying operated algebras, we construct free objects in the category of Ω-cocycle DN-bialgebras. We introduce the notation of a DN-associative Yang-Baxter equation (AYBE) and show that the dendriform-Nijenhuis bialgebra offers an algebraic framework of the DN-associative Yang-Baxter equation. We construct a Leroux's TD operator from a solution of the DN-AYBE. We also give two different ways to derive Lie algebras from quasitriangular DN-bialgebras. Finally, we classify the solutions of the DN-AYBE in the unitary algebras of dimensions two and three over the field of complex numbers.

In this paper, we further Meirong Zhang, et al.'s work by computing the number of weighted eigenvalues for Sturm-Liouville equations, equipped with general integrable potentials and Dirac weights, under Dirichlet boundary condition. We show that, for a Sturm-Liouville equation with a general integrable potential, if its weight is a positive linear combination of $n$ Dirac Delta functions, then it has at most $n$ (may be less than $n$, or even be $0$) distinct real Dirichlet eigenvalues, or every complex number is a Dirichlet eigenvalue; in particular, under some sharp condition, the number of Dirichlet eigenvalues is exactly $n$. Our main method is to introduce the concepts of characteristics matrix and characteristics polynomial for Sturm-Liouville problem with Dirac weights, and put forward a general and direct algorithm used for computing eigenvalues. As an application, a class of inverse Dirichelt problems for Sturm-Liouville equations involving single Dirac distribution weights is studied., 23 pages

A complete description of multipliers of Fourier–Haar series in Lorentz spaces is given. Necessary and sufficient conditions for sequences of complex numbers to belong to the class $$m(L_{p,r}\to L_{q,s})$$ of multipliers of Fourier–Haar series are obtained.

В статье рассматриваются различные формы записи комплексных и гиперкомплексных чисел: алгебраическая (стандартная); в парах; матричная форма. Приведены примеры вычислений в перечисленных способах записи чисел.

The list Λ = { λ 1 , λ 2 , … , λ n } , of complex numbers, is said to be realizable if it is the spectrum of an entrywise nonnegative matrix A. If A is diagonalizable, Λ is said to be diagonalizably realizable. Λ is universally realizable if it is realizable for each possible Jordan canonical form allowed by Λ. Here, new realizability criteria are introduced. A diagonalizable version of a perturbation result by Rado is also proved. It allows to construct diagonalizable nonnegative matrices with prescribed spectrum. Criteria to decide the universal realizability of spectra are also established.

Let [Formula: see text] be a complex number, and a class of non-semisimple and non-solvable rank two Lie conformal algebras [Formula: see text] are introduced. In this paper, conformal derivations, conformal quasiderivations, generalized conformal derivations and conformal biderivations of [Formula: see text] are studied. Besides, central extensions and conformal modules of rank one of [Formula: see text] are determined.

We establish a formula for computing the unramified Brauer group of tame conic bundle threefolds in characteristic 2. The formula depends on the arrangement and residue double covers of the discriminant components, the latter being governed by Artin-Schreier theory (instead of Kummer theory in characteristic not 2). We use this to give new examples of threefold conic bundles defined over the integers that are not stably rational over the complex numbers., 29 pages; v2: added Prop. 3.11, removed assumption of rationality for the base surface of the conic bundles, other small improvements

Let $\varrho$ be a complex number and let $f$ be a multiplicative arithmetic function whose Dirichlet series takes the form $\zeta(s)^\varrho G(s)$, where $G$ is associated to a multiplicative function $g$. The classical Selberg-Delange method furnishes asymptotic estimates for averages of $f$ under assumptions of either analytic continuation for $G$, or absolute convergence of a finite number of derivatives of $G(s)$ at $s=1$. We consider different set of hypotheses, not directly comparable to the previous ones, and investigate how they can yield sharp asymptotic estimates for the averages of~$f$.

