Search

A complete invariant defined for (closed, connected, orientable) 3-manifolds is an invariant defined for the 3-manifolds such that any two 3-manifolds with the same invariant are homeomorphic. Further, if the 3-manifold itself is reconstructed from the data of the complete invariant, then it is called a characteristic invariant defined for the 3-manifolds. In previous papers by the first author, a characteristic lattice point invariant and a characteristic rational invariant defined for the 3-manifolds were constructed which also produced a smooth real function with the definition interval ( − 1 , 1 ) as a characteristic invariant defined for the 3-manifolds. In this paper, a complex number-valued characteristic invariant for the 3-manifolds whose norm is smaller than or equal to one half is introduced by using an embedding of a set of lattice points called the ADelta set, distinct from the PDelta set, into the set of complex numbers. The distributive situation for the invariants of the 3-manifolds of lengths up to 10 is plotted in the complex number plane with radius smaller than or equal to one half. By using this complex number-valued characteristic invariant, a holomorphic function with the unit open disk as the definition domain which is called the characteristic quantity function is constructed as a characteristic invariant defined for the 3-manifolds.

本文给出了非均匀复数的定义，非均匀复解析函数的定义和性质，建立了非均匀复解析函数与偏微分方程的关系。 In this paper, we introduce the definitions of heterogeneous complex number and heterogeneous complex function, and then discuss the property of heterogeneous complex function. We establish the relationship between heterogeneous complex analytic function and partial differential equation system.

In this study, the planar kinematics has been studied in a generalized complex plane which is a geometric representation of the generalized complex number system. Firstly, the planar kinematic formulas with one parameter for homothetic motions in the generalized complex plane have been mentioned briefly. Then, the Steiner area formula given areas of the trajectories drawn by the points taken in a generalized complex plane have been obtained during the one-parameter planar homothetic motion. Finally, the Holditch theorem, which gives the relationship between these areas of trajectories, has been expressed for homothetic motions in a generalized complex plane. So, this theorem obtained in this study is the most general form of all Holditch theorems obtained so far.

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

We obtain $q$-analogues of several series for powers of $\pi$. For example, the identity $$\sum_{k=0}^\infty\frac{(-1)^k}{(2k+1)^3}=\frac{\pi^3}{32}$$ has the following $q$-analogue: \begin{equation*} \sum_{k=0}^\infty(-1)^k\frac{q^{2k}(1+q^{2k+1})}{(1-q^{2k+1})^3}=\frac{(q^2;q^4)_{\infty}^2(q^4;q^4)_{\infty}^6} {(q;q^2)_{\infty}^4}, \end{equation*} where $q$ is any complex number with $|q, Comment: 10 pages. Add Theorem 1.2

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~$f$.

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.

PurposeThe purpose of this current paper is to deal with the study of non-constant entire solutions of some non-linear complex differential equations in connection to Brück conjecture, by using the theory of complex differential equation. The results generalize the results due to Pramanik et al.Design/methodology/approach39B32, 30D35.FindingsIn the current paper, we mainly study the Brück conjecture and the various works that confirm this conjecture. In our study we find that the conjecture can be generalized for differential monomials under some additional conditions and it generalizes some works related to the conjecture. Also we can take the complex number a in the conjecture to be a small function. More precisely, we obtain a result which can be restate in the following way: Let f be a non-constant entire function such that σ2(f)<∞, σ2(f) is not a positive integer and δ(0, f)>0. Let M[f] be a differential monomial of f of degree γM and α(z), β(z)∈S(f) be such that max{σ(α), σ(β)} <σ(f). If M[f]+β and fγM−α share the value 0 CM, then M[f]+βfγM−α=c,where c≠0 is a constant.Originality/valueThis is an original work of the authors.

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.

