Matroids, Complexity and Computation

<p>The node deletion problem on graphs is: given a graph and integer k, can we delete no more than k vertices to obtain a graph that satisfies some property π. Yannakakis showed that this problem is NP-complete for an infinite family of well- defined properties. The edge deletion problem and matroid deletion problem are similar problems where given a graph or matroid respectively, we are asked if we can delete no more than k edges/elements to obtain a graph/matroid that satisfies a property π. We show that these problems are NP-hard for similar well-defined infinite families of properties. In 1991 Vertigan showed that it is #P-complete to count the number of bases of a representable matroid over any fixed field. However no publication has been produced. We consider this problem and show that it is #P-complete to count the number of bases of matroids representable over any infinite fixed field or finite fields of a fixed characteristic. There are many different ways of describing a matroid. Not all of these are polynomially equivalent. That is, given one description of a matroid, we cannot create another description for the same matroid in time polynomial in the size of the first description. Due to this, the complexity of matroid problems can vary greatly depending on the method of description used. Given one description a problem might be in P while another description gives an NP-complete problem. Based on these interactions between descriptions, we create and study the hierarchy of all matroid descriptions and generalize this to all descriptions of countable objects.</p>

Matroids, Complexity and Computation

<p>The node deletion problem on graphs is: given a graph and integer k, can we delete no more than k vertices to obtain a graph that satisfies some property π. Yannakakis showed that this problem is NP-complete for an infinite family of well- defined properties. The edge deletion problem and matroid deletion problem are similar problems where given a graph or matroid respectively, we are asked if we can delete no more than k edges/elements to obtain a graph/matroid that satisfies a property π. We show that these problems are NP-hard for similar well-defined infinite families of properties. In 1991 Vertigan showed that it is #P-complete to count the number of bases of a representable matroid over any fixed field. However no publication has been produced. We consider this problem and show that it is #P-complete to count the number of bases of matroids representable over any infinite fixed field or finite fields of a fixed characteristic. There are many different ways of describing a matroid. Not all of these are polynomially equivalent. That is, given one description of a matroid, we cannot create another description for the same matroid in time polynomial in the size of the first description. Due to this, the complexity of matroid problems can vary greatly depending on the method of description used. Given one description a problem might be in P while another description gives an NP-complete problem. Based on these interactions between descriptions, we create and study the hierarchy of all matroid descriptions and generalize this to all descriptions of countable objects.</p>

Generic cuspidal representations of 𝑈(2, 1)

Let 𝐹 be a non-Archimedean local field, and let 𝜎 be a non-trivial Galois involution with fixed field F 0 F_{0} . When the residue characteristic of F 0 F_{0} is odd, using the construction of cuspidal representations of classical groups by Stevens, we classify generic cuspidal representations of U ⁢ ( 2 , 1 ) ⁢ ( F / F 0 ) U(2,1)(F/F_{0}) .

Role of Magnetite Nanoparticles Size and Concentration on Hyperthermia under Various Field Frequencies and Strengths

Magnetite (Fe3O4) nanoparticles were synthesized using the chemical coprecipitation method. Several nanoparticle samples were synthesized by varying the concentration of iron salt precursors in the solution for the synthesis. Two batches of nanoparticles with average sizes of 10.2 nm and 12.2 nm with nearly similar particle-size distributions were investigated. The average particle sizes were determined from the XRD patterns and TEM images. For each batch, several samples with different particle concentrations were prepared. Morphological analysis of the samples was performed using TEM. The phase and structure of the particles of each batch were studied using XRD, selected area electron diffraction (SAED), Raman and XPS spectroscopy. Magnetic hysteresis loops were obtained using a Lakeshore vibrating sample magnetometer (VSM) at room temperature. In the two batches, the particles were found to be of the same pure crystalline phase of magnetite. The effects of particle size, size distribution, and concentration on the magnetic properties and magneto thermic efficiency were investigated. Heating profiles, under an alternating magnetic field, were obtained for the two batches of nanoparticles with frequencies 765.85, 634.45, 491.10, 390.25, 349.20, 306.65, and 166.00 kHz and field amplitudes of 100, 200, 250, 300 and 350 G. The specific absorption rate (SAR) values for the particles of size 12.2 nm are higher than those for the particles of size 10.2 nm at all concentrations and field parameters. SAR decreases with the increase of particle concentration. SAR obtained for all the particle concentrations of the two batches increases almost linearly with the field frequency (at fixed field strength) and nonlinearly with the field amplitude (at fixed field frequency). SAR value obtained for magnetite nanoparticles with the highest magnetization is 145.84 W/g at 765.85 kHz and 350 G, whereas the SAR value of the particles with the least magnetization is 81.67 W/g at the same field and frequency.

