Let
The classic Impagliazzo–Nisan–Wigderson (INW) pseudorandom generator (PRG) (STOC ‘94) for spacebounded computation uses a seed of length
 NSFPAR ID:
 10527806
 Publisher / Repository:
 Springer Science + Business Media
 Date Published:
 Journal Name:
 Algorithmica
 Volume:
 86
 Issue:
 10
 ISSN:
 01784617
 Format(s):
 Medium: X Size: p. 31533185
 Size(s):
 p. 31533185
 Sponsoring Org:
 National Science Foundation
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Abstract M (x ) denote the largest cardinality of a subset of on which the Euler totient function$$\{n \in \mathbb {N}: n \le x\}$$ $\{n\in N:n\le x\}$ is nondecreasing. We show that$$\varphi (n)$$ $\phi \left(n\right)$ for all$$M(x) = \left( 1+O\left( \frac{(\log \log x)^5}{\log x}\right) \right) \pi (x)$$ $M\left(x\right)=\left(1,+,O,\left(\frac{{(loglogx)}^{5}}{logx}\right)\right)\pi \left(x\right)$ , answering questions of Erdős and Pollack–Pomerance–Treviño. A similar result is also obtained for the sum of divisors function$$x \ge 10$$ $x\ge 10$ .$$\sigma (n)$$ $\sigma \left(n\right)$ 
A<sc>bstract</sc> In this paper we explore
pp →W ^{±}(ℓ ^{±}ν )γ to in the SMEFT expansion. Calculations to this order are necessary to properly capture SMEFT contributions that grow with energy, as the interference between energyenhanced SMEFT effects at$$ \mathcal{O}\left(1/{\Lambda}^4\right) $$ $O\left(1/{\Lambda}^{4}\right)$ and the Standard Model is suppressed. We find that there are several dimension eight operators that interfere with the Standard Model and lead to the same energy growth, ~$$ \mathcal{O}\left(1/{\Lambda}^2\right) $$ $O\left(1/{\Lambda}^{2}\right)$ , as dimension six squared. While energyenhanced SMEFT contributions are a main focus, our calculation includes the complete set of$$ \mathcal{O}\left({E}^4/{\Lambda}^4\right) $$ $O\left({E}^{4}/{\Lambda}^{4}\right)$ SMEFT effects consistent with U(3)^{5}flavor symmetry. Additionally, we include the decay of the$$ \mathcal{O}\left(1/{\Lambda}^4\right) $$ $O\left(1/{\Lambda}^{4}\right)$W ^{±}→ ℓ ^{±}ν , making the calculation actually . As such, we are able to study the impact of nonresonant SMEFT operators, such as$$ \overline{q}{q}^{\prime}\to {\ell}^{\pm}\nu \gamma $$ $\overline{q}{q}^{\prime}\to {\ell}^{\pm}\mathrm{\nu \gamma}$$$ \left({L}^{\dagger }{\overline{\sigma}}^{\mu }{\tau}^IL\right)\left({Q}^{\dagger }{\overline{\sigma}}^{\nu }{\tau}^IQ\right) $$ $\left({L}^{\u2020}{\overline{\sigma}}^{\mu}{\tau}^{I}L\right)\left({Q}^{\u2020}{\overline{\sigma}}^{\nu}{\tau}^{I}Q\right)$B _{μν}, which contribute to directly and not to$$ \overline{q}{q}^{\prime}\to {\ell}^{\pm}\nu \gamma $$ $\overline{q}{q}^{\prime}\to {\ell}^{\pm}\mathrm{\nu \gamma}$ . We show several distributions to illustrate the shape differences of the different contributions.$$ \overline{q}{q}^{\prime}\to {W}^{\pm}\gamma $$ $\overline{q}{q}^{\prime}\to {W}^{\pm}\gamma $ 
Abstract Let
X be ann element point set in thek dimensional unit cube where$$[0,1]^k$$ ${[0,1]}^{k}$ . According to an old result of Bollobás and Meir (Oper Res Lett 11:19–21, 1992) , there exists a cycle (tour)$$k \ge 2$$ $k\ge 2$ through the$$x_1, x_2, \ldots , x_n$$ ${x}_{1},{x}_{2},\dots ,{x}_{n}$n points, such that , where$$\left( \sum _{i=1}^n x_i  x_{i+1}^k \right) ^{1/k} \le c_k$$ ${\left({\sum}_{i=1}^{n},{{x}_{i}{x}_{i+1}}^{k}\right)}^{1/k}\le {c}_{k}$ is the Euclidean distance between$$xy$$ $xy$x andy , and is an absolute constant that depends only on$$c_k$$ ${c}_{k}$k , where . From the other direction, for every$$x_{n+1} \equiv x_1$$ ${x}_{n+1}\equiv {x}_{1}$ and$$k \ge 2$$ $k\ge 2$ , there exist$$n \ge 2$$ $n\ge 2$n points in , such that their shortest tour satisfies$$[0,1]^k$$ ${[0,1]}^{k}$ . For the plane, the best constant is$$\left( \sum _{i=1}^n x_i  x_{i+1}^k \right) ^{1/k} = 2^{1/k} \cdot \sqrt{k}$$ ${\left({\sum}_{i=1}^{n},{{x}_{i}{x}_{i+1}}^{k}\right)}^{1/k}={2}^{1/k}\xb7\sqrt{k}$ and this is the only exact value known. Bollobás and Meir showed that one can take$$c_2=2$$ ${c}_{2}=2$ for every$$c_k = 9 \left( \frac{2}{3} \right) ^{1/k} \cdot \sqrt{k}$$ ${c}_{k}=9{\left(\frac{2}{3}\right)}^{1/k}\xb7\sqrt{k}$ and conjectured that the best constant is$$k \ge 3$$ $k\ge 3$ , for every$$c_k = 2^{1/k} \cdot \sqrt{k}$$ ${c}_{k}={2}^{1/k}\xb7\sqrt{k}$ . Here we significantly improve the upper bound and show that one can take$$k \ge 2$$ $k\ge 2$ or$$c_k = 3 \sqrt{5} \left( \frac{2}{3} \right) ^{1/k} \cdot \sqrt{k}$$ ${c}_{k}=3\sqrt{5}{\left(\frac{2}{3}\right)}^{1/k}\xb7\sqrt{k}$ . Our bounds are constructive. We also show that$$c_k = 2.91 \sqrt{k} \ (1+o_k(1))$$ ${c}_{k}=2.91\sqrt{k}\phantom{\rule{0ex}{0ex}}(1+{o}_{k}\left(1\right))$ , which disproves the conjecture for$$c_3 \ge 2^{7/6}$$ ${c}_{3}\ge {2}^{7/6}$ . Connections to matching problems, power assignment problems, related problems, including algorithms, are discussed in this context. A slightly revised version of the Bollobás–Meir conjecture is proposed.$$k=3$$ $k=3$ 
Abstract Recent spectacular advances by AI programs in 3D structure predictions from protein sequences have revolutionized the field in terms of accuracy and speed. The resulting “folding frenzy” has already produced predicted protein structure databases for the entire human and other organisms’ proteomes. However, rapidly ascertaining a predicted structure’s reliability based on measured properties in solution should be considered. Shapesensitive hydrodynamic parameters such as the diffusion and sedimentation coefficients (
,$${D_{t(20,w)}^{0}}$$ ${D}_{t(20,w)}^{0}$ ) and the intrinsic viscosity ([$${s_{{\left( {{20},w} \right)}}^{{0}} }$$ ${s}_{\left(20,w\right)}^{0}$η ]) can provide a rapid assessment of the overall structure likeliness, and SAXS would yield the structurerelated pairwise distance distribution functionp (r ) vs.r . Using the extensively validated UltraScan SOlution MOdeler (USSOMO) suite, a database was implemented calculating from AlphaFold structures the corresponding ,$${D_{t(20,w)}^{0}}$$ ${D}_{t(20,w)}^{0}$ , [$${s_{{\left( {{20},w} \right)}}^{{0}} }$$ ${s}_{\left(20,w\right)}^{0}$η ],p (r ) vs.r , and other parameters. Circular dichroism spectra were computed using the SESCA program. Some of AlphaFold’s drawbacks were mitigated, such as generating whenever possible a protein’s mature form. Others, like the AlphaFold direct applicability to singlechain structures only, the absence of prosthetic groups, or flexibility issues, are discussed. Overall, this implementation of the USSOMOAF database should already aid in rapidly evaluating the consistency in solution of a relevant portion of AlphaFold predicted protein structures. 
A<sc>bstract</sc> We develop Standard Model Effective Field Theory (SMEFT) predictions of
σ ( →$$ \mathcal{GG} $$ $\mathrm{GG}$h ), Γ(h → ), Γ($$ \mathcal{GG} $$ $\mathrm{GG}$h → ) to incorporate full two loop Standard Model results at the amplitude level, in conjunction with dimension eight SMEFT corrections. We simultaneously report consistent Γ($$ \mathcal{AA} $$ $\mathrm{AA}$h → ) results including leading QCD corrections and dimension eight SMEFT corrections. This extends the predictions of the former processes Γ$$ \overline{\Psi}\Psi $$ $\overline{\Psi}\Psi $, σ to a full set of corrections at and$$ \mathcal{O}\left({\overline{v}}_T^2/{\varLambda}^2{\left(16{\pi}^2\right)}^2\right) $$ $O\left({\overline{v}}_{T}^{2}/{\Lambda}^{2}{\left(16{\pi}^{2}\right)}^{2}\right)$ , where$$ \mathcal{O}\left({\overline{v}}_T^4/{\Lambda}^4\right) $$ $O\left({\overline{v}}_{T}^{4}/{\Lambda}^{4}\right)$ is the electroweak scale vacuum expectation value and Λ is the cut off scale of the SMEFT. Throughout, cross consistency between the operator and loop expansions is maintained by the use of the geometric SMEFT formalism. For Γ($$ {\overline{v}}_T $$ ${\overline{v}}_{T}$h → ), we include results at$$ \overline{\Psi}\Psi $$ $\overline{\Psi}\Psi $ in the limit where subleading$$ \mathcal{O}\left({\overline{v}}_T^2/{\Lambda}^2\left(16{\pi}^2\right)\right) $$ $O\left({\overline{v}}_{T}^{2}/{\Lambda}^{2}\left(16{\pi}^{2}\right)\right)$m _{Ψ}→ 0 corrections are neglected. We clarify how gauge invariant SMEFT renormalization counterterms combine with the Standard Model counter terms in higher order SMEFT calculations when the Background Field Method is used. We also update the prediction of the total Higgs width in the SMEFT to consistently include some of these higher order perturbative effects.