Odd Degree Research Articles

Optimization on the Euclidean Unit Sphere

We consider the problem of minimizing a continuously differentiable function f of m linear forms in n variables on the Euclidean unit sphere. We show that this problem is equivalent to minimizing the same function of related m linear forms (but now in m variables) on the Euclidean unit ball. When the linear forms are known, this results in a drastic reduction in problem size whenever m ≪ n and allows to solve potentially large scale non-convex such problems. We also provide a test to detect when a polynomial is a polynomial in a fixed number of forms. Finally, we identify two classes of functions with no spurious local minima on the sphere: (i) quasi-convex polynomials of odd degree and (ii) nonnegative and homogeneous functions. Finally, odd degreed forms have only nonpositive local minima and at most (d − 1) m are strictly negative.

Redesign Arabic Calligraphy Outline by using Bezier, Wang-Ball and Said-Ball Curves

In the history of the revitalization of traditional art in the modern world, Arabic calligraphy was one of the art phenomena. This research suggests using Bezier curves, Wang Ball curves, and Said-Ball curves to improve Arabic calligraphy. Our goal in calligraphy is to achieve a smooth beautifying contour of a character. The calligraphy outline, which was largely Arabic, was frequently applied in Bezier curve degrees two and three. As a result, the degree elevation for each curve was increased until it reached degree four, allowing for greater Bezier curve study. This paper presents the comparisons between those three curves by using the Arabic calligraphy "Aamin" as the subject. Curve fitting, curve manipulation, mixing, and merging of curves were all examples of how Bezier curves were employed in Computer Aided Design. Ball curves, also known as a generalized ball basis, are a type of curve that has arbitrary even and odd degrees. There were two types of generalized ball curves: Wang-ball curves and Said-ball curves, each with their own special position parameter. The best curves of the Bezier, Wang-Ball, and Said-Ball curves were combined to create "Aamin" calligraphy. All of the results were programmed into the MATLAB software and graphed. For the redesign “Aamin” calligraphy Bezier curve computes the best time compared to Wang-Ball and Said-Ball curve. The computation time in MATLAB software for the degree 2, degree 3 and degree 4 are 1.48 seconds, 1.52 seconds and 2.50 seconds respectively. The higher the degree elevation, the longer the computation time to generate "Aamin" calligraphy, according to these findings. This research provides the fundamental design and empirical findings in the process of improving the production of computerized Arabic calligraphy.

Hodge conjecture for the moduli space of semi-stable sheaves over a nodal curve

In this article, we prove the Hodge conjecture for a desingularization of the moduli space of rank 2, semi-stable, torsion-free sheaves with fixed odd degree determinant over a very general irreducible nodal curve of genus at least 2. We also compute the algebraic Poincaré polynomial of the associated cohomology ring.

On correctness of conditions for the CSIDH algorithm implementation on Edwards curves

Incorrect formulation and incorrect solution of the problem of the CSIDH algorithm implementation on Edwards curves was revealed in one of the famous works. The purpose of this paper is to present a detailed critique of such concept with a proof of its inconsistency. Specific properties of three non-isomorphic classes of super singular curves in the generalized Edwards form are considered: full, quadratic, and twisted Edwards curves. Conditions for existence of curves of all 3 classes with order of curves over a prime field are determined. The implementation of the CSIDH algorithm on isogenies of odd prime degrees is based on the use of quadratic twist pairs of elliptic curves. To this end, the CSIDH algorithm can be built both on complete Edwards curves with quadratic twist within this class, and on quadratic and twisted Edwards curves forming pairs of quadratic twist. In contrast to this, the authors of a well-known work are trying to prove theorems that state that there is a solution within one class of curves with a parameter which is a square. The critical analysis of theorems, lemmas, erroneous statements in this work is carried out. Theorem 2 on quadratic twist in classes of Edwards curves is proved. The CSIDH algorithm modification based on isogenies of quadratic and twisted Edwards curves is presented. To illustrate the correct solution of the problem, an example of Alice and Bob calculations in the secret sharing scheme according to the CSIDH algorithm is considered for.

Generators of the cohomology ring, after Newstead

The moduli space of rank22stable, locally-free sheaves with fixed odd degree determinant over a smooth, projective curve is a classical object. In the early 1970s, Newstead gave the generators of the cohomology ring of this moduli space. There are generalizations of this result to higher rank, but nothing is known for the case when the underlying curve is singular. In this article, we generalize Newstead’s result to the case when the underlying curve is irreducible, nodal. We show that the generators of the cohomology ring in the nodal curve case arise naturally as degeneration of Newstead’s generators in the smooth curve case.

