Real dimension of the Lie algebra of S-skew-Hermitian matrices

Let $M_n(\mathbb{C})$ be the set of $n\times n$ matrices over the complex numbers. Let $S \in M_n(\mathbb{C})$. A matrix $A\in M_n(\mathbb{C})$ is said to be $S$-skew-Hermitian if $SA^*=-AS$ where $A^*$ is the conjugate transpose of $A$. The set $\mathfrak{u}_S$ of all $S$-skew-Hermitian matrices is a Lie algebra. In this paper, we give a real dimension formula for $\mathfrak{u}_S$ using the Jordan block decomposition of the cosquare $S(S^*)^{-1}$ of $S$ when $S$ is nonsingular.

Dimensions of ‘self-affine sponges’ invariant under the action of multiplicative integers

Let $m_1 \geq m_2 \geq 2$ be integers. We consider subsets of the product symbolic sequence space $(\{0,\ldots ,m_1-1\} \times \{0,\ldots ,m_2-1\})^{\mathbb {N}^*}$ that are invariant under the action of the semigroup of multiplicative integers. These sets are defined following Kenyon, Peres, and Solomyak and using a fixed integer $q \geq 2$ . We compute the Hausdorff and Minkowski dimensions of the projection of these sets onto an affine grid of the unit square. The proof of our Hausdorff dimension formula proceeds via a variational principle over some class of Borel probability measures on the studied sets. This extends well-known results on self-affine Sierpiński carpets. However, the combinatoric arguments we use in our proofs are more elaborate than in the self-similar case and involve a new parameter, namely $j = \lfloor \log _q ( {\log (m_1)}/{\log (m_2)} ) \rfloor $ . We then generalize our results to the same subsets defined in dimension $d \geq 2$ . There, the situation is even more delicate and our formulas involve a collection of $2d-3$ parameters.

An Upper Bound for the w-Weak Global Dimension of Pullbacks

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] an ideal of [Formula: see text]. We introduce and study the [Formula: see text]-weak global dimension [Formula: see text] of the factor ring [Formula: see text]. Let [Formula: see text] be a [Formula: see text]-linked extension of [Formula: see text], and we also introduce the [Formula: see text]-weak global dimension [Formula: see text] of [Formula: see text]. We show that the ring [Formula: see text] with [Formula: see text] is exactly a field and the ring [Formula: see text] with [Formula: see text] is exactly a [Formula: see text]. As an application, we give an upper bound for the [Formula: see text]-weak global dimension of a Cartesian square [Formula: see text]. More precisely, if [Formula: see text] is [Formula: see text]-linked over [Formula: see text], then [Formula: see text]. Furthermore, for a Milnor square [Formula: see text], we obtain [Formula: see text].

Relative convergence speed of Lüroth expansions and Hausdorff dimension

For any [Formula: see text] in [Formula: see text], let [Formula: see text] be the Lüroth expansion of [Formula: see text]. In this paper, we study the relative convergence speed of its convergents [Formula: see text] to the rate of growth of digits in the Lüroth expansion of an irrational number. For any [Formula: see text] in [Formula: see text], the sets [Formula: see text] and [Formula: see text] are proved to be of same Hausdorff dimension [Formula: see text]. Furthermore, for any [Formula: see text] in [Formula: see text] with [Formula: see text], the Hausdorff dimension of the set [Formula: see text] [Formula: see text] is proved to be either [Formula: see text] or [Formula: see text] according as [Formula: see text] or not.

A bound for the class of nilpotent symplectic alternating algebras

We continue developing the theory of nilpotent symplectic alternating algebras. The algebras of upper bound nilpotent class, that we call maximal algebras, have been introduced and well studied. In this paper, we continue with the external case problem of algebras of minimal nilpotent class. We show the existence of a subclass of algebras over a field [Formula: see text] that is of certain lower bound class that depends on the dimension only. This suggests a minimal bound for the class of nilpotent algebras of dimension [Formula: see text] of rank [Formula: see text] over any field.

