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In the developing theory of infinitedimensional quantum channels the relevance of the energyconstrained diamond norms was recently corroborated both from physical and informationtheoretic points of view. In this paper we study necessary and sufficient conditions for differentiability with respect to these norms of the strongly continuous semigroups of quantum channels (quantum dynamical semigroups). We show that these conditions can be expressed in terms of the generator of the semigroup. We also analyze conditions for representation of a strongly continuous semigroup of quantum channels as an exponential series converging w.r.t. the energyconstrained diamond norm. Examples of semigroups having such a representation are presented.
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We construct a nontrivial type of 1step exceptional BannaiIto polynomials which satisfy discrete orthogonality by using a generalized Darboux transformation. In this generalization, the Darboux transformed BannaiIto operator is directly obtained through an intertwining relation. Moreover, the seed solution, which consists of a gauge factor and a polynomial part, plays an important role in the construction of these 1step exceptional BannaiIto polynomials. And we show that there are 8 classes of gauge factors. We also provide the eigenfunctions of the corresponding multiplestep exceptional BannaiIto operator which can be expressed as a 3 x 3 determinant.
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A graph $G$ is said to be chordal if it has no induced cycles of length four or more. In a recent preprint Culbertson, Guralnik, and Stiller give a new characterization of chordal graphs in terms of sequences of what they call `edgeerasures'. In this note we show that these moves are in fact equivalent to a linear quotient ordering on $I_{\overline{G}}$, the edge ideal of the complement graph $\overline G$. Known results imply that $I_{\overline G}$ has linear quotients if and only if $G$ is chordal, and hence this recovers an algebraic proof of their characterization. We investigate higherdimensional analogues of this result, and show that in fact linear quotients for more general circuit ideals of $d$clutters can be characterized in terms of removing exposed circuits in the complement clutter. Restricting to properly exposed circuits can be characterized by a homological condition. This leads to a notion of higher dimensional chordal clutters which borrows from commutative algebra
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Given two endomorphisms $\tau_1,\tau_2$ of $\mathbb{C}^m$ with $m \ge 2n$ and a general $n$dimensional subspace $\mathcal{V} \subset \mathbb{C}^m$, we provide eigenspace conditions under which $\tau_1(v_1)=\tau_2(v_2)$ for $v_1,v_2 \in \mathcal{V}$ can only be true if $v_1=v_2$. As a special case, we recover the result of Unnikrishnan et al. in which $\tau_1,\tau_2$ are permutations composed with coordinate projections.
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In electricity markets with a dualpricing scheme for balancing energy, controllable production units typically participate in the balancing market as "active" actors by offering regulating energy to the system, while renewable stochastic units are treated as "passive" participants that create imbalances and are subject to less competitive prices. Against this background, we propose an innovative market framework whereby the participant in the balancing market is allowed to act as an active agent (i.e., a provider of regulating energy) in some trading intervals and as a passive agent (i.e., a user of regulating energy) in some others. To illustrate and evaluate the proposed market framework, we consider the case of a virtual power plant (VPP) that trades in a twosettlement electricity market composed of a dayahead and a dualprice balancing market. We formulate the optimal market offering problem of the VPP as a threestage stochastic program, where uncertainty is in the dayahead el
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In this paper we are concerned with the analysis of heavytailed data when a portion of the extreme values is unavailable. This research was motivated by an analysis of the degree distributions in a large social network. The degree distributions of such networks tend to have power law behavior in the tails. We focus on the Hill estimator, which plays a starring role in heavytailed modeling. The Hill estimator for this data exhibited a smooth and increasing "sample path" as a function of the number of upper order statistics used in constructing the estimator. This behavior became more apparent as we artificially removed more of the upper order statistics. Building on this observation we introduce a new version of the Hill estimator. It is a function of the number of the upper order statistics used in the estimation, but also depends on the number of unavailable extreme values. We establish functional convergence of the normalized Hill estimator to a Gaussian process. An estimation proc
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We recall Charles Babbage's 1819 criterion for primality, based on simultaneous congruences for binomial coefficients, and extend it to a leastprimefactor test. We also prove a partial converse of his nonprimality test, based on a single congruence. Two problems are posed. Along the way we encounter Bachet, Bernoulli, Bezout, Euler, Fermat, Kummer, Lagrange, Lucas, Vandermonde, Waring, Wilson, Wolstenholme, and several contemporary mathematicians.
