How many digits are needed?

Let $X_1,X_2,...$ be the digits in the base-$q$ expansion of a random variable $X$ defined on $[0,1)$ where $q\ge2$ is an integer. For $n=1,2,...$, we study the probability distribution $P_n$ of the (scaled) remainder $\sum_{k=n+1}^\infty X_qq^{n-k}$: If $X$ has an absolutely continuous CDF then $P_n$ converges in the total variation metric to Lebesgue measure on the unit interval; under certain smoothness conditions we establish exponentially fast convergence of $P_n$ and its PDF $f_n$; and we give examples of these results. The results are extended to the case of a multivariate random variable defined on a unit cube.Comment: 15 pages, 2 figure

Classical scattering with long range forces

We discuss the classical two-body scattering problem for potentials which decrease at infinity like r −α , 1≧α>0. We prove existence and uniqueness theorems for scattering orbits parametrized by their asymptotic data. Wave operators are constructed and their properties discussed. We also discuss and prove cluster properties of the S -operator.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46509/1/220_2005_Article_BF01646193.pd

On the connectedness structure of the Coulomb S -matrix

The forward direction singularity of the non-relativistic Coulomb S -matrix is examined and discussed. The relativistic Coulomb S -matrix to order α is shown to have a similar singularity.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46508/1/220_2005_Article_BF01646192.pd

31st Annual Meeting and Associated Programs of the Society for Immunotherapy of Cancer (SITC 2016) : part two

Background The immunological escape of tumors represents one of the main ob- stacles to the treatment of malignancies. The blockade of PD-1 or CTLA-4 receptors represented a milestone in the history of immunotherapy. However, immune checkpoint inhibitors seem to be effective in specific cohorts of patients. It has been proposed that their efficacy relies on the presence of an immunological response. Thus, we hypothesized that disruption of the PD-L1/PD-1 axis would synergize with our oncolytic vaccine platform PeptiCRAd. Methods We used murine B16OVA in vivo tumor models and flow cytometry analysis to investigate the immunological background. Results First, we found that high-burden B16OVA tumors were refractory to combination immunotherapy. However, with a more aggressive schedule, tumors with a lower burden were more susceptible to the combination of PeptiCRAd and PD-L1 blockade. The therapy signifi- cantly increased the median survival of mice (Fig. 7). Interestingly, the reduced growth of contralaterally injected B16F10 cells sug- gested the presence of a long lasting immunological memory also against non-targeted antigens. Concerning the functional state of tumor infiltrating lymphocytes (TILs), we found that all the immune therapies would enhance the percentage of activated (PD-1pos TIM- 3neg) T lymphocytes and reduce the amount of exhausted (PD-1pos TIM-3pos) cells compared to placebo. As expected, we found that PeptiCRAd monotherapy could increase the number of antigen spe- cific CD8+ T cells compared to other treatments. However, only the combination with PD-L1 blockade could significantly increase the ra- tio between activated and exhausted pentamer positive cells (p= 0.0058), suggesting that by disrupting the PD-1/PD-L1 axis we could decrease the amount of dysfunctional antigen specific T cells. We ob- served that the anatomical location deeply influenced the state of CD4+ and CD8+ T lymphocytes. In fact, TIM-3 expression was in- creased by 2 fold on TILs compared to splenic and lymphoid T cells. In the CD8+ compartment, the expression of PD-1 on the surface seemed to be restricted to the tumor micro-environment, while CD4 + T cells had a high expression of PD-1 also in lymphoid organs. Interestingly, we found that the levels of PD-1 were significantly higher on CD8+ T cells than on CD4+ T cells into the tumor micro- environment (p < 0.0001). Conclusions In conclusion, we demonstrated that the efficacy of immune check- point inhibitors might be strongly enhanced by their combination with cancer vaccines. PeptiCRAd was able to increase the number of antigen-specific T cells and PD-L1 blockade prevented their exhaus- tion, resulting in long-lasting immunological memory and increased median survival

Characterization of random variables with stationary digits

Let $q\ge2$ be an integer, $\{X_n\}_{n\geq 1}$ a stochastic process with state space $\{0,\ldots,q-1\}$, and $F$ the cumulative distribution function (CDF) of $\sum_{n=1}^\infty X_n q^{-n}$. We show that stationarity of $\{X_n\}_{n\geq 1}$ is equivalent to a functional equation obeyed by $F$ and use this to characterize the characteristic function of $X$ and the structure of $F$ in terms of its Lebesgue decomposition. More precisely, while the absolutely continuous component of $F$ can only be the uniform distribution on the unit interval, its discrete component can only be a countable convex combination of certain explicitly computable CDFs for probability distributions with finite support. We also show that $\mathrm{d} F$ is a Rajchman measure if and only if $F$ is the uniform CDF on $[0,1]$.Comment: 20 pages and 1 figur