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We show how the onemode pseudobosonic ladder operators provide concrete examples of nilpotent Lie algebras of dimension five. It is the first time that an algebraicgeometric structure of this kind is observed in the context of pseudobosonic operators. Indeed we don't find the well known Heisenberg algebras, which are involved in several quantum dynamical systems, but different Lie algebras which may be decomposed in the sum of two abelian Lie algebras in a prescribed way. We introduce the notion of semidirect sum (of Lie algebras) for this scope and find that it describes very well the behaviour of pseudobosonic operators in many quantum models.
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We introduce a framework for calculating sparse approximations to signals based on elements of continuous wavelet systems. The method is based on an extension of the continuous wavelet theory. In the new theory, the signal space is embedded in larger "abstract" signal space, which we call the windowsignal space. There is a canonical extension of the wavelet transform on the windowsignal space, which is an isometric isomorphism from the windowsignal space to a space of functions on phase space. Hence, the new framework is called a waveletPlancherel theory, and the extended wavelet transform is called the waveletPlancherel transform. Since the waveletPlancherel transform is an isometric isomorphism, any operation on phase space can be pulledback to an operation in the windowsignal space. Using this pull back property, it is possible to pull back a search for big wavelet coefficients to the windowsignal space. We can thus avoid inefficient calculations on phase space, performing
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We represent the Lebesgue measure on the unit interval as a boundary measure of the Farey tree and show that this representation has a certain symmetry related to the tree automorphism induced by Dyer's outer automorphism of the group PGL(2,Z). Our approach gives rise to three new measures on the unit interval which are possibly of arithmetic significance.
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As a simplified model for subsurface flows elliptic equations may be utilized. Insufficient measurements or uncertainty in those are commonly modeled by a random coefficient, which then accounts for the uncertain permeability of a given medium. As an extension of this methodology to flows in heterogeneous\fractured\porous media, we incorporate jumps in the diffusion coefficient. These discontinuities then represent transitions in the media. More precisely, we consider a second order elliptic problem where the random coefficient is given by the sum of a (continuous) Gaussian random field and a (discontinuous) jump part. To estimate moments of the solution to the resulting random partial differential equation, we use a pathwise numerical approximation combined with multilevel Monte Carlo sampling. In order to account for the discontinuities and improve the convergence of the pathwise approximation, the spatial domain is decomposed with respect to the jump positions in each sample, leadin
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We prove characterization theorems for relative entropy (also known as KullbackLeibler divergence), qlogarithmic entropy (also known as Tsallis entropy), and qlogarithmic relative entropy. All three have been characterized axiomatically before, but we show that earlier proofs can be simplified considerably, at the same time relaxing some of the hypotheses.
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The logarithmic strain measures $\lVert\log U\rVert^2$, where $\log U$ is the principal matrix logarithm of the stretch tensor $U=\sqrt{F^TF}$ corresponding to the deformation gradient $F$ and $\lVert\,.\,\rVert$ denotes the Frobenius matrix norm, arises naturally via the geodesic distance of $F$ to the special orthogonal group $\operatorname{SO}(n)$. This purely geometric characterization of this strain measure suggests that a viable constitutive law of nonlinear elasticity may be derived from an elastic energy potential which depends solely on this intrinsic property of the deformation, i.e. that an energy function $W\colon\operatorname{GL^+}(n)\to\mathbb{R}$ of the form \begin{equation} W(F)=\Psi(\lVert\log U\rVert^2) \tag{1} \end{equation} with a suitable function $\Psi\colon[0,\infty)\to\mathbb{R}$ should be used to describe finite elastic deformations. However, while such energy functions enjoy a number of favorable properties, we show that it is not possible to find a strictly m
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This article is a follow up of our submitted paper [11] in which a decomposition of the Richards equation along two soil layers was discussed. A decomposed problem was formulated and a decoupling and linearisation technique was presented to solve the problem in each time step in a fixed point type iteration. This article extends these ideas to the case of twophase in porous media and the convergence of the proposed domain decomposition method is rigorously shown.
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This article develops a duality principle for a semilinear model in micromagnetism. The results are obtained through standard tools of convex analysis and the Legendre transform concept. We emphasize the dual variational formulation presented is concave and suitable for numerical computations. Moreover, sufficient conditions of optimality are also established.
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Given a hypergraph $H = (V,E)$ and an integer parameter $k$, a coloring of $V$ is said to be $k$conflictfree ($k$CF in short) if for every hyperedge $S \in E$, there exists a color with multiplicity at most $k$ in $S$. A $k$CF coloring of a graph is a $k$CF coloring of the hypergraph induced by the (closed or punctured) neighborhoods of its vertices. The special case of $1$CF coloring of general graphs and hypergraphs has been studied extensively. In this paper we study $k$CF coloring of graphs and hypergraphs. First, we study the nongeometric case and prove that any hypergraph with $n$ vertices and $m$ hyperedges can be $k$CF colored with $\tilde{O}(m^{\frac{1}{k+1}})$ colors. This bound, which extends theorems of Cheilaris and of Pach and Tardos (2009), is tight, up to a logarithmic factor. Next, we study {\em string graphs}. We consider several families of string graphs on $n$ vertices for which the $1$CF chromatic number w.r.t. punctured neighborhoods is $\Omega(\sqrt{n})
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A supervised learning algorithm searches over a set of functions $A \to B$ parametrised by a space $P$ to find the best approximation to some ideal function $f\colon A \to B$. It does this by taking examples $(a,f(a)) \in A\times B$, and updating the parameter according to some rule. We define a category where these update rules may be composed, and show that gradient descentwith respect to a fixed step size and an error function satisfying a certain propertydefines a monoidal functor from a category of parametrised functions to this category of update rules. This provides a structural perspective on backpropagation, as well as a broad generalisation of neural networks.
