Given a suitable solution
We explore properties of the family sizes arising in a linear birth process with immigration (BI). In particular, we study the correlation of the number of families observed during consecutive disjoint intervals of time. Letting
- NSF-PAR ID:
- 10490097
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Journal of Mathematical Biology
- Volume:
- 88
- Issue:
- 3
- ISSN:
- 0303-6812
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract V (t ,x ) to the Korteweg–de Vries equation on the real line, we prove global well-posedness for initial data . Our conditions on$$u(0,x) \in V(0,x) + H^{-1}(\mathbb {R})$$ V do include regularity but do not impose any assumptions on spatial asymptotics. We show that periodic profiles satisfy our hypotheses. In particular, we can treat localized perturbations of the much-studied periodic traveling wave solutions (cnoidal waves) of KdV. In the companion paper Laurens (Nonlinearity. 35(1):343–387, 2022.$$V(0,x)\in H^5(\mathbb {R}/\mathbb {Z})$$ https://doi.org/10.1088/1361-6544/ac37f5 ) we show that smooth step-like initial data also satisfy our hypotheses. We employ the method of commuting flows introduced in Killip and Vişan (Ann. Math. (2) 190(1):249–305, 2019.https://doi.org/10.4007/annals.2019.190.1.4 ) where . In that setting, it is known that$$V\equiv 0$$ is sharp in the class of$$H^{-1}(\mathbb {R})$$ spaces.$$H^s(\mathbb {R})$$ -
Abstract It has been recently established in David and Mayboroda (Approximation of green functions and domains with uniformly rectifiable boundaries of all dimensions.
arXiv:2010.09793 ) that on uniformly rectifiable sets the Green function is almost affine in the weak sense, and moreover, in some scenarios such Green function estimates are equivalent to the uniform rectifiability of a set. The present paper tackles a strong analogue of these results, starting with the “flagship degenerate operators on sets with lower dimensional boundaries. We consider the elliptic operators associated to a domain$$L_{\beta ,\gamma } =- {\text {div}}D^{d+1+\gamma -n} \nabla $$ with a uniformly rectifiable boundary$$\Omega \subset {\mathbb {R}}^n$$ of dimension$$\Gamma $$ , the now usual distance to the boundary$$d < n-1$$ given by$$D = D_\beta $$ for$$D_\beta (X)^{-\beta } = \int _{\Gamma } |X-y|^{-d-\beta } d\sigma (y)$$ , where$$X \in \Omega $$ and$$\beta >0$$ . In this paper we show that the Green function$$\gamma \in (-1,1)$$ G for , with pole at infinity, is well approximated by multiples of$$L_{\beta ,\gamma }$$ , in the sense that the function$$D^{1-\gamma }$$ satisfies a Carleson measure estimate on$$\big | D\nabla \big (\ln \big ( \frac{G}{D^{1-\gamma }} \big )\big )\big |^2$$ . We underline that the strong and the weak results are different in nature and, of course, at the level of the proofs: the latter extensively used compactness arguments, while the present paper relies on some intricate integration by parts and the properties of the “magical distance function from David et al. (Duke Math J, to appear).$$\Omega $$ -
Abstract We consider the problem of covering multiple submodular constraints. Given a finite ground set
N , a weight function ,$$w: N \rightarrow \mathbb {R}_+$$ r monotone submodular functions over$$f_1,f_2,\ldots ,f_r$$ N and requirements the goal is to find a minimum weight subset$$k_1,k_2,\ldots ,k_r$$ such that$$S \subseteq N$$ for$$f_i(S) \ge k_i$$ . We refer to this problem as$$1 \le i \le r$$ Multi-Submod-Cover and it was recently considered by Har-Peled and Jones (Few cuts meet many point sets. CoRR.arxiv:abs1808.03260 Har-Peled and Jones 2018) who were motivated by an application in geometry. Even with$$r=1$$ Multi-Submod-Cover generalizes the well-known Submodular Set Cover problem (Submod-SC ), and it can also be easily reduced toSubmod-SC . A simple greedy algorithm gives an approximation where$$O(\log (kr))$$ and this ratio cannot be improved in the general case. In this paper, motivated by several concrete applications, we consider two ways to improve upon the approximation given by the greedy algorithm. First, we give a bicriteria approximation algorithm