Explicit Baker–Campbell–Hausdorff–Dynkin formula for spacetime via geometric algebra

We present a compact Baker–Campbell–Hausdorff–Dynkin formula for the composition of Lorentz transformations [Formula: see text] in the spin representation (a.k.a. Lorentz rotors) in terms of their generators [Formula: see text]: [Formula: see text] This formula is general to geometric algebras (a.k.a. real Clifford algebras) of dimension [Formula: see text], naturally generalizing Rodrigues’ formula for rotations in [Formula: see text]. In particular, it applies to Lorentz rotors within the framework of Hestenes’ spacetime algebra, and provides an efficient method for composing Lorentz generators. Computer implementations are possible with a complex [Formula: see text] matrix representation realized by the Pauli spin matrices. The formula is applied to the composition of relativistic 3-velocities yielding simple expressions for the resulting boost and the concomitant Wigner angle.

Classification of singular differential invariants in (1+3)-dimensional space and integrability

Singularity is one of the important features in invariant structures in several physical phenomena reflected often in the associated invariant differential equations. The classification problem for singular differential invariants in (1+3)-dimensional space associated with Lie algebras of dimension 4 is investigated. The formulation of singular invariants for a Lie algebra of dimension [Formula: see text] possessed by the underlying system of three second-order ordinary differential equations is studied in detail and the corresponding canonical forms for these systems are deduced. Furthermore, the categorization of singular invariants on the basis of conditional singularity, weak uncoupling, weak linearization, partial uncoupling and partial linearization are described for the underlying canonical forms. In addition, those cases of classified canonical forms are also mentioned which do not lead to singular invariant systems of three second-order ODEs for a Lie algebra of dimension 4. The integrability aspect of these classified singular-invariant systems in (1+3)-dimensional space is discussed in a detailed manner for a Lie algebra of dimension 4. Finally, two physical systems from mechanics are presented to illustrate the utilization of the physical aspect of these singular invariants.

Artin–Schelter regular algebras of dimension five with three generators

In this paper, we investigate Artin–Schelter regular algebras of dimension [Formula: see text] with three generators in degree [Formula: see text] under the hypothesis that [Formula: see text], in which the degree types of the relations for the number of the generating relations less than five can be determined. We prove that the only possible degree type of three generating relations is [Formula: see text] and the only possible degree type of four generating relations is [Formula: see text].

On the global existence and time-decay rates for a parabolic–hyperbolic model arising from chemotaxis

In this paper, we are concerned with the study of the Cauchy problem for a parabolic–hyperbolic model arising from chemotaxis in any dimension [Formula: see text]. We first prove the global existence of the model in [Formula: see text] critical regularity framework with respect to the scaling of the associated equations. Furthermore, under a mild additional decay assumption involving only the low frequencies of the data, we also establish the time-decay rates for the constructed global solutions. Our analyses mainly rely on Fourier frequency localization technology and on a refined time-weighted energy inequalities in different frequency regimes.

PERMEABILITY MODELS FOR TWO-PHASE FLOW IN FRACTAL POROUS-FRACTURE MEDIA WITH THE TRANSFER OF FLUIDS FROM POROUS MATRIX TO FRACTURE

Study of transport mechanism of two-phase flow through porous-fracture media is of considerable importance to deeply understand geologic behaviors. In this work, to consider the transfer of fluids, the analytical models of dimensionless relative permeabilities for the wetting and non-wetting phases flow are proposed based on the fractal geometry theory for porous media. The proposed models are expressed as functions of micro-structural parameters of the porous matrix and fracture, such as the fractal dimension ([Formula: see text] for pore area, the fractal dimensions [Formula: see text] for wetting phase and for non-wetting phase, porosity ([Formula: see text], the total saturations ([Formula: see text], the porous matrix saturation ([Formula: see text] of the wetting and non-wetting phases, fractal dimension ([Formula: see text] for tortuosity of tortuous capillaries, as well as the ratio ([Formula: see text] of the maximum pore size in porous matrix to fracture aperture. The ratio ([Formula: see text] has a significant impact on the relative permeabilities and total saturations of wetting phases. The results reveal that the flow contribution of wetting phase from the porous matrix to both the seepage behavior of the fracture and total wetting phase saturation can be neglected as [Formula: see text]. The models may shed light on the fundamental mechanisms of the wetting and non-wetting phase flow in porous-fracture media with fluid transfer.