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

In this paper new method is introduced for complex numbers; this method does not include imaginary number “i” but produces the same results that occur in Addition, Subtraction, Multiplication & Division of complex numbers, also proof of Eluer Formula eiθ and De Moivre theorem without using imaginary number “i”. Furthermore placing the light on the square root of a negative number, the square root of a negative number is equal to the square root of the same positive real number but with an angle of 90 degree to real line. The intention is that there’s nothing mystical about imaginary number “i”. The square root of minus one is just as real as any other number. Complex numbers exists without imaginary number “i”. This paper is limited to the results which are already established.

This work presents a simple study of stochastic arithmetic complex number operators for addition and multiplication. Their usage is demonstrated by design of a sum of product circuit As the stochastic complex number operators need more control random streams than stochastic rational number operators, we optimized the number of random generators used in the real circuit. In the end our sum of product circuit contains two LFSRs thus we analyzed the impact of the choice of the seeds for the LFSRs on the quality of the calculated results. By using exhaustive search over the LFSR state space we were able to reduce the output RMSE by 34% in comparison to choice of the equally spaced seeds over the LFSR state space.

paper introduces a novel method for complex number representation. The proposed Redundant Complex Binary Number System (RCBNS) is developed by combining a Redundant Binary Number and a complex number in base (-1+j). Donald [1] and Walter Penny [2,3] represented complex numbers using base –j and (-1+j) in the classified algorithmic models. A Redundant Complex Binary Number System consists of both real and imaginary-radix number systems that form a redundant integer digit set. This system is formed by using complex radix of (-1+j) and a digit set of á= 3, where á assumes a value of -3, -2, -1, 0, 1, 2, 3. The arithmetic operations of complex numbers with this system treat the real and imaginary parts as one unit. The carry-free addition has the advantage of Redundancy in number representation in the arithmetic operations. Results of the arithmetic operations are in the RCBNS form. The two methods for conversion from the RCBNS form to the standard binary number form have been presented. In this paper the RCBNS reduces the number of steps required to perform complex number arithmetic operations, thus enhancing the speed.

We establish some core inequalities for complex bounded sequences using the concept of statistical(C,1)summability.

In this paper, we suggested and analyzed a new higher-order iterative algorithm for solving nonlinear equation $$g(x)=0$$ g ( x ) = 0 , $$g:{\mathbb {R}}\longrightarrow {\mathbb {R}}$$ g : R ⟶ R , which is free from derivative by using the approximate version of the first derivative, and we studied the basins of attraction for the proposed iterative algorithm to find complex roots of complex functions $$g:{\mathbb {C}}\longrightarrow {\mathbb {C}}$$ g : C ⟶ C . To show the effectiveness of the proposed algorithm for the real and the complex domains, the numerical results for the considered examples are given and graphically clarified. The basins of attraction of the existing methods and our algorithm are offered and compared to clarify their performance. The proposed algorithm satisfied the condition such that $$|x_{m}-\alpha | | x m - α | < 1.0 × 10 - 15 , as well as the maximum number of iterations is less than or equal to 3, so the proposed algorithm can be applied to efficiently solve numerous type non-linear equations.

Some new results for finding all convenient moduli m for a complex number-theoretic transform with given transform length n and given primitive nth root of unity modulo m are presented. The main result is based on the prime factorization for values of cyclotomic polynomials in the ring of Gaussian integers.

Consider the universal gate set for quantum computing consisting of the gates X, CX, CCX, omega^dagger H, and S. All of these gates have matrix entries in the ring Z[1/2,i], the smallest subring of the complex numbers containing 1/2 and i. Amy, Glaudell, and Ross proved the converse, i.e., any unitary matrix with entries in Z[1/2,i] can be realized by a quantum circuit over the above gate set using at most one ancilla. In this paper, we give a finite presentation by generators and relations of U_n(Z[1/2,i]), the group of unitary nxn-matrices with entries in Z[1/2,i]., Comment: In Proceedings QPL 2021, arXiv:2109.04886