The unwinding number of a complex number was introduced to process automatic computations involving complex numbers and multi-valued complex functions, and has been successfully applied to computations involving branches of the Lambert W function. In this partly expository note we discuss the unwinding number from a purely topological perspective, and link it to the classical winding number of a curve in the complex plane. We also use the unwinding number to give a representation of the branches $$W_k$$ W k of the Lambert W function as a line integral.

We suggest an algorithm for finding the group of birational automorphisms of a hyperelliptic curve y2 = p(x), p ∈ ℚ[x], over the field of complex numbers based on power series expansion. We present an implementation of this algorithm in the computer algebra system Sage and examples of its application. Computer experiments show that the algorithm does not lead to unexpectedly cumbersome calculations. The format of the group representation allows one to use the embedded Sage tools for dealing with finite groups of low order. Bibliography: 20 titles.

The [Formula: see text]-algebra of periods was introduced by Kontsevich and Zagier as complex numbers whose real and imaginary parts are values of absolutely convergent integrals of [Formula: see text]-rational functions over [Formula: see text]-semi-algebraic domains in [Formula: see text]. The Kontsevich–Zagier period conjecture affirms that any two different integral expressions of a given period are related by a finite sequence of transformations only using three rules respecting the rationality of the functions and domains: additions of integrals by integrands or domains, change of variables and Stokes formula. In this paper, we prove that every non-zero real period can be represented as the volume of a compact [Formula: see text]-semi-algebraic set obtained from any integral representation by an effective algorithm satisfying the rules allowed by the Kontsevich–Zagier period conjecture.

We give an algebraic proof for which log del Pezzo pairs of Maeda type are K-polystable or not. If the base field is the complex number field, then the result is already known by Li and Sun., 30 pages

Let $\mathcal{C}$ be a fusion category over an algebraically closed field $\mathbb{k}$ of arbitrary characteristic. Two numerical invariants of $\mathcal{C}$ , that is, the Casimir number and the determinant of $\mathcal{C}$ are considered in this paper. These two numbers are both positive integers and admit the property that the Grothendieck algebra $(\mathcal{C})\otimes_{\mathbb{Z}}K$ over any field K is semisimple if and only if any of these numbers is not zero in K. This shows that these two numbers have the same prime factors. If moreover $\mathcal{C}$ is pivotal, it gives a numerical criterion that $\mathcal{C}$ is nondegenerate if and only if any of these numbers is not zero in $\mathbb{k}$ . For the case that $\mathcal{C}$ is a spherical fusion category over the field $\mathbb{C}$ of complex numbers, these two numbers and the Frobenius–Schur exponent of $\mathcal{C}$ share the same prime factors. This may be thought of as another version of the Cauchy theorem for spherical fusion categories.

We present an eight-dimensional even sub-algebra of the 2^4 = 16-dimensional associative Clifford algebra Cl(4,0) and show that its eight-dimensional elements denoted as X and Y respect the norm relation ||XY|| = ||X|| ||Y||, thus forming an octonion-like but associative normed division algebra, where the norms are calculated using the fundamental geometric product instead of the usual scalar product so that the underlying coefficient algebra resembles that of split complex numbers instead of reals. The corresponding 7-sphere has a topology that differs from that of octonionic 7-sphere.

The derivation of integrals in the table of Gradshteyn and Ryzhik in terms of closed form solutions is always of interest. We evaluate several of these definite integrals of the form $\int_{0}^{\infty}\ln^k(\alpha y)\ln(R(y))dy$ in terms of a special function, where $R(y)$ is a general function and $k$ and $\alpha$ are arbitrary complex numbers.

We say that a list Λ = { λ 1 , λ 2 , … , λ n } of complex numbers is realizable if it is the spectrum of an entrywise nonnegative matrix A. We say that Λ is universally realizable (UR) if Λ is realizable for each possible Jordan canonical form allowed by Λ. The universal realizability problem of spectra (URP) has been solved for certain kind of real and complex spectra, mainly spectra in the left half-plane. In this paper we look for extending the kind of spectra that are UR by given new sufficient conditions for the URP to have a solution. In particular, we show that some realizability criteria for the real and symmetric nonnegative inverse eigenvalue problem are realizability criteria for the URP as well.