Let $ f(z) $ be a transcendental meromorphic function of finite order and $ c\in\Bbb{C} $ be a nonzero constant. For any $ n\in\Bbb{N}^{+} $, suppose that $ P(z, f) $ is a difference polynomial in $ f(z) $ such as $ P(z, f) = a_{n}f(z+nc)+a_{n-1}f(z+(n-1)c)+\cdots+a_{1}f(z+c)+a_{0}f(z) $, where $ a_{k} (k = 0, 1, 2, \cdots, n) $ are not all zero complex numbers. In this paper, the authors investigate the uniqueness problems of $ P(z, f) $.

Let Tc := D(x - c)((x - c)D + 2II) be a second-order linear differential operator, where c is an arbitrary complex number, $$D: = \frac{d}{{dx}}$$ and II represents the identity on the linear space of polynomials with complex coefficients. The aim of this paper is to describe all of the Tc-classical orthogonal polynomials. Two canonical situations appear: the Laguerre $$\{L_n^{(2)}\}_{n\geq0}$$ and the Jacobi $$\{P_n^{(\alpha-2,2)}\}_{n\geq0}$$

Let μ1 be a complex number in the numerical range W(A) of a normal matrix A. In the case when no eigenvalues of A lie in the interior of W(A), we identify the smallest convex region containing all ...

A Gaussian Pisot number is an algebraic integer with modulus greater than one whose other conjugates, over the quadratic field Q ( − 1 ) , are of modulus less than one. Given a nonreal algebraic number α , with modulus greater than one, we prove that there is a nonzero complex number λ such that the series ∑ n ∈ N ( Re ( λ α n ) 2 + Im ( λ α n ) 2 ) converges (resp. such that the sequence ( { Re ( λ α n ) } , { Im ( λ α n ) } ) n ∈ N has a unique limit point), if and only if α is a Gaussian Pisot number (resp. α is a Gaussian Pisot number satisfying P ( 1 ) ≥ 2 or deg ( P ) = 2 , where P is the minimal polynomial of α over Q ).

All the roots of the general nth degree trinomial admit certain convenient representations in terms of the Lambert and Euler series for the asymmetric and symmetric cases of the trinomial equation, respectively. Previously, various methods have been used to provide the proofs for the general terms of these two series. Taking n to be any real or complex number, we presently give an alternative proof using the Bell (or exponential) polynomials. The ensuing series is summed up yielding a single, compact, explicit, analytical formula for all the trinomial roots as the confluent Fox–Wright function $${}_1\Psi _1$$ . Moreover, we also derive a slightly different, single formula of the trinomial root raised to any power (real or complex number) as another $${}_1\Psi _1$$ function. Further, in this study, the logarithm of the trinomial root is likewise expressed through a single, concise series with the binomial expansion coefficients or the Pochhammer symbols. These findings are anticipated to be of considerable help in various applications of trinomial roots. Namely, several properties of the $${}_1\Psi _1$$ function can advantageously be employed for its implementations in practice. For example, the simple expressions for the asymptotic limits of the $${}_1\Psi _1$$ function at both small and large values of the independent variable can be used to readily predict, by analytical means, the critical behaviors of the studied system in the two extreme conditions. Such limiting situations can be e.g. at the beginning of the time evolution of a system, and in the distant future, if the independent variable is time, or at low and high doses when the independent variable is radiation dose, etc. The present analytical solutions for the trinomial roots are numerically illustrated in the genome multiplicity corrections for survival of synchronous cell populations after irradiation.

In this paper, using the first-order Nedelec conforming edge element space of the second type, we develop and analyze a continuous interior penalty finite element method (CIP-FEM) for the time-harmonic Maxwell equation in the three-dimensional space. Compared with the standard finite element methods, the novelty of the proposed method is that we penalize the jumps of the tangential component of its vorticity field. It is proved that if the penalty parameter is a complex number with negative imaginary part, then the CIP-FEM is well-posed without any mesh constraint. The error estimates for the CIP-FEM are derived. Numerical experiments are presented to verify our theoretical results.

