Papers

Consider the heat equation: \begin{equation}\label{eq:00} \begin{cases} \displaystyle du(t,x)=\frac12\Delta u(t,x)\,dt+b(u(t,x))\,dt, &t>0,\ x\in D,\\[1mm] u(t,x)=0,&t\ge0,\ x\in\partial D, u(0,x)=u_0(x),&x\in D. \end{cases} \end{equation} It is well known that superlinear growth of the coefficient $b$ may lead to finite-time blow-up. In this paper, we show that, for every locally Lipschitz reaction coefficient $b:\R\to\R$, there always exists a locally Lipschitz noise coefficient $\sigma:\R\to\R$ such that the stochastic heat equation \begin{equation}\label{eq:01} \begin{cases} \displaystyle du(t,x)=\frac12\Delta u(t,x)\,dt+b(u(t,x))\,dt+\sigma(u(t,x))\,dB_t, &t>0,\ x\in D,\\[1mm] u(t,x)=0,&t\ge0,\ x\in\partial D, u(0,x)=u_0(x),&x\in D. \end{cases} \end{equation} has a unique global solution for every $u_0\in C_c(D)$. Here $(B_t)_{t\ge0}$ is a real-valued Brownian motion. In particular, if $b=0$, the equation has a global solution for every locally Lipschitz $\sigma$, without any growth restriction at infinity. The proof uses reflected scalar barriers, a Lyapunov estimate, truncation, and comparison techniques.

We derive a first-order representation of Black-Scholes implied variance in a continuous local martingale model. Total implied variance is the conditional expectation of the quadratic variation of the log price given its terminal value, up to a smaller-order term, for bounded standardized log-strikes. The framework incorporates small volatility-of-volatility, fast mean-reverting, and short-maturity asymptotics.

We give a local proof of Langlands's Second Main Lemma for local epsilon factors over nonarchimedean local fields. The lemma compares the induction expressions attached to compatible characters of distinct intermediate fields of prime degree in a bicyclic extension of prime-square degree. It is one of the main identities used in Langlands's construction of epsilon factors of local Weil representations. The proof builds on the work of Dwork, Lakkis, and Langlands. The main new point in the present paper is the wild dyadic case, i.e. biquadratic extensions over a nonarchimedean local field of residue characteristic two. This completes the proof of the Second Main Lemma, including the equal-characteristic case. Finally, together with the local proof of the First Main Lemma and Langlands's remaining reductions, we obtain a purely local construction of local epsilon factors envisioned by Langlands.

Let $Y_1,\ldots,Y_n$ be independent symmetric random variables with log-concave tails. We give a dimension-free characterization of the expected supremum of the canonical process $X_x=\sum_{i=1}^n x_iY_i$ without any $\Delta_2$ or regular-growth assumption on the coordinate tails. The characterization is governed by scale-dependent intrinsic costs determined directly by the convex tail potentials. It has an information-theoretic rate-distortion formulation as well as two equivalent majorizing-measure formulations on the original index set, one multiscale and one based on a single probability measure. More strongly, the comparison holds for every prescribed law of the index. In particular, the Gaussian majorizing-measure theorem of Talagrand and the Bernoulli theorem of Bednorz-Latała are recovered from the same formula, through quadratic and truncated quadratic costs, respectively. The new geometry is necessary: once the $\Delta_2$ growth condition is removed, the usual chaining functional based on increment moments can exceed the expected supremum by an unbounded factor. The proof is based on a uniform convex-order truncation principle for symmetric exponentials and a dyadic decomposition of the tail slopes that transfers simultaneously the random process and its intrinsic cost, with universal constants.

Shubin, Vakilian, and Wolff proved a Fourier uncertainty principle for sets of sufficiently small local density at the reciprocal scale and asked whether every density below one is admissible. We answer this question negatively in dimension one by constructing sequences of pairs of $1/2$-density thin sets and unit vectors whose total position and Fourier mass outside these sets tends to zero. This obstruction persists for every scaled reciprocal profile $\rho_\kappa(x)=\min\{1,\kappa/|x|\}$, $\kappa>0$. On the other hand, for $0<\kappa\le1$, the uncertainty estimate holds whenever the density $$ \varepsilon<\frac{1}{2(1+32\kappa)}. $$ Thus the critical density tends to $1/2$ as $\kappa\downarrow 0$. The obstruction uses odd Gaussian packets to transfer norm bounds from a free-group model. The positive estimate uses a Fourier-complementary anti-Wick operator and quadratic straightening of the reciprocal geometry.

