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In the tight-binding approximation, an Iwatsuka magnetic field is modeled by a function on $\mathbb{Z}^2$ with constant, but distinct values in the two parts of the lattice separated by a straight line of slope $\alpha\in [-\infty,\infty]$. In this paper, the $K$-theory of the magnetic $C^*$-algebras generated by an Iwatsuka magnetic field for any possible $\alpha$ is computed. One interesting aspect concerns the analysis of the behavior of the system in the transition from rational to irrational $\alpha$. It turns out that when $\alpha$ is irrational, the magnetic hull associated with the flux operator forms a Cantor set. On the other hand, for rational $\alpha$ this set coincides with the two-point compactification of $\mathbb{Z}$. This characterization, along with the use of the Pimsner-Voiculescu exact sequence, is the main ingredient for the computation of the $K$-theory. Once the $K$-theory is known, with the use of the index theory one can deduce the bulk-interface correspondence for tight-binding Hamiltonians subjected to an Iwatsuka magnetic field. Notably, it occurs that the topological quantization of the interface currents remains independent of the slope $\alpha$.

We introduce approximation functions of $li(x)$ for all $x\ge e$: (1) $\displaystyle li_{\underline{\omega},\alpha}(x) = \frac{x}{\log(x)}\left( \alpha\frac{\underline{m}!}{\log^{\underline{m}}(x)} + \sum_{k=0}^{\underline{m}-1}\frac{k!}{\log^{k}(x)} \right)$, and (2) $\displaystyle li_{\overline{\omega},\beta}=\frac{x}{\log(x)}\left( \beta\frac{\overline{m}!}{\log^{\overline{m}}(x)} + \sum_{k=0}^{\overline{m}-1}\frac{k!}{\log^{k}(x)} \right)$ with $0 < \omega < 1$ a real number, $\alpha \in \{ 0, \underline{\kappa}\log(x) \}$, $\underline{m} = \lfloor \underline{\kappa}\log(x) \rfloor$, $\beta \in \{ \overline{\kappa}\log(x), 1 \}$, $\overline{m} = \lfloor \overline{\kappa}\log(x) \rfloor$, and $\underline{\kappa} < \overline{\kappa}$ the solutions of $\kappa(1-\log(\kappa)) = \omega$. Since the error of approximating $li(x)$ using Stieltjes asymptotic series $\displaystyle li_{*}(x) = \frac{x}{\log(x)}\sum_{k=0}^{n-1}\frac{k!}{\log^{k}(x)} + (\log(x)-n)\frac{xn!}{\log^{n+1}(x)}$, with $\displaystyle n = \lfloor \log(x) \rfloor$ for all $x\ge e$, satisfies $\displaystyle |\varepsilon(x)| = |li(x)-li_{*}(x)| \le 1.265692883422\ldots$, by using Stirling's approximation and some facts about $\log(x)$ and floor functions, we show that $\displaystyle \varepsilon_{0}(x) = li(x) - li_{\underline{1/2},0}(x)$, $\displaystyle \underline{\varepsilon}(x) = li(x) - li_{\underline{1/2},\underline{\kappa}\log(x)}(x)$, $\displaystyle \overline{\varepsilon}(x) = li(x) - li_{\overline{1/2},\overline{\kappa}\log(x)}(x)$, and $\varepsilon_{1}(x) = li_{\overline{1/2},1}(x) - li(x)$ belong to $O\left(\sqrt{\frac{x}{\log(x)}}\right)$. Moreover, we conjecture that $li_{0}(x) \le \pi(x) \le li_{1}(x)$ and $\underline{li}(x) \le \pi(x) \le \overline{li}(x)$ for all $x \ge e$, here $\pi(x)$ is the prime counting function and we show that if one of those conjectures is true then the Riemann Hypothesis is true.

Differential privacy (DP) is a widely used approach for mitigating privacy risks when training machine learning models on sensitive data. DP mechanisms add noise during training to limit the risk of information leakage. The scale of the added noise is critical, as it determines the trade-off between privacy and utility. The standard practice is to select the noise scale in terms of a privacy budget parameter $\epsilon$. This parameter is in turn interpreted in terms of operational attack risk, such as accuracy, or sensitivity and specificity of inference attacks against the privacy of the data. We demonstrate that this two-step procedure of first calibrating the noise scale to a privacy budget $\epsilon$, and then translating $\epsilon$ to attack risk leads to overly conservative risk assessments and unnecessarily low utility. We propose methods to directly calibrate the noise scale to a desired attack risk level, bypassing the intermediate step of choosing $\epsilon$. For a target attack risk, our approach significantly decreases noise scale, leading to increased utility at the same level of privacy. We empirically demonstrate that calibrating noise to attack sensitivity/specificity, rather than $\epsilon$, when training privacy-preserving ML models substantially improves model accuracy for the same risk level. Our work provides a principled and practical way to improve the utility of privacy-preserving ML without compromising on privacy.