We provide a graphical treatment of SAT and #SAT on equal footing. Instances of #SAT can be represented as tensor networks in a standard way. These tensor networks are interpreted by diagrams of the ZH-calculus: a system to reason about tensors over C in terms of diagrams built from simple generators, in which computation may be carried out by transformations of diagrams alone. In general, nodes of ZH diagrams take parameters over C which determine the tensor coefficients; for the standard representation of #SAT instances, the coefficients take the value 0 or 1. Then, by choosing the coefficients of a diagram to range over B, we represent the corresponding instance of SAT. Thus, by interpreting a diagram either over the boolean semiring or the complex numbers, we instantiate either the decision or counting version of the problem. We find that for classes known to be in P, such as 2SAT and #XORSAT, the existence of appropriate rewrite rules allows for efficient simplification of the diagram, producing the solution in polynomial time. In contrast, for classes known to be NP-complete, such as 3SAT, or #P-complete, such as #2SAT, the corresponding rewrite rules introduce hyperedges to the diagrams, in numbers which are not easily bounded above by a polynomial. This diagrammatic approach unifies the diagnosis of the complexity of CSPs and #CSPs and shows promise in aiding tensor network contraction-based algorithms., Comment: In Proceedings QPL 2020, arXiv:2109.01534

'2(-a2 ? 12 + c2) + b'2(a2-b2 ? c2) + c'2(a2 + 12 c2)b > 16FF' (1) with equality if and only if the triangles ABC and A'B'C' are similar. Carlitz has verified (1) algebraically in [3] and given a further discussion in [4]. We now present another verification of (1) by complex-number computation. Set the vertex C of triangle ABC on the origin of the complex plane, and denote the other vertices by complex numbers A and B, the same symbols of these vertices for simplification. Deal with triangle A'B'C' similarly. Hence a = IBI, b = IAl, c = IA BI, a' = IB'I, b' = IA'I, c' = IA' B'I. Therefore

Let χ be a primitive Dirichlet character modulo k ≥ 3. In this paper, we define complex numbers associated with χ, which we denote by Cr(χ) (r = 0, 1,…), and we discuss their properties and their relationships with the generalized Bernoulli numbers.

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.

The paper presents algorithms for the generation of Residue Number System (RNS) triples with $SQ=2^{k}-1$ and quadruples with $SQ=2^{k}$ for some k. Triples and quadruples allow us to design efficient hardware implementations of non-modular operations in RNS such as division, sign detection, comparison of numbers, reverse conversion with using of a diagonal function from requiring division with the remainder by the diagonal module SQ. Division with a remainder in the general case is the most complex arithmetic operation in computer technology. However, the consideration of special cases can significantly simplify this operation and increase the efficiency of hardware implementation. We show that there are 5573 good RNS triples (2301 even and 2372 odd) with elements less than 10 000, as the values of SQ vary from $2^{5}-1$ to $2^{27}-1$ . In contrast, RNS quadruples with $SQ=2^{k}$ seem to be quite rare. Restricting our search to sums of the elements in a quadruple less than 4000 we find that exactly 31 such quadruples exist. Their values of SQ vary between 220 and 230 with always even exponent. We suggest the measure of RNS balance and find perfectly balanced RNS among triples according to this measure. We demonstrate the advantages of more balanced quadruples by means of hardware implementation.

Mutually orthogonal complementary codes have found many practical applications in wireless communications owing to their perfect correlation properties. However, the set size of a mutually orthogonal complementary code set (MOCCS) is upper bounded by the number of constituent sequences in each complementary code. As a result, the number of users in the communication system is limited. To overcome this limitation, quasi-complementary sequence sets (QCSSs) were proposed, which include low correlation zone complementary sequence sets (LCZ-CSSs) and low correlation complementary sequence sets (LC-CSSs). In this paper, aperiodic quasi-complementary sequences are our main interest. Constructions of aperiodic LC-CSSs and aperiodic LCZ-CSSs over the complex roots of unity are proposed. With these constructions, asymptotically optimal aperiodic QCSSs can be obtained.

A complex unit gain graph ( T -gain graph) Φ = ( G , φ ) is a graph where the function φ assigns a unit complex number to each orientation of an edge of G, and its inverse is assigned to the opposite orientation. A T -gain graph is balanced if the product of the edge gains of each oriented cycle (if any) is 1. We propose two notions of gain distance matrices D max ( Φ ) and D min ( Φ ) of a T -gain graph Φ, for any ordering ‘ D max ( Φ ) = D min ( Φ ) holds for the standard ordering of the vertices if and only if the same holds for any ordering of the vertices, and we call such T -gain graphs as distance compatible gain graphs. We characterize the distance compatible gain graphs whose gain distance matrices are cospectral with the distance matrix of the underlying graph. Besides, we introduce the notion of positively weighted T -gain graphs and establish an equivalent condition for the balance of a T -gain graph. Acharya's and Stanic's spectral criteria for balance are deduced as a consequence. Besides, we obtain some spectral characterizations for the balance of a T -gain graph in terms of the gain distance matrices. Finally, we characterize the distance compatible bipartite T -gain graphs. We show a T -gain graph Φ is distance compatible if and only if every block of Φ is distance compatible.

