Classification of AS-regular algebras is one of the major projects in non-commutative algebraic geometry. In this paper, we will study when given AS-regular algebras are graded Morita equivalent. In particular, for every geometric AS-regular algebra A, we define another graded algebra A, and show that if two geometric AS-regular algebras A and A' are graded Morita equivalent, then A and A' are isomorphic as graded algebras. We also show that the converse holds in many three-dimensional cases. As applications, we apply our results to Frobenius Koszul algebras and Beilinson algebras.

We prove that the Hatcher–Quinn and Wall invariants of a self-transverse immersion f: Nn ↬ M2n coincide. That is, we construct an isomorphism between their target groups, which carries one onto the other. We also employ methods of normal bordism theory to investigate the Hatcher–Quinn invariant of an immersion f: Nn ↬ M2n−1.

Some recent results of Khukhro and Makarenko on the existence of characteristic -subgroups of finite index in a group G, for certain varieties , are used to obtain generalisations of some well-known results in the literature pertaining to groups G, in which all proper subgroups satisfy some condition or other related to the property ‘soluble-by-finite’. In addition, a partial generalisation is obtained for the aforementioned results on the existence of characteristic subgroups.

It is shown that for the Laplacian masa in the free group factors, the orthocomplement of the associated Jones' projection is an infinite direct sum of coarse bimodules.

In this paper, we study the K-envelopes of the real interpolation methods with function space parameters in the sense of Brudnyi and Kruglyak [Y. A. Brudnyi and N. Ja. Kruglyak, Interpolation functors and interpolation spaces (North-Holland, Amsterdam, Netherlands, 1991)]. We estimate the upper bounds of the K-envelopes and the interpolation norms of bounded operators for the KΦ-methods in terms of the fundamental function of the rearrangement invariant space related to the function space parameter Φ. The results concerning the quasi-power parameters and the growth/continuity envelopes in function spaces are obtained.

We prove, using purely combinatorial methods, that there is a pairing

\begin{linenomath} $$\Sym^a\Q^2\times\Sym^a\Q^2\lra\Q$$ \end{linenomath}

We give an easy diagrammatical description of the parabolic Kazhdan–Lusztig polynomials for the Weyl group Wn of type Dn with parabolic subgroup of type An and consequently an explicit counting formula for the dimension of morphism spaces between indecomposable projective objects in the corresponding category . As a by-product we categorify irreducible Wn-modules corresponding to the pairs of one-line partitions. Finally, we indicate the motivation for introducing the combinatorics by connections to the Springer theory, the category of perverse sheaves on isotropic Grassmannians, and to the Brauer algebras, which will be treated in two subsequent papers of the second author.

In this paper we investigate the following functional inequality

$ \begin{eqnarray*} \| f(x-y-z) - f(x-y+z) + f(y) +f(z)\| \leq \|f(x+y-z) - f(x)\| \end{eqnarray*}$

The mean value inequality is characteristic for upper semi-continuous functions to be subharmonic. Quasinearly subharmonic functions generalise subharmonic functions. We find the necessary and sufficient conditions under which subsets of balls are big enough for the characterisation of non-negative, quasinearly subharmonic functions by mean value inequalities. Similar result is obtained also for generalised mean value inequalities where, instead of balls, we consider arbitrary bounded sets, which have non-void interiors and instead of the volume of ball some functions depending on the radius of this ball.

A module M over a commutative ring R has an almost trivial dual if there is no homomorphism from M onto a free R-module of countable infinite rank. Using a new combinatorial principle (the ℵn-Black Box), which is provable in ordinary set theory, we show that for every natural number n, there exist arbitrarily large ℵn-free R-modules with almost trivial duals, when R is a complete discrete valuation domain. A corresponding result for torsion modules is also obtained.

The notion of ring endomorphisms having large images is introduced. Among others, injectivity and surjectivity of such endomorphisms are studied. It is proved, in particular, that an endomorphism σ of a prime one-sided noetherian ring R is injective whenever the image σ(R) contains an essential left ideal L of R. If, in addition, σ(L)=L, then σ is an automorphism of R. Examples showing that the assumptions imposed on R cannot be weakened to R being a prime left Goldie ring are provided. Two open questions are formulated.

About 15 years ago, Lutwak and Zhang (E. Lutwak and G. Zhang, Blaschke–Santalo inequalities, J. Differ. Geom. 47 (1997), 1–16) introduced the notion of Lp moment bodies and established important volume inequalities for them, which were recently generalized by Haberl and Schuster (C. Haberl and E. Schuster, General Lp affine isoperimetric inequalities, J. Differ. Geom. 83 (2009), 1–26). In this paper, we establish new Brunn–Minkowski-type inequalities with respect to Blaschke Lp harmonic addition for the quermassintegrals and dual quermassintegrals of Lp moment bodies.

We prove uniqueness of positive solutions for the boundary value problems

\[ \{\begin{array}{ll} -\Delta u=\lambda f(u)\ \ &\text{in}\Omega, \ \ \ \ \ u=0 &\text{on \partial \Omega, \]

The present paper gives an example of a rigid spherical cone-manifold and that of a flexible one, which are both Seifert fibred.

Considered here is the pullback attractor of the process associated with the first initial boundary value problem for the non-autonomous semilinear degenerate parabolic equation

\begin{linenomath} u_t-\text{div}(\sigma(x)\nabla u)+f(u)=g(x,t) \end{linenomath}

$D_0^2(\Omega,\sigma)$

A group G is n-central if Gn ≤ Z(G), that is the subgroup of G generated by n-powers of G lies in the centre of G. We investigate pk-central groups for p a prime number. For G a finite group of exponent pk, the covering group of G is pk-central. Using this we show that the exponent of the Schur multiplier of G is bounded by $p^{\lceil \frac{c}{p-1} \rceil}$ , where c is the nilpotency class of G. Next we give an explicit bound for the order of a finite pk-central p-group of coclass r. Lastly, we establish that for G, a finite p-central p-group, and N, a proper non-maximal normal subgroup of G, the Tate cohomology Hn(G/N, Z(N)) is non-trivial for all n. This final statement answers a question of Schmid concerning groups with non-trivial Tate cohomology.

Let R be a commutative Noetherian ring, is an ideal of R and M is an R-module. We intend to establish the dual of Grothendieck's Vanishing Theorem for local homology modules. We conjecture that =0 for all i>magRM. We prove this in several cases.

We give the necessary and sufficient conditions for Lagrangian submanifolds in Kähler manifolds to be biharmonic. We classify biharmonic PNMC Lagrangian H-umbilical submanifolds in the complex space forms. Furthermore, we classify biharmonic PNMC Lagrangian surfaces in the two-dimensional complex space forms.