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We characterize the support projection of a state evolving under the action of a with unbounded generator represented in the generalized GKSL form and we give another proof of the classical Lévy-Austin-Ornstein theorem. [ABSTRACT FROM PUBLISHER]

Classical harmonic analysis says that the spaces of homogeneous harmonic polynomials (solutions of the Laplace equation) are irreducible modules of the corresponding orthogonal Lie group (algebra) and the whole polynomial algebra is a free module over the invariant polynomials generated by harmonic polynomials. In this paper, we first establish two-parameter ℤ2-graded supersymmetric oscillator generalizations of the above theorem for the Lie superalgebra gl(n | m). Then we extend the result to two-parameter ℤ-graded supersymmetric oscillator generalizations of the above theorem for the Lie superalgebras osp(2n | 2m) and osp(2n + 1 | 2m). [ABSTRACT FROM AUTHOR]

In the mid 1980s, Lichtenstein, Pnueli, and Zuck proved a classical theorem stating that every formula of Past LTL (the extension of Linear Temporal Logic (LTL) with past operators) is equivalent to a formula of the form \(\bigwedge _{i=1}^n {\mathbf {G}}{\mathbf {F}}\varphi _i \vee {\mathbf {F}}{\mathbf {G}}\psi _i\) , where φi and ψi contain only past operators. Some years later, Chang, Manna, and Pnueli built on this result to derive a similar normal form for LTL. Both normalization procedures have a non-elementary worst-case blow-up, and follow an involved path from formulas to counter-free automata to star-free regular expressions and back to formulas. We improve on both points. We present direct and purely syntactic normalization procedures for LTL, yielding a normal form very similar to the one by Chang, Manna, and Pnueli, that exhibit only a single exponential blow-up. As an application, we derive a simple algorithm to translate LTL into deterministic Rabin automata. The algorithm normalizes the formula, translates it into a special very weak alternating automaton, and applies a simple determinization procedure, valid only for these special automata. [ABSTRACT FROM AUTHOR]

A recent theorem by T. Barthel, M. Hausmann, N. Naumann, T. Nikolaus, J. Noel, and N. Stapleton says that if A is a finite abelian p-group of rank r, then any finite A-space X which is acyclic in the nth Morava K-theory with n ⩾ r will have its subspace XA of fixed points acyclic in the (n - r)th Morava Ktheory. This is a chromatic homotopy version of P. A. Smith's classical theorem that if X is acyclic in mod p homology, then so is XA. The main purpose of this paper is to give an elementary proof of this new theorem that uses minimal background, and follows, as much as possible, the reasoning in standard proofs of the classical theorem. We also give a new fixed point theorem for finite dimensional, but possibly infinite, A-CW complexes, which suggests some open problems. [ABSTRACT FROM AUTHOR]

Using elementary probabilistic methods, in particular a variant of the Weak Law of Large Numbers related to the Bernoulli distribution, we prove that for every infinite compact spaces K and L the product K\times L admits a sequence \langle \mu _n\colon n\in \mathbb {N}\rangle of normalized signed measures with finite supports which converges to 0 with respect to the weak* topology of the dual Banach space C(K\times L)^*. Our approach is completely constructive—the measures \mu _n are defined by an explicit simple formula. We also show that this result generalizes the classical theorem of Cembranos [Proc. Amer. Math. Soc. 91 (1984), pp. 556–558] and Freniche [Math. Ann. 267 (1984), pp. 479–486] which states that for every infinite compact spaces K and L the Banach space C(K\times L) contains a complemented copy of the space c_0. [ABSTRACT FROM AUTHOR]

