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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

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!

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.

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

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.

We study projections of matrix algebras over complex or real numbers, to (partially) classify the (generically non-degenerate) projections (in our sense) and their spaces algebraically and geometrically, in the three by three, and four by four matrix cases, and in the general matrix case., 紀要論文

With increased interests of neural networks, hardware implementations of neural networks have been investigated. Researchers pursue low hardware cost by using different technologies such as stochastic computing (SC) and quantization. More specifically, the quantization is able to reduce total number of trained weights and results in low hardware cost. SC aims to lower hardware costs substantially by using simple gates instead of complex arithmetic operations. However, the advantages of both quantization and SC in neural networks are not well investigated. In this article, we propose a new stochastic multiplier with simple CMOS transistors called the stochastic hybrid multiplier for quantized neural networks. The new design uses the characteristic of quantized weights and tremendously reduces the hardware cost of neural networks. Experimental results indicate that our stochastic design achieves about 7.7x energy reduction compared to its counterpart binary implementation while maintaining slightly higher recognition error rates than the binary implementation. Compared to previous stochastic neural network implementations, our work derives at least 4x, 9x, and 10x reduction in terms of area, power, and energy, respectively.

We classify real families of minimal degree rational curves that cover an embedded rational surface. A corollary is that if the projective closure of a smooth surface is not biregular isomorphic to the projective closure of the unit-sphere, then the set of minimal degree rational curves that cover the surface is either empty or of dimension at most two. Moreover, if these curves are of minimal degree over the real numbers, but not over the complex numbers, then almost all the curves are smooth. Our methods lead to an algorithm that takes as input a real surface parametrization and outputs all real families of rational curves of lowest possible degree that cover the image surface.

We prove a complex polynomial of degree $n$ has at most $\lceil n/2 \rceil$ attractive fixed points lying on a line. We also consider the general case., 4 pages

In this paper, a new lemma is proposed to study the stability of a fractional order complex chaotic system without dividing the complex number into real and imaginary parts. The proving process of the new lemma combines the fundamental properties of the complex field and the fractional order extension of the Lyapunov direct method. It extends the fractional order extension of the Lyapunov direct method from the real number field to the complex number field. Based on the new lemma, we propose a new impulsive synchronization scheme for fractional order complex chaotic systems. The numerical simulation results also show the validity of our conclusion.

In this paper, we study the representation theory of the Fuss–Catalan algebras, FC n ( a , b ) . We prove that this algebra is cellular with a cellular basis and forms a tower of recollement, as defined by Cox, Martin, Parker, and Xi [7] , and hence, it is quasi-hereditary algebra if a , b are non-zero complex numbers.

The Nonnegative Inverse Eigenvalue Problem (NIEP) is the problem of determining necessary and sufficient conditions for a list of $n$ complex numbers to be the spectrum of an entry--wise nonnegative matrix of dimension $n$. This is a very difficult and long standing problem and has been solved only for $n\leq 4$. In this paper, the NIEP for a particular class of nonnegative matrices, namely Leslie matrices, is considered. Leslie matrices are nonnegative matrices, with a special zero--pattern, arising in the Leslie model, one of the best known and widely used models to describe the growth of populations. The lists of nonzero complex numbers that are subsets of the spectra of Leslie matrices are fully characterized. Moreover, the minimal dimension of a Leslie matrix having a given list of three numbers among its spectrum is provided. This result is partially extended to the case of lists of $n > 2$ real numbers.

The problem of the existence and construction of nonnegative matrices with prescribed eigenvalues and diagonal entries is an important inverse problem, interesting by itself, but also necessary to apply a perturbation result, which has played an important role in the study of certain nonnegative inverse spectral problems. A number of partial results about the problem have been published by several authors, mainly by H. \v{S}migoc. In this paper, the relevance of a Brauer's result, and its implication for the nonnegative inverse eigenvalue problem with prescribed diagonal entries is emphasized. As a consequence, given a list of complex numbers of \v{S}migoc type, or a list $\Lambda = \left\{\lambda _{1},\ldots ,\lambda _{n} \right \}$ with $\operatorname{Re}\lambda _{i}\leq 0,$ $\lambda _{1}\geq -\sum\limits_{i=2}^{n}\lambda _{i}$, and $\left\{-\sum\limits_{i=2}^{n}\lambda _{i},\lambda _{2},\ldots ,\lambda _{n} \right\}$ being realizable; and given a list of nonnegative real numbers $% \Gamma = \left\{\gamma _{1},\ldots ,\gamma _{n} \right\}$, the remarkably simple condition $\gamma _{1}+\cdots +\gamma _{n} = \lambda _{1}+\cdots +\lambda _{n}$ is necessary and sufficient for the existence and construction of a realizing matrix with diagonal entries $\Gamma .$ Conditions for more general lists of complex numbers are also given.