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For each $p>1$ and each positive integer $m$ we give intrinsic characterizations of the restriction of the homogeneous Sobolev space $L^m_p(R)$ to an arbitrary closed subset $E$ of the real line. We show that the classical one dimensional Whitney extension operator is "universal" for the scale of $L^m_p(R)$ spaces in the following sense: for every $p\in(1,\infty]$ it provides almost optimal $L^m_p$extensions of functions defined on $E$. The operator norm of this extension operator is bounded by a constant depending only on $m$. This enables us to prove several constructive $L^m_p$extension criteria expressed in terms of $m^{th}$ order divided differences of functions.
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Optimal control problems are inherently hard to solve as the optimization must be performed simultaneously with updating the underlying system. Starting from an initial guess, Howard's policy improvement algorithm separates the step of updating the trajectory of the dynamical system from the optimization and iterations of this should converge to the optimal control. In the discrete spacetime setting this is often the case and even rates of convergence are known. In the continuous spacetime setting of controlled diffusion the algorithm consists of solving a linear PDE followed by maximization problem. This has been shown to converge, in some situations, however no global rate of is known. The first main contribution of this paper is to establish global rate of convergence for the policy improvement algorithm and a variant, called here the gradient iteration algorithm. The second main contribution is the proof of stability of the algorithms under perturbations to both the accuracy of t
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This is a supplement for "Pearls in graph theory"  a textbook written by Nora Hartsfield and Gerhard Ringel. We discuss bounds on Ramsey numbers, the probabilistic method, deletioncontraction formulas, the matrix theorem, chromatic polynomials, the marriage theorem and its relatives, the Rado graph, and generating functions.
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The class of split matroids arises by putting conditions on the system of split hyperplanes of the matroid base polytope. It can alternatively be defined in terms of structural properties of the matroid. We use this structural description to give an excluded minor characterisation of the class.
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In this paper we study the $(2+1)$dimensional DiracDunkl oscillator coupled to an external magnetic field. Our Hamiltonian is obtained from the standard Dirac oscillator coupled to an external magnetic field by changing the partial derivatives by the Dunkl derivatives. We solve the DunklKleinGordontype equations in polar coordinates in a closed form. The angular part eigenfunctions are given in terms of the JacobiDunkl polynomials and the radial functions in terms of the Laguerre functions. Also, we compute the energy spectrum of this problem and show that, in the nonrelativistic limit, it properly reduces to the Hamiltonian of the two dimensional harmonic oscillator.
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We prove that the known formulae for computing the optimal number of maximally entangled pairs required for entanglementassisted quantum errorcorrecting codes (EAQECCs) over the binary field holds for codes over arbitrary finite fields as well. We also give a GilbertVarshamov bound for EAQECCs and constructions of EAQECCs coming from punctured selforthogonal linear codes which are valid for any finite field.
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We discuss generalized partition function of 2d CFTs decorated by higher qKdV charges on thermal cylinder. We propose that in the large central charge limit qKdV charges factorize such that generalized partition function can be rewritten in terms of auxiliary noninteracting bosons. The explicit expression for the generalized free energy is readily available in terms of the boson spectrum, which can be deduced from the conventional thermal expectation values of qKdV charges. In other words, the picture of the auxiliary noninteracting bosons allows extending thermal onepoint functions to the full nonperturbative generalized partition function. We verify this conjecture for the first seven qKdV charges using recently obtained pertrubative results and find corresponding contributions to the auxiliary boson masses. We further extend these results by conjecturing the full spectrum of bosons and find an exact expression for the generalized partition function as a function of infinite towe
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Dyson's celebrated constant term conjecture ({\em J. Math. Phys.}, 3 (1962): 140156) states that the constant term in the expansion of $\prod_{1\leqq i\neq j\leqq n} (1x_i/x_j)^{a_j}$ is the multinomial coefficient $(a_1 + a_2 + \cdots + a_n)!/ (a_1! a_2! \cdots a_n!)$. The definitive proof was given by I. J. Good ({\em J. Math. Phys.}, 11 (1970) 1884). Later, Andrews extended Dyson's conjecture to a $q$analog ({\em The Theory and Application of Special Functions}, (R. Askey, ed.), New York: Academic Press, 191224, 1975.) In this paper, closed form expressions are given for the coefficients of several other terms in the Dyson product, and are proved using an extension of Good's idea. Also, conjectures for the corresponding $q$analogs are supplied. Finally, perturbed versions of the $q$Dixon summation formula are presented.
