Search

Over function fields of p -adic curves, we construct stably rational varieties in the form of homogeneous spaces of SL n with semisimple simply connected stabilizers and we show that strong approximation away from a non-empty set of places fails for such varieties. The construction combines the Lichtenbaum duality and the degree 3 cohomological invariants of the stabilizers. We then establish a reciprocity obstruction which accounts for this failure of strong approximation. We show that this reciprocity obstruction to strong approximation is the only one for counterexamples we constructed, and also for classifying varieties of tori. We also show that this reciprocity obstruction to strong approximation is compatible with known results for tori. At the end, we explain how a similar point of view shows that the reciprocity obstruction to weak approximation is the only one for classifying varieties of tori over p -adic function fields. [ABSTRACT FROM AUTHOR]

We have classified in [7] all linear control systems and studied their controllability and control sets on the homogeneous spaces L ﹨ H of the 3-dimensional Heisenberg group H through its closed subgroups L. In this paper, we characterize the controllability and control sets of the induced linear control systems in the homogeneous spaces left. In particular, we focus on the singularity of the induced drift vector fields. Quite a technical analysis with many cases reveals the control sets. We give some geometric illustrations. [ABSTRACT FROM AUTHOR]

We study structurable algebras and their associated Freudenthal triple systems over commutative rings. The automorphism groups of these triple systems are exceptional groups of type E 7 , and we realize groups of type E 6 as centralizers. When 6 is invertible, we further give a geometric description of homogeneous spaces of type E 7 / E 6 , and show that they parametrize principal isotopes of Brown algebras. As opposed to the situation over fields, we show that such isotopes may be non-isomorphic. [ABSTRACT FROM AUTHOR]

On a general open set of the euclidean space, we study the relation between the embedding of the homogeneous Sobolev space D 0 1 , p into L q and the summability properties of the distance function. We prove that, in the superconformal case (i.e. when 푝 is larger than the dimension), these two facts are equivalent, while in the subconformal and conformal cases (i.e. when 푝 is less than or equal to the dimension), we construct counterexamples to this equivalence. In turn, our analysis permits to study the asymptotic behavior of the positive solution of the Lane–Emden equation for the 푝-Laplacian with sub-homogeneous right-hand side, as the exponent 푝 diverges to ∞. The case of first eigenfunctions of the 푝-Laplacian is included, as well. As particular cases of our analysis, we retrieve some well-known convergence results, under optimal assumptions on the open sets. We also give some new geometric estimates for generalized principal frequencies. [ABSTRACT FROM AUTHOR]

Let (X , d , μ) be a space of homogeneous type in the sense of Coifman and Weiss. In this paper, we first establish several weighted norm estimates for various maximal functions. Then we show the weighted boundedness of the fractional integral I β associated with admissible functions and its commutators. Similarly to I β , corresponding results for Calderón–Zygmund operators T associated with admissible functions are also included in this article. [ABSTRACT FROM AUTHOR]

We present a new family of quantum Weyl algebras where the polynomial part is the quantum analog of functions on homogeneous spaces corresponding to symmetric matrices, skew symmetric matrices, and the entire space of matrices of a given size. The construction uses twisted tensor products and their deformations combined with invariance properties derived from quantum symmetric pairs. These quantum Weyl algebras admit U q (gl N) -module algebra structures compatible with standard ones on the polynomial part, have relations that are expressed nicely via matrices, and are closely related to an algebra arising in the theory of quantum bounded symmetric domains. [ABSTRACT FROM AUTHOR]

This paper is dedicated to investigating the Lp-bounds of wave operators W±(H, Δ²) associated with fourth-order Schrödinger operators H = Δ² + V on R³ with real potentials satisfying |V(x)| < (x)-µ for some µ > 0. A recent work by Goldberg and Green (2021) has demonstrated that wave operators W±(H, Δ³) are bounded on Lp (R³) for all 1 < p ∞ under the condition that µ > 9 and zero is a regular point of H. In the paper, we aim to further establish endpoint estimates for W±(H, Δ²) in two significant ways. First, we provide counterexamples to illustrate the unboundedness of W±(H, Δ²) on the endpoint spaces L¹ (R³) and L∞ (R³) for non-zero compactly supported potentials V. Second, we establish weak (1,1) estimates for the wave operators W± (H, Δ²) and their dual operators W±(H, Δ²)* in the case where zero is a regular point and µ > 11. These estimates depend critically on the singular integral theory of Calderón--Zygmund on a homogeneous space (X, dω) with a doubling measure dω. [ABSTRACT FROM AUTHOR]

