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The q-rung orthopair fuzzy sets (q-ROFSs) are increasingly valuable to express fuzzy and vague information, as the generalization of intuitionistic fuzzy sets (IFSs) and Pythagorean fuzzy sets (PFSs). In this paper, we propose complex $q$-rung orthopair fuzzy sets (C$q$-ROFSs) as a new tool to deal with vagueness, uncertainty and fuzziness by extending the range of membership and non-membership function of $q$-ROFS from real to complex number with the unit disc. We develop some new complex $q$-rung orthopair fuzzy Hamacher operations and complex $q$-rung orthopair fuzzy Hamacher aggregation operators, i.e., the complex $q$-rung orthopair fuzzy Hamacher weighted average (C$q$-ROFHWA) operator, and the complex $q$-rung orthopair fuzzy Hamacher weighted geometric (C$q$-ROFHWG) operator. Subsequently, we introduce the innovative concept of a complex $q$-rung orthopair fuzzy graphs based on Hamacher operator called complex $q$-rung orthopair fuzzy Hamacher graphs (C$q$-ROFHGs) and determine its energy and Randi'{c} energy. In particular, we present the energy of a splitting C$q$-ROFHG and shadow C$q$-ROFHG. Further, we describe the notions of complex $q$-rung orthopair fuzzy Hamacher digraphs (C$q$-ROFHDGs). Finally, a numerical instance related to the facade clothing systems selection is presented to demonstrate the validity of the proposed concepts in decision making (DM).

As the existing complex number Nth root computation methods are relatively discrete, we propose a general method and architecture based on coordinate rotation digital computer (CORDIC) to compute arbitrary complex number Nth root for the first time. Our method performs the tasks of computing complex modulus, complex phase angle, real Nth root, sine function and cosine function, which can be implemented by circular CORDIC, linear CORDIC and hyperbolic CORDIC. Based on these CORDICs, our proposed architecture can not only improve the hardware efficiency just through shift-add operations, but also flexibly adjust the precision and the input range of complex number Nth root. To prove its feasibility, we conduct a software simulation and implement an example circuit in hardware. Under the TSMC 28nm CMOS technology, we synthesize it and get the report that it has the area of 6561μm2 and the power of 3.95mW at the frequency of 1.5GHz.

We analyze the mathematical structure of the classical Grover’s algorithm and put it within the framework of linear algebra over the complex numbers. We also generalize it in the sense, that we are seeking not the one ‘chosen’ element (sometimes called a ‘solution’) of the dataset, but a set ofmsuch ‘chosen’ elements (out of$$n>m)$$n>m). Besides, we do not assume that the so-called initial superposition is uniform. We assume also that we have at our disposal an oracle that ‘marks,’ by a suitable phase change$$\varphi $$φ, all these ‘chosen’ elements. In the first part of the paper, we construct a unique unitary operator that selects all ‘chosen’ elements in one step. The constructed operator is uniquely defined by the numbers$$\varphi $$φand$$\alpha $$αwhich is a certain function of the coefficients of the initial superposition. Moreover, it is in the form of a composition of two so-called reflections. The result is purely theoretical since the phase change required to reach this heavily depends on$$\alpha $$α. In the second part, we construct unitary operators having a form of composition of two or more reflections (generalizing the constructed operator) given the set of orthogonal versors. We find properties of these operations, in particular, their compositions. Further, by considering a fixed, ‘convenient’ phase change$$\varphi ,$$φ,and by sequentially applying the so-constructed operator, we find the number of steps to find these ‘chosen’ elements with great probability. We apply this knowledge to study the generalizations of Grover’s algorithm ($$m=1,\phi =\pi $$m=1,ϕ=π), which are of the form, the found previously, unitary operators.

