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So I learned something new today. Back in the early and mid90s, IBM tried to build a PClike platform and ecosystem around its PowerPC processor. They called it the PowerPC Reference Platform, or PReP, and with it, you could build what were effectively PC clones with PowerPC processors, ready to run a number of operating systems, including AIX, Windows NT, OS/2, and Apple's failed Taligent project. None of this is news to me. What is news to me, however, is that aside from a number of desktop PReP machines, IBM also developed and sold a number of PReP laptops under the ThinkPad brand. Sometime in 1994, IBM started working on a prototype mobile system named Woodfield and designated as type 6020. Very little is known about this system; it was never officially announced or sold. On June 19, 1995, IBM announced the ThinkPad 850 and 820 (announcement letters 195178 and 195179, respectively) with a planned availability date of July 24, 1995. The ThinkPad 820 designation was typ
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The problems of partially observed major minor LQG and nonlinear mean field game (PO MM LQG MFG) problems where it is assumed the major agent's state is partially observed by each minor agent, and the major agent completely observes its own state have been analysed in the literature. In this paper, PO MM LQG MFG problems with general information patterns are studied where (i) the major agent has partial observations of its own state, and (ii) each minor agent has partial observations of its own state and the major agent's state. The assumption of partial observations by all agents leads to a new situation involving the recursive estimation by each minor agent of the major agent's estimate of its own state. For the general case of indefinite LQG MFG systems, the existence of $\epsilon$Nash equilibria together with the individual agents' control laws yielding the equilibria are established via the Separation Principle.
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The connection matrix is a powerful algebraic topological tool from Conley index theory that captures relationships between isolated invariant sets. Conley index theory is a topological generalization of Morse theory in which the connection matrix subsumes the role of the Morse boundary operator. Over the last few decades, the ideas of Conley have been cast into a purely computational form. In this paper we introduce a computational, categorical framework for the connection matrix theory. This contribution transforms the computational Conley theory into a computational homological theory for dynamical systems. More specifically, within this paper we have two goals: 1) We cast the connection matrix theory into appropriate categorical, homotopytheoretic language. We identify objects of the appropriate categories which correspond to connection matrices and may be computed within the computational Conley theory paradigm by using the technique of reductions. 2) We describe an algorithm for
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Traditionally, there are several polynomial algorithms for linear programming including the ellipsoid method, the interior point method and other variants. Recently, Chubanov [Chubanov, 2015] proposed a projection and rescaling algorithm, which has become a potentially \emph{practical} class of polynomial algorithms for linear feasibility problems and also for the general linear programming. However, the Chubanovtype algorithms usually perform much better on the infeasible instances than on the feasible instances in practice. To explain this phenomenon, we derive a new theoretical complexity bound for the infeasible instances based on the condition number, which shows that algorithms can indeed run much faster on infeasible instances in certain situations. In order to speed up the feasible instances, we propose a \emph{Polynomialtime PrimalDual Projection} algorithm (called PPDP) by explicitly developing the dual algorithm. The numerical results validate that our PPDP algorithm achi
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We consider solving the surface Helmholtz equation on a smooth two dimensional surface embedded into a three dimensional space meshed with tetrahedra. The mesh does not respect the surface and thus the surface cuts through the elements. We consider a Galerkin method based on using the restrictions of continuous piecewise linears defined on the tetrahedra to the surface as trial and test functions.Using a stabilized method combining Galerkin least squares stabilization and a penalty on the gradient jumps we obtain stability of the discrete formulation under the condition $h k < C$, where $h$ denotes the mesh size, $k$ the wave number and $C$ a constant depending mainly on the surface curvature $\kappa$, but not on the surface/mesh intersection. Optimal error estimates in the $H^1$ and $L^2$norms follow.
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The stable center conjecture asserts that the space of stable distributions in the Bernstein center of a reductive padic is closed under convolution. It is closely related to the notion of an Lpacket and endoscopy theory. We describe a categorical approach to the depth zero part of the conjecture. As an illustration of our method, we show that the Bernstein projector to the depth zero spectrum is stable.
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We study a categorical construction called the cobordism category, which associates to each Waldhausen category a simplicial category of cospans. We prove that this construction is homotopy equivalent to Waldhausen's $S_{\bullet}$construction and therefore it defines a model for Waldhausen $K$theory. As an example, we discuss this model for $A$theory and show that the cobordism category of homotopy finite spaces has the homotopy type of Waldhausen's $A(*)$. We also review the canonical map from the cobordism category of manifolds to $A$theory from this viewpoint.
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It has been discovered that linear codes may be described by binomial ideals. This makes it possible to study linear codes by commutative algebra and algebraic geometry methods. In this paper, we give a decoding algorithm for binary linear codes by utilizing the Groebner bases of the associated ideals.
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In this note, we establish an expression of the Loewner energy of a Jordan curve on the Riemann sphere in terms of Werner's measure on simple loops of SLE$_{8/3}$ type. The proof is based on a formula for the change of the Loewner energy under a conformal map that is reminiscent of the restriction properties derived for SLE processes.
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We design and analyze solution techniques for a linearquadratic optimal control problem involving the integral fractional Laplacian. We derive existence and uniqueness results, first order optimality conditions, and regularity estimates for the optimal variables. We propose two strategies to discretize the fractional optimal control problem: a semidiscrete approach where the control is not discretized  the socalled variational discretization approach  and a fully discrete approach where the control variable is discretized with piecewise constant functions. Both schemes rely on the discretization of the state equation with the finite element space of continuous piecewise polynomials of degree one. We derive a priori error estimates for both solution techniques. We illustrate the theory with twodimensional numerical tests.