Predicting student performance is crucial for both preventing failure and enabling personalized teaching-and-learning strategies. The digitalization of educational institutions has led to the collection of extensive student learning data over the years. Current research primarily focuses on short-term data, e.g. a single year or semester. In contrast, long-term data has the potential to offer a deeper insight into student behavior, thereby increasing the accuracy of predictions. However, the direct application of long-term data in prediction models assumes consistent data distributions over time. In the real world, evolutions in course content and structure can lead to variations in feature spaces (heterogeneity) and distribution shifts across different academic years, compromising the effectiveness of prediction models. To address these challenges, we introduce the Learning Ability Self-Adaptive Algorithm (LASA), which can adapt to the evolving feature spaces and distributions encountered in long-term data. LASA comprises two primary components: Learning Ability Modeling (LAM) and Long-term Distribution Alignment (LTDA). LAM assumes that students' responses to exercises are samples from distributions that are parameterized by their learning abilities. It then estimates these parameters from the heterogeneous student exercise response data, thereby creating a new homogeneous feature space to counteract the heterogeneity present in long-term data. Subsequently, LTDA employs multiple asymmetric transformations to align distributions of these new features across different years, thus mitigating the impact of distribution shifts on the model's performance. With these steps, LASA can generate well-aligned features with meaningful semantics. Furthermore, we propose an interpretable prediction framework including three components, i.e. LASA, a base classifier for outcome predictions, and Shapley Additive Explanations (SHAP) for elucidating the impact of specific features on student performance. Our exploration of long-term student data covers an eight-year period (2016-2023) from a face-to-face course at Tsinghua University. Comprehensive experiments demonstrate that leveraging long-term data significantly enhances prediction accuracy compared to short-term data, with LASA achieving up to a 7.9% increase. Moreover, when employing long-term data, LASA outperforms state-of-the-art models, ProbSAP and SFERNN, by an average accuracy improvement of 6.8% and 6.4%, respectively. We also present interpretable insights for pedagogical interventions based on a quantitative analysis of feature impacts on student performance. To the best of our knowledge, this study is the first to investigate student performance prediction in long-term data scenarios, addressing a significant gap in the literature. [ABSTRACT FROM AUTHOR]

The paper considers an axisymmetric stress state of a homogeneous elastic space with a circular disc-shaped crack, one of the edges of which is pressed into a cylindrical circular stamp with static friction. It is assumed that the contact zone is considered under the generalized law of dry friction, i.e. tangential contact stresses are proportional to normal contact pressure, while the proportionality coefficient depends on the radial coordinates of the points of the contacting surfaces and is directly proportional to them. Considering the fact that in this case the Abel images of contact stresses are also related in a similar way, the solution of the problem, with the help of rotation operators and theory of analytical functions, is reduced to an inhomogeneous Riemann problem for two functions and the closed solution in quadratures is constructed. A numerical analysis was carried out and regularities of changes in both normal and shear real contact stresses, as well as rigid displacement of the stamp depending on the physical and geometric parameters were revealed. [ABSTRACT FROM AUTHOR]

We first study the existence, uniqueness and well-posedness of a general class of integro-differential diffusion equation on L p ( G ) ( 1 < p < + ∞, G is a graded Lie group). We show the explicit solution of the considered equation. The equation involves a nonlocal in time operator (with a general kernel) and a positive Rockland operator acting on G. Also, we provide L p ( G ) − L q ( G ) ( 1 < p ⩽ 2 ⩽ q < + ∞) norm estimates and time decay rate for the solutions. In fact, by using some contemporary results, one can translate the latter regularity problem to the study of boundedness of its propagator which strongly depends on the traces of the spectral projections of the Rockland operator. Moreover, in many cases, we can obtain time asymptotic decay for the solutions which depends intrinsically on the considered kernel. As a complement, we give some norm estimates for the solutions in terms of a homogeneous Sobolev space in L 2 ( G ) that involves the Rockland operator. We also give a counterpart of our results in the setting of compact Lie groups. Illustrative examples are also given. [ABSTRACT FROM AUTHOR]