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This is a course of lectures given for students of the Regional Mathematical Center of the Novosibirsk State University from October 20 to November 3, 2017. The course is devoted to some geometric problems of ramified coverings of the Riemann sphere. A special attention is payed to compact surfaces of genus one (complex tori). In the first section we give a short introduction to the theory of elliptic functions. Section 2 is devoted to oneparametric families of holomorphic and meromorphic functions. We recall the role of such families on Loewner's equation in solving some problems of the theory of univalent functions. Further we deduce a system of ODEs expressing dependence of critical points of a family of rational functions from their critical values. This gives an approximate method to find a conformal mapping of the Riemann sphere onto a given simplyconnected compact Riemann surface over the sphere. Thereafter a similar problem is solved for elliptic functions uniformizing comple
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In this paper we present explicit estimate for Lipschitz constant of solution to a problem of calculus of variations. The approach we use is due to Gamkrelidze and is based on the equivalence of the problem of calculus of variations and a timeoptimal control problem. The obtained estimate is used to compute complexity bounds for a pathfollowing method applied to a convex problem of calculus of variations with polyhedral endpoint constraints.
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We present exact analytical results for the Caputo fractional derivative of a wide class of elementary functions, including trigonometric and inverse trigonometric, hyperbolic and inverse hyperbolic, Gaussian, quartic Gaussian, and Lorentzian functions. These results are especially important for multiscale physical systems, such as porous materials, disordered media, and turbulent fluids, in which transport is described by fractional partial differential equations. The exact results for the Caputo fractional derivative are obtained from a single generalized Euler's integral transform of the generalized hypergeometric function with a powerlaw argument. We present a proof of the generalized Euler's integral transform and directly apply it to the exact evaluation of the Caputo fractional derivative of a broad spectrum of functions, provided that these functions can be expressed in terms of a generalized hypergeometric function with a powerlaw argument. We determine that the Caputo fr
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We are dealing with the question whether every group or semigroup action (with some additional property) on a continuum (with some additional property) has a fixed point. One of such results was given in 2009 by Shi and Sun. They proved that every nilpotent group action on a uniquely arcwise connected continuum has a fixed point. We are seeking for this type of results with e.g. commutative, compact or torsion groups and semigroups acting on dendrites, dendroids, $\lambda$dendroids and uniquely arcwise connected continua. We prove that every continuous action of a compact or torsion group on a uniquely arcwise connected continuum has a fixed point. We also prove that every continuous action of a compact and commutative semigroup on a uniquely arcwise connected continuum has a fixed point.
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This paper studies a twohop decodeandforward underlay cognitive radio system with interference alignment technique. An energyconstrained relay node harvests the energy from the interference signals through a powersplitting (PS) relaying protocol. Firstly, the beamforming matrices design for the primary and secondary networks is demonstrated. Then, a bit error rate (BER) performance of the system under perfect and imperfect channel state information (CSI) scenarios for PS protocol is calculated. Finally, the impact of the CSI mismatch parameters on the BER performance is simulated.
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In this paper, we consider a twohop amplifyandforward (AF) relaying system, where the relay node is energyconstrained and harvests energy from the source node. In the literature, there are three main energyharvesting (EH) protocols, namely, timeswitching relaying (TSR), powersplitting (PS) relaying (PSR) and ideal relaying receiver (IRR). Unlike the existing studies, in this paper, we consider $\alpha$$\mu$ fading channels. In this respect, we derive accurate unified analytical expressions for the ergodic capacity for the aforementioned protocols over independent but not identically distributed (i.n.i.d) $\alpha$$\mu$ fading channels. Three special cases of the $\alpha$$\mu$ model, namely, Rayleigh, NakagamimandWeibull fading channels were investigated. Our analysis is verified through numerical and simulation results. It is shown that finding the optimal value of the PS factor for the PSR protocol and the EH time fraction for the TSR protocol is a crucial step in achieving
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In this paper, we establish equivariant mirror symmetry for the weighted projective line. This extends the results by B. Fang, C.C. Liu and Z. Zong, where the projective line was considered [13]. More precisely, we prove the equivalence of the Rmatrices for Amodel and Bmodel for large radius limit, and establish isomorphism for $R$matrices for general radius. We further demonstrate the graph sum of higher genus cases for both models to be the same, hence establish equivariant mirror symmetry for the weighted projective line.
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This paper investigates an energyefficient nonorthogonal transmission design problem for two downlink receivers that have strict reliability and finite blocklength (latency) constraints. The Shannon capacity formula widely used in traditional designs needs the assumption of infinite blocklength and thus is no longer appropriate. We adopt the newly finite blocklength coding capacity formula for explicitly specifying the tradeoff between reliability and code blocklength. However, conventional successive interference cancellation (SIC) may become infeasible due to heterogeneous blocklengths. We thus consider several scenarios with different channel conditions and with/without SIC. By carefully examining the problem structure, we present in closedform the optimal power and code blocklength for energyefficient transmissions. Simulation results provide interesting insights into conditions for which nonorthogonal transmission is more energy efficient than the orthogonal transmission suc
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Recently, Yamanaka and Yamashita (2017) proposed the socalled positively homogeneous optimization problems, which generalize many important problems, in particular the absolutevalue and the gauge optimizations. They presented a closed dual formulation for these problems, proving weak duality results, and showing that it is equivalent to the Lagrangian dual under some conditions. In this work, we focus particularly in optimization problems whose objective functions and constraints consist of some gauge and linear functions. Through the positively homogeneous framework, we prove that both weak and strong duality results hold. We also discuss necessary and sufficient optimality conditions associated to these problems. Finally, we show that it is possible to recover primal solutions from KarushKuhnTucker points of the dual formulation.