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We establish existence of the etainvariant as well as of the AtiyahPatodiSinger and the CheegerGromov rhoinvariants for a class of Dirac operators on an incomplete edge space. Our analysis applies in particular to the signature, the GaussBonnet and the spin Dirac operator. We derive an analogue of the AtiyahPatodiSinger index theorem for incomplete edge spaces and their noncompact infinite Galois coverings with edge singular boundary. Our arguments employ microlocal analysis of the heat kernel asymptotics on incomplete edge spaces and the classical argument of AtiyahPatodiSinger. As an application, we discuss stability results for the two rhoinvariants we have defined.
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If spectrum of a Schr\"{o}dinger oparator with a nonHermitian potential contains a spectral singularity (SS), the latter requires exact matching of the parameters characterizing the potential. We provide a necessary and sufficient condition for a potential to have a SS at a given wavelength. It is shown that potentials with SS at prescribed wavelengths can be obtained by a simple and effective procedure. In particular, the developed approach allows one to obtain potentials with several SSs or with SSs of the second order and potentials obeying a given symmetry, say, PTsymmetric potentials. Also, the problem can be solved when it is required to obtain a potential obeying a given symmetry, say, $\mathcal{PT}$symmetric potential. We illustrate all opportunities with examples. We also describe splitting of the SSs of the second order, under change of the potential parameters, and discuss possibilities of experimental observation of SSs of different orders.
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Beck's distributive laws provide sufficient conditions under which two monads can be composed, and monads arising from distributive laws have many desirable theoretical properties. Unfortunately, finding and verifying distributive laws, or establishing if one even exists, can be extremely difficult and errorprone. We develop generalpurpose techniques for showing when there can be no distributive law between two monads. Two approaches are presented. The first widely generalizes ideas from a counterexample attributed to Plotkin, yielding generalpurpose theorems that recover the previously known situations in which no distributive law can exist. Our second approach is entirely novel, encompassing new practical situations beyond our generalization of Plotkin's approach. It negatively resolves the open question of whether the list monad distributes over itself. Our approach adopts an algebraic perspective throughout, exploiting a syntactic characterization of distributive laws. This appr
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In this paper, sufficient conditions are given for the existence of limiting distribution of a conservative affine process on the canonical state space $\mathbb{R}_{\geqslant0}^{m}\times\mathbb{R}^{n}$, where $m,\thinspace n\in\mathbb{Z}_{\geqslant0}$ with $m+n>0$. Our main theorem extends and unifies some known results for OUtype processes on $\mathbb{R}^{n}$ and onedimensional CBI processes (with state space $\mathbb{R}_{\geqslant0}$). To prove our result, we combine analytical and probabilistic techniques; in particular, the stability theory for ODEs plays an important role.
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We shall generalize the notion of a Laver table to algebras which may have many generators, several fundamental operations, fundamental operations of arity higher than 2, and to algebras where only some of the operations are selfdistributive or where the operations satisfy a generalized version of selfdistributivity. These algebras mimic the algebras of rankintorank embeddings $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ in the sense that composition and the notion of a critical point make sense for these sorts of algebras.
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The classical Eulerian polynomials $A_n(t)$ are known to be gamma positive. Define the positive Eulerian polynomial $\mathsf{AExc^{+}}_n(t)$ as the polynomial obtained when we sum excedances over the alternating group. We show that $\mathsf{AExc^{+}}_n(t)$ is gamma positive iff $n \geq 5$ and $n \equiv 1$ (mod 2). When $n \geq 4$, and $n \equiv 0$ (mod 2) we show that $\mathsf{AExc^{+}}_n(t)$ can be written as a sum of two gamma positive polynomials. Similar results are shown when we consider the positive typeD and typeD Eulerian polynomials. Finally, we show gamma positivity results when we sum excedances over derangements with positive and negative sign. Our main resuls is that the polynomial obtained by summing excedance over a conjugacy class indexed by $\lambda$ is gamma positive.