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We analyse features of the patterns formed from a simple model for a martensitic phase transition. This is a fragmentation model that can be encoded by a general branching random walk. An important quantity is the distribution of the lengths of the interfaces in the pattern and we establish limit theorems for some of the asymptotics of the interface profile. We are also able to use a general branching process to show almost sure power law decay of the number of interfaces of at least a certain size. We discuss the numerical aspects of determining the behaviour of the density profile and power laws from simulations of the model.
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The Weinstein operator has several applications in pure and applied Mathematics especially in Fluid Mechanics and satisfies some uncertainty principles similar to the Euclidean Fourier transform. The aim of this paper is establish a generalization of uncertainty principles for Weinstein transform in $L_\alpha^p$norm. Firstly, we extend the HeisenbergPauliWeyl uncertainty principle to more general case. Then we establish three continuous uncertainty principles of concentration type. The first and the second uncertainty principles are $L_\alpha^p$ versions and depend on the sets of concentration $\Omega$ and $\Sigma$, and on the time function $\varphi$. However, the third uncertainty principle is also $L_\alpha^p$ version depends on the sets of concentration and he is independent on the band limited function $\varphi$. These $L_\alpha^p$DonohoStarktype inequalities generalize the results obtained in the case $p=q=2$.
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We describe main issues and design principles of an efficient implementation, tailored to recent generations of Nvidia Graphics Processing Units (GPUs), of an Algebraic Multigrid (AMG) preconditioner previously proposed by one of the authors and already available in the opensource package BootCMatch: Bootstrap algebraic multigrid based on Compatible weighted Matching for standard CPU. The AMG method relies on a new approach for coarsening sparse symmetric positive definite (spd) matrices, named "coarsening based on compatible weighted matching". It exploits maximum weight matching in the adjacency graph of the sparse matrix, driven by the principle of compatible relaxation, providing a suitable aggregation of unknowns which goes beyond the limits of the usual heuristics applied in the current methods. We adopt an approximate solution of the maximum weight matching problem, based on a recently proposed parallel algorithm, referred as the Suitor algorithm, and show that it allow us to o
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A path (cycle) in a $c$edgecolored multigraph is alternating if no two consecutive edges have the same color. The problem of determining the existence of alternating Hamiltonian paths and cycles in $2$edgecolored multigraphs is an $\mathcal{NP}$complete problem and it has been studied by several authors. In BangJensen and Gutin's book "Digraphs: Theory, Algorithms and Applications", it is devoted one chapter to survey the last results on this topic. Most results on the existence of alternating Hamiltonian paths and cycles concern on complete and bipartite complete multigraphs and a few ones on multigraphs with high monochromatic degrees or regular monochromatic subgraphs. In this work, we use a different approach imposing local conditions on the multigraphs and it is worthwhile to notice that the class of multigraphs we deal with is much larger than, and includes, complete multigraphs, and we provide a full characterization of this class. Given a $2$edgecolored multigraph $G$,
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We use BrascampLieb's inequality to obtain new decoupling inequalities for general Gaussian vectors, and for stationary cyclic Gaussian processes. In the second case, we use a version by Bump and Diaconis of the strong Szego limit theorem. This extends results of Klein, Landau and Shucker.
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Let $(X,{\mathcal A},\mu)$ be a probability space and let $S\colon X\to X$ be a measurable transformation. Motivated by the paper of K. Nikodem [Czechoslovak Math. J. 41(116) (4) (1991) 565569], we concentrate on a functional equation generating measures that are absolutely continuous with respect to $\mu$ and $\varepsilon$invariant under $S$. As a consequence of the investigation, we obtain a result on the existence and uniqueness of solutions $\varphi\in L^1([0,1])$ of the functional equation $$ \varphi(x)=\sum_{n=1}^{N}f_n'(x)\varphi(f_n(x))+g(x), $$ where $g\in L^1([0,1])$ and $f_1,\ldots,f_N\colon[0,1]\to[0,1]$ are functions satisfying some extra conditions.
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Let $p\equiv 4,7\mod 9$ be a rational prime number such that $3\mod p$ is not a cubic residue. In this paper we prove the 3part of the product of the full BSD conjectures for $E_p$ and $E_{3p^3}$ is true using an explicit GrossZagier formula, where $E_p: x^3+y^3=p$ and $E_{3p^2}: x^3+y^3=3p^2$ are the elliptic curves related to the Sylvester conjecture and cube sum problems.
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In this paper we obtain a very general GaussGreen formula for weakly differentiable functions and sets of finite perimeter. This result is obtained by revisiting Anzellotti's pairing theory and by characterizing the measure pairing $(\boldsymbol{A}, Du)$ when $\boldsymbol{A}$ is a bounded divergence measure vector field and $u$ is a bounded function of bounded variation.
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Given an open subset $\Omega$ of a Banach space and a Lipschitz function $u_0: \overline{\Omega} \to \mathbb{R},$ we study whether it is possible to approximate $u_0$ uniformly on $\Omega$ by $C^k$smooth Lipschitz functions which coincide with $u_0$ on the boundary $\partial \Omega$ of $\Omega$ and have the same Lipschitz constant as $u_0.$ As a consequence, we show that every $1$Lipschitz function $u_0: \overline{\Omega} \to \mathbb{R},$ defined on the closure $\overline{\Omega}$ of an open subset $\Omega$ of a finite dimensional normed space of dimension $n \geq 2$, and such that the Lipschitz constant of the restriction of $u_0$ to the boundary of $\Omega$ is less than $1$, can be uniformly approximated by differentiable $1$Lipschitz functions $w$ which coincide with $u_0$ on $\partial \Omega$ and satisfy the equation $\ D w\_* =1$ almost everywhere on $\Omega.$ This result does not hold in general without assumption on the restriction of $u_0$ to the boundary of $\Omega$.
