On the Diophantine Equation 2^a3^b + 2^c3^d = 2^e3^f + 2^g3^h

This paper is a continuation of [1], in which I studied Harvey Friedman's problem of whether the function f(x,y) = x^2 + y^3 satisfies any identities; however, no knowledge of [1] is necessary to understand this paper. We will break the exponential Diophantine equation 2^a3^b + 2^c3^d = 2^e3^f + 2^g3^h into subcases that are easier to analyze. Then we will solve an equation obtained by imposing a restriction on one of these subcases, after which we will solve a generalization of this equation.Comment: This 6-page paper is the second part of an honors thesis I have written as an undergraduate at UC Berkele

Identities of the Function f(x,y) = x^2 + y^3

Harvey Friedman asked in 1986 whether the function f(x,y) = x^2 + y^3 on the real plane R^2 satisfies any identities; examples of identities are commutativity and associativity. To solve this problem of Friedman, we must either find a nontrivial identity involving expressions formed by recursively applying f to a set of variables {x_1,x_2, ..., x_n} that holds in the real numbers or to prove that no such identities hold. In this paper, we will solve certain special cases of Friedman's problem and explore the connection between this problem and certain Diophantine equations.Comment: This 18-page paper is the first part of an honors thesis I have written as an undergraduate at UC Berkele

Generalizations of an Expansion Formula for Top to Random Shuffles

In the top to random shuffle, the first a cards are removed from a deck of n cards 12 \cdots n and then inserted back into the deck. This action can be studied by treating the top to random shuffle as an element B_a, which we define formally in Section 2, of the algebra Q[S_n]. For a = 1, Adriano Garsia in "On the Powers of Top to Random Shuffling" (2002) derived an expansion formula for B_1^k for k \leq n, though his proof for the formula was non-bijective. We prove, bijectively, an expansion formula for the arbitrary finite product B_{a_1}B_{a_2} \cdots B_{a_k} where a_1, \ldots, a_k are positive integers, from which an improved version of Garsia's aforementioned formula follows. We show some applications of this formula for B_{a_1}B_{a_2} \cdots B_{a_k}, which include enumeration and calculating probabilities. Then for an arbitrary group G we define the group of G-permutations S_n^G := G \wr S_n and further generalize the aforementioned expansion formula to the algebra Q[S_n^G] for the case of finite G, and we show how other similar expansion formulae in Q[S_n] can be generalized to Q[S_n^G].Comment: 15 page

Connecting kk-Naples parking functions and obstructed parking functions via involutions

Parking functions were classically defined for $n$ cars attempting to park on a one-way street with $n$ parking spots, where cars only drive forward. Subsequently, parking functions have been generalized in various ways, including allowing cars the option of driving backward. The set $PF_{n,k}$ of $k$-Naples parking functions have cars who can drive backward a maximum of $k$ steps before driving forward. A recursive formula for $|PF_{n,k}|$ has been obtained, though deriving a closed formula for $|PF_{n,k}|$ appears difficult. In addition, an important subset $B_{n,k}$ of $PF_{n,k}$, called the contained $k$-Naples parking functions, has been shown, with a non-bijective proof, to have the same cardinality as that of the set $PF_n$ of classical parking functions, independent of $k$. In this paper, we study $k$-Naples parking functions in the more general context of $m$ cars and $n$ parking spots, for any $m \leq n$. We use various parking function involutions to establish a bijection between the contained $k$-Naples parking functions and the classical parking functions, from which it can be deduced that the two sets have the same number of ties. Then we extend this bijection to inject the set of $k$-Naples parking functions into a certain set of obstructed parking functions, providing an upper bound for the cardinality of the former set.Comment: 14 pages; corrected a mistake in the last part of v1 regarding the factor of 1/2 in the upper bounds obtaine