Let k be a nonzero complex number. In this paper, we consider a k-circulant matrix whose first row is $$(Q_{1},Q_{2},\ldots ,Q_{n})$$ , where $$Q_{n}$$ is the nth Pell–Lucas number. The formulas for the eigenvalues of such matrix are obtained. Namely, the result which can be obtained from the result of Theorem 7. (Yazlik and Taskara, J Inequal Appl 2013:394, 2013) is improved. The obtained formulas for the eigenvalues of a k-circulant matrix involving the Pell–Lucas numbers show that the result of Theorem 8. (Jing, Li and Shen, WSEAS Trans Math 12(3):341-351, 2013) (i.e. Theorem 8. (Yazlik and Taskara 2013)) is not always applicable. The Euclidean norm of such matrix is determined. The upper and lower bounds for the spectral norm of a k-circulant matrix whose first row is $$(Q_{1}^{-1},Q_{2}^{-1},\ldots ,Q_{n}^{-1})$$ are also investigated. The obtained results are illustrated by examples. As a consequence of the previous results, the eigenvalues, the determinant, the Euclidean norm of a k-circulant matrix whose first row is $$(q_{1},q_{2},\ldots ,q_{n})$$ , where $$q_{n}$$ is the nth modified Pell number, are presented. Also, the upper and lower bounds for the spectral norm of a k-circulant matrix whose first row is $$(q_{1}^{-1},q_{2}^{-1},\ldots ,q_{n}^{-1})$$ are given

Made available in DSpace on 2021-06-25T10:19:10Z (GMT). No. of bitstreams: 0 Previous issue date: 2021-04-15 We describe the set [X,Y] of path-components of pointed mapping spaces M⁎(X,Y), where X is chosen to be the reduced kth suspension EkFPm of a projective space FPm and Y is a sphere Sn or FPn for F=R,C, the fields of real or complex numbers, respectively. In particular, the cohomotopy sets πn(EkFPm) are studied for k≥0 and certain m,n≥1. Faculty of Mathematics and Computer Science University of Warmia and Mazury, Słoneczna 54 Street Instituto de Geociências e Ciências Exatas UNESP–Univ Estadual Paulista, Av. 24A, 1515, Bela Vista Departamento de Matemática Universidade Federal de São Carlos Centro de Ciências Exatas e Tecnologia, CP 676 Instituto de Geociências e Ciências Exatas UNESP–Univ Estadual Paulista, Av. 24A, 1515, Bela Vista

This article concerns the problems of synchronization in a fixed time or prespecified time for memristive complex-valued neural networks (MCVNNs), in which the state variables, activation functions, rates of neuron self-inhibition, neural connection memristive weights, and external inputs are all assumed to be complex-valued. First, the more comprehensive fixed-time stability theorem and more accurate estimations on settling time (ST) are systematically established by using the comparison principle. Second, by introducing different norms of complex numbers instead of decomposing the complex-valued system into real and imaginary parts, we successfully design several simpler discontinuous controllers to acquire much improved fixed-time synchronization (FXTS) results. Third, based on similar mathematical derivations, the preassigned-time synchronization (PATS) conditions are explored by newly developed new control strategies, in which ST can be prespecified and is independent of initial values and any parameters of neural networks and controllers. Finally, numerical simulations are provided to illustrate the effectiveness and superiority of the improved synchronization methodology.

For regular continued fraction, if a real number [Formula: see text] and its rational approximation [Formula: see text] satisfying [Formula: see text], then, after deleting the last integer of the partial quotients of [Formula: see text], the sequence of the remaining partial quotients is a prefix of that of [Formula: see text]. In this paper, we show that the situation is completely different if we consider the Hurwitz continued fraction expansions of a complex number and its rational approximations. More specifically, we consider the set [Formula: see text] of complex numbers which are well approximated with the given bound [Formula: see text] and have quite different Hurwitz continued fraction expansions from that of their rational approximations. The Hausdorff and packing dimensions of such set are determined. It turns out that its packing dimension is always full for any given approximation bound [Formula: see text] and its Hausdorff dimension is equal to that of the [Formula: see text]-approximable set [Formula: see text] of complex numbers. As a consequence, we also obtain an analogue of the classical Jarník theorem in real case.