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We prove, using variational methods, the existence in dimension two of positive vector ground states solutions for the BoseEinstein type systems \begin{equation} \begin{cases} \Delta u+\lambda_1u=\mu_1u(e^{u^2}1)+\beta v\left(e^{uv}1\right) \text{ in } \Omega, &\\ \Delta v+\lambda_2v=\mu_2v(e^{v^2}1)+\beta u\left(e^{uv}1\right)\text{ in } \Omega, &\\ u,v\in H^1_0(\Omega) \end{cases} \end{equation} where $\Omega$ is a bounded smooth domain, $\lambda_1,\lambda_2>\Lambda_1$ (the first eigenvalue of $(\Delta,H^1_0(\Omega))$, $\mu_1,\mu_2>0$ and $\beta$ is either positive (small or large) or negative (small). The nonlinear interaction between two Bose fluids is assumed to be of critical exponential type in the sense of J. Moser. For `small' solutions the system is asymptotically equivalent to the corresponding one in higher dimensions with powerlike nonlinearities.
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We consider the problem of private linear computation (PLC) in a distributed storage system. In PLC, a user wishes to compute a linear combination of $f$ messages stored in noncolluding databases while revealing no information about the coefficients of the desired linear combination to the databases. In extension of our previous work we employ linear codes to encode the information on the databases. We show that the PLC capacity, which is the ratio of the desired linear function size and the total amount of downloaded information, matches the maximum distance separable (MDS) coded capacity of private information retrieval for a large class of linear codes that includes MDS codes. In particular, the proposed converse is valid for any number of messages and linear combinations, and the capacity expression depends on the rank of the coefficient matrix obtained from all linear combinations.
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The performance enhancements observed in various models of continuous quantum thermal machines have been linked to the buildup of coherences in a preferred basis. But, is this connection always an evidence of 'quantumthermodynamic supremacy'? By force of example, we show that this is not the case. In particular, we compare a powerdriven threelevel quantum refrigerator with a fourlevel combined cycle, partly driven by power and partly by heat. We focus on the weak driving regime and find the fourlevel model to be superior since it can operate in parameter regimes in which the threelevel model cannot, it may exhibit a larger cooling rate, and, simultaneously, a better coefficient of performance. Furthermore, we find that the improvement in the cooling rate matches the increase in the stationary quantum coherences exactly. Crucially, though, we also show that the thermodynamic variables for both models follow from a classical representation based on graph theory. This implies that w
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A successive cancellation list (SCL) decoder with limited list size for polar codes can not be analyzed as a successive cancellation (SC) decoder, nor as a maximum likelihood (ML) decoder, due to the complicated decoding errors caused by path elimination. To address this issue, an analytical tool, named as cluster pairwise error probability (CPEP), is proposed in this paper to measure the competitiveness of the correct path against the error paths in an SCL decoder. It is shown that the sum of CPEPs over error paths could be used as an indicator of the probability of correct path being eliminated from the decoder list. Then, we use CPEP to explain the error performance gain of paritycheckconcatenated (PCC) polar code, and apply CPEP as the optimization criterion in the construction of PCC polar codes, aiming to reduce the elimination probability of the correct path in an SCL decoder with limited list size. Simulation results show that the constructed CRCPCC polar codes outperform th
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We study numerical invariants of identities of finitedimensional solvable Lie superalgebras. We define new series of finitedimensional solvable Lie superalgebras $L$ with nonnilpotent derived subalgebra $L'$ and discuss their codimension growth. For the first algebra of this series we prove the existence and integrality of $exp(L)$.
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We study a natural growth process with competition, which was recently introduced to analyze MDLA, a challenging model for the growth of an aggregate by diffusing particles. The growth process consists of two firstpassage percolation processes $\text{FPP}_1$ and $\text{FPP}_\lambda$, spreading with rates $1$ and $\lambda>0$ respectively, on a graph $G$. $\text{FPP}_1$ starts from a single vertex at the origin $o$, while the initial configuration of $\text{FPP}_\lambda$ consists of infinitely many \emph{seeds} distributed according to a product of Bernoulli measures of parameter $\mu>0$ on $V(G)\setminus \{o\}$. $\text{FPP}_1$ starts spreading from time 0, while each seed of $\text{FPP}_\lambda$ only starts spreading after it has been reached by either $\text{FPP}_1$ or $\text{FPP}_\lambda$. A fundamental question in this model, and in growth processes with competition in general, is whether the two processes coexist (i.e., both produce infinite clusters) with positive probabilit
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Electrical impedance tomography (EIT) is highly affected by modeling errors regarding electrode positions and the shape of the imaging domain. In this work, we propose a new inclusion detection technique that is completely independent of such errors. Our new approach is based on a combination of frequencydifference and ultrasound modulated EIT measurements.
