The existence of solutions for boundary value problem of fractional hybrid differential equations

In this paper, we study the existence of solutions for the boundary value problem of fractional hybrid differential equations$D_{0^{+}}^{\alpha }\left[\frac{x(t)}{f(t\text{,}x(t))}\right]+g(t\text{,}x(t))=0\text{,}0<t<1\text{,}$$x(0)=x(1)=0\text{,}$where $1<\alpha ?2$ is a real number, $D_{0^{+}}^{\alpha }$ is the Riemann–Liouville fractional derivative. By a fixed point theorem in Banach algebra due to Dhage, an existence theorem for fractional hybrid differential equations is proved under mixed Lipschitz and Carathéodory conditions. As an application, examples are presented to illustrate the main results.     

On the class group and S-class group of formal power series rings

Let ${\text{Cl}}_t(A)$ denote the t-class group of an integral domain A. P. Samuel has established that if A is a Krull domain then the mapping ${\text{Cl}}_t(A)\to {\text{Cl}}_t(A?X?)$, is injective and if A is a regular UFD, then ${\text{Cl}}_t(A)\to {\text{Cl}}_t(A?X?)$, is bijective. Later, L. Claborn extended this result in case A is a regular Noetherian domain. In the first part of this paper we prove that the mapping ${\text{Cl}}_t(A)\to {\text{Cl}}_t(A?X?)$; $[I]?[{(I.A?X?)}_t]$ is an injective homomorphism and in case of an integral domain A such that each υ-invertible υ-ideal of A has υ-finite type, we give an equivalent condition for ${\text{Cl}}_t(A)\to {\text{Cl}}_t(A?X?)$, to be bijective, thus generalizing the result of Claborn. In the second part of this paper, we define the S-class group of an integral domain A: let S be a (not necessarily saturated) multiplicative subset of an integral domain A. Following [11], a nonzero fractional ideal I of A is S-principal if there exist an $s\in S$ and $a\in I$ such that $sI?aA?I$. The S-class group of A, S-${\text{Cl}}_t(A)$ is the group of fractional t-invertible t-ideals of A under t-multiplication modulo its subgroup of S-principal t-invertible t-ideals of A. We generalize some known results developed for the classic contexts of Krull and PυMD domain and we investigate the case of isomorphism S-${\text{Cl}}_t(A)?S$-${\text{Cl}}_t(A?X?)$.     

The inverse of nonsymmetric two-level Toeplitz operator matrices

The Gohberg–Semencul formula allows one to express the entries of the inverse of a Toeplitz matrix using only a few entries (the first row and the first column) of the inverse matrix, under some nonsingularity condition. In this paper we will provide a two variable generalization of the Gohberg–Semencul formula in the case of a nonsymmetric two-level Toeplitz matrix with a symbol of the form $f(z_1\text{,}{z}_2)=\frac{1}{\overline{P(z_1\text{,}{z}_2)}Q(z_1\text{,}{z}_2)}$ where $P(z_1\text{,}{z}_2)$ and $Q(z_1\text{,}{z}_2)$ are stable polynomials of two variables. We also consider the case of operator valued two-level Toeplitz matrices. In addition, we propose an equation solver involving two-level Toeplitz matrices. Numerical results are included.     

The finite-step realizability of the joint spectral radius of a pair of d×d matrices one of which being rank-one

We study the finite-step realizability of the joint/generalized spectral radius of a pair of real square matrices ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$, one of which has rank 1, where $2?d<+\infty$. Let $\rho (A)$ denote the spectral radius of a square matrix A. Then we prove that there always exists a finite-length word $(i_1^1\text{,}\dots \text{,}{i}_m^1)\in \{1\text{,}2{\}}^m$, for some finite $m?1$, such that$\sqrt[m]{\rho \left({\mathrm{S}}_{i_1^1}?{\mathrm{S}}_{i_m^1}\right)}={\sup }_{n?1}\left\{{\displaystyle \underset{(i_1\text{,}\dots \text{,}{i}_n)\in \{1\text{,}2{\}}^n}{\max }}\sqrt[n]{\rho ({\mathrm{S}}_{i_1}?{\mathrm{S}}_{i_n})}\right\}\text{.}$In other words, there holds the spectral finiteness property for $\{{\mathrm{S}}_1\text{,}{\mathrm{S}}_2\}$. Explicit formula for computation of the joint spectral radius is derived. This implies that the stability of the switched system induced by $\{{\mathrm{S}}_1\text{,}{\mathrm{S}}_2\}$ is algorithmically decidable in this case.     