The problem of Zarankiewicz asks for the maximum number of edges in a bipartite graph on n vertices which does not contain the complete bipartite graph K k , k as a subgraph. A classical theorem due to Kővári, Sós, and Turán says that this number of edges is O n 2 - 1 / k . An important variant of this problem is the analogous question in bipartite graphs with VC-dimension at most d, where d is a fixed integer such that k ≥ d ≥ 2 . A remarkable result of Fox et al. (J. Eur. Math. Soc. (JEMS) 19:1785–1810, 2017) with multiple applications in incidence geometry shows that, under this additional hypothesis, the number of edges in a bipartite graph on n vertices and with no copy of K k , k as a subgraph must be O n 2 - 1 / d . This theorem is sharp when k = d = 2 , because by design any K 2 , 2 -free graph automatically has VC-dimension at most 2, and there are well-known examples of such graphs with Ω n 3 / 2 edges. However, it turns out this phenomenon no longer carries through for any larger d. We show the following improved result: the maximum number of edges in bipartite graphs with no copies of K k , k and VC-dimension at most d is o (n 2 - 1 / d) , for every k ≥ d ≥ 3 . [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence theorem for longest paths in sub-Lorentzian problems, which generalizes the classical theorem for globally hyperbolic Lorentzian manifolds. We specifically address the case of invariant structures on homogeneous spaces, as the conditions for the existence theorem in this case can be significantly simplified. In particular, it turns out that longest paths exist for any left-invariant sub-Lorentzian structures on Carnot groups. [ABSTRACT FROM AUTHOR]

Let H be a monoid and pH be the unique extension of the identity map on H to a monoid homomorphism F(H)-H, where we denote by F(X) the free monoid on a set X. Given A⊆H, an A-word z (i.e., an element of F(A)) is minimal if πH(z)≠πH(z′) for every permutation z′ of a proper subword of z. The minimal A-elasticity of H is then the supremum of all rational numbers m/n with m, n∊N+ such that there exist minimal A-words a and b of length m and n, resp., with πH(a)=πH(b). Among other things, we show that if H is commutative and A is finite, then the minimal A-elasticity of H is finite. This provides a non-trivial generalization of the finiteness part of a classical theorem of Anderson et al. from the case where H is cancellative, commutative, and finitely generated modulo units, and A is the set A (H) of atoms of H. We also demonstrate that commutativity is somewhat essential here, by proving the existence of an atomic, cancellative, finitely generated monoid with trivial group of units whose minimal A (H)-elasticity is infinite. [ABSTRACT FROM AUTHOR]

We consider the problem of finding nearly optimal solutions of optimization problems with random objective functions. Such problems arise widely in the theory of random graphs, theoretical computer science, and statistical physics. Two concrete problems we consider are (a) optimizing the Hamiltonian of a spherical or Ising p-spin glass model and (b) finding a large independent set in a sparse Erd\H os-Renyi graph. The following families of algorithms are considered: (a) low-degree polynomials of the input-a general framework that captures many prior algorithms; (b) low-depth Boolean circuits; (c) the Langevin dynamics algorithm, a canonical Monte Carlo analogue of the gradient descent algorithm. We show that these families of algorithms cannot have high success probability. For the case of Boolean circuits, our results improve the state-of-the-art bounds known in circuit complexity theory (although we consider the search problem as opposed to the decision problem). Our proof uses the fact that these models are known to exhibit a variant of the overlap gap property (OGP) of near-optimal solutions. Specifically, for both models, every two solutions whose objectives are above a certain threshold are either close to or far from each other. The crux of our proof is that the classes of algorithms we consider exhibit a form of stability (noise-insensitivity): a small perturbation of the input induces a small perturbation of the output. We show by an interpolation argument that stable algorithms cannot overcome the OGP barrier. The stability of Langevin dynamics is an immediate consequence of the well-posedness of stochastic differential equations. The stability of low-degree polynomials and Boolean circuits is established using tools from Gaussian and Boolean analysis-namely hypercontractivity and total influence, as well as a novel lower bound for random walks avoiding certain subsets, which we expect to be of independent interest. In the case of Boolean circuits, the result also makes use of Linial--Mansour--Nisan's classical theorem. Our techniques apply more broadly to low influence functions, and we expect that they may apply more generally. [ABSTRACT FROM AUTHOR]

