An Extension of a Simply Inequality Between von Neumann–Jordan and James Constants in Banach Spaces

For a non-trivial Banach space X, let J(X), CNJ(X), C_(NJ)~(p)(X) respectively stand for the James constant, the von Neumann–Jordan constant and the generalized von Neumann–Jordan constant recently inroduced by Cui et al. In this paper, we discuss the relation between the James and the generalized von Neumann–Jordan constants, and establish an inequality between them: C_(NJ)~(p)(X) ≤J(X) with p ≥ 2, which covers the well-known inequality CNJ(X) ≤ J(X). We also introduce a new constant, from which we establish another inequality that extends a result of Alonso et al.     

Birational Darboux Coordinates on (Co)Adjoint Orbits of GL(N, ℂ)

The set of all linear transformations with a fixed Jordan structure J is a symplectic manifold isomorphic to the coadjoint orbit O(J) of the general linear group GL(N, C). Any linear transformation can be projected along its eigenspace onto a coordinate subspace of complementary dimension. The Jordan structure \(\tilde J\) of the image under the projection is determined by the Jordan structure J of the preimage; consequently, the projection \(O\left( J \right) \to O\left( {\tilde J} \right)\) is a mapping of symplectic manifolds.     

Permutation modules of profinite groups

Given an arbitrary profinite group G and a commutative domain R, we define the notion of permutation RG-module which generalizes the known notion from the representation theory of profinite groups. We establish an independence theorem of such a module as an R-module over a ring of scalars.     

Schwarz problem for first-order elliptic systems on the plane

We consider the Schwarz problem for J-analytic functions for the case in which the Jordan basis Q of the matrix J contains complex conjugate vectors. Conditions on the matrix Q are obtained under which there exists a unique solution of the Schwarz problem in Hölder classes.     

ABELIAN BALANCED HERMITIAN STRUCTURES ON UNIMODULAR LIE ALGEBRAS

Let g be a 2n-dimensional unimodular Lie algebra equipped with a Hermitian structure (J; F) such that the complex structure J is abelian and the fundamental form F is balanced. We prove that the holonomy group of the associated Bismut connection reduces to a subgroup of SU(n – k), being 2k the dimension of the center of g. We determine conditions that allow a unimodular Lie algebra to admit this particular type of structures. Moreover, we give methods to construct them in arbitrary dimensions and classify them if the Lie algebra is 8-dimensional and nilpotent.     

Slices and Levels of Extensions of the Minimal Logic

We consider two classifications of extensions of Johansson’s minimal logic J. Logics and then calculi are divided into levels and slices with numbers from 0 to ω. We prove that the first classification is strongly decidable over J, i.e., from any finite list Rul of axiom schemes and inference rules, we can effectively compute the level number of the calculus (J + Rul). We prove the strong decidability of each slice with finite number: for each n and arbitrary finite Rul, we can effectively check whether the calculus (J + Rul) belongs to the nth slice.     

The 3-Isometric Lifting Theorem

An operator T on Hilbert space is a 3-isometry if \({T^{*n}T^{n}= I +n B_1 +n^{2} B_2}\) is quadratic in n. An operator J is a Jordan operator if J = U + N where U is unitary, N ² = 0 and U and N commute. If T is a 3-isometry and \({c > 0,}\) then \({I-c^{-2} B_{2} + sB_{1} + s^{2}B_2}\) is positive semidefinite for all real s if and only if it is the restriction of a Jordan operator J = U + N with the norm of N at most c. As a corollary, an analogous result for 3-symmetric operators, due to Helton and Agler, is recovered.     