This paper studies the error introduced ($\displaystyle \varepsilon(x)$) by the Stieltjes asymptotic approximation series $\displaystyle li_{*}(x) = \frac{x}{\log(x)}\sum_{k=0}^{n-1}\frac{k!}{\log^{k}(x)} + (\log(x)-n)\frac{xn!}{\log^{n+1}(x)}$ to the logarithmic integral function $li(x)$ with $\displaystyle n = \lfloor \log(x) \rfloor$ for all $x\ge e$. For this purpose, this paper uses some relations between term $\displaystyle \frac{n!}{\log^{n}(x)}$ and Stirling's approximation formula. In particular, this paper establishes two non-asymptotic lower and upper bounds for $\varepsilon(e^n)$ with $n \ge 1$ and $1 \le m \le n$: (i) $\displaystyle \varepsilon(e^m) + \frac{\sqrt{2\pi}}{8}\sum_{k=m+1}^{n} \frac{1}{k^{\frac{3}{2}}} \le \varepsilon(e^n) \le \varepsilon(e^{m}) + \frac{\sqrt{2\pi}}{4}\sum_{k=m+1}^{n} \frac{k+2}{k^{\frac{5}{2}}}$ (ii) $\displaystyle L_m - \frac{\sqrt{2\pi}}{4\sqrt{n+1}} \le \varepsilon_{n} \le R_m - \frac{\sqrt{2\pi}}{2\sqrt{n}} - \frac{\sqrt{2\pi}}{3\sqrt{n^3}}$. Here, $\displaystyle L_m = \varepsilon_{m} + \frac{\sqrt{2\pi}}{4\sqrt{m+1}}$ and $\displaystyle R_m = \varepsilon_{m} + \frac{\sqrt{2\pi}}{2\sqrt{m}} + \frac{\sqrt{2\pi}}{3\sqrt{m^3}}$ Moreover, this paper shows that if $\displaystyle |\varepsilon(x)| \le \frac{\sqrt{2\pi}}{\sqrt{\log(x)}}$ then $li(e^n) \le li_{*}(e^n)$ for all $n \ge 1$. Finally, this paper establishes non-asymptotic lower and upper bounds for $\varepsilon(x)$ with $x \ge e$: $\displaystyle L_m - \frac{\sqrt{2\pi}}{4\sqrt{n+1}} -\frac{\sqrt{2\pi}}{36\sqrt{n^3}} < \varepsilon(x) \le R_m - \frac{\sqrt{2\pi}}{2\sqrt{n}} - \frac{\sqrt{2\pi}}{3\sqrt{n^3}}$

The optimization of chemical processes is challenging due to the nonlinearities arising from process physics and discrete design decisions. In particular, optimal synthesis and design of chemical processes can be posed as a Generalized Disjunctive Programming (GDP) superstructure problem. Various solution methods are available to address these problems, such as reformulating them as Mixed-Integer Nonlinear Programming (MINLP) problems; nevertheless, algorithms explicitly designed to solve the GDP problem and potentially leverage its structure remain scarce. This paper presents the Logic-based Discrete-Steepest Descent Algorithm (LD-SDA) as a solution method for GDP problems involving ordered Boolean variables. The LD-SDA reformulates these ordered Boolean variables into integer decisions called external variables. The LD-SDA solves the reformulated GDP problem using a two-level decomposition approach where the upper-level subproblem determines external variable configurations. Subsequently, the remaining continuous and discrete variables are solved as a subproblem only involving those constraints relevant to the given external variable arrangement, effectively taking advantage of the structure of the GDP problem. The advantages of LD-SDA are illustrated through a batch processing case study, a reactor superstructure, a distillation column, and a catalytic distillation column, and its open-source implementation is available online. The results show convergence efficiency and solution quality improvements compared to conventional GDP and MINLP solvers.

We prove an analogue of Miller's stable splitting of the unitary group $U(m)$ for spaces of commuting elements in $U(m)$. After inverting $m!$, the space $\text{Hom}(\mathbb{Z}^n,U(m))$ splits stably as a wedge of Thom-like spaces of bundles of commuting varieties over certain partial flag manifolds. Using Steenrod operations we prove that our splitting does not hold integrally. Analogous decompositions for symplectic and orthogonal groups as well as homological results for the one-point compactification of the commuting variety in a Lie algebra are also provided.

Let $G$ be an infinite residually finite group. We show that for every minimal equicontinuous Cantor system $(Z,G)$ with a free orbit, and for every minimal extension $(Y,G)$ of $(Z,G)$, there exist a minimal almost 1-1 extension $(X,G)$ of $(Z,G)$ and a Borel equivariant map $\psi:Y\to X$ that induces an affine bijection $\psi^*$ between $M(Y,G)$ and $M(X,G)$, the spaces of invariant probability measures of $(Y,G)$ and $(X,G)$, respectively. If $Y$ is a Cantor set, then $(Y,G)$ and $(X,G)$ are Borel isomorphic, i.e., $\psi^*$ is also a homeomorphism. As an application, we show that the family of Toeplitz subshifts is a test for amenability for residually finite groups, i.e., a residually finite group $G$ is amenable if and only if every Toeplitz $G$-subshift has invariant probability measures.