Sums of Dirac masses and conditioned ubiquity

Multifractal formalisms hold for certain classes of atomless measures μ obtained as limits of multiplicative processes. This naturally leads us to ask whether non trivial discontinuous measures obey such formalisms. This is the case for a new kind of measures, whose construction combines additive and multiplicative chaos. This class is defined by ${\nu }_{\gamma \text{,}\sigma }={\sum }_{j?1}{b}^{?j\gamma }/j^2{\sum }_{k=0}^{b^j?1}\mu ([k{b}^{?j}\text{,}{(k+1)b^{?j}))}^{\sigma }{\delta }_{k{b}^{?j}}$ ($\text{supp}(\mu )=[0\text{,}1]\text{,}b$ integer $?2\text{,}\gamma ?0\text{,}\sigma ?1$). Under suitable assumptions on the initial measure μ, ${\nu }_{\gamma \text{,}\sigma }$ obeys some multifractal formalisms. Its Hausdorff multifractal spectrum $h?d_{{\nu }_{\gamma \text{,}\sigma }}(h)$ is composed of a linear part for h smaller than a critical value $h_{\gamma \text{,}\sigma }$, and then of a concave part when $h?h_{\gamma \text{,}\sigma }$. The same properties hold for the Hausdorff spectrum of some function series $f_{\gamma \text{,}\sigma }$ constructed according to the same scheme as ${\nu }_{\gamma \text{,}\sigma }$. These phenomena are the consequences of new results relating ubiquitous systems to the distribution of the mass of μ. To cite this article: J. Barral, S. Seuret, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

New identities for 7-cores with prescribed BG-rank

Let $\pi$ be a partition. BG-rank$(\pi )$ is defined as an alternating sum of parities of parts of $\pi$ [A. Berkovich, F.G. Garvan, On the Andrews-Stanley refinement of Ramanujan's partition congruence modulo 5 and generalizations, Trans. Amer. Math. Soc. 358 (2006) 703–726. [1]]. Berkovich and Garvan [The BG-rank of a partition and its applications, Adv. in Appl. Math., to appear in $〈$http://arxiv.org/abs/math/0602362$〉$] found theta series representations for the t-core generating functions ${\sum }_{n?0}{a}_{t,j}(n)q^n$, where $a_{t,j}(n)$ denotes the number of t-cores of n with $\text{BG-rank}=j$. In addition, they found positive eta-quotient representations for odd t-core generating functions with extreme values of BG-rank. In this paper we discuss representations of this type for all 7-cores with prescribed BG-rank. We make an essential use of the Ramanujan modular equations of degree seven [B.C. Berndt, Ramanujan's Notebooks, Part III, Springer, New York, 1991] to prove a variety of new formulas for the 7-core generating function$\prod _{j?1}\frac{(1-q^{7j}{)}^7}{(1-q^j)}\text{.}$These formulas enable us to establish a number of striking inequalities for $a_{7,j}(n)$ with $j=-1,0,1,2$ and $a_7(n)$, such as$a_7(2n+2)?2a_7(n),a_7(4n+6)?10a_7(n)\text{.}$Here $a_7(n)$ denotes a number of unrestricted 7-cores of n. Our techniques are elementary and require creative imagination only.     

An explicit counterexample for the Lp-maximal regularity problem

In this short Note we give a self-contained example of a consistent family of holomorphic semigroups $(T_p{(t))}_{t?0}$ such that $(T_p{(t))}_{t?0}$ does not have maximal regularity for $p>2$. This answers negatively the open question whether maximal regularity extrapolates from $L^2$ to the $L^p$-scale.