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Let η ( s ) = ∑ n = 1 ∞ ( − 1 ) n + 1 n − s be the alternating zeta function. For a real number τ we define certain complex numbers b M , m ( τ ) and consider finite Dirichlet series υ M ( τ , s ) = ∑ m = 1 M b M , m ( τ ) m − s and η N ( τ , s ) = ∑ M = 1 N υ M ( τ , s ) . Computations demonstrate some remarkable properties of these finite Dirichlet series, but nothing was supported by a proof so far. First, numerical data show that η N ( τ , s ) approximates η ( s ) with high accuracy for s in the vicinity of 1 / 2 + i τ ; this allows one to surmise that (*) η ( s ) = ∑ M = 1 ∞ υ M ( τ , s ) . Moreover, it looks that lim M → ∞ ⁡ m υ M ( τ , 1 − σ + i t ) ‾ m υ M ( τ , σ + i t ) = η ( σ + i t ) m η ( 1 − σ + i t ) ‾ m ; in other words, the individual summands in expected expansion ( ⁎ ) satisfy with an increasing accuracy a counterpart of the classical functional equation. Let ϒ M ( τ , σ + i t ) = υ M ( τ , σ + i t ) / η ( σ + i t ) . When M, τ, and either σ or t are fixed, and the fourth parameter varies, the plot of ϒ M ( τ , σ + i t ) on the complex plane contains numerous almost ideally circular arcs with geometrical parameters closely related to the non-trivial zeros of the zeta function.

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.

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.

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.

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) $.

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

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

In this paper we describe two methods, both inspired from Godsil-McKay switching on simple graphs, to build cospectral gain graphs whose gain group consists of the complex numbers of modulus 1 (the circle group). The results obtained here can be also applied to the Hermitian matrix of mixed graphs.

Recently, some uniqueness theorems about meromorphic functions f(z) concerning their derivatives $$f'(z)$$ and shifts $$f(z+c)$$ with three CM sharing values have been obtained. In this paper, we continue to study this topic. We consider not only high order derivatives instead of just $$f'(z)$$ , but also IM sharing value instead of CM sharing value. In fact, we mainly prove that for a non-constant meromorphic function f(z) of hyper order strictly less than 1, if $$f^{(k)}(z)$$ and $$f(z+c)$$ share $$0,\infty $$ CM and 1 IM, then $$f^{(k)}(z)\equiv f(z+c)$$ , where c is a non-zero finite complex number. Our main theorem generalizes and greatly improves the related result due to Qi–Li–Yang. In addition, we give some discussion of this issue and obtain a uniqueness theorem concerning defective values in Sect. 3.

In this paper we present a simple spectral theory of polynomially bounded functions on the half line, and then apply it to study the asymptotic behavior of solutions of fractional differential equations of the form $$D^{\alpha }_Cu(t)=Au(t)+f(t)$$ , where $$D^{\alpha }_Cu(t)$$ is the derivative of the function u in Caputo’s sense, A is generally an unbounded closed operator, f is polynomially bounded. Our main result claims that if u is a mild solution of the Cauchy problem such that $$\lim _{h\downarrow 0} \sup _{t\ge 0} \Vert u(t+h)-u(t)\Vert /(1+t)^n=0$$ , and $$\sup _{t\ge 0} \Vert u(t)\Vert /(1+t)^n

Let sl N ˆ ( C q ) be the core of extended affine Lie algebra of type A N − 1 coordinated by the rational quantum 2-torus C q . In this paper, we first prove that for any complex number l, the category of restricted sl N ˆ ( C q ) -modules of level l is canonically isomorphic to the category of twisted modules for the vertex algebra V C g ˆ ( l , 0 ) arising from a conformal Lie algebra C g , where C g ˆ is isomorphic to a toroidal Lie algebra. Then we prove that for any nonnegative integer l, the integrable restricted sl N ˆ ( C q ) -modules of level l are exactly the twisted modules for the quotient vertex algebra L C g ˆ ( l , 0 ) of V C g ˆ ( l , 0 ) . Finally, we classify irreducible graded twisted L C g ˆ ( l , 0 ) -modules.

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

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 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.

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~$f$.

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

Complex uncertain variables are measurable functions from an uncertainty space to the set of complex numbers and are used to model complex uncertain quantities. This paper introduces the convergence concepts of convergence almost surely (a.s.), convergence in measure, convergence in mean, convergence in distribution and convergence uniformly almost surely complex uncertain double sequences. In addition, relationships among the introduced classes of sequences have been introduced.

In this article, under certain restrictions, the convergence rate of type II Hermite–Pade approximants (including nondiagonal ones) for a system $$\{{}_{1}F_{1}(1,\gamma;\lambda_{j}z)\}_{j=1}^{k}$$ , consisting of degenerate hypergeometric functions is found, when $$\{\lambda_{j}\}_{j=1}^{k}$$ are different complex numbers, and $$\gamma\in\mathbb{C}\setminus\{0,-1,-2,...\}$$ . Without the indicated restrictions, similar statements were obtained for approximants of the indicated type, provided that the numbers $$\{\lambda_{j}\}_{j=1}^{k}$$ are the roots of the equation $$\lambda^{k}=1$$ . The theorems proved in this paper complement and generalize the results obtained earlier by other authors.

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.

In this paper, we investigate the growth of solutions of the differential equation: $$\begin{aligned} f^{\left( k\right) }+A_{k-1}\left( z\right) \exp \left\{ \frac{a_{k-1}}{ \left( z_{0}-z\right) ^{n}}\right\} f^{\left( k-1\right) }+\cdots +A_{0}\left( z\right) \exp \left\{ \frac{a_{0}}{\left( z_{0}-z\right) ^{n}}\right\} f=0, \end{aligned}$$ where $$A_{j}\left( z\right) $$ are analytic functions in the closed complex plane except at $$z_{0}$$ and $$a_{j}\left( j=0,\ldots ,k-1\right) $$ are distinct complex numbers. Other linked cases have been studied.