The Siegel-Jacobi space is a non-symmetric homogeneous space which is very important geometrically and arithmetically. In this short paper, we propose the basic problems in the geometry of the Siegel-Jacobi space. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the auto-ignition process of hydrogen in a hot oxidizer stream composed of oxygen–nitrogen and oxygen–water vapor with nitrogen/water vapor mass fractions in a range of 0.1–0.9. The temperature of the oxidizer varies from 1100 K to 1500 K and the temperature of hydrogen is assumed to be 300 K. The research is performed in 1D mixture fraction space and in a forced homogeneous isotropic turbulent (HIT) flow field. In the latter case, the Large Eddy Simulation (LES) method combined with the Eulerian Stochastic Field (ESF) combustion model is applied. The results obtained in mixture fraction space aim to determine the most reactive mixture fraction, maximum flame temperature, and dependence on the scalar dissipation rate. Among others, we found that the ignition in H 2 - O 2 - H 2 O mixtures occurs later than in H 2 - O 2 - N 2 mixtures, especially at low oxidizer temperatures. On the other hand, for a high oxidizer temperature, the ignitability of H 2 - O 2 - H 2 O mixtures is extended, i.e., the ignition occurs for a larger content of H 2 O and takes place faster. The 3D LES-ESF results show that the ignition time is virtually independent of initial conditions, e.g., randomness of an initial flow field and turbulence intensity. The latter parameter, however, strongly affects the flame evolution. It is shown that the presence of water vapor decreases ignitability and makes flames more prone to extinction. [ABSTRACT FROM AUTHOR]

We give a new formula for the Lichnerowicz Laplacian on normal homogeneous spaces in terms of Casimir operators. We derive some practical estimates and apply them to the known list of non-symmetric, compact, simply connected homogeneous spaces G / H with 퐺 simple whose standard metric is Einstein. This yields many new examples of Einstein metrics which are stable in the Einstein–Hilbert sense, which have long been lacking in the positive scalar curvature setting. [ABSTRACT FROM AUTHOR]

Let 퐺 be a real semisimple Lie group with finite centre and without compact factors, Q < G a parabolic subgroup and 푋 a homogeneous space of 퐺 admitting an equivariant projection on the flag variety G / Q with fibres given by copies of lattice quotients of a semisimple factor of 푄. Given a probability measure 휇, Zariski-dense in a copy of H = SL 2 ⁡ (R) in 퐺, we give a description of 휇-stationary probability measures on 푋 and prove corresponding equidistribution results. Contrary to the results of Benoist–Quint corresponding to the case G = Q , the type of stationary measures that 휇 admits depends strongly on the position of 퐻 relative to 푄. We describe possible cases and treat all but one of them, among others using ideas from the works of Eskin–Mirzakhani and Eskin–Lindenstrauss. [ABSTRACT FROM AUTHOR]

In this paper, we examine the geodesics on adjoint orbits of SL (n , ℝ) that are equipped with SO (n) -invariant metrics, where SO (n) is the maximal compact subgroup. Our primary technique involves translating this problem into a geometric problem in the tangent bundle of certain SO (n) -flag manifolds and describing the geodesic equations with respect to the Sasaki metric on the tangent bundle. Additionally, we utilize tools from Lie Theory to obtain explicit descriptions of families of geodesics. We provide a detailed analysis of the case for SL (2 , ℝ). [ABSTRACT FROM AUTHOR]

We present the notion of nonabelian descent type, which classifies torsors up to twisting by a Galois cocycle. This relies on the previous construction of kernels and nonabelian Galois 2-cohomology due to Springer and Borovoi. The necessity of descent types arises in the context of the descent theory where no torsors are given a priori , for example, when we wish to study the arithmetic properties such as the Brauer–Manin obstruction to the Hasse principle on homogeneous spaces without rational points. This new definition also unifies the types by Colliot-Thélène–Sansuc, the extended types by Harari–Skorobogatov, and the finite descent types by Harpaz–Wittenberg. [ABSTRACT FROM AUTHOR]

We determine all homogeneous metrics which are critical for some quadratic curvature functional in dimension four. [ABSTRACT FROM AUTHOR]

In this paper, firstly we study geodesic vectors for the m-th root homogeneous Finsler space admitting (α, β)-type. Then we obtain the necessary and sufficient condition for an arbitrary non-zero vector to be a geodesic vector for the m-th root homogeneous Finsler metric under mild conditions. Finally, we consider a quartic homogeneous Finsler metric on a simply connected nilmanifold of dimension five equipped with an invariant Riemannian metric and an invariant vector field. We study its geodesic vectors and classify the set of all the homogeneous geodesics on 5-dimensional nilmanifolds. [ABSTRACT FROM AUTHOR]