In this paper, we transfer the approach of finding the explicit solutions to linear impulsive differential equations (LIDEs) to quaternion-valued case. We develop the similar idea to find the solutions to LIDEs and give the details to derive the representations of the solutions to quaternion-valued LIDEs in the sense of complex numbers and quaternion. In addition, we present that two formulas are equivalent to each other. Finally, some examples are given to illustrate the theory approach.

A systematic method to facilitate the solution of a $$(n+1)$$ th order differential equation is introduced. This set of ordinary differential equations is unique in the sense that it defines a hierarchy whose solution can be manipulated to solve a partial differential hierarchy, called Burgers’ hierarchy. The latter forms a highly nonlinear evolutionary class of equations. Our approach can be described as in the following two ways. Firstly, we construct the one-parameter Lie group of transformations that leave the ordinary differential equation invariant, and find that they are universally derived from the $$(n+1)$$ complex roots of a certain polynomial. Secondly, we prove that the exact solution of the hierarchy for all values of n, is given by an integration formula. An important consequence of this study, is the extension of the formulaic method to exactly solve the entire Burgers’ hierarchy. We exemplify the advantageous nature of our results through solving various members of the hierarchy.

It is a known uniqueness result that if f is a transcendental meromorphic function of finite order with two Borel excetional values $$a \ne \infty , b$$ and its first order difference operator $${\triangle }_c f\not \equiv 0$$ , for some complex number c, and if f and $${\triangle }_c f$$ share a, b CM, then $$a=0, b=\infty $$ and $$f=\exp (Az+B)$$ , where $$A\ne 0, B \in {\mathbb {C}} $$ . This type of results has its origin dating back to Csillag-Tumura’s uniqueness theorems. In this paper, by using completely different methods, we shall show that the result holds for arbitrary higher order difference operators. Examples are provided to show that this result is not valid for meromorphic functions with infinite order, which also shows a distinction between the derivatives and the difference operators of meromorphic functions, in view of Csillag-Tumura type uniqueness theorems.

It is well established the complex exponential and logarithm are multivalued functions, both failing to maintain most identities originally valid over the positive integers domain. Moreover the general case of complex logarithm, with a complex base, is hardly mentionned in mathematic litterature. We study the exponentiation and logarithm as binary operations where all operands are complex. In a redefined complex number system using an extension of the C field, hereafter named E, we proove both operations always produce single value results and maintain the validity of identities such as logu (w v) = logu (w) + logu (v) where u, v, w in E. There is a cost as some algebraic properties of the addition and subtraction will be diminished, though remaining valid to a certain extent. In order to handle formulas in a C and E dual number system, we introduce the notion of set precision and set truncation. We show complex numbers as defined in C are insufficiently precise to grasp all subtleties of some complex operations, as a result multivaluation, identity failures and, in specific cases, wrong results are obtained when computing exclusively in C. A geometric representation of the new complex number system is proposed, in which the complex plane appears as an orthogonal projection, and where the complex logarithm an exponentiation can be simply represented. Finally we attempt an algebraic formalization of E.

We give a simple proof of the Siebeck–Marden theorem. We use only basic facts of geometry and complex numbers.

The subject of complex numbers issue is very significant because of its wide utility, especially in the engineering circuits representation. In this paper, a modern method to find the square root of the complex number has been analyzed, and some examples on the subject were presented.

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.