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A graph $G$ is said to be distance magic if there exists a bijection $f:V\rightarrow \{1,2, \ldots , v\}$ and a constant {\sf k} such that for any vertex $x$, $\sum_{y\in N(x)} f(y) ={\sf k}$, where $N_(x)$ is the set of all neighbours of $x$. In this paper we shall study distance magic labelings of graphs obtained from four graph products: cartesian, strong, lexicographic, and cronecker. We shall utilise magic rectangle sets and magic column rectangles to construct the labelings.
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A unified treatment for fast and spectrally accurate evaluation of electrostatic potentials subject to periodic boundary conditions in any or none of the three space dimensions is presented. Ewald decomposition is used to split the problem into a real space and a Fourier space part, and the FFT based Spectral Ewald (SE) method is used to accelerate the computation of the latter. A key component in the unified treatment is an FFT based solution technique for the freespace Poisson problem in three, two or one dimensions, depending on the number of nonperiodic directions. The cost of calculations is furthermore reduced by employing an adaptive FFT for the doubly and singly periodic cases, allowing for different local upsampling rates. The SE method will always be most efficient for the triply periodic case as the cost for computing FFTs will be the smallest, whereas the computational cost for the rest of the algorithm is essentially independent of the periodicity. We show that the cost
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Given a $C^k$smooth closed embedded manifold $\mathcal N\subset{\mathbb R}^m$, with $k\ge 2$, and a compact connected smooth Riemannian surface $(S,g)$ with $\partial S\neq\emptyset$, we consider $\frac 12$harmonic maps $u\in H^{1/2}(\partial S,\mathcal N)$. These maps are critical points of the nonlocal energy \begin{equation}E(f;g):=\int_S\big\nabla\widetilde u\big^2\,d\text{vol}_g,\end{equation} where $\widetilde u$ is the harmonic extension of $u$ in $S$. We express the energy as a sum of the $\frac 12$energies at each boundary component of $\partial S$ (suitably identified with the circle $\mathcal S^1$), plus a quadratic term which is continuous in the $H^s(\mathcal S^1)$ topology, for any $s\in\mathbb R$. We show the $C^{k1,\delta}$ regularity of $\frac 12$harmonic maps. We also establish a connection between free boundary minimal surfaces and critical points of $E$ with respect to variations of the pair $(f,g)$, in terms of the Teichm\"uller space of $S$.
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Maxstable processes are very relevant for the modelling of spatial extremes. In this paper, we focus on some processes belonging to the class of spacetime maxstable models introduced in Embrechts et al. (2016). The mentioned processes are Markov chains with state space the space of continuous functions from the unit sphere of $\mathbb{R}^3$ to $(0, \infty)$. We show that these Markov chains are geometrically ergodic. An interesting feature lies in the fact that the previously mentioned state space is not locally compact, making the classical methodology to be found, e.g., in Meyn and Tweedie (2009), inapplicable. Instead, we use the fact that the state space is Polish and apply results on Markov chains with Polish state spaces presented in Hairer (2010).
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We introduce new Langevintype equations describing the rotational and translational motion of rigid bodies interacting through conservative and nonconservative forces, and hydrodynamic coupling. In the absence of nonconservative forces the Langevintype equations sample from the canonical ensemble. The rotational degrees of freedom are described using quaternions, the lengths of which are exactly preserved by the stochastic dynamics. For the proposed Langevintype equations, we construct a weak 2nd order geometric integrator which preserves the main geometric features of the continuous dynamics. The integrator uses Verlettype splitting for the deterministic part of Langevin equations appropriately combined with an exactly integrated OrnsteinUhlenbeck process. Numerical experiments are presented to illustrate both the new Langevin model and the numerical method for it, as well as to demonstrate how inertia and the coupling of rotational and translational motion can introduce qualit
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We provide an algorithm to solve the word problem in all fundamental groups of closed 3manifolds; in particular, we show that these groups are autostackable. This provides a common framework for a solution to the word problem in any closed 3manifold group using finite state automata. We also introduce the notion of a group which is autostackable respecting a subgroup, and show that a fundamental group of a graph of groups whose vertex groups are autostackable respecting any edge group is autostackable. A group that is strongly coset automatic over an autostackable subgroup, using a prefixclosed transversal, is also shown to be autostackable respecting that subgroup. Building on work by Antolin and Ciobanu, we show that a finitely generated group that is hyperbolic relative to a collection of abelian subgroups is also strongly coset automatic relative to each subgroup in the collection. Finally, we show that fundamental groups of compact geometric 3manifolds, with boundary consistin
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We propose strategies to estimate and make inference on key features of heterogeneous effects in randomized experiments. These key features include best linear predictors of the effects using machine learning proxies, average effects sorted by impact groups, and average characteristics of most and least impacted units. The approach is valid in high dimensional settings, where the effects are proxied by machine learning methods. We postprocess these proxies into the estimates of the key features. Our approach is agnostic about the properties of the machine learning estimators used to produce proxies, and it completely avoids making any strong assumption. Estimation and inference relies on repeated data splitting to avoid overfitting and achieve validity. Our variational inference method is shown to be uniformly valid and quantifies the uncertainty coming from both parameter estimation and data splitting. In essence, we take medians of pvalues and medians of confidence intervals, resul
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We show that in the category of groups, every singlygenerated class which is closed under isomorphisms, direct limits and extensions is also singlygenerated under isomorphisms and direct limits, and in particular is coreflective. We also establish several new relations between singlygenerated closed classes.