Let k be a nonzero complex number. We consider a k-circulant matrix whose first row is (J1, J2, . . . , Jn), where Jn is the n th Jacobsthal number, and obtain the formulae for the eigenvalues of such matrix improving the formula which can be obtained from the result of Y. Yazlik and N. Taskara [J. Inequal. Appl. 2013, 2013:394, Theorem 7]. The obtained formulae for the eigenvalues of a k-circulant matrix involving the Jacobsthal numbers show that the result of Z. Jiang, J. Li, and N. Shen [WSEAS Trans. Math. 12 (2013), no. 3, 341–351, Theorem 10] is not always applicable. The Euclidean norm of such matrix is determined. We also consider a k-circulant matrix whose first row is (J −1 1 , J−1 2 , . . . , J−1 n ) and obtain the upper and lower bounds for its spectral norm

In the present paper, we study frames for finite-dimensional vector spaces over an arbitrary field. We develop a theory of dual frames in order to obtain and study the different representations of the elements of the vector space provided by a frame. We relate the introduced theory with the classical one of dual frames for Hilbert spaces and apply it to study dual frames for three types of vector spaces: for vector spaces over conjugate closed subfields of the complex numbers (in particular, for cyclotomic fields), for metric vector spaces, and for ultrametric normed vector spaces over complete non-archimedean valued fields. Finally, we consider the matrix representation of operators using dual frames and its application to the solution of operators equations in a Petrov-Galerkin scheme.

We study some classical identities for multiple zeta values and show that they still hold for zeta functions built on the zeros of an arbitrary function. We introduce the complementary zeta function of a system, which naturally occurs when lifting identities for multiple zeta values to identities for quasisymmetric functions., Comment: 22 pages, 3rd and final version

Let p n : C → C p_n:\mathbb {C} \rightarrow \mathbb {C} be a random complex polynomial whose roots are sampled i.i.d. from a radial distribution 2 π r u ( r ) d r 2\pi r u(r) dr in the complex plane. A natural question is how the distribution of roots evolves under repeated (say n / 2 − n/2- times) differentiation of the polynomial. We conjecture a mean-field expansion for the evolution of ψ ( s ) = u ( s ) s \psi (s) = u(s) s : ∂ ψ ∂ t = ∂ ∂ x ( ( 1 x ∫ 0 x ψ ( s ) d s ) − 1 ψ ( x ) ) . \begin{equation*} \frac {\partial \psi }{\partial t} = \frac {\partial }{\partial x} \left ( \left ( \frac {1}{x} \int _{0}^{x} \psi (s) ds \right )^{-1} \psi (x) \right ). \end{equation*} The evolution of ψ ( s ) ≡ 1 \psi (s) \equiv 1 corresponds to the evolution of random Taylor polynomials p n ( z ) = ∑ k = 0 n γ k z k k ! where γ k ∼ N C ( 0 , 1 ) . \begin{equation*} p_n(z) = \sum _{k=0}^{n}{ \gamma _k \frac {z^k}{k!}} \quad \text {where} \quad \gamma _k \sim \mathcal {N}_{\mathbb {C}}(0,1). \end{equation*} We discuss some numerical examples suggesting that this particular solution may be stable. We prove that the solution is linearly stable. The linear stability analysis reduces to the classical Hardy integral inequality. Many open problems are discussed.

In this work, we consider "finite bandwidth" reproducing kernel Hilbert spaces which have orthonormal bases of the form $f_n(z)=z^n \prod_{j=1}^J \left( 1 - a_{n}w_j z \right)$, where $w_1 ,w_2, \ldots w_J $ are distinct points on the circle $\mathbb{T}$ and $\{ a_n \}$ is a sequence of complex numbers with limit $1$. We provide general conditions based on a matrix recursion that guarantee such spaces contain a functional multiple of the Hardy space. Then we apply this general method to obtain strong results for finite bandwidth spaces when $\lim_{n\rightarrow \infty} n (1-a_n)=p$. In particular, we show that point evaluation can be extended boundedly to precisely $J$ additional points on $\mathbb{T}$ and we obtain an explicit functional decomposition of these spaces for $p>1/2$ in analogy with a previous result in the tridiagonal case due to Adams and McGuire. We also prove that multiplication by $z$ is a bounded operator on these spaces and that they contain the polynomials., 19 pages with references

