We apply multigrade efficient congruencing to estimate Vinogradov’s integral of degree $k$ for moments of order $2s$ , establishing strongly diagonal behaviour for $1\leqslant s\leqslant \frac{1}{2}k(k+1)-\frac{1}{3}k+o(k)$ . In particular, as $k\rightarrow \infty$ , we confirm the main conjecture in Vinogradov’s mean value theorem for a proportion asymptotically approaching $100\%$ of the critical interval $1\leqslant s\leqslant \frac{1}{2}k(k+1)$ .

A self-adjoint first-order system with Hermitian π-periodic potential Q(z), integrable on compact sets, is considered. It is shown that all zeros of are double zeros if and only if this self-adjoint system is unitarily equivalent to one in which Q(z) is π/2-periodic. Furthermore, the zeros of are all double zeros if and only if the associated self-adjoint system is unitarily equivalent to one in which Q(z) = σ 2 Q(z)σ 2. Here, Δ denotes the discriminant of the system and σ 0, σ 2 are Pauli matrices. Finally, it is shown that all instability intervals vanish if and only if Q = rσ 0 + qσ 2, for some real-valued π-periodic functions r and q integrable on compact sets.

We study functions whose time-frequency content are concentrated in a compact region in phase space using time-frequency localization operators as a main tool. We obtain approximation inequalities for such functions using a finite linear combination of eigenfunctions of these operators, as well as a local Gabor system covering the region of interest. These would allow the construction of modified time-frequency dictionaries concentrated in the region.

Let X be a smooth rational surface. We calculate a differential graded (DG) quiver of a full exceptional collection of line bundles on X obtained by an augmentation from a strong exceptional collection on the minimal model of X. In particular, we calculate canonical DG algebras of smooth toric surfaces.

We investigate a system of singular–degenerate parabolic equations with non-local terms, which can be regarded as a spatially heterogeneous competition model of Lotka–Volterra type. Applying the Leray–Schauder fixed-point theorem, we establish the existence of coexistence periodic solutions to the problem, which, together with the existing literature, gives a complete picture for such a system for all parameters.

Cloneable sets of states in C *-algebras are characterized in terms of strong orthogonality of states. Moreover, the relation between strong cloning and distinguishability of states is investigated together with some additional properties of strong cloning in abelian C *-algebras.

We present a full classification of the short-time behaviour of the interfaces and local solutions to the nonlinear parabolic p-Laplacian type reaction–diffusion equation of non-Newtonian elastic filtration:

$$\begin{equation*}u_t-\Big(|u_x|^{p-2}u_x\Big)_x+bu^{\beta}=0, \ p>2, \beta >0.\end{equation*}$$

We present a first example of a flag vector of a polyhedral sphere that is not the flag vector of any polytope. Namely, there is a unique $3$ -sphere with the parameters $(f_{0},f_{1},f_{2},f_{3};f_{02})=(12,40,40,12;120)$ , but this sphere is not realizable by a convex $4$ -polytope. The $3$ -sphere, which is $2$ -simple and $2$ -simplicial, was found by Werner [Linear constraints on face numbers of polytopes. PhD Thesis, TU Berlin, Germany, 2009]; we present results of a computer enumeration which imply that the sphere with these parameters is unique. We prove that it is non-polytopal in two ways: first, we show that it has no oriented matroid, and thus it is not realizable; this proof was found by computer, but can be verified by hand. The second proof is again a computer-based oriented matroid proof and shows that for exactly one of the facets this sphere does not even have a diagram based on this facet. Using the non-polytopality, we finally prove that the sphere is not even embeddable as a polytopal complex.

A graph G is H-saturated if it contains no copy of H as a subgraph but the addition of any new edge to G creates a copy of H. In this paper we are interested in the function satt (n,p), defined to be the minimum number of edges that a Kp -saturated graph on n vertices can have if it has minimum degree at least t. We prove that satt (n,p) = tn − O(1), where the limit is taken as n tends to infinity. This confirms a conjecture of Bollobás when p = 3. We also present constructions for graphs that give new upper bounds for satt (n,p).

We provide new quantitative versions of Helly’s theorem. For example, we show that for every family $\{P_{i}:i\in I\}$ of closed half-spaces in $\mathbb{R}^{n}$ such that $P=\bigcap _{i\in I}P_{i}$ has positive volume, there exist $s\leqslant \unicode[STIX]{x1D6FC}n$ and $i_{1},\ldots ,i_{s}\in I$ such that

$$\begin{eqnarray}\operatorname{vol}_{n}(P_{i_{1}}\cap \cdots \cap P_{i_{s}})\leqslant (Cn)^{n}\operatorname{vol}_{n}(P),\end{eqnarray}$$

$\unicode[STIX]{x1D6FC},C>0$

Let ℱ be a family of graphs and let d be large enough. For every d-regular graph G, we study the existence of a spanning ℱ-free subgraph of G with large minimum degree. This problem is well understood if ℱ does not contain bipartite graphs. Here we provide asymptotically tight results for many families of bipartite graphs such as cycles or complete bipartite graphs. To prove these results, we study a locally injective analogue of the question.