The purpose of this article is to consider a special class of combinatorial problems, the so called Prouhet-Tarry-Escot problem, solution of which is realized by constructing finite sequences of ±1. For example, for fixed p ∈ ℕ, is well known the existence of n p ∈ ℕ with the property: any set of n p consecutive integers can be divided into 2 sets, with equal sums of its pth-powers. The considered property remains valid also for sets of finite arithmetic progressions of complex numbers.

The transition matrix of a graph G corresponding to the adjacency matrix A is defined by H ( t ) ≔ exp − i t A , where t ∈ R . The graph is said to exhibit pretty good state transfer between a pair of vertices u and v if there exists a sequence t k of real numbers such that lim k → ∞ H ( t k ) e u = γ e v , where γ is a complex number of unit modulus. We present a class of circulant graphs admitting pretty good state transfer. Also we find some circulant graphs not exhibiting pretty good state transfer. This generalizes several pre-existing results on circulant graphs admitting pretty good state transfer.

The main purpose of this paper is to determine all matrix representations of the real numbers. It is shown that every such representation is completely reducible, while all non-trivial irreducible representations must be of 2-dimensional and can be expressed in a unique form. It is found that those representations are essentially determined by the ways of embedding the real numbers into the complex numbers. This results in a one-to-one correspondence between the equivalent classes of irreducible representations and the equivalent classes of homomorphisms from the real number field to the complex number field. The matrix representations of the complex numbers are also determined.

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.

Most mathematician, have accepted that a constant divided by zero is undefined. However, accepting this situation is an unsatisfactory solution to the problem as division by zero has arisen frequently enough in mathematics and science to warrant some serious consideration. The aim of this paper was to propose and prove the existence of a new number set in which division by zero is well defined. To do this, the paper first uses set theory to develop the idea of unstructured numbers and uses this new number to create a new number set called “Semi-structured Complex Number set” (Ś). It was then shown that a semi-structured complex number is a three-dimensional number which can be represented in the xyz-space with the x-axis being the real axis, the y-axis the imaginary axis and the z-axis the unstructured axis. A unit of rotation p was defined that enabled rotation of a point along the xy-, xz- and yz- planes. The field axioms were then used to show that the set is a “complete ordered field” and hence prove its existence. Examples of how these semi-structured complex numbers are used algebraically are provided. The successful development of this proposed number set has implications not just in the field of mathematics but in other areas of science where division by zero is essential.

Let P ( z ) be a polynomial of degree n and for any complex number α , let D α P ( z ) : = n P ( z ) + ( α − z ) P ′ ( z ) denote the polar derivative of P ( z ) with respect to α . In this paper, we present an integral inequality for the polar derivative of a polynomial. Our theorem includes as special cases several interesting generalisations and refinements of Erdox–Lax theorem.

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

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

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

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.

Let q a prime power and ${\mathbb F}_q$ the finite field of q elements. We study the analogues of Mahler’s and Koksma’s classifications of complex numbers for power series in ${\mathbb F}_q((T^{-1}))$ . Among other results, we establish that both classifications coincide, thereby answering a question of Ooto.

Planar graphs play an effective role in many practical applications where the crossing of edges becomes problematic. This paper aims to investigate the complex q-rung orthopair fuzzy (CQROF) planar graphs (CQROFPGs). In a CQROFPG, the nodes and edges are based on complex QROF information that represents the uncertain knowledge in the range of unit circles in terms of complex numbers. The motivation in discussing such a topic is the wide flexibility of QROF information in the expression of uncertain knowledge compared to intuitionistic and Pythagorean fuzzy settings. We discussed the complex QROF graphs (CQROFGs), complex QROF multigraphs (CQROFMGs), and related terms followed by examples. Furthermore, the notion of strength and planarity index (PI) of the CQROFPGs is defined and exemplified followed by a study of strong and weak edges. We further defined the notion of complex QROF face (CQROFF) and complex QROF dual graph (CQROFDG) and exemplified these concepts. A study of isomorphism, coweak and weak isomorphism, is set up, and some results relating to the CQROFPG and isomorphisms are explored using examples. Furthermore, the problem of short circuits that results due to crossing is discussed because of the proposed study where an algorithm based on complex QROF (CQROF) information is presented for reducing the crossing in networks. Some advantages of the projected study over the previous study are observed, and some future study is predicted.

