Papers

Let $m\geq 1$ be a fixed integer, $a$ an integer satisfying $(a,m)=1$, and $z\geq 1$ a real parameter. Denote by $\omega_{z}(n;m,a)$ the number of distinct prime divisors $p$ of $n$ satisfying $p\equiv a\, (m)$ and $p\leq z$. We study an asymptotic behaviour of $\sum_{n\leq x}\left(\omega_{z}(n;m,a)-\frac{1}{\varphi(m)}\log\log z\right)^{k}$ as $x\to\infty$ for a wide range of positive integer $k\geq 2$, where $\varphi(\cdot)$ is the Euler function. Following a method of Granville and Soundararajan we lead an asymptotic formula for the above. Also, we investigate $\sum_{n\leq x}\left(\omega(n;m,a)-\frac{1}{\varphi(m)}\log\log x\right)^{k}$, where $\omega(n;m,a)$ denotes the number of distinct prime divisors $p$ of $n$ such that $p\equiv a\, (m)$.

We study regular $\mathbb{N}$-graded vertex operator algebras $V=\bigoplus_{n\geq 0}V_n$ whose weight-zero algebra $V_0$ is a nontrivial finite-dimensional local Gorenstein algebra, asking which structural features of strongly rational vertex operator algebras persist without the CFT-type condition $V_0=\mathbb{C}{\bf 1}$. For semisimple Lie subalgebras of the left Leibniz algebra $V_1$, assuming $\ker\left(\left.L(-1)\right|_{V_0}\right)=\mathbb{C}\mathbf{1}$, the product $u_1v$ induces an invariant symmetric bilinear form. Under $C_2$-cofiniteness and a nondegeneracy condition, each simple component with nonzero form generates an affine vertex operator algebra at positive integral level and acts integrably on $V$. When $V_1$ is solvable, the Frobenius structure of $V_0$ yields a distinguished nondegenerate subspace $M\subset V_1$. Under a quasi-primary condition, $M$ is abelian and generates a Heisenberg vertex operator algebra. With additional semisimplicity, full-rank integrality, and cocycle compatibility assumptions, $V$ contains a conformally embedded lattice vertex operator algebra $V_K$, where $K$ is positive-definite and even, with rank $\dim_{\mathbb{C}}M$ and minimum norm at least $4$. Conformally shifted lattice vertex operator algebras provide explicit models illustrating the distinction among weight-one Lie, mode-generated Lie, and lattice structures. They also show that regularity alone does not ensure semisimplicity of arbitrary Heisenberg zero-mode actions, so the additional lattice-theorem hypotheses represent genuine structural obstructions.

At an $O(2)$-equivariant Hopf bifurcation, the real parts $\zeta_R$ and $\xi_R$ of the two cubic normal-form coefficients govern selection between traveling and standing waves. In a fourth-order regularized $p$-system, $\xi_R$ was observed to vanish and the cubic nonlinearity to shift only frequencies. We show that both observations are structural: variational nonlinearities enforce them even when the linear part is dissipative and non-Hamiltonian. We consider a class of two-component PDEs on the circle whose quadratic and cubic nonlinearities are generated by reflection-invariant local Hamiltonians of the fields and finitely many of their spatial derivatives, through a common constant-coefficient Poisson operator. Then $\xi_R=0$ for every admissible linear part, while $\zeta_R$ depends only on the quadratic nonlinearity. For variational nonlinearities the direct cubic terms are purely imaginary. The mixed cascade term loses its real part because its second harmonic has zero temporal frequency and the resolvent keeps its parity, while the self-cascade at the doubled critical frequency keeps a real part. When $\zeta_R\ne0$, the bifurcating traveling waves are saddles, while the standing waves are supercritical and orbitally asymptotically stable for $\zeta_R<0$ and subcritical and unstable for $\zeta_R>0$. When the nonlinearities are the Poisson operator applied to polynomials in the fields alone, and its symbol does not vanish at the critical and second-harmonic wavenumbers, cancellation for every admissible linear part conversely characterizes variationality. In a dissipatively regularized Boussinesq family, as dissipation tends to zero, $\zeta_R$ vanishes linearly off resonance and diverges inversely at the 2:1 resonance. The Hamiltonian limit of the cubic coefficient is therefore singular.