Let k ⩾ 3 be a fixed integer and let Zk (G) be the number of k-colourings of the graph G. For certain values of the average degree, the random variable Zk (G(n, m)) is known to be concentrated in the sense that $\tfrac{1}{n}(\ln Z_k(G(n,m))-\ln\Erw[Z_k(G(n,m))])$ converges to 0 in probability (Achlioptas and Coja-Oghlan, Proc. FOCS 2008). In the present paper we prove a significantly stronger concentration result. Namely, we show that for a wide range of average degrees, $\tfrac{1}{\omega}(\ln Z_k(G(n,m))-\ln\Erw[Z_k(G(n,m))])$ converges to 0 in probability for any diverging function $\omega=\omega(n)\ra\infty$ . For k exceeding a certain constant k 0 this result covers all average degrees up to the so-called condensation phase transitiond k,con , and this is best possible. As an application, we show that the experiment of choosing a k-colouring of the random graph G(n,m) uniformly at random is contiguous with respect to the so-called ‘planted model’.

Let $\widetilde{\text{Sp}}(2n)$ be the metaplectic covering of $\text{Sp}(2n)$ over a local field of characteristic zero. The core of the theory of endoscopy for $\widetilde{\text{Sp}}(2n)$ is the geometric transfer of orbital integrals to its elliptic endoscopic groups. The dual of this map, called the spectral transfer, is expected to yield endoscopic character relations which should reveal the internal structure of $L$-packets. As a first step, we characterize the image of the collective geometric transfer in the non-archimedean case, then reduce the spectral transfer to the case of cuspidal test functions by using a simple stable trace formula. In the archimedean case, we establish the character relations and determine the spectral transfer factors by rephrasing the works by Adams and Renard.

Classical ergodic theory for integer-group actions uses entropy as a complete invariant for isomorphism of IID (independent, identically distributed) processes (a.k.a. product measures). This theory holds for amenable groups as well. Despite recent spectacular progress of Bowen, the situation for non-amenable groups, including free groups, is still largely mysterious. We present some illustrative results and open questions on free groups, which are particularly interesting in combinatorics, statistical physics and probability. Our results include bounds on minimum and maximum bisection for random cubic graphs that improve on all past bounds.

Given an undirected graph G, let us randomly orient G by tossing independent (possibly biased) coins, one for each edge of G. Writing a → b for the event that there exists a directed path from a vertex a to a vertex b in such a random orientation, we prove that for any three vertices s, a and b of G, we have ℙ(s → a ∩ s → b) ⩾ ℙ(s → a) ℙ(s → b).

We determine the Galois representations inside the $\ell$-adic cohomology of some quaternionic and related unitary Shimura varieties at ramified places. The main results generalize the previous works of Reimann and Kottwitz in this setting to arbitrary levels at $p$, and confirm the expected description of the cohomology due to Langlands and Kottwitz.

In this article, we use Mellin transforms to derive alternative results for option pricing and implied volatility estimation when the underlying asset price is governed by jump-diffusion dynamics. The current well known results are restrictive since the jump is assumed to follow a predetermined distribution (e.g., lognormal or double exponential). However, the results we present are general since we do not specify a particular jump-diffusion model within the derivations. In particular, we construct and derive an exact solution to the option pricing problem in a general jump-diffusion framework via Mellin transforms. This approach of Mellin transforms is further extended to derive a Dupire-like partial integro-differential equation, which ultimately yields an implied volatility estimator for assets subjected to instantaneous jumps in the price. Numerical simulations are provided to show the accuracy of the estimator.

When a towed sonar array is straight, it has the difficulty that it cannot distinguish a contact on the left from one at the same angle on the right. When the array is not straight and its shape known, we calculate the probability that the left–right ambiguity is resolved correctly, using the Neyman–Pearson hypothesis testing framework, assuming a delay-sum beamformer, a single-frequency contact, and Gaussian noise. We also initially consider the noise field to be uncorrelated and show that the evaluation of the probability of correct resolution reduces to evaluating a one-dimensional integral. We find, as expected, that the probability increases, as the signal-to-noise ratio and the lateral deviation of the array from straight increase. For demonstration purposes, we also evaluate the probability of correct resolution for two representative shapes the array might assume in practice. Finally, we consider a more realistic, correlated noise field and we show that the initial assumption of an uncorrelated noise field provides a good approximation when the lateral deviation of the array is sufficiently large.