This paper is a continuation of the research of our previous work [5] and considers quaternionic generalized Caratheodory functions and the related family of generalized positive functions. It is addressed to a wide audience which includes researchers in complex and hypercomplex analysis, in the theory of linear systems, but also electric engineers. For this reason it includes some results on generalized Caratheodory functions and their factorization in the classic complex case which might be of independent interest. An important new result is a pseudo-spectral factorization and we also discuss some interpolation problems in the class of quaternionic generalized positive functions.

We study cubic surfaces as symmetric tensors of format $4 \times 4 \times 4$. We consider the non-symmetric tensor rank and the symmetric Waring rank of cubic surfaces, and show that the two notions coincide over the complex numbers. The corresponding algebraic problem concerns border ranks. We show that the non-symmetric border rank coincides with the symmetric border rank for cubic surfaces. As part of our analysis, we obtain minimal ideal generators for the symmetric analogue to the secant variety from the salmon conjecture. We also give a test for symmetric rank given by the non-vanishing of certain discriminants. The results extend to order three tensors of all sizes, implying the equality of rank and symmetric rank when the symmetric rank is at most seven, and the equality of border rank and symmetric border rank when the symmetric border rank is at most five. We also study real ranks via the real substitution method., 16 pages

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.

In this paper, we show that for a twice differentiable function g having countable zeros and for Lebesgue almost every $$\beta >1$$ β > 1 , the sequence $$(e^{2\pi i \beta ^ng(\beta )})_{n\in {\mathbb {N}}}$$ ( e 2 π i β n g ( β ) ) n ∈ N is orthogonal to all topological dynamical systems of zero entropy. To this end, we define the Chowla property and the Sarnak property for numerical sequences taking values 0 or complex numbers of modulus 1. We prove that the Chowla property implies the Sarnak property and show that for Lebesgue almost every $$\beta >1$$ β > 1 , the sequence $$(e^{2\pi i \beta ^n})_{n\in {\mathbb {N}}}$$ ( e 2 π i β n ) n ∈ N shares the Chowla property. It is also discussed whether the samples of a given random sequence have the Chowla property almost surely. Some dependent random sequences having almost surely the Chowla property are constructed.

Since (±i)2 = –1, there are two time-harmonic conventions, namely, exp(+iωt) and exp(–iωt) with ω and t for angular frequency and time, respectively. Both conventions are mathematically valid. However, a question may still be asked whether a specific acoustic problem can favor one or the other convention. This letter reports one such example in which the use of exp(–iωt) can be preferable when the principal value of the square root of a complex number is considered as an argument of the Hankel functions in the context of 2.5-dimensional problems.

Let $p,x$ be real numbers, and $s$ be a complex number, with $\Re(s)>1-r$, $p\geq 1$, and $x+1>0$. The zeta function $Z^{\bf\alpha}_p(s;x)$ is defined by $$ Z^{\bf\alpha}_p(s;x) =\frac{1}{\Gamma(s)}\int^\infty_0 \frac{e^{-xt}} {e^t-1}\,Li_{\bf{\alpha}}\left(\frac{1-e^{-t}}p\right) t^{s-1}\,dt, $$ where ${\bf\alpha}=(\alpha_1,\ldots,\alpha_r)$ is a $r$-tuple positive integers, and $Li_{\bf{\alpha}}(z)$ is the one-variable multiple polylogarithms. Since $Z^{\bf\alpha}_1(s;0)=\xi(\bf\alpha;s)$, we call this function as a generalized Arakawa-Kaneko zeta function. In this paper, we investigate the properties and values of $Z^{\bf\alpha}_p(s;x)$ with different values $s$, $x$, and $p$. We then give some applications on them., Comment: 20 pages