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We consider links that are alternating on surfaces embedded in a compact 3manifold. We show that under mild restrictions, the complement of the link decomposes into simpler pieces, generalising the polyhedral decomposition of alternating links of Menasco. We use this to prove various facts about the hyperbolic geometry of generalisations of alternating links, including weakly generalised alternating links described by the first author. We give diagrammatical properties that determine when such links are hyperbolic, find the geometry of their checkerboard surfaces, bound volume, and exclude exceptional Dehn fillings.
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In Kac's classification of finitedimensional Lie superalgebras, the contragredient ones can be constructed from Dynkin diagrams similar to those of the simple finitedimensional Lie algebras, but with additional types of nodes. For example, $A(n1,0) = \mathfrak{sl}(1n)$ can be constructed by adding a "gray" node to the Dynkin diagram of $A_{n1} = \mathfrak{sl}(n)$, corresponding to an odd null root. The Cartan superalgebras constitute a different class, where the simplest example is $W(n)$, the derivation algebra of the Grassmann algebra on $n$ generators. Here we present a novel construction of $W(n)$, from the same Dynkin diagram as $A(n1,0)$, but with additional generators and relations.
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For a domain $D \subset \mathbb C^n$, $n \ge 2$, let $F^k_D(z)=K_D(z)\lambda\big(I^k_D(z)\big)$, where $K_D(z)$ is the Bergman kernel of $D$ along the diagonal and $\lambda\big(I^k_D(z)\big)$ is the Lebesgue measure of the Kobayashi indicatrix at the point $z$. This biholomorphic invariant was introduced by B\l ocki and in this note, we study the boundary behaviour of $F^k_D(z)$ near a finite type boundary point where the boundary is smooth, pseudoconvex with the corank of its Levi form being at most $1$. We also compute its limiting behaviour near the boundary of certain other basic classes of domains.
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Facial recognition has to be regulated to protect the public, says AI repor
1207 MIT Technology 1818 
On a reduced analytic space $X$ we introduce the concept of a generalized cycle, which extends the notion of a formal sum of analytic subspaces to include also a form part. We then consider a suitable equivalence relation and corresponding quotient $\mathcal{B}(X)$ that we think of as an analogue of the Chow group and a refinement of de Rham cohomology. This group allows us to study both global and local intersection theoretic properties. We provide many $\mathcal{B}$analogues of classical intersection theoretic constructions: For an analytic subspace $V\subset X$ we define a $\mathcal{B}$Segre class, which is an element of $\mathcal{B}(X)$ with support in $V$. It satisfies a global King formula and, in particular, its multiplicities at each point coincide with the Segre numbers of $V$. When $V$ is cut out by a section of a vector bundle we interpret this class as a MongeAmp\`eretype product. For regular embeddings we construct a $\mathcal{B}$analogue of the Gysin morphism.
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In this paper we present novel $ADE$ correspondences by combining an earlier induction theorem of ours with one of Arnold's observations concerning Trinities, and the McKay correspondence. We first extend Arnold's indirect link between the Trinity of symmetries of the Platonic solids $(A_3, B_3, H_3)$ and the Trinity of exceptional 4D root systems $(D_4, F_4, H_4)$ to an explicit Clifford algebraic construction linking the two ADE sets of root systems $(I_2(n), A_1\times I_2(n), A_3, B_3, H_3)$ and $(I_2(n), I_2(n)\times I_2(n), D_4, F_4, H_4)$. The latter are connected through the McKay correspondence with the ADE Lie algebras $(A_n, D_n, E_6, E_7, E_8)$. We show that there are also novel indirect as well as direct connections between these ADE root systems and the new ADE set of root systems $(I_2(n), A_1\times I_2(n), A_3, B_3, H_3)$, resulting in a web of threeway ADE correspondences between three ADE sets of root systems.
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Seeking compliance with Linux's new Code of Conduct, Intel software engineer Jarkko Sakkinen recently requested comments on a set of changes to kernel code comments which Neowin described as "replacing the Fword with 'hug'. " 80 comments quickly followed on the Linux Kernel Maintainer's List: Several contributors responded to the alterations calling them insane. One wondered if Sakkinen was just trying to make a joke, and another called it censorship and said he'd refuse to apply any sort of patches like this to the code he's in charge of... Some of the postchange comments read "Some Athlon laptops have really hugged PST tables", "If you don't see why, please stay the hug away from my code", and "Only Sun can take such nice parts and hug up the programming interface". Eventually LWN.net publisher Jonathan Corbet deflated most of the controversy by pointing out that Linux's new Code of Conduct applies to future comments but clearly indicates that it does not apply explicitly to pas
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A major difference between Go 1 and Go 2 is who is going to influence the design and how decisions are made. Go 1 was a small team effort with modest outside influence; Go 2 will be much more communitydriven. After almost 10 years of exposure, we have learned a lot about the language and libraries that we didn't know in the beginning, and that was only possible through feedback from the Go community. The Go team s revealing some things about the future of the programming language.
