Prime graphs of finite groups with a small number of triangles

The prime graph \(\Delta (G)\) of a finite group \(G\) is a graph whose vertices are the primes which divide the degrees of some irreducible complex characters of \(G\) and two distinct primes \(p\) and \(q\) are joined by an edge if the product \(pq\) divides some character degree of \(G\) . In this paper, we determine the upper bounds for the numbers of vertices of the prime graphs of finite groups which possess a small number of triangles. In some cases, we study the structure of such finite groups and their prime graphs in detail.     

Generalized Random Spectral Measures

In an attempt to examine the random version of the spectral theorem, the notion of random spectral measures and generalized random spectral measures are introduced and investigated. It is shown that each generalized random spectral measure on $(\mathbb C ,\mathcal{B}(\mathbb C ))$ admits a modification which is a random spectral measure.     

Circuits of Each Length in Tournaments

A tournament is a directed graph whose underlying graph is a complete graph. A circuit is an alternating sequence of vertices and arcs of the form v ₁, a ₁, v ₂, a ₂, v ₃, . . . , v _n-1, a _n-1, v _n in which vertex v _n  = v ₁, arc a _i  = v _i v _i+1 for i = 1, 2, . . . , n?1, and \({a_i \neq a_j}\) if \({i \neq j}\) . In this paper, we shall show that every tournament T _n in a subclass of tournaments has a circuit of each length k for \({3 \leqslant k \leqslant \theta(T_n)}\) , where \({\theta(T_n) = \frac{n(n-1)}{2}-3}\) if n is odd and \({\theta(T_n) = \frac{n(n-1)}{2}-\frac{n}{2}}\) otherwise. Note that a graph having θ(G) > n can be used as a host graph on embedding cycles with lengths larger than n to it if congestions are allowed only on vertices.     

Extracting informative variables in the validation of two-group causal relationship

The validation of causal relationship between two groups of multivariate time series data often requires the precedence knowledge of all variables. However, in practice one finds that some variables may be negligible in describing the underlying causal structure. In this article we provide an explicit definition of “non-informative variables” in a two-group causal relationship and introduce various automatic computer-search algorithms that can be utilized to extract informative variables based on a hypothesis testing procedure. The result allows us to represent a simplified causal relationship by using minimum possible information on two groups of variables.     

The weak conjecture on spherical classes

Let be the mod 2 Steenrod algebra. We construct a chain-level representation of the dual of Singer's algebraic transfer, which maps Singer's invariant-theoretic model of the dual of the Lambda algebra, , to and is the inclusion of the Dickson algebra, , into . This chain-level representation allows us to confirm the weak conjecture on spherical classes (see [9]), assuming the truth of (1) either the conjecture that the Dickson invariants of at least k = 3 variables are homologically zero in }, (2) or a conjecture on ${\mathcal{A}}$ -decomposability of the Dickson algebra in $\Gamma_k^{\wedge}$. We prove the conjecture in item (1) for k = 3 and also show a weak form of the conjecture in item (2). Received November 27, 1996; in final form March 6, 1998     

Rhodamine‐Ethylenediol,A Novel Vital Fluorescent Probe for Labelling Alkaline Phosphatase‐Rich Organelles