Let $\{P_j\}$ be a sequence of projections onto the closed subspaces $\mathcal{M}_j$ of a Hilbert space $\mathscr{H}$. Consider an infinite sequence $P_{i_1}, P_{i_2}, \ldots$ with each $P_{i_n} \in \{P_1, P_2, \ldots\}$, possibly repeating in some order or randomly. The question is: Under what conditions does the sequence $\{P_{i_n} \cdots P_{i_2} P_{i_1} x\}_{n=1}^{\infty}$ converge strongly or weakly to $Px$ for every $x \in \mathscr{H}$, where $P$ is the projection onto the intersection $\mathcal{M} = \bigcap_{i=1}^{\infty} \mathcal{M}_i$? In this paper, we present some results concerning random products of countably infinitely many projections $\{P_j\}_{j=1}^{\infty}$ that incorporates the notion of an infinite-periodic function. More precisely, we introduce a new class of functions $\sigma \colon \mathbb{N} \to \mathbb{N}$, called infinite-periodic functions, and rigorously show that the sequence $\{T_n x\}$ defined by \[ T_1 := P_{\sigma(1)}\quad \mbox{and} \quad T_n := P_{\sigma(n)} T_{n-1} \quad \text{for all } n \geq 2, \] for $x \in \mathscr{H}$, converges weakly to $Px$, where $P$ is the projection onto $\bigcap_{j=1}^{\infty} \mathcal{R}(P_j)$. We also provide some technical examples to illustrate our results.

We review the definitions of the Rumin complex and analytic contact torsion for contact manifolds. We then compute the contact torsion for symmetric contact manifolds. To this end, we use a representation-theoretic description of the relevant operators and spaces of differential forms. Our computation of the contact torsion for symmetric contact manifolds relies on a recent result by Rumin that reduces the problem to harmonic forms. Following a construction by Boothby and Wang, the contact manifolds under consideration are total spaces of $S^1$-principal bundles over Kähler manifolds. This allows us to interpret the harmonic forms as Dolbeault cohomology with coefficients in a certain holomorphic line bundle of the base Kähler manifold, thereby simplifying the computation by allowing us to use results from complex geometry. Furthermore, we generalise Rumin's result to the equivariant case and, using the previous considerations, determine the equivariant contact torsion for the case of isolated fixed points.

I consider vortex solitons in two-dimensional star networks created by several one-dimensional line waveguide arrays (or rays) brought in close proximity. First waveguide in each array belongs to the ring of certain radius and its modal field overlaps with modal fields of waveguides from other rays laying on this ring, as well as with modal fields of waveguides in this line array. This creates a system with discrete rotational symmetry that is capable of supporting various localized linear vortex states due to local coupling between waveguides on the central ring, where line arrays approach each other. Localized vortex states emerge from the continuum of the states extended along the rays of the structure upon variation of the radius of the central ring. I show how different vortex modes with progressively increasing topological charges emerge when the number of rays in structure and, therefore, its discrete rotational symmetry increases. Each such linear localized mode gives rise to a family of thresholdless vortex solitons in nonlinear optical medium, whose topological charges are limited by the discrete rotational symmetry of the system. Vortex solitons in star networks exhibit qualitatively different behavior with increasing power depending on whether the propagation constant of linear vortex that gives rise to a family of solitons is located above or below the band of extended states. Stability properties of vortex solitons nontrivially depend on their topological charge and discrete rotational symmetry of the system, showing very different picture from usual ring-like waveguide arrays.