In the paper "A Chaotic Search for $i$"~(\cite{strang1991chaotic}), Strang completely explained the behaviour of Newton's method when using real initial guesses on $f(x) = x^{2}+1$, which has only a pair of complex roots $\pm i$. He explored an exact symbolic formula for the iteration, namely $x_{n}=\cot{ \left( 2^{n} \theta_{0} \right) }$, which is valid in exact arithmetic. In this paper, we extend this to to $k^{th}$ order Householder methods, which include Halley's method, and to the secant method. Two formulae, $x_{n}=\cot{ \left( \theta_{n-1}+\theta_{n-2} \right) }$ with $\theta_{n-1}=\mathrm{arccot}{\left(x_{n-1}\right)}$ and $\theta_{n-2}=\mathrm{arccot}{\left(x_{n-2}\right)}$, and $x_{n}=\cot{ \left( (k+1)^{n} \theta_{0} \right) }$ with $\theta_{0} = \mathrm{arccot}(x_{0})$, are provided. The asymptotic behaviour and periodic character are illustrated by experimental computation. We show that other methods (Schr\"{o}der iterations of the first kind) are generally not so simple. We also explain an old method that can be used to allow Maple's \textsl{Fractals[Newton]} package to visualize general one-step iterations by disguising them as Newton iterations., Comment: 22 pages, 11 figures

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.

We prove new exponents for the energy version of the Erd\H{o}s-Szemer\'edi sum-product conjecture, raised by Balog and Wooley. They match the previously established milestone values for the standard formulation of the question, both for general fields and the special case of real or complex numbers, and appear to be the best ones attainable within the currently available technology. Further results are obtained about multiplicative energies of additive shifts and a strengthened energy version of the "few sums, many products" inequality of Elekes and Ruzsa. The latter inequality enables us to obtain a minor improvement of the state-of the art sum-product exponent over the reals due to Konyagin and the second author, up to $\frac{4}{3}+\frac{1}{1509}$. An application of energy estimates to an instance of arithmetic growth in prime residue fields is presented., Comment: 25 pages. A new best sum-product exponent is attained over the reals via a new corollary which locally improves the two recent sum-product papers by Konyagin and Shkredov

Hyers–Ulam stability of the sequence { z n } n ∈ N satisfying the difference equation z n + 1 = g ( z n ) where g ( z ) = a z + b c z + d with complex numbers a, b, c and d is defined. Let g be loxodromic Mobius map, that is, g satisfies that a d − b c = 1 and a + d ∈ C ∖ [ − 2 , 2 ] . Hyers–Ulam stability holds if the initial point of { z n } n ∈ N is in the exterior of avoided region, which is the union of the certain disks of g − n ( ∞ ) for all n ∈ N .

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.

In single-degree-of-freedom (DOF) mechanisms, velocity coefficients (VCs) and their 1st derivative with respect to the generalized coordinate (acceleration coefficients (ACs)) only depend on the mechanism configuration. In addition, if the mechanism is a planar linkage (i.e., a mechanism containing only lower pairs), complex numbers are an easy-to-use tool for writing linkage's loop equations that are formally differentiable and are the only constraint equations of linkages. Actually, the use of complex numbers for systematically writing the constraint equations of planar mechanisms is often limited to linkages. The possible presence of higher pairs in these mechanisms is usually managed through either equivalent linkages or apparent velocity/acceleration equations. Both these methods are simple to implement for a single mechanism configuration, but become cumbersome when continuous motion has to be analyzed. Other approaches use ad hoc auxiliary equations. Here, first, a general notation that brings to select particular auxiliary equations is proposed; then, such notation is adopted to propose an algorithm that systematically uses VCs and ACs for solving the kinematic-analysis problems of single-DOF planar mechanisms. The proposed notation and algorithm, over being efficient enough for constituting the kinematic block of any dynamic model of these mechanisms, are easy to present planar kinematics, with the complex-number method extended to higher pairs, in graduate and/or undergraduate courses.