Let $G$ be a non-amenable countable group. We show that every almost automorphic $G$-action on a compact Hausdorff space, with a maximal equicontinuous factor whose phase space is a Cantor set, admits invariant probability measures (this partially answers a question posed by Veech). In particular, every Toeplitz $G$-subshift has a non-empty space of invariant measures, meaning that this family of subshifts is not a test for amenability for countable groups. We prove that almost one-to-one extensions without measures ensure the existence of symbolic almost one-to-one extensions with equal characteristics. As a consequence, we obtain the most general result of this paper. Finally, as a corollary of our results, we deduce that the class of Toeplitz subshifts is not dense in the space of infinite transitive subshifts of $\Sigma^G$, unlike $G=\mathbb{Z}$.

In this short note we show that the path-connected component of the identity of the derived subgroup of a compact Lie group consists just of commutators. We also discuss an application of our main result to the homotopy type of the classifying space for commutativity for a compact Lie group whose path-connected component of the identity is abelian.

For each countable residually finite group $G$, we present examples of irregular Toeplitz subshifts in $\{0,1\}^G$ that are topo-isomorphic extensions of its maximal equicontinuous factor. To achieve this, we first establish sufficient conditions for Toeplitz subshifts to have invariant probability measures as limit points of periodic invariant measures of $\{0,1\}^G$. Next, we demonstrate that the set of Toeplitz subshifts satisfying these conditions is non-empty. When the acting group $G$ is amenable, this construction provides non-regular extensions of totally disconnected metric compactifications of $G$ that are (Weyl) mean-quicontinuous dynamical systems.

We prove a sharp quantitative version of the Faber--Krahn inequality for the short-time Fourier transform (STFT). To do so, we consider a deficit $\delta(f;\Omega)$ which measures by how much the STFT of a function $f\in L^2(\mathbb R)$ fails to be optimally concentrated on an arbitrary set $\Omega\subset \mathbb R^2$ of positive, finite measure. We then show that an optimal power of the deficit $\delta(f;\Omega)$ controls both the $L^2$-distance of $f$ to an appropriate class of Gaussians and the distance of $\Omega$ to a ball, through the Fraenkel asymmetry of $\Omega$. Our proof is completely quantitative and hence all constants are explicit. We also establish suitable generalizations of this result in the higher-dimensional context.

We construct a family of infinite degree tt-rings, giving a negative answer to an open question by P. Balmer.

Let $G$ be a countable residually finite group (for instance $\mathbb{F}_2$) and let $\overleftarrow{G}$ be a totally disconnected metric compactification of $G$ equipped with the action of $G$ by left multiplication. For every $r\geq 1$ we construct a Toeplitz $G$-subshift $(X,\sigma,G)$, which is an almost one-to-one extension of $\overleftarrow{G}$, having $r$ ergodic measures $\nu_1, \cdots,\nu_r$ such that for every $1\leq i\leq r$ the measure-theoretic dynamical system $(X,\sigma,G,\nu_i)$ is isomorphic to $\overleftarrow{G}$ endowed with the Haar measure. The construction we propose is general (for amenable and non-amenable residually finite groups), however, we point out the differences and obstructions that could appear when the acting group is not amenable.

The quantization scheme in probability theory deals with finding a best approximation of a given probability distribution by a probability distribution that is supported on finitely many points. In this paper, first we state and prove a theorem, and then give a conjecture. We verify the conjecture by a few examples. Assuming that the conjecture is true, for a set of discrete distributions on the set of natural numbers we have calculated the optimal sets of $n$-means and the $n$th quantization errors for all positive integers $n$. In addition, the quantization dimension is also calculated.

We introduce the stable module $\infty$-category for groups of type $\Phi$ as an enhancement of the stable category defined by N. Mazza and P. Symonds. For groups of type $\Phi$ which act on a tree, we show that the stable module $\infty$-category decomposes in terms of the associated graph of groups. For groups which admit a finite-dimensional cocompact model for the classifying space for proper actions, we exhibit a decomposition in terms of the stable module $\infty$-categories of their finite subgroups. We use these decompositions to provide methods to compute the Picard group of the stable module category. In particular, we provide a description of the Picard group for countable locally finite $p$-groups.

Counting the number of prime numbers up to a certain natural number and describing the asymptotic behavior of such a counting function has been studied by famous mathematicians like Gauss, Legendre, Dirichlet, and Euler. The prime number theorem determines that such asymptotic behavior is similar to the asymptotic behavior of the number divided by its natural logarithm. In this paper, we take advantage of a multiplicative representation of a number and the properties of the logarithm function to express the prime number theorem in terms of primorial numbers, and n-primorial totative numbers. A primorial number is the multiplication of the first n prime numbers while n-primorial totatives are the numbers that are coprime to the n-th primorial number. By doing this we can define several different functions that can be used to approximate the behavior of the prime counting function asymptotically.

Prime numbers have attracted the attention of mathematiciansand enthusiasts for millenniums due to their simple definition and remarkable properties. In this paper, we study primorial numbers (the product of the first prime numbers) to define primorial sets, primorial intervals, primorial tables, and primorial totative numbers. We establish relationships between prime numbers and primorial totative numbers and between admissible k-tuples of prime numbers and admissible k-tuples of primorial totative. Finally, we study the Goldbach conjecture and derive four Goldbach conjectures using primordial intervals, twin, cousin, and sexy prime numbers.

