Two-sided bounds for the logarithmic capacity of multiple intervals

Potential theory on the complement of a subset of the real axis attracts much attention in both function theory and applied sciences. This paper discusses one aspect of the theory — the logarithmic capacity of closed subsets of the real line. We give simple but precise upper and lower bounds for the logarithmic capacity of multiple intervals and a lower bound valid also for closed sets comprising an infinite number of intervals. Using some known methods to compute the exact values of capacity, we demonstrate graphically how our estimates compare with them. The main machinery behind our results are the separating transformation and dissymmetrization developed by V. N. Dubinin and a version of the latter due to K. Haliste, as well as some classical symmetrization and projection results for the logarithmic capacity. The results of the paper improve some previous achievements by A. Yu. Solynin and K. Shiefermayr.     

A Shear Compression Disk Specimen with Controlled Stress Triaxiality under Quasi-Static Loading

This paper introduces a double shear axisymmetric specimen (Shear Compression Disk) and the methodology to extract flow and fracture properties of ductile materials, under various stress triaxiality levels. A thorough numerical investigation of the experimental set-up is performed, which reveals that the stresses are quite uniformly distributed in the gauge section during all the stages of the test. The attainable level of stress triaxiality (with pressures of up to 1.9 GPa) ranges from −0.1 to 1, which can be adjusted by a proper choice of geometrical parameters of the specimen. The methodology is implemented to quasi-static experiments on 4340 Steel and Aluminum 7075-T651 specimens. The flow properties are compared to those obtained by upsetting cylinders and show a very good agreement. For these materials it is observed that, contrary to the fracture strain, the flow properties are quite insensitive to the level of stress triaxiality. The fracture strain of the aluminum alloy increases with triaxiality and may be fitted with an exponential polynomial of the type suggested by [27]. These examples demonstrate the potential of the new specimen to obtain flow and fracture properties of ductile materials under controlled triaxiality.     

Quantum walk with jumps

We analyze a special class of 1-D quantum walks (QWs) realized using optical multi-ports. We assume non-perfect multi-ports showing errors in the connectivity, i.e. with a small probability the multi-ports can connect not to their nearest neighbor but to another multi-port at a fixed distance – we call this a jump. We study two cases of QW with jumps where multiple displacements can emerge at one timestep. The first case assumes time-correlated jumps (static disorder). In the second case, we choose the positions of jumps randomly in time (dynamic disorder). The probability distributions of position of the QW walker in both instances differ significantly: dynamic disorder leads to a Gaussian-like distribution, while for static disorder we find two distinct behaviors depending on the parity of jump size. In the case of even-sized jumps, the distribution exhibits a three-peak profile around the position of the initial excitation, whereas the probability distribution in the odd case follows a Laplace-like discrete distribution modulated by additional (exponential) peaks for long times. Finally, our numerical results indicate that by an appropriate mapping a universal functional behavior of the variance of the long-time probability distribution can be revealed with respect to the scaled average of jump size.     

Study of dynamic end effects in an elastic strip with sliding boundary conditions

The dynamic response of an elastic, semi-infinite strip with sliding surfaces, subjected to various forms of end excitations, was solved analytically employing the property of bi-orthogonality of wave modes. An explicit relation between the amplitudes of evanescent waves and the form of the excitation was obtained. Quantitative measure for dynamic end effects was suggested, termed Saint-Venant ratio (SVR). It was shown that two qualities of that ratio are useful for monitoring the health of structural joints (SHM): being that ratio not affected by the intensity of the end excitation and its high sensitivity to small variations in the form of the excitation. The axial behavior of the strip subjected to several forms of end excitations was further used to demonstrate the validity of a previously suggested dynamic version of Saint-Venant’s principle.     

Log-convexity and log-concavity of hypergeometric-like functions

We find sufficient conditions for log-convexity and log-concavity for the functions of the forms a?∑fkk(a)xk, a?∑fkΓ(a+k)xk and a?∑fkxk/k(a). The most useful examples of such functions are generalized hypergeometric functions. In particular, we generalize the Turán inequality for the confluent hypergeometric function recently proved by Barnard, Gordy and Richards and log-convexity results for the same function recently proved by Baricz. Besides, we establish a reverse inequality which complements naturally the inequality of Barnard, Gordy and Richards. Similar results are established for the Gauss and the generalized hypergeometric functions. A conjecture about monotonicity of a quotient of products of confluent hypergeometric functions is made.     

Crutch or Catalyst: Teachers' Beliefs and Practices Regarding Calculator use in Mathematics Instruction

This study investigated K‐12 teachers' beliefs and reported teaching practices regarding calculator use in their mathematics instruction. A survey was administered to more than 800 elementary, middle and high school teachers in a large metropolitan area to address the following questions: (a) what are the beliefs and practices of mathematics teachers regarding calculator use? and (b) how do these beliefs and practices differ among teachers in three grade bands? Factor analysis of 20 Likert scale items revealed four factors that accounted for 54% of the variance in the ratings. These factors were named Catalyst Beliefs, Teacher Knowledge, Crutch Beliefs, and Teacher Practices. Compared to elementary teachers, high school teachers were significantly higher in their perception of calculator use as a catalyst in mathematics instruction. However, the higher the grade level of the teacher, the higher the mean score on the perception that calculator use may be a way of getting answers without understanding mathematical processes. The mean scores for teachers in all three grade bands indicated agreement that students can learn mathematics through calculator use and using calculators in instruction will lead to better student understanding and make mathematics more interesting. The survey results shed light on teachers' self reported beliefs, knowledge, and practices in regard to consistency with elements of the National Council of Teachers of Mathematics Principles and Standards for School Mathematics (2000) technology principle and the NCTM use of technology position paper (2003). This study extended previous research on teachers' beliefs regarding calculator use in classrooms by examining and comparing the results of teacher surveys across three grade bands.     

Cell docking inside microwells within reversibly sealed microfluidic channels for fabricating multiphenotype cell arrays

We present a soft lithographic method to fabricate multiphenotype cell arrays by capturing cells within an array of reversibly sealed microfluidic channels. The technique uses reversible sealing of elastomeric polydimethylsiloxane (PDMS) molds on surfaces to sequentially deliver various fluids or cells onto specific locations on a substrate. Microwells on the substrate were used to capture and immobilize cells within low shear stress regions inside channels. By using an array of channels it was possible to deposit multiple cell types, such as hepatocytes, fibroblasts, and embryonic stem cells, on the substrates. Upon formation of the cell arrays on the substrate, the PDMS mold could be removed, generating a multiphenotype array of cells. In addition, the orthogonal alignment and subsequent attachment of a secondary array of channels on the patterned substrates could be used to deliver fluids to the patterned cells. The ability to position many cell types on particular regions within a two dimensional substrate could potentially lead to improved high-throughput methods applicable to drug screening and tissue engineering.