Symplectic geometry on moduli spaces of <Emphasis Type="Italic">J</Emphasis>-holomorphic curves

Let (M, ω) be a symplectic manifold, and Σ a compact Riemann surface. We define a 2-form \({\omega_{\mathcal{S}_{i}(\Sigma)}}\) on the space \({\mathcal{S}_{i}(\Sigma)}\) of immersed symplectic surfaces in M, and show that the form is closed and non-degenerate, up to reparametrizations. Then we give conditions on a compatible almost complex structure J on (M, ω) that ensure that the restriction of \({\omega_{\mathcal{S}_{i}(\Sigma)}}\) to the moduli space of simple immersed J-holomorphic Σ-curves in a homology class \({A \in {H}_2(M,\,\mathbb{Z})}\) is a symplectic form, and show applications and examples. In particular, we deduce sufficient conditions for the existence of J-holomorphic Σ-curves in a given homology class for a generic J.     

The existence of <Emphasis Type="Italic">J</Emphasis>-holomorphic curves in almost Hermitian manifolds

In this paper, we investigate the existence of J-holomorphic curves on almost Hermitian manifolds. Let (M, g, J, F) be an almost Hermitian manifold and \(f:\Sigma \rightarrow M\) be an injective immersion. We prove that if the \(L_p\) functional has a critical point or a stable point in the same almost Hermitian class, then the immersion is J-holomorphic.     

Depth and regularity of powers of sums of ideals

Given arbitrary homogeneous ideals I and J in polynomial rings A and B over a field k, we investigate the depth and the Castelnuovo–Mumford regularity of powers of the sum \(I+J\) in \(A \otimes _k B\) in terms of those of I and J. Our results can be used to study the behavior of the depth and regularity functions of powers of an ideal. For instance, we show that such a depth function can take as its values any infinite non-increasing sequence of non-negative integers.     

The general rigidity result for bundles of <Emphasis Type="Italic">A</Emphasis>-covelocities and <Emphasis Type="Italic">A</Emphasis>-jets

Let M be an m-dimensional manifold and A = D _k ^r /I = R⊕N _A a Weil algebra of height r. We prove that any A-covelocity T _x ^A f ∈ T _x ^A *M, x ∈ M is determined by its values over arbitrary max{width A,m} regular and under the first jet projection linearly independent elements of T _x ^A M. Further, we prove the rigidity of the so-called universally reparametrizable Weil algebras. Applying essentially those partial results we give the proof of the general rigidity result T ^A *M ? T ^r *M without coordinate computations, which improves and generalizes the partial result obtained in Tomá? (2009) from m ≥ k to all cases of m.We also introduce the space J ^A (M,N) of A-jets and prove its rigidity in the sense of its coincidence with the classical jet space J ^r (M,N).     

Reversibility of Rings with Respect to the Jacobson Radical

Let R be a ring with identity and J(R) denote the Jacobson radical of R. A ring R is called J-reversible if for any a, \(b \in R\), \(ab = 0\) implies \(ba \in J(R)\). In this paper, we give some properties of J-reversible rings. We prove that some results of reversible rings can be extended to J-reversible rings for this general setting. We show that J-quasipolar rings, local rings, semicommutative rings, central reversible rings and weakly reversible rings are J-reversible. As an application it is shown that every J-clean ring is directly finite.     

<Emphasis Type="Italic">p</Emphasis>-Adic solid-on-solid model on a Cayley tree

We consider a p-adic solid-on-solid (SOS) model with a nearest-neighbor coupling, m+1 spins, and a coupling constant J ∈ Q _p on a Cayley tree. We find conditions under which a phase transition does not occur in the model. We show that if p | m + 1 for some J, then a phase transition occurs. Moreover, we formulate a criterion for the boundedness of p-adic Gibbs measures for the (m+1)-state SOS model.     

Similarity Results for Operators of Class <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>

If T is a multiplicity-free contraction of class C ₀ with minimal function m _T , then it is quasisimilar to the Jordan block S(m _T ). In case m _T is a Blaschke product with simple roots forming a Carleson sequence, we show that the relation between T and S(m _T ) can be strengthened to similarity. Under the additional assumption that u(T) has closed range for every inner divisor \({u\in H^\infty}\) of m _T , the result also holds in the more general setting where the roots have bounded multiplicities.     