We improve the existing lower bounds on the order of counterexamples to a conjecture by P. Schmid, determine some properties of the possible counterexamples of minimum order for each prime, and the isomorphism type of the center of the Frattini subgroup for the counterexamples of order 256. We also show that nonabelian metacyclic p-groups, nonabelian groups of maximal nilpotency class and 2-groups of coclass two satisfy the conjecture.

We address the numerical simulation of periodic solids (phononic crystals) within the framework of couple stress elasticity. The additional terms in the elastic potential energy lead to dispersive behavior in shear waves, even in the absence of material periodicity. To study the bulk waves in these materials, we establish an action principle in the frequency domain and present a finite element formulation for the wave propagation problem related to couple stress theory subject to an extended set of Bloch-periodic boundary conditions. A major difference from the traditional finite element formulation for phononic crystals is the appearance of higher-order derivatives. We solve this problem with the use of a Lagrange-multiplier approach. After presenting the variational principle and general finite element treatment, we particularize it to the problem of finding dispersion relations in elastic bodies with periodic material properties. The resulting implementation is used to determine the dispersion curves for homogeneous and porous couple stress solids, in which the latter is found to exhibit an interesting bandgap structure.

Gomez proposes a formal and systematic approach for characterizing stochastic global optimization algorithms. Using it, Gomez formalizes algorithms with a fixed next-population stochastic method, i.e., algorithms defined as stationary Markov processes. These are the cases of standard versions of hill-climbing, parallel hill-climbing, generational genetic, steady-state genetic, and differential evolution algorithms. This paper continues such a systematic formal approach. First, we generalize the sufficient conditions convergence lemma from stationary to non-stationary Markov processes. Second, we develop Markov kernels for some selection schemes. Finally, we formalize both simulated-annealing and evolutionary-strategies using the systematic formal approach.

Let $G$ be a compact connected Lie group and $n\geqslant 1$ an integer. Consider the space of ordered commuting $n$-tuples in $G$, $Hom(\mathbb{Z}^n,G)$, and its quotient under the adjoint action, $Rep(\mathbb{Z}^n,G):=Hom(\mathbb{Z}^n,G)/G$. In this article we study and in many cases compute the homotopy groups $\pi_2(Hom(\mathbb{Z}^n,G))$. For $G$ simply--connected and simple we show that $\pi_2(Hom(\mathbb{Z}^2,G))\cong \mathbb{Z}$ and $\pi_2(Rep(\mathbb{Z}^2,G))\cong \mathbb{Z}$, and that on these groups the quotient map $Hom(\mathbb{Z}^2,G)\to Rep(\mathbb{Z}^2,G)$ induces multiplication by the Dynkin index of $G$. More generally we show that if $G$ is simple and $Hom(\mathbb{Z}^2,G)_{1}\subseteq Hom(\mathbb{Z}^2,G)$ is the path--component of the trivial homomorphism, then $H_2(Hom(\mathbb{Z}^2,G)_{1};\mathbb{Z})$ is an extension of the Schur multiplier of $\pi_1(G)^2$ by $\mathbb{Z}$. We apply our computations to prove that if $B_{com}G_{1}$ is the classifying space for commutativity at the identity component, then $\pi_4(B_{com}G_{1})\cong \mathbb{Z}\oplus \mathbb{Z}$, and we construct examples of non-trivial transitionally commutative structures on the trivial principal $G$-bundle over the sphere $\mathbb{S}^{4}$.

The major difficulty in Multi-objective Optimization Evolutionary Algorithms (MOEAs) is how to find an appropriate solution that is able to converge towards the true Pareto Front with high diversity. Most existing methodologies, which have demonstrated their niche on various practical problems involving two and three objectives, face significant challenges in the dependency of the selection of the EA parameters. Moreover, the process of setting such parameters is considered time-consuming, and several research works have tried to deal with this problem. This paper proposed a new Multi-objective Algorithm as an extension of the Hybrid Adaptive Evolutionary algorithm (HAEA) called MoHAEA. MoHAEA allows dynamic adaptation of the application of operator probabilities (rates) to evolve with the solution of the multi-objective problems combining the dominance- and decomposition-based approaches. MoHAEA is compared with four states of the art MOEAs, namely MOEA/D, pa$\lambda$-MOEA/D, MOEA/D-AWA, and NSGA-II on ten widely used multi-objective test problems. Experimental results indicate that MoHAEA outperforms the benchmark algorithms in terms of how it is able to find a well-covered and well-distributed set of points on the Pareto Front.

For $G$ a finite group, a normalized 2-cocycle $\alpha\in Z^{2}\big(G,{\mathbb S}^{1}\big)$ and $X$ a $G$-space on which a normal subgroup $A$ acts trivially, we show that the $\alpha$-twisted $G$-equivariant $K$-theory of $X$ decomposes as a direct sum of twisted equivariant $K$-theories of $X$ parametrized by the orbits of an action of $G$ on the set of irreducible $\alpha$-projective representations of $A$. This generalizes the decomposition obtained in [Gómez J.M., Uribe B., Internat. J. Math. 28 (2017), 1750016, 23 pages, arXiv:1604.01656] for equivariant $K$-theory. We also explore some examples of this decomposition for the particular case of the dihedral groups $D_{2n}$ with $n\ge 2$ an even integer.