Denote by $\omega(n)$ the number of distinct prime divisors of the natural number $n$. In 2007, Granville and Soundararajan gave a quite new method to compute the higher moments $\sum_{n\leq x}(\omega(n)-\log\log x)^{k}$, for a wide range of integers $k\geq 2$. In this notes, we shall apply the method for $\omega_{z}(n)$ which denotes the number of distinct prime divisors $\leq z$ of $n$. Especially, for odd integers $k\geq 3$, we lead asymptotic formulas for $\sum_{n\leq x}(\omega_{z}(n)-\log\log z)^{k}$, under certain restrictions on $k$, $z$, and $x$. Also, we refer to a framework of the approach.

For all $0< s \le 1$, we provide a new deterministic construction of Cantor sets whose Fourier dimension and Hausdorff dimension are both equal to $s$. The construction is based on a straightforward Cantor-Moran construction with contraction ratios given by reciprocals of integers. The key tool to obtain the fast Fourier decay is due to the Weil bound in analytic number theory. Furthermore, we show that the natural equal-weighted Cantor-Moran measure is the desired measure admitting the near optimal Fourier decay and the measure admits an exponential orthonormal basis $\{e^{2\pi i \lambda x}: \lambda\in \Lambda\}$ for its $L^2$ space. This gives the first examples of singular Salem spectral measures in ${\mathbb R}^1$.

We construct a family of finite local rings whose annihilating-ideal graphs are naturally described by orthogonality of subspaces of $\mathbb{F}_2^n$. For $n=4$ we determine the clique and chromatic numbers exactly and obtain \[ \omega(\AG(R_4))=5<6=\chi(\AG(R_4)). \] Thus $\AG(R_4)$ is not weakly perfect, and the Behboodi–Rakeei conjecture fails for non-reduced commutative rings.

Ozaki proved that every finite $p$-group occurs as the Galois group of a maximal unramified $p$-extension of a number field. Hajir, Maire and Ramakrishna made this theorem effective, obtaining a base-field degree of order $|G|$. In this article, we reduce that degree by taking the Frattini structure of $G$ into account. More precisely, for an odd prime $p$ and a finite $p$-group $G$ of order $p^n$, we prove \[ \tau_p(G) \leq p^{\ell_\Phi(G)+ \left\lceil\log_p\left(\binom{n+2}{2}+1\right)\right\rceil}. \] Here $\tau_p(G)$ is the least degree of a number field realizing $G$ as its $p$-class tower group, and $\ell_\Phi(G)$ is the iterated Frattini length of $G$. Thus, for groups of bounded Frattini length, the order-scale bound $p^n$ is replaced by the quadratic bound $O_p(n^2)$. For $E_m=(\Z/p\Z)^m$, we further prove the sharp estimate $\tau_p(E_m)\asymp_p m^2$.

We give a negative answer to Problem 17.39 of the Kourovka Notebook. For every odd prime power $q$ and every $m\geq1$, we construct a finite metabelian group $G_{m,q}$ with $Z_\infty(G_{m,q})=1$ and $|\Phi(G_{m,q})|=q$ for which the least number of system normalizers with trivial intersection is $m+1$. Thus the number is unbounded even when the order of the Frattini subgroup is fixed.