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As of today, Fedora 27 will not be getting any more updates, including security updates. Users should be planning to upgrade more or less immediately. "Fedora 28 will continue to receive updates until 4 weeks after the release of Fedora 30. The maintenance schedule of Fedora releases is documented on the Fedora Project wiki. The Fedora Project wiki also contains instructions on how to upgrade from a previous release of Fedora to a version receiving updates."
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A logistics division of DHL announced today that it will invest $300 million to modernize 60 percent of its warehouses in North America with more IoT sensors and robots. Robotic process automation and software made to reduce workflow interruptions will also play a role. VentureBeat reports: Such technology is already in operation in 85 DHL facilities, or roughly 20 percent of warehouses across North America. Funding announced today will bring emerging technology to 350 of DHL Supply Chain's 430 operating sites. The company has more than 35,000 employees in North America. Conversations are ongoing with more than 25 robotics and process automation industry leaders, DHL Supply Chain president of retail Jim Gehr said. DHL Supply Chain warehouse robots will work primarily with unitpicking operations and will be able to complete a range of tasks, from collaborative piece picking to shuttling items across a factory to following human packers.
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Internal Facebook documents seized by British lawmakers suggest that the social media giant once considered selling access to user data, according to extracts obtained by the Wall Street Journal. Back in April, Facebook CEO Mark Zuckerberg told congress unequivocally that, "We do not sell data." But these documents suggest that it was something that the company internally considered doing between 2012 and 2014, while the company struggled to generate revenue after its IPO. This just goes to show that no matter what promises a company makes, once the shareholders come knocking, they'll disregard all promises, morals, and values they claim to have.
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George H.W. Bush, the 41st president of the United States, has passed away tonight at the age of 94. As The Washington Post reports, he was "the last veteran of World War II to serve as president, he was a consummate public servant and a statesman who helped guide the nation and the world out of a fourdecade Cold War that had carried the threat of nuclear annihilation." From the report: Although Mr. Bush served as president three decades ago, his values and ethic seem centuries removed from today's acrid political culture. His currency of personal connection was the handwritten letter  not the social media blast. He had a competitive nature and considerable ambition that were not easy to discern under the sheen of his New England politesse and his earnest generosity. He was capable of running hardedge political campaigns, and took the nation to war. But his principal achievements were produced at negotiating tables. Despite his grace, Mr. Bush was an easy subject for caricature. He
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The millimeter wave (mmWave) bands and other high frequencies above 6~GHz have emerged as a central component of FifthGeneration (5G) cellular standards to deliver high data rates and ultralow latency. A key challenge in these bands is blockage from obstacles, including the human body. In addition to the reduced coverage, blockage can result in highly intermittent links where the signal quality varies significantly with motion of obstacles in the environment. The blockages have widespread consequences throughout the protocol stack including beam tracking, link adaptation, cell selection, handover and congestion control. Accurately modeling these blockage dynamics is therefore critical for the development and evaluation of potential mmWave systems. In this work, we present a novel spatial dynamic channel sounding system based on phased array transmitters and receivers operating at 60 GHz. Importantly, the sounder can measure multiple directions rapidly at high speed to provide detaile
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We analyze the forward performance process in a general semimartingale market accounting for portfolio constraints, when investor's preferences are homothetic. We provide necessary and sufficient conditions for the construction of such a performance process, and establish its connection to the solution of an infinitehorizon quadratic backward stochastic differential equation (BSDE) driven by a semimartingale. We prove the existence and uniqueness of a solution to our infinitehorizon BSDE using techniques based on Jacod's decomposition and an extended argument of the comparison principle for finitehorizon BSDEs. We show the equivalence between the factor representation of the BSDE solution and the smooth solution to the illposed partial integraldifferential HamiltonJacobiBellman (HJB) equation arising in the extended semimartingale factor framework. Our study generalizes existing results on forward performance in Brownian settings, and shows that timemonotone processes are prese