Here, we discuss the generalized Bernstein-Vazirani algorithm for determining a complex number string. The generalized algorithm presented here has the following structure. Given the set of complex values {a1, a2, a3,…, aN} and a special function $$ g:\mathbf{C}\to \mathbf{C} $$, we determine N real parts of values of the function l(a1), l(a2), l(a3),…, l(aN) and N imaginary parts of values of the function h(a1), h(a2), h(a3),…, h(aN) simultaneously. That is, we determine the N complex values g(aj) = l(aj) + ih(aj) simultaneously. We mention the two computing can be done in parallel computation method simultaneously. The speed of determining the string of complex values is shown to outperform the best classical case by a factor of N. Additionally, we propose a method for calculating many different matrices A, B, C,... into g(A), g(B), g(C),... simultaneously. The speed of solving the problem is shown to outperform the classical case by a factor of the number of the elements of them. We hope our discussions will give a first step to the quantum simulation problem.

In this paper, we find the principal asymptotic term at infinity of a certain fundamental system of solutions to the equation l2n[y] = λy of order 2n, where l2n is the product of second-order linear differential expressions and λ is a fixed complex number. We assume that the coefficients of these differential expressions are not necessarily smooth but have a prescribed power growth at infinity. The asymptotic formulas obtained are applied for the problem on the defect index of differential operators in the case where l2n is a symmetric (formally self-adjoint) differential expression.

In this paper, we construct a family of vertex operator algebras VJ,r parameterized by a non-zero complex number r, whose Griess algebras V2 are isomorphic to Hermitian Jordan algebras J of type C ...

We provide a rigorous account of the equivalence between the complex-valued widely linear estimation method and the quaternion involution widely linear estimation method with their vector-valued real linear estimation counterparts. This is achieved by an account of degrees of freedom and by providing matrix mappings between a complex variable and an isomorphic bivariate real vector, and a quaternion variable versus a quadri-variate real vector. Furthermore, we show that the parameters in the complex-valued linear estimation method, the complex-valued widely linear estimation method, the quaternion linear estimation method, the quaternion semi-widely linear estimation method, and the quaternion involution widely linear estimation method include distinct geometric structures imposed on complex numbers and quaternions, respectively, whereas the real-valued linear estimation methods do not exhibit any structure. This key difference explains, both in theoretical and practical terms, the advantage of estimation in division algebras (complex, quaternion) over their multivariate real vector counterparts. In addition, we discuss the computational complexities of the estimators of the hypercomplex widely linear estimation methods.

We show that the generalized resolvent of a linear pencil in a unital Banach algebra over the field of complex numbers is analytic on an open annular region of the complex plane if and only if the coefficients of the Laurent series expansion satisfy a system of left and right fundamental equations and are geometrically bounded. Our analysis includes the case where the resolvent has an isolated essential singularity at the origin. We find a closed form for the resolvent and use the fundamental equations to establish key spectral separation properties when the resolvent has only a finite number of isolated singularities. We show that our results can be used to solve an infinite system of ordinary differential equations and to solve the generalized Sylvester equation. We also show that our results can be extended to polynomial pencils Refereed/Peer-reviewed

Let C be the field of complex numbers and n and k natural numbers satisfying k≥2 and n ⩾ k+3. Let Pnk(C) be the subset of nth symmetric product (Pk(C))(n) of Pk(C) with itself consisting of...

This paper investigates the axial vibration of single-walled carbon nanotubes (SWCNTs) with fractional damping based on doublet mechanics. The Kelvin–Vigot model is used to incorporate damping effect for CNTs. By solving the equation of motion, the relation between natural frequency with scale parameter and fractional order is derived in the axial mode of vibration. It is shown that fractional order and scale parameter play significant roles in the axial vibration behavior of SWCNTs. Such effects decrease the natural frequency compared to the predictions of the classical continuum mechanics models and also ignores the damping effects. These effects on the natural frequency are more apparent in higher mode numbers and lower tube lengths and radii. Results for complex roots of characteristic equation obtained for a SWCNT without viscoelastic foundation, where imaginary parts represent damped frequencies, were compared with the results found from molecular mechanics simulations and a good agreement was achieved.

