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Consider the polynomial ring in countably infinitely many variables over a field of characteristic zero, together with its natural action of the infinite general linear group G. We study the algebraic and homological properties of finitely generated modules over this ring that are equipped with a compatible G-action. We define and prove finiteness properties for analogues of Hilbert series, systems of parameters, depth, local cohomology, Koszul duality, and regularity. We also show that this category is built out of a simpler, more combinatorial, quiver category which we describe explicitly. Our work is motivated by recent papers in the literature which study finiteness properties of infinite polynomial rings equipped with group actions. (For example, the paper by Church, Ellenberg and Farb on the category of FI-modules, which is equivalent to our category.) Along the way, we see several connections with the character polynomials from the representation theory of the symmetric groups. Several examples are given to illustrate that the invariants we introduce are explicit and computable. [ABSTRACT FROM AUTHOR]

The 4-dimensional Sklyanin algebras, over ℂ., A(E, τ), are constructed from an elliptic curve E and a translation automorphism τ of E. The Klein vierergruppe Γ acts as graded algebra automorphisms of A(E, τ). There is also an action of Γ as automorphisms of the matrix algebra M2(ℂ.) making it isomorphic to the regular representation. The main object of study in this paper is the invariant subalgebra à := A(E, τ)⊗M2(ℂ))Γ. Like A(E, τ), à is noetherian, generated by 4 degree-one elements modulo six quadratic relations, Koszul, Artin-Schelter regular of global dimension 4, has the same Hilbert series as the polynomial ring on 4 variables, satisfies the x condition, and so on. These results are special cases of general results proved for a triple (A, T, H) consisting of a Hopf algebra H, an (often graded) H-comodule algebra A, and an H-torsor T. Those general results involve transferring properties between A, A ⊗ T, and (A ⊗ T)coH. We then investigate à from the point of view of non-commutative projective geometry. We examine its point modules, line modules, and a certain quotient ... := Ã/(Θ,Θ') where Θ and Θ' are homogeneous central elements of degree two. In doing this we show that à differs from A in interesting ways. For example, the point modules for A are parametrized by E and 4 more points, whereas à has exactly 20 point modules. Although ... is not a twisted homogeneous coordinate ring in the sense of Artin and Van den Bergh, a certain quotient of the category of graded ...-modules is equivalent to the category of quasi-coherent sheaves on the curve E/E[2] where E[2] is the 2-torsion subgroup. We construct line modules for à that are parametrized by the disjoint union (E/(ξ1)) ⊔ (E/(ξ2)) ⊔ (E/(ξ3)) of the quotients of E by its three subgroups of order 2. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the behavior of the unique solution uδ ∈ H01 (Ω), as δ → 0, to the equation div(sδA∇uδ) + k2s0Σuδ = s0 f in Ω, where Ω is a smooth connected bounded open subset of ℝd with d = 2 or 3, f ∈ L2(Ω), k is a non-negative constant, A is a uniformly elliptic matrixvalued function, Σ is a real function bounded above and below by positive constants, and sδ is a complex function whose real part takes the values 1 and −1 and whose imaginary part is positive and converges to 0 as δ goes to 0. This is motivated from a result of Nicorovici, McPhedran, and Milton; another motivation is the concept of complementary media. After introducing the reflecting complementary media, complementary media generated by reflections, we characterize f for which ||uδ|| H1(Ω) remains bounded as δ goes to 0. For such an f, we also show that uδ converges weakly in H1(Ω) and provide a formula to compute the limit. [ABSTRACT FROM AUTHOR]