On negatively curved compact manifolds, it is possible to associate to every closed form a bounded cocycle – hence a bounded cohomology class – via integration over straight simplices. The kernel of this map is contained in the space of exact forms. We show that in degree 2 this kernel is trivial, in contrast with higher degree. In other words, exact non-zero 2-forms define non-trivial bounded cohomology classes. This result is the higher dimensional version of a classical theorem by Barge and Ghys [Invent. Math. 92 (1988), pp. 509–526] for surfaces. As a consequence, one gets that the second bounded cohomology of negatively curved manifolds contains an infinite dimensional space, whose classes are explicitly described by integration of forms. This also showcases that some recent results by Marasco [Proc. Amer. Math. Soc. 151 (2023), pp. 2707–2715] can be applied in higher dimension to obtain new non-trivial results on the vanishing of certain cup products and Massey products. Some other applications are discussed. [ABSTRACT FROM AUTHOR]

Bohr's classical theorem and its generalizations are now active areas of research and have been the source of investigations in numerous function spaces. For harmonic mappings of the form f = h + g ¯ , we obtain an improved version of Bohr inequality for K-quasiregular harmonic mappings in the shifted disk Ω γ = { z ∈ C : | z + γ 1 - γ | < 1 1 - γ , γ ∈ [ 0 , 1) } which contains the unit disk D . In addition, we obtain sharp Bohr inequalities for class of K-quasiconformal harmonic mappings with bounded analytic part. [ABSTRACT FROM AUTHOR]

Consider a commutative monoid (M , + , 0) and a biadditive binary operation μ : M × M → M. We will show that under some additional general assumptions, the operation μ is automatically both associative and commutative. The main additional assumption is localizability of μ , which essentially means that a certain canonical order on M is compatible with adjoining some multiplicative inverses of elements of M. As an application we show that a division ring F is commutative provided that for all a ∈ F there exists a natural number k such that a − k is not a sum of products of squares. This generalizes the classical theorem that every archimedean ordered division ring is commutative to a more general class of formally real division rings that do not necessarily allow for an archimedean (total) order. Similar results about automatic associativity and commutativity are well-known for special types of partially ordered extended rings ("extended" in the sense that neither associativity nor commutativity of the multiplication is required by definition), namely in the uniformly bounded and the lattice-ordered cases, i.e. for (extended) f -rings. In these cases the commutative monoid in question is the positive cone of the partially ordered extended ring. We also discuss how these classical results can be obtained from our main theorem. [ABSTRACT FROM AUTHOR]

We prove that for every path H $H$, and every integer d $d$, there is a polynomial f $f$ such that every graph G $G$ with chromatic number greater than f(t) $f(t)$ either contains H $H$ as an induced subgraph, or contains as a subgraph the complete d $d$‐partite graph with parts of cardinality t $t$. For t=1 $t=1$ and general d $d$ this is a classical theorem of Gyárfás, and for d=2 $d=2$ and general t $t$ this is a theorem of Bonamy et al. [ABSTRACT FROM AUTHOR]

We prove that for any graph G of maximum degree at most Δ, the zeros of its chromatic polynomial χ G (x) (in C) lie inside the disc of radius 5.94Δ centered at 0. This improves on the previously best known bound of approximately 6.91Δ. We also obtain improved bounds for graphs of high girth. We prove that for every g there is a constant K g such that for any graph G of maximum degree at most Δ and girth at least g , the zeros of its chromatic polynomial χ G (x) lie inside the disc of radius K g Δ centered at 0, where K g is the solution to a certain optimization problem. In particular, K g < 5 when g ≥ 5 and K g < 4 when g ≥ 25 and K g tends to approximately 3.86 as g → ∞. Key to the proof is a classical theorem of Whitney which allows us to relate the chromatic polynomial of a graph G to the generating function of so-called broken-circuit-free forests in G. We also establish a zero-free disc for the generating function of all forests in G (aka the partition function of the arboreal gas) which may be of independent interest. [ABSTRACT FROM AUTHOR]

In this note we give a short proof of a classical theorem in the theory of Fourier integrals. [ABSTRACT FROM AUTHOR]

