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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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Let $G$ be a semisimple algebraic group over a field of characteristic $p > 0$. We prove that the dual Weyl modules for $G$ all have $p$filtrations when $p$ is not too small. Moreover, we give applications of this theorem to $p^n$filtrations for $n > 1$, to modules containing the Steinberg module as a tensor factor, and to the Donkin conjecture on modules having $p$filtrations.
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A graph $G$ is called edgemagic if there is a bijective function $f$ from the set of vertices and edges to the set $\{1,2,\ldots,V(G)+E(G)\}$ such that the sum $f(x)+f(xy)+f(y)$ for any $xy$ in $E(G)$ is constant. Such a function is called an edgemagic labelling of G and the constant is called the valence of $f$. An edgemagic labelling with the extra property that $f(V(G))= \{1,2,\ldots,V(G)\}$ is called super edgemagic. In this paper, we establish a relationship between the valences of (super) edgemagic labelings of certain types of bipartite graphs and the existence of a particular type of decompositions of such graphs.
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A convex optimizationbased method is proposed to numerically solve dynamic programs in continuous state and action spaces. This approach using a discretization of the state space has the following salient features. First, by introducing an auxiliary optimization variable that assigns the contribution of each grid point, it does not require an interpolation in solving an associated Bellman equation and constructing a control policy. Second, the proposed method allows us to solve the Bellman equation with a desired level of precision via convex programming in the case of linear systems and convex costs. We can also construct a control policy of which performance converges to the optimum as the grid resolution becomes finer in this case. Third, when a nonlinear controlaffine system is considered, the convex optimization approach provides an approximate control policy with a provable suboptimality bound. Fourth, for general cases, the proposed convex formulation of dynamic programming op
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FirstFit is a greedy algorithm for partitioning the elements of a poset into chains. Let $\textrm{FF}(w,Q)$ be the maximum number of chains that FirstFit uses on a $Q$free poset of width $w$. A result due to Bosek, Krawczyk, and Matecki states that $\textrm{FF}(w,Q)$ is finite when $Q$ has width at most $2$. We describe a family of posets $\mathcal{Q}$ and show that the following dichotomy holds: if $Q\in\mathcal{Q}$, then $\textrm{FF}(w,Q) \le 2^{c(\log w)^2}$ for some constant $c$ depending only on $Q$, and if $Q\not\in\mathcal{Q}$, then $\textrm{FF}(w,Q) \ge 2^w  1$.
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Computationally efficient numerical methods for highorder approximations of convolution integrals involving weakly singular kernels find many practical applications including those in the development of fast quadrature methods for numerical solution of integral equations. Most fast techniques in this direction utilize uniform grid discretizations of the integral that facilitate the use of FFT for $O(n\log n)$ computations on a grid of size $n$. In general, however, the resulting error converges slowly with increasing $n$ when the integrand does not have a smooth periodic extension. Such extensions, in fact, are often discontinuous and, therefore, their approximations by truncated Fourier series suffer from Gibb's oscillations. In this paper, we present and analyze an $O(n\log n)$ scheme, based on a Fourier extension approach for removing such unwanted oscillations, that not only converges with highorder but is also relatively simple to implement. We include a theoretical error analys
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We present a hybrid spectral elementFourier spectral method for solving the coupled system of NavierStokes and CahnHilliard equations to simulate wallbounded twophase flows in a threedimensional domain which is homogeneous in at least one direction. Fourier spectral expansions are employed along the homogeneous direction and $C^0$ highorder spectral element expansions are employed in the other directions. A critical component of the method is a strategy we developed in a previous work for dealing with the variable density/viscosity of the twophase mixture, which makes the efficient use of Fourier expansions in the current work possible for twophase flows with different densities and viscosities for the two fluids. The attractive feature of the presented method lies in that the twophase computations in the threedimensional space are transformed into a set of decoupled twodimensional computations in the planes of the nonhomogeneous directions. The overall scheme consists of
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In this paper we study class $\mathcal{S}^+$ of univalent functions $f$ such that $\frac{z}{f(z)}$ has real and positive coefficients. For such functions we give estimates of the FeketeSzeg\H{o} functional and sharp estimates of their initial coefficients and logarithmic coefficients. Also, we present necessary and sufficient conditions for $f\in \mathcal{S}^+$ to be starlike of order $1/2$.