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In this paper, we establish a generalized Taylor expansion of a given function $f$ in the form $\displaystyle{f(x) = \sum_{j=0}^m c_j^{\alpha,\rho}\left(x^\rhoa^\rho\right)^{j\alpha} + e_m(x)}$ \noindent with $m\in \mathbb{N}$, $c_j^{\alpha,\rho}\in \mathbb{R}$, $x>a$ and $0< \alpha \leq 1$. In case $\rho = \alpha = 1$, this expression coincides with the classical Taylor formula. The coefficients $c_j^{\alpha,\rho}$, $j=0,\dots,m$ as well as an estimation of $e_m(x)$ are given in terms of the generalized Caputotype fractional derivatives. Some applications of these results for approximation of functions and for solving some fractional differential equations in series form are given in illustration.
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In this paper we propose a welljustified synthetic approach of the projective space. We define the concepts of plane and space of incidence and also the Gallucci's axiom as an axiom of the projective space. To this purpose we prove from our axioms, the theorems of Desargues, PappusPascal and the fundamental theorem of projectivities, respectively.
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We study the N=1 supersymmetric solutions of D=11 supergravity obtained as a warped product of fourdimensional antideSitter space with a sevendimensional Riemannian manifold M. Using the octonion bundle structure on M we reformulate the Killing spinor equations in terms of sections of the octonion bundle on M. The solutions then define a single complexified G2structure on M or equivalently two real G2structures. We then study the torsion of these G2structures and the relationships between them.
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A family $F$ of sets is said to satisfy the $(p,q)$property if among any $p$ sets of $F$ some $q$ intersect. The celebrated $(p,q)$theorem of Alon and Kleitman asserts that any family of compact convex sets in $\mathbb{R}^d$ that satisfies the $(p,q)$property for some $q \geq d+1$, can be pierced by a fixed number $f_d(p,q)$ of points. The minimum such piercing number is denoted by $HD_d(p,q)$. Already in 1957, Hadwiger and Debrunner showed that whenever $q>\frac{d1}{d}p+1$ the piercing number is $HD_d(p,q)=pq+1$; no exact values of $HD_d(p,q)$ were found ever since. While for an arbitrary family of compact convex sets in $\mathbb{R}^d$, $d \geq 2$, a $(p,2)$property does not imply a bounded piercing number, such bounds were proved for numerous specific families. The beststudied among them is axisparallel rectangles in the plane. Wegner and (independently) Dol'nikov used a $(p,2)$theorem for axisparallel rectangles to show that $HD_{\mathrm{rect}}(p,q)=pq+1$ holds for all
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We study an evolution equation that is the gradient flow in the $2$Wasserstien metric of a nonconvex functional for densities in $\mathbb{R}^n$ with $n \geq 3$. Like the PatlackKellerSegel system on $\mathbb{R}^2$, this evolution equation features a competition between the dispersive effects of diffusion, and the accretive effects of a concentrating drift. We determine a parameter range in which the diffusion dominates, and all mass leaves any fixed compact subset of $\mathbb{R}^n$ at an explicit polynomial rate.
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Natural calamities and disasters disrupt the conventional communication setups and the wireless bandwidth becomes constrained. A safe and costeffective solution for communication and data access in such scenarios is long needed. LightFidelity (LiFi) which promises wireless access to data at high speeds using visible light can be a good option. Visible light being safe to use for wireless access in such affected environments also provides illumination. Importantly, when a LiFi unit is attached to an air balloon and a network of such LiFi balloons are coordinated to form a LiFi balloon network, data can be accessed anytime and anywhere required and hence many lives can be tracked and saved. We propose this idea of a LiFi balloon and give an overview of its design using the Philips LiFi hardware. Further, we propose the concept of a balloon network and coin it with an acronym, the LiBNet. We consider the balloons to be arranged as a homogeneous Poisson point process in the LiBNet
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We study integrals of the form $\int_{\Omega}f\left( d\omega_1 , \ldots , d\omega_m \right), $ where $m \geq 1$ is a given integer, $1 \leq k_{i} \leq n$ are integers and $\omega_{i}$ is a $(k_{i}1)$form for all $1 \leq i \leq m$ and $ f:\prod_{i=1}^m \Lambda^{k_i}\left( \mathbb{R}^{n}\right) \rightarrow\mathbb{R}$ is a continuous function. We introduce the appropriate notions of convexity, namely vectorial ext. one convexity, vectorial ext. quasiconvexity and vectorial ext. polyconvexity. We prove weak lower semicontinuity theorems and weak continuity theorems and conclude with applications to minimization problems. These results generalize the corresponding results in both classical vectorial calculus of variations and the calculus of variations for a single differential form.