In this paper, we first create a network model using a complex number-weighted node, where the real number-weighted network model is its special case. According to the "send effect a " and "receive effect b " of node measures, the complex number weight can be determined as a +i b . Then, we build a complex vector space on the complex plane, and the real network can be mapped into the effect field of this space, where the effect importance (I) and classification of the node may be determined (Importance on Left, Minimum, Importance on Right). Finally, we present four theorems that trigger the evolutionary dynamics of network systems when the node measure or topological structure is changed, so that the conjunction principle and "entangled effect" between the topological structure and kinetic characteristics are revealed. As a result, a new path is discovered to explain some real phenomena in nature or human society, such as the mechanism of the "stronger the strong" and "win-win" cooperation. The method may be applied to the analysis of various network models including regular networks, stochastic networks, and small-world networks, which may exhibit special explanatory capabilities.

We present the zeta-expansion as a complex version of the well-known beta-expansion. It allows us to expand complex numbers with respect to a complex base by using integer digits. Our concepts fits into the framework of the recently published rotational beta-expansions. But we also establish relations with piecewise affine maps of the torus and with shift radix systems.

A generalized polygon is an ordered set of vertices. This notion generalizes the concept of the boundary of a polygonal shape because self-intersections are allowed. In this paper we study the problem of matching generalized polygons under affine transformations. Our approach is based on invariants. Firstly we associate an ordered set of complex numbers with each polygon and construct a collection of complex scalar functions on the space of plane polygons. These invariant functions are defined as quotients of the so-called Fourier descriptors, also known as discrete Fourier transforms.Each one of these functions is invariant under similarity transformations; that is, the function associates the same complex number to similar polygons. Moreover, if two polygons are affine related (one of them is the image of the other under an affine transformation), the pseudo-hyperbolic distance between their associated values is a constant that depends only on the affine transformation involved, but independent of the polygons.More formally, given a collection { Z 1 , Z 2 , ź , Z m } of n-sided polygons in the plane and a query polygon W, we give algorithms to find all Z ź such that f ( Z ź ) = W + Δ W , where f is an unknown affine transformation and Δ W = ( Δ w 1 , ź , Δ w n ) with | Δ w k | ź ź , where ź is certain tolerance.

Complex fuzzy set (CFS) is a recent development in the field of fuzzy set (FS) theory. The significance of CFS lies in the fact that CFS assigned membership grades from a unit circle in the complex plane, i.e., in the form of a complex number whose amplitude term belongs to a [ 0 , 1 ] interval. The interval-valued complex fuzzy set (IVCFS) is one of the extensions of the CFS in which the amplitude term is extended from the real numbers to the interval-valued numbers. The novelty of IVCFS lies in its larger range comparative to CFS. We often use fuzzy distance measures to solve some problems in our daily life. Hence, this paper develops some series of distance measures between IVCFSs by using Hamming and Euclidean metrics. The boundaries of these distance measures for IVCFSs are obtained. Finally, we study two geometric properties include rotational invariance and reflectional invariance of these distance measures.

This paper gives the first algorithm for finding a set of natural $\epsilon$-clusters of complex zeros of a triangular system of polynomials within a given polybox in $\mathbb{C}^n$, for any given $\epsilon>0$. Our algorithm is based on a recent near-optimal algorithm of Becker et al (2016) for clustering the complex roots of a univariate polynomial where the coefficients are represented by number oracles. Our algorithm is numeric, certified and based on subdivision. We implemented it and compared it with two well-known homotopy solvers on various triangular systems. Our solver always gives correct answers, is often faster than the homotopy solver that often gives correct answers, and sometimes faster than the one that gives sometimes correct results., Comment: Research report V6: description of the main algorithm updated

We report an ongoing work on clustering algorithms for complex roots of a univariate polynomial p of degree d with real or complex coefficients. As in their previous best subdivision algorithms our root-finders are robust even for multiple roots of a polynomial given by a black box for the approximation of its coefficients, and their complexity decreases at least proportionally to the number of roots in a region of interest (ROI) on the complex plane, such as a disc or a square, but we greatly strengthen the main ingredient of the previous algorithms. We build the foundation for a new counting test that essentially amounts to the evaluation of a polynomial p and its derivative \(p'\), which is a major benefit, e.g., for sparse polynomials p. Moreover with evaluation at about \(\log (d)\) points (versus the previous record of order d) we output correct number of roots in a disc whose contour has no roots of p nearby. Our second and less significant contribution concerns subdivision algorithms for polynomials with real coefficients. Our tests demonstrate the power of the proposed algorithms.