In this paper we present necessary and sufficient conditions for a graded (trimmed) double Ore extension to be a graded (quasi-commutative) skew PBW extension. Using this fact, we prove that a graded skew PBW extension $A = \sigma(R)\langle x_1,x_2 \rangle$ of an Artin-Schelter regular algebra $R$ is Artin-Schelter regular. As a consequence, every graded skew PBW extension $A = \sigma(R)\langle x_1,x_2 \rangle$ of a connected skew Calabi-Yau algebra $R$ of dimension $d$ is skew Calabi-Yau of dimension $d+2$.

We study the fixed points of the universal G-equivariant n-dimensional complex vector bundle and obtain a decomposition formula in terms of twisted equivariant universal complex vector bundles of smaller dimension. We use this decomposition to describe the fixed points of the complex equivariant K-theory spectrum and the equivariant unitary bordism groups for adjacent families of subgroups.

We present a extension of the classical open mapping principle and Effros' theorem for Polish group actions to the context of partial group actions.

We consider twisted equivariant K--theory for actions of a compact Lie group $G$ on a space $X$ where all the isotropy subgroups are connected and of maximal rank. We show that the associated rational spectral sequence à la Segal has a simple $E_2$--term expressible as invariants under the Weyl group of $G$. Namely, if $T$ is a maximal torus of $G$, they are invariants of the $\pi_1(X^T)$-equivariant Bredon cohomology of the universal cover of $X^T$ with suitable coefficients. In the case of the inertia stack $\Lambda Y$ this term can be expressed using the cohomology of $Y^T$ and algebraic invariants associated to the Lie group and the twisting. A number of calculations are provided. In particular, we recover the rational Verlinde algebra when $Y=\{*\}$.

We provide a sufficient condition for a topological partial action of a Hausdorff group on a metric space is continuous, provide that it is separately continuous.

As we know, some global optimization problems cannot be solved using analytic methods, so numeric/algorithmic approaches are used to find near to the optimal solutions for them. A stochastic global optimization algorithm (SGoal) is an iterative algorithm that generates a new population (a set of candidate solutions) from a previous population using stochastic operations. Although some research works have formalized SGoals using Markov kernels, such formalization is not general and sometimes is blurred. In this paper, we propose a comprehensive and systematic formal approach for studying SGoals. First, we present the required theory of probability (\sigma-algebras, measurable functions, kernel, markov chain, products, convergence and so on) and prove that some algorithmic functions like swapping and projection can be represented by kernels. Then, we introduce the notion of join-kernel as a way of characterizing the combination of stochastic methods. Next, we define the optimization space, a formal structure (a set with a \sigma-algebra that contains strict \epsilon-optimal states) for studying SGoals, and we develop kernels, like sort and permutation, on such structure. Finally, we present some popular SGoals in terms of the developed theory, we introduce sufficient conditions for convergence of a SGoal, and we prove convergence of some popular SGoals.

For G a finite group and X a G-space on which a normal subgroup A acts trivially, we show that the G-equivariant K-theory of X decomposes as a direct sum of twisted equivariant K-theories of X parametrized by the orbits of the conjugation action of G on the irreducible representations of A. The twists are group 2-cocycles which encode the obstruction of lifting an irreducible representation of A to the subgroup of G which fixes the isomorphism class of the irreducible representation.

We prove central limit theorems (CLT) for empirical processes of extreme values cluster functionals as in Drees and Rootzén (2010). We use coupling properties enlightened for Dedecker \& Prieur's $\tau-$dependence coefficients in order to improve the conditions of dependence and continue to obtain these CLT. The assumptions are precisely set for particular processes and cluster functionals of interest. The number of excesses provides a complete example of a cluster functional for a simple non-mixing model (AR(1)-process) for which ours results are definitely needed. We also give the expression explicit the covariance structure of limit Gaussian process. Also we include in this paper some results of Drees (2011) for the extremal index and some simulations for this index to demonstrate the accuracy of this technique.

This paper aims to validate the $\beta$-Ginibre point process as a model for the distribution of base station locations in a cellular network. The $\beta$-Ginibre is a repulsive point process in which repulsion is controlled by the $\beta$ parameter. When $\beta$ tends to zero, the point process converges in law towards a Poisson point process. If $\beta$ equals to one it becomes a Ginibre point process. Simulations on real data collected in Paris (France) show that base station locations can be fitted with a $\beta$-Ginibre point process. Moreover we prove that their superposition tends to a Poisson point process as it can be seen from real data. Qualitative interpretations on deployment strategies are derived from the model fitting of the raw data.

Using a construction derived from the descending central series of the free groups, we produce filtrations by infinite loop spaces of the classical infinite loop spaces $BSU$, $BU$, $BSO$, $BO$, $BSp$, $BGL_{\infty}(R)^{+}$ and $Q_0(\mathbb{S}^{0})$. We show that these infinite loop spaces are the zero spaces of non-unital $E_\infty$-ring spectra. We introduce the notion of $q$-nilpotent K-theory of a CW-complex $X$ for any $q\ge 2$, which extends the notion of commutative K-theory defined by Adem-Gómez, and show that it is represented by $\mathbb Z\times B(q,U)$, were $B(q,U)$ is the $q$-th term of the aforementioned filtration of $BU$. For the proof we introduce an alternative way of associating an infinite loop space to a commutative $\mathbb{I}$-monoid and give criteria when it can be identified with the plus construction on the associated limit space. Furthermore, we introduce the notion of a commutative $\mathbb{I}$-rig and show that they give rise to non-unital $E_\infty$-ring spectra.