Let U n be a complex unitary group and D n be its subgroup that consists of n × n diagonal matrices whose diagonal elements are complex numbers of modulus 1. In this paper we present some general results on Jordan triple product maps between U n and U m . As a main result we present the general form of all (continuous) Jordan triple product maps from the group U n to the group D m . Also, we characterize the general form of all (continuous) maps between the groups D n and D m that preserve the Jordan triple product. These are the (continuous) maps φ : D n → D m which satisfy φ ( V W V ) = φ ( V ) φ ( W ) φ ( V ) , V , W ∈ D n .

We generalized the concepts in probability of rough lacunary statistical by introducing the diference operator of fractional order, where is a proper fraction and = (mnk ) is anyxed sequence of nonzero real or complex numbers. We study some properties of this operator involving lacunary sequence and arbitrary sequence p = (prst) of strictly positive real numbers and investigate the topological structures of related triple diference sequence spaces. The main focus of the present paper is to generalized rough lacunary statistical of triple diference sequence spaces and investigate their topological structures as well as some inclusion concerning the operator

The Geometric Algebra formalism opens the door to developing a theory deeper than conventional quantum mechanics. Generalizations, stemming from implementation of complex numbers as geometrically feasible objects in three dimensions, unambiguous definition of states, observables, measurements, Maxwell equations solution in that terms, bring into reality a kind of physical fields, states in the suggested theory, spreading through the whole three-dimensional space and values of the time parameter. The fields can be modified instantly in all points of space and time values, thus eliminating the concept of cause and effect and perceiving of one-directional time.

In this note, for any pair of natural numbers (n,k), n≥3, k≥1, and 2k

We show that symmetric and positive profiles of ground-state standing-wave of the non-linear Schr\"odinger equation are non-degenerate and unique up to a translation of the argument and multiplication by complex numbers in the unit sphere. The non-linear term is a combination of two or three pure-powers. The class of non-linearities satisfying the mentioned properties can be extended beyond two or three power combinations. Specifically, it is sufficient that an Euler differential inequality is satisfied and that a certain auxiliary function is such that the first local maximum is also an absolute maximum., Comment: 12 pages, 4 figures, AMS Special Session on "Spectral Calculus and Quasilinear Partial Differential Equations" at the Joint Mathematics Meeting

Of great importance for modern technology are the issues of the dynamic behavior of machines and structures. Given the trends of today such as new information technologies, the transformation of design processes; creation of CAD systems (automation systems for design work), issues of improving methods for solving dynamics problems are of particular importance.Numerical modeling has become an integral part of the study of the most complex processes for the most complex physical models, and the Finite Element Method has become the main numerical method for their study. These trends will inevitably be accompanied by a significant increase in the number of parameters for determining the state of the object. In particular, in the dynamics of machines, a significant number of degrees of freedom. As a result, the appearance of the problem of a multiple increase in the size of the task. The solution of the eigenvalue problem (EigenValue) can be considered as an important component in the construction of numerical-analytical approaches that are alternative to simple step-by-step integration schemes of the Runge-Kutta type in large-scale problems. Certain benefits can be obtained if the linear transformation matrices are first reduced to a simple form. This approach is widely used in dynamics (modal analysis). In this paper, we propose a scheme for a step-by-step numerical analysis of the structure of the matrix K in a problem and a scheme for constructing a Jordan basis for the general case of roots of a characteristic polynomial (for real and complex roots, simple and multiple). As the starting form, the standard eigenvalue problem with the matrix K previously reduced to the Schur form (block-triangular shape matrix) is adopted. The scheme is accompanied by the solution of test cases.