This research relies on a two‐stage content analysis of Reddit to test the assumption that by emphasizing othering and polarizing opinions, homogeneous ideological spaces trigger impolite and uncivil comments towards others, as opposed to spaces that enable cross partisan‐ideological exposure, which overall are seen as positive for democracy. It examines subreddits with varying probability of occurrence of echo chamber effects to measure the prevalence and the targets of incivility and studies the interactions in which these comments emerge in discussions about immigration. Findings show that incivility is higher in subreddits that bring together users with the same ideological and political profile, in which echo chamber effects are more likely to occur, but the redditors’ reactions are similar in both types of environments. The targets of incivility vary according to the political orientation of the comments and the political context of the country. [ABSTRACT FROM AUTHOR]

The main focus of this contribution is on the harmonic Bergman spaces ℬαp on the q-homogeneous tree 픛q endowed with a family of measures σα that are constant on the horocycles tangent to a fixed boundary point and turn out to be doubling with respect to the corresponding horocyclic Gromov distance. A central role is played by the reproducing kernel Hilbert space ℬα2 for which we find a natural orthonormal basis and formulae for the kernel. We also consider the atomic Hardy space and the bounded mean oscillation space. Appealing to an adaptation of Calderón–Zygmund theory and to standard boundedness results for integral operators on Lαp spaces with Hörmander-type kernels, we determine the boundedness properties of the Bergman projection. This work was inspired by [J. M. Cohen, F. Colonna, M. A. Picardello and D. Singman, Bergman spaces and Carleson measures on homogeneous isotropic trees, Potential Anal. 44(4) (2016) 745–766, doi:10.1007/s11118-015-9529-7; F. De Mari, M. Monti and M. Vallarino, Harmonic Bergman projectors on homogeneous trees, Potential Anal. 61 (2024) 153–182]. [ABSTRACT FROM AUTHOR]

Selective oxidation of hydrocarbons using molecular oxygen as sole oxidant under mild conditions remains a challenging task. In this context, metal‐organic frameworks (MOFs) have been widely used in various oxidation reactions due to their porosity, high surface area and designability. However, the slow diffusion of substrates/products in micropores of three‐dimensional (3D) bulk MOFs hinders the efficient catalytic performance of such materials. Herein an ultrathin two‐dimensional (2D) porphyrin‐based Zr‐MOF nanosheet (Zr‐TCPP) is synthesized through modulator‐control strategy. Subsequently, various metal ions are anchored into the porphyrin ring by post synthesis modification to afford a series of 2D Zr‐TCPP(M) (M=Mn, Fe, Co, Ni, Cu and Zn). Various structural characterization techniques indicate Zr‐TCPP(M) is nanoflower structure with ultrathin nanoplate petals which provides fully exposed accessible active sites. Among them, Zr‐TCPP(Fe) shows excellent catalytic performance in styrene epoxidation reactions and benzylic C−H oxidation reactions using O2 as sole oxidant under ambient temperature and pressure. The remarkable activity arises from high density of exposed porphyrin‐Fe active sites, low diffusion barriers for substrates and products, as well as a similar homogeneous reaction space. Furthermore, Zr‐TCPP(Fe) nanosheet is easily recycled by centrifugation and reused at least five times without significant loss of catalytic activity. [ABSTRACT FROM AUTHOR]

A topological space is said to be homogeneous if for any there exists a self-homeomorphism of such that . We propose a method for constructing topological spaces representable as a union of but not fewer homogeneous subspaces, where is an arbitrary given positive integer. Further, we present a solution of a similar problem for the case of infinitely many summands. [ABSTRACT FROM AUTHOR]

This paper investigates some well-posedness issues of the fractional inhomogeneous Schrödinger equation i u ˙ - (- Δ) γ u = ± | x | ρ | u | p - 1 u , where 0 < γ < 1 and ρ < 0 . Here, one considers the inter-critical regime 0 < s c : = N 2 - 2 γ + ρ p - 1 < γ , where s c is the energy critical exponent, which is the only one real number satisfying ‖ κ 2 γ + ρ p - 1 u 0 (κ ·) ‖ H ˙ s c = ‖ u 0 ‖ H ˙ s c . In order to avoid a loss of regularity in Strichartz estimates, one assumes that the datum is spherically symmetric. First, using a sharp Gagliardo–Nirenberg-type estimate, one develops a local theory in the space H ˙ γ ∩ H ˙ s c . Then, one investigates the L N (p - 1) ρ + 2 γ concentration of finite-time blow-up solutions bounded in H ˙ s c . Finally, one proves the existence of non-global solutions with negative energy. Since one considers the homogeneous Sobolev space H ˙ s c , the main difficulty here is to avoid the mass conservation law. [ABSTRACT FROM AUTHOR]

