Physical Consequences of Complex Dimensions of Fractals

It has recently been realized that fractals may be characterized by complex dimensions, arising from complex poles of the corresponding zeta function, and we show here that these lead to oscillatory behavior in various physical quantities. We identify the physical origin of these complex poles as the exponentially large degeneracy of the iterated eigenvalues of the Laplacian, and discuss applications in quantum mesoscopic systems such as oscillations in the fluctuation $\Sigma^2 (E)$ of the number of levels, as a correction to results obtained in Random Matrix Theory. We present explicit expressions for these oscillations for families of diamond fractals, also studied as hierarchical lattices.Comment: 4 pages, 3 figures; v2: references added, as published in Europhysics Letter

Existence of a Meromorphic Extension of Spectral Zeta Functions on Fractals

We investigate the existence of the meromorphic extension of the spectral zeta function of the Laplacian on self-similar fractals using the classical results of Kigami and Lapidus (based on the renewal theory) and new results of Hambly and Kajino based on the heat kernel estimates and other probabilistic techniques. We also formulate conjectures which hold true in the examples that have been analyzed in the existing literature

Spectral renormalization group theory on networks

Discrete amorphous materials are best described in terms of arbitrary networks which can be embedded in three dimensional space. Investigating the thermodynamic equilibrium as well as non-equilibrium behavior of such materials around second order phase transitions call for special techniques. We set up a renormalization group scheme by expanding an arbitrary scalar field living on the nodes of an arbitrary network, in terms of the eigenvectors of the normalized graph Laplacian. The renormalization transformation involves, as usual, the integration over the more "rapidly varying" components of the field, corresponding to eigenvectors with larger eigenvalues, and then rescaling. The critical exponents depend on the particular graph through the spectral density of the eigenvalues.Comment: 17 pages, 3 figures, presented at the Continuum Models and Discrete Systems (CMDS-12), 21-25 Feb 2011, Saha Institute of Nuclear Physics, Kolkata, Indi

Spectral analysis on infinite Sierpinski fractafolds

A fractafold, a space that is locally modeled on a specified fractal, is the fractal equivalent of a manifold. For compact fractafolds based on the Sierpinski gasket, it was shown by the first author how to compute the discrete spectrum of the Laplacian in terms of the spectrum of a finite graph Laplacian. A similar problem was solved by the second author for the case of infinite blowups of a Sierpinski gasket, where spectrum is pure point of infinite multiplicity. Both works used the method of spectral decimations to obtain explicit description of the eigenvalues and eigenfunctions. In this paper we combine the ideas from these earlier works to obtain a description of the spectral resolution of the Laplacian for noncompact fractafolds. Our main abstract results enable us to obtain a completely explicit description of the spectral resolution of the fractafold Laplacian. For some specific examples we turn the spectral resolution into a "Plancherel formula". We also present such a formula for the graph Laplacian on the 3-regular tree, which appears to be a new result of independent interest. In the end we discuss periodic fractafolds and fractal fields

Laplace Operators on Fractals and Related Functional Equations

We give an overview over the application of functional equations, namely the classical Poincar\'e and renewal equations, to the study of the spectrum of Laplace operators on self-similar fractals. We compare the techniques used to those used in the euclidean situation. Furthermore, we use the obtained information on the spectral zeta function to define the Casimir energy of fractals. We give numerical values for this energy for the Sierpi\'nski gasket

Phase I study of dose-escalated paclitaxel, ifosfamide, and cisplatin (PIC) combination chemotherapy in advanced solid tumours

Based on the already known in vitro synergy between paclitaxel (taxol), cisplatin and oxazophosphorine cytostatics and the broad spectrum of activity of the above drugs we sought to evaluate the paclitaxel (taxol)-ifosfamide-cisplatin (PIC) combination in the outpatient setting in individuals with a variety of advanced solid tumours. Cohorts of patients were entered into six successive dose levels (DLs) with drug doses ranging as follows: paclitaxel 135–215 mg m−2day 1 – (1 h infusion), ifosfamide 4.5–6.0 g m−2(total dose) – divided over days 1 and 2, and cisplatin 80–100 mg m−2(total) – divided over days 1 and 2. Granulocyte colony-stimulating factor was given from day 5 to 14. Forty-two patients were entered. Eighteen patients had 2–8 cycles of prior chemotherapy with no taxanes or ifosfamide (cisplatin was allowed). The regimen was tolerated with outpatient administration in 36/42 patients. Toxicities included: grade 4 neutropenia for ≤ 5 days in 27% of cycles; 5 episodes of febrile neutropenia in three patients at DL-III, -V and -VI. Grade 3/4 thrombocytopenia and cumulative grade 3 anaemia were seen in 7% and 13% of cycles respectively. Three cases of severe grade 3 neuromotor/sensory neuropathy were recorded at DL-II, -III, and -V, all after cycle 3. The maximum tolerated dose was not formally reached at DL-V, but because of progressive anaemia and asthenia/fatigue, it was decided to test a new DL-VI with doses of paclitaxel 200 mg m−2, ifosfamide 5.0 g m−2and cisplatin 100 mg m−2; this appeared to be tolerable and is recommended for further phase II testing. The response rate was 47.5% (complete response + partial response: 20/42). The PIC regimen appears to be feasible and safe in the outpatient setting. Care should be paid to neurotoxicity. Phase II studies are starting in non-small-cell lung cancer, ovarian cancer and head and neck cancer at DL-VI. © 2000 Cancer Research Campaig