A pseudo-Euclidean Malcev algebra is a Malcev algebra equipped with a non-degenerate symmetric bilinear form. In this paper, we introduce a curvature operator for pseudo-Euclidean Malcev algebras, generalizing the notion of curvature for pseudo-Euclidean Lie algebras. We define flat pseudo-Euclidean Malcev algebras and show that every flat Euclidean solvable Malcev algebra which is also a Lie algebra remains flat in the classical Lie algebra curvature sense. Furthermore, we develop the flat double extension construction for flat pseudo-Euclidean Malcev algebras and prove that every flat Lorentzian Malcev algebra with a degenerate center can be obtained via the flat double extension of a flat Euclidean Malcev algebra. Moreover, we demonstrate that all flat Lorentzian nilpotent Malcev algebras arise from the flat double extension of a Euclidean abelian Lie algebras. Finally, we establish that any flat Lorentzian nilpotent Malcev algebra is necessarily a Lie algebra and is flat in the classical Lie algebra curvature sense.

This is the second article of a sequence of research on the local rigidity of constant mean curvature (CMC) hypersurfaces in space forms. In the previous one, we studied the local rigidity of CMC hypersurfaces whose the number of the distinct principal curvatures satisfies $g\leq 3$. In this paper, we study the local rigidity of CMC hypersurfaces with $g\geq 4$. When $g>4$, we prove that if the $k$-order mean curvatures $H_k$, $(k=2,\cdots, g-1)$ are constant and there exist enough multiple principal curvatures, then the CMC hypersurface is an isoparametric hypersurface. When $g=4$, if $H_2$ and $H_3$ are constant, then the CMC hypersurface is an isoparametric hypersurface.

This work studies subspace embeddings obtained by two normalized real Walsh transforms, two independent sign diagonals, and uniform coordinate sampling without replacement. The main result shows that the prescribed sample size $k=\min\{n,\lceil Cr/\varepsilon^2\rceil\}$, for a universal constant $C$, suffices to preserve all squared norms on each fixed $r$-dimensional subspace within $1\pm\varepsilon$ with probability at least $0.99$. The result holds for every ambient Walsh dimension and all ranks, and answers Problem TR-01 in the Open Problems in Numerical Linear Algebra repository. The proof controls joint entry cumulants of the transformed projection through connected graph contractions and Walsh character identities. These estimates then bound the expected trace of even powers of a product of centered projections. A two-projection decomposition converts this estimate into control of both spectral edges. Bernoulli sampling at arbitrary densities and a deterministic upper bound near full sampling yield the prescribed number of coordinates.

Let $(\mathscr{E}, \langle \cdot, \cdot\rangle)$ be a Hilbert $C^*$-module over a $C^*$-algebra $\mathfrak{A}$. The space of adjointable operators on $\mathscr{E}$ is denoted by $\mathcal{L}\left(\mathscr{E}\right)$. The sets of all states and pure states on $\mathfrak{A}$ are denoted by $\mathcal{S}\left(\mathfrak{A}\right)$ and $\mathcal{P}\left( \mathfrak{A}\right)$, respectively. For $\tau\in\mathcal{S}\left( \mathfrak{A}\right)$, let us define $\mathcal{N}^{\mathscr{E}}_{\tau}:=\left\lbrace x\in\mathscr{E}:\tau\left( \left\langle x,x\right\rangle\right)=0 \right\rbrace$. The Hilbert completion of ${\mathscr{E}}/{\mathcal{N}^{\mathscr{E}}_{\tau}}$ is denoted by $\mathscr{E}_{\tau}$. For $T\in\mathcal{L}(\mathscr{E})$, the operator $T_{\mathscr{E}_\tau}\in\mathbb{B}\left( \mathscr{E}_\tau\right)$, is defined by $T_{\mathscr{E}_\tau}\left(x+\mathcal{N}^{\mathscr{E}}_{\tau}\right)=Tx+\mathcal{N}^{\mathscr{E}}_{\tau}$. In this paper, we show that $\mathscr{E}_{\tau}={\mathscr{E}}/{\mathcal{N}^{\mathscr{E}}_{\tau}}$ when $\mathfrak{A}$ either is a $C^*$-algebra of compact operators or is commutative. We introduce a quantity in the context of Hilbert $C^*$-modules, denoted by $\pi^{\mathscr{E}}_k(\cdot)$ for $k\geq1$. We prove that $\pi^{\mathscr{E}}_k(T)\leq\sup_{\tau\in\mathcal{P}\left( \mathfrak{A}\right)}\left\|T_{\mathscr{E}_\tau}\right\|_{\left(k\right)}\leq{\pi}^{\mathscr{E}^{\sharp}}_k(T_{\mathscr{E}^{\sharp}})$ for every $T\in\mathcal{L}\left(\mathscr{E}\right)$, where the space $\mathscr{E}^{\sharp}$ is constructed as the extension of $\mathscr{E}$ by the embedding of $\mathfrak{A}$ into its enveloping von Neumann algebra $\mathfrak{A}^{**}$.