Generalized (Jordan) left derivations on rings associated with an element of rings

In this paper, we introduce a new notion of generalized (Jordan) left derivation on rings as follows: let R be a ring, an additive mapping F : R → R is called a generalized (resp. Jordan) left derivation if there exists an element w ∈ R such that F(xy) = xF(y) + yF(x) + yxw (resp. F(x ²) = 2xF(x) + x ² w) for all x, y ∈ R. Then, some related properties and results on generalized (Jordan) left derivation of square closed Lie ideals are obtained.     

Generalized <Emphasis Type="Italic">SV</Emphasis>-modules

Given an arbitrary quasiprojective right R-module P, we prove that every module in the category σ(P) is weakly regular if and only if every module in σ(M/I(M)) is lifting, where M is a generating object in σ(P). In particular, we describe the rings over which every right module is weakly regular.     

Classification of rings with toroidal Jacobson graph

Let R be a commutative ring with nonzero identity and J(R) the Jacobson radical of R. The Jacobson graph of R, denoted by J_R, is defined as the graph with vertex set RJ(R) such that two distinct vertices x and y are adjacent if and only if 1 ? xy is not a unit of R. The genus of a simple graph G is the smallest nonnegative integer n such that G can be embedded into an orientable surface S_n. In this paper, we investigate the genus number of the compact Riemann surface in which J_R can be embedded and explicitly determine all finite commutative rings R (up to isomorphism) such that J_R is toroidal.     

Classification of finite commutative rings with planar,toroidal, and projective line graphs associated with Jacobson graphs

Let R be a commutative ring with nonzero identity and J(R) be the Jacobson radical of R. The Jacobson graph of R, denoted by J _R , is a graph with vertex-set R J(R), such that two distinct vertices a and b in R J(R) are adjacent if and only if 1 ? ab is not a unit of R. Also, the line graph of the Jacobson graph is denoted by L(J _R ). In this paper, we characterize all finite commutative rings R such that the graphs L(J _R ) are planar, toroidal or projective.     

The column-sufficiency and row-sufficiency of the linear transformation on Hilbert spaces

Given a real Hilbert space H with a Jordan product and \({\Omega\subset H}\) being the Lorentz cone, \({q\in H}\), and let T : H → H be a bounded linear transformation, the corresponding linear complementarity problem is denoted by LCP(T, Ω, q). In this paper, we introduce the concepts of the column-sufficiency and row-sufficiency of T. In particular, we show that the row-sufficiency of T is equivalent to the existence of the solution of LCP(T, Ω, q) under an operator commutative condition; and that the column-sufficiency along with cross commutative property is equivalent to the convexity of the solution set of LCP(T, Ω, q). In our analysis, the properties of the Jordan product and the Lorentz cone in H are interconnected.     

Adjoining batch Markov arrival processes of a Markov chain

A batch Markov arrival process (BMAP) X* = (N, J) is a 2-dimensional Markov process with two components, one is the counting process N and the other one is the phase process J. It is proved that the phase process is a time-homogeneous Markov chain with a finite state-space, or for short, Markov chain. In this paper, a new and inverse problem is proposed firstly: given a Markov chain J, can we deploy a process N such that the 2-dimensional process X* = (N, J) is a BMAP? The process X* = (N, J) is said to be an adjoining BMAP for the Markov chain J. For a given Markov chain the adjoining processes exist and they are not unique. Two kinds of adjoining BMAPs have been constructed. One is the BMAPs with fixed constant batches, the other one is the BMAPs with independent and identically distributed (i.i.d) random batches. The method we used in this paper is not the usual matrix-analytic method of studying BMAP, it is a path-analytic method. We constructed directly sample paths of adjoining BMAPs. The expressions of characteristic (D _k , k = 0, 1, 2 · · ·) and transition probabilities of the adjoining BMAP are obtained by the density matrix Q of the given Markov chain J. Moreover, we obtained two frontal Theorems. We present these expressions in the first time.