Papers

We study the learning dynamics of fine-tuning a policy model on self-generated and reward-weighted data, with particular focus on a generalized version of REINFORCE – referred to as RE(S) – that updates the rollout distribution once every $S \ge 1$ gradient steps. Prior work in bandits and reinforcement learning has developed rich theory for policy gradient methods, and on-policy sampling (i.e., a small $S$, ideally $1$) is often viewed as crucial to their success; yet in prominent application like post-training large language models, reward-guided self-training has proved to be effective even when the rollout distribution is updated infrequently, but theoretical understanding remains limited for the convergence properties of these off-policy methods. To bridge these gaps, we develop a unified theory for RE(S) that covers the full spectrum of $S \ge 1$: it can be interpreted as a stage-wise optimization process, where each stage takes $S$ gradient steps for minimizing the Kullback-Leibler distance to a fixed reward-weighted rollout distribution. For multi-arm bandits with softmax policies, our in-depth analysis and numerical experiments reveal three key findings: (1) for any fixed $S$, RE(S) enjoys global convergence to the optimal policy as the number of rollout distribution updates $B = \lfloor T / S \rfloor \rightarrow \infty$, where $T$ denotes the number of gradient steps; (2) we prove tight two-sided bounds showing that the suboptimality gap of RE(S) achieves an asymptotic $\Theta(1 / T)$ convergence rate, while $S$ only affects the length of a burn-in phase; (3) when initialized at a weak policy with a small optimal-action probability, RE(1) gets trapped around suboptimal policies for a long period, whereas RE(S) with a suitable $S$ avoids the detour and achieves significantly faster convergence to the global optimum, highlighting the benefits of off-policyness in this case.

Let $S$ be a smooth, affine noetherian $\mathbb{C}$-scheme, and let $X/S$ be a smooth, quasi-projective family of Calabi–Yau $d$-folds. We prove that the derived moduli stack of flat families of properly supported coherent sheaves on $X/S$ has a relative $(2-d)$-shifted symplectic structure by generalizing a construction of Brav–Dyckerhoff. The main points are to show that the category $\mathrm{IndCoh}(X)$ is a smooth, compact $\mathrm{QCoh}(S)$-module and to identify our moduli stack of interest as a substack of the moduli of pseudo-perfect objects in $\mathrm{IndCoh}(X)$. Finally, we discuss two examples of families of smooth, quasi-projective Calabi–Yau threefolds that are potentially useful for applications in enumerative geometry.

We first study the part size statistic in cylindric partitions, listing some specializations of a crude but general formula. Then, we establish bijections between ordinary partitions and cylindric partitions of small profiles with additional restrictions. Lastly, we extend earlier formulas for generating functions of cylindric partitions into distinct parts and functional equations of unrestricted cylindric partition generating functions, involving the part size statistic.

O'Grady's generalized Franchetta conjecture asks whether any codimension two cycle on the universal polarized K3 surfaces restricts to a multiple of the Beauville–Voisin class on a given K3 surface. We apply Bergeron–Li's result on the cohomology of universal K3 surfaces to give an affirmative answer to this conjecture when the universal K3 surface is rationally connected, including the case of genus 22.

Let $d_1,d_2,\ldots, d_r\ge 2$ be $m$-th power free positive integers. In this paper, we provide explicit representations of all irreducible $\mathbb{Z}_+$-modules over $\mathbb{Z}[\sqrt[m]{d_1},\sqrt[m]{d_2},\ldots,\sqrt[m]{d_r}]$, where $\mathbb{Z}_+$ denotes the semiring of non-negative integers.

We study reduced solutions of Toda systems generated by spherical metrics for an arbitrary complex simple Lie algebra. The main result is an intrinsic centralizer parametrization: for a spherical metric, the reduced family is the positive Iwasawa slice cut out by the centralizer of the principal image of closure of monodromy group. As an application, we transfer existing spherical-metric existence results to the diagonal Toda source tuple.