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The paper is devoted to the development of a comprehensive calculus for directional limiting normal cones, subdifferentials and coderivatives in finite dimensions. This calculus encompasses the whole range of the standard generalized differential calculus for (nondirectional) limiting notions and relies on very weak (nonrestrictive) qualification conditions having also a directional character. The derived rules facilitate the application of tools exploiting the directional limiting notions to difficult problems of variational analysis including, for instance, various stability and sensitivity issues. This is illustrated by some selected applications in the last part of the paper.
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Given an element of the Bloch group of a number field~$F$ and a natural number~$m$, we construct an explicit unit in the field $F_m=F(e^{2 \pi i/m})$, welldefined up to $m$th powers of nonzero elements of~$F_m$. The construction uses the cyclic quantum dilogarithm, and under the identification of the Bloch group of~$F$ with the $K$group $K_3(F)$ gives an explicit formula for a certain abstract Chern class from~$K_3(F)$. The units we define are conjectured to coincide with numbers appearing in the quantum modularity conjecture for the Kashaev invariant of knots (which was the original motivation for our investigation), and also appear in the radial asymptotics of Nahm sums near roots of unity. This latter connection is used to prove one direction of Nahm's conjecture relating the modularity of certain $q$hypergeometric series to the vanishing of the associated elements in the Bloch group of~$\overline{\mathbb{Q}}$.
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A branching L\'evy process can be seen as the continuoustime version of a branching random walk. It describes a particle system on the real line in which particles move and reproduce independently one of the others, in a Poissonian manner. Just as for L\'evy processes, the law of a branching L\'evy process is determined by its characteristic triplet $(\sigma^2,a,\Lambda)$, where the L\'evy measure $\Lambda$ describes the intensity of the Poisson point process of births and jumps. We establish a version of Biggins' theorem on martingale convergence for branching random walks in this framework. That is we provide necessary and sufficient conditions in terms of the characteristic triplet $(\sigma^2,a,\Lambda)$ for additive martingales of branching L\'evy processes to have a nondegenerate limit. The proof is adapted from the spinal decomposition argument of Lyons.
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In this paper, we show that, if a group $G$ acts geometrically on a geodesically complete CAT(0) space $X$ which contains at least one point with a CAT(1) neighborhood, then $G$ must be either virtually cyclic or acylindrically hyperbolic. As a consequence, the fundamental group of a compact Riemannian manifold whose sectional curvature is nonpositive everywhere and negative in at least one point is either virtually cyclic or acylindrically hyperbolic. This statement provides a precise interpretation of an idea expressed by Gromov in his paper Asymptotic invariants of infinite groups.
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Frustration index is a key measure for analysing signed networks that has been underused due to its computational complexity. We use an optimisationbased method to analyse frustration as a global structural property of signed networks under different contexts. We provide exact numerical results on social and biological signed networks as well as networks of formal alliances and antagonisms between countries and financial portfolio networks. Molecular graphs of carbon and spin glass models are among other networks that we consider to discuss contextdependent interpretations of the frustration index. The findings unify the applications of a graphtheoretical measure in understanding signed networks. Highlights:  Models involving signed networks in 6 different disciplines are analysed  An optimisation model is used for a wide range of computational experiments  Values of the frustration index in large graphs are computed exactly and efficiently  Applications of signed network balanc
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The aim of this paper is to give the characterisation of the L p Range (p $\ge$ 2) of the Generalised Poisson Transform of the Hyperbolic space B(H n), (n $\ge$ 2), over the classical field of the quaternions H. Namely, if f is an hyperfunction in the boundary of B(H n), then we show that f is in L p ($\partial$B(H n)) if and only if it's generalised poisson transform satisfy an Hardy type growth condition. An explicit expression of the generalized spherical functions is given. Mathematics Subject Classification (2010). Primary 22E46; Secondary 33Cxx.
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We establish a new characterization of the Choquet order on the space of probability measures on a compact convex set. This characterization is dilationtheoretic, meaning that it relates to the representation theory of positive linear maps on the C*algebra of continuous functions on the set. We develop this connection between Choquet theory and the theory of operator algebras, and utilize it to establish Arveson's hyperrigidity conjecture for function systems. This yields a significant strengthening of \v{S}a\v{s}kin's approximation theorem for positive maps on commutative C*algebras that is valid in the nonmetrizable setting and does not require the range of the maps to be commutative. We also obtain an extension of Cartier's theorem on dilation of measures that is valid in the nonmetrizable setting.
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This paper considers a twohop network architecture known as a combination network, where a layer of relay nodes connects a server to a set of end users. In particular, a new model is investigated where the intermediate relays employ caches in addition to the end users. First, a new centralized coded caching scheme is developed that utilizes maximum distance separable (MDS) coding, jointly optimizes cache placement and delivery phase, and enables decomposing the combination network into a set virtual multicast subnetworks. It is shown that if the sum of the memory of an end user and its connected relay nodes is sufficient to store the database, then the server can disengage in the delivery phase and all the end users' requests can be satisfied by the caches in the network. Lower bounds on the normalized delivery load using genieaided cutset arguments are presented along with second hop optimality. Next recognizing the information security concerns of coded caching, this new model is
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We use methods for computing Picard numbers of reductions of K3 surfaces in order to study the decomposability of Jacobians over number fields and the variance of MordellWeil ranks of families of Jacobians over different ground fields. For example, we give examples of surfaces whose Picard numbers jump in rank at all primes of good reduction using MordellWeil groups of Jacobians and show that the genus of curves over number fields whose Jacobians are isomorphic to a product of elliptic curves satisfying certain reduction conditions is bounded. The isomorphism result addresses the number field analogue of some questions of Ekedahl and Serre on decomposability of Jacobians of curves into elliptic curves.