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Computation biology helps to understand all processes in organisms from interaction of molecules to complex functions of whole organs. Therefore, there is a need for mathematical methods and models that deliver logical explanations in a reasonable time. For the last few years there has been a growing interest in biological theory connected to finite fields: the algebraic modeling tools used up to now are based on Gr\"obner bases or Boolean group. Let $n$ variables representing gene products, changing over the time on $p$ values. A Polynomial dynamical system (PDS) is a function which has several components, each one is a polynom with $n$ variables and coefficient in the finite field $Z/pZ$ that model the evolution of gene products. We propose herein a method using algebraic separators, which are special polynomials abundantly studied in effective Galois theory. This approach avoids heavy calculations and provides a first Polynomial model in linear time.
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Linear Hamiltonian systems with timedependent coefficients are of importance to nonlinear Hamiltonian systems, accelerator physics, plasma physics, and quantum physics. It is shown that the solution map of a linear Hamiltonian system with timedependent coefficients can be parameterized by an envelope matrix $w(t)$, which has a clear physical meaning and satisfies a nonlinear envelope matrix equation. It is proved that a linear Hamiltonian system with periodic coefficients is stable iff the envelope matrix equation admits a solution with periodic $\sqrt{w^{\dagger}w}$ and a suitable initial condition. The mathematical devices utilized in this theoretical development with significant physical implications are timedependent canonical transformations, normal forms for stable symplectic matrices, and horizontal polar decomposition of symplectic matrices. These tools systematically decompose the dynamics of linear Hamiltonian systems with timedependent coefficients, and are expected to b
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While the definition of a fractional integral may be codified by Riemann and Liouville, an agreedupon fractional derivative has eluded discovery for many years. This is likely a result of integral definitions including numerous constants of integration in their results. An elimination of constants of integration opens the door to an operator that reconciles all known fractional derivatives and shows surprising results in areas unobserved before, including the appearance of the Riemann Zeta Function and fractional Laplace and Fourier Transforms. A new class of functions, known as Zero Functions and closely related to the Dirac Delta Function, are necessary for one to perform elementary operations of functions without using constants. The operator also allows for a generalization of the Volterra integral equation, and provides a method of solving for Riemann's "complimentary" function introduced during his research on fractional derivatives.
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In The Delta Conjecture (arxiv:1509.07058), Haglund, Remmel and Wilson introduced a four variable $q,t,z,w$ Catalan polynomial, so named because the specialization of this polynomial at the values $(q,t,z,w) = (1,1,0,0)$ is equal to the Catalan number $\frac{1}{n+1}\binom{2n}{n}$. We prove the compositional version of this conjecture (which implies the noncompositional version) that states that the coefficient of $s_{r,1^{nr}}$ in the expression $\Delta_{h_\ell} \nabla C_\alpha$ is equal to a weighted sum over decorated Dyck paths.
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In this paper we study a subFinsler geometric problem on the freenilpotent group of rank 2 and step 3. Such a group is also called Cartan group and has a natural structure of Carnot group, which we metrize considering the $\ell_\infty$ norm on its first layer. We adopt the point of view of timeoptimal control theory. We characterize extremal curves via Pontryagin maximum principle. We describe abnormal and singular arcs, and construct the bangbang flow.
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We consider the dynamic multichannel access problem, which can be formulated as a partially observable Markov decision process (POMDP). We first propose a modelfree actorcritic deep reinforcement learning based framework to explore the sensing policy. To evaluate the performance of the proposed sensing policy and the framework's tolerance against uncertainty, we test the framework in scenarios with different channel switching patterns and consider different switching probabilities. Then, we consider a timevarying environment to identify the adaptive ability of the proposed framework. Additionally, we provide comparisons with the DeepQ network (DQN) based framework proposed in [1], in terms of both average reward and the time efficiency.