Let $G$ be a graph of order $n$ with adjacency matrix $A_G$, let $\mathbf e$ denote the all-one vector, and let$W_G=[\mathbf e,A_G\mathbf e,\ldots,A_G^{n-1}\mathbf e]$ be its walk matrix. We consider the case $\operatorname{rank}W_G=n-2$, the first corank for which distinct graphs can have the same walk matrix. We give a complete structural description of such pairs. More precisely, if $G$ and $H$ are distinct graphs on the same labelled vertex set and $\operatorname{rank}W_G=n-2$, then $W_G=W_H$ if and only if $H$ is obtained from $G$ by a reciprocal Wang–Qiu–Hu (WQH) switching. In this case, $A_G-A_H=uv^T+vu^T$, where $u,v\in\{0,\pm1\}^n$ have disjoint supports and form a basis of $\ker W_G^T$. We also determine the minimum order at which a non-isomorphic corank-two walk mate can occur: no such pair exists for $n\le9$, so the previously known $10$-vertex example is sharp. Starting from a labelled realization of that pair, we use singleton union and join operations, together with the graph coronal, to construct connected non-isomorphic pairs with equal corank-two walk matrices for every $n\ge10$.

We consider the moduli stack $\mathcal{H}_{g,d}$ parametrising maps $\theta\colon C\to L$ of degree $d$ between complex curves of genera $g$ and $0$. We provide a splitting of the stable rational cohomology of $\mathcal{H}_{g,d}$, for $g\to\infty$, in terms of degrees of intermediate covers, and identify each direct summand in terms of factorisation homology. For $2\le d\le 5$, we further interpret the stable rational cohomology in terms of configuration spaces with labels in partial abelian monoids; we show in particular the existence of stable odd-dimensional cohomology classes for $d=4,5$.

The index of a given monic irreducible polynomial $f(x)\in\mathbb Z[x]$ having a root $\theta $ is the index of $\mathbb Z[\theta]$ in the ring of algebraic integers of $\mathbb Q(\theta).$ Further, $f(x)$ is monogenic if its index is $1$. In this article, we prove that the index of $f(x)$ always divides the index of $f(g(x)),$ where $f(x), g(x)\in \mathbb Z[x]$. Additionally, we give the precise power of the index of $f(x)$ dividing the index of $f(g(x))$. Also, we provide a necessary condition for the monogenity of $f(g(x))$. As an application of our results, we prove that the $n$-fold composition $f^n(x)=(f\circ f\cdots\circ f)(x)$ of a polynomial $f(x)$ is monogenic for all $n\in\mathbb{N}$ if and only if the sequence $\langle I_n\rangle$ is convergent, where $I_n$ is the index of $f^n(x)$. Moreover, we show that for a given $n\in\mathbb{N}$, if $I_n$ is $m^{\text{th}}$ power free integer, then the $n$-tower $\mathbb Q\subseteq K_1 \subseteq\cdots \subseteq K_{n-1} \subseteq K_n,$ defined by $n$ iterates of a polynomial $f(x)\in\mathbb Z[x]$ contains at least $\max\{\lfloor n-\log_{\deg f}m \rfloor,0\}$ monogenic number fields.

We study the approximation of finite-energy Sobolev functions by bounded Sobolev functions for integral functionals associated with possibly signed Carathéodory integrands. The approximating sequence is required to converge strongly in the Sobolev space, while the corresponding energy densities converge in \(L^1\). Ordinary truncations may fail because they replace the gradient by zero on the tails, where the integrand need not be integrable. Our main contribution is a general method based on spatially dependent barriers whose gradients can be adapted to the structure of the Lagrangian.

We study the approximation of finite-energy Sobolev functions by bounded Sobolev functions for integral functionals associated with possibly signed Carathéodory integrands. The approximating sequence is required to converge strongly in the Sobolev space, while the corresponding energy densities converge in \(L^1\). Ordinary truncations may fail because they replace the gradient by zero on the tails, where the integrand need not be integrable. Our main contribution is a general method based on spatially dependent barriers whose gradients can be adapted to the structure of the Lagrangian.

We provide a tutorial overview of receding horizon control across deterministic, stochastic, multi-agent, and game-theoretic settings. We contrast the minimizer of an optimal control problem with the equilibrium of a game in terms of cost and constraint handling to motivate the difference of MPC schemes involving multiple agents/players. Interestingly, dissipativity theory and the turnpike property have turned out to be a unifying thread and system-theoretic backbone across all these schemes. This paper is the first to give an overview of results, summarizing fundamental insights, drawing parallels, and highlighting technical differences in dissipativity and turnpike analysis across economic, game theoretic, and stochastic settings. We also explore recent trends in numerical methods for games. We conclude by pointing to a list of open challenges.