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Lie group methods are applied to the timedependent, monoenergetic neutron diffusion equation in materials with spatial and time dependence. To accomplish this objective, the underlying 2nd order partial differential equation (PDE) is recast as an exterior differential system so as to leverage the isovector symmetry analysis approach. Some of the advantages of this method as compared to traditional symmetry analysis approaches are revealed through its use in the context of a 2nd order PDE. In this context, various material properties appearing in the mathematical model (e.g., a diffusion coefficient and macroscopic cross section data) are left as arbitrary functions of space and time. The symmetry analysis that follows is restricted to a search for translation and scaling symmetries; consequently the Lie derivative yields specific material conditions that must be satisfied in order to maintain the presence of these important similarity transformations. The principal outcome of this wor
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Graded bundles are a particularly nice class of graded manifolds and represent a natural generalisation of vector bundles. By exploiting the formalism of supermanifolds to describe Lie algebroids we define the notion of a weighted $A$connection on a graded bundle. In a natural sense weighted $A$connections are adapted to the basic geometric structure of a graded bundle in the same way as linear $A$connections are adapted to the structure of a vector bundle. This notion generalises directly to multigraded bundles and in particular we present the notion of a biweighted $A$connection on a double vector bundle. We prove the existence of such adapted connections and use them to define (quasi)actions of Lie algebroids on graded bundles.
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We give a constructive proof of the existence of the almost revlex ideal $J\subset K[x_1,\dots,x_n]$ with the same Hilbert function of a complete intersection defined by $n$ forms of degrees $d_1\leq \dots \leq d_n$, when for every $i\geq 4$ the degrees satisfy the condition $d_i\geq \bar u_{i1}+1=\min\Bigl\{\Big\lfloor\frac{\sum_{j=1}^{i1}d_ji+1}{2}\Big\rfloor, \sum_{j=1}^{i2} d_ji+2\Bigr\}+1$. The further property that, for every $t\geq \bar u_n+1$, all terms of degree $t$ outside $J$ are divisible by the last variable has an important role in our inductive and constructive proof, which is different from the more general construction given by Pardue in 2010.
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We prove the existence of a subsonic weak solution $({\bf u}, \rho, p)$ to steady Euler system in a twodimensional infinitely long nozzle when prescribing the value of the entropy $(= \frac{p}{\rho^{\gamma}})$ at the entrance by a piecewise $C^2$ function with a discontinuity at a point. Due to the variable entropy condition with a discontinuity at the entrance, the corresponding solution has a nonzero vorticity and contains a contact discontinuity $x_2=g_D(x_1)$. We construct such a solution via Helmholtz decomposition. The key step is to decompose the RankineHugoniot conditions on the contact discontinuity via Helmholtz decomposition so that the compactness of approximated solutions can be achieved. Then we apply the method of iteration to obtain a piecewise smooth subsonic flow with a contact discontinuity and nonzero vorticity. We also analyze the asymptotic behavior of the solution at far field.
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This work serves as a primer to our efforts in arriving at convergence estimates for the fixed stress split iterative scheme for single phase flow coupled with small strain anisotropic poroelastoplasticity. The fixed stress split iterative scheme solves the flow subproblem with stress tensor fixed using a mixed finite element method, followed by the poromechanics subproblem using a conforming Galerkin method in every coupling iteration at each time step. The coupling iterations are repeated until convergence and Backward Euler is employed for time marching. The convergence analysis is based on studying the equations satisfied by the difference of iterates to show that the iterative scheme is contractive.
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We compute the asymptotic growth rate of the number N(C, R) of closed geodesics of length less than R in a connected component C of a stratum of quadratic differentials. We prove that for any 0 < \theta < 1, the number of closed geodesics of length at most R that spend at least \thetafraction of time outside of a compact subset of C is exponentially smaller than N(C, R). The theorem follows from a lattice counting statement. For points x, y in the moduli space M of Riemann surfaces, and for 0 < \theta < 1, we find an upperbound for the number of geodesic paths of length less than R in C which connect a point near x to a point near y and spend a \thetafraction of the time outside of a compact subset of C.
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We prove that under certain conditions, phase separation is enough to sustain a regime in which current flows along the concentration gradient, a phenomenon which is known in the literature as \textit{uphill diffusion}. The model we consider here is a version of that proposed in [G. B. Giacomin, J. L. Lebowitz, Phase segregation dynamics in particle system with long range interactions, Journal of Statistical Physics 87(1) (1997): 3761], which is the continuous mesoscopic limit of a $1d$ discrete Ising chain with a Kac potential. The magnetization profile lies in the interval $\left[\varepsilon^{1},\varepsilon^{1}\right]$, $\varepsilon>0$, staying in contact at the boundaries with infinite reservoirs of fixed magnetization $\pm\mu$, $\mu\in(m^*\left(\beta\right),1)$, where $m^*\left(\beta\right)=\sqrt{11/\beta}$, $\beta>1$ representing the inverse temperature. At last, an external field of Heavisidetype of intensity $\kappa>0$ is introduced. According to the axiomatic non
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We introduce the cycle intersection graph of a graph, an adaptation of the cycle graph of a graph, and use the structure of these graphs to prove an upper bound for the decycling number of all even graphs. This bound is shown to be significantly better when an even graph admits a cycle decomposition in which any two cycles intersect in at most one vertex. Links between the cycle rank of the cycle intersection graph of an even graph and the decycling number of the even graph itself are found. The problem of choosing an ideal cycle decomposition is addressed and is presented as an optimization problem over the space of cycle decompositions of even graphs.
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This paper is devoted for classifying all capable Heisenberg Lie superalgebras where we have shown that there exists at least one capable Lie superalgebra of corank $\geq 4$. This paper can be thought up as a super symmetric extension of a recent result by Peyman Niroomand, Mohesen Parvizi, Francesco G. Russo [20] who classify all capable Heisenberg Lie algebras where they have shown that there exists atleast one capable Lie algebra of arbitrary corank.