For every complex number $x$, let $\Vert x\Vert_{\mathbb{Z}}:=\min\{|x-m|:\ m\in\mathbb{Z}\}$. Let $K$ be a number field, let $k\in\mathbb{N}$, and let $\alpha_1,\ldots,\alpha_k$ be non-zero algebraic numbers. In this paper, we completely solve the problem of the existence of $\theta\in (0,1)$ such that there are infinitely many tuples $(n,q_1,\ldots,q_k)$ satisfying $\Vert q_1\alpha_1^n+\ldots+q_k\alpha_k^n\Vert_{\mathbb{Z}}

It is well known that for two univariate polynomials over the complex number field the number of their common roots is equal to the order of their resultant. In this paper, we show that this fundamental relationship still holds for the tropical polynomials under suitable adaptation of the notion of order, if the roots are simple and non-zero., Ministerio de Economía y Competitividad

In this paper, we study the fattening effect of points over the complex numbers for del Pezzo surfaces $$\mathbb {S}_r$$ S r arising by blowing-up of $$\mathbb {P}^2$$ P 2 at r general points, with $$ r \in \{1, \dots , 8 \}$$ r ∈ { 1 , ⋯ , 8 } . Basic questions when studying the problem of points fattening on an arbitrary variety are what is the minimal growth of the initial sequence and how are the sets on which this minimal growth happens characterized geometrically. We provide a complete answer for del Pezzo surfaces.

In this paper, we consider the inverse eigenvalue problem for a Stieltjes string subject to the Robin condition at the left end and a damping condition at the right end with mixed data, which consists of the values of a part of masses and lengths and a subset of its spectrum. We use Krein-Nudelman's interpolation formula for a rational S -function to deal with our problem. We present a decomposition of the S 0 -function associated with the Stieltjes string, which makes it possible to use Krein-Nudelman's interpolation formula. Necessary and sufficient conditions are given for existence and uniqueness of solution to the inverse problem such that a finite number of simple not purely imaginary complex numbers lying in the open upper half-plane are a subset of the spectrum of a damped Stieltjes string.

For any semifield K we define a K-form of a partial flag manifold of a semisimple group G of simply laced type over the complex numbers. The definition is in terms of the theory of canonical bases., Comment: 6 pages

We present a full list of all representations of the special linear group SLnover the complex numbers with complete intersection invariant ring of homological dimension greater than or equal to two, completing the classification of Shmelkin. For this task, we combine three techniques. Firstly, the graph method for invariants of SLndeveloped by the author to compute invariants, covariants and explicit forms of syzygies. Secondly, a new algorithm for finding a monomial order such that a certain basis of an ideal is a Gröbner basis with respect to this order, in between usual Gröbner basis computation and computation of the Gröbner fan. Lastly, a modification of an algorithm by Xin for MacMahon partition analysis to compute Hilbert series.

International audience; Rigid graph theory is an active area with many open problems, especially regarding embeddings in R^d or other manifolds, and tight upper bounds on their number for a given number of vertices. Our premise is to relate the number of embeddings to that of solutions of a well-constrained algebraic system and exploit progress in the latter domain. In particular, the system's complex solutions naturally extend the notion of real embeddings, thus allowing us to employ bounds on complex roots. We focus on multihomogeneous Bézout (m-Bézout) bounds of algebraic systems since they are fast to compute and rather tight for systems exhibiting structure as in our case. We introduce two methods to relate such bounds to combinatorial properties of minimally rigid graphs in C^d and S^d. The first relates the number of graph orientations to the m-Bézout bound, while the second leverages a matrix permanent formulation. Using these approaches we improve the best known asymptotic upper bounds for planar graphs in dimension 3, and all minimally rigid graphs in dimension d ≥ 5, both in the Euclidean and spherical case. Our computations indicate that m-Bézout bounds are tight for embeddings of planar graphs in S^2 and C36. We exploit Bernstein's second theorem on the exactness of mixed volume, and relate it to the m-Bézout bound by analyzing the associated Newton polytopes. We reduce the number of checks required to verify exactness by an exponential factor, and conjecture further that it suffices to check a linear instead of an exponential number of cases overall.