We derive families of Newton-like inequalities involving the elementary symmetric functions of sets of self-conjugate complex numbers in the right half-plane. These are the first known inequalities of this type which are independent of the proximity of the complex numbers to the real axis.

In this article, we investigate bicomplex triple Laplace transform in the framework of bicomplexified frequency domain with Region of Convergence (ROC), which is generalization of complex triple Laplace transform. Bicomplex numbers are pairs of complex numbers with commutative ring with unity and zero-divisors, which describe physical interpretation in four dimensional spaces and provide large class of frequency domain. Also, we derive some basic properties and inversion theorem of triple Laplace transform in bicomplex space. In this technique, we use idempotent representation methodology of bicomplex numbers, which play vital role in proving our results. Consequently, the obtained results can be highly applicable in the fields of Quantum Mechanics, Signal Processing, Electric Circuit Theory, Control Engineering, and solving differential equations. Application of bicomplex triple Laplace transform has been discussed in finding the solution of third-order partial differential equation of bicomplex-valued function.

The Victoria equation, a generalization of De Moivre’s formula in 1+n dimensional scator algebra, is inverted to obtain the roots of a scator. For the qth root in S1+n of a real or a scator number, there are qn possible roots. For n=1, the usual q complex roots are obtained with their concomitant cyclotomic geometric interpretation. For n≥2, in addition to the previous roots, new families arise. These roots are grouped according to two criteria: sets satisfying Abelian group properties under multiplication and sets catalogued according to director conjugation. The geometric interpretation is illustrated with the roots of unity in S1+2.

It is well known that Galilean conformal algebras play important roles in the nonrelativistic anti-de Sitter/conformal field theory correspondence. The finite Lie conformal algebras P G ( a , b ) of planar Galilean type can be viewed as Lie conformal analogues of certain planar Galilean conformal algebras. In this paper, we classify finite irreducible conformal modules over P G ( a , b ) for all complex numbers a and b.

Frobenius' Theorem states that, besides the fields of real and complex numbers, the algebra of quaternions H is the only finite-dimensional real division algebra. We first give a short elementary p...

We determine the rational divisor class group of the moduli spaces of smooth pointed hyperelliptic curves and of their Deligne-Mumford compactification, over the field of complex numbers., Comment: Accepted version

Given two algebras and , if lies in the Zariski closure of the orbit , we say that is a degeneration of . We denote this by . Degenerations (or contractions) were widely applied to a range of physical and mathematical point of view. The most well-known example oriented to the application on degenerations is limiting process from quantum mechanics to classical mechanics under that corresponds to the contraction of the Heisenberg algebras to the abelian ones of the same dimension. Research on degenerations of Lie, Leibniz and other classes of algebras are very active. Throughout the paper we are dealing with mathematical background with abstract algebraic structures. The present paper is devoted to the degenerations of low-dimensional nilpotent Leibniz algebras over the field of complex numbers. Particularly, we focus on the classification of three-dimensional nilpotent Leibniz algebras. List of invariance arguments are provided and its dimensions are calculated in order to find the possible degenerations between each pair of algebras. We show that for each possible degenerations, there exists construction of parameterized basis on parameter We proof the non-degeneration case for mentioned classes of algebras by providing some reasons to reject the degenerations. As a result, we give complete list of degenerations and non-degenerations of low-dimensional complex nilpotent Leibniz algebras. In future research, from this result we can find its rigidity and irreducible components.