State-dependent Riccati equation (SDRE) control requires repeated online solution of continuous-time algebraic Riccati equations (CAREs). A projected-channel $H_\infty$ formulation preserves the physical actuator and sensor dimensions of under-actuated, partially observed systems. We derive a computable sufficient attenuation bound for positive-semidefinite stabilizing Riccati solutions and construct an update that enforces their spectral-radius coupling condition. A discrete Kalman recursion supplies the scheduling state estimates. The two CAREs arising in controller synthesis are solved by a structure-preserving doubling algorithm (SDA) or a warm-started Newton–Kleinman iteration equipped with a doubling Lyapunov solver. Numerical studies use a twelve-state F-16 model with complete aerodynamic forces, moments, and trim, together with quadrotor spiral tracking. With instantaneous command updates, all three CARE backends give essentially the same closed-loop response. In the tracking simulations, measured computation times determine when new control inputs are applied, and the previous inputs are held during computation. SDA and Newton reduce airspeed and pitch errors in all six paired aircraft tracking runs. In the faster quadrotor task with limited computational resources, SDA and Newton complete all paired runs, while runs using MATLAB icare terminate early.

An exact Max-Cut semidefinite relaxation can have optima outside the convex hull of optimal cut matrices. We study when this hull equals the optimal face and how to recognize equality. For rational weights, the unique optimal dual slack is recoverable in polynomial time, but its kernel may contain directions no optimal matrix uses. Under strict complementarity, recognition and recovery of all optimal cuts on equality instances take deterministic polynomial time. At dual rank one, this hypothesis is unnecessary. Recognition is polynomial-time Turing equivalent to exact rational semidefinite feasibility, even on simple unweighted graphs. Testing whether a supplied cut matrix is the unique optimum is many-one equivalent, even on connected simple graphs with a unique optimal cut. NP-completeness of recognition would imply NP equals coNP. Replication realizes doubly nonnegative matrices with positive diagonal and rational kernel as sign-spanned optimal dual slacks, preserving rank, complete positivity, and cp-rank.

Let $\overline{\mathfrak{F}}$ be the group class obtained from the class $\mathfrak{F}$ of finite groups by iterated applications of Kropholler's operation ${\scriptstyle{{\bf LH}}}$ and Talelli's operation $\Phi$. We describe the structure of the cokenels of acyclic complexes of projective modules over the group algebra of $\overline{\mathfrak{F}}$-groups with coefficients in a commutative ring. If $G$ is an $\overline{\mathfrak{F}}$-group, we show that any acyclic complex of projective (respectively, flat) $\mathbb{Q}G$-modules is contractible (respectively, pure acyclic). If $G$ is torsion-free, the same conclusions hold for acyclic complexes of projective or flat $\mathbb{Z}G$-modules. Analogous results are valid for acyclic complexes of injective modules. We present some applications regarding modules that admit complete resolutions, groups with periodic cohomology after some steps and the relation between Gorenstein and ordinary homological dimensions for modules over group algebras.

We show the existence of infinitely many singular positive forward self-similar solutions for the semilinear heat equation $\partial_t u = \Delta u + u^p$ for $x \in \R^N$, $t > 0$, where $N \geq 3$ and $N/(N-2) \leq p \leq (N+2)/(N-2)$. We also investigate the asymptotic behavior and the bound of the profiles of the solutions as $|x| \to 0$ and $|x| \to \infty$, respectively. Furthermore, we show nonuniqueness of positive soutions of the Cauchy problem for the equation with $p = N/(N-2)$ by using singular self-similar solutions.

Let $f_{0}$ be a plane curve singularity. We study the jump of the Milnor number of a homogeneous singularity $f_{0}$ of order $n$ in the class of linear deformations $f_{s}=f_{0}+sg$ of $f_0$ with fixed order $ord f_{s}=j<n$ for $s\neq0$, where $g$ is holomorphic and $ord g=j$.