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This work focuses on finding optimal locations for charging stations for oneway electric car sharing programs. The relocation of vehicles by a service staff is generally required in vehicle sharing programs in order to correct imbalances in the network. We seek to limit the need for vehicle relocation by strategically locating charging stations given estimates of traffic flow. A mixedinteger linear programming formulation is presented with a large number of potential charging station locations. A column generation approach is used which finds an optimal set of locations for the continuous relaxation of our problem. Results of a numerical experiment using real traffic and geographic information system location data show that our formulation significantly increases the balanced flow across the network, while our column generation technique was found to produce a superior solution in much shorter computation time compared to solving the original formulation with all possible station loc
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We investigate fractional colorings of graphs in which the amount of color given to a vertex depends on local parameters, such as its degree or the clique number of its neighborhood; in a \textit{fractional $f$coloring}, vertices are given color from the $[0, 1]$interval and each vertex $v$ receives at least $f(v)$ color. By Linear Programming Duality, all of the problems we study have an equivalent formulation as a problem concerning weighted independence numbers. However, these problems are most natural in the framework of fractional coloring, and the concept of coloring is crucial to most of our proofs. Our results and conjectures considerably generalize many wellknown fractional coloring results, such as the fractional relaxation of Reed's Conjecture, Brooks' Theorem, and Vizing's Theorem. Our results also imply previously unknown bounds on the independence number of graphs. We prove that if $G$ is a graph and $f(v) \leq 1/(d(v) + 1/2)$ for each $v\in V(G)$, then either $G$ has
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We investigate a subalgebra of the TemperleyLieb algebra called the JonesWenzl algebra, which is obtained by action of certain JonesWenzl projectors. This algebra arises naturally in applications to conformal field theory and statistical physics. It is also the commutant (centralizer) algebra of the Hopf algebra $U_q(\mathfrak{sl}_2)$ on its typeone modules  this fact is a generalization of the $q$SchurWeyl duality of Jimbo. In this article, we find two minimal generating sets for the JonesWenzl algebra. In special cases, we also find all of the independent relations satisfied by these generators.
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We consider the Graph Isomorphism problem for classes of graphs characterized by two forbidden induced subgraphs $H_1$ and $H_2$. By combining old and new results, Schweitzer settled the computational complexity of this problem restricted to $(H_1,H_2)$free graphs for all but a finite number of pairs $(H_1,H_2)$, but without explicitly giving the number of open cases. Grohe and Schweitzer proved that Graph Isomorphism is polynomialtime solvable on graph classes of bounded cliquewidth. By combining previously known results for Graph Isomorphism with known results for boundedness of cliquewidth, we reduce the number of open cases to 14. By proving a number of new results we then further reduce this number to seven. By exploiting the strong relationship between Graph Isomorphism and cliquewidth, we also prove that the class of $(\mbox{gem},P_1+2P_2)$free graphs has unbounded cliquewidth. This reduces the number of open cases for boundedness of cliquewidth for $(H_1,H_2)$free grap
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In this article we investigate partially truncated correlation functions (PTCF) of infinite continuous systems of classical point particles with pair interaction. We derive KirkwoodSalsburg (KS)type equations for the PTCF and write the solutions of these equations as a sum of contributions labelled by certain special graphs (forests), the connected components of which are tree graphs. We generalize the method introduced by Minlos and Pogosyan in the case of truncated correlations. These solutions make it possible to derive strong cluster properties for PTCF which were obtained earlier for lattice spin systems.
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In this article we give normal forms in a neighbourhood of a compact orbit of a Poisson Lie group action on a $b$symplectic manifold. In particular we establish cotangent models for Poisson group actions on $b$Poisson manifolds and a $b$symplectic slice theorem. We examine interesting particular instances of PoissonLie group actions on $b$symplectic manifolds. Also, we revise the notion of cotangent lift and twisted $b$cotangent lift introduced in \cite{km} and provide a generalization of the twisted $b$cotangent lift to higher dimensional torus actions. We introduce the notion of $b$Lie group and the associated $b$symplectic structures in its $b$cotangent bundle together with their reduction theory.
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We present a new penalty term approximating the CiarletNe\v{c}as condition (global invertibility of deformations) as a soft constraint for hyperelastic materials. For nonsimple materials including a suitable higher order term in the elastic energy, we prove that the penalized functionals converge to the original functional subject to the CiarletNe\v{c}as condition. Moreover, the penalization can be chosen in such a way that all low energy deformations, selfinterpenetration is completely avoided even for sufficiently small finite values of the penalization parameter. We also present numerical experiments in 2d illustrating our theoretical results.
