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We consider general mixed p-spin mean field spin glass models and provide a method to prove that the spectral gap of the Dirichlet form associated with the Gibbs measure is of order one at sufficiently high temperature. Our proof is based on an iteration scheme relating the spectral gap of the N-spin system to that of suitably conditioned subsystems. [ABSTRACT FROM AUTHOR]

Let (K,d) be a connected compact metric space and p\in (1, \infty). Under the assumption of Kigami [ Conductive homogeneity of compact metric spaces and construction of p-energy , Memoirs of the European Mathematical Society, vol. 5, Europea Mathematical Society (EMS), Berline, 2023, Assumption 2.15] and the conductive p-homogeneity, we show that \mathcal {W}^p(K)\subset C(K) holds if and only if p>\operatorname {dim}_{AR}(K,d), where \mathcal {W}^p(K) is Kigami's (1,p)-Sobolev space and \operatorname {dim}_{AR}(K,d) is the Ahlfors regular dimension. [ABSTRACT FROM AUTHOR]

In the framework of quasi‐regular strongly local Dirichlet form (E,D(E))$(\mathcal {E},D(\mathcal {E}))$ on L2(X;m)$L^2(X;\mathfrak {m})$ admitting minimal E$\mathcal {E}$‐dominant measure μ$\mu$, we construct a natural p$p$‐energy functional (Ep,D(Ep))$(\mathcal {E}^{\,p},D(\mathcal {E}^{\,p}))$ on Lp(X;m)$L^p(X;\mathfrak {m})$ and (1,p)$(1,p)$‐Sobolev space (H1,p(X),∥·∥H1,p)$(H^{1,p}(X),\Vert \cdot \Vert _{H^{1,p}})$ for p∈]1,+∞[$p\in]1,+\infty [$. In this paper, we establish the Clarkson‐type inequality for (H1,p(X),∥·∥H1,p)$(H^{1,p}(X),\Vert \cdot \Vert _{H^{1,p}})$. As a consequence, (H1,p(X),∥·∥H1,p)$(H^{1,p}(X),\Vert \cdot \Vert _{H^{1,p}})$ is a uniformly convex Banach space, hence it is reflexive. Based on the reflexivity of (H1,p(X),∥·∥H1,p)$(H^{1,p}(X),\Vert \cdot \Vert _{H^{1,p}})$, we prove that (generalized) normal contraction operates on (Ep,D(Ep))$(\mathcal {E}^{\,p},D(\mathcal {E}^{\,p}))$, which has been shown in the case of various concrete settings, but has not been proved for such a general framework. Moreover, we prove that (1,p)$(1,p)$‐capacity Cap1,p(A)<∞${\rm Cap}_{1,p}(A)<\infty$ for open set A$A$ admits an equilibrium potential eA∈D(Ep)$e_A\in D(\mathcal {E}^{\,p})$ with 0≤eA≤1$0\le e_A\le 1$ m$\mathfrak {m}$‐a.e. and eA=1$e_A=1$ m$\mathfrak {m}$‐a.e. on A$A$. [ABSTRACT FROM AUTHOR]

We prove that certain closable derivations on the GNS Hilbert space associated with a non-tracial weight on a von Neumann algebra give rise to GNS-symmetric semigroups of contractive completely positive maps on the von Neumann algebra. [ABSTRACT FROM AUTHOR]

We construct symmetric self-similar Dirichlet forms on unconstrained Sierpinski carpets, which are natural extension of planar Sierpinski carpets by allowing the small cells to live off the 1/k grids. The intersection of two cells can be a line segment of irrational length, and the non-diagonal assumption is dropped in this recurrent setting. [ABSTRACT FROM AUTHOR]

The goal of this paper is to establish Green function estimates for a class of purely discontinuous symmetric Markov processes with jump kernels degenerate at the boundary and critical killing potentials. The jump kernel and the killing potential depend on several parameters. We establish sharp two-sided estimates on the Green functions of these processes for all admissible values of the parameters involved. Depending on the regions where the parameters belong, the estimates on the Green functions are different. In fact, the estimates have three different forms. As applications, we prove that the boundary Harnack principle holds in certain region of the parameters and fails in some other region of the parameters. Together with the main results of our previous paper [Potential Anal., online, 2021], we completely determine the region of the parameters where the boundary Harnack principle holds. [ABSTRACT FROM AUTHOR]

Let |$({\mathcal{M}},d,\mu)$| be the metric measure space with a Dirichlet form |$\mathscr{E}$|⁠. In this paper, we obtain that the Campanato function and the Lipschitz function do always coincide. Our approach is based on the harmonic extension technology, which extends a function u on |${\mathcal{M}}$| to its Poisson integral P t u on |${\mathcal{M}}\times\mathbb{R}_+$|⁠. With this tool in hand, we can utilize the same Carleson measure condition of the Poisson integral to characterize its Campanato/Lipschitz trace, and hence, they are equivalent to each other. This equivalence was previously obtained by Macías–Segovia [Adv. Math. 1979]. However, we provide a new proof, via the boundary value problem for the elliptic equation. This result indicates the famous saying of Stein–Weiss at the beginning of Chapter II in their book [Princeton Mathematical Series, No. 32, 1971]. [ABSTRACT FROM AUTHOR]

