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Abstract Let$$f$$ $f$be an analytic polynomial of degree at most$$K-1$$ $K-1$. A classical inequality of Bernstein compares the supremum norm of$$f$$ $f$over the unit circle to its supremum norm over the sampling set of the$$K$$ $K$-th roots of unity. Many extensions of this inequality exist, often understood under the umbrella of Marcinkiewicz–Zygmund-type inequalities for$$L^{p},1\le p\leq \infty $$ $L^p,1\le p\le \infty$norms. We study dimension-free extensions of these discretization inequalities in the high-dimension regime, where existing results construct sampling sets with cardinality growing with the total degree of the polynomial. In this work we show that dimension-free discretizations are possible with sampling sets whose cardinality is independent of$$\deg (f)$$ $deg\left(f\right)$and is instead governed by the maximumindividualdegree of$$f$$ $f$;i.e., the largest degree of$$f$$ $f$when viewed as a univariate polynomial in any coordinate. For example, we find that for$$n$$ $n$-variate analytic polynomials$$f$$ $f$of degree at most$$d$$ $d$and individual degree at most$$K-1$$ $K-1$,$$\|f\|_{L^{\infty }(\mathbf{D}^{n})}\leq C(X)^{d}\|f\|_{L^{\infty }(X^{n})}$$ ${\parallel f\parallel }_{L^{\infty }\left(D^n\right)}\le C{\left(X\right)}^d{\parallel f\parallel }_{L^{\infty }\left(X^n\right)}$for any fixed$$X$$ $X$in the unit disc$$\mathbf{D}$$ $D$with$$|X|=K$$ $\left|X\right|=K$. The dependence on$$d$$ $d$in the constant is tight for such small sampling sets, which arise naturally for example when studying polynomials of bounded degree coming from functions on products of cyclic groups. As an application we obtain a proof of the cyclic group Bohnenblust–Hille inequality with an explicit constant$$\mathcal{O}(\log K)^{2d}$$ $O{(logK)}^{2d}$. 

Recent efforts in Analysis of Boolean Functions aim to extend core results to new spaces, including to the slice binom([n],k), the hypergrid [K]ⁿ, and noncommutative spaces (matrix algebras). We present here a new way to relate functions on the hypergrid (or products of cyclic groups) to their harmonic extensions over the polytorus. We show the supremum of a function f over products of the cyclic group {exp(2π i k/K)}_{k = 1}^K controls the supremum of f over the entire polytorus ({z ∈ ℂ:|z| = 1}ⁿ), with multiplicative constant C depending on K and deg(f) only. This Remez-type inequality appears to be the first such estimate that is dimension-free (i.e., C does not depend on n). This dimension-free Remez-type inequality removes the main technical barrier to giving 𝒪(log n) sample complexity, polytime algorithms for learning low-degree polynomials on the hypergrid and low-degree observables on level-K qudit systems. In particular, our dimension-free Remez inequality implies new Bohnenblust-Hille-type estimates which are central to the learning algorithms and appear unobtainable via standard techniques. Thus we extend to new spaces a recent line of work [Eskenazis and Ivanisvili, 2022; Huang et al., 2022; Volberg and Zhang, 2023] that gave similarly efficient methods for learning low-degree polynomials on the hypercube and observables on qubits. An additional product of these efforts is a new class of distributions over which arbitrary quantum observables are well-approximated by their low-degree truncations - a phenomenon that greatly extends the reach of low-degree learning in quantum science [Huang et al., 2022]. 

During mitosis, cells round up and utilize the interphase adhesion sites within the fibrous extracellular matrix (ECM) as guidance cues to orient the mitotic spindles. Here, using suspended ECM-mimicking nanofiber networks, we explore mitotic outcomes and error distribution for various interphase cell shapes. Elongated cells attached to single fibers through two focal adhesion clusters (FACs) at their extremities result in perfect spherical mitotic cell bodies that undergo significant 3-dimensional (3D) displacement while being held by retraction fibers (RFs). Increasing the number of parallel fibers increases FACs and retraction fiber-driven stability, leading to reduced 3D cell body movement, metaphase plate rotations, increased interkinetochore distances, and significantly faster division times. Interestingly, interphase kite shapes on a crosshatch pattern of four fibers undergo mitosis resembling single-fiber outcomes due to rounded bodies being primarily held in position by RFs from two perpendicular suspended fibers. We develop a cortex–astral microtubule analytical model to capture the retraction fiber dependence of the metaphase plate rotations. We observe that reduced orientational stability, on single fibers, results in increased monopolar mitotic defects, while multipolar defects become dominant as the number of adhered fibers increases. We use a stochastic Monte Carlo simulation of centrosome, chromosome, and membrane interactions to explain the relationship between the observed propensity of monopolar and multipolar defects and the geometry of RFs. Overall, we establish that while bipolar mitosis is robust in fibrous environments, the nature of division errors in fibrous microenvironments is governed by interphase cell shapes and adhesion geometries. 

Ovarian cancer is routinely diagnosed long after the disease has metastasized through the fibrous submesothelium. Despite extensive research in the field linking ovarian cancer progression to increasingly poor prognosis, there are currently no validated cellular markers or hallmarks of ovarian cancer that can predict metastatic potential. To discern disease progression across a syngeneic mouse ovarian cancer progression model, here we fabricated extracellular matrix mimicking suspended fiber networks: cross-hatches of mismatch diameters for studying protrusion dynamics, aligned same diameter networks of varying interfiber spacing for studying migration, and aligned nanonets for measuring cell forces. We found that migration correlated with disease while a force-disease biphasic relationship exhibited F-actin stress fiber network dependence. However, unique to suspended fibers, coiling occurring at the tips of protrusions and not the length or breadth of protrusions displayed the strongest correlation with metastatic potential. To confirm that our findings were more broadly applicable beyond the mouse model, we repeated our studies in human ovarian cancer cell lines and found that the biophysical trends were consistent with our mouse model results. Altogether, we report complementary high throughput and high content biophysical metrics capable of identifying ovarian cancer metastatic potential on a timescale of hours.