for$$k = \sum _i k_i$$ Multi-Submod-Cover that covers each constraint to within a factor of while incurring an approximation of$$(1-1/e-\varepsilon )$$ in the cost. Second, we consider the special case when each$$O(\frac{1}{\epsilon }\log r)$$ is a obtained from a truncated coverage function and obtain an algorithm that generalizes previous work on partial set cover ($$f_i$$ Partial-SC ), covering integer programs (CIPs ) and multiple vertex cover constraints Bera et al. (Theoret Comput Sci 555:2–8 Bera et al. 2014). Both these algorithms are based on mathematical programming relaxations that avoid the limitations of the greedy algorithm. We demonstrate the implications of our algorithms and related ideas to several applications ranging from geometric covering problems to clustering with outliers. Our work highlights the utility of the high-level model and the lens of submodularity in addressing this class of covering problems. -
Abstract Let us fix a prime
p and a homogeneous system ofm linear equations for$$a_{j,1}x_1+\dots +a_{j,k}x_k=0$$ with coefficients$$j=1,\dots ,m$$ . Suppose that$$a_{j,i}\in \mathbb {F}_p$$ , that$$k\ge 3m$$ for$$a_{j,1}+\dots +a_{j,k}=0$$ and that every$$j=1,\dots ,m$$ minor of the$$m\times m$$ matrix$$m\times k$$ is non-singular. Then we prove that for any (large)$$(a_{j,i})_{j,i}$$ n , any subset of size$$A\subseteq \mathbb {F}_p^n$$ contains a solution$$|A|> C\cdot \Gamma ^n$$ to the given system of equations such that the vectors$$(x_1,\dots ,x_k)\in A^k$$ are all distinct. Here,$$x_1,\dots ,x_k\in A$$ C and are constants only depending on$$\Gamma $$ p ,m andk such that . The crucial point here is the condition for the vectors$$\Gamma in the solution$$x_1,\dots ,x_k$$ to be distinct. If we relax this condition and only demand that$$(x_1,\dots ,x_k)\in A^k$$ are not all equal, then the statement would follow easily from Tao’s slice rank polynomial method. However, handling the distinctness condition is much harder, and requires a new approach. While all previous combinatorial applications of the slice rank polynomial method have relied on the slice rank of diagonal tensors, we use a slice rank argument for a non-diagonal tensor in combination with combinatorial and probabilistic arguments.$$x_1,\dots ,x_k$$ -
Abstract We continue the program of proving circuit lower bounds via circuit satisfiability algorithms. So far, this program has yielded several concrete results, proving that functions in
and other complexity classes do not have small circuits (in the worst case and/or on average) from various circuit classes$\mathsf {Quasi}\text {-}\mathsf {NP} = \mathsf {NTIME}[n^{(\log n)^{O(1)}}]$ , by showing that$\mathcal { C}$ admits non-trivial satisfiability and/or$\mathcal { C}$ # SAT algorithms which beat exhaustive search by a minor amount. In this paper, we present a new strong lower bound consequence of having a non-trivial# SAT algorithm for a circuit class . Say that a symmetric Boolean function${\mathcal C}$ f (x 1,…,x n ) issparse if it outputs 1 onO (1) values of . We show that for every sparse${\sum }_{i} x_{i}$ f , and for all “typical” , faster$\mathcal { C}$ # SAT algorithms for circuits imply lower bounds against the circuit class$\mathcal { C}$ , which may be$f \circ \mathcal { C}$ stronger than itself. In particular:$\mathcal { C}$ # SAT algorithms forn k -size -circuits running in 2$\mathcal { C}$ n /n k time (for allk ) implyN E X P does not have -circuits of polynomial size.$(f \circ \mathcal { C})$ # SAT algorithms for -size$2^{n^{{\varepsilon }}}$ -circuits running in$\mathcal { C}$ time (for some$2^{n-n^{{\varepsilon }}}$ ε > 0) implyQ u a s i -N P does not have -circuits of polynomial size.$(f \circ \mathcal { C})$ Applying
# SAT algorithms from the literature, one immediate corollary of our results is thatQ u a s i -N P does not haveE M A J ∘A C C 0∘T H R circuits of polynomial size, whereE M A J is the “exact majority” function, improving previous lower bounds againstA C C 0[Williams JACM’14] andA C C 0∘T H R [Williams STOC’14], [Murray-Williams STOC’18]. This is the first nontrivial lower bound against such a circuit class.