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Let $U^_q = U^_q(\mathfrak g)$ be the negative part of the quantum group associated to a finite dimensional simple Lie algebra $\mathfrak g$, and $\sigma : \mathfrak g \to \mathfrak g$ be the automorphism obtained from the diagram automorphism. Let $\mathfrak g^{\sigma}$ be the fixed point subalgebra of $\mathfrak g$, and put $\underline U^_q = U^_q(\mathfrak g^{\sigma})$. Let $B$ be the canonical basis of $U_q^$ and $\underline B$ the canonical basis of $\underline U_q^$. $\sigma$ induces a natural action on $B$, and we denote by $B^{\sigma}$ the set of $\sigma$fixed elements in $B$. Lusztig proved that there exists a canonical bijection $B^{\sigma} \simeq \underline B$ by using geometric considerations. In this paper, we construct such a bijection in an elementary way. We also consider such a bijection in the case of certain affine quantum groups, by making use of PBWbases constructed by Beck and Nakajima.
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We consider a second order differential operator $\mathscr{A}$ on an (typically unbounded) open and Dirichlet regular set $\Omega\subset \mathbb{R}^d$ and subject to nonlocal Dirichlet boundary conditions of the form \[ u(z) = \int_\Omega u(x)\mu (z, dx) \quad \mbox{ for } z\in \partial \Omega. \] Here, $\mu : \partial\Omega \to \mathscr{M}(\Omega)$ is a $\sigma (\mathscr{M}(\Omega), C_b(\Omega))$continuous map taking values in the probability measures on $\Omega$. Under suitable assumptions on the coefficients in $\mathscr{A}$, which may be unbounded, we prove that a realization $A_\mu$ of $\mathscr{A}$ subject to the nonlocal boundary condition, generates a (not strongly continuous) semigroup on $L^\infty(\Omega)$. We also establish a sufficient condition for this semigroup to be Markovian and prove that in this case, it enjoys the strong Feller property. We also study the asymptotic behavior of the semigroup.
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We study the distinguishability of a particular type of maximally entangled states  the "ququadququad" states which are tensor products of Bell states in $\mathbb{C}^4\otimes\mathbb{C}^4$. We first prove that any three orthogonal ququadququad maximally entangled states can be distinguished with LOCC. Then we use a new approach of semidefinite program to construct all sets of four ququadququad orthogonal maximally entangled states that are PPTindistinguishable and we find some interesting sets of six states having interesting property of distinguishability. Also, we show that our approach of the optimization problem can make some computational complex problem more tractable.
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It is the purpose of this article to outline a course that can be given to engineers looking for an understandable mathematical description of the foundations of distribution theory and the necessary functional analytic methods. Arguably, these are needed for a deeper understanding of basic questions in signal analysis. Objects such as the Dirac delta and Dirac comb require a proper definition, and it should be possible to explain how one can reconstruct a bandlimited function from its samples by means of simple series expansions. It should also be useful for graduate students who want to see how functional analysis can help to understand fairly practical problems, or teachers who want to offer a course related to the "Mathematical Foundations of Signal Processing". The course requires only an understanding of the basic terms from linear functional analysis, namely Banach spaces and their duals, bounded linear operators and a simple version of weak$^{*}$convergence. As a matter of fa
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This paper addresses the trajectory tracking control problem for underactuated VTOL UAVs. According to the different actuation mechanisms, the most common UAV platforms can achieve only a partial decoupling of attitude and position tasks. Since position tracking is of utmost importance for applications involving aerial vehicles, we propose a control scheme in which position tracking is the primary objective. To this end, this work introduces the concept of attitude planner, a dynamical system through which the desired attitude reference is processed to guarantee the satisfaction of the primary objective: the attitude tracking task is considered as a secondary objective which can be realized as long as the desired trajectory satisfies specific trackability conditions. Two numerical simulations are performed by applying the proposed control law to a hexacopter with and without tilted propellers, which accounts for unmodeled dynamics and external disturbances not included in the control d
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We give formulae for the multiplicities of eigenvalues of generalized rotation operators in terms of generalized FrobeniusSchur indicators in a semisimple spherical tensor category $\mathcal{C}$. In particular, this implies that the entire collection of rotation eigenvalues for a fusion category can be computed from the fusion rules and the traces of rotation at finitely many tensor powers. We also establish a rigidity property for FS indicators of fusion categories with a given fusion ring via Jones's theory of planar algebras. If $\mathcal{C}$ is also braided, these formulae yield the multiplicities of eigenvalues for a large class of braids in the associated braid group representations. When $\mathcal{C}$ is modular, this allows one to determine the eigenvalues and multiplicities of braids in terms of just the $S$ and $T$ matrices.
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We consider two models of a compressible inviscid isentropic twofluid flow. The first one describes the liquidgas twophase flow. The second one can describe the mixture of two fluids of different densities or the mixture of fluid and particles. Introducing an entropylike function, we reduce the equations of both models to a symmetric form which looks like the compressible Euler equations written in the nonconservative form in terms of the pressure, the velocity and the entropy. Basing on existing results for the Euler equations, this gives a number of instant results for both models. In particular, we conclude that all compressive shock waves in these models exist locally in time. For the 2D case, we make the conclusion about the localintime existence of vortex sheets under a "supersonic" stability condition. In the sense of a much lower regularity requirement for the initial data, our result for 2D vortex sheets essentially improves a recent result for vortex sheets in the liqui
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Let U(N) be the quasisplit unitary group in N variables for a quadratic unramified extension of padic fields. We compute the characters of simple supercuspidal representations of twisted GL(N) and U(N). Comparing them by the endoscopic character relation, we determine the liftings of simple supercuspidal representations of U(N) to GL(N), under the assumption that p is not equal to 2.