It is well known that solutions to the compressible Euler equations can develop singularities in finite time. In this paper, we carry out a detailed analysis on behaviors of solutions up to the time of the first singularity for the one-dimensional compressible Euler equations with general smooth initial data. Our main results consist of three parts. First, for an open dense set of $C^3$ initial data, we show that the solution of Euler equations is twice continuously differentiable except at most finitely many points when the first singularity happens, using Thom's Transversality Theorem. Second, for any initial data in the open dense set of $C^3$ functions given in the first result, we provide the precise asymptotic description of the solution in a semi-neighborhood in the $(x,t)$-plane of each singular point at the time of the first singularity, and verify that the solution has a cusp-type singularity with Hölder exponent $1/3$ at each singular point. The proofs of the first two results are based on the representation of the solution in terms of a semilinear system. Third, for smooth initial data with small BV norm, we construct two Finsler type optimal transport metrics, then under these metrics show that the solution depends Lipschitz continuously on the initial data up to the time of the first singularity, with uniformly bounded Lipschitz constants. In particular, the $C^{1/3}$ generic singularity is stable in this sense. On the other hand, since our first two results hold for an open and dense set of initial data, any Hölder continuous cusp singularity with exponent other than $1/3$ is unstable under initial perturbations.

In this note, we study volume comparison for 3-manifolds with 2-Ricci curvature lower bound. We establish monotonicity and rigidity for geodesic sphere area and ball volume, and derive the pointed Gromov-Hausdorff precompactness under a uniform lower bound on the conjugate radius for 3-manifolds with 2-Ricci curvature lower bound. We also discuss the almost preservation of 2-Ricci curvature lower bound and the local non-collapsing propagation along Ricci flow. Finally, we point out a topological rigidity result on nonnegative scalar curvature.

We show that there exist two separable real Banach spaces $X$ and $Y$ such that, for every $\varepsilon>0$, there is a standard exact $\varepsilon$-isometry $f_\varepsilon: X\to Y$ whose distortion of every distance is between $0$ and $\varepsilon$, whereas no isometric embedding of $X$ into $Y$ exists, even without a linearity assumption. This gives a negative answer to \cite[Problem 1 and Problem 2]{cheng4} of Cheng and Zhou. The construction combines a snowflaked Lipschitz-free space with the Schur-property theorem of Kalton and the isometric linearization theorem of Godefroy and Kalton.

Let $M$ be a $(0,\omega_1+1)$-iterable mouse with no largest cardinal. Let $\gamma\leq\kappa$ be uncountable cardinals of $M$, with $\kappa$ regular in $M$. Then $M$ satisfies $\diamondsuit_{\kappa\gamma}^+$, and if $\kappa$ is non-ineffable in $M$ then $M$ satisfies $\diamondsuit_{\kappa\kappa}^+$. Suppose that either $\omega_1^M=\omega_1$ or $M$ satisfies "I am $(0,\omega_1+1)$-iterable". Let $\kappa\geq\omega_1^M$ be a cardinal of $M$. Then $M$ satisfies $\diamondsuit^*_{\kappa\omega_1^M}$.