It is well known that $\zeta(2)=\pi^2/6$ as discovered by Euler. In this paper we present the following two $q$-analogues of this celebrated formula: $$\sum_{k=0}^\infty\frac{q^k(1+q^{2k+1})}{(1-q^{2k+1})^2}=\prod_{n=1}^\infty\frac{(1-q^{2n})^4}{(1-q^{2n-1})^4}$$ and $$\sum_{k=0}^\infty\frac{q^{2k-\lfloor(-1)^kk/2\rfloor}}{(1-q^{2k+1})^2} =\prod_{n=1}^\infty\frac{(1-q^{2n})^2(1-q^{4n})^2}{(1-q^{2n-1})^2(1-q^{4n-2})^2},$$ where $q$ is any complex number with $|q

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

We formalize algebraic numbers in Isabelle/HOL. Our development serves as a verified implementation of algebraic operations on real and complex numbers. We moreover provide algorithms that can identify all the real or complex roots of rational polynomials, and two implementations to display algebraic numbers, an approximative version and an injective precise one. We obtain verified Haskell code for these operations via Isabelle's code generator. The development combines various existing formalizations such as matrices, Sturm's theorem, and polynomial factorization, and it includes new formalizations about bivariate polynomials, unique factorization domains, resultants and subresultants.

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.

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}}

Simple finite dimensional Kantor triple systems over the complex numbers are classified in terms of Satake diagrams. We prove that every simple and linearly compact Kantor triple system has finite dimension and give an explicit presentation of all the classical and exceptional systems., Comment: 46 pages, 3 tables; v2: Major revision of the introduction; v3: Final version to appear in Journal of Algebra

We report on a verification of the Fundamental Theorem of Algebra in ACL2(r). The proof consists of four parts. First, continuity for both complex-valued and real-valued functions of complex numbers is defined, and it is shown that continuous functions from the complex to the real numbers achieve a minimum value over a closed square region. An important case of continuous real-valued, complex functions results from taking the traditional complex norm of a continuous complex function. We think of these continuous functions as having only one (complex) argument, but in ACL2(r) they appear as functions of two arguments. The extra argument is a "context", which is uninterpreted. For example, it could be other arguments that are held fixed, as in an exponential function which has a base and an exponent, either of which could be held fixed. Second, it is shown that complex polynomials are continuous, so the norm of a complex polynomial is a continuous real-valued function and it achieves its minimum over an arbitrary square region centered at the origin. This part of the proof benefits from the introduction of the "context" argument, and it illustrates an innovation that simplifies the proofs of classical properties with unbound parameters. Third, we derive lower and upper bounds on the norm of non-constant polynomials for inputs that are sufficiently far away from the origin. This means that a sufficiently large square can be found to guarantee that it contains the global minimum of the norm of the polynomial. Fourth, it is shown that if a given number is not a root of a non-constant polynomial, then it cannot be the global minimum. Finally, these results are combined to show that the global minimum must be a root of the polynomial. This result is part of a larger effort in the formalization of complex polynomials in ACL2(r)., In Proceedings ACL2 2018, arXiv:1810.03762

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

Let G = G L n ( C ) , the general linear group over the complex numbers, and let B be the set of invertible upper triangular matrices in G . Let b = Lie ( B ) . For μ : T ∗ ( b × C n ) → b ∗ , where b ∗ ≅ g ∕ u and u being strictly upper triangular matrices in g = Lie ( G ) , we prove that the Hamiltonian reduction μ − 1 ( 0 ) r s s ∕ ∕ B of the extended regular semisimple locus b r s s of the Borel subalgebra is smooth, affine, reduced, and scheme-theoretically isomorphic to a dense open locus of C 2 n . We also show that the B -invariant functions on the regular semisimple locus of the Hamiltonian reduction of b × C n arise as the trace of a certain product of matrices.