In this paper, we consider the parabolic–elliptic Keller–Segel system, which is coupled to the incompressible Navier–Stokes equations through transportation and friction. It is shown that when the system is diffused by Lévy motion, the well‐posedness of the mild solution to the corresponding Cauchy problem in homogeneous Besov spaces is established by means of the Banach fixed point theorem. Furthermore, we prove the Lorentz regularity in time direction and the maximal regularity of solutions. In addition, we obtain the additional regularity and explore the time decay property of global mild solutions. [ABSTRACT FROM AUTHOR]

In this paper, we give the flag curvature formula of general (α, β)-metrics of Berwald type. We study conformally related (α, β)-metrics, especially general (α, β)-metrics that are conformally related to invariant (α, β)-metrics. Also, a necessary and sufficient condition for a Finsler metric conformally related to an (α, β)-metric is given and conformally related Douglas Randers metrics are studied. Finally, we present some examples of conformally related (α, β)-metrics. [ABSTRACT FROM AUTHOR]

In this study, some new necessary and sufficient conditions for a two-weight, weak-type maximal inequality of the form φ 1 (λ) ∫ { x ∈ X : M f (x) > λ } ϱ (x) d μ (x) ≤ c ∫ X φ 2 c | f (x) | σ (x) d μ (x) are obtained in Orlicz classes, where M f is a Hardy–Littlewood maximal function defined on homogeneous spaces and ϱ is a weight function. [ABSTRACT FROM AUTHOR]

The weak-type BMO-regularity property for couples of quasi-Banach lattices of measurable functions was recently introduced as a suitable substitute for the usual BMO-regularity in connection with characterization of the K-closedness of Hardy-type spaces on the unit circle and stability for the real interpolation. It was characterized in terms of the BMO-regularity of couples ((X, Y)α,p, (X, Y)β,q), 0 < α < β < 1, of the real interpolation spaces. In the present note, a natural characterization of this property similar to that of BMO-regularity for couples of Banach lattices (X, Y) in terms of the BMO-regularity of X′Y is extended to couples of lattices of measurable functions on homogeneous type spaces. We also derive equivalent conditions corresponding to the limit case where α = 0. [ABSTRACT FROM AUTHOR]

The traditional approach to the exponential synthesis problem for the space of solutions of a homogeneous convolution-type equation in a convex domain assumes that this space is invariant under some differential operator. This assumption makes it possible to reduce the problem of exponential synthesis to the problem of spectral synthesis. Is this assumption due to the method used to solve the problem, or is the invariance of the solution space necessary for a positive answer to the exponential synthesis problem? To answer this question, we consider special equations of the convolution type, the equations with q-sided convolution. For these equations, we show that the requirement of invariance for the solution space is necessary and cannot be omitted if we assume that the solution space admits exponential synthesis with a free choice of the convex region and the characteristic function of the equation. [ABSTRACT FROM AUTHOR]

The physical capabilities of a reconfigurable assembly system (RAS) increase the agility and responsiveness of the system in highly volatile market conditions. However, achieving optimal RAS utilization entails solving complex optimization problems effectively and efficiently. These optimizations often define homogenous sets of problem instances. While algorithm configuration in such homogeneous contexts traditionally adopts a "one-size-fits-all" approach, recent studies have shown the potential of per-instance algorithm configuration (PIAC) methods in these settings. In this work, we evaluate and compare the performance of different PIAC methods in this context, namely Hydra—a state-of-the-art PIAC method—and a simpler case-based reasoning (CBR) approach. We evaluate the impact of the tuning time budget and/or the number of unique problem instances used for training on each of the method's performance and robustness. Our experiments show that whilst Hydra fails to improve upon the default algorithm configuration, the CBR method can lead to 16% performance increase using as few as 100 training instances. Following these findings, we evaluate Hydra's methodology when applied to homogenous instance spaces. This analysis shows the limitations of Hydra's inference mechanisms in these settings and showcases the advantages of distance-based approaches used in CBR. [ABSTRACT FROM AUTHOR]

In this paper, the authors prove that bilinear pseudo-differential operator... [ABSTRACT FROM AUTHOR]

In this paper, state space representation based approach to relative stability of Linear Time Invariant (LTI) systems is discussed. Specifically, given a bound on the output, relative Bounded Input-Bounded Output (BIBO) stability of LTI systems is discussed. The bound on dominant pole is first determined using input-output description of LTI system. Using state space representation of homogeneous (no external input) LTI system, the components of state vector are upper bound. The results are applied to Continuous Time Markov Chains (i.e. homogeneous stochastic linear systems). Finally, using state space representation of arbitrary (with external input) Linear Time Invariant systems, the location of dominant pole is bounded. Using the upper bound, the location of line (parallel to imaginary-axis jω and to the left of it) is utilized for determining relative stability. [ABSTRACT FROM AUTHOR]