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Differential calculus on Euclidean spaces has many generalisations. In particular, on a set $X$, a diffeological structure is given by maps from open subsets of Euclidean spaces to $X$, a differential structure is given by maps from $X$ to $\mathbb{R}$, and a Fr\"{o}licher structure is given by maps from $\mathbb{R}$ to $X$ as well as maps from $X$ to $\mathbb{R}$. We illustrate the relations between these structures through examples.
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Until recently, the only known method of finding the roots of polynomials over prime power rings, other than fields, was brute force. One reason for this is the lack of a division algorithm, obstructing the use of greatest common divisors. Fix a prime $p \in \mathbb{Z}$ and $f \in ( \mathbb{Z}/p^n \mathbb{Z} ) [x]$ any nonzero polynomial of degree $d$ whose coefficients are not all divisible by $p$. For the case $n=2$, we prove a new efficient algorithm to count the roots of $f$ in $\mathbb{Z}/p^2\mathbb{Z}$ within time polynomial in $(d+\operatorname{size}(f)+\log{p})$, and record a concise formula for the number of roots, formulated by Cheng, Gao, Rojas, and Wan.
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The main theme of this thesis is the development of computational methods for classes of infinitedimensional optimization problems arising in optimal control and information theory. The first part of the thesis is concerned with the optimal control of discretetime continuous space Markov decision processes (MDP). The second part is centred around two fundamental problems in information theory that can be expressed as optimization problems: the channel capacity problem as well as the entropy maximization subject to moment constraints.
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Let L be an ample line bundle on a smooth projective variety X over a nonarchimedean field K. For a continuous metric on L, we show in the following two cases that the semipositive envelope is a continuous semipositive metric on L and that the nonarchimedean MongeAmp\`ere equation has a solution. First, we prove it for curves using results of Thuillier. Second, we show it under the assumption that X is a surface defined geometrically over the function field of a curve over a perfect field k of positive characteristic. The second case holds in higher dimensions if we assume resolution of singularities over k. The proof follows a strategy from Boucksom, Favre and Jonsson, replacing multiplier ideals by test ideals. Finally, the appendix by Burgos and Sombra provides an example of a semipositive metric whose retraction is not semipositive. The example is based on the construction of a toric variety which has two SNCmodels which induce the same skeleton but different retraction maps.
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Nous rappelons des versions \covs de la th\'eorie de la dimension de Krull dans les anneaux commutatifs et dans les \trdisz, dont les bases ont \'et\'e pos\'ees par Joyal, Espan\~ol et les deux auteurs. Nous montrons sur les exemples de la dimension des alg\`ebres \pfz, du Going Up, du Going Down, % du Principal Ideal Theorem, \ldots que cela nous permet de donner une version \cov des grands th\'eor\`emes classiques, et par cons\'equent de r\'ecup\'erer un contenu calculatoire explicite lorsque ces th\'eor\`emes abstraits sont utilis\'es pour d\'emontrer l'existence d'objets concrets. Nous pensons ainsi mettre en oeuvre une r\'ealisation partielle du programme de Hilbert pour l'alg\`ebre abstraite classique. We present constructive versions of Krull's dimension theory for commutative rings and distributive lattices. The foundations of these constructive versions are due to Joyal, Espan\~ol and the authors. We show that this gives a constructive version of basic classical theorems (dime
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We show that the only projective smooth rational surfaces which admit a constant scalar curvature K\"ahler metric for all polarisations are the projective plane and the quadric surface. In particular, we show that all rational surfaces other than those two admit a destabilising slope test configuration, as introduced by Ross and Thomas. We further show that all Hirzebruch surfaces other than the quadric surface and all rational surfaces with Picard rank $3$ do not admit a constant scalar curvature K\"ahler metric for any polarisation.
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We investigate the combinatorial game Slime Trail.This game is played on a graph with a starting piece in a node. Each player's objective is to reach one of their own goal nodes. Every turn the current player moves the piece and deletes the node they came from. We show that the game is PSPACEcomplete when played on a planar graph.
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Of concern is the study of the space of curves in homogeneous spaces. Motivated by applications in shape analysis we identify two curves if they only differ by their parametrization and/or a rigid motion. For curves in Euclidean space the SquareRootVelocityFunction (SRVF) allows to define and efficiently compute a distance on this infinite dimensional quotient space. In this article we present a generalization of the SRVF to curves in homogeneous spaces. We prove that, under mild conditions on the curves, there always exist optimal reparametrizations realizing the quotient distance and demonstrate the efficiency of our framework in selected numerical examples.
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In this paper, we consider a concentration of measure problem on Riemannian manifolds with boundary. We study concentration phenomena of nonnegative $1$Lipschitz functions vanishing on the boundary. In order to capture such phenomena, we introduce a new invariant called the observable inscribed radius that measures the difference between such $1$Lipschitz functions and zero. We examine its basic properties, and formulate a comparison theorem under a lower Ricci curvature bound, and a lower mean curvature bound for the boundary.