In this paper, novel methods for the evaluation of digital-filter stability are investigated. The methods are based on phase analysis of a complex function in the characteristic equation of a digital filter. It allows for evaluating stability when a characteristic equation is not based on a polynomial. The operation of these methods relies on sampling the unit circle on the complex plane and extracting the phase quadrant of a function value for each sample. By calculating function-phase quadrants, regions in the immediate vicinity of unstable roots (i.e., zeros), called candidate regions, are determined. In these regions, both real and imaginary parts of complex-function values change signs. Then, the candidate regions are explored. When the sizes of the candidate regions are reduced below an assumed accuracy, then filter instability is verified with the use of discrete Cauchy’s argument principle. Three different algorithms of the unit-circle sampling are benchmarked, i.e., global complex roots and poles finding (GRPF) algorithm, multimodal genetic algorithm with phase analysis (MGA-WPA), and multimodal particle swarm optimization with phase analysis (MPSO-WPA). The algorithms are compared in four benchmarks for integer- and fractional-order digital filters and systems. Each algorithm demonstrates slightly different properties. GRPF is very fast and efficient; however, it requires an initial number of nodes large enough to detect all the roots. MPSO-WPA prevents missing roots due to the usage of stochastic space exploration by subsequent swarms. MGA-WPA converges very effectively by generating a small number of individuals and by limiting the final population size. The conducted research leads to the conclusion that stochastic methods such as MGA-WPA and MPSO-WPA are more likely to detect system instability, especially when they are run multiple times. If the computing time is not vitally important for a user, MPSO-WPA is the right choice, because it significantly prevents missing roots.

In the case of many low-cycle multiaxial fatigue criteria, we encounter a mathematical problem of adding vectors of normal and shear strains. Typically, the problem of defining an equivalent strain is solved by weighting factors. Unfortunately, this ignores the fact that these vectors represent other physical quantities: the normal strain is a longitudinal strain, and the shear strain is a rotation angle. Therefore, the goal of the present work was to propose a method of combining different types of strains by adopting a system of complex numbers. The normal strain was defined as the real part and the shear strain was defined as the imaginary part. Using this approach, simple load states, such as pure bending and pure torsion, have been transformed into an expression for equivalent strain identical to the previously proposed criteria defined by Macha.

Complex numbers are widely used in both classical and quantum physics, and are indispensable components for describing quantum systems and their dynamical behavior. Recently, the resource theory of imaginarity has been introduced, allowing for a systematic study of complex numbers in quantum mechanics and quantum information theory. In this work we develop theoretical methods for the resource theory of imaginarity, motivated by recent progress within theories of entanglement and coherence. We investigate imaginarity quantification, focusing on the geometric imaginarity and the robustness of imaginarity, and apply these tools to the state conversion problem in imaginarity theory. Moreover, we analyze the complexity of real and general operations in optical experiments, focusing on the number of unfixed wave plates for their implementation. We also discuss the role of imaginarity for local state discrimination, proving that any pair of real orthogonal pure states can be discriminated via local real operations and classical communication. Our study reveals the significance of complex numbers in quantum physics, and proves that imaginarity is a resource in optical experiments., 13 pages, 2 figures. Accompanying article for arXiv:2007.14847

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.

If f is an entire function and a is a complex number, a is said to be an asymptotic value of f if there exists a path $$\gamma $$ from 0 to infinity such that $$f(z) - a$$ tends to 0 as z tends to infinity along $$\gamma $$ . The Denjoy–Carleman–Ahlfors Theorem asserts that if f has n distinct asymptotic values, then the rate of growth of f is at least order n/2, mean type. A long-standing problem asks whether this conclusion holds for entire functions having n distinct asymptotic (entire) functions, each of growth at most order 1/2, minimal type. In this paper conditions on the function f and associated asymptotic paths are obtained that are sufficient to guarantee that f satisfies the conclusion of the Denjoy–Carleman–Ahlfors Theorem. In addition, for each positive integer n, an example is given of an entire function of order n having n distinct, prescribed asymptotic functions, each of order less than 1/2.