In this work, we generalize Kac's original many-particle binary stochastic model to derive a space homogeneous Boltzmann equation that includes a linear combination of higher-order collisional terms. First, we prove an abstract theorem about convergence from a finite hierarchy to an infinite hierarchy of coupled equations. We apply this convergence theorem on hierarchies for marginals corresponding to the generalized Kac model mentioned above. As a corollary, we prove propagation of chaos for the marginals associated to the generalized Kac model. In particular, the first marginal converges towards the solution of a Boltzmann equation including interactions up to a finite order and whose collision kernel is of Maxwell type with cut-off. [ABSTRACT FROM AUTHOR]

For linear control systems with bounded control range, chain controllability properties are analyzed. It is shown that there exists a unique chain control set and that it equals the sum of the control set around the origin and the center Lyapunov space of the homogeneous part. For the proof, the linear control system is extended to a bilinear control system on an augmented state space. This system induces a control system on projective space. For the associated control flow, attractor-repeller decompositions are used to show that the control system on projective space has a unique chain control set that is not contained in the equator. It is given by the image of the chain control set of the original linear control system. [ABSTRACT FROM AUTHOR]

We study the limiting distribution of dense orbits of a lattice subgroup |$\Gamma \leq \textrm{SL}(m+1,\mathbb{R})$| acting on |$H\backslash \textrm{SL}(m+1,\mathbb{R})$|⁠ , with respect to a filtration of growing norm balls. The novelty of our work is that the groups |$H$| we consider have infinitely many non-trivial connected components. For a specific such |$H$|⁠ , the homogeneous space |$H\backslash G$| identifies with |$X_{m,m+1}$|⁠ , a moduli space of rank |$m$| -discrete subgroups in |$\mathbb{R}^{m+1}$|⁠. This study is motivated by the work of Shapira-Sargent who studied random walks on |$X_{2,3}$|⁠. [ABSTRACT FROM AUTHOR]

We prove two rigidity results on holomorphic isometries into homogeneous Kähler manifolds. The first shows that a Kähler-Ricci soliton induced by the homogeneous metric of the Kähler product of a special generalized flag manifold (i.e. a flag of classical type or integral type) with a bounded homogeneous domain is trivial, i.e. Kähler-Einstein. In the second one we prove that: (i) a flat space is not relative to the Kähler product of a special generalized flag manifold with a homogeneous bounded domain, (ii) a special generalized flag manifold is not relative to the Kähler product of a flat space with a homogeneous bounded domain and (iii) a homogeneous bounded domain is not relative to the Kähler product of a flat space with a special generalized flag manifold. Our theorems strongly extend the results of Cheng and Hao [Ann. Global Anal. Geom. 60 (2021), pp. 167–180], Cheng, Di Scala, and Yuan [Internat. J. Math. 28 (2017), p. 1750027], Loi and Mossa [Proc. Amer. Math. Soc. 149 (2021), pp. 4931–4941], Loi and Mossa [Proc. Amer. Math. Soc. 151 (2023), pp. 3975–3984] and Umehara [Tokyo J. Math. 10 (1987), pp. 203–214]. [ABSTRACT FROM AUTHOR]

In this paper, we prove an existence result for a quasilinear elliptic variational inequality of the form u ∈ K ⊂ V : 0 ∈ - Δpu+aF(u) + ...IK(u) ∈ V* in the whole ℝ N under bilateral constraints K given by K = {v ∈ V : ϕ (x) ≤ v(x) ≤ Ψ(x) a.e. in ℝ N}, where Δp is the p-Laplacian, the underlying solution space V is the homogeneous Sobolev space (also called Beppo-Levi space) V = D1,p(ℝN) with 1 < p < N, and IK : V → ℝ U {+∞} is the indicator functional corresponding to K with its subdifferential ...IK. The lower order Nemytskij operator F is generated by a Carathéodory function f : ℝN x ℝ → ℝ, and the measurable and bounded coefficient α is supposed to decay like |x|- (N+α) at infinity. The growth conditions that we impose on f are such that the operator -- Δp + aF : V → V*, in general, is not coercive with respect to K which prevents us from applying standard existence results. Another difficulty, which arises due to the lack of compact embedding of V into Lq(ℝN) spaces, needs to be overcome in an appropriate way. Without assuming additional assumptions such as the existence of sub- and supersolutions, we are able not only to prove the existence of solutions, but also show the compactness of the set of all solutions in V. Finally, an extension of the theory is established, which allows us to deal with noncoercive bilateral variational-hemivariational inequalities in ℝN. The proof of our main existence result is based on a modified penalty approach. [ABSTRACT FROM AUTHOR]

