Papers

Complete negative resolution of the canonical complex positive-definite weighted singular-value log-majorization statement, using an exact rational adaptation and proved genuine CFC, singular-value and Euclidean-norm semantics.

Proves a complete negative answer to the original labeled real Hadamard counting conjecture. A genuine finite matrix subtype and an injective restricted doubling construction give m! H(m)^2 ≤ H(2m), the exact source power-of-two lower bound, and a counterexample for every …

For a Hermitian tridiagonal $T$ and $0<\delta<1/2$, the note gives deterministic constructions, on two explicit classes, of a real diagonal $D$ with $\|D\|_2\le\delta$ that separates the eigenvalues of $T+D$. Write $T=\operatorname{diag}(t_i)+R$. If the absolute off-diagonal row sums of $R$ are …

For real $a<b$ and every $n\ge2$, the maximum spread over real symmetric matrices with entries in $[a,b]$ equals the maximum over the $2^{2n-1}$ signed-threshold endpoint matrices $\frac{a+b}2\mathbf 1\mathbf 1^T+\frac{b-a}2S$ with $S_{ij}=d_id_j\varepsilon_{\min(i,j)}$. The proof perturbs the linear functional $xx^T-yy^T$ given by …

For normal $A,B\in\mathbb C^{n\times n}$ with $A$ having at most two distinct eigenvalues (either matrix may play this role), the spectral-norm distance $\delta_n(A,B)=\min_U\|A-U^*BU\|_2$ between their unitary orbits equals the bottleneck distance $d_\infty(A,B)$, the least over bijections of the eigenvalue multisets …

We give an explicit randomized algorithm, in an exact-real arithmetic model in which every input entry may be read, for $Ax = b$ with $A \in \mathbb{R}^{n \times n}$, $\|A\|_2 = 1$, $b \ne 0$ and $0 < \varepsilon < …

For a sparse nonsingular matrix with at most $k$ singular-value outliers and tail condition number at most $\kappa$, conjugate gradients on the implicit normal equations gives a deterministic $O(N[k+1+\kappa\log(2/\varepsilon)])$ arithmetic bound. We give its residual-norm proof and identify regimes where …

The conjectured $4/\pi$ subset bound is false. Let $\omega=e^{2\pi i/3}$ and $L=\{\omega^a+10^{-3}\omega^b:0\le a,b\le2\}$. At degree $k=4$, this nine-point set satisfies $M_4(L)/\max_{S\subset L,\ |S|=5}M_4(S)>13/10>4/\pi$. The proof is analytic and covers every subset by its cluster occupancy. An additional exact-arithmetic certificate checks …

Formalizes the full negative answer to the canonical inheritance conjecture with actual ordered tensor contractions, real nonzero H-eigenpairs, universal lower positivity, an explicit upper negative pair and unconditional lower-pair existence from the intermediate value theorem.

Formalizes the complete negative answer to the corrected canonical conjecture using complex positive-definite matrices, genuine CFC real powers, full characteristic-root lists and exact log-majorization. Proved spectral and Euclidean norm bridges connect a small exact rational certificate to the original target.

Proves the complete canonical discrete-convexity assertion for (j+1)e_(j+1)/e_j over every positive real spectrum. Actual subset sums, polynomial derivatives, full root multiplicities and an exact nonnegative pair certificate connect the proof to the original ratios.

For rational $A$ and $b$ such that every matrix $A-\operatorname{diag}(d)$, $d\in[-1,1]^n$, is nonsingular, the note gives a deterministic algorithm of polynomial bit complexity that computes the exact solution of $Ax-|x|=b$ when $A$ is lower Hessenberg, including zero superdiagonal entries and …

For rational $A$ and $b$ such that every $A-\operatorname{diag}(d)$, $d\in[-1,1]^n$, is nonsingular, the note gives a deterministic algorithm of polynomial bit complexity that computes the exact solution of $Ax-|x|=b$ when the leading $(n-1)\times(n-1)$ block of $A$ is lower triangular with …

Let the checkerboard vertex matrices $A^-$ and $A^+$ of a real interval matrix $[L,U]$ be nonsingular sign regular with a common signature. The note proves that every matrix in $[L,U]$ is nonsingular sign regular with that signature whenever the fixed …

Proves the original function-class equivalence, genuine finite-spectrum CFC and half-line extension bridges, exact polynomial identities for two rational projections, their positive quadratic-form obstruction, and the full negation of the universal complex PSD unitary-orbit assertion.

Proves the genuine CFC square-root modulus identities, an exact rational three-dimensional witness, a nonzero common orthogonal vector for arbitrary complex constraints, all-unitary quadratic bounds, and the full negation of the original universal factor-sqrt(2) PSD domination conjecture.

Proves genuine complex matrix moduli, positive-integer CFC root-sequence limits at all witness matrices, the true spectral-norm topology bridge, an all-unitary trace obstruction, and the full original constant-one conjecture negation.