We construct a strongly local regular Dirichlet form on the golden ratio Sierpinski gasket, which is a self-similar set without a finitely ramified cell structure, via a study on the trace of an electrical network on an infinite graph. The Dirichlet form is the unique one that is self-similar in the sense of an infinite iterated function system, and is decimation invariant with respect to a graph-directed construction. The proof is based on a fixed point problem of a renormalization map, inspired by Sabot's celebrated work for finitely ramified fractals. Lastly, the Hunt process associated with the Dirichlet form satisfies a two-sided sub-Gaussian heat kernel estimate. [ABSTRACT FROM AUTHOR]

We show the existence and uniqueness as well as boundedness of weak solutions to linear elliptic equations with L^2-drifts of negative divergence and singular zero-order terms which are positive. Our main target is to show the L^r-contraction properties of the unique weak solutions. Indeed, using the Dirichlet form theory, we construct a sub-Markovian C_0-resolvent of contractions and identify it to the weak solutions. Furthermore, we derive an L^1-stability result through an extended version of the L^1-contraction property. [ABSTRACT FROM AUTHOR]

For a Dirichlet series of form F(s) = exp{sλ1} + P+∞ k=2 fk exp{sλk} (1) abcolutely convergent in the half-plane Π0 = {s: Re s < 0} new sufficient conditions for the close-to-pseudoconvexity are found and the obtained result is applied to studying of solutions linear differential equations of second order with exponential coefficients. In particular, are proved the following statements: 1) Let λk = λk-1 + λ1 and fk > 0 for all k ≥ 2. If 1 ≤ λ2f2/λ1 ≤ 2 and λkfk-λk+1fk+1 ↘ q ≥ 0 as k → +∞ then function of form (1) is close-to-pseudoconvex in Π0 (Theorem 3). This theorem complements Alexander's criterion obtained for power series. 2) If either -h2 ≤ γ ≤ 0 or γ = h2 then differential equation (1-ehs)2w′′-h(1-e2hs)w′+γe2hs = 0 (h > 0, γ ∈ R) has a solution w = F of form (1) with the exponents λk = kh and the the abscissa of absolute convergence σa = 0 that is close-to-pseudoconvex in Π0 (Theorem 4). [ABSTRACT FROM AUTHOR]

Let λ = (λn)n∈N0 be a non-negative sequence increasing to +∞, τ (λ) = limn→∞(ln n/λn), and D0(λ) be the class of all Dirichlet series of the form F(s) = P∞ n=0 an(F)esλn absolutely convergent in the half-plane Re s < 0 with an(F) ̸= 0 for at least one integer n ≥ 0. Also, let α be a continuous function on [x0,+∞) increasing to +∞, β be a continuous function on [a, 0) such that β(σ) → +∞ as σ ↑ 0, and γ be a continuous positive function on [b, 0). In the article, we investigate the growth of a Dirichlet series F ∈ D0(λ) depending on the behavior of the sequence (|an(F)|) in terms of its α, β, γ-orders determined by the equalities... where μ(σ) = max{|an(F)|eσλn: n ≥ 0} and M(σ) = sup{|F(s)|: Re s = σ} are the maximal term and the supremum modulus of the series F, respectively. In particular, if for every fixed t > 0 we have α(tx) ∼ α(x) as x → +∞, β(tσ) ∼ t-ρβ(σ) as σ ↑ 0 for some fixed ρ > 0, 0 < limσ↑0 γ(tσ)/γ(σ) ≤ limσ↑0 γ(tσ)/γ(σ) < +∞, Φ(σ) = α-1(β(σ))/γ(σ) for all σ ∈ [σ0, 0), eΦ(x) = max{xσ -Φ(σ): σ ∈ [σ0, 0)} for all x ∈ R, and ΔΦ(λ) = limn→∞(-ln n/eΦ(λn)), then: (a) for each Dirichlet series F ∈ D0(λ) we have... (b) if τ (λ) > 0, then for each p0 ∈ [0,+∞] and any positive function Ψ on [c, 0) there exists a Dirichlet series F ∈ D0(λ) such that R∗ α,β,γ(F) = p0 and M(σ, F) ≥ Ψ(σ) for all σ ∈ [σ0, 0); (c) if τ (λ) = 0, then (Rα,β,γ(F))1/ρ ≤ (R∗ α,β,γ(F))1/ρ + ΔΦ(λ) for every Dirichlet series F ∈ D0(λ); (d) if τ (λ) = 0, then for each p0 ∈ [0,+∞] there exists a Dirichlet series F ∈ D0(λ) such that R∗ α,β,γ(F) = p0 and (Rα,β,γ(F))1/ρ = (R∗ α,β,γ(F))1/ρ + ΔΦ(λ). [ABSTRACT FROM AUTHOR]