This paper investigates the $c$-differential uniformity of power functions over finite fields, an important class of cryptographic functions with favorable differential properties. Specifically, for finite fields $\mathbb{F}_q$ satisfying $q-1=en$ with $e\ge 3$ and $e\mid n$, we prove that there exists $c\in\mathbb{F}_q\setminus\{0,1,\epsilon,\dots,\epsilon^{e-1}\}$, where $\epsilon$ is an $e$-th primitive root of unity in $\mathbb{F}_q^*$, the constructed power functions $f(x)=x^{ln+1}$ with $1\le l\le e-1$ and $\gcd(l,e)=1$ satisfy the upper bound $\Delta(f,c)\le e$, provided that certain cyclotomic conditions hold. We show that our conditions are mild; namely, such power functions can be constructed over infinitely many extension fields $\mathbb{F}_q$ of $\mathbb{F}_p$ for any given $e\ge 3$ and prime $p$ with $p\nmid e$. Furthermore, based on the Weil bound for multiplicative character sums, we prove that the obtained upper bound is tight for sufficiently large $q$, demonstrating the optimality of our results. We also analyze the special case $c=-1$ and derive simplified explicit conditions. In particular, we explicitly characterize the admissible parameters for the case $e=3$ and present concrete function examples for practical validation.

Motivated by the work of M. L. Lapidus and M. van Frankenhuisjen on fractal explicit formulas for generalized fractal strings, we develop in this paper a distributional fractal Taylor's formula with error term and an exact distributional fractal Taylor series representation for generalized fractal strings. The explicit formulas that the aforementioned authors developed provide a way of expressing a generalized fractal string $\eta$ in terms of the underlying fractal complex dimensions, under suitable growth assumptions on its geometric zeta function, $\zeta_{\eta}$, known as the languidity assumptions. This involves a sum indexed by the complex dimensions of the generalized fractal string, summing over all residues of $\zeta_{\eta}$, multiplied by the Mellin transform of a tempered or Schwartz test function $\phi$, denoted by $\widetilde{\phi}$. Under languidity assumptions, there is an error term present. Under stronger assumptions, that is, in the case of strong languidity, no error term is present and the resulting fractal explicit formula is said to be exact. In this paper, we express the residue term in the fractional distributional explicit formulas as the fractional $(-\omega)^{\mathrm{th}}$-ordered distributional derivative of the Dirac $\delta$ distribution, denoted by $Y_{\omega}$, applied to modified test functions of the form $\phi(x) \ln^{k-1}(x)$, where $k$ runs from $1$ to the multiplicity of the pole $\omega$ of $\zeta_{\eta}$. The coefficients in this sum are written in terms of the coefficients in the principal part of the Laurent expansion at $\omega$ of the geometric zeta function $\zeta_{\eta}$, multiplied by the gamma function evaluated at $\omega$. The results obtained in this paper contribute to the broader program of characterizing fractals in terms of fractal Taylor series expansions involving their underlying fractal complex dimensions.

To meet the increasing demands for high data rates and large capacity, next generation wireless communication systems require transceivers equipped with a large number of antennas. Massive multiple-input multiple-output (MIMO) with metasurface antennas has emerged as a promising solution. In this paper, we investigate the channel estimation problem for the emerging reconfigurable intelligent metasurface antenna (RIMSA) array systems. Specifically, we develop a two-stage refinement (TSR) channel estimation method based on the compressed sensing (CS) principle. In the first stage, we exploit the antenna structure of RIMSA to receive pilots by setting identical phase response vectors across all RIMSAs. In this manner, the coherence of the measurement matrix under the CS framework is reduced and the channel estimation performance is improved. However, this special design introduces channel direction-of-arrival (DoA) estimation ambiguity and yields an ambiguous candidate DoA set. In the second stage, we optimize the phase responses of the metamaterial elements to resolve the ambiguity and accurately estimate the DoAs and channel coefficients. Overall, the estimation accuracy is improved in the first stage at the cost of ambiguity, and this ambiguity is eliminated in the second stage. We illustrate the performance advantages of the TSR method by presenting the numerical results and comparing it with the existing methods.