Complex rational curves have been used to represent circular splines as well as many classical curves including epicycloids, cardioids, Joukowski profiles, and the lemniscate of Bernoulli. Complex rational curves are known to have low degree (typically half the degree of the corresponding rational planar curve), circular precision, invariance with respect to Mobius transformations, special implicit forms, an easy detection procedure, and a fast algorithm for computing their μ-bases. But only certain very special rational planar curves are also complex rational curves. To construct a wider collection of curves with similar appealing properties, we generalize complex rational curves to hyperbolic and parabolic rational curves by invoking the hyperbolic and parabolic numbers. We show that the special properties of complex rational curves extend to these hyperbolic and parabolic rational curves. We also provide examples to flesh out the theory.

Let a1,a2, …,an, and λ be complex numbers, and let p1,p2, …,pn be measurable complex-valued functions on ℝ+ (:= [0, + ∞)) such that

In many technical applications, non-linearities play an important role and must be taken into account as early as the design phase. In the field of dynamics and vibration, nonlinear effects can dramatically change the forced response behavior. For example, several vibration states can be possible for one excitation frequency. The Harmonic Balance Method (HBM) is widely used to solve differential equations resulting from nonlinear dynamic problems. Periodic stationary oscillations are efficiently approximated by a truncated Fourier series, where the Fourier coefficients are to be calculated from a nonlinear algebraic system of equations. As in linear theory, the harmonic signals can be defined with real or complex arithmetic and so HBM can be applied in real or complex form. However, since the resulting equation system for the Fourier coefficients is usually solved numerically with Newton’s method, one encounters a problem when using the complex representation. For the complex residual function, the complex derivative simply does not exist, which is why it must be separated into real and imaginary parts. The present paper deals with the problem of non-differentiable complex functions by first briefly discussing the history of complex numbers. Complex differentiability is explained using simple mathematical examples. For a Duffing oscillator and for a Friction oscillator it is shown in detail that no complex derivative exists for the associated HBM residual function. Furthermore, general amplitude-dependent nonlinear terms are shown to end up in complex non-differentiability with the need of splitting the problem into real and imaginary parts.

Let $$\mathcal {S}^*_{SG}=\{f\in \mathcal {A}:zf'(z)/f(z)\prec 2/(1+e^{-z})\}$$. For this class, several radius estimates and coefficient bounds are obtained as well as structural formula, growth theorem, distortion theorem and inclusion relations are established. Further, let p be an analytic function such that $$p(0)=1$$. Sharp bounds on $$\beta \in \mathbb {R}$$ are determined for various first-order differential subordinations such as $$1+\beta zp'(z)/p^k(z)$$, $$p(z)+\beta zp'(z)/p^k(z)\prec 2/(1+e^{-z})$$ to imply that $$p(z)\prec (1+Az)/(1+Bz)$$, where $$-1\le B

The famous theorem of Van Aubel for quadrilaterals postulates that if squares are built externally on the sides of any quadrilateral, then the two segments that join the opposing centers of these squares are congruent and orthogonal. Inspired by this result and also by the results of Krishna, in this article we will prove the following result of plane geometry: each octagon is associated with a parallelogram, in some cases the parallelogram in question can be degenerate at a point or a segment. This is possible because of complex numbers and basics of analytical geometry.

We study a semilinear wave equation whose linear part corresponds to the linear Klein–Gordon equation in the non-relativistic limit, augmented with a nonlinearity that is Frechet-differentiable over the complex numbers. We show that this equation possesses an almost invariant manifold in phase space that generalizes the slow manifold which is known to exist for finite-dimensional Galerkin truncations of the system. This manifold is shown to be almost invariant to any algebraic order and can be constructed in the H s − 1 × H s phase space of the equation uniformly in the order of the approximation. In particular, we prove that the dynamics on this “slow manifold” shadows orbits of the full system over a finite interval of time.