A set of positive integers $A$ is called a $B_{h}[g]$ set if there are at most $g$ different sums of $h$ elements from $A$ with the same result. This definition has a generalization to abelian groups and the main problem related to this kind of sets, is to find $B_{h}[g]$ maximal sets i.e. those with larger cardinality. We construct $B_{h}[g]$ modular sets from $B_{h}$ modular sets using homomorphisms and analyze the constructions of $B_{h}$ sets by Bose and Chowla, Ruzsa, and Gómez and Trujillo look at for the suitable homomorphism that allows us to preserve the cardinal of this types of sets.

We evaluated the performance of the classical and spectral finite element method in the simulation of elastodynamic problems. We used as a quality measure their ability to capture the actual dispersive behavior of the material. Four different materials are studied: a homogeneous non-dispersive material, a bilayer material, and composite materials consisting of an aluminum matrix and brass inclusions or voids. To obtain the dispersion properties, spatial periodicity is assumed so the analysis is conducted using Floquet-Bloch principles. The effects in the dispersion properties of the lumping process for the mass matrices resulting from the classical finite element method are also investigated, since that is a common practice when the problem is solved with explicit time marching schemes. At high frequencies the predictions with the spectral technique exactly match the analytical dispersion curves, while the classical method does not. This occurs even at the same computational demands. At low frequencies however, the results from both the classical (consistent or mass-lumped) and spectral finite element coincide with the analytically determined curves. Surprisingly, at low frequencies even the results obtained with the artificial diagonal mass matrix from the classical technique exactly match the analytic dispersion curves.

Drees and Rootzén [2010] have proven central limit theorems (CLT) for empirical processes of extreme values cluster functionals built from $\beta$-mixing processes. The problem with this family of $\beta$-mixing processes is that it is quite restrictive, as has been shown by Andrews [1984]. We expand this result to a more general dependent processes family, known as weakly dependent processes in the sense of Doukhan and Louhichi [1999], but in finite-dimensional convergence (fidis). We show an example where the application of the CLT-fidis is sufficient in several cases, including a small simulation of the extremogram introduced by Davis and Mikosch [2009] to confirm the efficacy of our result.

The descriptions (up to isomorphism) of naturally graded $p$-filiform Leibniz algebras and $p$-filiform ($p\leq 3$) Leibniz algebras of maximum length are known. In this paper we study the gradation of maximum length for $p$-filiform Leibniz algebras. The present work aims at the classification of complex $p$-filiform ($p \geq 4$) Leibniz algebras of maximum length.

This work completes the study of the solvable Leibniz algebras, more precisely, it completes the classification of the $3$-filiform Leibniz algebras of maximum length \cite3-filiform. Moreover, due to the good structure of the algebras of maximum length, we also tackle some of their cohomological properties. Our main tools are the previous result of Cabezas and Pastor \citePastor, the construction of appropriate homogeneous basis in the considered connected gradation and the computational support provided by the two programs implemented in the software \textitMathematica.

In this article we consider a space B_comG assembled from commuting elements in a Lie group G first defined in [Adem, Cohen, Torres-Giese 2012]. We describe homotopy-theoretic properties of these spaces using homotopy colimits, and their role as a classifying space for transitionally commutative bundles. We prove that ZxB_comU is a loop space and define a notion of commutative K-theory for bundles over a finite complex X which is isomorphic to [X,ZxB_comU]. We compute the rational cohomology of B_comG for G equal to any of the classical groups U(n), SU(n) and Sp(n), and exhibit the rational cohomologies of B_comU, B_comSU and B_comSp as explicit polynomial rings.

Let n>0 be an integer and let B_n denote the hyperoctahedral group of rank n. The group B_n acts on the polynomial ring Q[x_1,...,x_n,y_1,...,y_n] by signed permutations simultaneously on both of the sets of variables x_1,...,x_n and y_1,...,y_n. The invariant ring M^B_n:=Q[x_1,...,x_n,y_1,...,y_n]^B_n is the ring of diagonally signed-symmetric polynomials. In this article we provide an explicit free basis of M^B_n as a module over the ring of symmetric polynomials on both of the sets of variables x_1^2,..., x^2_n and y_1^2,..., y^2_n using signed descent monomials.

The type-I intermittency route to (or out of) chaos is investigated within the Horizontal Visibility graph theory. For that purpose, we address the trajectories generated by unimodal maps close to an inverse tangent bifurcation and construct, according to the Horizontal Visibility algorithm, their associated graphs. We show how the alternation of laminar episodes and chaotic bursts has a fingerprint in the resulting graph structure. Accordingly, we derive a phenomenological theory that predicts quantitative values of several network parameters. In particular, we predict that the characteristic power law scaling of the mean length of laminar trend sizes is fully inherited in the variance of the graph degree distribution, in good agreement with the numerics. We also report numerical evidence on how the characteristic power-law scaling of the Lyapunov exponent as a function of the distance to the tangent bifurcation is inherited in the graph by an analogous scaling of the block entropy over the degree distribution. Furthermore, we are able to recast the full set of HV graphs generated by intermittent dynamics into a renormalization group framework, where the fixed points of its graph-theoretical RG flow account for the different types of dynamics. We also establish that the nontrivial fixed point of this flow coincides with the tangency condition and that the corresponding invariant graph exhibit extremal entropic properties.