We consider a class of plane orthotropic deformations of the form ex = σx + a12σy, γxy = 2(p − a12)Txy, ey = a12σx + σy, where σx, Txy, σy and$$ {\upvarepsilon}_x\frac{\upgamma_{xy}}{2},{\upvarepsilon}_Y $$ are components of the stress tensor and the deformation tensor, respectively, real parameters p and a12 satisfy the inequalities: -1 < p < 1, -1 < a12 < p. A class of solutions of the Lame equilibrium system for displacements is built in the form of linear combinations of components of “analytic” functions which take values in commutative and associative two-dimensional algebras with unity over the field of complex numbers.

This note is devoted to a weaker version of Bishop property $$\beta$$ at a given complex number. We show in particular that this notion is a regularity and hence the induced spectrum satisfies all classical properties of the spectrum.

After a general discussion of group actions, orbifolds, and "weak orbifolds" this note will provide elementary introductions to two basic moduli spaces over the real or complex numbers: First the moduli space of effective divisors with finite stabilizer on the projective space ${\mathbb P}^1$ modulo the group ${\rm PGL}_2$ of projective transformations of ${\mathbb P}^1$; and then the moduli space of effective 1-cycles with finite stabilizer on ${\mathbb P}^2$ modulo the group ${\rm PGL}_3$ of projective transformations of ${\mathbb P}^2$.

Clifford algebra includes the real and complex numbers. The hyperbolic numbers also belong to Clifford algebra. In recent years, neural networks (NNs) are extended using Clifford algebra, and hyperbolic NNs have been proposed. Since the hyperbolic numbers have zero divisors, it is difficult to analyze the hyperbolic NNs. Thus, the reducibilities of hyperbolic NNs have never been revealed. In this work, the reducibilities of hyperbolic NNs are studied. The reducibilities are tightly related to learning process. In the real-valued and complex-valued NNs, there exist three types of reducibilities. In the hyperbolic NNs, there exists another type of reducibilities, and it has been difficult to determine all the reducibilities of hyperbolic NNs. It is proved that hyperbolic NNs have another reducibility, referred to as hyperbola-reducibility, and all the reducibities of hyperbolic NNs are determined. In addition, the inherent singularities of hyperbolic NNs are revealed. These facts are expected to improve the learning process of hyperbolic NNs in future.

Let $\mathcal{F}$ be a singular codimension one holomorphic foliation on a compact complex manifold $X$ of dimension at least three such that its singular set has codimension at least two. In this paper, we determine Lehmann-Suwa residues of $\mathcal{F}$ as multiples of complex numbers by integration currents along irreducible complex subvarieties of $X$. We then prove a formula that determines the Baum-Bott residue of simple almost Liouvillian foliations of codimension one, in terms of Lehmann-Suwa residues, generalizing a result of Marco Brunella. As an application, we give sufficient conditions for the existence of dicritical singularities of a singular real-analytic Levi-flat hypersurface $M\subset X$ tangent to $\mathcal{F}$.

Let (u??) be a double sequence of real or complex numbers which is (C; 1; 1) summable to a finite limit. We obtain some Tauberian conditions of slow decreasing or oscillating types in terms of the generator sequences in certain senses under which P-convergence of a double sequence (u??) follows from its (C,1,1) summability. We give Tauberian theorems in which Tauberian conditions are of Hardy and Landau types as special cases of our results. We present some Tauberian conditions in terms of the de la Vall?e Poussin means of double sequences under which P-convergence of a double sequence (u??) follows from its (C,1,1) summability. Moreover, we give analogous results for (C,1,0) and (C,0,1) summability methods.

International audience; The aim of this paper is to study, for a given sequence $(\rho_n)_{n\ge1}$ of complex numbers, the class of Hilbert space operators possessing $(\rho_n)$-unitary dilations. This is the class of bounded linear operators $T$ acting on a Hilbert space $H$, whose iterates $T^n$ can be represented as $T^n=\rho_n P_H U^n|_H$, $n \ge 1$, for some unitary operator $U$ acting on a larger Hilbert space, containing $H$ as a closed subspace. Here $P_H$ is the projection from this larger space onto $H$. The case when all $\rho_n$'s are equal to a positive real number $\rho$ leads to the class $C_{\rho}$ introduced in the 1960s by Foias and Sz.-Nagy, while the case when all $\rho_n$'s are positive real numbers has been previously considered by several authors. Some applications and examples of operators possessing $(\rho_n)$-unitary dilations, showing a behavior different from the classical case, are given in this paper.