In this paper, we investigate the analytic properties of the Hardy-type operator $-\Delta + \mu H_0$ on the $d$-dimensional integer lattice graph $\mathbb{Z}^d$, where $\mu \geq -1$ and $H_0$ is a critical Hardy potential at infinity–arising naturally from discrete Hardy inequalities. We derive sharp fixed-pole Green kernel estimates on $\mathbb{Z}^d$ and obtain a necessary and sufficient weighted summability condition characterizing the solvability of the nonnegative Poisson problem associated with this operator. Building upon this classification, we provide a complete characterization of the nonexistence of positive solutions to the semilinear elliptic inequality $$ -\Delta u + \mu H_0 u \geq W u^p \quad \text{in } \mathbb{Z}^d, $$ where $p > 0$, $d \geq 3$, $\mu \geq -1$, and $H_0, W \in C(\mathbb{Z}^d)$ are strictly positive potentials satisfying the asymptotic behaviors $H_0(x) \sim |x|^{-2}$ and $W(x) \sim |x|^{\theta}$ as $|x| \to \infty$, with $\theta \in \mathbb{R}$.

For an odd prime $p$, let $A_j$ be the $\omega^j$-eigenspace of the $p$-primary class group of $\mathbb Q(\zeta_p)$. Fix an even integer $d\ge4$, put $N=(p-1)/d$, and let $U_d=(\mathbb Z/d\mathbb Z)^\times$. For a relative density-one set of primes $p\equiv d+1\pmod{2d}$, we prove that the odd block $\bigoplus_{a\in U_d}A_{aN}$ has order at most $p^{\varphi(d)/2-1}$, and reflection shows that the even block $\bigoplus_{a\in U_d}A_{p-aN}$ has $p$-rank at most $\varphi(d)/2-1$. For each $d\in\{4,6\}$, both blocks vanish for a relative density-one set of primes $p\equiv d+1\pmod{2d}$; in particular, the even components vanish in accordance with Vandiver's conjecture. For each $d\in\{8,10,12\}$, the odd block has order at most $p$ for a relative density-one set of primes in the same progression, so every summand $A_{aN}$, $a\in U_d$, is cyclic, in accordance with Iwasawa's cyclicity conjecture. The proof uses an atomless limiting law for products of Dirichlet $L$-values, integrality of generalized Bernoulli norms, the relative class-number formula and reflection. A separate exact computation proves $A_{34}=0$ for every odd prime $p$; a single-file PARI/GP program reproduces the calculation.

This work initiates a systematic development of conjugation theory within the framework of Hilbert $C^*$-modules, extending classical complex symmetric operator theory beyond the Hilbert space setting. We first explore the foundational structure of $*$-conjugate-automorphisms $\sharp$ on $C^*$-algebras and characterize their existence through isomorphisms with opposite algebras. We show that the set $\mathbb{L}_c(\mathscr{E})$ of all adjointable conjugate-linear operators on a Hilbert $C^*$-module $\mathscr{E}$ possesses the structure of a Hilbert $\mathbb{L}(\mathscr{E})$-module. We also introduce the key notion of $\sharp$-conjugation on a Hilbert $C^*$-module and fully characterize such modules in terms of real forms and real $C^*$-algebras. Moreover, we investigate the geometric properties of $\sharp$-conjugate duals. A central achievement is the development of polar decomposition theory for semiregular linear (conjugate-linear) operators acting on a Hilbert $C^*$-module equipped with a $\sharp$-conjugation $C$, thereby extending the celebrated Garcia–Putinar theorem [Trans. Amer. Math. Soc. 359 (2007), 3913–3931]. As a consequence, we show that any $C$-symmetric unitary operator factors as a product of two $\sharp$-conjugations. A comparative analysis of alternative definitions shows that requiring the naive adjoint condition $\langle Cx,y\rangle = \langle x,Cy\rangle^*$ would force the underlying $C^*$-algebra to be essentially commutative, thus justifying the adopted approach. We supply various examples and counterexamples that extensively illustrate our results.