We prove that any complete, uniformly elliptic Weingarten surface in Euclidean 3-space whose Gauss map image omits an open hemisphere is a cylinder or a plane. This generalizes a classical theorem by Hoffman, Osserman and Schoen for constant mean curvature surfaces. In particular, this proves that planes are the only complete, uniformly elliptic Weingarten multigraphs. We also show that this result holds for a large class of non-uniformly elliptic Weingarten equations. In particular, this solves in the affirmative the Bernstein problem for entire graphs for that class of elliptic equations. To obtain these results, we prove that planes are the only complete multigraphs with quasiconformal Gauss map and bounded second fundamental form. [ABSTRACT FROM AUTHOR]

In this paper we present a monotonicity which extends a classical theorem of A. Schur comparing the chord length of a convex plane curve with that of a space curve of smaller curvature. The main result establishes a Schur's Theorem for spherical curves, which extends the well-known Cauchy's Arm Lemma. We also extend the result by allowing the curve with the smaller curvature being any curve in 핊 n . [ABSTRACT FROM AUTHOR]

In this paper the generalization of the converse of Baer's theorem for two-nilpotent variety of class row (n,m) is carried out. Baer proved that finiteness of G/Zn(G) implies that γn+1(G) is finite. Hekster proved the converse of the Baer's theorem with the assumption that G can be finitely generated. The Baer's theorem can be considered as a result of a classical theorem by Schur denoting that finiteness of G/Z(G) leads to the finiteness of G ′. The converse of the Baer's theorem has been proved conditionally by Taghavi et al. (2019), as well. In the Main Theorem, we prove that, if γm,n(G) ∩ Zn,m(G) = 1 and γm,n+i(G) is finite for some n, i,m ≥ 0. Then G/Zn,m(G) is finite in which γm,n(G) and Zn,m(G) denote verbal and marginal subgroups with respect to two-nilpotent variety of class row (n,m). Thus the generalization of the converse of Baer's theorem for two-nilpotent variety of groups valids by considering i = 0. In this article some other results are attained by the converse of the Baer's theorem. It is also concluded that when n = m = 1. Similar results are obtained for variety of the soluble groups. In addition, the converse of the Schur's theorem which proved by Halasi and Podoski is concluded in this paper, for two-nilpotent variety. We have also obtained some similar results of Chakaneh et al. (2019) for (n,m)-isoclinic family of groups and (1,m)-stem groups. [ABSTRACT FROM AUTHOR]

A variational problem for a conditional extremum of the Mayer type on the construction of the distribution law of the normal component of the velocity of injection of a cooled gas into a turbulent boundary layer under a supersonic flow regime that provides the minimum value of the convective heat flux transmitted from the hot gas to the streamlined surface is considered. The isoperimetric condition is the power of the injection control system, calculated taking into account Darcy's filtration law. To solve the optimal problem, we use the first integral for the conjugate system with respect to Lagrange multipliers, obtained earlier by the authors using the classical theorem of E. Noether on invariant variational problems and the Lie-Ovsyannikov infinitesimal apparatus. A.A. Dorodnitsyn's method of generalized integral relations, which has proven itself well in calculating the characteristics of boundary layers under various flow regimes, is used for the calculations. A computational experiment conducted for the case of a flow around a sphere shows the effectiveness of the optimal control law found in comparison with a uniform injection: the gain in the value of the minimized functional is 16.8%. The novelty of the study lies in the development of a method for solving the variational problem using the first integral for the conjugate system, as well as the Dorodnitsyn's method of generalized integral relations. The scientific significance of the study lies in the development of the theory of an optimally controlled boundary layer under a turbulent flow regime in supersonic gas flows. The results obtained may be of interest in the design of systems for the active thermal protection of surfaces in high-velocity gas flows. [ABSTRACT FROM AUTHOR]

Let \omega and \nu be radial weights on the unit disc of the complex plane such that \omega admits the doubling property \sup _{0\le r<1}\frac {\int _r^1 \omega (s)\,ds}{\int _{\frac {1+r}{2}}^1 \omega (s)\,ds}<\infty. Consider the one weight inequality \begin{equation} \|P_\omega (f)\|_{L^p_\nu }\le C\|f\|_{L^p_\nu },\quad 1