In this note we study topological invariants of the spaces of homomorphisms Hom(\pi,G), where \pi is a finitely generated abelian group and G is a compact Lie group arising as an arbitrary finite product of the classical groups SU(r), U(q) and Sp(k).

Let G denote a compact connected Lie group with torsion-free fundamental group acting on a compact space X such that all the isotropy subgroups are connected subgroups of maximal rank. Let $T\subset G$ be a maximal torus with Weyl group W. If the fixed-point set $X^T$ has the homotopy type of a finite W-CW complex, we prove that the rationalized complex equivariant K-theory of X is a free module over the representation ring of G. Given additional conditions on the W-action on the fixed-point set $X^T$ we show that the equivariant K-theory of X is free over R(G). We use this to provide computations for a number of examples, including the ordered n-tuples of commuting elements in G with the conjugation action.

In the present work we consider the electromagnetic wave equation in terms of the fractional derivative of the Caputo type. The order of the derivative being considered is 0 <\gamma<1. A new parameter \sigma, is introduced which characterizes the existence of the fractional components in the system. We analyze the fractional derivative with respect to time and space, for \gamma = 1 and \gamma = 1/2 cases.

We prove the uniqueness of twisted K-theory in both the real and complex cases using the computation of the K-theories of Eilenberg-MacLane spaces due to Anderson and Hodgkin. As an application of our method, we give some vanishing results for actions of Eilenberg-MacLane spaces on K-theory spectra.

In this paper we present the classification of a subclass of naturally graded Leibniz algebras. These $n$-dimensional Leibniz algebras have the characteristic sequence equal to (n-3,3). For this purpose we use the software Mathematica.

In this paper the space of almost commuting elements in a Lie group is studied through a homotopical point of view. In particular a stable splitting after one suspension is derived for these spaces and their quotients under conjugation. A complete description for the stable factors appearing in this splitting is provided for compact connected Lie groups of rank one.By using symmetric products, the colimits $\Rep(\Z^n, SU)$, $\Rep(\Z^n,U)$ and $\Rep(\Z^n, Sp)$ are explicitly described as finite products of Eilenberg-MacLane spaces.

The $n$-dimensional $p$-filiform Leibniz algebras of maximum length have already been studied with $0\leq p\leq 2$. For Lie algebras whose nilindex is equal to $n-2$ there is only one characteristic sequence, $(n-2,1,1)$, while in Leibniz theory we obtain two possibilities: $(n-2,1,1)$ and $(n-2,2)$. The first case (the 2-filiform case) is already known. The present paper deals with the second case, i.e., quasi-filiform non Lie Leibniz algebras of maximum length. Therefore this work completes the study of maximum length of Leibniz algebras with nilindex $n-p$ with $0 \leq p \leq 2$.

Let G be a compact Lie group, and consider the variety Hom(Z^k,G) of representations of Z^k into G. We view this as a based space by designating the trivial representation to be its base point. We prove that the fundamental group of this space is naturally isomorphic to \pi_1(G)^k.

In this work we investigate the derivations of $n-$dimensional complex evolution algebras, depending on the rank of the appropriate matrices. For evolution algebra with non-singular matrices we prove that the space of derivations is zero. The spaces of derivations for evolution algebras with matrices of rank $n-1$ are described.

The paper is devoted to the study of finite dimensional complex evolution algebras. The class of evolution algebras isomorphic to evolution algebras with Jordan form matrices is described. For finite dimensional complex evolution algebras the criteria of nilpotency is established in terms of the properties of corresponding matrices. Moreover, it is proved that for nilpotent $n-$dimensional complex evolution algebras the possible maximal nilpotency index is $1+2^{n-1}.$ The criteria of planarity for finite graphs is formulated by means of evolution algebras defined by graphs.

Let $\phi:\Z/p\to GL_{n}(\Z)$ denote an integral representation of the cyclic group of prime order $p$. This induces a $\Z/p$-action on the torus $X=\R^{n}/\Z^{n}$. The goal of this paper is to explicitly compute the cohomology groups $H^{*}(X/\Z/p;\Z)$ for any such representation. As a consequence we obtain an explicit calculation of the integral cohomology of the classifying space associated to the family of finite subgroups for any crystallographic group $\Gamma =\Z^n\rtimes\Z/p$ with prime holonomy.

In this note we show that there are no higher twistings for the Borel cohomology theory associated to G-equivariant K-theory over a point and for a compact Lie group G. Therefore, twistings over a point for this theory are classified by the group H^1(BG,Z/2)\xH^3(BG,Z)

In here we define the concept of fibered symmetric bimonoidal categories. These are roughly speaking fibered categories D->C whose fibers are symmetric monoidal categories parametrized by C and such that both D and C have a further structure of a symmetric monoidal category that satisfy certain coherences that we describe. Our goal is to show that we can correspond to a fibered symmetric bimonoidal category an E_∞-ring spectrum in a functorial way.