Let $q$ be a prime power. We give a construction of Butson Hadamard matrices $BH((F_q\times F_q,+),lcm(6,d))$ for any divisor $d>1$ of $q-1$. Most character values of the group ring elements corresponding to these matrices involve products of two Gauss sums over $F_q$ (with non-quadratic Gauss sums occurring) and this property distinguishes our result from all previously known constructions of group invariant Butson Hadamard matrices. In fact, our $BH((F_q\times F_q,+),lcm(6,d))$ matrices arise as a special case of a more general construction based on what we call “Butson-Jacobsthal functions” that resembles the construction of Hadamard matrices from Jacobsthal matrices.

Diffusion models have demonstrated strong power in generative modeling tasks across multiple domains, exhibiting a remarkable capability of learning complex distributions from samples. In this paper, we leverage such capability to design an efficient universal diffusion-based solver for parameterized black-box optimizations (BBO), where the optimizer has only black-box access to queries of the objective function at the learning stage, yet is able to reduce the additional computational cost at the inference stage for each BBO instance while also capturing the potential multi-modal landscape of non-convex objectives. To cast our formulation as a compatible generative modeling task, we introduce the notion of minimization policy as a new solution concept, which defines a sampling distribution over the solutions that should concentrate around the minimizer set for each BBO instance. We then propose Diffusion-based Iterative Minimization Policy Learning (D-IMPL), a practical generative-model-based solver for solving parameterized BBOs that employs diffusion models to learn a minimization policy, whose density is proportional to the exponential of the negated objective values, thereby amortizing the computational costs across different BBO instances. Furthermore, we demonstrate the performance of our D-IMPL algorithm by establishing a sample complexity guarantee showing that a $\delta$-approximate minimization policy can be effectively learned within $O(\log(1/\delta))$ iterations, and by extensive empirical evaluations over a range of constrained and unconstrained BBO tasks.

This paper investigates the continuous dependence (stability) of solutions to the barotropic compressible Navier–Stokes equations in critical Besov spaces. Using the Lagrangian approach developed in \cite{Danchin2014}, it was shown that, for \(1<p<2d\), the flow map $(a_0,u_0)\mapsto (\bar a,\bar u)=(a\circ X,u\circ X)$ is Lipschitz continuous from $\dot B_{p,1}^{d/p}\times \dot B_{p,1}^{d/p-1}$ into $\mathcal{C}([0,T];\dot B_{p,1}^{d/p})\times E_p(T)$. However, this result does not directly imply the corresponding continuous dependence in the original Eulerian coordinates because of the low regularity of the critical initial data. Previously, only for $p<d$, continuous dependence was known only in certain lower-regularity spaces with a loss of one derivative relative to the natural solution spaces arising essentially as a by-product of the uniqueness argument. We close this gap and prove that for $1<p<2d$ the flow map $(a_0,u_0)\mapsto(a,u)$ is continuous (not Lipschitz continuous) from $\dot B_{p,1}^{d/p}\times\dot B_{p,1}^{d/p-1}$ to $\mathcal{C}([0,T];\dot B_{p,1}^{d/p})\times E_p(T)$ in Eulerian coordinates without any loss of regularity, which together with the known existence and uniqueness theory \cite{Danchin2014} completes Hadamard well-posedness in the critical spaces.

For a finite abelian group $G$, let $\mu_2(G)$ be the minimum number of positive eigenvalues of an entrywise positive real symmetric group-circulant matrix whose entrywise square is positive definite. We prove that $\mu_2(G)\le5$ precisely when $|G|\le15$, and that $\mu_2(G)=6$ precisely when $|G|=16$ or $G$ is cyclic of order $18$, $20$ or $22$. For the noncyclic groups of order at most fifteen, the minimum is three for $C_2^2$ and five otherwise. If the positive Fourier support of such a matrix is $\{0,\pm r_1,\ldots,\pm r_d\}$, where $d\ge1$ and all $2d+1$ displayed elements are distinct, then there are integers $\alpha_1,\ldots,\alpha_d$, not all zero, such that $\sum_i\alpha_i r_i=0$ and $\sum_i|\alpha_i|\le\max\{5,d+2\}$. The bound five is sharp for $d=2$ and $d=3$.