Complex-order differintegral (COD) is the extended version of fractional-order one in which the differintegral order can be a complex number rather than a real number. In comparison with fractional-order differintegral (FOD), the distinguishing feature of the COD is that the phase slope of its Bode diagram is a function of imaginary part of the complex order of the COD. In this paper, by the use of this property of the COD, a robust control system is proposed. The design procedure and the realization of the proposed COD-based closed-loop control system are discussed. Since the phase of COD’s frequency response is a nonsymmetric function of frequency, stability analysis of the proposed control system is considered a problematic task. It is proven that for the stability of the control system, it is essential that the COD be applied in a limited frequency band that is derived by the use of the Nyquist stability criterion. Finally, some numerical examples are given to demonstrate the validity and superiority of the proposed complex-order control system.

We describe how we have acquired geometrical models of many periodic tilings of regular polygons from two large collections of images. These models are based on a simplification of the representation recently proposed by us that uses complex numbers. We also describe an algorithm for deciding when two representations give the same tiling, which was used to identify coincidences in these collections.

We consider Berry's random planar wave model (1977) for a positive Laplace eigenvalue $E>0$, both in the real and complex case, and prove limit theorems for the nodal statistics associated with a smooth compact domain, in the high-energy limit ($E\to \infty$). Our main result is that both the nodal length (real case) and the number of nodal intersections (complex case) verify a Central Limit Theorem, which is in sharp contrast with the non-Gaussian behaviour observed for real and complex arithmetic random waves on the flat $2$-torus, see Marinucci et al. (2016) and Dalmao et al. (2016). Our findings can be naturally reformulated in terms of the nodal statistics of a single random wave restricted to a compact domain diverging to the whole plane. As such, they can be fruitfully combined with the recent results by Canzani and Hanin (2016), in order to show that, at any point of isotropic scaling and for energy levels diverging sufficently fast, the nodal length of any Gaussian pullback monochromatic wave verifies a central limit theorem with the same scaling as Berry's model. As a remarkable byproduct of our analysis, we rigorously confirm the asymptotic behaviour for the variances of the nodal length and of the number of nodal intersections of isotropic random waves, as derived in Berry (2002)., 51 pages

Let P, Q be fixed rank-one idempotents of Mn(C), the algebra of n×n matrices over the complex numbers. In this paper, we will study a bijective linear map f:Mn(C)→Mn(C) satisfying the property f(X)...

The present work is an analytical study of the influence of geometrical parameters, such as length, thickness and immersion of the plate, on the reflection coefficient of a regular wave for an immersed horizontal plate in the presence of a uniform current with the same direction as the propagation of the incident regular wave. This study was performed using the linearized potential theory with the evanescent modes while searching for complex roots to the dispersion equation that are neither pure real nor pure imaginary. The results show that the effects of the immersion and the relative length on the reflection coefficient of the plate are accentuated by the presence of the current, whereas the plate thickness practically does not have an effect if it is relatively small.

This paper continues the study of combinatorial properties of binary functions — that is, functions f : 2 E → ℂ such that f ( 0 ) = 1 , where E is a finite set. Binary functions have previously been shown to admit families of transforms that generalise duality, including a trinity transform, and families of associated minor operations that generalise deletion and contraction, with both these families parameterised by the complex numbers. Binary function representations exist for graphs (via the indicator functions of their cutset spaces) and indeed arbitrary matroids (as shown by the author previously). In this paper, we characterise degenerate elements – analogues of loops and coloops – in binary functions, with respect to any set of minor operations from our complex-parameterised family. We then apply this to study the relationship between binary functions and Tutte’s alternating dimaps, which also support a trinity transform and three associated minor operations. It is shown that only the simplest alternating dimaps have binary representations of the form we consider, which seems to be the most direct type of representation. The question of whether there exist other, more sophisticated types of binary function representations for alternating dimaps is left open.