A Dosimetric Study of Volumetric Arc Modulation with RapidArc Versus Intensity-modulated Radiotherapy inCervical Cancer Patients

Introduction: The method of radiotherapy has moved away from two-dimensional and three-dimensional conformal radiotherapy towards Volumetric-Modulated Arc Therapy (VMAT) for advanced carcinomas. VMAT treatments often result in significant clinical advantage, particularly when concave dose distributions are required as is often the situation since these tumours are in close proximity to several critical structures. Aim:To investigate the potential clinical role of volumetric arc therapy on cervical cancer patients and its comparison with fixed-field Intensity-Modulated Radiotherapy (IMRT) was used as a benchmark. Materials and Methods: Retrospectively, radiotherapy treatment plans of fifteen cervical cancer patients were selected for this study. These patients were previously treated with sliding window IMRT techniques during January 2020 to November 2020. For dosimetric comparison of sliding window IMRT techniques with RapidArc, a new set of plans were created using VMAT/RapidArc technique. For each patient two plans were generated and in this way total 30 plans were analysed. The prescription dose to Planning Target Volume (PTV) was 50.4 Gy in 28 fractions (1.8Gy/fraction) for the 6 MV photon beam. Comparison of each plan done on the basis of Organs At Risk (OARs) sparing, coverage index (C), Conformity Index (CI), Homogeneity Index (HI), dose Gradient Index (GI), and Unified Dosimetry Index (UDI). This study utilised UDI scoring for evaluation and comparison of RapidArc and IMRT plans. Treatment Time (TT) for patient comfort and the number of Monitor Units (MUs) for long-term side-effects was also taken into consideration. A paired two-tailed t-test was executed for the dosimetric study of volumetric arc modulation with RapidArc and its comparison with the IMRT technique in the radiotherapy treatment of cervical cancer patients. All the collected data was analysed using Statistical Package for Social Sciences (SPSS) version 20.0. The (p-value<0.05) was contemplated for the level of statistical significance. Results: Comparable target coverage and better sparing of OARs were achieved with the RapidArc technique in comparison to IMRT. As was evident with results of present study, the values of CI (1.55±0.07), HI (1.07±0.07), GI (0.98±0.01) and UDI (1.25±0.11) of RapidArc technique showed significant difference from respective values of IMRT Technique (1.67±0.06, 1.10±0.06, 0.96±0.01 and 1.38±0.13). Values of MUs (1560.47±52.16) and treatment time (3.71±0.73 mins) were significantlly high in IMRT technique as compared to RapidArc technique (542.33±51.09 and 2.39±0.35 mins respectively). Conclusion: From this study, it is clear that a similar planning goal can be achieved by RapidArc in comparison to fixed-field IMRT with less normal organ toxicity. RapidArc is a faster and precise treatment technique. The most significant change comes to see in the number of MUs and TT, which is much lesser in RapidArc.

On a rationality problem for fields of cross-ratios II

Let k be a field, $x_1, \dots , x_n$ be independent variables and let $L_n = k(x_1, \dots , x_n)$ . The symmetric group $\operatorname {\Sigma }_n$ acts on $L_n$ by permuting the variables, and the projective linear group $\operatorname {PGL}_2$ acts by $$ \begin{align*} \begin{pmatrix} a & b \\ c & d \end{pmatrix}\, \colon x_i \longmapsto \frac{a x_i + b}{c x_i + d} \end{align*} $$ for each $i = 1, \dots , n$ . The fixed field $L_n^{\operatorname {PGL}_2}$ is called “the field of cross-ratios”. Given a subgroup $S \subset \operatorname {\Sigma }_n$ , H. Tsunogai asked whether $L_n^S$ rational over $K_n^S$ . When $n \geqslant 5,$ the second author has shown that $L_n^S$ is rational over $K_n^S$ if and only if S has an orbit of odd order in $\{ 1, \dots , n \}$ . In this paper, we answer Tsunogai’s question for $n \leqslant 4$ .