Let
 Award ID(s):
 1902756
 NSFPAR ID:
 10439299
 Date Published:
 Journal Name:
 Advanced Nonlinear Studies
 Volume:
 23
 Issue:
 1
 ISSN:
 21690375
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Abstract denote the standard Haar system on [0, 1], indexed by$$(h_I)$$ $\left({h}_{I}\right)$ , the set of dyadic intervals and$$I\in \mathcal {D}$$ $I\in D$ denote the tensor product$$h_I\otimes h_J$$ ${h}_{I}\otimes {h}_{J}$ ,$$(s,t)\mapsto h_I(s) h_J(t)$$ $(s,t)\mapsto {h}_{I}\left(s\right){h}_{J}\left(t\right)$ . We consider a class of twoparameter function spaces which are completions of the linear span$$I,J\in \mathcal {D}$$ $I,J\in D$ of$$\mathcal {V}(\delta ^2)$$ $V\left({\delta}^{2}\right)$ ,$$h_I\otimes h_J$$ ${h}_{I}\otimes {h}_{J}$ . This class contains all the spaces of the form$$I,J\in \mathcal {D}$$ $I,J\in D$X (Y ), whereX andY are either the Lebesgue spaces or the Hardy spaces$$L^p[0,1]$$ ${L}^{p}[0,1]$ ,$$H^p[0,1]$$ ${H}^{p}[0,1]$ . We say that$$1\le p < \infty $$ $1\le p<\infty $ is a Haar multiplier if$$D:X(Y)\rightarrow X(Y)$$ $D:X\left(Y\right)\to X\left(Y\right)$ , where$$D(h_I\otimes h_J) = d_{I,J} h_I\otimes h_J$$ $D({h}_{I}\otimes {h}_{J})={d}_{I,J}{h}_{I}\otimes {h}_{J}$ , and ask which more elementary operators factor through$$d_{I,J}\in \mathbb {R}$$ ${d}_{I,J}\in R$D . A decisive role is played by theCapon projection given by$$\mathcal {C}:\mathcal {V}(\delta ^2)\rightarrow \mathcal {V}(\delta ^2)$$ $C:V\left({\delta}^{2}\right)\to V\left({\delta}^{2}\right)$ if$$\mathcal {C} h_I\otimes h_J = h_I\otimes h_J$$ $C{h}_{I}\otimes {h}_{J}={h}_{I}\otimes {h}_{J}$ , and$$I\le J$$ $\leftI\right\le \leftJ\right$ if$$\mathcal {C} h_I\otimes h_J = 0$$ $C{h}_{I}\otimes {h}_{J}=0$ , as our main result highlights: Given any bounded Haar multiplier$$I > J$$ $\leftI\right>\leftJ\right$ , there exist$$D:X(Y)\rightarrow X(Y)$$ $D:X\left(Y\right)\to X\left(Y\right)$ such that$$\lambda ,\mu \in \mathbb {R}$$ $\lambda ,\mu \in R$ i.e., for all$$\begin{aligned} \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}\mathcal {C})\text { approximately 1projectionally factors through }D, \end{aligned}$$ $\begin{array}{c}\lambda C+\mu (\phantom{\rule{0ex}{0ex}}\text{Id}\phantom{\rule{0ex}{0ex}}C)\phantom{\rule{0ex}{0ex}}\text{approximately 1projectionally factors through}\phantom{\rule{0ex}{0ex}}D,\end{array}$ , there exist bounded operators$$\eta > 0$$ $\eta >0$A ,B so thatAB is the identity operator ,$${{\,\textrm{Id}\,}}$$ $\phantom{\rule{0ex}{0ex}}\text{Id}\phantom{\rule{0ex}{0ex}}$ and$$\Vert A\Vert \cdot \Vert B\Vert = 1$$ $\Vert A\Vert \xb7\Vert B\Vert =1$ . Additionally, if$$\Vert \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}\mathcal {C})  ADB\Vert < \eta $$ $\Vert \lambda C+\mu (\phantom{\rule{0ex}{0ex}}\text{Id}\phantom{\rule{0ex}{0ex}}C)ADB\Vert <\eta $ is unbounded on$$\mathcal {C}$$ $C$X (Y ), then and then$$\lambda = \mu $$ $\lambda =\mu $ either factors through$${{\,\textrm{Id}\,}}$$ $\phantom{\rule{0ex}{0ex}}\text{Id}\phantom{\rule{0ex}{0ex}}$D or .$${{\,\textrm{Id}\,}}D$$ $\phantom{\rule{0ex}{0ex}}\text{Id}\phantom{\rule{0ex}{0ex}}D$ 