A phase I/II study of gemcitabine and fractionated cisplatin in an outpatient setting using a 21-day schedule in patients with advanced and metastatic bladder cancer

A randomised phase III trial of MVAC (methotrexate, vincristine, doxorubicin, cisplatin) vs gemcitabine and cisplatin (GC) (G 1000 mg m(-2) days 1, 8, and 15 plus C 70 mg m(-2) day 2, q 4 wks) indicated GC had similar efficacy and lower toxicity (JCO 2000). Significant haematologic toxicities in the GC arm occurred on day 15, necessitating dose adjustments in 37% of cycles. We conducted a phase I/II dose escalation trial using GC on a 21-day cycle, with G and C split between days 1 and 8. The objective of the study to define maximum-tolerated dose and dose-limiting toxicity (DLT), objective response rate, and overall survival. In all, 32 patients with locally advanced, relapsed, or metastatic disease received: dose level 1, G/C 1000/35; level 2, 1100/35; level 3, 1200/35; level 4, 1200/45 mg m(-2) (G and C given on days 1 and 8 every 3 wks). A total of 19 patients had glomerular filtration rate <60 ml min(-1) and 19 patients had metastatic disease. Dose-limiting toxicity was haematologic (grade 4 thrombocytopenia) at dose level 2. Of 151 cycles, at day 15, platelets were <100 in 61 cycles; neutrophils <0.5, platelets <50 in 26 cycles. Only seven cycles were deferred due to haematological toxicity; four for renal toxicity (chemotherapy instituted posthydration). Overall response rate was 65.5% on an intention-to-treat analysis (75% [21/28] for assessable patients), with four complete responses (12.5%) and 17 partial responses (53%). After the median follow-up of 17.2 months (range 13.1-32.4 months), 12 patients remain alive. The overall median survival was 16 months (range 10.1-26.6 months). G plus C every 3 weeks is active and well tolerated in an outpatient setting, even in patients receiving prior platinum-based regimens and with poor renal reserve

Adaptive intrapatient dose escalation of cisplatin in combination with low-dose vp16 in patients with nonsmall cell lung cancer

The objective of this phase II and pharmacologic study was to explore the feasibility toxicity and activity of adaptive intrapatient dose escalation of cisplatin in a dose-intensive weekly schedule using predefined levels of exposure, with the ultimate aim to improve the antitumour activity of the therapy in patients with nonsmall cell lung cancer (NSCLC). Platinum DNA-adduct levels in peripheral white blood cells during treatment were used as the primary parameter for adaptive dosing. If DNA-adduct levels were not available, the area under the concentration-time curve (AUC) of unbound platinum in plasma was used for dose adaptation. Target levels for DNA-adducts and AUC have been defined in a previously performed pharmacologic study. The feasibility of adaptive dosing was tested in 76 patients with stage IIIB and IV NSCLC, who were planned to receive 6 weekly courses of cisplatin at a starting dose of 70 mg m-2, together with daily low oral dose of 50 mg VP16. In total, 37 patients (49%) who were given more than one course received a dose increase varying from 10 to 55%. The majority of patients reached the defined target levels by a dose increase during course two. Relevant grade 2 neurotoxicity was observed in eight (10%) patients and reversible ototoxicity grade 2 in 14 (18%) patients. The strategy of adaptive intrapatient dose adjustment of cisplatin is practically feasible in a research setting even when results for dose adaptation have to be reported within a short time-period of I week. The toxicity appeared to be manageable in this cohort of patients. In some patients, exposure after the standard dose was substantially lower than the defined target level and significant dose escalations of more than 50% had to be applied. The response rate (RR) was relatively high: overall 40% (29 out of 72 patients) partial remission (PR), in patients with stage IIIB the RR was 60% (15 out of 25 patients) and with stage IV 30% (14 out of 47 patients). Randomised studies are needed to determine whether the adaptive dosing strategy results in better efficacy than standard dosing