Motivated by problems where jumps across lower dimensional subsets and sharp transitions across interfaces are of interest, this paper studies the properties of fractional bounded variation (BV)-type spaces. Two different natural fractional analogs of classical BV are considered: BVa, a space induced from the Riesz-fractional gradient that has been recently studied by Comi-Stefani; and bva, induced by the Gagliardo-type fractional gradient often used in Dirichlet forms and Peridynamics--this one is naturally related to the Caffarelli-Roquejoffre-Savin fractional perimeter. Our main theoretical result is that the latter bva actually corresponds to the Gagliardo-Slobodeckij space Wα>1. As an application, using the properties of these spaces, novel image denoising models are introduced and their corresponding Fenchel predual formulations are derived. The latter requires density of smooth functions with compact support. We establish this density property for convex domains. [ABSTRACT FROM AUTHOR]

The tensorization problem for Sobolev spaces asks for a characterization of how the Sobolev space on a product metric measure space X × Y can be determined from its factors. We show that two natural descriptions of the Sobolev space from the literature coincide, W 1,2(X × Y) = J 1,2 (X, Y), thus settling the tensorization problem for Sobolev spaces in the case p =2, when X and Y are infinitesimally quasi-Hilbertian, i.e., the Sobolev space W 1,2 admits an equivalent renorming by a Dirichlet form. This class includes in particular metric measure spaces X, Y of finite Hausdorff dimension as well as infinitesimally Hilbertian spaces. More generally, for p ε (1,∞) we obtain the norm-one inclusion ||f|| J1,p(X,Y) ≤ ||f|| W1,p(X×Y) and show that the norms agree on the algebraic tensor product W1,p(X) ⊗ W 1,p(Y) ⊂ W 1,p(X × Y). When p = 2 and X and Y are infinitesimally quasi-Hilbertian, standard Dirichlet forms theory yields the density of W 1,2(X) ⊗ W 1,2(Y) in J 1,2(X, Y), thus implying the equality of the spaces. Our approach raises the question of the density of W 1,p(X) ⊗ W 1,p(Y) in J 1,p(X, Y) in the general case. [ABSTRACT FROM AUTHOR]

In this paper, we provide relations among the following properties: the tail triviality of a probability measure μ on the configuration space Υ ; the finiteness of a suitable L 2 -transportation-type distance d ¯ Υ ; the irreducibility of local μ -symmetric Dirichlet forms on Υ . As an application, we obtain the ergodicity (i.e., the convergence to the equilibrium) of interacting infinite diffusions having logarithmic interaction and arising from determinantal/permanental point processes including sine 2 , Airy 2 , Bessel α , 2 ( α ≥ 1 ), and Ginibre point processes. In particular, the case of the unlabelled Dyson Brownian motion is covered. For the proof, the number rigidity of point processes in the sense of Ghosh–Peres plays a key role. [ABSTRACT FROM AUTHOR]

We establish the existence of a scaling limit \mathcal {E}_p of discrete p-energies on the graphs approximating a generalized Sierpiński carpet for p > d_{\mathrm {ARC}}, where d_{\mathrm {ARC}} is the Ahlfors regular conformal dimension of the underlying generalized Sierpiński carpet. Furthermore, the function space \mathcal {F}_{p} defined as the collection of functions with finite p-energies is shown to be a reflexive and separable Banach space that is dense in the set of continuous functions with respect to the supremum norm. In particular, (\mathcal {E}_2, \mathcal {F}_2) recovers the canonical regular Dirichlet form constructed by Barlow and Bass [Ann. Inst. H. Poincaré Probab. Statist. 25 (1989), pp. 225–257] or Kusuoka and Zhou [Probab. Theory Related Fields 93 (1992), pp. 169–196]. We also provide \mathcal {E}_{p}-energy measures associated with the constructed p-energy and investigate its basic properties like self-similarity and chain rule. [ABSTRACT FROM AUTHOR]

Albeverio, Kondratiev, and Röckner have introduced a type of differential geometry, which we call lifted geometry, for the configuration space ΓX of any manifold X. The name comes from the fact that various elements of the geometry of ΓX are constructed via lifting of the corresponding elements of the geometry of X. In this note, we construct a general algebraic framework for lifted geometry which can be applied to various "infinite dimensional spaces" associated to X. In order to define a lifted geometry for a "space", one dose not need any topology or local coordinate system on the space. As example and application, lifted geometry for spaces of Radon measures on X, mappings into X, embedded submanifolds of X, and tilings on X, are considered. The gradient operator in the lifted geometry of Radon measures is considered. Also, the construction of a natural Dirichlet form associated to a random measure is discussed. It is shown that Stokes' Theorem appears as "differentiability" of "boundary operator" in the lifted geometry of spaces of submanifolds. It is shown that (generalized) action functionals associated with Lagrangian densities on X form the algebra of smooth functions in a specific lifted geometry for the path-space of X. [ABSTRACT FROM AUTHOR]