Proves admissibility of the genuine Euclidean operator norm, exact CFC geometric-mean identities at positive definite complex matrices, a strict counterexample for every positive outer parameter, and the full original universal conjecture's negation.

For $n\ge3$ and a positive spectrum $\lambda$, let $x_1(\lambda)$ be the largest, over orthogonal $V$, of the least trace error of a one-column Nyström approximation of $K_V=V^{\mathsf T}\operatorname{diag}(\lambda)V$, and let $y_1(\lambda)=2e_2(\lambda)/e_1(\lambda)$. The note proves $x_1(\lambda)=y_1(\lambda)=\sum_j\lambda_j-\sum_j\lambda_j^2/\sum_j\lambda_j$ for every positive spectrum, so …

For every fixed column sparsity, aspect ratio, and positive singular-value threshold, we prove that a uniformly sampled proportional-size column subset of any sufficiently redundant signed sparse dictionary has a positive fraction of singular values below that threshold, except with exponentially …

Proves genuine continuous-functional-calculus bridges, positive-real determinant semantics, an admissible exact complex-matrix witness, and the negation of the full original conjecture with arbitrary nonnegative real exponents.

Proves the genuine finite randomized LU process and its normalized history law, derives the actual singular-value tail and expected squared Frobenius error at an exact matrix witness, and refutes the full original universal bound.

Proves an admissible real symmetric order-three counterexample using actual matrix inverses and the maximum absolute row-sum norm, then refutes the complete original universal conjecture including its equality clause.

Proves an actual complex Hermitian positive semidefinite Gram witness, evaluates its full and subset-preserving permutation sums, and refutes the complete original universal inequality by an exact strict gap.

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 …

For every $d\ge1$ and every fixed nonempty compact family $\mathcal M\subseteq\mathbb C^{d\times d}$ there are $C,r>0$ such that every nonempty compact $\mathcal N$ with $d_H(\mathcal M,\mathcal N)<r$ satisfies $\widehat\rho(\mathcal N)\ge\widehat\rho(\mathcal M)-C\,d_H(\mathcal M,\mathcal N)$. This is the pointwise one-sided Lipschitz lower …

Let $H(n)$ count labeled real Hadamard matrices of order $n$. An injective construction gives $H(2m)\ge(2m-1)!!\,H(m)^2$: given a perfect matching of the $2m$ row positions and Hadamard matrices $A,B$ of order $m$, the two rows of the $i$th pair are $(A_{i,\cdot},B_{i,\cdot})$ …

For block size $b = 2$ and $d = 3$ blocks, the manuscript gives deterministic diagonal data with pairwise disjoint spectra, $\Lambda_1 = 2I_2$, $\Lambda_2 = \operatorname{diag}(\varepsilon, 2\varepsilon)$ and $\Lambda_3 = \operatorname{diag}(0, 1)$, for which the random interpolation constant $\chi_{\mathrm{mono}}\chi_{\mathrm{coef}}$ …

The manuscript proves two bounds conjectured by Austin and Trefethen for trigonometric interpolation of degree $N$ in $m = 2N + 1$ points, each displaced from the equispaced grid by at most $\alpha$ grid spacings: the Lebesgue constant is at …

For $0 < \delta < 1$, the note compares compositions of odd cubic polynomials $a x + b x^3$ with arbitrary real polynomials computed from $1$ and $x$ with at most $m$ multiplications, as approximations of $\operatorname{sign}(x)$ on $[-1, -\delta] …

There is a nonconstant smooth one-parameter family of complex $9\times9$ matrices whose members have the same singular values for every complex scalar shift, while no two distinct members are unitarily similar; each member is nilpotent with Jordan type $(3,3,3)$. Consequently …

Let $H_n$ be Hermitian $n \times n$ matrices whose eigenvalues are distributed, in the Weyl sense, as a real measurable function $f$ on $[0, 1]$, and let $E_n$ be complex $n \times n$ matrices with trace norm $o(n)$. The manuscript …

The manuscript proves that for every $n \ge 3$ the variety of singular complex $n \times n$ matrices with $x_{11} = 0$ has Euclidean distance degree $5n - 7$ for the bilinear squared Frobenius distance, counting critical points on its …

For every $n\ge2$ and $C\in\{0,I_n\}$, a matrix $G$ with independent standard normal entries satisfies $\Pr\{\rho_{\rm PP}(C+G)>\tfrac12(3/2)^{n-1}\}\ge2^{-n^2(n^2+n+5)}$, where $\rho_{\rm PP}$ is the growth factor of Gaussian elimination with partial pivoting in exact arithmetic. Consequently no positive constants $c_1,c_2$ give $\Pr\{\rho_{\rm PP}(\bar …