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In this paper, we study energyefficient resource allocation in distributed antenna system (DAS) with wireless power transfer, where timedivision multiple access (TDMA) is adopted for downlink multiuser information transmission. In particular, when a user is scheduled to receive information, other users harvest energy at the same time using the same radiofrequency (RF) signal. We consider two types of energy efficiency (EE) metrics: usercentric EE (UCEE) and networkcentric EE (NCEE). Our goal is to maximize the UCEE and NCEE, respectively, by optimizing the transmission time and power subject to the energy harvesting requirements of the users. For both UCEE and NCEE maximization problems, we transform the nonconvex problems into equivalently tractable problems by using suitable mathematical tools and then develop iterative algorithms to find the globally optimal solutions. Simulation results demonstrate the superiority of the proposed methods compared with the benchmark schem
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We study the problem of minimizing the energy function $M^p(m,n) := \min \sum_{1\le i<j\le m} \langle v_i, v_j\rangle^p$, where $v_i$ are unit vectors in $F^n$, $F=\mathbb R$ or $\mathbb C$, $m,n,p>0$ are integers and $p$ is even. This problem has implications on finding nice polyhedra in projective spaces, and on quantum random access codes. We conduct experimental search in the complex case which suggests nice patterns on the minimum values. In some cases($p=2$ and partially $n=2$) we supply analytical proofs and give full descriptions of the minimal configurations. We also show that as $m\to \infty$, nearly equidistributed configurations points nearly give the minimal values we expect from our patterns.
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In this work we combing models of disease dynamics and economic production, and we show the potential implications of this for demonstrating the importance of savings for buffering an economy during the period of an epidemic. Finding an explicit function that relates poverty and the production of a community is an almost impossible task because of the number of variables and parameters that should be taken into account. However, studying the dynamics of an endemic disease in a region that affects its population, and therefore its ability to work, is an honest approach to understanding this function. We propose a model, perhaps the simplest, that couples two dynamics, the dynamics of an endemic disease and the dynamics of a closed economy of products and goods that the community produces in the epidemic period. Some of the results of this study are expected and known in the literature but some others are not. We highlight three of them: the interdependence that exists between health and
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We introduce the extension groups between atoms in an abelian category. For a locally noetherian Grothendieck category, the localizing subcategories closed under injective envelopes are characterized in terms of those extension groups. We also introduce the virtual dual of the extension groups between atoms to measure the global dimension of the category. A new topological property of atom spectra is revealed and it is used to relate the projective dimensions of atoms with the KrullGabriel dimensions. As a byproduct of the topological observation, we show that there exists a spectral space that is not homeomorphic to the atom spectrum of any abelian category.
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We give a qualitative description of extremals for the simplest version of Morrey's inequality. Our theory is based on exploiting the invariances of this inequality, studying the equation satisfied by extremals and the observation that extremals are optimal for a related convex minimization problem.
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We study the problem of decomposing a nonnegative polynomial as an exact sum of squares (SOS) in the case where the associated semidefinite program is feasible but not strictly feasible (for example if the polynomial has real zeros). Computing symbolically roots of the original polynomial and applying facial reduction techniques, we can solve the problem algebraically or restrict to a subspace where the problem becomes strictly feasible and a numerical approximation can be rounded to an exact solution. As an application, we study the problem of determining when can a rational polynomial that is a sum of squares of polynomials with real coefficients be written as sum of squares of polynomials with rational coefficients, and answer this question for some previously unknown cases. We first prove that if $f$ is the sum of two squares with coefficients in an algebraic extension of ${\mathbb Q}$ of odd degree, then it can always be decomposed as a rational SOS. For the case of more than two
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Following the work of Altmann and Hausen we give a combinatorial description in terms for smooth Fano threefolds admitting a 2torus action. We show that a whole variety of properties and invariants can be read off from this description. As an application we prove and disprove the existence of KahlerEinstein metrics for some of these Fano threefolds, calculate their Cox rings and some of their toric canonical degenerations.
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Persistent homology is a popular data analysis technique that is used to capture the changing topology of a filtration associated with some simplicial complex $K$. These topological changes are summarized in persistence diagrams. We propose two contraction operators which when applied to $K$ and its associated filtration, bound the perturbation in the persistence diagrams. The first assumes that the underlying space of $K$ is a $2$manifold and ensures that simplices are paired with the same simplices in the contracted complex as they are in the original. The second is for arbitrary $d$complexes, and bounds the bottleneck distance between the initial and contracted $p$dimensional persistence diagrams. This is accomplished by defining interleaving maps between persistence modules which arise from chain maps defined over the filtrations. In addition, we show how the second operator can efficiently compose across multiple contractions. We conclude with experiments demonstrating the seco
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Let $\Gamma_{w}$ be a noncofinite Hecke triangle group with cusp width $w>2$ and let $\varrho\colon\Gamma_w\to U(V)$ be a finitedimensional unitary representation of $\Gamma_w$. In this note we announce a new fractal upper bound for the Selberg zeta function of $\Gamma_{w}$ twisted by $\varrho$. In strips parallel to the imaginary axis and bounded away from the real axis, the Selberg zeta function is bounded by $\exp\left( C_{\varepsilon} \vert s\vert^{\delta + \varepsilon} \right)$, where $\delta = \delta_{w}$ denotes the Hausdorff dimension of the limit set of $\Gamma_{w}$. This bound implies fractal Weyl bounds on the resonances of the Laplacian for all geometrically finite surfaces $X=\widetilde{\Gamma}\backslash\mathbb{H}$ where $\widetilde{\Gamma}$ is a finite index, torsionfree subgroup of $\Gamma_w$.
