Finite-temperature form factors in the free Majorana theory

We study the large distance expansion of correlation functions in the free massive Majorana theory at finite temperature, alias the Ising field theory at zero magnetic field on a cylinder. We develop a method that mimics the spectral decomposition, or form factor expansion, of zero-temperature correlation functions, introducing the concept of "finite-temperature form factors". Our techniques are different from those of previous attempts in this subject. We show that an appropriate analytical continuation of finite-temperature form factors gives form factors in the quantization scheme on the circle. We show that finite-temperature form factor expansions are able to reproduce expansions in form factors on the circle. We calculate finite-temperature form factors of non-interacting fields (fields that are local with respect to the fundamental fermion field). We observe that they are given by a mixing of their zero-temperature form factors and of those of other fields of lower scaling dimension. We then calculate finite-temperature form factors of order and disorder fields. For this purpose, we derive the Riemann-Hilbert problem that completely specifies the set of finite-temperature form factors of general twist fields (order and disorder fields and their descendants). This Riemann-Hilbert problem is different from the zero-temperature one, and so are its solutions. Our results agree with the known form factors on the circle of order and disorder fields.Comment: 40 pp.; v2: 42 pp., refs and acknowledgment added, typos corrected, description of general matrix elements corrected and extended; v3: 47 pp., appendix adde

Extension of the enhanced assumed strain method based on the structure of polyconvex strain‐energy functions

In this work, two well-known approaches for mixed finite elements are combined to render three novel classes of elements. First, the widely used enhanced assumed strain (EAS) method is considered. Its key idea is to enhance a compatible kinematic field with an incompatible part. The second concept is a framework for mixed elements inspired by polyconvex strain-energy functions, in which the deformation gradient, its cofactor and determinant are three principal kinematic fields. The key idea for the novel elements is to treat enhancement of those three fields separately. This approach leads to a plethora of novel enhancement strategies and promising mixed finite elements. Some key properties of the newly proposed mixed approaches are that they are based on a Hu-Washizu type variational functional, fulfill the patch test, are frame-invariant, can be constructed completely locking free and show no spurious hourglassing in elasticity. Furthermore, they give additional insight into the mechanisms of standard EAS elements. Extensive numerical investigations are performed to assess the elements\u27 behavior in elastic and elasto-plastic simulations

On the existence of primitive completely normal bases of finite fields

Let $\mathbb{F}_q$ be the finite field of characteristic $p$ with $q$ elements and $\mathbb{F}_{q^n}$ its extension of degree $n$. We prove that there exists a primitive element of $\mathbb{F}_{q^n}$ that produces a completely normal basis of $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$, provided that $n=p^{\ell}m$ with $(m,p)=1$ and $q>m$

Further results on the Morgan-Mullen conjecture

Let $\mathbb{F}_q$ be the finite field of characteristic $p$ with $q$ elements and $\mathbb{F}_{q^n}$ its extension of degree $n$. The conjecture of Morgan and Mullen asserts the existence of primitive and completely normal elements (PCN elements) for the extension $\mathbb{F}_{q^n}/\mathbb{F}_q$ for any $q$ and $n$. It is known that the conjecture holds for $n \leq q$. In this work we prove the conjecture for a larger range of exponents. In particular, we give sharper bounds for the number of completely normal elements and use them to prove asymptotic and effective existence results for $q\leq n\leq O(q^\epsilon)$, where $\epsilon=2$ for the asymptotic results and $\epsilon=1.25$ for the effective ones. For $n$ even we need to assume that $q-1\nmid n$.Comment: arXiv admin note: text overlap with arXiv:1709.0314

Finite energy shifts in SU(n) supersymmetric Yang-Mills theory on T^3xR at weak coupling

We consider a semi-classical treatment, in the regime of weak gauge coupling, of supersymmetric Yang-Mills theory in a space-time of the form T^3xR with SU(n)/Z_n gauge group and a non-trivial gauge bundle. More specifically, we consider the theories obtained as power series expansions around a certain class of normalizable vacua of the classical theory, corresponding to isolated points in the moduli space of flat connections, and the perturbative corrections to the free energy eigenstates and eigenvalues in the weakly interacting theory. The perturbation theory construction of the interacting Hilbert space is complicated by the divergence of the norm of the interacting states. Consequently, the free and interacting Hilbert furnish unitarily inequivalent representation of the algebra of creation and annihilation operators of the quantum theory. We discuss a consistent redefinition of the Hilbert space norm to obtain the interacting Hilbert space and the properties of the interacting representation. In particular, we consider the lowest non-vanishing corrections to the free energy spectrum and discuss the crucial importance of supersymmetry for these corrections to be finite.Comment: 31 pages, 1 figure, v4 Minor changes, references correcte

Presentations of Galois groups of maximal extensions with restricted ramification

Motivated by the work of Lubotzky, we use Galois cohomology to study the difference between the number of generators and the minimal number of relations in a presentation of the Galois group $G_S(k)$ of the maximal extension of a global field $k$ that is unramified outside a finite set $S$ of places, as $k$ varies among a certain family of extensions of a fixed global field $Q$. We prove a generalized version of the global Euler-Poincar\'{e} Characteristic, and define a group $B_S(k,A)$, for each finite simple $G_S(k)$-module $A$, to generalize the work of Koch about the pro-$\ell$ completion of $G_S(k)$ to study the whole group $G_S(k)$. In the setting of the nonabelian Cohen-Lenstra heuristics, we prove that the objects studied by the Liu--Wood--Zureick-Brown conjecture are always achievable by the random group that is constructed in the definition the probability measure in the conjecture.Comment: Comments are welcom