By Cartan’s version of Nevanlinna’s theory, we prove the following result: let m and n be two positive integers satisfying $$n\ge 2+m,$$ n ≥ 2 + m , let $$p\not \equiv 0$$ p ≢ 0 be a polynomial, let $$\eta \ne 0$$ η ≠ 0 be a finite complex number, let $$\omega _{1}, \omega _{2}, \ldots , \omega _{m}$$ ω 1 , ω 2 , … , ω m be m distinct finite nonzero complex numbers, and let $$H_{j}$$ H j be either exponential polynomials of degree less than q, or an ordinary polynomial in z for $$0\le j\le m$$ 0 ≤ j ≤ m , such that $$H_{j}\not \equiv 0$$ H j ≢ 0 for $$1\le j\le m.$$ 1 ≤ j ≤ m . Suppose that $$f\not \equiv \infty $$ f ≢ ∞ is a meromorphic solution of the difference equation: $$\begin{aligned} f^n(z)+p(z)f(z+\eta )&=H_0(z)+H_1(z)e^{\omega _{1}z^{q}}+H_2(z)e^{\omega _{2}z^{q}}\\&\quad +\cdots +H_m(z)e^{\omega _{m}z^{q}}, \end{aligned}$$ f n ( z ) + p ( z ) f ( z + η ) = H 0 ( z ) + H 1 ( z ) e ω 1 z q + H 2 ( z ) e ω 2 z q + ⋯ + H m ( z ) e ω m z q , such that the hyper-order of f satisfies $$\rho _2(f) ρ 2 ( f ) < 1 . Then, f reduces to a transcendental entire function, such that either $$n=m+2$$ n = m + 2 with $$H_0\not \equiv 0$$ H 0 ≢ 0 and $$\lambda (f)=\rho (f)=q,$$ λ ( f ) = ρ ( f ) = q , or $$m=2,$$ m = 2 , $$H_0=0$$ H 0 = 0 and: $$\begin{aligned} f(z)=\frac{H_1(z-\eta )e^{\omega _{1}(z-\eta )^{q}}}{p(z-\eta )} \end{aligned}$$ f ( z ) = H 1 ( z - η ) e ω 1 ( z - η ) q p ( z - η ) with $$\begin{aligned} H^n_1(z)=p^n(z)H_2(z+\eta )e^{\omega _2P_{q-1}(z)}\quad \text {and}\quad P_{q-1}(z)=\sum \limits _{k=1}^q\left( {\begin{array}{c}q\\ k\end{array}}\right) \eta ^kz^{q-k}. \end{aligned}$$ H 1 n ( z ) = p n ( z ) H 2 ( z + η ) e ω 2 P q - 1 ( z ) and P q - 1 ( z ) = ∑ k = 1 q q k η k z q - k . This result improves Theorems 1.1 and 1.3 from [19] by removing some assumptions of theirs. An example is provided to show that some results obtained in this paper, in a sense, are the best possible.

Finding arbitrary roots of polynomials is a fundamental problem in various areas of science and engineering. A myriad of methods was suggested to address this problem, such as the sequential Newton’s method and the Durand–Kerner (D–K) simultaneous iterative method. The sequential iterative methods, on the one hand, need to use a deflation procedure in order to compute approximations to all the roots of a given polynomial, which can produce inaccurate results due to the accumulation of rounding errors. On the other hand, the simultaneous iterative methods require good initial guesses to converge. However, Artificial Neural Networks (ANNs) are widely known by their capacity to find complex mappings between the dependent and independent variables. In view of this, this paper aims to determine, based on comparative results, whether ANNs can be used to compute approximations to the real and complex roots of a given polynomial, as an alternative to simultaneous iterative algorithms like the D–K method. Although the results are very encouraging and demonstrate the viability and potentiality of the suggested approach, the ANNs were not able to surpass the accuracy of the D–K method. The results indicated, however, that the use of the approximations computed by the ANNs as the initial guesses for the D–K method can be beneficial to the accuracy of this method.