The cube polynomial $C_G(x)$ generates the number of $k$-cubes on a graph $G$. As a subclass of partial cubes, the tope graphs of lopsided sets (LOPs) generalize daisy cubes and median graphs. In this paper, we prove that every tope graph of a LOP shares its cube polynomial with some daisy cube, thereby answering affirmatively a problem posed earlier by the authors. Furthermore, we present explicit expressions for the cube polynomials of tope graphs of LOPs: $C_G(x)=f_\mathcal{K}(x+1)$, where $f_\mathcal{K}(x)$ is the $f$-polynomial of the cubical complex $\mathcal{K}$ of $G$. Finally, we show that a class $\mathcal{G}$ of partial cubes is the class of tope graphs of LOPs if and only if the following equivalent conditions hold: (a) $\mathcal{G}$ is the maximal pc-minor-closed class such that every graph in $\mathcal{G}$ has the same cube polynomial as some daisy cube; (b) for every $G\in\mathcal{G}$, each antipodal subgraph of $G$ has the same cube polynomial as some daisy cube.

We prove that Sierpiński carpet parameters are dense in the bifurcation loci of some natural families of finite type transcendental entire functions having exactly one active critical orbit. Moreover, some specific transcendental entire functions with Sierpiński carpet Julia sets are constructed such that they contain either an attracting basin, a parabolic basin, a Siegel disk or wandering domains. The proofs are mainly based on applying the polynomial-like renormalization and some results on the local connectivity of transcendental Julia sets obtained before.

We study the diffusion limit of the classical solution to the relativistic Boltzmann equation with the initial data near a global relativistic Maxwellian. By using spectral analysis, we establish the convergence of the classical solution to the relativistic Boltzmann equation to that of the incompressible Navier-Stokes system, and give for the first time the optimal convergence rate and a precise estimate of the initial layer. Moreover, through a refined analysis of the linear collision operator $L$, combined with Duhamel's principle, we obtain the existence and uniqueness of a global strong solution to the relativistic Boltzmann equation, as well as a convergence rate independent of the speed of light $\mathbf{c}.$

Let $U$ be an affine log Calabi–Yau variety. Although special and finitely generated valuations are defined using a log CY-Fano compactification of $U$, we prove that the corresponding skeleta in the dual complex are independent of this choice, and hence invariant under $\mathrm{Aut}(U)$. We also show that a finitely generated valuation is special if and only if it is maximal with respect to a natural partial order induced by regular functions on $U$. We then consider the affine log Calabi–Yau threefold obtained as the complement of the Markov cubic surface in $\mathbb{A}^3$. We describe the action of the three Vieta involutions on the special skeleton and relate it to the $(\infty,\infty,\infty)$-triangle reflection group on the hyperbolic plane. We also show that, for every finite triangulation of the dual complex, the special skeleton fails to be locally closed on some simplex. Via the cone construction, this provides a counterexample to a conjecture of the author and Xu.

This paper establishes the global existence of classical solutions to the initial-boundary value problem for the planar four-velocity Broadwell model in a multidimensional setting. While uniqueness was already established in our previous work (which addressed existence under the assumption of sufficiently small, bounded, and $C^{1}$-regular initial and boundary data), the objective of the present study is to completely lift this size restriction. By extending the previously developed fixed-point framework, we prove that the methodology remains highly effective for arbitrary initial and boundary data. Within this generalized setting, both the solutions and their first-order partial derivatives are shown to be well-defined and to exist globally for all times.

Let \(\Gm=\prod_{k\ge0}\mathbb Z_{m_k}\) be a Vilenkin group that is not necessarily bounded, i.e., \(\sup_k m_k=\infty\). We prove that, for every UMD Banach space \(X\) and every \(1<p<\infty\), the Vilenkin partial-sum operators are uniformly bounded on \(L^p(\Gm;X)\), with a bound depending only on \(p\) and the UMD constant of \(X\), and not on \(\mathbf m\). This resolves an open problem arising from the work of Clément et al.~\cite{ClementDePagterSukochevWitvliet2000} and later recorded explicitly in the book of Hytönen et al.~\cite[p.~362]{HNVWI}. The proof reduces the partial-sum estimate, via a Paley conjugation identity and a tangent-sequence decoupling argument, to a decoupling inequality for Fourier projections on finite cyclic groups, which appears to be new. The same approach also yields \(\mathcal R\)-boundedness for the family of partial-sum operators associated with the finer block decomposition, thereby resolving another related problem communicated to us by Fedor Sukochev.