We review some works on non-classical boundary conditions, and loading devices inspired by discussions with late Professor J.L. Ericksen. For a circular contact area between a body and a linearly elastic, isotropic and homogeneous half space as a loading device, we estimate the constant in the classical Winkler's expression of the normal surface traction at a point of the contact region in terms of the normal to the surface displacement of the same point. We also provide expressions for the tangential tractions at points of the common interface between the body and the loading device. The surface tractions are in general functionals of the contact surface displacements and their tangential derivatives. It is shown that many boundary conditions fall outside the usual definition of natural boundary conditions and may require special treatments in numerically solving boundary-value problems. [ABSTRACT FROM AUTHOR]

Let X be a space of homogeneous type. Assume that L is a non negative, selfadjoint operator on L 2 (X) satisfying the sub-Gaussian upper bounds. In this paper, we prove that ‖ (I + L) - n / 2 e i τ L f ‖ 1 , ∞ ≤ C (1 + | τ |) n / 2 ‖ f ‖ 1 , ∀ τ ∈ R. By interpolation, we obtain ‖ (I + L) - n | 1 / p - 1 / 2 | e i τ L f ‖ p ≤ C (1 + | τ |) n | 1 / p - 1 / 2 | ‖ f ‖ p , ∀ τ ∈ R , 1 < p < ∞. [ABSTRACT FROM AUTHOR]

We say that a sequence of proper geodesic spaces Xn ​ consists of almost homogeneous spaces if there is a sequence of discrete groups of isometries Gn ​ ≤ Iso(Xn) with diam(Xn/Gn) → 0 as n → ∞. We show that if a sequence (Xn, pn) of pointed almost homogeneous spaces converges in the pointed Gromov-Hausdorff sense to a space (X, p), then X is a nilpotent locally compact group equipped with an invariant geodesic metric. Under the above hypotheses, we show that if X is semi-locally-simply-connected, then it is a nilpotent Lie group equipped with an invariant sub-Finsler metric, and for n large enough, π1(X) is a subgroup of a quotient of π1(Xn). [ABSTRACT FROM AUTHOR]

We consider families of conical Kähler–Einstein metrics on rank one horosymmetric Fano manifolds, with decreasing cone angles along a codimension one orbit. At the limit angle, which is positive, we show that the metrics, restricted to the complement of that orbit, converge to (the pull‐back of) the Kähler–Einstein metric on the basis of the horosymmetric homogeneous space, which is a projective homogeneous space. Then we show that, on the symmetric space fibers, the rescaled metrics converge to Stenzel's Ricci flat Kähler metric. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the 3D incompressible magnetohydrodynamic equations and establish several logarithmically improved regularity criteria in the homogeneous Besov space. These criteria are based on certain assumptions made in the velocity field and pressure, respectively. The first result solely based on the velocity field indicates that the velocity field plays a dominant role in determining the regularity of weak solutions, when compared with the magnetic field. [ABSTRACT FROM AUTHOR]

In this paper we study the harmonic map heat flow on the euclidean space R d and we show an unconditional uniqueness result for maps with small initial data in the homogeneous Besov space B ˙ p , ∞ d p (R d) where d < p < ∞. As a consequence we obtain decay rates for solutions of the harmonic map flow of the form ‖ ∇ u (t) ‖ L ∞ (R d) ≤ C t − 1 2 . Additionally, under the assumption of a stronger spatial localization of the initial conditions, we show that the temporal decay happens in a self-similar way. We also explain that similar results hold for the biharmonic map heat flow and the semilinear heat equation with a power-type nonlinearity. [ABSTRACT FROM AUTHOR]

In this paper, the local existence and uniqueness of weak solutions to a d-dimensional non-resistive MHD equations in homogeneous Besov spaces are studied. Specifically we obtain the local existence of a weak solution $ (u,b) $ (u , b) of the non-resistive MHD equations for the initial data $ u_0\in \dot B^{\frac {d}{p}-1}_{p,1}(\mathbb {R}^{d}) $ u 0 ∈ B ˙ p , 1 d p − 1 (R d) and $ b_0\in \dot B^{\frac {d}{p}}_{p,1}(\mathbb {R}^{d}) $ b 0 ∈ B ˙ p , 1 d p (R d) with $ 1\le p \le \infty $ 1 ≤ p ≤ ∞ , and the uniqueness of the weak solution when $ 1\le p\le 2d $ 1 ≤ p ≤ 2 d. Compared with the previous results for the non-resistive MHD equations, in the local existence part, the range of p extends to $ 1\le p \le \infty $ 1 ≤ p ≤ ∞ from $ 1\le p\le 2d $ 1 ≤ p ≤ 2 d , but the uniqueness of the solution requires $ 1\le p\le 2d $ 1 ≤ p ≤ 2 d yet. [ABSTRACT FROM AUTHOR]