Extending a classical theorem of Sperner, we characterize the integers m such that there exists a maximal antichain of size m in the Boolean lattice Bn, that is, the power set of [ n ] : = { 1 , 2 , ... , n } , ordered by inclusion. As an important ingredient in the proof, we initiate the study of an extension of the Kruskal-Katona theorem which is of independent interest. For given positive integers t and k, we ask which integers s have the property that there exists a family F of k-sets with | F | = t such that the shadow of F has size s, where the shadow of F is the collection of (k − 1)-sets that are contained in at least one member of F . We provide a complete answer for t ≤ k + 1 . Moreover, we prove that the largest integer which is not the shadow size of any family of k-sets is 2 k 3 / 2 + 8 4 k 5 / 4 + O (k) . [ABSTRACT FROM AUTHOR]

The goal of this note is to provide a general lower bound on the number of even values of the Fourier coefficients of an arbitrary eta-quotient F , over any arithmetic progression. Namely, if g a , b (x) denotes the number of even coefficients of F in degrees n ≡ b (mod a) such that n ≤ x , then we show that g a , b (x) / x is unbounded for x large. Note that our result is very close to the best bound currently known even in the special case of the partition function p (n) (namely, x log log x , proven by Bellaïche and Nicolas in 2016). Our argument substantially relies upon, and generalizes, Serre's classical theorem on the number of even values of p (n) , combined with a recent modular-form result by Cotron et al. on the lacunarity modulo 2 of certain eta-quotients. Interestingly, even in the case of p (n) first shown by Serre, no elementary proof is known of this bound. At the end, we propose an elegant problem on quadratic representations, whose solution would finally yield a modular form-free proof of Serre's theorem. [ABSTRACT FROM AUTHOR]

We prove a version of Quillen's stratification theorem in equivariant homotopy theory for a finite group G , generalizing the classical theorem in two directions. Firstly, we work with arbitrary commutative equivariant ring spectra as coefficients, and secondly, we categorify it to a result about equivariant modules. Our general stratification theorem is formulated in the language of equivariant tensor-triangular geometry, which we show to be tightly controlled by the non-equivariant tensor-triangular geometry of the geometric fixed points. We then apply our methods to the case of Borel-equivariant Lubin–Tate E -theory E n _ , for any finite height n and any finite group G , where we obtain a sharper theorem in the form of cohomological stratification. In particular, this provides a computation of the Balmer spectrum as well as a cohomological parametrization of all localizing ⊗-ideals of the category of equivariant modules over E n _ , thereby establishing a finite height analogue of the work of Benson, Iyengar, and Krause in modular representation theory. [ABSTRACT FROM AUTHOR]

An equitable coloring of a graph G is a proper vertex coloring of G such that the sizes of any two color classes differ by at most one. In the paper, we pose a conjecture that offers a gap-one bound for the smallest number of colors needed to equitably color every block graph. In other words, the difference between the upper and the lower bounds of our conjecture is at most one. Thus, in some sense, the situation is similar to that of chromatic index, where we have the classical theorem of Vizing and the Andersen–Goldberg–Seymour conjecture for multigraphs. The results obtained in the paper support our conjecture. More precisely, we verify it in the class of block graphs in which each vertex belongs to a maximum independent set. We also show that the conjecture is true for block graphs which contain a vertex that does not lie in an independent set of size larger than two. Finally, we verify the conjecture for some symmetric-like block graphs. In order to derive our results we obtain structural characterizations of block graphs from these classes. [ABSTRACT FROM AUTHOR]