Viro–Zvonilov Inequalities for Flexible Curves on an Almost Complex Four-Dimensional Manifold

The restrictions on the topology of nonsingular plane projective real algebraic curves of odd degree, obtained by O. Viro and the author in the paper published in the early 90s, are extended to flexible curves lying on an almost complex four-dimensional manifold. Some examples of real algebraic surfaces and real curves on them prove the sharpness of the obtained inequalities. In addition, it is proved that a compact Lie group smooth action can be lifted to a cyclic branched covering space $ \tilde{X} $ over a closed four-dimensional manifold, and a sufficient condition for $ H_1(\tilde{X})=0 $ was found.

Weighted isogeometric collocation based on Spline Projectors

In this paper, we propose a novel version of weighted isogeometric collocation that is especially suited for adaptive THB-spline refinement.It is well known that the choice of the collocation nodes is crucial to ensure stability and good approximation properties, especially when adaptive refinement is performed. In order to address this issue, we make use of a particular class of locally supported quasi-interpolant schemes to propose the new method of Weighted Isogeometric Collocation based on Spline Projectors (WICSP).We show that WICSP performs well in the case of tensor-product spline discretizations, both with respect to the rate of convergence and computational complexity. In particular, we observe experimentally an optimal rate of convergence for odd degree basis functions and obtain a dimension-independent computational complexity O(np) for matrix-free applications, similar to other approaches such as weighted quadrature (Calabrò et al., 2017) and clustered collocation (Montardini et al., 2017).We explore how these results extend to the case of adaptively refined THB-spline discretizations. We observe that WICSP is compatible with THB-spline refinement, exhibiting good accuracy and low computational costs. In fact, we get a complexity of O(np2d) for matrix assembly and O(npd) for the matrix-free case. This compares well with the available methods for isogeometric THB-spline discretizations.

Embeddings of Maximal Tori in Classical Groups, Odd Degree Descent and Hasse Principles

The aim of this paper is to revisit the question of local–global principles for embeddings of étale algebras with involution into central simple algebras with involution over global fields of characteristic not 2. A necessary and sufficient condition is given in Bayer-Fluckiger et al. (J Eur Math Soc 20:137–163, 2018). In the present paper, we give a simpler description of the obstruction group. It is also shown that if the étale algebra is a product of pairwise linearly disjoint field extensions, then the Hasse principle holds, and that if an embedding exists after an odd degree extension, then it also exists over the global field itself. An appendix gives a generalization of this later result, in the framework of a question of Burt Totaro.

Critical functions and inf-sup stability of Crouzeix-Raviart elements

In this paper, we prove that Crouzeix-Raviart finite elements of polynomial order p≥5, p odd, are inf-sup stable for the Stokes problem on triangulations. For p≥4, p even, the stability was proved by Á. Baran and G. Stoyan in 2007 by using the macroelement technique, a dimension formula, the concept of critical points in a triangulation and a representation of the corresponding critical functions. Baran and Stoyan proved that these critical functions belong to the range of the divergence operator applied to Crouzeix-Raviart velocity functions and the macroelement technique implies the inf-sup stability.The generalization of this theory to cover odd polynomial orders p≥5 is involved; one reason is that the macroelement classes, which have been used for even p, are unsuitable for odd p. In this paper, we introduce a new and simple representation of non-conforming Crouzeix-Raviart basis functions of odd degree. We employ only one type of macroelement and derive representations of all possible critical functions. Finally, we show that they are in the range of the divergence operator applied to Crouzeix-Raviart velocities from which the stability of the discretization follows.

Stabilization of polynomial systems in ℝ3 via homogeneous feedback

Abstract In this paper, we study the stabilization problem of a class of polynomial systems of odd degree in dimension three. The constructed stabilizing feedback is homogeneous and guarantee the homogeneity of the closed loop system.mynotered In the end of the paper, we show the efficiency of such a study in the local stabilization of nonlinear systems affine in control.

