Narrow Results

In this paper, we study the notions of Browder operator and Browder S-spectrum of bounded right linear operator defined over the right quaternionic Hilbert space. Some properties of Browder operator and stability of the ascent, descent and Browder S-spectrum have been investigated in the right quaternionic setting. We also characterize the property of invariant Browder operators and study the spectral mapping theorem of Browder S-spectrum for self-adjoint operators in quaternionic setting. This investigation concludes by exploring the Browder S-spectrum of the sum of two bounded linear operators.

Gait assessment is important for identification of potential faller among the elderly populations. Slope walking is associated with fall risk factor and elderly women have higher fall rate compared with elderly men. Therefore, this study investigated gait characteristics of elderly women in overground and slope walkway conditions. Thirty healthy elderly women (15 younger-elderly women and 15 older-elderly women) walked along the linear walkway including three walking conditions (overground, ascent and descent conditions). Temporal gait variables and normalized peak vertical GRF (ground reaction force) variables were derived from commercial motion analysis software. Repeated-measures analysis of variance (ANOVA) was evaluated to compare mean differences of the three conditions and mean difference between younger and older elderly women. All gait characteristics were significantly different from the slope walking conditions ($p<0.05$). Elderly women walked with longer loading response and mid stance phase during descent walking. Also, ascent walking induced a longer terminal stance phase. Interactions of age and walkway conditions were also significant in vertical GRF, where older-elderly women were greater than younger-elderly women in ascent walkway condition ($p<0.01$) and in descent walkway condition ($p=0.05$). These findings suggest that specific-walkway condition should be considered for fall prevention and clinical interventions in elderly women.

We describe a qualitative improvement to the algorithms for performing $2$-descents to obtain information regarding the Mordell–Weil rank of a hyperelliptic Jacobian. The improvement has been implemented in the Magma Computational Algebra System and as a result, the rank bounds for hyperelliptic Jacobians are now sharper and have the conjectured parity.

This paper is a continuation of our previous one under the same title. In both articles, we study the hyperelliptic curves $C_a:y^2=x^5+ax$ defined over $\mathbb{Q}$, and their Jacobians $J_a$ (without loss of generality a is a nonzero 8th power free integer). Previously, we considered the case when the polynomial $x^4+a$ is irreducible in $\mathbb{Q}[x]$ and obtained (under certain conditions on the quartic field $\mathbb{Q}(\sqrt[4]{-a})$) upper bounds for the $rank{J}_a(\mathbb{Q})$; in particular, we found infinite subfamilies of $J_a$ with rank zero. Now we consider all cases when $x^4+a$ is reducible in $\mathbb{Q}[x]$ and prove analogous results. First we obtain (under mild conditions on some quadratic fields) upper bounds for the ranks in a rather general situation, then we restrict to a several infinite subfamilies of $J_a$ (when 2a has two primes divisors) and get the best possible bounds or even the exact value of rank (if it is zero). We deduce as conclusions the complete lists of rational points on $C_a$ in such cases.

By computing the completely bounded norm of the flip map on the Haagerup tensor product $C_0{Y}_1{\otimes }_{C_{0}X}{C}_0{Y}_2$ associated to a pair of continuous mappings of locally compact Hausdorff spaces $Y_1\to X\leftarrow Y_2$, we establish a simple characterization of the Beck-Chevalley condition for base change of operator modules over commutative $C^{\ast }$-algebras, and a descent theorem for continuous fields of Hilbert spaces.

Let $\mathcal{ℬ}(X)$ be the algebra of all bounded linear operators on an infinite-dimensional complex or real Banach space $X$. We prove that a bijective bicontinuous map $\mathrm{\Phi }$ on $\mathcal{ℬ}(X)$ preserves the difference of group invertible operators in both directions if and only if $\mathrm{\Phi }$ is either of the form $\mathrm{\Phi }(T)=\alpha AT{A}^{-1}+\mathrm{\Phi }(0)$ or of the form $\mathrm{\Phi }(T)=\alpha B{T}^{\ast }{B}^{-1}+\mathrm{\Phi }(0)$, where $\alpha$ is a nonzero scalar, $A:X\to X$ and $B:X^{\ast }\to X$ are two bounded invertible linear or conjugate linear operators.

The purpose of this paper is to present new additive results for $n$-quasi-isometries operators on Hilbert spaces. In particular Precisely, we focus on the study of the descent of a $n$-quasi-isometry operator. Some spectral properties for this class of operators and decomposition theorems are also given. Part of the results proved in this paper improve and generalize some results known for isometries and quasi-isometries.

We consider the number of passes a permutation needs to take through a stack if we only pop the appropriate output values and start over with the remaining entries in their original order. We define a permutation $\pi$ to be $k$-pass sortable if $\pi$ is sortable using $k$ passes through the stack. Permutations that are $1$-pass sortable are simply the stack sortable permutations as defined by Knuth. We define the permutation class of $2$-pass sortable permutations in terms of their basis. We also show all $k$-pass sortable classes have finite bases by giving bounds on the length of a basis element of the permutation class for any positive integer $k$. Finally, we define the notion of tier of a permutation $\pi$ to be the minimum number of passes after the first pass required to sort $\pi$. We then give a bijection between the class of permutations of tier $t$ and a collection of integer sequences studied by Parker [The combinatorics of functional composition and inversion, PhD thesis, Brandeis University (1993)]. This gives an exact enumeration of tier $t$ permutations of a given length and thus an exact enumeration for the class of $(t+1)$-pass sortable permutations. Finally, we give a new derivation for the generating function in [S. Parker, The combinatorics of functional composition and inversion, PhD thesis, Brandeis University (1993)] and an explicit formula for the coefficients.