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We establish a functional limit theorem for joint laws of occupation time processes of infinite ergodic transformations, in the sense of strong distributional convergence. Our limit theorem is a functional and jointdistributional extention both of Aaronson's limit theorem of DarlingKac type, and of Thaler's generalized arcsine laws of Lamperti type, at the same time. We apply it to obtain a functional limit theorem for joint laws of sojourns of interval maps near and away from indifferent fixed points. For the proof, we represent occupation times in terms of excursion lengths, and show a functional convergence of excursion lengths to stable L\'evy processes.
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We consider the deformation theory of two kinds of geometric objects: foliations on one hand, presymplectic forms on the other. For each of them, we prove that the geometric notion of equivalence given by isotopies agrees with the algebraic notion of gauge equivalence obtained from the $L_{\infty}$algebras governing these deformation problems.
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In this note we construct a "restriction" map from the cocenter of a reductive group G over a local nonarchimedean field F to the cocenter of a Levi subgroup. We show that the dual map corresponds to parabolic induction and deduce that parabolic induction preserves stability. We also give a new (purely geometric) proof that the character of normalized parabolic induction does not depend on a parabolic subgroup. In the appendix, we use a similar argument to extend a theorem of LusztigSpaltenstein on induced unipotent classes to all infinite fields.
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In this review paper I present two geometric constructions of distinguished nature, one is over the field of complex numbers $\mathbb{C}$ and the other one is over the two elements field $\mathbb{F}_2$. Both constructions have been employed in the past fifteen years to describe two quantum paradoxes or two resources of quantum information: entanglement of pure multipartite systems on one side and contextuality on the other. Both geometric constructions are linked to representation of semisimple Lie groups/algebras. To emphasize this aspect one explains on one hand how wellknown results in representation theory allows one to see all the classification of entanglement classes of various tripartite quantum systems ($3$ qubits, $3$ fermions, $3$ bosonic qubits...) in a unified picture. On the other hand, one also shows how some weight diagrams of simple Lie groups are encapsulated in the geometry which deals with the commutation relations of the generalized $N$Pauli group.
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In this paper, we establish global $C^{1+\alpha,\frac{1+\alpha}{2}}$ estimates for solutions of the linearized parabolic MongeAmp$\grave{e}$re equation $$\mathcal{L}_\phi u(x,t):=u_t\,\mathrm{det}D^2\phi(x)+\mathrm{tr}[\Phi(x) D^2 u]=f(x,t)$$ under appropriate conditions on the domain, MongeAmp$\grave{e}$re measures, boundary data and $f$, where $\Phi:=\mathrm{det}(D^2\phi)(D^2\phi)^{1}$ is the cofactor of the Hessian of $D^2\phi$.
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In this paper, we establish global $W^{2,p}$ estimates for solutions of the linearized MongeAmp$\grave{e}$re equation $$\mathcal{L}_{\phi}u:=\mathrm{tr}[\Phi D^2 u]=f,$$ where the density of the MongeAmp$\grave{e}$re measure $g:=\mathrm{det}D^2\phi$ satisfies a $\mathrm{VMO}$type condition, and $\Phi:=(\mathrm{det}D^2\phi)(D^2\phi)^{1}$ is the cofactor matrix of $D^2\phi$.
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We study the connection between the BaumConnes conjecture for an ample groupoid $G$ with coefficient $A$ and the K\"unneth formula for the Ktheory of tensor products by the crossed product $A\rtimes_r G$. To do so we develop the machinery of GoingDown functors for ample groupoids. As an application we prove that both the uniform Roe algebra of a coarse space which uniformly embeds into a Hilbert space and the maximal Roe algebra of a space admitting a fibred coarse embedding into a Hilbert space satisfy the K\"unneth formula. We also provide a stability result for the K\"unneth formula using controlled Ktheory, and apply it to give an example of a space that does not admit a coarse embedding into a Hilbert space, but whose uniform Roe algebra satisfies the K\"unneth formula. As a byproduct of our methods, we also prove a permanence property for the BaumConnes conjecture with respect to equivariant inductive limits of the coefficient algebra.
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We consider the discrete, fractional operator $\left(L_a^\nu x\right) (t) := \nabla [p(t) \nabla_{a^*}^\nu x(t)] + q(t) x(t1)$ involving the nabla Caputo fractional difference, which can be thought of as an analogue to the selfadjoint differential operator. We show that solutions to difference equations involving this operator have expected properties, such as the form of solutions to homogeneous and nonhomogeneous equations. We also give a variation of constants formula via a Cauchy function in order to solve initial value problems involving $L_a^\nu$. We also consider boundary value problems of any fractional order involving $L_a^\nu$. We solve these BVPs by giving a definition of a Green's function along with a corresponding Green's Theorem. Finally, we consider a (2,1) conjugate BVP as a special case of the more general Green's function definition.