We study Ehrhart coefficients, the integer decomposition property, and unimodular triangulations for two families of Hermite normal form simplices considered by Bruckamp, Caicedo, and Juhnke. The first family consists of simplices of the form $S_{\boldsymbol{a}}=\mathrm{conv}(0,e_1,\ldots,e_{d-1},\boldsymbol{a})$, where $\boldsymbol{a}=(a_1,\ldots,a_{d-1},N)$, and the second consists of the simplices $T_{d,N}=\mathrm{conv}\bigl(0,e_1,\ldots,e_{d-2},(d-2,\ldots,d-2,d-1,0),(1,\ldots,1,N)\bigr)$. For the family $T_{d,N}$, we prove unimodality of the Ehrhart coefficients in every dimension and determine the log-concave and real-rooted cases. For the subfamily $S_{\boldsymbol{a}}$ with $\boldsymbol{a}=(N-q,\ldots,N-q,N)$, we characterize both the integer decomposition property and the existence of a regular unimodular triangulation via a congruence condition on a negative continued fraction. We also give an arithmetic test for the integer decomposition property and finite triangulation tests for $S_{\boldsymbol{a}}$ with arbitrary $\boldsymbol{a}$. These results address three open questions of Bruckamp, Caicedo, and Juhnke.

The study of monogenic polynomials is a classical problem in algebraic number theory. Existing criteria for deciding whether a polynomial is monogenic typically rely on discriminant computations together with methods such as Dedekind's criterion, Newton polygons, or valuation-theoretic techniques. In this paper, we develop a general local criterion for the $p$-maximality of orders generated by roots of arbitrary monic irreducible polynomials. As an application, we apply this to irreducible polynomials of the type $f(X)=X^n+A(BX+1)^m,$ where $1\le m<n$, $\gcd(n,mB)=1$, and $A,B\in\mathbb{Z}\setminus\{0\}$. We show that $f$ is monogenic if and only if both $A~\text{and}~n^n+(-1)^{n+m}B^n(n-m)^{\,n-m}m^mA$ are square-free. This provides a new proof of the main theorem of \cite{KK}, thereby proving Lenny Jones' conjecture \cite[Conjecture 4.1]{LJ}. Furthermore, we obtain explicit infinite families of irreducible non-monogenic polynomials, including trinomial, quadrinomial, and power-compositional families.

Stokes-Brinkman-Darcy (SBD) models connect free flow and porous flow through a Brinkman layer of finite thickness. The proposed method for solving these coupled equations consists of a mixed first-order formulation, shared interface traces and hard traction constraints, and a pressure correction based on boundary mass balance. Independent stresses, Darcy flux and pressure-gradient auxiliaries avoid second derivatives of network outputs. Boundary liftings and shared traces impose exterior and interface kinematic conditions exactly, while a stress map enforces native Brinkman-Darcy force balance. An integral momentum relation determines the pressure correction after training, without reference-pressure labels. Across two manufactured solutions and three permeabilities, mean corrected upper-pressure relative $L^2$ errors range from 0.87% to 2.58%; correction reduces both upper-pressure $L^2$ errors and the full Brinkman-pressure $H^1$ error in every run. Paired controls assess the contributions of interface sharing and traction enforcement. A non-manufactured filtration problem tests mixed-boundary adaptation against an independently refined finite-element reference, with all four runs satisfying the prescribed field and physical-consistency criteria.

Motivated by the work of Ohira and the advantages of the offset linear canonical transform (OLCT) over the Fourier transform (FT), this paper proposes an OLCT based framework for solving a class of delay differential equations. By exploiting the operational properties of the OLCT, the original delay differ- ential equation is transformed into a Volterra-type delay integral equation in the transform domain and solved numerically using Brunners method of steps . An explicit analytical solution is derived for a special case to investigate the effect of the delay parameter. A unified transform-domain formulation is also established by relating the proposed OLCT approach to its Fourier transform counterpart. Numerical and graphical results demonstrate the accuracy and effectiveness of the proposed method and validate it through comparisons with the Fourier transform based formulation of Ohira . The proposed framework may provide a promising foundation for solving non linear delay differential equations involving transcendental terms, as both the OLCT and the resulting Volterra integral equation possess the essential properties required to handle non linearities and transcendental terms.

Dunfield, Friedl, and Jackson showed that if the $SL(2, \mathbb{C})$-character variety of a knot has an irreducible curve component that contains the character of an irreducible representation and a character with nonmonic twisted Alexander polynomial, then this component has only finitely many characters with monic twisted Alexander polynomials. In this paper, we give explicit upper bounds on the number of such characters in terms of a presentation of the knot group. In particular, we give upper bounds in terms of the crossing number of the knot.

