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We generalize the measurement using an expanded concept of cover, in order to provide a new approach to size of set other than cardinality. The generalized measurement has application backgrounds such as a generalized problem in dimension reduction, and has reasons from the existence of the minimum of both the positive size and the positive graduation, i.e., both the minimum is the size of the set ${0}$. The minimum of positive graduation in actual measurement provides the possibility that an object cannot be partitioned arbitrarily, e.g., an interval $[0, 1]$ cannot be partitioned by arbitrarily infinite times to keep compatible with the minimum of positive size. For the measurement on size of set, it can be assumed that this minimum is the size of ${0}$, in symbols $|{0}|$ or graduation 1. For a set $S$, we generalize any graduation as the size of a set $C_i$ where $\exists x \in S (x \in C_i)$, and $|S|$ is represented by a pair, in symbols $(C, N(C))$, where ${C} = \cup {C_i}$ and $N(C)$ is a set function on $C_i$, with $C_i$ independent of the order $i$ and $N(C)$ reflecting the quantity of $C_i$. This pair is a generalized form of box-counting dimension. The yielded size satisfies the properties of outer measure in general cases, and satisfies the properties of measure in the case of graduation 1; while in the reverse view, measure is a size using the graduation of size of an interval. As for cardinality, the yielded size is a one-to-one correspondence where only addition is allowable, a weak form of cardinality, and rewrites Continuum Hypothesis using dimension as $\omega \dot |{0,1}| = 1$. In the reverse view, cardinality of a set is a size in the graduation of the set. The generalized measurement provides a unified approach to dimension, measure, cardinality and hence infinity., Comment: 10 pages, 0 figures, using amsart

The temperature-dependent transverse mechanical properties of single-walled nanotubes are studied using a molecular mechanics approach. The stretching and bond angle force constants describing the mechanical behaviour of the sp^{2} bonds are resolved in the temperature range between 0 K and 1600 K, allowing to identify a temperature dependence of the nanotubes wall thickness. We observe a decrease of the stiffness properties (axial and shear Young's modulus) with increasing temperatures, and an augmentation of the transverse Poisson's ratio, with magnitudes depending on the chirality of the nanotube. Our closed-form predictions compare well with existing Molecular Dynamics simulations., Comment: 15 pages, 4 figures. Accepted for Applied Physics Letters

Based on the eigenfunction symmetry constraint of the q-deformed modified KP hierarchy, a q-deformed mKP hierarchy with self-consistent sources ($q$-mKPHSCSs) is constructed. The q-mKPHSCSs contains two types of q-deformed mKP equation with self-consistent sources. By combination of the dressing method and the method of variation of constants, a generalized dressing approach is proposed to solve the q-deformed KP hierarchy with self-consistent sources ($q$-KPHSCSs). Using the gauge transformation between the $q$-KPHSCSs and the $q$-mKPHSCSs, the q-deformed Wronskian solutions for the $q$-KPHSCSs and the $q$-mKPHSCSs are obtained. The one-soliton solutions for the q-deformed KP (mKP) equation with a source are given explicitly., Comment: 17 pages, no figue, to be published in J. Phys. A