In this paper by means of contour integration we will evaluate definite integrals of the form $ \begin{equation*} \int_{0}^{1}\left(\ln^k(ay)-\ln^k\left(\frac{a}{y}\right)\right)R(y)dy \end{equation*} $ in terms of a special function, where $R(y)$ is a general function and $k$ and $a$ are arbitrary complex numbers. These evaluations can be expressed in terms of famous mathematical constants such as the Euler's constant $\gamma$ and Catalan's constant $C$. Using derivatives, we will also derive new integral representations for some Polygamma functions such as the Digamma and Trigamma functions along with others.

Geometric algebras known as a generalization of Grassmann algebras complex numbers and quaternions are presented by Clifford (1878). Geometric algebra describing the geometric symmetries of both physical space and spacetime is a strong language for physics. Groups generated from `Clifford numbers` are firstly defined by Lipschitz (1886). They are used for defining rotations in a Euclidean space. In this work, Clifford algebra is identified. The energy of classic particles with Clifford algebra are defined. This calculations are applied to some Archimedean solids. Also, the vertices of Archimedean solids presented in the Cartesian coordinates are calculated.

The purpose of this paper is to study stable representations of partially ordered sets (posets) and compare it to the well known theory for quivers. In particular, we prove that every indecomposable representation of a poset of finite type is stable with respect to some weight and construct that weight explicitly in terms of the dimension vector. We show that if a poset is primitive then Coxeter transformations preserve stable representations. When the base field is the field of complex numbers we establish the connection between the polystable representations and the unitary $\chi$-representations of posets. This connection explains the similarity of the results obtained in the series of papers., Comment: Updated version. Appendix A was rewritten. Some corrections following the referee reports

The study of singularities of algebraic or analytic spaces over the field of complex numbers is a traditional subject, but, in contrast, the parallel theory over arbitrary algebraically closed fields is still poor and there are lots of interesting questions to be answered. Our main concern here is the study of the Milnor number μ of an isolated hypersurface singularity which is defined as the codimension of the ideal generated by the partial derivatives of a power series that represents locally the hypersurface. This is an important topological invariant of the singularity over the complex numbers, but its behavior changes dramatically when the base field has positive characteristic, in which case it may be infinite and depends upon the local equation of the hypersurface, not being an intrinsic invariant. In this paper we will study the variation of the Milnor number in terms of the local equations, showing that its minimum value has an intrinsic meaning and give necessary and sufficient conditions for its invariance. We also relate the smoothness of the generic fiber of an isolated hypersurface singularity deformation with the finiteness of this number, connecting it to a Bertini type result. At the end, we will show how, in arbitrary characteristic, one defines the sequence μ ⁎ of Milnor numbers of sections of a hypersurface singularity by general linear spaces of increasing codimension.

In this study, we generalize the well-known formulae of De-Moivre and Euler of complex numbers to dual-complex numbers. Furthermore, we investigate the roots and powers of a dual-complex number by using these formulae. Consequently, we give some examples to illustrate the main results in this paper.

Let $p$ be an analytic polynomial on the unit disk. We obtain a necessary and sufficient condition for Toeplitz operators with the symbol $\overline{z}+p$ to be invertible on the Bergman space when all coefficients of $p$ are real numbers. Furthermore, we establish several necessary and sufficient, easy-to-check conditions for Toeplitz operators with the symbol $\overline{z}+p$ to be invertible on the Bergman space when some coefficients of $p$ are complex numbers.