The Interconnectedness Coefficient (IC) is a bounded semi-local graph-theoretic node measure designed to identify connector vertices between cohesive network regions. Such connector vertices, also referred to as bridging nodes, may mediate between locally cohesive regions even when they are neither hubs nor themselves highly clustered. The IC preferentially assigns high values to weakly clustered focal vertices whose adjacent vertices remain strongly clustered after exclusion of the focal connection. Candidate vertices are required to have degree at least two. The construction is partition-free, uses information within radius two, and requires no predefined community or module partition. The range and extremal properties of the score are derived analytically. Exact graph families isolate its maximal response to fully cohesive branches, its controlled response to a single cohesion defect, its invariance under a cohesion-free hub extension, and a sharp fragmentation threshold. A separate application to a Human Interactome Map reveals pronounced degree-dependent stabilization of IC values near the network's mean clustering level. This behavior follows directly from the multiplicative definition. If focal clustering tends to zero while mean leave-one-out cohesion in the neighborhood stabilizes, the IC converges to that neighborhood-cohesion level. Among the highly ranked IC vertices are proteins with established interface, scaffold, and adaptor roles in molecular complexes. The IC is therefore positioned as a semi-local connector measure for cohesive network regions.

Consider Gaussian denoising with a fixed prior and varying noise variance. When the prior has a density, maximum a posteriori estimation uses its negative logarithm as a fixed regularizer; only the weight of the quadratic data term changes with the noise variance. We ask whether a fixed regularizer can also recover the posterior mean. Although a suitable regularizer exists at each variance, we prove that no single regularizer works at two distinct positive variances for any compactly supported prior other than a point mass. This holds even for nonconvex and nonsmooth regularizers. A residual computed from two denoiser evaluations gives positive lower bounds on the uniform approximation error and excess mean squared error of estimators using a common regularizer. Numerical examples illustrate these bounds and the cost of reusing a regularizer across noise levels.

This work studies the stability and convergence of augmented primal-dual dynamics when constraint values are estimated from samples. Unbiased constraint observations can produce a biased augmented multiplier signal, shifting the equilibria of the mean dynamics. For componentwise inequalities, we give a necessary and sufficient condition for preserving the Karush-Kuhn-Tucker (KKT) equilibria and construct a convex example with a locally exponentially stable equilibrium that violates complementarity. To address this bias, constraint values are estimated recursively before forming the augmented multiplier signal. For smooth convex conic problems, a joint energy analysis establishes boundedness of the primal, dual, and estimation states, vanishing estimation error, and almost sure convergence of the primal-dual iterates to a single KKT point under global regularity and bounded conditional second moments. The result allows nonunique solutions and multipliers while keeping the number of samples per iteration fixed. Numerical studies illustrate the predicted equilibrium bias and examine convergence with nonunique KKT points and nonlinear constraints.

We prove the all-genus conifold gap conjecture for the Gromov–Witten (GW) theories of the quintic threefold, other smooth Fermat Calabi–Yau threefold hypersurfaces, and local $\mathbb P^2$, as well as for the FJRW theories associated with these hypersurfaces. Our proof uses Mixed Spin P-field (MSP) theory and analogous master-space constructions. The central object is the cone-vertex theory. We express this theory as a translation of the point CohFT via a formal Laplace transform and prove a universal conifold gap theorem for the resulting translated theories. The master-space construction then transfers the gap from the cone-vertex theory to the GW and FJRW theories.

We show that for every finite group $G$ and every integer $n$, the Nil group $\mathrm{NK}_n(\mathbb{Z}[G])$ of the integral group ring of $G$ has finite exponent, and give a bound depending on n and the order $|G|$. This bound is explicit when every prime divisor of $|G|$ is at least $3+n/2$. More generally, our finite exponent result applies to $\mathrm{NK}_n(R[G])$ whenever $R$ is a regular, torsion-free, Noetherian commutative ring such that $R/p$ is regular for every prime $p$ that divides $|G|$. In the Appendix, M. Morrow provides an alternative approach and also shows that the Nil group $\mathrm{NK}_n(X)$ of an excellent Noetherian scheme $X$, with $X[1/p]$ regular, is annihilated by a finite power of $p$, provided $X$ admits a suitable resolution of singularities. By extending his argument to certain noncommutative rings, we also show that, for $R$ as above and $\alpha$ any automorphism of $G$, the Farrell Nil groups $\mathrm{NK}_n(R[G],\alpha)$ have finite exponent. As a consequence, for every virtually cyclic group $\Gamma$, the group $\mathrm{K}_n(\mathbb{Z}[\Gamma])$ is a direct sum of a finitely generated abelian group and an infinite countable direct sum of copies of a finite abelian group.