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Let $G$ be a nontrivial connected, edgecolored graph. An edgecut $S$ of $G$ is called a rainbow cut if no two edges in $S$ are colored with a same color. An edgecoloring of $G$ is a rainbow disconnection coloring if for every two distinct vertices $s$ and $t$ of $G$, there exists a rainbow cut $S$ in $G$ such that $s$ and $t$ belong to different components of $G\setminus S$. For a connected graph $G$, the {\it rainbow disconnection number} of $G$, denoted by $rd(G)$, is defined as the smallest number of colors such that $G$ has a rainbow disconnection coloring by using this number of colors. In this paper, we show that for a connected graph $G$, computing $rd(G)$ is NPhard. In particular, it is already NPcomplete to decide if $rd(G)=3$ for a connected cubic graph. Moreover, we prove that for a given edgecolored (with an unbounded number of colors) connected graph $G$ it is NPcomplete to decide whether $G$ is rainbow disconnected.
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The Go Blog looks forward to version 2 of the Go language. "A major difference between Go 1 and Go 2 is who is going to influence the design and how decisions are made. Go 1 was a small team effort with modest outside influence; Go 2 will be much more communitydriven. After almost 10 years of exposure, we have learned a lot about the language and libraries that we didn’t know in the beginning, and that was only possible through feedback from the Go community."
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We compute the Dolbeault cohomology of certain domains contained in Cousin groups defined by lattices which satisfy a strong dispersiveness condition. As a consequence we obtain a description of the Dolbeault cohomology of OeljeklausToma manifolds and in particular the fact that the Hodge decomposition holds for their cohomology.
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We show a method to build new examples of Lie algebras admitting LCS or LCK structures starting with a smaller dimensional Lie algebra endowed with a LCS or LCK structure respectively, and a suitable representation. We also study the existence of lattices in the associated simply connected Lie groups in order to obtain compact examples of manifolds admitting these kind of structures. Finally we show that the Lie algebra underlying of the well known OeljesklausToma solvmanifold can me reobtained using our construction.
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We study the extremal particles of the twodimensional Coulomb gas with confinement generated by a radially symmetric positive background in the determinantal case and the zeros of the corresponding random polynomials. We show that when the background is supported on the unit disk, the point process of the particles outside of the disk converges towards a universal point process, i.e. that does not depend on the background. This limiting point process may be seen as the determinantal point process governed by the Bergman kernel on the complement of the unit disk. It has an infinite number of particles and its maximum is a heavy tailed random variable. To prove this convergence we study the case where the confinement is generated by a positive background outside of the unit disk. For this model we show that the point process of the particles inside the disk converges towards the determinantal point process governed by the Bergman kernel on the unit disk. In the case where the background
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We give an overview of the general framework of forms of Bak, Tits and Wall, when restricting to vector spaces over fields, including its relationship to the classical notions of Hermitian, alternating and quadratic forms. We then prove a version of Witt's lemma in this context, showing in particular that the action of the group of isometries of a space equipped with a form is transitive on isometric subspaces.
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We geometrically describe a canonical sequence of modular blowups of the relative Picard stack of the Artin stack of prestable genus two curves. The final blowup stack locally diagonalizes certain tautological derived objects. This implies a resolution of the primary component of the moduli space of genus two stable maps to projective space and meanwhile makes the whole moduli space admit only normal crossing singularities. Our approach should extend to higher genera.