Representing the conditional independences present in a multivariate random vector via graphs has found widespread use in applications, and such representations are popularly known as graphical models or Markov random fields. These models have many useful properties, but their fundamental attractive feature is their ability to reflect conditional independences between blocks of variables through graph separation, a consequence of the equivalence of the pairwise, local, and global Markov properties demonstrated by Pearl and Paz (1985). Modern-day applications often necessitate working with either an infinite collection of variables (such as in a spatial-temporal field) or approximating a large high-dimensional finite stochastic system with an infinite-dimensional system. However, it is unclear whether the conditional independences present in an infinite-dimensional random vector or stochastic process can still be represented by separation criteria in an infinite graph. In light of the advantages of using graphs as tools to represent stochastic relationships, we undertake in this paper a general study of infinite graphical models. First, we demonstrate that näıve extensions of the assumptions required for the finite case results do not yield equivalence of the Markov properties in the infinite-dimensional setting, thus calling for a more in-depth analysis. To this end, we proceed to derive general conditions which do allow representing the conditional independence in an infinite-dimensional random system by means of graphs, and our results render the result of Pearl and Paz as a special case of a more general phenomenon. We conclude by demonstrating the applicability of our theory through concrete examples of infinite-dimensional graphical models. [ABSTRACT FROM AUTHOR]

We present two simple descriptions of the denominator vectors of the cluster variables of a cluster algebra of finite type, with respect to any initial cluster seed: one in terms of the compatibility degrees between almost positive roots defined by S. Fomin and A. Zelevinsky, and the other in terms of the root function of a certain subword complex. These descriptions only rely on linear algebra. They provide two simple proofs of the known fact that the d-vector of any non-initial cluster variable with respect to any initial cluster seed has non-negative entries and is different from zero. [ABSTRACT FROM AUTHOR]

We consider the non-local operator of variable order as follows: Lf(x) = ... (f(x + z) -- f(x) -- (∇ f(x), z) 1{|z|≤1})n(x, z)/... dz, where α(x) ∈ [α0, α2] for any x ∈ Rd and some constants 0 < α0 ≤ α2 < 2 and n(x, z) is a positive Borel measurable function bounded from above and below. Under mild continuity conditions on α(x) and n(x, z), we establish the H¨older regularity for the associated semigroups. The proof is based on a new construction of the coupling by reflection for non-local operator L, and the results successfully apply to both stable-like processes in the sense of Bass and time-change of symmetric stable processes. [ABSTRACT FROM AUTHOR]

We consider the ɸ4 model in one space dimension with propagation speeds that are small deviations from a constant function. In the constantspeed case, a stationary solution called the kink is known explicitly, and the recent work of Kowalczyk, Martel, and Muñoz established the asymptotic stability of the kink with respect to odd perturbations in the natural energy space. We show that a stationary kink solution exists also for our class of nonconstant propagation speeds, and extend the asymptotic stability result by taking a perturbative approach to the method of Kowalczyk, Martel, and Muñoz. This requires an understanding of the spectrum of the linearization around the variable-speed kink. [ABSTRACT FROM AUTHOR]

We prove the global Calderón-Zygmund estimates for second order parabolic equations in nondivergence form in weighted variable exponent Lebesgue spaces. We assume that the associated variable exponent is log-Hölder continuous, the weight is of a certain Muckenhoupt class with respect to the variable exponent, the coefficients of the equation are the functions of small bonded mean oscillation, and the underlying domain is a C1,1-domain. [ABSTRACT FROM AUTHOR]

We introduce two q-analogues of the 2D-Hermite polynomials which are functions of two complex variables. We derive explicit formulas, orthogonality relations, raising and lowering operator relations, generating functions, and Rodrigues formulas for both families. We also introduce a q-2D analogue of the disk polynomials (Zernike polynomials) and derive similar formulas for them as well, including evaluating certain connection coefficients. Some of the generating functions may be related to Rogers-Ramanujan type identities. [ABSTRACT FROM AUTHOR]

We extend the theory of Rubio de Francia extrapolation, including off-diagonal, limited range and A∞ extrapolation, to the weighted variable Lebesgue spaces Lp(·)(w). As a consequence we are able to show that a number of different operators from harmonic analysis are bounded on these spaces. The proofs of our extrapolation results are developed in a way that outlines a general approach to proving extrapolation theorems on other Banach function spaces. [ABSTRACT FROM AUTHOR]