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We propose a superfield method to construct offshell actions for Nextended conformal supergravity theories in three spacetime dimensions. It makes use of the superform technique to engineer supersymmetric invariants. The method is specifically applied to the case of N=1 conformal supergravity and provides a new realization for the actions of conformal and topologically massive supergravities.
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We generalize the classic convergence rate theory for subgradient methods to apply to nonLipschitz functions via a new measure of steepness. For the deterministic projected subgradient method, we derive a global $O(1/\sqrt{T})$ convergence rate for any function with at most exponential growth. Our approach implies generalizations of the standard convergence rates for gradient descent on functions with Lipschitz or H\"older continuous gradients. Further, we show a $O(1/\sqrt{T})$ convergence rate for the stochastic projected subgradient method on functions with at most quadratic growth, which improves to $O(1/T)$ under strong convexity.
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This paper presents optimization issues of energy detection (ED) thresholds in cooperative spectrum sensing (CSS) with regard to general Gaussian noise. Enhanced ED thresholds are proposed to overcome sensitivity of multiple noise uncertainty. Twosteps decision pattern and convex samples thresholds have been put forward under Gaussian noise uncertainty. Through deriving the probability of detection (Pd) and the probability of false alarm (Pf ) for independent and identical distribution (i.i.d.) SUs, we obtain lower total error rate (Qe) with proposed ED thresholds at low signaltonoiseratio (SNR) condition. Furthermore, simulation results show that proposed schemes outperform most other noise uncertainty plans.
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We fix a counting function of multiplicities of algebraic points in a projective hypersurface over a number field, and take the sum over all algebraic points of bounded height and fixed degree. An upper bound for the sum with respect to this counting function will be given in terms of the degree of the hypersurface, the dimension of the singular locus, the upper bounds of height, and the degree of the field of definition.
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Let $G_{\mathbb{R}}$ be a simple real linear Lie group with maximal compact subgroup $K_{\mathbb{R}}$ and assume that ${\rm rank}(G_\mathbb{R})={\rm rank}(K_\mathbb{R})$. In \cite{MPVZ} we proved that for any representation $X$ of GelfandKirillov dimension $\frac{1}{2}\dim(G_{\mathbb{R}}/K_{\mathbb{R}})$, the polynomial on the dual of a compact Cartan subalgebra given by the dimension of the Dirac index of members of the coherent family containing $X$ is a linear combination, with integer coefficients, of the multiplicities of the irreducible components occurring in the associated cycle. In this paper we compute these coefficients explicitly.
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The interleaving distance is arguably the most prominent distance measure in topological data analysis. In this paper, we provide bounds on the computational complexity of determining the interleaving distance in several settings. We show that the interleaving distance is NPhard to compute for persistence modules valued in the category of vector spaces. In the specific setting of multidimensional persistent homology we show that the problem is at least as hard as a matrix invertibility problem. Persistence modules valued in the category of sets are also studied. As a corollary, we obtain that the isomorphism problem for Reeb graphs is graph isomorphism complete.
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This paper is intended as the first step of a programme aiming to prove in the long run the longconjectured closedness under holomorphic deformations of compact complex manifolds that are bimeromorphically equivalent to compact K\"ahler manifolds, known as Fujiki {\it class} ${\cal C}$ manifolds. Our main idea is to explore the link between the {\it class} ${\cal C}$ property and the closed positive currents of bidegree $(1,\,1)$ that the manifold supports, a fact leading to the study of semicontinuity properties under deformations of the complex structure of the dual cones of cohomology classes of such currents and of Gauduchon metrics. Our main finding is a new class of compact complex, possibly nonK\"ahler, manifolds defined by the condition that every Gauduchon metric be strongly Gauduchon (sG), or equivalently that the Gauduchon cone be small in a certain sense. We term them sGG manifolds and find numerical characterisations of them in terms of certain relations between various
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We give a characterization of common zeros of a sequence of univariate polynomials $W_n(z)$ defined by a recurrence of order two with polynomial coefficients, and with $W_0(z)=1$. Real common zeros for such polynomials with real coefficients are studied further. This paper contributes to the study of root distribution of recursive polynomial sequences.
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We know that for a finite field $F$, every function on $F$ can be given by a polynomial with coefficients in $F$. What about the converse? i.e. if $R$ is a ring (not necessarily commutative or with unity) such that every function on $R$ can be given by a polynomial with coefficients in $R$, can we say $R$ is a finite field ? We show that the answer is yes, and that in fact it is enough to only require that all bijections be given by polynomials. If we allow our rings to have unity, we show that the property that all characteristic functions can be given by polynomials actually characterizes finite fields and if we moreover allow our rings to be commutative, then to characterize finite fields, it is enough that some special characteristic function be given by a polynomial (with coefficients even in some extension ring). Motivated by this, we determine all commutative rings with unity which admits a characteristic function which can be given by some polynomial with coefficients in the ri
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We propose an extension of a special form of gradient descent  in the literature known as linearised Bregman iteration  to a larger class of nonconvex functionals. We replace the classical (squared) two norm metric in the gradient descent setting with a generalised Bregman distance, based on a proper, convex and lower semicontinuous functional. The proposed algorithm is a generalisation of numerous wellknown optimisation methods. Its global convergence is proven for functions that satisfy the Kurdyka\L ojasiewicz property. Examples illustrate that for suitable choices of Bregman distances this method  in contrast to traditional gradient descent  allows iterating along regular solutionpaths. The effectiveness of the linearised Bregman iteration in combination with early stopping is illustrated for the applications of parallel magnetic resonance imaging, blind deconvolution as well as image classification.