A<sc>bstract</sc> We study the “three particle coupling”
, in 2$$ {\Gamma}_{11}^1\left(\xi \right) $$ ${\Gamma}_{11}^{1}\left(\xi \right)$d Ising Field Theory in a magnetic field, as the function of the scaling parameterξ :=h /(−m )^{15/8}, wherem ∼T _{c}−T andh ∼H are scaled deviation from the critical temperature and scaled external field, respectively. The “φ ^{3}coupling” is defined in terms of the residue of the 2 → 2 elastic scattering amplitude at its pole associated with the lightest particle itself. We limit attention to the HighTemperature domain, so that$$ {\Gamma}_{11}^1 $$ ${\Gamma}_{11}^{1}$m is negative. We suggest “standard analyticity”: , as the function of$$ {\left({\Gamma}_{11}^1\right)}^2 $$ ${\left({\Gamma}_{11}^{1}\right)}^{2}$u :=ξ ^{2}, is analytic in the whole complexu plane except for the branch cut from – ∞ to –u _{0}≈ – 0. 03585, the latter branching point –u _{0}being associated with the YangLee edge singularity. Under this assumption, the values of at any complex$$ {\Gamma}_{11}^1 $$ ${\Gamma}_{11}^{1}$u are expressed through the discontinuity across the branch cut. We suggest approximation for this discontinuity which accounts for singular expansion of near the YangLee branching point, as well as its known asymptotic at$$ {\Gamma}_{11}^1 $$ ${\Gamma}_{11}^{1}$u → +∞ . The resulting dispersion relation agrees well with known exact data, and with numerics obtained via Truncated Free Fermion Space Approach. This work is part of extended project of studying the Smatrix of the Ising Field Theory in a magnetic field. 
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$\mathsf {Quasi}\text {}\mathsf {NP} = \mathsf {NTIME}[n^{(\log n)^{O(1)}}]$ $\mathrm{Quasi}\mathrm{NP}=\mathrm{NTIME}\left[{n}^{{\left(\mathrm{log}n\right)}^{O\left(1\right)}}\right]$ , by showing that$\mathcal { C}$ $C$ admits nontrivial satisfiability and/or$\mathcal { C}$ $C$# SAT algorithms which beat exhaustive search by a minor amount. In this paper, we present a new strong lower bound consequence of having a nontrivial# SAT algorithm for a circuit class . Say that a symmetric Boolean function${\mathcal C}$ $C$f (x _{1},…,x _{n}) issparse if it outputs 1 onO (1) values of . We show that for every sparse${\sum }_{i} x_{i}$ ${\sum}_{i}{x}_{i}$f , and for all “typical” , faster$\mathcal { C}$ $C$# SAT algorithms for circuits imply lower bounds against the circuit class$\mathcal { C}$ $C$ , which may be$f \circ \mathcal { C}$ $f\circ C$stronger than itself. In particular:$\mathcal { C}$ $C$# SAT algorithms forn ^{k}size circuits running in 2^{n}/$\mathcal { C}$ $C$n ^{k}time (for allk ) implyN E X P does not have circuits of polynomial size.$(f \circ \mathcal { C})$ $(f\circ C)$# SAT algorithms for size$2^{n^{{\varepsilon }}}$ ${2}^{{n}^{\epsilon}}$ circuits running in$\mathcal { C}$ $C$ time (for some$2^{nn^{{\varepsilon }}}$ ${2}^{n{n}^{\epsilon}}$ε > 0) implyQ u a s i N P does not have circuits of polynomial size.$(f \circ \mathcal { C})$ $(f\circ C)$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], [MurrayWilliams STOC’18]. This is the first nontrivial lower bound against such a circuit class. 