We formulate a class of "homogeneous" Dirichlet forms (DF) that aims to explore those forms that do not satisfy the conventional energy self-similar identity (degenerate DFs). This class of DFs has been studied in Hambly Jones (J. Theoret. Probab., 15, 285–322 2002), Hambly and Kumagai (Potential Anal., 8, 359–397 1998), Hambly and Yang (J. Fractal Geom., 6, 1–51 2019) and Hattori et al. (Probab. Theory Related Fields, 100, 85–116 1994) in connection with the asymptotically one-dimensional diffusions on the Sierpinski gaskets (SG) and their generalizations. In this paper, we give a systematic study of such DFs and their spectral properties. We also emphasize the construction of some new homogeneous DFs. Moreover, a basic assumption on the resistance growth that was required in Hambly and Kumagai (Potential Anal., 8, 359–397 1998) to investigate the heat kernel and the existence of the "non-fixed point" limiting diffusion is verified analytically. [ABSTRACT FROM AUTHOR]

Consider the semigroup of random walk on a complete graph, which we call the Potts semigroup. Diaconis and Saloff-Coste (Ann Appl Probab 6(3):695–750, 1996) computed the maximum ratio between the relative entropy and the Dirichlet form, obtaining the constant α 2 in the 2-log-Sobolev inequality (2-LSI). In this paper, we obtain the best possible non-linear inequality relating entropy and the Dirichlet form (i.e., p-NLSI, p ≥ 1 ). As an example, we show α 1 = 1 + 1 + o (1) log q . Furthermore, p-NLSIs allow us to conclude that for q ≥ 3 , distributions that are not a product of identical distributions can have slower speed of convergence to equilibrium, unlike the case q = 2 . By integrating the 1-NLSI we obtain new strong data processing inequalities (SDPI), which in turn allows us to improve results of Mossel and Peres (Ann Appl Probab 13(3):817–844, 2003) on reconstruction thresholds for Potts models on trees. A special case is the problem of reconstructing color of the root of a q-colored tree given knowledge of colors of all the leaves. We show that to have a non-trivial reconstruction probability the branching number of the tree should be at least log q log q - log (q - 1) = (1 - o (1)) q log q. This recovers previous results (of Sly in Commun Math Phys 288(3):943–961, 2009 and Bhatnagar et al. in SIAM J Discrete Math 25(2):809–826, 2011) in (slightly) more generality, but more importantly avoids the need for any coloring-specific arguments. Similarly, we improve the state-of-the-art on the weak recovery threshold for the stochastic block model with q balanced groups, for all q ≥ 3 . To further show the power of our method, we prove optimal non-reconstruction results for a broadcasting on trees model with Gaussian kernels, closing a gap left open by Eldan et al. (Combin Probab Comput 31(6):1048–1069, 2022). These improvements advocate information-theoretic methods as a useful complement to the conventional techniques originating from the statistical physics. [ABSTRACT FROM AUTHOR]

Let p ∈ N, s=(s1,...,sp) ∈ Cp, h=(h1,...,hp) ∈ R+p, (n)=(n1,...,np) ∈ Np and the sequences λ(n)=(λn1(1),...,λnp(p)) are such that 0<λ(j)1<λk(j)<λk+1(j)↑+∞as k→∞ for every j=1,...,p. For a=(a1,...,ap) and c=(c1,...,cp) let (a,c)=a1c1+...+apcp, and we say that a>c if aj>cj for all 1≤j≤p. For a multiple Dirichlet series ... absolutely converges in Π0p = {s: Res < 0}, concepts of pseudostarlikeness and pseudoconvexity are introduced and criteria for pseudostarlikeness and the pseudoconvexity are proved. Using the obtained results, we investigated neighborhoods of multiple Dirichlet series, Hadamard compositions, and properties of solutions of some differential equations. [ABSTRACT FROM AUTHOR]