We prove effective density of random walks on homogeneous spaces, assuming that the underlying measure is supported on matrices generating a dense subgroup and having algebraic entries. The main novelty is an argument passing from high dimension to effective equidistribution in the setting of random walks on homogeneous spaces, exploiting the spectral gap of the associated convolution operator. [ABSTRACT FROM AUTHOR]

The wonderful compactification X m of a symmetric homogeneous space of type AIII (2 , m) for each m ≥ 4 is Fano, and its blowup Y m along the unique closed orbit is Fano if m ≥ 5 and Calabi–Yau if m = 4. Using a combinatorial criterion for K-polystability of smooth Fano spherical varieties obtained by Delcroix, we prove that X m admits a Kähler–Einstein metric for each m ≥ 4 and Y m admits a Kähler–Einstein metric if and only if m = 4 , 5. [ABSTRACT FROM AUTHOR]

We investigate variational, Lie and Killing symmetries of the Lagrangian of an essential class of four-dimensional (pseudo-)Riemannian manifolds, i.e., non-reductive homogeneous spaces. Moreover, a Lie algebra analysis is shown. With the help of a result of the Noether's theorem, we have presented the expressions for conservation laws corresponding to all variational symmetries. [ABSTRACT FROM AUTHOR]

An analytical solution to the dynamic response of a functionally graded cylindrical tube embedded in a full homogeneous space subjected to elastic SH waves is presented using the wavefunction expansion method. The shear modulus of the functionally graded tube is assumed to have a power-law variation with radius. The wave fields in the tube and unbounded surrounding medium are derived in a form of wavefunction series with unknown coefficients. The wave field coefficients are obtained by satisfying the boundary conditions at the inner and outer tube surfaces. The influences of the inhomogeneity exponent, shear modulus and thickness of the functionally graded tube, and the frequency of the incident SH waves on the dynamic stress concentration factor (DSCF) of the tube are systematically discussed. It is concluded that the application of functionally graded materials (FGMs) with a positive inhomogeneity exponent tends to weaken the DSCF along the inner surface of the tube significantly but intensify that along the outer surface slightly, which is beneficial to the long-term use of the tube. [ABSTRACT FROM AUTHOR]

We investigate three aspects of weak* convergence of the n-step distributions of random walks on finite volume homogeneous spaces $ G/\Gamma $ G / Γ of semisimple real Lie groups. First, we look into the obvious obstruction to the upgrade from Cesàro to non-averaged convergence: periodicity. We give examples where it occurs and conditions under which it does not. In a second part, we prove convergence towards Haar measure with exponential speed from almost every starting point. Finally, we establish a strong uniformity property for the Cesàro convergence towards Haar measure for uniquely ergodic random walks. [ABSTRACT FROM AUTHOR]

In this paper, we apply a result of the classification of a compact cohomogeneity one Riemannian manifold with a compact Lie group G to obtain a classification of compact cohomogeneity one locally conformal Kähler manifolds. In particular, we prove that the compact complex manifold is a complex one-dimensional torus bundle over a projective rational homogeneous, or cohomogeneity one manifold except of a class of manifolds with a generalized Hopf surface bundle over a projective rational homogeneous space. Additionally, it is a homogeneous compact complex manifold under the complexification G C of the given compact Lie group G under an extra condition that the related closed one form is cohomologous to zero on the generic G orbit. Moreover, the semi-simple part S of the Lie group action has hypersurface orbits, i.e., it is of cohomogeneity one with respect to the semi-simple Lie group S in that special case. [ABSTRACT FROM AUTHOR]

We show that, for any n\geq 2, there exists a homogeneous space of dimension d=8n-4 with metrics of Ric_{\frac {d}{2}-5}>0 if n\neq 3 and Ric_6>0 if n=3 which evolve under the Ricci flow to metrics whose Ricci tensor is not (d-4)-positive. Consequently, Ricci flow does not preserve a range of curvature conditions that interpolate between positive sectional and positive scalar curvature. This extends a theorem of Böhm and Wilking [Geom. Funct. Anal. 17 (2007), pp. 665–681] in the case of n=2. [ABSTRACT FROM AUTHOR]