Zebrafish have received considerable attention as an organism‐based model in the development of pharmacological agents.^1,2 Many small molecules applied to zebrafish show important behaviours and may constitute new kinds of markers for clinical purposes.³ Analysis of these molecules can facilitate the development of useful tools for monitoring environmental changes.⁴ Many chemicals that are toxic to the environment are known to influence the sensory systems of humans⁵ and fish.⁶ One important sensory system in all fish is the lateral line organ,⁷ which is readily accessible for the assessment of environmental changes.⁸ Neuromasts, which are located on the surface of the fish body, are one of the major components of the lateral lines of the zebrafish.⁹ Copper‐enriched water is known to affect the olfactory system in fish. Therefore, small molecules that induce specific patterns in the neuromasts of zebrafish should provide an important animal model with which to explore the effects of environmental changes on the sensory system.^10,11 Recently, chemical sensors based on the rhodamine skeleton¹² have been designed to specifically detect metal ions, such as Cu(II)¹³ and Fe(III)/Hg(II),¹⁴ in zebrafish. However, there has been no report of these rhodamine derivatives used in the specific recognition of the sensory system of zebrafish. Commonly, the sensory system is studied with antibody staining assays of scarified fish. Here, we report that a new rhodamine derivative can be used as a fluorescent chemical probe to visualize the neuromasts and intestinal villi of living zebrafish. Based on the specific recognition of this area in zebrafish, we narrowed the possible enzymes targeted by this rhodamine probe to alkaline phosphatase and confirmed this with a binding assay. It is a well‐recognized challenge to develop a fluorescent chemical probe that specifically recognizes a particular enzyme. Furthermore, the transfer of phosphate groups to certain enzymes can activate their catalytic reactivity, triggering a cascade reaction in a signal transduction pathway. The alkaline‐phosphatase‐specific recognition by this rhodamine derivative may be applicable to clinical purposes.     

Electronic quantum motions in hydrogen molecule ion

The hydrogen molecule ion is a two‐center force system expressed under the prolate spheroidal coordinates, whose quantum motions and quantum trajectories have never been addressed in the literature before. The momentum operators in this coordinate system are derived for the first time from the Hamilton equations of motion and used to construct the Hamiltonian operator. The resulting Hamiltonian comprises a kinetic energy T and a total potential V_Total consisting of the Coulomb potential and a quantum potential. It is shown that the participation of the quantum potential and the accompanied quantum forces in the force interaction within H₂⁺ is essential to develop an electronic motion consistent with the prediction of the probability density function |Ψ|². The motion of the electron in H₂⁺ can be either described by the Hamilton equations derived from the Hamiltonian H = T_K + V_Total or by the Lagrange equations derived from the Lagrangian H = T_K ? V_Total. Solving the equations of motion with different initial positions, we show that the solutions yield an assembly of electronic quantum trajectories whose distribution and concentration reconstruct the σ and π molecular orbitals in H₂⁺. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Characterizing fracture energy of proton exchange membranes using a knife slit test

Pinhole formation in proton exchange membranes (PEM) may be caused by a process of flaw formation and crack propagation within membranes exposed to cyclic hygrothermal loading. Fracture mechanics can be used to characterize the propagation process, which is thought to occur in a slow, time‐dependent manner under cyclic loading conditions, and believed to be associated with limited plasticity. The intrinsic fracture energy has been used to characterize the fracture resistance of polymeric material with limited viscoelastic and plastic dissipation, and has been found to be associated with long‐term durability of polymeric materials. Insight into this limiting value of fracture energy may be useful in characterizing the durability of proton exchange membranes, including the formation of pinhole defects. In an effort to collect fracture data with limited plasticity, a knife slit test was adapted to measure fracture energies of PEMs, resulting in fracture energies that were two orders of magnitude smaller than those obtained with other fracture test methods. The presence of a sharp knife blade reduces crack tip plasticity, providing fracture energies that may be more representative of the intrinsic fracture energies of the thin membranes. Three commercial PEMs were tested to evaluate their fracture energies (G_c) at temperatures ranging from 40 to 90 °C and humidity levels varying from dry to 90% relative humidity (RH). Experiments were also conducted with membrane specimens immersed in water at various temperatures. The time temperature moisture superposition principle was applied to generate fracture energy master curves plotted as a function of reduced cutting rate based on the humidity and temperature conditions of the tests. The shift with respect to temperature and humidity suggests that the slitting process is viscoelastic in nature. Also such shifts were found to be consistent with those obtained from constitutive tests such as stress relaxation. The fracture energy is more sensitive to temperature than on humidity. The master curves converge at the lowest reduced cutting rates, suggesting similar intrinsic fracture energies; but diverge at higher reduced cutting rates to significantly different fracture energies. Although the relationship between G_c and ultimate mechanical durability has not been established, the test method may hold promise for investigating and comparing membrane resistance to failure in fuel cell environments. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 48: 333–343, 2010