We define rigorously a solution to the fourth-order total variation flow equation in R n . If n ≥ 3 , it can be understood as a gradient flow of the total variation energy in D − 1 , the dual space of D 0 1 , which is the completion of the space of compactly supported smooth functions in the Dirichlet norm. However, in the low dimensional case n ≤ 2 , the space D − 1 does not contain characteristic functions of sets of positive measure, so we extend the notion of solution to a larger space. We characterize the solution in terms of what is called the Cahn-Hoffman vector field, based on a duality argument. This argument relies on an approximation lemma which itself is interesting. We introduce a notion of calibrability of a set in our fourth-order setting. This notion is related to whether a characteristic function preserves its form throughout the evolution. It turns out that all balls are calibrable. However, unlike in the second-order total variation flow, the outside of a ball is calibrable if and only if n ≠ 2. If n ≠ 2 , all annuli are calibrable, while in the case n = 2 , if an annulus is too thick, it is not calibrable. We compute explicitly the solution emanating from the characteristic function of a ball. We also provide a description of the solution emanating from any piecewise constant, radially symmetric datum in terms of a system of ODEs. We define rigorously a solution to the fourth-order total variation flow equation in . If , it can be understood as a gradient flow of the total variation energy in , the dual space of , which is the completion of the space of compactly supported smooth functions in the Dirichlet norm. However, in the low dimensional case , the space does not contain characteristic functions of sets of positive measure, so we extend the notion of solution to a larger space. We characterize the solution in terms of what is called the Cahn-Hoffman vector field, based on a duality argument. This argument relies on an approximation lemma which itself is interesting. We introduce a notion of calibrability of a set in our fourth-order setting. This notion is related to whether a characteristic function preserves its form throughout the evolution. It turns out that all balls are calibrable. However, unlike in the second-order total variation flow, the outside of a ball is calibrable if and only if . If , all annuli are calibrable, while in the case , if an annulus is too thick, it is not calibrable. We compute explicitly the solution emanating from the characteristic function of a ball. We also provide a description of the solution emanating from any piecewise constant, radially symmetric datum in terms of a system of ODEs. [ABSTRACT FROM AUTHOR]

We show L r (ℝ d , μ) -uniqueness for any r ∈ (1 , 2 ] and the essential self-adjointness of a Dirichlet operator L f = Δ f + 〈 1 ρ ∇ ρ , ∇ f 〉 , f ∈ C 0 ∞ (ℝ d) with d ≥ 3 and μ = ρ d x. In particular, ∇ ρ is allowed to be in L loc d (ℝ d , ℝ d) or in L loc 2 + (ℝ d , ℝ d) for some > 0 , while ρ is required to be locally bounded below and above by strictly positive constants. The main tools in this paper are elliptic regularity results for divergence and non-divergence type operators and basic properties of Dirichlet forms and their resolvents. [ABSTRACT FROM AUTHOR]

This paper addresses the boundary output feedback stabilisation of a general class of 1-D reaction–diffusion PDEs with delayed boundary measurement. The output takes the form of either a Dirichlet or Neumann trace. The output delay can be arbitrarily large. The control strategy is composed of a finite-dimensional observer that is used to observe a delayed version of the first modes of the PDE and a predictor component that is employed to obtain the control input to be applied at the current time. For any given value of the output delay, we assess the stability of the resulting closed-loop system provided the order of the observer is selected large enough. Taking advantage of this result, we discuss the extension of the control strategy to the case of simultaneous input and output delays. [ABSTRACT FROM AUTHOR]

In this note we study the eigenvalue growth of infinite graphs with discrete spectrum. We assume that the corresponding Dirichlet forms satisfy certain Sobolev-type inequalities and that the total measure is finite. In this sense, the associated operators on these graphs display similarities to elliptic operators on bounded domains in the continuum. Specifically, we prove lower bounds on the eigenvalue growth and show by examples that corresponding upper bounds cannot be established. [ABSTRACT FROM AUTHOR]

This study aimed to verify the applicability of the community of inquiry (CoI) survey instrument in MOOC involving 1,186 college students from 11 different disciplines in China. Exploratory factor analysis was used to explore potential factor structure models, and confirmatory factor analysis was utilized to verify the four-factor structure obtained from exploratory factor analysis. The original three- and new six-factor structure models were also included in the study. Confirmatory factor analysis results indicating that all three models fit very well with the data. Then Chi-square difference test was used to select the optimal model. Results indicate that the six-factor structure model with teaching presence, social presence, cognitive presence, design and organization, affective expression, and resolution is the optimal one, with good convergent and discriminant validity. Especially, the chi-square difference results indicate that design and organization can be significantly distinguished from teaching presence, whereas affective expression can be significantly distinguished from social presence, and resolution can be significantly distinguished from cognitive presence. Based on these findings, the present study argues that the six-factor structure model can provide a better understanding for the fine design and implementation of MOOC. [ABSTRACT FROM AUTHOR]

This study performed a scoping review of the literature concerning the use of technology in mathematics education published between January 1981 and March 2022 to explore research trends. After the defined filtering process, we retrieved 2,433 articles from Web of Science, ERIC, and PsycInfo databases and employed Latent Dirichlet Allocation (LDA) topic modeling to extract key terms and topics from the selected articles. The analysis focused on the four aspects: (a) evolution of research trends of technology use in mathematics education, (b) frequently used words, (c) latent research topics, and (d) research trends for particular topics. The findings revealed a steady increase in research interest, and the combination of frequently used words in the article abstracts suggests popular research topics that have been studied during the set period. The results of LDA identified seven research topics that were not precisely aligned with those identified in prior studies on mathematics education or educational technology. This implied technology integration into mathematics education as a distinctive research area. Over time, the seven topics showed different research trends (stable, fluctuating, increasing, and decreasing). We discussed plausible reasons for these varied patterns and proposed implications based on the research findings. [ABSTRACT FROM AUTHOR]