Multicomplex and multidual numbers are two generalizations of complex numbers with multiple imaginary axes, useful for numerical computation of derivatives with machine precision. The similarities between multicomplex and multidual algebras allowed us to create a unified library to use either one for sensitivity analysis. This library can be used to compute arbitrary order derivates of functions of a single variable or multiple variables. The storage of matrix representations of multicomplex and multidual numbers is avoided using a combination of one-dimensional resizable arrays and an indexation method based on binary bitwise operations. To provide high computational efficiency and low memory usage, the multiplication of hypercomplex numbers up to sixth order is carried out using a hard-coded algorithm. For higher hypercomplex orders, the library uses by default a multiplication method based on binary bitwise operations. The computation of algebraic and transcendental functions is achieved using a Taylor series approximation. Fortran and Python versions were developed, and extensions to other languages are self-evident.

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.

To a pair (X, f), where X is a compact ANR and f : X → 𝕊1 is a continuous angle valued map, a field κ, and a nonnegative integer r, one assigns a finite configuration of complex numbers z with multiplicities δfr (z) and a finite configuration of free κ[t−1, t]-modules $$ {\hat{\delta}}_r^f $$ of rank $$ {\delta}_r^f $$ (z) indexed by the same numbers z. This is in analogy with the configuration of eigenvalues and of generalized eigenspaces of a linear operator in a finite-dimensional complex vector space. The configuration $$ {\delta}_r^f $$ refines the Novikov–Betti number in dimension r, and the configuration $$ {\hat{\delta}}_r^f $$ refines the Novikov homology in dimension r associated with the cohomology class defined by f. In the case of the field κ = C, the configuration $$ {\hat{\delta}}_r^f $$ provides by “von-Neumann completion” of a configuration $$ {\hat{\hat{\delta}}}_r^f $$ of mutually orthogonal closed Hilbert submodules of the L2-homology of the infinite cyclic cover of X determined by the map f, which is an L∞(𝕊1)-Hilbert module.

We introduce and study the Hyers–Ulam stability (HUS) of a Cayley quantum (q-difference) equation of first order, where the constant coefficient is allowed to range over the complex numbers. In particular, if this coefficient is non-zero, then the quantum equation has Hyers–Ulam stability for certain values of the Cayley parameter, and we establish the best (minimal) HUS constant in terms of the coefficient only, independent of q and the Cayley parameter. If the Cayley parameter equals one half, then there is no Hyers–Ulam stability for any coefficient value in the complex plane.

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

The problem of determining $DS_n$, the complex numbers that occur as an eigenvalue of an $n$-by-$n$ doubly stochastic matrix, has been a target of study for some time. The Perfect-Mirsky region, $PM_n$, is contained in $DS_n$, and is known to be exactly $DS_n$ for $n \leq 4$, but strictly contained within $DS_n$ for $n = 5$. Here, we present a Boundary Conjecture that asserts that the boundary of $DS_n$ is achieved by eigenvalues of convex combinations of pairs of (or single) permutation matrices. We present a method to efficiently compute a portion of $DS_n$, and obtain computational results that support the Boundary Conjecture. We also give evidence that $DS_n$ is equal to $PM_n$ for certain $n > 5$.

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 ${\mathbb P}^2$.

Let G L n ( C ) denote the set of n-by-n nonsingular matrices with entries from the field C of complex numbers. For any S ∈ G L n ( C ) , define the map ψ S : G L n ( C ) → G L n ( C ) by ψ S ( A ) = S A − 1 ‾ S − 1 . A matrix A ∈ G L n ( C ) is said to be ψ S -orthogonal if ψ S ( A ) = A − 1 . A ψ S -orthogonal H is called a ψ S -symmetry if rank ( H − I ) = 1 . We give conditions on S so that ψ S -symmetries exist. Moreover, we determine conditions on S such that the ψ S -symmetries generate the ψ S -orthogonal matrices, and the minimum number of ψ S -symmetries needed to express a ψ S -orthogonal.