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Let $\L$ be a Schr\"odinger operator of the form $\L=\Delta+V$ acting on $L^2(\mathbb R^n)$ where the nonnegative potential $V$ belongs to the reverse H\"older class $B_q$ for some $q\geq n.$ Let $L^{p,\lambda}(\mathbb{R}^{n})$, $0\le \lambda<n$ denote the Morrey space on $\mathbb{R}^{n}$. In this paper, we will show that a function $f\in L^{2,\lambda}(\mathbb{R}^{n})$ is the trace of the solution of ${\mathbb L}u=u_{t}+{\L}u=0, u(x,0)= f(x),$ where $u$ satisfies a Carlesontype condition \begin{eqnarray*} \sup_{x_B, r_B} r_B^{\lambda}\int_0^{r_B^2}\int_{B(x_B, r_B)} \nabla u(x,t)^2 {dx dt} \leq C <\infty. \end{eqnarray*} Conversely, this Carlesontype condition characterizes all the ${\mathbb L}$carolic functions whose traces belong to the Morrey space $L^{2,\lambda}(\mathbb{R}^{n})$ for all $0\le \lambda<n$. This result extends the analogous characterization founded by Fabes and Neri for the classical BMO space of John and Nirenberg.
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In earlier work (arXiv:1707.04927) the authors obtained formulas for the probability in the asymmetric simple exclusion process that at time $t$ a particle is at site $x$ and is the beginning of a block of $L$ consecutive particles. Here we consider asymptotics. Specifically, for the KPZ regime with step initial condition, we determine the conditional probability (asymptotically as $t\rightarrow\infty$) that a particle is the beginning of an $L$block, given that it is at site $x$ at time $t$. Using duality between occupied and unoccupied sites we obtain the analogous result for a gap of $G$ unoccupied sites between the particle at $x$ and the next one.
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It is known that for a conditional quasigreedy basis $\mathcal{B}$ in a Banach space $\mathbb{X}$, the associated sequence $(k_{m}[\mathcal{B}])_{m=1}^{\infty}$ of its conditionality constants verifies the estimate $k_{m}[\mathcal{B}]=\mathcal{O}(\log m)$ and that if the reverse inequality $\log m =\mathcal{O}(k_m[\mathcal{B}])$ holds then $\mathbb{X}$ is nonsuperreflexive. However, in the existing literature one finds very few instances of nonsuperreflexive spaces possessing quasigreedy basis with conditionality constants as large as possible. Our goal in this article is to fill this gap. To that end we enhance and exploit a combination of techniques developed independently, on the one hand by Garrig\'os and Wojtaszczyk in [Conditional quasigreedy bases in Hilbert and Banach spaces, Indiana Univ. Math. J. 63 (2014), no. 4, 10171036] and, on the other hand, by Dilworth et al. in [On the existence of almost greedy bases in Banach spaces, Studia Math. 159 (2003), no. 1, 67101], an
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We describe in this survey several results relating Fractal Geometry, Dynamical Systems and Diophantine Approximations, including a description of recent results related to geometrical properties of the classical Markov and Lagrange spectra and generalizations in Dynamical Systems and Differential Geometry.
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Receiver diversity methods play a key role in combating the detrimental effects of fading in wireless communication and other applications. Commonly used linear diversity methods include maximal ratio combining, equal gain combining and antenna selection combining. A novel linear combining method is proposed where a universal orthogonal dimensionreducing spacetime transformation is applied prior to quantization of the signals. The scheme may be considered as the counterpart of Alamouti modulation, and more generally of orthogonal spacetime block codes. The scheme is wellsuited to reducedcomplexity multiple receiveantenna analogtodigital conversion of narrowband signals. It also provides a method to achieve diversityenhanced relaying of communication signals, for multiuser detection at a remote terminal, minimizing the required bandwidth used in the links between relays and terminal.
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In our recent publication we have proposed a new methodology for determination of the twoterm Machinlike formula for pi with small arguments of the arctangent function of kind $$ \frac{\pi }{4} = {2^{k  1}}\arctan \left( {\frac{1}{{{\beta _1}}}} \right) + \arctan \left( {\frac{1}{{{\beta _2}}}} \right), $$ where $k$ and ${\beta _1}$ are some integers and ${\beta _2}$ is a rational number, dependent upon ${\beta _1}$ and $k$. Although ${1/\left\beta _2\right}$ may be significantly smaller than ${1/\beta _1}$, the large numbers in the numerator and denominator of $\beta_2$ decelerate the computation. In this work we show how this problem can be effectively resolved by the NewtonRaphson iteration method.
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The discretetime Distributed Bayesian Filtering (DBF) algorithm is presented for the problem of tracking a target dynamic model using a timevarying network of heterogeneous sensing agents. In the DBF algorithm, the sensing agents combine their normalized likelihood functions in a distributed manner using the logarithmic opinion pool and the dynamic average consensus algorithm. We show that each agent's estimated likelihood function globally exponentially converges to an error ball centered on the joint likelihood function of the centralized multisensor Bayesian filtering algorithm. We rigorously characterize the convergence, stability, and robustness properties of the DBF algorithm. Moreover, we provide an explicit bound on the time step size of the DBF algorithm that depends on the timescale of the target dynamics, the desired convergence error bound, and the modeling and communication error bounds. Furthermore, the DBF algorithm for linearGaussian models is cast into a modified
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