Covers in hypergraphs are frequently studied to capture various forms of dependence between hyperedges. For example, even covers–which check if each vertex appears in an even number of hyperedges–have found much success recently in the study of locally decodable codes. Inspired by a recently-emerging line of work on the non-redundancy of constraint satisfaction problems (CSPs), we introduce and study two novel families of covers of hypergraphs which are stricter than even covers: Abelian covers and Catalan covers. Abelian covers are similar to even covers, except that arithmetic is now done over the integers rather than modulo 2, allowing us to capture dependences over arbitrary Abelian groups. Catalan covers capture the behavior of non-Abelian groups by only allowing local cancellations in a sequence of hyperedges. We prove three main results about Abelian and Catalan covers. First, using tools from lattice theory, we show that any $r$-uniform hypergraph with $n$ vertices and $n \log(r)$ hyperedges has an Abelian cover. Second, using tools from algebraic topology, we show that in any $3$-uniform hypergraph, Abelian covers and Catalan covers are equivalent; thereby showing that Catalan covers emerge after $O(n)$ hyperedges in $3$-uniform hypergraphs. Finally, using the theory of nilpotent groups, we show that there exists a $4$-uniform hypergraph which has an Abelian cover but not a Catalan cover. Collectively, these results exactly characterize the reach that Abelian covers have in deducing dependences in hypergraphs. As our primary application, we show that any arity-$3$ CSP with an infinite-domain Mal'tsev extension has linear non-redundancy. This implies near optimal streaming, sparsification, and kernelization algorithms for this family of CSPs. Previously, such a result was only known for the much simpler case of arity-$2$ CSPs.

This paper studies efficient waveform design for MIMO-OFDM dual-functional radar-communication (DFRC) systems with 1-bit digital-to-analog converters (DACs), which remains underexplored in the literature. We formulate the DFRC waveform design problem by maximizing the trace of the target angular Fisher information matrix (FIM), subject to symbol-error probability constraints realized via constructive interference (CI) and a time-domain 1-bit transmit alphabet constraint. The resulting problem is a large-scale nonlinear integer programming problem. We develop an inexact alternating direction method of multipliers (ADMM) algorithm with alternating time- and frequency-domain updates. The proposed scheme enables subcarrier-wise optimization in the frequency domain and closed-form primal and dual updates in the time domain. In particular, the CI constraints are deliberately retained in both the time- and frequency-domain subproblems to promote the feasibility of intermediate iterates and improve the convergence behavior of ADMM. Simulation results demonstrate promising DFRC performance.

For a $d$-dimensional probability vector $\mathbf{h}=(h_1,\dots, h_d)$, let $(S^{\mathbf{h}}_n)_{n\geq 0}$ be a nearest-neighbor random walk on $\mathbb{Z}^d$ such that at each step, it moves to one of the two nearest neighbors in the $i$-th dimension with probability $\frac{1}{2} h_i$ ($i=1,\dots, d$). Let $T^{\mathbf{h}}=\inf\{n\geq 1: S^{\mathbf{h}}_n=(0,\dots,0)\}$, the first return time to the origin. For two $d$-dimensional probability vectors $\mathbf{h}'$ and $\mathbf{h}''$ with the former majorizing the latter, we show that $T^{\mathbf{h}'}$ is stochastically smaller than $T^{\mathbf{h}''}$. In particular, the first return time for the $d$-dimensional simple random walk is stochastically larger than $T^{\mathbf{h}}$ for all $d$-dimensional probability vectors $\mathbf{h}$.

HARS (Hybrid Adaptive Reliability-Aware and Syndrome-Aware) is a bit-flipping decoder that combines local channel reliability with parameter selection from the initial syndrome. A bounded check reliability modifies the parity-check weight, while the initial syndrome weight selects the base weight, reliability coefficient, threshold decay factor, and perturbation amplitude for each frame. Computing these quantities at initialization supports synchronous multi-bit updates with a simple iterative datapath. For rate-one-half PEGReg and WiMAX codes over the binary-input additive white Gaussian noise channel, HARS reduces both bit-error rate (BER) and mean iteration count relative to a calibrated SM-NGDBF baseline. The PEGReg BER at 3.25 dB is approximately one-sixth of the comparison value; the WiMAX BER at 2.75 dB is reduced by 59%. A seven-fractional-bit FPGA implementation shares 132 variable-node lanes among four frame contexts. Precomputed check weights, encoded threshold states, and parallel perturbation generation support one group per clock during sustained decoding. On an XC7A200T at 60 MHz, for preloaded 128-frame batches with a ready receiver, measured coded throughputs are 49.7 Mb/s at 2.50 dB and 361.9 Mb/s at 4.00 dB, including initialization and output transfers.