We introduce a combinatorial definition of chromatic symmetric functions for annular webs. We prove their symmetry by constructing a web analogue of the Shareshian–Wachs involution and show that they coincide with the symmetric functions associated to annular webs via Turaev's isomorphism. We then derive explicit formulas for their hook Schur coefficients. We also introduce web LLT functions, whose hook Schur coefficients admit positive Laurent-polynomial formulas. These formulas yield a combinatorial expression for the coefficients of the HOMFLY–PT polynomial of an annular web.

We establish an exact product principle for the Euclidean operator norm of differentials of harmonic maps. For a bounded domain \(G\subset\R^m\) and \(p\in G\), let \(M_G(p)\) denote the supremum of \(\|dF_0\|\) over harmonic maps \(F:\D\to G\) with \(F(0)=p\). For bounded domains \(G_j\subset\R^{m_j}\), we prove \[ M_{G_1\times\cdots\times G_N}(p_1,\ldots,p_N)^2 =\sum_{j=1}^N M_{G_j}(p_j)^2. \] The theorem separates the geometry of the individual factors from the Euclidean geometry of the product: the factor extremal constants combine by a sum-of-squares law, while equality is governed by a single compatibility condition, namely a common maximizing direction for the component differentials. Neither convexity nor attainment of the factor suprema is required. Combining the product principle with the sharp interval and disk factor problems yields exact operator-norm estimates and all equality cases for harmonic maps into boxes and polydiscs. In both families, for every extremal map, the real differential at the origin has one-dimensional image. The same factor constants also define coordinatewise metrics for which the harmonic contraction estimate is sharp when the source disk is equipped with its Poincaré metric of curvature \(-1\). For boxes, the resulting metric is complete and equals twice the restriction of the Kobayashi-Royden metric of the product of vertical strips. For polydiscs, the harmonic product metric is pointwise maximal among contracting metrics of the form \(\max_j a_j(p)|v_j|\), with \(a_j(p)>0\). It is strictly smaller than twice the Kobayashi-Royden metric on every nonzero tangent vector, and its induced path metric is incomplete.

For a dissection $D$ of the unit square into $n$ nondegenerate triangles, let $R(D)=\max_i a_i-\min_i a_i, \Delta(n)=\inf_D R(D).$ We prove that this infimum is attained for every $n\ge2$, and determine the exact minima for $n=5$ and $n=7$, allowing T-junctions. For five triangles, $\Delta(5)=\frac{5\sqrt5-11}{8};$ equality holds precisely when three areas equal $(3-\sqrt5)/4$ and two equal $(3\sqrt5-5)/8$. For seven triangles, $\Delta(7)=r_7$, where $r_7$ is the unique root in $(0,1/4900)$ of $864r^4+2160r^3-6060r^2+4972r-1.$ Every minimizer has four areas $(1+3r_7)/7$ and three areas $(1-4r_7)/7$, although its geometry need not be unique. The proofs combine finite combinatorial classification with exact symbolic and integer-interval certificates. For nine triangles, a tilted-strip construction gives the explicit algebraic upper bound $\Delta(9)\le 0.0001273496861283553341\ldots,$ which is the exact minimum within that topology. Conversely, every dissection in the complete single-cap two-rail zig-zag family, with arbitrary continuous areas, has range greater than $1/3500$; hence a global minimizer must lie outside that family. The exact value of $\Delta(9)$ remains open.

