Cubic Formula Size Lower Bounds Based on Compositions with Majority

We define new functions based on the Andreev function and prove that they require n^{3}/polylog(n) formula size to compute. The functions we consider are generalizations of the Andreev function using compositions with the majority function. Our arguments apply to composing a hard function with any function that agrees with the majority function (or its negation) on the middle slices of the Boolean cube, as well as iterated compositions of such functions. As a consequence, we obtain n^{3}/polylog(n) lower bounds on the (non-monotone) formula size of an explicit monotone function by combining the monotone address function with the majority function

Native defects in the Co2_2TiZZ (Z=Z= Si, Ge, Sn) full Heusler alloys: formation and influence on the thermoelectric properties

We have performed first-principles investigations on the native defects in the full Heusler alloys Co$_2$Ti$Z$ ($Z$ one of the group IV elements Si, Ge, Sn), determining their formation energies and how they influence the transport properties. We find that Co vacancies (Vc) in all compounds and the Ti$_\text{Sn}$ anti-site exhibit negative formation energies. The smallest positive values occur for Co in excess on anti-sites (Co$_Z$ or Co$_\text{Ti}$) and for Ti$_Z$. The most abundant native defects were modeled as dilute alloys, treated with the coherent potential approximation in combination with the multiple-scattering theory Green function approach. The self-consistent potentials determined this way were used to calculate the residual resistivity via the Kubo-Greenwood formula and, based on its energy dependence, the Seebeck coefficient of the systems. The latter is shown to depend significantly on the type of defect, leading to variations that are related to subtle, spin-orbit coupling induced, changes in the electronic structure above the half-metallic gap. Two of the systems, Vc$_\text{Co}$ and Co$_Z$, are found to exhibit a negative Seebeck coefficient. This observation, together with their low formation energy, offers an explanation for the experimentally observed negative Seebeck coefficient of the Co$_2$Ti$Z$ compounds as being due to unintentionally created native defects

New Tolerance Factor to Predict the Stability of Perovskite Oxides and Halides

Predicting the stability of the perovskite structure remains a longstanding challenge for the discovery of new functional materials for many applications including photovoltaics and electrocatalysts. We developed an accurate, physically interpretable, and one-dimensional tolerance factor, {\tau}, that correctly predicts 92% of compounds as perovskite or nonperovskite for an experimental dataset of 576 $ABX_3$ materials ($\textit{X} =$ $O^{2-}$, $F^-$, $Cl^-$, $Br^-$, $I^-$) using a novel data analytics approach based on SISSO (sure independence screening and sparsifying operator). {\tau} is shown to generalize outside the training set for 1,034 experimentally realized single and double perovskites (91% accuracy) and is applied to identify 23,314 new double perovskites ($A_2$$\textit{BB'}$$X_6$) ranked by their probability of being stable as perovskite. This work guides experimentalists and theorists towards which perovskites are most likely to be successfully synthesized and demonstrates an approach to descriptor identification that can be extended to arbitrary applications beyond perovskite stability predictions

Magnetization reversal in mixed ferrite-chromite perovskites with non magnetic cation on the A-site

In this work, we have performed Monte Carlo simulations in a classical model for RFe$_{1-x}$Cr$_x$O$_3$ with R=Y and Lu, comparing the numerical simulations with experiments and mean field calculations. In the analyzed compounds, the antisymmetric exchange or Dzyaloshinskii-Moriya (DM) interaction induced a weak ferromagnetism due to a canting of the antiferromagnetically ordered spins. This model is able to reproduce the magnetization reversal (MR) observed experimentally in a field cooling process for intermediate $x$ values and the dependence with $x$ of the critical temperatures. We also analyzed the conditions for the existence of MR in terms of the strength of DM interactions between Fe$^{3+}$ and Cr$^{3+}$ ions with the x values variations.Comment: 8 pages, 7 figure

A New Framework for Kernelization Lower Bounds: The Case of Maximum Minimal Vertex Cover

In the Maximum Minimal Vertex Cover (MMVC) problem, we are given a graph G and a positive integer k, and the objective is to decide whether G contains a minimal vertex cover of size at least k. Motivated by the kernelization of MMVC with parameter k, our main contribution is to introduce a simple general framework to obtain lower bounds on the degrees of a certain type of polynomial kernels for vertex-optimization problems, which we call {lop-kernels}. Informally, this type of kernels is required to preserve large optimal solutions in the reduced instance, and captures the vast majority of existing kernels in the literature. As a consequence of this framework, we show that the trivial quadratic kernel for MMVC is essentially optimal, answering a question of Boria et al. [Discret. Appl. Math. 2015], and that the known cubic kernel for Maximum Minimal Feedback Vertex Set is also essentially optimal. On the positive side, given the (plausible) non-existence of subquadratic kernels for MMVC on general graphs, we provide subquadratic kernels on H-free graphs for several graphs H, such as the bull, the paw, or the complete graphs, by making use of the Erd?s-Hajnal property in order to find an appropriate decomposition. Finally, we prove that MMVC does not admit polynomial kernels parameterized by the size of a minimum vertex cover of the input graph, even on bipartite graphs, unless NP ? coNP / poly. This indicates that parameters smaller than the solution size are unlike to yield polynomial kernels for MMVC

Dolní odhady velikosti Booleovských formulí

Cı́lem této práce je studovat metody konstrukce dolnı́ch odhadů velikosti Booleovských formulı́. Soustředı́me se zde předevšı́m na formálnı́ mı́ry složitosti, přičemž zobecnı́me známou Krapchenkovu mı́ru na třı́du grafových měr, které následně studujeme. Zabýváme se také dalšı́m z hlavnı́ch přı́stupů, využı́vajı́cı́ náhodné restrikce Booleovských funkcı́. Na závěr zmı́nı́me program pro nalezenı́ super-polynomiálnı́ch odhadů založený na KRW doměnce. 1The aim of this thesis is to study methods of constructing lower bounds on Boolean formula size. We focus mainly on formal complexity measures, gener- alizing the well-known Krapchenko measure to a class of graph measures, which we thereafter study. We also review one of the other main approaches, using ran- dom restrictions of Boolean functions. This approach has yielded the currently largest lower bounds. Finally, we mention a program for finding super-polynomial bounds based on the KRW conjecture. 1Department of LogicKatedra logikyFaculty of ArtsFilozofická fakult