Let $f$ be a polynomial function over the complex numbers and let $\phi$ be a smooth function over $\mathbb{C}$ with compact support. When $f$ is non-degenerate with respect to its Newton polyhedron, we give an explicit list of candidate poles for the complex local zeta function attached to $f$ and $\phi$. The provided list is given just in terms of the normal vectors to the supporting hyperplanes of the Newton polyhedron attached to $f$. More precisely, our list does not contain the candidate poles coming from the additional vectors required in the regular conical subdivision of the first orthant, and necessary in the study of local zeta functions through resolution of singularities. Our results refine the corresponding results of Varchenko and generalize the results of Denef and Sargos in the real case, to the complex setting., Comment: 19 pages. Comments welcome!

Let f be a nonconstant meromorphic function of lower order µ (f) > 1/2 in ℂ, and let aj (j = 1, 2, 3) be three distinct finite complex numbers. We show that there exists an angular domain D = {z: α ≤ arg z ≤ β}, where 0 < β − α ≤ 2π, such that if f share aj (j = 1, 2, 3) CM with its k-th linear differential polynomial L[f] in D, then f = L[f]. This generalizes the corresponding results from Frank and Schwick, Zheng and Li-Liu-Yi.

We show that a$\mathbb{P}$-object and simple configurations of$\mathbb{P}$-objects have a formal derived endomorphism algebra. Hence the triangulated category (classically) generated by such objects is independent of the ambient triangulated category. We also observe that the category generated by the structure sheaf of a smooth projective variety over the complex numbers only depends on its graded cohomology algebra.

In this paper, we prove an exponential integral formula for the Fourier transform of Bessel functions over complex numbers, along with a radial exponential integral formula. The former will enable us to develop the complex spectral theory of the relative trace formula for the Shimura–Waldspurger correspondence and extend the Waldspurger formula from totally real fields to arbitrary number fields.

Sibling curves are a new way to represent the zeroes of any complexvalued function. Interesting results are already known for the sibling curves of quadratic polynomials. In this article a complete investigation of sibling curves of cubic polynomials is given. This article concludes by providing a new general result for the sibling curves of complex polynomials. Mathematics Subject Classification (2010): Primary: 12D10; Secondary: 30A99 Keywords: Complex numbers, polynomials, cubic polynomials, sibling curves

We give a diagrammatic calculus for the intertwiners between permutation supermodules of the Sergeev superalgebra over the complex numbers. We also give a diagrammatic basis for the space o...

Let $X$ be a smooth projective Fano variety over the complex numbers. We study the moduli spaces of rational curves on $X$ using the perspective of Manin's Conjecture. In particular, we bound the dimension and number of components of spaces of rational curves on $X$. We propose a Geometric Manin's Conjecture predicting the growth rate of a counting function associated to the irreducible components of these moduli spaces., Comment: 29 pages; a minor gap in Section 7 has been addressed. To appear in Compositio Mathematica

In the paper we consider a certain analog of the Cauchy type integral taking values in a three-dimensional commutative algebra over the field of complex numbers with one-dimensional radical. We have established sufficient conditions for the existence of limiting values for such an integral. It is also shown that analogues of Sokhotskii–Plemelj formulas hold.

The purpose of this survey is to discuss Poisson and contact brackets and related infinite dimensional superalgebras. All vector spaces are considered over the field of complex numbers $${\mathbb {C}}$$ .

The derivative concept was defined by Newton and Leipzig. After these scientific, there are many approaches about the order of derivative, since derivative defined by Newton and Leipzig considered as order of 1. Many scientists such as Caputo, Riemann, etc. defined the fractional order derivative. Karci is one of them who defined fractional order derivative. was defined by Karci, and it is not a linear derivative operator; it is a non-linear derivative operator. In this paper, we verified the most important properties of . has got an α parameter and this parameter can be any complex number. The properties of , which are derivative of product, derivative of quotient, the chain rule, the relationship between and complex numbers, etc., were verified in this paper. The most of these properties were not satisfied by other definitions for fractional order derivatives such as Caputo, Riemann-Lioville, Euler, etc. Khallil and his friends also defined fractional order derivative in a special case. This derivative satisfies these properties for special functions; in general, this definition also does not satisfy these properties.