We develop an index policy for finite-horizon Bernoulli multi-armed bandits from minimax solutions to single-arm bandit (SAB) problems. Each SAB problem involves choosing between an unknown Bernoulli arm and a known reward. We show that minimizing worst-case regret of SAB problems over all non-anticipative policies admits an exact semi-infinite linear programming formulation. The resulting stopping policies offer a natural way to compare arms: the higher the known reward against which a policy continues sampling, the more promising the unknown arm. We turn this intuition into indices based on cumulative continuation probabilities, with a monotone adjustment and a reward-shortfall cap. By relating index errors to the regret of single-arm stopping policies, we establish a distribution-free regret bound of $4.45\sqrt{KT}+10.75K$ for $K$ arms and horizon $T$. This bound matches the minimax-optimal regret order established in the literature. The guarantee extends to rewards supported on $[0,1]$ through Bernoulli randomization. We also provide a finite-grid implementation with quantified approximation loss. In numerical experiments, the SAB-based index policy achieves lower worst-case regret than every tested benchmark policy across all evaluated numbers of arms and horizons, while closely matching the grid-based MAB minimax policy in the two-arm setting.

We study the global continuation of solution manifolds arising in dynamical systems. We present a rigorous continuation method based on a Chebyshev series expansion of the solution manifold. The branch is first approximated by a high-order Chebyshev interpolation polynomial, and an explicit error bound is then obtained by verifying the contraction of a quasi-Newton operator near this approximation. The contraction is formulated on a weighted $\ell^1$ space, giving a finer control than the typical $C^0$-error bound obtained from the uniform contraction theorem. In fact, the latter follows directly from our contraction operator. Furthermore, we discuss how our strategy applies naturally to pseudo-arclength continuation, where the continuation parameter fails to provide a valid local coordinate, and extends to multi-parameter continuation. Lastly, we detail two applications in which we compute a two-parameter family of steady-states for the Cahn–Hilliard equation, and a one-parameter family of steady-states undergoing saddle-node bifurcations for the Shigesada–Kawasaki–Teramoto system.

Given a unital inclusion of $C^*$-algebras $\mathcal C \subset \mathcal A$ and a unital inclusion $\mathcal C \subset \mathcal B$ into a von Neumann algebra $\mathcal B$, we investigate the extreme points of unital completely positive maps from $\mathcal A$ to $\mathcal B$ that fix $\mathcal C$ denoted by $UCP_\mathcal C(\mathcal A, \mathcal B)$. This space is obviously convex and $C^*$-convex with respect to the $C^*$-algebra $\mathcal C' \cap \mathcal B$. In this article we show that the $\mathcal C' \cap \mathcal B$-extreme points are exactly the $*$-homomorphisms that fix $\mathcal C$ when $\mathcal A$ is commutative and $\mathcal B$ has a normal faithful center valued trace. This generalizes a result due to Farenick and Morenz where $\mathcal C = \mathbb C 1$ and $\mathcal B = M_n(\mathbb C)$.

We construct explicit examples of skew braces $A$ for which the third term $A^3$ in the left series is not generated by the right-normed star products $a_1*(a_2*a_3)$ of length $3$ as a sub-skew brace.

We prove that, for any $f:S^2\to S^1$ and any $\varepsilon>0$, there exist orthogonal vectors $x,y\in S^2$ such that the length of the shortest arc between $f(x)$ and $f(y)$ is at most $\pi/2 +\varepsilon$. This proves a conjecture of Ghebleh from 2007 that the circular chromatic number of the real orthogonality graph is equal to four.

For infinite cardinals $\lambda \leq \kappa$, we calculate the calibers of topological spaces $X$ of cardinality $\kappa$ whose open sets are the subsets $A$ satisfying $|X \setminus A| < \lambda$. We also prove that the set of calibers of $X$ coincides with the set of calibers of the Vietoris hyperspace $\mathcal{H}(X)$, where the space $\mathcal{H}(X)$ has as its underlying set a collection of subsets of $X$ that includes all its finite subsets and is contained within the collection of its closed subsets.