The nonnegative inverse eigenvalue problem is the problem of determining necessary and sufficient conditions for a multiset of complex numbers to be the spectrum of a nonnegative real matrix of size equal to the cardinality of the multiset itself. The problem is longstanding and proved to be very difficult so that several variations have been defined by considering particular classes of multisets and nonnegative real matrices. In this paper, a novel variation of the problem is proposed. This variation is motivated by a practical application in the positive realization problem, that is the problem of characterizing existence and minimality of a positive state--space representation of a given transfer function.

We show that, apart from some obvious exceptions, the number of trinomials vanishing at given complex numbers is bounded by an absolute constant. When the numbers are algebraic, we also bound effectively the degrees and the heights of these trinomials., Inaccuracies corrected following the referee's suggestions

The pslq algorithm computes integer relations for real numbers and Gaussian integer relations for complex numbers. We endeavour to extend pslq to find integer relations consisting of algebraic integers from some quadratic extension fields (in both the real and complex cases). We outline the algorithm, discuss the required modifications for handling algebraic integers, problems that have arisen, experimental results, and challenges to further work.

The article shows materials to the question about algebraic features of the genetic code and about the dictatorial influence of the DNA and RNA molecules on the whole organism. Presented results testify in favor that the genetic code is an algebraic code related with a wide class of algebraic codes, which are a basis of noise-immune coding of information in communication technologies. Structural features of the genetic systems are associated with hypercomplex double (or hyperbolic) numbers and with bisymmetric doubly stochastic matrices. The received results confirm that represented matrix approaches are effective for modeling genetic phenomena and revealing the interconnections of structures of biological bodies at various levels of their organization. This allows one to think that living organisms are algebraically encoded entities where structures of genetic molecules have the dictatorial influence on inherited structures of the whole organism. New described algebraic approaches and results are discussed.

International audience; Let f be a polynomial (or polynomial system) with all simple roots. The root separation of f is the minimum of the pair-wise distances between the complex roots. A root separation bound is a lower bound on the root separation. Finding a root separation bound is a fundamental problem, arising in numerous disciplines. We present two new root separation bounds: one univariate bound, and one multivariate bound. The new bounds improve on the old bounds in two ways: (1) The new bounds are usually significantly bigger (hence better) than the previous bounds. (2) The new bounds scale correctly, unlike the previous bounds. Crucially, the new bounds are not harder to compute than the previous bounds.

This paper extends three results from classical finite frame theory over real or complex numbers to binary frames for the vector space ${\mathbb Z}_2^d$. Without the notion of inner products or order, we provide an analog of the "fundamental inequality" of tight frames. In addition, we prove the binary analog of the characterization of dual frames with given inner products and of general frames with prescribed norms and frame operator., Comment: Version 2 of this paper corrects a mistake in the last sentence of the paragraph following Theorem 2.4. The mistake remains in the published version (Involve); however, it is not consequential

The sweeping development of mathematics is largely due to the introduction of complex numbers. Although complex numbers are just absurd notations for most people, complex numbers play essential roles in engineering fields. When mathematicians began to take an investigation of complex numbers, human beings enter a marvelous world. Complex numbers convince scientists that our world is magical, full of wonderful insights and even miraculous. In this paper, I first review several basic properties of complex numbers. The set of complex numbers is a group not only under the operation of the multiplication but under the operation of addition. Then, I show a visualisation of complex numbers through building a bijective connection between complex numbers and points on the complex plane. I also give several alternative expression forms of complex numbers, namely the trigonometric form and the general form. By invoking the arithmetic properties of complex numbers, I prove two trigonometric identities.

A Gaussian integer is a complex number whose real and imaginary parts are both integers. Meanwhile, a sequence is defined as perfect if and only if it has an ideal periodic auto-correlation function. This paper proposes a method for constructing sparse perfect Gaussian integer sequences (SPGISs) in which most of the sequence elements are zero. The proposed SPGISs are obtained by linearly combining four base sequences or their cyclic-shift equivalents using nonzero Gaussian integer coefficients of equal magnitudes. Each base sequence contains four nonzero elements belonging to the set $\{\pm 1, \pm j\}$ . The number of nonzero elements of the constructed SPGISs depends on the choice of complex coefficients and cyclic shifts. However, each SPGIS has at most 16 nonzero elements, irrespective of the sequence length. A systematic investigation is performed into the properties of the SPGISs and their Fourier dual equivalents. Finally, a general expression is derived for a perfect Gaussian integer sequence (PGIS) of length $4n$ , where $n$ is any positive integer and most of the sequence elements are nonzero.