RANDOMIZATION OF CSIDH ALGORITHM ON QUADRATIC AND TWISTED EDWARDS CURVES

The properties of quadratic and twisted supersingular Edwards curves that form pairs of quadratic twist with order over a prime field are considered. A modification of the CSIDH algorithm based on odd degree isogenies of these curves is considered. A simple model for the implementation of the CSIDH algorithm in 3 minimal odd isogeny degrees 3, 5, 7, with the prime field modulus and the order of supersingular curves is constructed. At the precipitation stage, the parameters of isogenic chains of all degrees for these two classes of supersingular Edwards curves are calculated and tabulated. An example of the implementation of the CSIDH algorithm as a non-interactive secret sharing scheme based on the secret and public keys of Alice and Bob is given. A new randomized CSIDH algorithm with a random equiprobable choice of one of the curves of these two classes at each step of the isogeny chain is proposed. The choice of the degree of each isogeny is randomized. The operation of the randomized algorithm by an example is illustrated. This algorithm as a possible alternative to "CSIDH with constant time" is considered. A combination of the two approaches is possible to counter side channel attacks. Estimates of the probability of a successful side-channel attack in a randomized algorithm are given. It is noted that all calculations in the CSIDH algorithm necessary to calculate the shared secret are reduced only to calculating the parameter of the isogenic curve and are performed by field and group operations, in particular, scalar point multiplications and doubling points of the isogeny kernel. In the new algorithm we propose to abandon the calculation of the isogenic function of random point , which significantly speeds up the algorithm.

HOW TO CONSTRUCT CSIDH ON QUADRATIC AND TWISTED EDWARDS CURVES

In one of the famous works, an incorrect formulation and an incorrect solution of the implementation problem of the CSIDH algorithm on Edwards curves is discovered. A detailed critique of this work with a proof of the fallacy of its concept is given. Specific properties of three non-isomorphic classes of supersingular curves in the generalized Edwards form is considered: complete, quadratic, and twisted Edwards curves. Conditions for the existence of curves of all classes with the order p+1 of curves over a prime field are determined. The implementation of the CSIDH algorithm on isogenies of odd prime degrees based on the use of quadratic twist pairs of elliptic curves. To this end, the CSIDH algorithm can be construct both on complete Edwards curves with quadratic twist within this class, and on quadratic and twisted Edwards curves forming pairs of quadratic twist. In contrast to this, the authors of a well-known work are trying to prove theorems with statement about existing a solution within one class of curves with a parameter that is a square. The critical analysis of theorems, lemmas, and erroneous statements in this work is given. Theorem 2 on quadratic twist in classes of Edwards curves is proved. A modification of the CSIDH algorithm based on isogenies of quadratic and twisted Edwards curves is presented. To illustrate the correct solution of the problem, an example of Alice and Bob calculations in the secret sharing scheme according to the CSIDH algorithm is considered.

A new realisation of the i-quantum group U(n)

We follow the approach developed in [3] and modified in [7] to investigate a new realisation for the i -quantum groups U ȷ ( n ) , building on the multiplication formulas discovered in [2, Lem. 3.2] . This allows us to present U ȷ ( n ) via a basis and multiplication formulas by generators. We also establish a surjective algebra homomorphism from a Lusztig type form U Z ȷ ( n ) of U ȷ ( n ) to the integral q -Schur algebras S Z ȷ ( n , r ) of type B . Thus, base change to a field k allows us to relate representations of the i -quantum hyperalgebras U k ȷ ( n ) to representations of finite orthogonal groups of odd degree in non-defining characteristics. This generalises part of Dipper–James' type A theory to the type B case.

Using incomplete polynomial functions of the odd degree n and their inverses for data encryption and decryption

The need and the demand to hide the content of written messages from prying eyes has arisen as long as writing itself has existed. Particularly, recent times the protection of information is considered to be quite crucial due to the increased use and development of streaming applications.Therefore, the urging of having more sophisticated, stronger and hard to break data encryption and decryption systems is increasing. In order to attain these requirements, cryptography plays an important role, where many researchers have come up with different proposals and developed algorithms that have helped out a little in ensuring the confidentiality, integrity and authentication of the given information. However, even the internet security through modern cryptography is quite complex and depends on the difficulty of certain computational problems in mathematics.Modern systems of encryption are based on complex mathematical algorithms and carry out a combination of symmetric and asymmetric key encryption schemes to secure communication. For those a significant background in algebra, number theory and geometry is required. In this paper a new cryptosystem is presented, which is based on compound commutative functions such as exponential and logarithmic functions. After the necessary theoretical deliberation in this paper, we will provide the encryption and decryption approach accompanied with relevant examples.