Identification of Poinsettia Cultivars Using RAPD Markers

Randomly amplified polymorphic DNA (RAPD) techniques were used to compare the DNA from leaf tissues of nine commercial poinsettia (Euphorbia pulcherrima Wild ex Klotzsch) cultivars. Amplification occurred in 57 out of 60 (95%) tested primers. Nine primers that revealed polymorphisms among cultivars were selected for further evaluation. Forty-eight RAPD bands were scored from these primers, and 33 (69%) were polymorphic. All tested cultivars could be discriminated with seven bands generated from primers OPB7 and OPC13. Results of a UPGMA cluster analysis and principal components analysis placed the nine cultivars into two groups: one group consisted of `Jingle Bells\u27, `Supjibi\u27, and `V-17 Angelika\u27, the other of `V-14 Glory\u27, `Red Sails\u27, `Jolly Red\u27, and `Freedom\u27. `Lilo Red\u27 and `Pink Peppermint\u27 belonged to the latter group, but were relatively distant from other cultivars in that group. These results indicate that RAPDs are efficient for identification of poinsettia cultivars and for determination of the genetic relationships among cultivars

Use of multiobjective genetic algorithms to optimize inter-vehicle active suspensions

This paper studies inter-vehicle active suspensions for railway vehicles and presents an optimization process for the design of vertical active suspension controllers using multiobjective genetic algorithms. A three-vehicle train set is used in the study and two active control schemes are considered primarily to provide the best improvement in the passenger ride quality. The first scheme uses only actuators placed between adjacent vehicles while the second adds two actuators between bogie and vehicle body at either end of the train set in addition to the inter-vehicle actuators. The development of the control laws is assisted by the use of genetic algorithms to achieve the 'best' compromise of different design criteria, especially that between the ride quality and the suspension deflections. The study shows that, when the control laws for the proposed active schemes are optimized, a significant improvement in the vertical ride quality on random tracks is obtained and in the mean time the suspension deflections can be kept within their allowed clearance when the vehicles run on to a gradient

Heterogeneous reactions of particulate matter-bound PAHs and NPAHs with NO3/N2O5, OH radicals, and O3 under simulated long-range atmospheric transport conditions: reactivity and mutagenicity.

The heterogeneous reactions of ambient particulate matter (PM)-bound polycyclic aromatic hydrocarbons (PAHs) and nitro-PAHs (NPAHs) with NO3/N2O5, OH radicals, and O3 were studied in a laboratory photochemical chamber. Ambient PM2.5 and PM10 samples were collected from Beijing, China, and Riverside, California, and exposed under simulated atmospheric long-range transport conditions for O3 and OH and NO3 radicals. Changes in the masses of 23 PAHs and 20 NPAHs, as well as the direct and indirect-acting mutagenicity of the PM (determined using the Salmonella mutagenicity assay with TA98 strain), were measured prior to and after exposure to NO3/N2O5, OH radicals, and O3. In general, O3 exposure resulted in the highest relative degradation of PM-bound PAHs with more than four rings (benzo[a]pyrene was degraded equally well by O3 and NO3/N2O5). However, NPAHs were most effectively formed during the Beijing PM exposure to NO3/N2O5. In ambient air, 2-nitrofluoranthene (2-NF) is formed from the gas-phase NO3 radical- and OH radical-initiated reactions of fluoranthene, and 2-nitropyrene (2-NP) is formed from the gas-phase OH radical-initiated reaction of pyrene. There was no formation of 2-NF or 2-NP in any of the heterogeneous exposures, suggesting that gas-phase formation of NPAHs did not play an important role during chamber exposures. Exposure of Beijing PM to NO3/N2O5 resulted in an increase in direct-acting mutagenic activity which was associated with the formation of mutagenic NPAHs. No NPAH formation was observed in any of the exposures of the Riverside PM. This was likely due to the accumulation of atmospheric degradation products from gas-phase reactions of volatile species onto the surface of PM collected in Riverside prior to exposure in the chamber, thus decreasing the availability of PAHs for reaction

Interaction of two lump solitons described by the Kadomtsev-Petviashvili I equation

The interaction of two lump solitons described by the Kadomtsev-Petviashvili I(KPI) equation is analyzed using both exact and numerical methods. The numerical method is based on a third order Runge-Kutta method, and a Crank- Nicholson scheme. The main characteristic of a direct interaction when the two lumps are initially aligned along the x-axis, is that they may separate in the y-direction, but then come back to the xaxis after collision; the dependence of the maximum separation in the y-direction on the relative velocity difference is described. Two lumps may also experience an abrupt phase change in the case of an oblique interaction