Continuous reducibility and dimension of metric spaces

If $(X,d)$ is a Polish metric space of dimension $0$, then by Wadge's lemma, no more than two Borel subsets of $X$ can be incomparable with respect to continuous reducibility. In contrast, our main result shows that for any metric space $(X,d)$ of positive dimension, there are uncountably many Borel subsets of $(X,d)$ that are pairwise incomparable with respect to continuous reducibility. The reducibility that is given by the collection of continuous functions on a topological space $(X,\tau)$ is called the \emph{Wadge quasi-order} for $(X,\tau)$. We further show that this quasi-order, restricted to the Borel subsets of a Polish space $(X,\tau)$, is a \emph{well-quasiorder (wqo)} if and only if $(X,\tau)$ has dimension $0$, as an application of the main result. Moreover, we give further examples of applications of the technique, which is based on a construction of graph colorings

On the suitability of power functions as S-boxes for symmetric cryptosystems

textI present some results towards a classification of power functions that are Almost Perfect Nonlinear (APN), or equivalently differentially 2-uniform, over F2n for infinitely many positive integers n. APN functions are useful in constructing S-boxes in AES-like cryptosystems. An application of a theorem by Weil [20] on absolutely irreducible curves shows that a monomial x m is not APN over F2n for all sufficiently large n if a related two variable polynomial has an absolutely irreducible factor defined over F2. I will show that the latter polynomial’s singularities imply that except in five cases, all power functions have such a factor. Three of these cases are already known to be APN for infinitely many fields. The last two cases are still unproven. Some specific cases of power functions have already been known to be APN over only finitely many fields, but they also follow from the results below.Mathematic

NLO Higgs Effective Field Theory and kappa-framework

A consistent framework for studying Standard Model deviations is developed. It assumes that New Physics becomes relevant at some scale beyond the present experimental reach and uses the Effective Field Theory approach by adding higher-dimensional operators to the Standard Model Lagrangian and by computing relevant processes at the next-to-leading order, extending the original kappa-framework.Comment: 33 pages + appendice

Modified mixed Tsirelson spaces

We study the modified and boundedly modified mixed Tsirelson spaces $T_M[({\cal F}_{k_n},\theta_n)_{n=1}^{\infty }]$ and $T_{M(s)}[({\cal F}_{k_n},\theta_n)_{n=1}^{\infty }]$ respectively, defined by a subsequence $({\cal F}_{k_n})$ of the sequence of Schreier families $({\cal F}_n)$. These are reflexive asymptotic $\ell_1$ spaces with an unconditio- nal basis $(e_i)_i$ having the property that every sequence $\{ x_i\}_{i=1}^n$ of normalized disjointly supported vectors contained in $\langle e_i\rangle_{i=n}^{\infty }$ is equivalent to the basis of $\ell_1^n$. We show that if $\lim\theta_n^{1/n}=1$ then the space $T[({\cal F}_n,\theta_n) _{n=1}^{\infty }]$ and its modified variations are totally incomparable by proving that $c_0$ is finitely disjointly representable in every block subspace of $T[({\cal F}_n, \theta_n)_{n=1}^{\infty }]$. Next, we present an example of a boundedly modified mixed Tsirelson space $X_{M(1),u}$ which is arbitrarily distortable. Finally, we construct a variation of the space $X_{M(1),u}$ which is hereditarily indecomposable

Magnetic vortex-antivortex crystals generated by spin-polarized current

We study vortex pattern formation in thin ferromagnetic films under the action of strong spin-polarized currents. Considering the currents which are polarized along the normal of the film plane, we determine the critical current above which the film goes to a saturated state with all magnetic moments being perpendicular to the film plane. We show that stable square vortex-antivortex superlattices (\emph{vortex crystals}) appears slightly below the critical current. The melting of the vortex crystal occurs with current further decreasing. A mechanism of current-induced periodic vortex-antivortex lattice formation is proposed. Micromagnetic simulations confirm our analytical results with a high accuracy.Comment: 12 pages, 11 figure

Quantum phase transitions in cascading gauge theory

We study a ground state of N=1 supersymmetric SU(K+P) x SU(K) cascading gauge theory of Klebanov et.al [1,2] on R x S^3 at zero temperature. A radius of S^3 sets a compactification scale mu. An interplay between mu and the strong coupling scale Lambda of the theory leads to an interesting pattern of quantum phases of the system. For mu > mu_cSB=1.240467(8)Lambda the ground state of the theory is chirally symmetric. At mu=mu_cSB the theory undergoes the first-order transition to a phase with spontaneous breaking of the chiral symmetry. We further demonstrate that the chirally symmetric ground state of cascading gauge theory becomes perturbatively unstable at scales below mu_c=0.950634(5)mu_cSB. Finally, we point out that for mu < 1.486402(5)Lambda the stress-energy tensor of cascading gauge theory can source inflation of a closed Universe.Comment: 62 pages, 9 figure

The Vietoris-Rips complexes of a circle

Given a metric space X and a distance threshold r>0, the Vietoris-Rips simplicial complex has as its simplices the finite subsets of X of diameter less than r. A theorem of Jean-Claude Hausmann states that if X is a Riemannian manifold and r is sufficiently small, then the Vietoris-Rips complex is homotopy equivalent to the original manifold. Little is known about the behavior of Vietoris-Rips complexes for larger values of r, even though these complexes arise naturally in applications using persistent homology. We show that as r increases, the Vietoris-Rips complex of the circle obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., until finally it is contractible. As our main tool we introduce a directed graph invariant, the winding fraction, which in some sense is dual to the circular chromatic number. Using the winding fraction we classify the homotopy types of the Vietoris-Rips complex of an arbitrary (possibly infinite) subset of the circle, and we study the expected homotopy type of the Vietoris-Rips complex of a uniformly random sample from the circle. Moreover, we show that as the distance parameter increases, the ambient Cech complex of the circle also obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., until finally it is contractible.Comment: Final versio