Let $L_8$ be the $8\times8$ unit lower triangular matrix with every entry below the diagonal equal to $-1$, and let $Q_8$ be its orthogonal factor in the QR factorization with positive diagonal in the triangular factor. Changing the $(8,2)$ entry …

For every $n\ge2$, every $\lambda\in\mathbb C\setminus\{0\}$ and every $1\le k<n$, the Jordan block $J_n(\lambda)=\lambda I+N$ satisfies $\max_{\|v\|_2=1}\min_{p\in\mathcal P_k}\|p(J_n(\lambda))v\|_2=\min_{p\in\mathcal P_k}\|p(J_n(\lambda))\|_2$, where $\mathcal P_k$ is the set of complex polynomials of degree at most $k$ with $p(0)=1$: worst-case and ideal GMRES coincide. …

Every hollow symmetric complex $3\times3$ matrix has the form $X(a,b,c)$ with off-diagonal entries $a,b,c$ and determinant $2abc$, so the variety of such matrices of rank at most two is the union of the three coordinate planes, and its smooth locus …

For a fixed real $\rho>0$, let $H_n(\rho)$ be the $2n\times2n$ block Toeplitz matrix with blocks $iB_k-\rho C_k$, $-1\le k\le2$, arising from cubic $C^1$ splines in time in a space-time discretisation of the Schrödinger equation. The matrices $H_n(\rho)$ are nonsingular for …

For the Gaussian trace estimator $T_m$ with $m$ samples and a nonzero real symmetric, possibly indefinite, matrix $A$ with $\lambda = \|A\|_2$, $\phi = \|A\|_F$ and $\rho = \phi^2/\lambda^2$, the manuscript proves that for every $\varepsilon \ge 2\lambda/m + \sqrt{2\phi^2/m …

Let $C,D,R,P\in\mathbb{C}^{n\times n}$ with $R,P$ Hermitian and $P+\lambda D^*+\lambda^{-1}D$ positive definite on the unit circle, and let $X_\eta$ be the stabilizing solution of $X+(C^*+i\eta D^*)X^{-1}(C+i\eta D)=R+i\eta P$. If $X_\eta$ tends to a finite nonsingular $X_0$ as $\eta\downarrow0$, the pencil $\lambda^2C^*-\lambda …

For every Segre–Veronese cone and every positive definite real inner product on its ambient partially symmetric tensor space, its Euclidean distance degree is at least the degree for the Frobenius inner product. The argument uses a basis of Chern–Schwartz–MacPherson classes …

An explicit smooth, irreducible, nondegenerate curve $X \subset \mathbb{P}^{11}$ and a point $p$ satisfy $\underline{R}_X(p) = 2$, $R_X(p) = 3$, and $R_{X \times X}(p \otimes p) = 9$. Thus the implication conjectured by Ballico, Bernardi, Gesmundo, Oneto and Ventura is …

For a bounded family of matrices of order $d$, with joint spectral radius one and individual spectral norms at most $L$, every product of length $n$ is bounded by $\Theta_d(Ln)^{d-1}$, where $\Theta_d$ depends only on $d$. The proof uses an …

For every real $\gamma\geq0$, a finite family of real matrices is constructed whose joint spectral radius is one and whose maximal norm of a product of length $n$ is bounded above and below by positive constant multiples of $n^\gamma$, at …

For every positive integer $r$, an explicit one-parameter family of real, entrywise-positive, positive-definite matrices of order $r+1$ makes the expected trace error of $r$ randomly pivoted Cholesky steps approach $2^r$ times the optimal rank-$r$ trace error. The same family makes …

The manuscript proves that the transfer holds with the universal constant $C = 11$: for real positive semidefinite $A$ and $\widehat A$, any ordered eigenbasis of $\widehat A$ used for both truncations, every continuous operator monotone $f : [0, \infty) …

For prescribed tensor-train rank bounds, a deterministic algorithm in the exact arithmetic/SVD model returns an approximation with squared error strictly smaller than $(d - 1)$ times the optimal squared error whenever that optimum is positive, and reconstructs exactly at zero …

For every odd order $m \ge 3$ and dimension $n \ge 2$, the generic ordinary rank, symmetric rank, ordinary border rank, symmetric border rank and Vandermonde rank of a complex Hankel tensor all equal $\lceil (m(n-1)+1)/2 \rceil$. The lower bound …

A logarithmic compactification of the critical-point correspondence on $\mathbb{P}^{m-1} \times \mathbb{P}^{n-1}$ gives a coefficient formula for its nonisotropic Rayleigh–Ritz discriminant. Its derivation counts ramification on a general parameter pencil, computes the boundary contributions by first-jet bundles, and gives an argument …

The manuscript proves that the angular condition number of the tensor rank decomposition has finite mean for random real tensors of order $d \ge 3$, mode sizes $n_j \ge 2$ and rank $r \ge 3$, in every format where complex …