Estimation of the computational cost of the CSIDH algorithm on supersingular twisted and quadratic Edwards curves

The properties of twisted and quadratic supersingular Edwards curves that form pairs of quadratic torsion with order p+1 over a prime field Fp are considered. A modification of the CSIDH algorithm based on the isogenies of these curves instead of the traditional arithmetic of curves in the Montgomery form is presented. The parameters of these two classes of supersingular Edwards curves for p=239 are calculated and tabulated. An example of the isogenies of these curves in the implementation of the CSIDH algorithm as a non-interactive secret sharing scheme based on the secret and public keys of Alice and Bob is given. It is shown that the sequences of parameters ±d(i) of isogeny chains for quadratic and twisted supersingular Edwards curves, respectively, have a reverse nature on the period of the sequence. A recurrent algorithm for calculating the coordinates of points that form the kernels of isogenies of odd degrees is proposed, and its implementation in various coordinate systems is considered. A comparative analysis of the cost of calculating the parameter d´ of the isogenic curve E´ using the Farashakhi-Hosseini (W : Z) - coordinates and classical projective coordinates (X : Y : Z) is given. It is noted that all calculations in the CSIDH algorithm necessary to calculate the shared secret dAB are reduced only to the calculation of the isogenic curve E´ parameter d´ and are performed by field operations and the scalar multiplication of the point. The controversial issue of refusal to calculate the isogenic function ϕ(R) of a curve point R in the CSIDH algorithm is discussed.

On Superspecial abelian surfaces over finite fields III

In the paper (J Math Soc Jpn 72(1):303–331, 2020), Tse-Chung Yang and the first two current authors computed explicitly the number \(|\mathrm {SSp}_2(\mathbb {F}_q)|\) of isomorphism classes of superspecial abelian surfaces over an arbitrary finite field \(\mathbb {F}_q\) of even degree over the prime field \(\mathbb {F}_p\). There it was assumed that certain commutative \(\mathbb {Z}_p\)-orders satisfy an étale condition that excludes the primes \(p=2, 3, 5\). We treat these remaining primes in the present paper, where the computations are more involved because of the ramification. This completes the calculation of \(|\mathrm {SSp}_2(\mathbb {F}_q)|\) in the even degree case. The odd degree case was previous treated by Tse-Chung Yang and the first two current authors in (Doc Math 21:1607–1643, 2016). To complete the proof of our main theorem, we give a classification of lattices over local quaternion Bass orders, which is a new input to our previous works.

The Basis Invariant of Generalized n-Cube Symmetries Group with Odd Degrees

The basis polynomial invariants with even degrees relatively to the symmetries group were described in cited literature. Here, the polynomial invariants with odd degrees are constructed. We give an explicit construction of all the basic polynomial invariants as algebra generators of odd degrees relatively to the symmetries group. All calculations are presented in detail.

A characterization of graphs with given maximum degree and smallest possible matching number: II

Let α′(G) be the matching number of a graph G. A characterization of the graphs with given maximum odd degree and smallest possible matching number is given by Henning and Shozi (2021) [13]. In this paper we complete our study by giving a characterization of the graphs with given maximum even degree and smallest possible matching number. In 2018 Henning and Yeo [10] proved that if G is a connected graph of order n, size m and maximum degree k where k≥4 is even, then α′(G)≥nk(k+1)+mk+1−1k(k+1), unless G is k-regular and n∈{k+1,k+3}. In this paper, we give a complete characterization of the graphs that achieve equality in this bound when the maximum degree k is even, thereby completing our study of graphs with given maximum degree and smallest possible matching number.

Fermat quartics with only trivial solutions in any odd degree number field

Fermat quartics with only trivial solutions in any odd degree number field

Minimal degree of standard identities of matrix algebras with symplectic graded involution

Let [Formula: see text] be a field of characteristic zero and [Formula: see text] the algebra of [Formula: see text] matrices over [Formula: see text]. By the classical Amitsur–Levitzki theorem, it is well known that [Formula: see text] is the smallest degree of a standard polynomial identity of [Formula: see text]. A theorem due to Rowen shows that when the symplectic involution [Formula: see text] is considered, the standard polynomial of degree [Formula: see text] in symmetric variables is an identity of [Formula: see text]. This means that when only certain kinds of matrices are considered in the substitutions, the minimal degree of a standard identity may not remain being the same. In this paper, we present some results about the minimal degree of standard identities in skew or symmetric variables of odd degree of [Formula: see text] in the symplectic graded involution case. Along the way, we also present the minimal total degree of a double Capelli polynomial identity in symmetric variables of [Formula: see text] with transpose involution.