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Given a large ensemble of interacting particles, driven by nonlocal interactions and localized repulsion, the meanfield limit leads to a class of nonlocal, nonlinear partial differential equations known as aggregationdiffusion equations. Over the past fifteen years, aggregationdiffusion equations have become widespread in biological applications and have also attracted significant mathematical interest, due to their competing forces at different length scales. These competing forces lead to rich dynamics, including symmetrization, stabilization, and metastability, as well as sharp dichotomies separating wellposedness from finite time blowup. In the present work, we review known analytical results for aggregationdiffusion equations and consider singular limits of these equations, including the slow diffusion limit, which leads to the constrained aggregation equation, as well as localized aggregation and vanishing diffusion limits, which lead to metastability behavior. We also revie
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In this paper we focus on a certain selfdistributive multiplication on coalgebras, which leads to socalled rack bialgebra. We construct canonical rack bialgebras (some kind of enveloping algebras) for any Leibniz algebra. Our motivation is deformation quantization of Leibniz algebras in the sense of [6]. Namely, the canonical rack bialgebras we have constructed for any Leibniz algebra lead to a simple explicit formula of the rackstarproduct on the dual of a Leibniz algebra recently constructed by Dherin and Wagemann in [6]. We clarify this framework setting up a general deformation theory for rack bialgebras and show that the rackstarproduct turns out to be a deformation of the trivial rack bialgebra product.
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The dominant theme of this thesis is the construction of matrix representations of finite solvable groups using a suitable system of generators. For a finite solvable group $G$ of order $N = p_{1}p_{2}\dots p_{n}$, where $p_{i}$'s are primes, there always exists a subnormal series: $\langle {e} \rangle = G_{o} < G_{1} < \dots < G_{n} = G$ such that $G_{i}/G_{i1}$ is isomorphic to a cyclic group of order $p_{i}$, $i = 1,2,\dots,n$. Associated with this series, there exists a system of generators consisting $n$ elements $x_{1}, x_{2}, \dots, x_{n}$ (say), such that $G_{i} = \langle x_{1}, x_{2}, \dots, x_{i} \rangle$, $i = 1,2,\dots,n$, which is called a "long system of generators". In terms of this system of generators and conjugacy class sum of $x_{i}$ in $G_{i}$, $i = 1,2, \dots, n$, we present an algorithm for constructing the irreducible matrix representations of $G$ over $\mathbb{C}$ within the group algebra $\mathbb{C}[G]$. This algorithmic construction needs the knowled
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Equipping a nonequivariant topological $E_\infty$operad with the trivial $G$action gives an operad in $G$spaces. For a $G$spectrum, being an algebra over this operad does not provide any multiplicative norm maps on homotopy groups. Algebras over this operad are called na\"{i}vecommutative ring $G$spectra. In this paper we take $G=SO(2)$ and we show that commutative algebras in the algebraic model for rational $SO(2)$spectra model rational na\"{i}vecommutative ring $SO(2)$spectra. In particular, this applies to show that the $SO(2)$equivariant cohomology associated to an elliptic curve $C$ from previous work of the second author is represented by an $E_\infty$ring spectrum. Moreover, the category of modules over that $E_\infty$ring spectrum is equivalent to the derived category of sheaves over the elliptic curve $C$ with the Zariski torsion point topology.
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We consider the inverse problem of recovering the magnetic and potential term of a magnetic Schr\"{o}dinger operator on certain compact Riemannian manifolds with boundary from partial Dirichlet and Neumann data on suitable subsets of the boundary. The uniqueness proof relies on proving a suitable Carleman estimate for functions which vanish only on a part of boundary and constructing complex geometric optics solutions which vanish on a part of the boundary.
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We review and apply CheegerGromov theory on $l^2$cohomology of infinite coverings of complete manifolds with bounded curvature and finite volume. Applications focus on $l^2$cohomology of (pullback of) harmonic Higgs bundles on some covering of Zariski open sets of K\"ahler manifolds. The $l^2$Dolbeault to DeRham spectral sequence of these Higgs bundles is seen to degenerate at $E_2$.
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We introduce approximation schemes for a type of countablyinfinitedimensional linear programs (CILPs) whose feasible points are unsigned measures and whose optimal values are bounds on the averages of these measures. In particular, we explain how to approximate the program's optimal value, optimal points, and minimal point (should one exist) by solving finitedimensional linear programs. We show that the approximations converge to the CILP's optimal value, optimal points, and minimal point as the size of the finitedimensional program approaches that of the CILP. Inbuilt in our schemes is a degree of error control: they yield lower and upper bounds on the optimal values and we give a simple bound on the approximation error of the minimal point. To motivate our work, we discuss applications of our schemes taken from the Markov chain literature: stationary distributions, occupation measures, and exit distributions.
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