A Graph Theoretical Approach to Network Encoding Complexity

Consider an acyclic directed network $G$ with sources $S_1, S_2,..., S_l$ and distinct sinks $R_1, R_2,..., R_l$. For $i=1, 2,..., l$, let $c_i$ denote the min-cut between $S_i$ and $R_i$. Then, by Menger's theorem, there exists a group of $c_i$ edge-disjoint paths from $S_i$ to $R_i$, which will be referred to as a group of Menger's paths from $S_i$ to $R_i$ in this paper. Although within the same group they are edge-disjoint, the Menger's paths from different groups may have to merge with each other. It is known that by choosing Menger's paths appropriately, the number of mergings among different groups of Menger's paths is always bounded by a constant, which is independent of the size and the topology of $G$. The tightest such constant for the all the above-mentioned networks is denoted by $\mathcal{M}(c_1, c_2,..., c_2)$ when all $S_i$'s are distinct, and by $\mathcal{M}^*(c_1, c_2,..., c_2)$ when all $S_i$'s are in fact identical. It turns out that $\mathcal{M}$ and $\mathcal{M}^*$ are closely related to the network encoding complexity for a variety of networks, such as multicast networks, two-way networks and networks with multiple sessions of unicast. Using this connection, we compute in this paper some exact values and bounds in network encoding complexity using a graph theoretical approach.Comment: 44 pages, 22 figure

Modeling of Transient Response of an Elastic Beam With Flexible Supports and Variable-Location Impact

This research is prerequisite to determining structural health and estimating wear-limited life of contact/impact machinery components. Prevention, or at least early notification, of impact-induced wear is essential for preventing economic loss and enhancing personnel safety. Thus, an efficient model which is discrete in time and continuous in space was undertaken; an euler-bernoulli beam with adjustable boundary conditions and variable impact is numerically studied under a pulse loading. Structural stiffness, material modulus, contact stiffness, contact location, damping ratio, pulse duration, clearance and boundary conditions are investigated. A reference system is used as the basis for parameter studies and solution convergence is examined for three boundary conditions. Overall numerical simulations show reasonable response for all the comparison of case studies. The contact location and clearance were found to be important factors due to their direct influence of mode shapes. One example application is illustrated, and comparisons show that considering possible boundary contact but not changing e provides better estimation. Experiments were carried out to verify the effects of influential parameters. Two beam specimens with difference slenderness were designed and examined under point contact/impact. A half-sine pulse excitation was applied through a mechanical shaker, and the deflection was captured by a high speed camera. Numerous test cases were conducted that varied pulse duration, pulse amplitude, clearance, and contact location. Decreasing the pulse duration lowers all deflection amplitudes, but the time in contact is insensitive. No gap causes the smallest beam response, and increasing clearance generates greater free deflection amplitude. Representative test cases were selected for validating the theoretical model. When comparing numerical simulation with experimental result for both specimens, satisfactory agreement for amplitude and duration can be reached even with raw input parameters of the cases without contactor. When there is a contactor, the model shows better prediction for the thick specimen with slenderness ratio of 0.0279 than the thin specimen with slenderness ratio of 0.0186. Contact stiffness and pulse amplitude are two possible sources of error. The contribution of this study is the incorporation of unique pulse loading, changeable boundary conditions, adjustable contact/impact situations, comprehensive parameter studies, and high speed photography

Build an Assessment Rubric of Student Creativity in Higher Education

[EN] Attention to student creativity has triggered a number of educational transformations in higher education. However, widespread measures of creativity in higher education are primary based on norm-referenced assessment, which provide minimal information of student performance against the learning development in creativity. In reponse to the lack of effective measures and criteria to link assessment with instruction in creativity education, this article discusses the process of building an assessment rubric of creativity based on the standards-referenced model. It is intended to help teachers and students better understand the learning objectives related to creativity, as well as to monitor and guide the development of student creativity. Xu, W.; Tognolini, J. (2022). Build an Assessment Rubric of Student Creativity in Higher Education. En 8th International Conference on Higher Education Advances (HEAd'22). Editorial Universitat Politècnica de València. 135-142. https://doi.org/10.4995/HEAd22.2022.1469513514

Dirichlet type extensions of Euler sums

In this paper, we study the alternating Euler $T$-sums and $\S$-sums, which are infinite series involving (alternating) odd harmonic numbers, and have similar forms and close relations to the Dirichlet beta functions. By using the method of residue computations, we establish the explicit formulas for the (alternating) linear and quadratic Euler $T$-sums and $\S$-sums, from which, the parity theorems of Hoffman's double and triple $t$-values and Kaneko-Tsumura's double and triple $T$-values are further obtained. As supplements, we also show that the linear $T$-sums and $\S$-sums are expressible in terms of colored multiple zeta values. Some interesting consequences and illustrative examples are presented