Asymptotic trace formula for the Hecke operators

Given integers $m$, $n$ and $k$, we give an explicit formula with an optimal error term (with square root cancelation) for the Petersson trace formula involving the $m$-th and $n$-th Fourier coefficients of an orthonormal basis of $S_k(N)^*$ (the weight $k$ newforms with fixed square-free level $N$) provided that $|4 \pi \sqrt{mn}- k|=o(k^{\frac{1}{3}})$. Moreover, we establish an explicit formula with a power saving error term for the trace of the Hecke operator $\mathcal{T}_n^*$ on $S_k(N)^*$ averaged over $k$ in a short interval. By bounding the second moment of the trace of $\mathcal{T}_{n}$ over a larger interval, we show that the trace of $\mathcal{T}_n$ is unusually large in the range $|4 \pi \sqrt{n}- k| = o(n^{\frac{1}{6}})$. As an application, for any fixed prime $p$ with $\gcd(p,N)=1$, we show that there exists a sequence $\{k_n\}$ of weights such that the error term of Weyl's law for $\mathcal{T}_p$ is unusually large and violates the prediction of arithmetic quantum chaos. In particular, this generalizes the result of Gamburd, Jakobson and Sarnak~\cite[Theorem 1.4]{Gamburd} with an improved exponent

Toll-подобные рецепторы

The study results of the last 10-12 years have established that the activity of the nonspecific immunological protection system is based on a limited set of specific membrane receptors, which have been called molecular ‘pattern recognition receptors’ (PRRs) that recognize pathogen associated molecular patterns (PAMPs). After the detection of PAMPs PRRs trigger an inflammatory response that leads to the destruction of pathogens. Toll-like receptors (TLRs) refer to PRRs, along with three other families: NOD-Like receptors, RIG-I-Like receptors and C-type Lectin receptors. In humans, TLRs are present on dendritic cells, macrophages, neutrophils, B-lymphocytes, mast cells, enterocytes, some cells of the central nervous system (astrocytes), hepatocytes, etc. The attempts have been made in order to obtain vaccines against microbes, as well as the vaccines designed to trigger or enhance an anti-tumor immune response by stimulating the TLRs and other similar receptors. The suppression or enhancement of TLRs activity is an important pathogenic mechanism, underlying many infectious or immunological diseases (including autoimmune). These mechanisms can be used for the further elaboration of new methods and algorithms for prevention, diagnosis and treatment of the mentioned diseases. Thus, not only strong fundamental data on the mechanisms of the innate immunity have been accumulated and the attempts to apply these results have been made, but also it has been managed to integrate the non-specific immunity mechanisms with the adaptive immunity and to found a new field in immunology - an integrated immunology, which creates the background for new therapeutic, diagnostic and preventive strategies in the future

Alternating electric fields (TTFields) inhibit metastatic spread of solid tumors to the lungs

Tumor treating fields (TTFields) are low intensity, intermediate frequency, alternating electric fields used to treat cancerous tumors. This novel treatment modality effectively inhibits the growth of solid tumors in vivo and has shown promise in pilot clinical trials in patients with advanced stage solid tumors. TTFields were tested for their potential to inhibit metastatic spread of solid tumors to the lungs in two animal models: (1) Mice injected with malignant melanoma cells (B16F10) into the tail vein, (2) New Zealand White rabbits implanted with VX-2 tumors within the kidney capsule. Mice and rabbits were treated using two-directional TTFields at 100–200 kHz. Animals were either monitored for survival, or sacrificed for pathological and histological analysis of the lungs. The total number of lung surface metastases and the absolute weight of the lungs were both significantly lower in TTFields treated mice then in sham control mice. TTFields treated rabbits survived longer than sham control animals. This extension in survival was found to be due to an inhibition of metastatic spread, seeding or growth in the lungs of TTFields treated rabbits compared to controls. Histologically, extensive peri- and intra-tumoral immune cell infiltration was seen in TTFields treated rabbits only. These results raise the possibility that in addition to their proven inhibitory effect on the growth of solid tumors, TTFields may also have clinical benefit in the prevention of metastatic spread from primary tumors