A moplex is a natural graph structure that arises when lifting Dirac's classical theorem from chordal graphs to general graphs. The notion is known to be closely related to lexicographic searches in graphs as well as to asteroidal triples, and has been applied in several algorithms related to graph classes, such as interval graphs, claw‐free, and diamond‐free graphs. However, while every noncomplete graph has at least two moplexes, little is known about structural properties of graphs with a bounded number of moplexes. The study of these graphs is, in part, motivated by the parallel between moplexes in general graphs and simplicial modules in chordal graphs: unlike in the moplex setting, properties of chordal graphs with a bounded number of simplicial modules are well understood. For instance, chordal graphs having at most two simplicial modules are intervals. In this work, we initiate an investigation of k $k$‐moplex graphs, which are defined as graphs containing at most k $k$ moplexes. Of particular interest is the smallest nontrivial case k=2 $k=2$, which forms a counterpart to the class of interval graphs. As our main structural result, we show that, when restricted to connected graphs, the class of 2‐moplex graphs is sandwiched between the classes of proper interval graphs and cocomparability graphs; moreover, both inclusions are tight for hereditary classes. From a complexity‐theoretic viewpoint, this leads to the natural question of whether the presence of at most two moplexes guarantees a sufficient amount of structure to efficiently solve problems that are known to be intractable on cocomparability graphs, but not on proper interval graphs. We develop new reductions that answer this question negatively for two prominent problems fitting this profile, namely, Graph Isomorphism and Max‐Cut. On the other hand, we prove that every connected 2‐moplex graph contains a Hamiltonian path, generalising the same property of connected proper interval graphs. Furthermore, for graphs with a higher number of moplexes, we lift the previously known result that graphs without asteroidal triples have at most two moplexes to the more general setting of larger asteroidal sets. [ABSTRACT FROM AUTHOR]

Locus computation is an essential issue in mathematics education, and a traditional feature of Dynamic Geometry software (DGS). Nowadays, the rising of new DGS programs, such as GeoGebra, merging DGS and Computer Algebra software (CAS) has fostered a combined approach to locus computation, quite performing in standard examples, but demanding an extended theoretical, and the related algorithmic counterpart, able to deal with less conventional situations. Here we formulate--and reflect about, yielding some proposals--on a few pending issues related to the protocols for the computation of parametric families of loci. Then we focus on a different source of difficulties, through the example of the very elementary and classical theorem of Thales. Thus, in the framework of the current development of automated deduction in geometry (ADG) tools, we will show how the automatic discovery of Thales's converse statement might require a locus computation that gives rise to an unexpected family of octic curves. Finally, we will exhibit how the handling (finding the equation, plotting, geometric characterization, etc.) of such curves requires the concourse of DGS and CAS programs, a mixed graphic-symbolic-numeric approach, and human-machine interaction, a cooperation that could be the basis towards achieving the required improvements concerning locus computation software. [ABSTRACT FROM AUTHOR]

Let G be a nontrivial transitive permutation group on a finite set Ω and recall that an element of G is a derangement if it has no fixed points. Derangements always exist by a classical theorem of Jordan, but there are so-called elusive groups that do not contain any derangements of prime order. In a recent paper, Burness and the author introduced the family of almost elusive groups, which contain a unique conjugacy class of derangements of prime order. In this paper, we complete the classification of the quasiprimitive almost elusive groups. [ABSTRACT FROM AUTHOR]

It follows from the classical theorem by Lebesgue (1940) on the structure of minor faces in 3-polytopes that every plane triangulation with minimum degree at least 4 has a 3-face for which the set of degrees of its vertices is majorized by one of the following sequences: (4, 4, ∞), (4, 5, 19), (4, 6, 11), (4, 7, 9), (5, 5, 9), (5, 6, 7). In 1999, Jendrol' gave the following description of faces: (4, 4, ∞), (4, 5, 13), (4, 6, 17), (4, 7, 8), (5, 5, 7), (5, 6, 6). Also, Jendrol' (1999) conjectured that there is a face of one of the types: (4, 4, ∞), (4, 5, 10), (4, 6, 15), (4, 7, 7), (5, 5, 7), (5, 6, 6). In 2002, Lebesgue's description was strengthened by Borodin to (4, 4, ∞), (4, 5, 17), (4, 6, 11), (4, 7, 8), (5, 5, 8), (5, 6, 6). In 2014, we obtained the following tight description, which, in particular, disproves the above mentioned conjecture by Jendrol': (4, 4, ∞), (4, 5, 11), (4, 6, 10), (4, 7, 7), (5, 5, 7), (5, 6, 6). Recently, we obtained another tight description: (4, 4, ∞ (4, 6, 10), (4, 7, 7), (5, 5, 8), (5, 6, 7). In 2014, we obtained the following tight description, which, in particular, disproves the above mentioned conjecture by Jendrol': (4, 4, ∞), (4, 5, 11), (4, 6, 10), (4, 7, 7), (5, 5, 7), (5, 6, 6). Recently, we obtained another tight description: (4, 4, ∞), (4, 6, 10), (4, 7, 7), (5, 5, 8), (5, 6, 7). [ABSTRACT FROM AUTHOR]