We prove two arithmetic nonexistence conditions for tight spherical $5$-designs in dimension $(2m+1)^2-2$. Let $m$ be a positive even integer satisfying $\nu_2(m(m+1))\leq5$, where $\nu_p(a)$ denotes the exponent of the prime $p$ in the positive integer $a$. We show that such a design does not exist if either (i) $m\equiv1\pmod3$ and $\nu_p(m(m+1))\leq1$ for every prime $p\geq7$ with $p\equiv3\pmod4$, or (ii) $m\equiv0\pmod3$, $\nu_3(m(m+1))=1$, and $\nu_p(m(m+1))\leq1$ for every prime $p\geq5$ with $p\not\equiv1\pmod{12}$. Together, these conditions recover the corresponding nonexistence results of Bannai, Munemasa, and Venkov and of Nebe and Venkov, and exclude parameters not covered by either earlier result, including $m=16,40,88,96,100$. The proofs use lattice theory to constrain the discriminant groups of the associated lattices, and derive contradictions through Gauss sums and determinant congruences.

For every $\alpha>1$ we construct a finite set $A\subset\mathbb{D}$ such that the zero-based invariant subspace $I_A$ fails the wandering subspace property. Consequently, the Beurling-type theorem holds on the weighted Bergman space $A^2_\alpha$ if and only if $-1<\alpha\le1$, confirming a conjecture of Shimorin. As a further application, we extend the prescribed-curvature construction of Hedenmalm and Perdomo to all $\alpha>1$, thereby replacing the constant $\alpha_0\approx1.04$ in their result by $1$.

Phase-field simulations of solid-state dewetting based on polynomial potentials may exhibit spurious bulk-diffusion coarsening, inconsistent with the surface-diffusion-dominated physics. To address this issue, we formulate a degenerate Cahn–Hilliard model with the logarithmic Flory–Huggins potential and dynamic contact line boundary conditions for the film–substrate–vapor triple junction. The logarithmic barrier confines the phase variable to its physical range and suppresses spurious coarsening at the continuum level. We develop a fully discrete, structure-preserving scheme that combines a Lagrange multiplier approach with an upwind finite-volume discretization and rigorously guarantees pointwise boundedness, mass conservation, and energy dissipation without artificial cut-offs or projections. A dimensional-splitting strategy further reduces computational cost while preserving these properties in each one-dimensional sweep. We also derive an explicit estimate of the equilibrium radius contraction caused by spontaneous film shrinking, showing that the logarithmic potential produces weaker spurious shrinkage than the polynomial potential. Numerical experiments confirm the analysis and demonstrate that, for small temperature parameters, the proposed scheme suppresses spurious coarsening and pinch-off while accurately reproducing surface-diffusion-dominated dynamics and relaxation toward equilibrium island structures.

Minimal surfaces bounded by cables or threads arise naturally in tensile architecture: a membrane under uniform tension takes the shape of a minimal surface, and its flexible, inextensible boundary lies along an asymptotic line of constant geodesic curvature. Such configurations have been studied experimentally since the 1960s at the Institute for Lightweight Structures in Stuttgart and more recently realized in gridshells built along networks of asymptotic and geodesic curves. Motivated by this architectural context, we construct two new families of embedded minimal surfaces bounded by finitely many asymptotic arcs of constant curvature, using the Plateau-conjugate method applied to the solution of a partially-free boundary problem for minimal disks that meet the unit sphere orthogonally along the free boundary component. For every integer $m\geq 3$ , we prove the existence of a one-parameter family of embedded minimal annuli, called tensile catenoids, whose boundary components each consist of $m$ asymptotic arcs of constant curvature that meet at cusps. As the parameter varies, the family degenerates from a planar configuration to a union of $m$ minimal disks. For every integer $k\geq 3$, we prove the existence of an embedded minimal surface with the topology of a sphere minus $k$ disks, called a tensile $k$-noid, each of whose $k$ boundary components consists of four asymptotic arcs joined by cusps. Both families are symmetric with respect to a horizontal plane and possess several vertical planes of symmetry. Embeddedness follows from showing that the fundamental piece obtained by conjugation is a graph contained in the region delimited by the planes of symmetry.