Consider the following time-dependent stable-like operator with drift: L t φ (x) = ∫ R d [ φ (x + z) − φ (x) − z (α) ⋅ ∇ φ (x) ] σ (t , x , z) ν α (d z) + b (t , x) ⋅ ∇ φ (x) , where d ⩾ 1 , ν α is an α -stable type Lévy measure with α ∈ (0 , 1 ] and z (α) = 1 α = 1 1 | z | ⩽ 1 z , σ is a real-valued Borel function on R + × R d × R d and b is an R d -valued Borel function on R + × R d. By using the Littlewood-Paley theory, we establish the well-posedness for the martingale problem associated with L t under the sharp balance condition α + β ⩾ 1 , where β is the Hölder index of b with respect to x. Moreover, we also study a class of stochastic differential equations driven by Markov processes with generators of the form L t. We prove the pathwise uniqueness of strong solutions for such equations when the coefficients are in certain Besov spaces. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish the results of a double Dirichlet average of the generalized Bessel-Maitland function by using fractional derivative. We obtain the solution in the compact form of a double Dirichlet average of the generalized Bessel-Maitland function. Further, several special cases involving a number of well-known functions such as the Bessel-Maitland function, Mittag Leffler functions, Wright-hypergeometric functions and H-functions etc. have been established. Numerous expansions of the generalised Bessel-Maitland function, which reduces to the Mittag-Leffler function, have been studied and applied to solve a wide range of problems in physics, biology, chemistry and engineering. In the context of fractional calculus, Bessel-Maitland function, Mittag-Leffler functions, hypergeometric function and H-functions etc. have been widely studied. Since Carlson first introduced the concept of the Dirichlet average and its various forms. The Dirichlet average of elementary function like power function, exponential function etc is given by many notable mathematicians. These averages have been investigated and utilized in a number of different fields. [ABSTRACT FROM AUTHOR]

Many studies have quantified the effects of TV ad spending or gross rating points on brand sales. Yet this effect is likely moderated by the different types of brand-related messages or cues (e.g., logo, brand attributes) embedded in the ads and by the ways (e.g., explicitly or implicitly) these cues are conveyed to TV audiences. The authors thus measure 17 cues often used within ads to build brand awareness (or salience) and brand image and investigate their influence on ad effectiveness. Technically, the study builds a dynamic model to quantify the effects of advertising on sales; builds a robust and interpretable (i.e., nonparametric and sparse) factor model that integrates correlated, left-censored branding cues; and then models the effects of advertising as a function of the factors identified by these cues. An analysis of 177 campaigns aired by 62 brands finds that salience cues (e.g., logo) and benefit and attribute messages moderate ad effectiveness. It also finds that explicit cues are more effective than implicit ones; nonetheless, the primary drivers of ad effectiveness are visual salience cues: the duration and frequency with which the logo and the duration with which the product are displayed. The study can thus suggest ways brand and ad agency managers can improve the effects of creative ad content on sales. [ABSTRACT FROM AUTHOR]

We consider the metric growth in Liouville quantum gravity (LQG) for γ ∈ (0 , 2) . We show that a process associated with the trace of the free field on the boundary of a filled LQG ball is stationary, for every γ ∈ (0 , 2) . The infinitesimal version of this stationarity combined with an explicit expression of the generator of the evolution of the trace field (h t) provides a formal invariance equation that a measure on trace fields must satisfy. When considering a modified process corresponding to an evolution of LQG surfaces, we prove that the invariance equation is satisfied by an explicit σ -finite measure on trace fields. This explicit measure on trace fields only corresponds to the pure gravity case. On the way to prove this invariance, we retrieve the specificity of both γ = 8 / 3 and of the LQG dimension d γ = 4 . In this case, we derive an explicit expression of the (nonsymmetric) Dirichlet form associated with the process (h t) and construct dynamics associated with its symmetric part. [ABSTRACT FROM AUTHOR]

We present conservativeness criteria for sub-Markovian semigroups generated by divergence type operators with specified infinitesimally invariant measures. The conservativeness criteria in this article are derived by L 1 -uniqueness and imply that a given infinitesimally invariant measure becomes an invariant measure. We explore further conditions on the coefficients of the partial differential operators that ensure the uniqueness of the invariant measure beyond the case where the corresponding semigroups are recurrent. A main observation is that for conservativeness and uniqueness of invariant measures in this article, no growth conditions are required for the partial derivatives related to the anti-symmetric matrix of functions C = (c i j ) 1 ≤ i , j ≤ d that determine a part of the drift coefficient. As stochastic counterparts, our results can be applied to show not only the existence of a pathwise unique and strong solution up to infinity to a corresponding Itô-SDE, but also the existence and uniqueness of invariant measures for the family of strong solutions. [ABSTRACT FROM AUTHOR]