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Hereditarily non uniformly perfect (HNUP) sets were introduced by Stankewitz, Sugawa, and Sumi in \cite{SSS} who gave several examples of such sets based on Cantor setlike constructions using nested intervals. We exhibit a class of examples in nonautonomous iteration where one considers compositions of polynomials from a sequence which is in general allowed to vary. In particular, we give a sharp criterion for when Julia sets from our class will be HNUP and we show that the maximum possible Hausdorff dimension of $1$ for these Julia sets can be attained. The proof of the latter considers the Julia set as the limit set of a nonautonomous conformal iterated function system and we calculate the Hausdorff dimension using a version of Bowen's formula given in the paper by RempeGillen and Urb\'{a}nski \cite{RU}
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Gallagher's theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. We provide a fibre refinement, solving a problem posed by Beresnevich, Haynes and Velani in 2015. Hitherto, this was only known on the plane, as previous approaches relied heavily on the theory of continued fractions. Using reduced successive minima in lieu of continued fractions, we develop the structural theory of Bohr sets of arbitrary rank, in the context of diophantine approximation. In addition, we generalise the theory and result to the inhomogeneous setting. To deal with this inhomogeneity, we employ diophantine transference inequalities in lieu of the three distance theorem.
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We exhibit three double octic CalabiYau threefolds over the certain quadratic fields and prove their modularity. The nonrigid threefold has two conjugate Hilbert modular forms of weight [4,2] and [2,4] attached while the two rigid threefolds correspond to a Hilbert modular form of weight [4,4] and to the twist of the restriction of a classical modular form of weight 4.
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It is noted that using complex Hessian equations and the concavity inequalities for elementary symmetric polynomials implies a generalized form of Hodge index inequality. Inspired by this result, using G{\aa}rding's theory for hyperbolic polynomials, we obtain a mixed Hodgeindex type theorem for classes of type $(1,1)$. The new feature is that this Hodgeindex type theorem holds with respect to mixed polarizations in which some satisfy particular positivity condition, but could be degenerate and even negative along some directions.
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We consider geothermal inverse problems and uncertainty quantification from a Bayesian perspective. Our goal is to make standard, 'outofthebox' Markov chain Monte Carlo (MCMC) sampling more feasible for complex simulation models. To do this, we first show how to pose the inverse and prediction problems in a hierarchical Bayesian framework. We then show how to incorporate socalled posterior model approximation error into this hierarchical framework, using a modified form of the Bayesian approximation error (BAE) approach. This enables the use of a 'coarse', approximate model in place of a finer, more expensive model, while also accounting for the additional uncertainty and potential bias that this can introduce. Our method requires only simple probability modelling and only modifies the target posterior  the same standard MCMC sampling algorithm can be used to sample the new target posterior. We show that our approach can achieve significant computational speedups on a geothermal
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For a Tychonoff space $X$ and a family $\lambda$ of subsets of $X$, we denote by $C_{\lambda}(X)$ the $T_1$space of all realvalued continuous functions on $X$ with the $\lambda$ open topology. A topological space is productively Lindel\"of if its product with every Lindel\"of space is Lindel\"of. A space is indestructibly productively Lindel\"of if it is productively Lindel\"of in any extension by countably closed forcing. In this paper, we study indestructibly productively Lindel\"of and Menger function space $C_{\lambda}(X)$.
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Simple inequalities are established for integrals of the type $\int_0^x \mathrm{e}^{\gamma t} t^{\nu} \mathbf{L}_\nu(t)\,\mathrm{d}t$, where $x>0$, $0\leq\gamma<1$, $\nu>\frac{3}{2}$ and $\mathbf{L}_{\nu}(x)$ is the modified Struve function of the first kind. In most cases, these inequalities are tight in certain limits. As a consequence we deduce a tight double inequality, involving the modified Struve function $\mathbf{L}_{\nu}(x)$, for a generalized hypergeometric function.
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We prove the instability of some families of Riemannian manifolds with nontrivial real Killing spinors. These include the invariant Einstein metrics on the AloffWallach spaces $N_{k, l}={\rm SU}(3)/i_{k, l}(S^{1})$ (which are all nearly ${\rm G}_2$ except $N_{1,0}$), and Sasaki Einstein circle bundles over certain irreducible Hermitian symmetric spaces. We also prove the instability of most of the simply connected nonsymmetric compact homogeneous Einstein spaces of dimensions $5, 6, $ and $7$, including the strict nearly K\"ahler ones (except ${\rm G}_2/{\rm SU}(3)$).
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In this paper, we establish interior $C^{1,\alpha}$ estimates for solutions of the linearized MongeAmp$\grave{e}$re equation $$\mathcal{L}_{\phi}u:=\mathrm{tr}[\Phi D^2 u]=f,$$ where the density of the MongeAmp$\grave{e}$re measure $g:=\mathrm{det}D^2\phi$ satisfies a $\mathrm{VMO}$type condition and $\Phi:=(\mathrm{det}D^2\phi)(D^2\phi)^{1}$ is the cofactor matrix of $D^2\phi$.
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Novel reconstruction methods for electrical impedance tomography (EIT) often require voltage measurements on currentdriven electrodes. Such measurements are notoriously difficult to obtain in practice as they tend to be affected by unknown contact impedances and require problematic simultaneous measurements of voltage and current. In this work, we develop an interpolation method that predicts the voltages on currentdriven electrodes from the more reliable measurements on currentfree electrodes for difference EIT settings, where a conductivity change is to be recovered from difference measurements. Our new method requires the apriori knowledge of an upper bound of the conductivity change, and utilizes this bound to interpolate in a way that is consistent with the special geometryspecific smoothness of difference EIT data. Our new interpolation method is computationally cheap enough to allow for realtime applications, and simple to implement as it can be formulated with the standar
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