Building on a construction of Weifeng Yang, we present a simplified counterexample to Rockafellar's sum conjecture on $c_0(\N_0\times\N)$. The central observation is geometric: the domain of a maximally monotone operator can have a nonconvex norm closure. In our example, this gives a short proof that the operator is not of type (FPV), which in turn implies the failure of the sum conjecture. We define an explicit operator using geometric series and provide a detailed proof of its maximal monotonicity. The example was developed with assistance from GPT-6 Astra. The purpose is to simplify and explain the construction, not to claim an independent counterexample mechanism.

Let $G$ be a finite group with $t$ conjugacy classes, and let $S_N$ be the symmetric group. Let $Z_t(N)$ be the number of zeros in the character table of the wreath product $G\wr S_N$. We prove \begin{equation*} Z_t(N)\ge \frac{2p_t(N)^{2}}{\log \frac{N}{t}}\left(1+O\left(\frac{\log \log \frac{N}{t}}{\log \frac{N}{t}}\right)\right), \end{equation*} where $p_t(N)$ is the number of $t$-multipartitions of $N$.

Let $b, p\in {\mathbb Z}$ with $b\ge 2$ and $p\ge 3$ a prime. If $b^{p-1}\equiv 1 \pmod{p^2}$, then $p$ is called a {\em generalized Wieferich prime base $b$}, or more succinctly, a {\em base-$b$ Wieferich prime}. When $b=2$, $p$ is also known simply as a Wieferich prime. We say that a monic polynomial $f(x)\in {\mathbb Z}[x]$ is monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots,\theta^{\deg(f)-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. Recently, necessary and sufficient conditions for the monogenicity of the trinomials $x^{2n}+bx^n+b$ were given that included base-$b$ Wieferich prime congruences, and also congruences involving a certain Lucas sequence. In this article, we prove a similar result for a different class of trinomials that does not rely on any congruence condition involving a Lucas sequence. Furthermore, we extend this result to a related class of $N$-nomials, for any $N\ge 4$.

We prove that for additive complements $A$ and $B$, if $$ \limsup_{x\to+\infty}\frac{A(x)B(x)}{x}<\frac43, $$ then $$ A(x)B(x)-x\rightarrow+\infty\qquad(x\to+\infty). $$ This improves the earlier upper bound $3-\sqrt{3}$ due to Fang and Chen.

We investigate the nonparametric estimation of the diffusion matrix in stochastic differential equations featuring multidimensional, strong mixing covariate processes. We propose a flexible statistical framework based on general function classes that does not require a linear basis representation, rendering our results directly applicable to deep neural network estimators. Our approach employs a two-step estimation procedure: constructing a preliminary nonparametric quasi-likelihood estimator and subsequently regularizing it via a $\beta$-Hölder class approximation. We establish general risk comparison theorems between empirical and generalization risks for arbitrary estimators without relying on a specific probabilistic structure of the underlying process. In diffusion matrix learning based on $n + 1$ observations over the time interval $[0,T]$, the derived upper bounds capture the intrinsic interplay between the $T$-rate, associated with the mixing behavior of the covariate process, and the intrinsic $n$-rate governing the volatility estimation.

We prove that, for any positive function $\Psi(t)\to\infty$, almost all positive integers $n$ satisfying $4\nmid n$ and $n\not\equiv7\pmod8$ can be written as $n=x^2+y^2+z^2$, where $x,y,z\in\mathbb{Z}$ and $1\leqslant z<\sqrt{\log n}\,\Psi(n)$. This improves Wooley's bound, and we show that the result is best possible. Our approach combines Hooley's weighted representation function for sums of two squares with Green's $W$-trick. We also give a necessary and sufficient condition, in terms of covering systems, for almost all odd integers to be represented as the sum of a squarefree number and a power of two, answering a question of Granville and Soundararajan.