In this paper the space of commuting elements in the central product $G_{m,p}$ of $m$ copies of the special unitary group $SU(p)$ is studied, where $p$ is a prime number. In particular, a computation for the number of path connected components of these spaces is given and the geometry of the moduli space $\Rep(\mathbb Z^n, G_{m,p})$ of flat principal $G_{m,p}$--bundles over the $n$--torus is completely described for all values of $n$, $m$ and $p$.

In this paper we investigate the description of the complex Leibniz superalgebras with nilindex n+m, where n and m ($m\neq 0$) are dimensions of even and odd parts, respectively. In fact, such superalgebras with characteristic sequence equal to $(n_1, ..., n_k | m_1, ..., m_s)$ (where $n_1+... +n_k=n, m_1+ ... + m_s=m$) for $n_1\geq n-1$ and $(n_1, ..., n_k | m)$ were classified in works \citeFilSup--\citeC-G-O-Kh1. Here we prove that in the case of $(n_1, ..., n_k| m_1, ..., m_s)$, where $n_1\leq n-2$ and $m_1 \leq m-1$ the Leibniz superalgebras have nilindex less than n+m. Thus, we complete the classification of Leibniz superalgebras with nilindex n+m.

In this work we investigate the complex Leibniz superalgebras with characteristic sequence $(n_1,...,n_k|m)$ and nilindex n+m, where $n=n_1+...+n_k,$ n and m (m is not equal to zero) are dimensions of even and odd parts, respectively. Such superalgebras with condition n_1 > n-2 were classified in \citeFilSup--\citeC-G-O-Kh. Here we prove that in the case $n_1 < n-1$ the Leibniz superalgebras have nilindex less than $n+m.$ Thus, we get the classification of Leibniz superalgebras with characteristic sequence $(n_1, ...,n_k|m)$ and nilindex n+m.

We present the description up to isomorphism of Leibniz superalgebras with characteristic sequence $(n|m_1,...,m_k)$ and nilindex $n+m,$ where $m=m_1+ >...+m_k,$ $n$ and $m$ ($m\neq 0$) are dimensions of even and odd parts, respectively.

Consider real bivariate polynomials f and g, respectively having 3 and m monomial terms. We prove that for all m>=3, there are systems of the form (f,g) having exactly 2m-1 roots in the positive quadrant. Even examples with m=4 having 7 positive roots were unknown before this paper, so we detail an explicit example of this form. We also present an O(n^11) upper bound for the number of diffeotopy types of the real zero set of an n-variate polynomial with n+4 monomial terms.

In this paper we prove that in classifying of complex filiform Leibniz algebras, for which its naturally graded algebra is non-Lie algebra, it suffices to consider some special basis transformations. Moreover, we establish a criterion whether given two such Leibniz algebras are isomorphic in terms of such transformations. The classification problem of filiform Leibniz algebras, for which its naturally graded algebras are non-Lie in an arbitrary dimension, is reduced to the investigation of the obtained conditions.

The aim of this work is to present the first problems that appear in the study of nilpotent Leibniz superalgebras. These superalgebras and so the problems, will be considered as a natural generalization of nilpotent Leibniz algebras and Lie superalgebras.

The aim of this work is to present the description up to isomorphism of Leibniz superalgebras with characteristic sequence (n | m-1, 1) and nilindex n+m.

The Leibniz algebras appear as a generalization of the Lie algebras \citeloday. The classification of naturally graded $p$-filiform Lie algebras is known \citeC-G-JM, \citeJ.Lie.Theory, \citeAJM, \citeVe. In this work we deal with the classification of 2-filiform Leibniz algebras. The study of $p$-filiform Leibniz non Lie algebras is solved for $p=0$ (trivial) and $p=1$ \citeOmirov1. In this work we get the classification of naturally graded non Lie 2-filiform Leibniz algebras.

The problem of known signal detection in Additive White Gaussian Noise is considered. In this paper a new detection algorithm based on Discrete Wavelet Transform pre-processing and threshold comparison is introduced. Current approaches described in [7] use the maximum value obtained in the wavelet domain for decision. Here, we use all available information in the wavelet domain with excellent results. Detector performance is presented in Probability of detection curves for a fixed probability of false alarm.

The occurrence of the September 28, 2004 Mw=6.0 mainshock at Parkfield, California, has significantly increased the mean and aperiodicity of the series of time intervals between mainshocks in this segment of the San Andreas fault. We use five different statistical distributions as renewal models to fit this new series and to estimate the time-dependent probability of the next Parkfield mainshock. Three of these distributions (lognormal, gamma and Weibull) are frequently used in reliability and time-to-failure problems. The other two come from physically-based models of earthquake recurrence (the Brownian Passage Time Model and the Minimalist Model). The differences resulting from these five renewal models are emphasized.

In this paper we give graphs with the largest known order for a given degree $\Delta$ and diameter $D$. The graphs are constructed from Moore bipartite graphs by replacement of some vertices by adequate structures. The paper also contains the latest version of the $(\Delta, D)$ table for graphs.