Introducing the notion of extended Schrödinger spaces, we define the criticality and subcriticality of Schrödinger forms in the manner similar to the recurrence and transience of Dirichlet forms. We show that a Schrödinger form is critical (resp. subcritical) if and only if there exists an excessive function of the associated Schrödinger semigroup and the Dirichlet form defined by h-transform of the excessive function is recurrent (resp. transient). We give an analytical condition for the subcriticality of Schrödinger forms in terms of the bottom of spectrum. We introduce a subclass {\mathcal {K}}_H of the local Kato class and show a Schrödinger form with potential in {\mathcal {K}}_H is critical. Critical Schrödinger forms lead us to critical Hardy-type inequalities. As an example, we treat fractional Schrödinger operators with potential in {\mathcal {K}}_H and reconsider the classical Hardy inequality by our approach. [ABSTRACT FROM AUTHOR]

Fields in the social sciences, such as education research, have started to expand the use of computer-based research methods to supplement traditional research approaches. Natural language processing techniques, such as topic modeling, may support qualitative data analysis by providing early categories that researchers may interpret and refine. This study contributes to this body of research and answers the following research questions: (RQ1) What is the relative coverage of the latent Dirichlet allocation (LDA) topic model and human coding in terms of the breadth of the topics/themes extracted from the text collection? (RQ2) What is the relative depth or level of detail among identified topics using LDA topic models and human coding approaches? A dataset of student reflections was qualitatively analyzed using LDA topic modeling and human coding approaches, and the results were compared. The findings suggest that topic models can provide reliable coverage and depth of themes present in a textual collection comparable to human coding but require manual interpretation of topics. The breadth and depth of human coding output is heavily dependent on the expertise of coders and the size of the collection; these factors are better handled in the topic modeling approach. [ABSTRACT FROM AUTHOR]

We give necessary and sufficient conditions for a regular semi-Dirichlet form to enjoy a new Feller type property, which we call weak Feller property. Our characterization involves potential theoretic as well as probabilistic aspects and seems to be new even in the symmetric case. As a consequence, in the symmetric case, we obtain a new variant of a decomposition principle of the essential spectrum for (the self-adjoint operators induced by) regular symmetric Dirichlet forms and a Persson type theorem, which applies e.g. to Cheeger forms on RCD ∗ spaces. [ABSTRACT FROM AUTHOR]

This paper is devoted to characterizing so-called order isomorphisms intertwining the L 2 -semigroups of two quasi-regular Dirichlet forms. We first show that every unitary order isomorphism intertwining semigroups is the composition of h-transformation and quasi-homeomorphism. In addition, under the assumption that the underlying spaces admit so-called irreducible decompositions for Dirichlet forms, every (not necessarily unitary) order isomorphism intertwining semigroups can be expressed as the composition of h-transformation, quasi-homeomorphism and multiplication by a certain step function. [ABSTRACT FROM AUTHOR]

We establish transience criteria for symmetric nonlocal Dirichlet forms on L2(Rd;dx)$L^2({\mathbb {R}}^d;{\rm d}x)$ in terms of the coefficient growth rates at infinity. Applying these criteria, we find a necessary and sufficient condition for recurrence of Dirichlet forms of symmetric stable‐like with unbounded/degenerate coefficients. This condition indicates that both of the coefficient growth rates of small and big jump parts affect the sample path properties of the associated symmetric jump processes. [ABSTRACT FROM AUTHOR]

This paper aims at proving the local boundedness and continuity of solutions of the heat equation in the context of Dirichlet spaces under some rather weak additional assumptions. We consider symmetric local regular Dirichlet forms, which satisfy mild assumptions concerning (1) the existence of cut‐off functions, (2) a local ultracontractivity hypothesis, and (3) a weak off‐diagonal upper bound. In this setting, local weak solutions of the heat equation, and their time derivatives, are shown to be locally bounded; they are further locally continuous, if the semigroup admits a locally continuous density function. Applications of the results are provided including discussions on the existence of locally bounded heat kernel; L∞$L^\infty$ structure results for ancient (local weak) solutions of the heat equation. [ABSTRACT FROM AUTHOR]

Dyson's model in infinite dimensions is a system of Brownian particles that interact via a logarithmic potential with an inverse temperature of β = 2. The stochastic process can be represented by the solution to an infinite-dimensional stochastic differential equation. The associated unlabeled dynamics (diffusion process) are given by the Dirichlet form with the sine2 point process as a reference measure. In a previous study, we proved that Dyson's model in infinite dimensions is irreducible, but left the ergodicity of the unlabeled dynamics as an open problem. In this paper, we prove that the unlabeled dynamics of Dyson's model in infinite dimensions are ergodic. [ABSTRACT FROM AUTHOR]