As we know that the power series method is a very effective method for solving the Ordinary differential equations (ODEs) which have variable coefficient, so in this paper we have studied how to solve second-order ordinary differential equation with variable coefficient at a singular point t = 0 and determined the form of second linearly independent solution. Based on the roots of initial equation there are real and complex cases. When the roots of initial equation are real then there are three kinds of second linearly independent solutions. If the roots of the initial equation are distinct complex numbers, then the solution is complex-valued.

We aim to get the step derivative of a complex function, as it derives the step derivative in the imaginary direction of a real function. Given that the step derivative of a complex function cannot be derived using i, which is used to derive the step derivative of a real function, we intend to derive the complex function using the base direction of the quaternion. Because many analytical studies on quaternions have been conducted, various examples can be presented using the expression of the elementary function of a quaternion. In a previous study, the base direction of the quaternion was regarded as the base separate from the basis of the complex number. However, considering the properties of the quaternion, we propose two types of step derivatives in this study. The step derivative is first defined in the j direction, which includes a quaternion. Furthermore, the step derivative in the j+k2 direction is determined using the rule between bases i, j, and k defined in the quaternion. We present examples in which the definition of the j-step derivative and (j,k)-step derivative are applied to elementary functions ez, sinz, and cosz.

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.

The classical theorems of Mittag-Leffler and Weierstrass show that when $\{\lambda_n\}$ is a sequence of distinct points in the open unit disk $\D$, with no accumulation points in $\D$, and $\{w_n\}$ is any sequence of complex numbers, there is an analytic function $\phi$ on $\D$ for which $\phi(\lambda_n) = w_n$. A celebrated theorem of Carleson \cite{MR117349} characterizes when, for a bounded sequence $\{w_n\}$, this interpolating problem can be solved with a bounded analytic function. A theorem of Earl \cite{MR284588} goes further and shows that when Carleson's condition is satisfied, the interpolating function $\phi$ can be a constant multiple of a Blaschke product. In this paper, we explore when the interpolating $\phi$ can be an outer function. We then use our results to refine a result of McCarthy \cite{MR1065054} and explore the common range of the co-analytic Toeplitz operators on a model space., Comment: 27 pages

Clifford algebras are used for constructing spin groups, and are therefore of particular importance in the theory of quantum mechanics. An algebraist’s perspective on the many subgroups and subalgebras of Clifford algebras may suggest ways in which they might be applied more widely to describe the fundamental properties of matter. I do not claim to build a physical theory on top of the fundamental algebra, and my suggestions for possible physical interpretations are indicative only, and may not work. Nevertheless, both the existence of three generations of fermions and the symmetry-breaking of the weak interaction seem to emerge naturally from an extension of the Dirac algebra from complex numbers to quaternions.

By giving the definition of the sum of a series indexed by a set on which a group acts, we prove that the sum of the series that defines the Riemann zeta function, the Epstein zeta function, and a few other series indexed by $\Z^k$ has an intrinsic meaning as a complex number, independent of the requirements of analytic continuation. The definition of the sum requires nothing more than algebra and the concept of absolute convergence. The analytical significance of the algebraically defined sum is then explained by an argument that relies on the Poisson formula for tempered distributions., Comment: Several recommendationsof the reviewer to improve readability of the paper have been adopted

We prove that for a large class of multiplicative functions, referred to as generalized divisor functions, it is possible to find a lower bound for the corresponding variance in arithmetic progressions. As a main corollary, we deduce such a result for any $\alpha$-fold divisor function, for any complex number $\alpha\not\in \{1\}\cup-\mathbb{N}$, even when considering a sequence of parameters $\alpha$ close in a proper way to $1$. Our work builds on that of Harper and Soundararajan, who handled the particular case of $k$-fold divisor functions $d_k(n)$, with $k\in\mathbb{N}_{\geq 2}$., Comment: Corrected some typos in the references