Викладено теорію аналітичних функцій та її застосування, зокрема, для обчислення інтегралів за допомогою лишків. Також розглянуто питання аналитичного продовження, теорію конфирмних відображень. Навчальний посібник «Кратні, криволінійні та поверхневі інтеграли, гармонічний аналіз. Частина II», призначений для вивчення диференціального та інтегрального числення аналітичних функцій, теорії степеневих рядів та рядів Лорана. Методи комплексного аналізу застосовуються для обчислення контурних, власних та невласних інтегралів, в теорії конформних відображень областей. Розглянуто також питання аналітичного продовження функцій. Посібник містить багато прикладів і питань для самоперевірки. У Доповненні детально розглянуто стереографічну проекцію та інверсію і визначено зв’язок між ними. Textbook "Multiple, curvilinear and surface integrals, harmonic analysis. Part II ", designed to study the differential and integral calculus of analytic functions, the theory of power series and Laurent series. Methods of complex analysis are used to calculate contour, eigenvalues ​​and eigenintegrals in the theory of conformal mappings of domains. The issue of analytical continuation of functions is also considered. The manual contains many examples and questions for self-examination. The Appendix details stereographic projection and inversion and the relationship between them. Учебное пособие «Кратные, криволинейные и поверхностные интегралы, гармонический анализ. Часть II», предназначенный для изучения дифференциального и интегрального исчисления аналитических функций, теории степенных рядов и рядов Лорана. Методы комплексного анализа применяются для вычисления контурных, собственных и несобственных интегралов, в теории конформных отображений областей. Рассмотрены также вопросы аналитического продолжения функций. Пособие содержит много примеров и вопросов для самопроверки. В Приложении подробно рассмотрены стереографическая проекция и инверсию и определена связь между ними.

In this article, we study the obstructions to the local-global principle for homogeneous spaces with connected or abelian stabilizers over finite extensions of the field $\mathbb{C}((x,y))$ of Laurent series in two variables over the complex numbers and over function fields of curves over $\mathbb{C}((t))$. We give examples that prove that the usual Brauer-Manin obstruction is not enough to explain the failure of the local-global principle, and we then construct a variant of this obstruction using torsors under quasi-trivial tori which turns out to work. In the end of the article, we compare this new obstruction to the descent obstruction with respect to torsors under tori. For that purpose, we use a result on towers of torsors, that is of independent interest and therefore is proved in a separate appendix., 22 pages, final version

Fuzzy sets and fuzzy logics are used to model events with imprecise, incomplete, and uncertain information. Researchers have developed numerous methods and techniques to cope with fuzziness or uncertainty. This research intends to introduce the novel concepts of complex neutrosophic relations (CNRs) and its types based on the idea of complex neutrosophic sets (CNSs). In addition, these concepts are supported by suitable examples. A CNR discusses the quality of a relationship using the degree of membership, the degree of abstinence, and the degree of nonmembership. Each of these degrees is a complex number from the unit circle in a complex plane. The real part of complex-valued degrees represents the amplitude term, while the imaginary part represents the phase term. This property empowers CNRs to model multidimensional variables. Moreover, some interesting properties and useful results have also been proved. Furthermore, the practicality of the proposed concepts is verified by an application, which discusses the use of the proposed concepts in statistical decision-making. Additionally, a comparative analysis between the novel concepts of CNRs and the existing methods is carried out.

It has long been known that the ellipse normally used to model the phase space extension of a beam in linear dynamics may be represented by a complex number which can be interpreted similar to a complex impedance in electrical circuits, so that classical electrical methods might be used for the design of such beam transport lines. However, this method has never been fully developed, and only the transport transformation of single particular elements, like drift spaces or quadrupoles, has been presented in the past. In this article, we complete the complex formalism of linear beam dynamics by obtaining a general differential equation and solving it, to show that the general transformation of a linear beam line is a complex Moebius transformation. This result opens the possibility of studying the effect of the beam line on complete regions of the complex plane and not only on a single point. Taking advantage of this capability of the formalism, we also obtain an important result in the theory of the transport through a periodic line, proving that the invariant points of the transformation are only a special case of a more general structure of the solution, which are the invariant circles of the one-period transformation. Among other advantages, this provides a new description of the betatron functions beating in case of a mismatched injection in a circular accelerator This work was supported in part by the Ministerio de Asuntos Economicos y Transformacion Digital (MINECO) under Grant DPI2017-82373-R and in part by Universidad del Pais Vasco/Euskal Herriko Univertsitatea (UPV/EHU) under Grant GIU18/196

Let $A$ be an algebra over the field of complex numbers with a (Hausdorff) topology given by a family $\Cal{Q}= \{q_\lambda |\lambda\in\Lambda\}$ of square preserving $r_\lambda$-homogeneous seminorms ($r_\lambda \in (0,1]$). We shall show that $(A,T(\Cal{Q}))$ is a locally m-convex algebra. Furthermore we shall show that $A$ is commutative.