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.

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.

In 2011, Heittokangas et al. (Complex Var Ellipt Equat 56(1–4):81–92, 2011) proved that if a non-constant finite order entire function f(z) and $$f(z+\eta )$$ share a, b, c IM, where $$\eta $$ is a finite non-zero complex number, while a, b, c are three distinct finite complex values, then $$f(z)=f(z+\eta )$$ for all $$z\in \mathbb {C}$$ . We prove that if a non-constant finite order entire function f and its n-th difference operator $$\Delta ^n_{\eta }f$$ share $$a_1$$ , $$a_2$$ , $$a_3$$ IM, where n is a positive integer, $$\eta \ne 0$$ is a finite complex value, while $$a_1$$ , $$a_2$$ , $$a_3$$ are three distinct finite complex values, then $$f=\Delta ^n_{\eta }f$$ . The main results in this paper also improve Theorems 1.1 and 1.2 from Li and Yi (Bull Korean Math Soc 53(4):1213–1235, 2016).

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.

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.

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.

Let H be a Hilbert $$C^*$$ -module, and let $$H_M$$ be the indefinite inner space induced by a self-adjointable and invertible operator M on H. Given weighted projections P and Q on $$H_M$$ , let $$S_{\lambda ,k}=(PQ)^k-\lambda (QP)^k$$ for a pair $$(k, \lambda )$$ , where k is a natural number and $$\lambda $$ is a complex number. It is proved that $$PQ-QP$$ is weighted Moore–Penrose invertible if and only if $$S_{\lambda ,k}$$ is weighted Moore–Penrose invertible for every pair $$(k, \lambda )$$ .

We introduce a concept of A-strongly statistical convergence for sequences of complex numbers, where A = (ank)n,k∈ℕ is an infinite matrix with nonnegative entries. A sequence (xn) is called strongly convergent to L if $$ \underset{n\to \infty }{\lim }{\sum}_{k=1}^{\infty }{a}_{nk}\left|{x}_k\right.-L\left|=0\right. $$ in the ordinary sense. In the definition of A-strongly statistical limit, we use the statistical limit instead of the ordinary limit with a convenient density. We study some densities and show that the (ank)-strongly statistical limit is an ( $$ {a}_{m_nk} $$ )-strong limit, where the density of the set {mn ∈ ℕ : n ∈ ℕ} is equal to 1. We introduce the notion of dense positivity for nonnegative sequences. A nonnegative sequence (rn) is dense positive provided the limit superior of a subsequence ( $$ {r}_{m_n} $$ ) is positive for all (mn) with density equal to 1. We show that the dense positivity of (rn) is a necessary and sufficient condition for the uniqueness of A-strongly statistical limit, where A = (ank) and rn = $$ {\sum}_{k=1}^{\infty }{a}_{nk} $$ . Furthermore, necessary conditions for the regularity, linearity and multiplicativity of the A-strongly statistical limit are established.

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.

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 $$\{P_n^{(\alpha-2,2)}\}_{n\geq0}$$

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 $n > 5$.

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.

We show that over the field of complex numbers, every homogeneous polynomial of degree d can be approximated (in the border complexity sense) by a depth-3 arithmetic circuit of top fan-in at most 2. This is quite surprising, since there exist homogeneous polynomials P on n variables of degree 2, such that any depth-3 arithmetic circuit computing P must have top fan-in at least Ω ( n ). As an application, we get a new tradeoff between the top fan-in and formal degree in an approximate analog of the celebrated depth reduction result of Gupta, Kamath, Kayal, and Saptharishi [7, 10]. Formally, we show that if a degree d homogeneous polynomial P can be computed by an arithmetic circuit of size s ≥ d , then for every t ≤ d , P is in the border of a depth-3 circuit of top fan-in s O( t ) and formal degree s O( d/t ) . To the best of our knowledge, the upper bound on the top fan-in in the original proof of Reference [7] is always at least s Ω (√d) , regardless of the formal degree.

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 ${\mathbb P}^2$.

Asymptotic formulas are obtained in the paper for x → +∞ for the fundamental system of solutions to the equation $$l(y): = {i^{2n + 1}}\{ {(q{y^{(n + 1)}})^{(n)}} + {(q{y^{(n)}})^{(n + 1)}}\} + py = \lambda y,\;\;\;\;\;\;x \in I: = [1, + \infty ),$$ where λ is a complex parameter. It is assumed that q is a positive continuously differentiable function, p has the form p = σ(k), 0 ≤ k ≤ n, where σ( is a locally integrable on I function, and the derivative is understood in the sense of the theory of distributions. In the case when k = 0 and λ ≠ 0, and the coefficients q and p of the expression l(y) are such that q =1/2 + q1, and q1, σ(= p) are integrable on I, restrictions on q1 and σ and for any 1 ≤ k ≤ n − 1. For k = n additional constraints arise on these functions. We consider separately the case when λ = 0. Asymptotic formulas were also obtained for solutions to the equation l(y) = λy under the condition $$q(x) = \alpha {x^{2n + 1 + \nu }}{(1 + r(x))^{ - 2}},\;\sigma (x) = {x^{k + \nu }}(\beta + s(x))$$ , where α ≠ 0 mid β are complex numbers, ν ⩾ 0, and the functions r and s satisfy certain conditions of integral decay.

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 ).

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.