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To every topologically transitive Cantor dynamical system $(X, \varphi)$ we associate a group $T(\varphi)$ acting faithfully by homeomorphism on the real line. It is defined as the group of homeomorphisms of the suspension flow of $(X, \varphi)$ which preserve every leaf and acts by dyadic piecewise linear homeomorphisms in the flow direction. We show that if $(X, \varphi)$ is minimal, the group $T(\varphi)$ is simple, and if $(X, \varphi)$ is a subshift the group $T(\varphi)$ is finitely generated. The proofs of these two statements are short and elementary, providing straightforward examples of finitely generated simple leftorderable groups. We show that if the system $(X, \varphi)$ is minimal, every action of the group $T(\varphi)$ on the circle has a fixed point, providing examples of so called "orderable monsters". We additionally have the following: for every subshift $(X, \varphi)$ the group $T(\varphi)$ does not have nontrivial subgroups with Kazhdan's property (T); for every
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The search for generating compatibility conditions (CC) for a given operator is a very recent problem met in General Relativity in order to study the Killing operator for various standard useful metrics (Minkowski, Schwarschild and Kerr). In this paper, we prove that the link existing between the lack of formal exactness of an operator sequence on the jet level, the lack of formal exactness of its corresponding symbol sequence and the lack of formal integrability (FI) of the initial operator is of a purely homological nature as it is based on the long exact connecting sequence provided by the socalled snake lemma. It is therefore quite difficult to grasp it in general and even more difficult to use it on explicit examples. It does not seem that any one of the results presented in this paper is known as most of the other authors who studied the above problem of computing the total number of generating CC are confusing this number with a kind of differential transcendence degree, also c
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In the framework of Density Functional Theory with Strongly Correlated Electrons we consider the so called bond dissociating limit for the energy of an aggregate of atoms. We show that the multimarginals optimal transport cost with Coulombian electronelectron repulsion may correctly describe the dissociation effect. The variational limit is completely calculated in the case of N=2 electrons. The theme of fractional number of electrons appears naturally and brings into play the question of optimal partial transport cost. A plan is outlined to complete the analysis which involves the study of the relaxation of optimal transport cost with respect to the weak* convergence of measures.
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In this paper we construct families of homology spheres which bound 4manifolds with intersection forms isomorphic to $E_8$. We show that these families have arbitrary large correction terms. This result says that among homology spheres, the difference of the maximal rank of minimal sublattice of definite filling and the maximal rank of even definite filling is arbitrarily large.
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We explore the representation theory of Renner monoids associated to classical groups and their Hecke algebras. In Cartan type $A_n$, the Hecke algebra is a natural deformation of the rook monoid algebra, and its representation theory has been studied extensively by Solomon and Halverson, among others. It is known that the character tables are block upper triangular, i.e. $M=AY=YB$ for some matrices $A$ and $B$. We compute the $A$ and $B$ matrices in Cartan type $B_n$ by using the results of Li, Li, and Cao to pursue analogous results to those of Solomon. We then compute some type $B_n$ Hecke algebra character values by using the same $B$ matrix as in the monoid case.
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We generalize DengDu's folding argument, for the bounded derived category D(Q) of an acyclic quiver Q, to the finite dimensional derived category D(Gamma Q) of the Ginzburg algebra Gamma Q associated to Q. We show that the Fstable category of D(Gamma Q) is equivalent to the finite dimensional derived category D(Gamma\SS) of the Ginzburg algebra Gamma\SS associated to the specie \SS, which is folded from Q. Then we show that, if (Q,\SS) is of Dynkin type, the principal component Stab_0 D(Gamma\SS) of the space of the stability conditions of D(Gamma\SS) is canonically isomorphic to the principal component Stab_0^F D(Gamma Q) of the space of Fstable stability conditions of D(Gamma Q). As an application, we show that, if (Q,\SS) is of type (A_3, B_2) or (D_4, G_2), the space Stab^N D(Gamma Q) of numerical stability conditions in Stab^0 D(Gamma Q), consists of Br Gamma Q/Br Gamma\SS many connected components, each of which is isomorphic to Stab^0 D(Gamma\SS) \cong Stab^F D(Gamma Q).
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In this article we study a class of delay differential equations with infinite delay in weighted spaces of uniformly continuous functions. We focus on the integrated semigroup formulation of the problem and so doing we provide an spectral theory. As a consequence we obtain a local stability result and a Hopf bifurcation theorem for the semiflow generated by such a problem
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We construct geometric models for the $\mathbb P^1$spectrum $M_{\mathbb P^1}(Y)$, which computes in GarkushaPanin's theory of framed motives \cite{GP14} a positively motivically fibrant $\Omega_{\mathbb P^1}$ replacement of $\Sigma_{\mathbb P^1}^\infty Y$ for a smooth scheme $Y\in \Sm_k$ over a perfect field $k$. Namely, we get the $T$spectrum in the category of pairs of smooth indschemes that defines $\mathbb P^1$spectrum of pointed sheaves termwise motivically equivalent to $M_{\mathbb P^1}(Y)$.
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