The aim of this article is to prove that diffusion processes in R d with a drift can be approximated by suitable Markov chains on n − 1 Z d . Moreover, we investigate sufficient conditions on the edge weights which guarantee convergence of the associated Markov chains to such Markov processes. Analogous questions are answered for a large class of nonsymmetric jump processes. The proofs of our results rely on regularity estimates for weak solutions to the corresponding nonsymmetric parabolic equations and Dirichlet form techniques. [ABSTRACT FROM AUTHOR]

We show that, for the Möbius function μ(n), we have ... μ(n) = o(xϑ) for any ϑ > 0.55. This improves on a result of Motohashi and Ramachandra from 1976, which is valid for ϑ > 7/12. Motohashi and Ramachandra's result corresponded to Huxley's 7/12 exponent for the prime number theorem in short intervals. The main new idea leading to the improvement is using Ramaré's identity to extract a small prime factor from the n-sum. The proof method also allows us to improve on an estimate of Zhan for the exponential sum of the Möbius function as well as some results on multiplicative functions and almost primes in short intervals. [ABSTRACT FROM AUTHOR]

In the ordinary theory of Sobolev spaces on domains of ℝn, the p -energy is defined as the integral of |∇ f |p. In this paper, we try to construct a p-energy on compact metric spaces as a scaling limit of discrete p-energies on a series of graphs approximating the original space. In conclusion, we propose a notion called conductive homogeneity under which one can construct a reasonable p-energy if p is greater than the Ahlfors regular conformal dimension of the space. In particular, if p = 2, then we construct a local regular Dirichlet form and show that the heat kernel associated with the Dirichlet form satisfies upper and lower sub-Gaussian type heat kernel estimates. As examples of conductively homogeneous spaces, we present new classes of square-based self-similar sets and rationally ramified Sierpmski crosses, where no diffusions were constructed before. [ABSTRACT FROM AUTHOR]

Let Hk denote the space of primitive holomorphic cusp forms of even integral weight k for the full modular group Γ = SL(2, ℤ). Denote by λ sym m f (n) the nth normalized coefficient of the Dirichlet expansion of the mth symmetric power L-function associated to f. In this paper, we establish the asymptotic formulas of sums of pairwise maxima concerning the normalized coefficients of symmetric power L-functions. We also establish similar results for the normalized coefficients of Rankin-Selberg L-functions L(symif × symjf, s) and L(symif × symjg, s) attached to f and g, respecitvely. As applications, we also consider the proportion of the sign changes among the difference of normalized coefficients associated with symmetric power L-functions and Rankin-Selberg L-functions attached to f and g, respectively. [ABSTRACT FROM AUTHOR]

The authors consider the existence, multiplicity, and uniqueness for polyharmonic problem with Navier boundary conditions. One of the interesting features in our proof is that we give a new attempt to consider the uniqueness of nontrivial solution for the above polyharmonic problem by using the theory of monotone mappings. This is probably the first time this theory is used to solve polyharmonic problems. Then we apply the fixed point theorems on cones to analyze the existence and multiplicity of positive solutions for the above polyharmonic problem. This is very difficult for partial differential equations, especially for polyharmonic equations. The main reason is that the Green's function for the above polyharmonic problem is unbounded. We overcome the difficulties by using some new techniques. The uniqueness of nontrivial solution and the existence of positive solutions for polyharmonic equations with Dirichlet boundary conditions are also considered. [ABSTRACT FROM AUTHOR]

We construct a Sierpinski carpet like fractal, on which a diffusion with sub-Gaussian heat kernel estimate does not exist, in contrast to previous researches on the existence of such diffusions, on the generalized Sierpinski carpets and recently introduced unconstrained Sierpinski carpets. [ABSTRACT FROM AUTHOR]

We study ergodic decompositions of Dirichlet spaces under intertwining via unitary order isomorphisms. We show that the ergodic decomposition of a quasi-regular Dirichlet space is unique up to a unique isomorphism of the indexing space. Furthermore, every unitary order isomorphism intertwining two quasi-regular Dirichlet spaces is decomposable over their ergodic decompositions up to conjugation via an isomorphism of the corresponding indexing spaces. [ABSTRACT FROM AUTHOR]

We consider weakly and strongly asymptotically mean value harmonic (amv-harmonic) functions on subriemannian and RCD settings. We demonstrate that, in non-collapsed RCD-spaces with vanishing metric measure boundary, Cheeger harmonic functions are weakly amv-harmonic and that, in Carnot groups, weak amv-harmonicity equivalently characterizes harmonicity in the sense of the sub-Laplacian. In homogeneous Carnot groups of step 2, we prove a Blaschke–Privaloff–Zaremba type theorem. Similar results are discussed in the settings of Riemannian manifolds and for Alexandrov surfaces. [ABSTRACT FROM AUTHOR]