The third Painlevé equation in its generic form, often referred to as PainlevéIII($D_6$), is given by $$ \frac{{\rm d}^2u}{{\rm d}x^2} =\frac{1}{u}\left(\frac{{\rm d}u}{{\rm d}x} \right)^2\frac{1}{x} \frac{{\rm d}u}{{\rm d}x} + \frac{\alpha u^2 + \beta}{x}+4u^3\frac{4}{u}, \qquad \alpha,\beta \in \mathbb C. $$ Starting from a generic initial solution $u_0(x)$ corresponding to parameters $\alpha$, $\beta$, denoted as the triple $(u_0(x),\alpha,\beta)$, we apply an explicit Bäcklund transformation to generate a family of solutions $(u_n(x),\alpha + 4n,\beta + 4n)$ indexed by $n \in \mathbb N$. We study the large $n$ behavior of the solutions $(u_n(x), \alpha + 4n, \beta + 4n)$ under the scaling $x = z/n$ in two different ways: (a) analyzing the convergence properties of series solutions to the equation, and (b) using a RiemannHilbert representation of the solution $u_n(z/n)$. Our main result is a proof that the limit of solutions $u_n(z/n)$ exists and is given by a solution of the degenerate PainlevéIII equation, known as PainlevéIII($D_8$), $$ \frac{{\rm d}^2U}{{\rm d}z^2} =\frac{1}{U}\left(\frac{{\rm d}U}{{\rm d}z}\right)^2\frac{1}{z} \frac{{\rm d}U}{{\rm d}z} + \frac{4U^2 + 4}{z}.$$ A notable application of our result is to rational solutions of PainlevéIII($D_6$), which are constructed using the seed solution $(1,4m,4m)$ where $m \in \mathbb C \setminus \big(\mathbb Z + \frac{1}{2}\big)$ and can be written as a particular ratio of Umemura polynomials. We identify the limiting solution in terms of both its initial condition at $z = 0$ when it is well defined, and by its monodromy data in the general case. Furthermore, as a consequence of our analysis, we deduce the asymptotic behavior of generic solutions of PainlevéIII, both $D_6$ and $D_8$ at $z = 0$. We also deduce the large $n$ behavior of the Umemura polynomials in a neighborhood of $z = 0$.

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$$L_{\beta ,\gamma } = {\text {div}}D^{d+1+\gamma n} \nabla $$ ${L}_{\beta ,\gamma}=\text{div}{D}^{d+1+\gamma n}\nabla $ with a uniformly rectifiable boundary$$\Omega \subset {\mathbb {R}}^n$$ $\Omega \subset {R}^{n}$ of dimension$$\Gamma $$ $\Gamma $ , the now usual distance to the boundary$$d < n1$$ $d<n1$ given by$$D = D_\beta $$ $D={D}_{\beta}$ for$$D_\beta (X)^{\beta } = \int _{\Gamma } Xy^{d\beta } d\sigma (y)$$ ${D}_{\beta}{\left(X\right)}^{\beta}={\int}_{\Gamma}{Xy}^{d\beta}d\sigma \left(y\right)$ , where$$X \in \Omega $$ $X\in \Omega $ and$$\beta >0$$ $\beta >0$ . In this paper we show that the Green function$$\gamma \in (1,1)$$ $\gamma \in (1,1)$G for , with pole at infinity, is well approximated by multiples of$$L_{\beta ,\gamma }$$ ${L}_{\beta ,\gamma}$ , in the sense that the function$$D^{1\gamma }$$ ${D}^{1\gamma}$ satisfies a Carleson measure estimate on$$\big  D\nabla \big (\ln \big ( \frac{G}{D^{1\gamma }} \big )\big )\big ^2$$ $D\nabla (ln(\frac{G}{{D}^{1\gamma}})){}^{2}$ . 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).$$\Omega $$ $\Omega $