In this paper,we present a theorem about stability of nonlinear fractional difference equation with Riemann-Liouvile difference operator. The result is a version of classical theorem on linear approximation and to derive them,we prove the variation of constants formula for nabla Riemann- Liouville fractional difference equations.We also present some results concerning the existence and uniqueness of the equation under consideration. [ABSTRACT FROM AUTHOR]

Let (M, g) be a complete Riemannian manifold of dimension n. We consider a Ricci curvature condition on M to prove a compactness theorem including a diameter estimate. It is an improvement of the classical theorem of Myers, and it is comparable with some other Myers type compactness theorems. [ABSTRACT FROM AUTHOR]

Schur's classical theorem states that for a group G, if G/Z(G) is finite, then G' is finite. Baer extended this theorem for the factor group G/Zn(G), inwhich Zn(G) is the n-th term of the upper central series of G. Hekster proved a converse of Baer's theorem as follows: If G is a finitely generated group such that γn+1(G) is finite, then G/Zn(G) is finite where γn+1(G) denotes the (n+1)st term of the lower central series of G. In this paper, we generalize this result by obtaining the same conclusion under the weaker hypothesis that G/Zn(G) is finitely generated. Furthermore, we show that the index of the subgroup Zn(G) is bounded by a precisely determined function of the order of γn+1(G). Moreover, we prove that the mentioned theorem of Hekster is also valid under a weaker condition that Z2n(G)/Zn(G) is finitely generated. Although in this case the bound for the order of γn+1(G) is not achieved. [ABSTRACT FROM AUTHOR]

For any f ∈ H p (D) (p > 0) , the Hardy space over the unit disk, we characterize completely the triple (α , p , q) ∈ (- 1 , ∞) × (0 , ∞) 2 such that the Cesàro partial sum σ n α f converges in norm in H q (D) , and in the Bergman space L a q (D) , respectively. This extends recent results of McNeal–Xiong and of Park–Zhao–Zhu, and complements a classical theorem of Hardy-Littlewood in 1934. [ABSTRACT FROM AUTHOR]

Similar to the Bers simultaneous uniformization, the product of the p-Weil–Petersson Teichmüller spaces for p ≥ 1 provides the coordinates for the space of p-Weil–Petersson embeddings γ of the real line R into the complex plane C . We prove the biholomorphic correspondence from this space to the p-Besov space of u = log γ ′ on R for p > 1 . From this fundamental result, several consequences follow immediately which clarify the analytic structures concerning parameter spaces of p-Weil–Petersson curves. Specifically, it implies that the correspondence of the Riemann mapping parameters to the arc-length parameters preserving the images of curves is a homeomorphism with bi-real-analytic dependence of the change of parameters. This is analogous to the classical theorem of Coifman and Meyer for chord-arc curves. [ABSTRACT FROM AUTHOR]

A classical theorem of A. D. Alexandrov says that a connected compact smooth hypersurface in Euclidean space with constant mean curvature must be a sphere. We give exposition to some results on symmetry properties of hypersurfaces with ordered mean curvature and associated variations of the Hopf Lemma. Some open problems will be discussed. [ABSTRACT FROM AUTHOR]