A Self-Linking Invariant of Virtual Knots

In this paper we introduce a new invariant of virtual knots and links that is non-trivial for infinitely many virtuals, but is trivial on classical knots and links. The invariant is initially be expressed in terms of a relative of the bracket polynomial and then extracted from this polynomial in terms of its exponents, particularly for the case of knots. This analog of the bracket polynomial will be denoted {K} (with curly brackets) and called the binary bracket polynomial. The key to the combinatorics of the invariant is an interpretation of the state sum in terms of 2-colorings of the associated diagrams. For virtual knots, the new invariant, J(K), is a restriction of the writhe to the odd crossings of the diagram (A crossing is odd if it links an odd number of crossings in the Gauss code of the knot. The set of odd crossings is empty for a classical knot.) For K a virtual knot, J(K) non-zero implies that K is non-trivial, non-classical and inequivalent to its planar mirror image. The paper also condsiders generalizations of the two-fold coloring of the states of the binary bracket to cases of three and more colors. Relationships with graph coloring and the Four Color Theorem are discussed.Comment: 36 pages, 22 figures, LaTeX documen

Virtual Knot Theory --Unsolved Problems

This paper is an introduction to the theory of virtual knots and links and it gives a list of unsolved problems in this subject.Comment: 33 pages, 7 figures, LaTeX documen

The Clifford group, stabilizer states, and linear and quadratic operations over GF(2)

We describe stabilizer states and Clifford group operations using linear operations and quadratic forms over binary vector spaces. We show how the n-qubit Clifford group is isomorphic to a group with an operation that is defined in terms of a (2n+1)x(2n+1) binary matrix product and binary quadratic forms. As an application we give two schemes to efficiently decompose Clifford group operations into one and two-qubit operations. We also show how the coefficients of stabilizer states and Clifford group operations in a standard basis expansion can be described by binary quadratic forms. Our results are useful for quantum error correction, entanglement distillation and possibly quantum computing.Comment: 9 page

Combining the Band-Limited Parameterization and Semi-Lagrangian Runge–Kutta Integration for Efficient PDE-Constrained LDDMM

The family of PDE-constrained Large Deformation Diffeomorphic Metric Mapping (LDDMM) methods is emerging as a particularly interesting approach for physically meaningful diffeomorphic transformations. The original combination of Gauss–Newton–Krylov optimization and Runge–Kutta integration shows excellent numerical accuracy and fast convergence rate. However, its most significant limitation is the huge computational complexity, hindering its extensive use in Computational Anatomy applied studies. This limitation has been treated independently by the problem formulation in the space of band-limited vector fields and semi-Lagrangian integration. The purpose of this work is to combine both in three variants of band-limited PDE-constrained LDDMM for further increasing their computational efficiency. The accuracy of the resulting methods is evaluated extensively. For all the variants, the proposed combined approach shows a significant increment of the computational efficiency. In addition, the variant based on the deformation state equation is positioned consistently as the best performing method across all the evaluation frameworks in terms of accuracy and efficiency

Modern Cryptography Volume 2

This open access book covers the most cutting-edge and hot research topics and fields of post-quantum cryptography. The main purpose of this book is to focus on the computational complexity theory of lattice ciphers, especially the reduction principle of Ajtai, in order to fill the gap that post-quantum ciphers focus on the implementation of encryption and decryption algorithms, but the theoretical proof is insufficient. In Chapter 3, Chapter 4 and Chapter 6, author introduces the theory and technology of LWE distribution, LWE cipher and homomorphic encryption in detail. When using random analysis tools, there is a problem of "ambiguity" in both definition and algorithm. The greatest feature of this book is to use probability distribution to carry out rigorous mathematical definition and mathematical demonstration for various unclear or imprecise expressions, so as to make it a rigorous theoretical system for classroom teaching and dissemination. Chapters 5 and 7 further expand and improve the theory of cyclic lattice, ideal lattice and generalized NTRU cryptography. This book is used as a professional book for graduate students majoring in mathematics and cryptography, as well as a reference book for scientific and technological personnel engaged in cryptography research