We give a complete characterization of polynomials in two complex variables that are cyclic with respect to the coordinate shifts acting on Dirichlet-type spaces in the bidisk, which include the Hardy space and the Dirichlet space of the bidisk. The cyclicity of a polynomial depends on both the size and nature of the zero set of the polynomial on the distinguished boundary. The techniques in the proof come from real analytic function theory, determinantal representations for polynomials, and harmonic analysis on curves. [ABSTRACT FROM AUTHOR]

We study the 1D Klein-Gordon equation with variable coefficient nonlinearity. This problem exhibits an interesting resonant interaction between the spatial frequencies of the nonlinear coefficients and the temporal oscillations of the solutions. In the case when only the cubic coefficients are variable we prove L∞ scattering and smoothness of the solution in weighted spaces with the help of both quadratic and cubic normal forms transformations. In the case of cubic interactions these normal forms appear to be novel. [ABSTRACT FROM AUTHOR]

A connection between the theory of formal groups and arithmetic number theory is established. In particular, it is shown how to construct general Almkvist-Meurman-type congruences for the universal Bernoulli polynomials that are related with the Lazard universal formal group (based on earlier works of the author). Their role in the theory of L-genera for multiplicative sequences is illustrated. As an application, sequences of integer numbers are constructed. New congruences are also obtained, useful to compute special values of a new class of Riemann-Hurwitz-type zeta functions. [ABSTRACT FROM AUTHOR]

Motivated by the representation theory of quivers with potential introduced by Derksen, Weyman and Zelevinsky and by work of Caldero and Chapoton, who gave explicit formulae for the cluster variables of cluster algebras of Dynkin type, we associate a Caldero-Chapoton algebra AA to any (possibly infinite-dimensional) basic algebra A. By definition, AA is (as a vector space) generated by the Caldero-Chapoton functions CA(M) of the decorated representations M of A. If A = P (Q, W) is the Jacobian algebra defined by a 2-acyclic quiver Q with non-degenerate potential W, then we have AQ Ç AA Ç AQup, where AQ and AQup are the cluster algebra and the upper cluster algebra associated to Q. The set BA of generic Caldero-Chapoton functions is parametrized by the strongly reduced components of the varieties of representations of the Jacobian algebra P (Q,W) and was introduced by Geiss, Leclerc and Schroer. Plamondon parametrized the strongly reduced components for finite-dimensional basic algebras. We generalize this to arbitrary basic algebras. Furthermore, we prove a decomposition theorem for strongly reduced components. We define BA for arbitrary A, and we conjecture that BA is a basis of the Caldero-Chapoton algebra AA. Thanks to the decomposition theorem, all elements of BA can be seen as generalized cluster monomials. As another application, we obtain a new proof for the sign-coherence of g-vectors. [ABSTRACT FROM AUTHOR]

The bivariate moment problem for a sequence β &#8801 β(6) of degree 6 remains unsolved, but we prove that if the associated 10 X 10 moment matrix M3(β) satisfies M3 ≥ 0 and rank M3 ≥ 6, then β admits a sequence of approximate representing measures, and β(5) has a representing measure. More generally, let Fd denote the closure of the positive flat moment matrices of degree 2d in n variables. Each matrix in Fd admits computable approximate representing measures, and in 2013, Jiawang Nie and the author began to study concrete conditions for membership in this class. Let β ≡ β(2d) = {βi}i∊zn+,∣i∣≤2d, β0 > 0, denote a real n-dimensional sequence of degree 2d. If the corresponding moment matrix Md = Md(β) is the limit of a sequence of positive flat moment matrices {M(k)d}, i.e., M(k)d ≥ and rank M(k) = rank M(k)d, then i) Md ≥ 0, ii) rank Md ≥ pd-1 = dim R [x1,...,xn]d-1, and iii) β(2d-1) admits a representing measure. We extend our earlier results by proving, conversely, that for n = 2, if Md satisfies certain positivity and rank conditions related to i)-iii), then Md is the limit of positive flat moment matrices. [ABSTRACT FROM AUTHOR]