Zhang Teng

Matrix Triangle Inequalities: Past, Present, and Future

Jean-Christophe Bourin, Teng Zhang

This survey provides a concise overview of matrix triangle inequalities. Classical topics, such as inequalities for unitarily invariant norms, are revisited from a different perspe…

The Bloch constant is strictly smaller than the Ahlfors--Grunsky value

Teng Zhang

We prove that the classical Bloch constant is strictly smaller than the Ahlfors--Grunsky value, disproving the famous Bloch constant conjecture. The Lean 4 formalization of the mai…

Borcea's 2-variance conjecture

Teng Zhang

In this paper, we prove Borcea's 2-variance conjecture. The Lean 4 formalization of the main results are also provided.

Baernstein's quasi-norm monotonicity conjecture for polynomials with unimodular zero

Teng Zhang

Let $m$ denote the normalized Haar measure on the unit circle $T$ . For $0 < r < \infty$ , define $∥ f ∥_{r} := (\int_{T} ∣ f ∣^{r} d m)^{1/ r}$ …

Holomorphic curves of finite lower order with few inflection points

Alexandre Eremenko, Teng Zhang

We prove a conjecture proposed by the first-named author in 1998. Let $f : C \to P^{n}$ be a transcendental linearly non-degenerate holomorphic curve of finite low…

A counterexample to Fuchs's conjecture

Alexandre Eremenko, Teng Zhang

For every $ρ$ in (0, 1/2), we construct an entire function F of order $ρ$ whose logarithmic derivative has zero as a deficient value. This disproves an old conjecture of W. H. J. F…

The $A$ -optimal design problem and the $S$ -matrix conjecture

Teng Zhang

The $S$ -matrix conjecture was formulated by Sloane and Harwit in 1976, motivated by an $A$ -optimal design problem arising in spectroscopy. It states that if $A$ is a nonsingular re…

Sendov's conjecture holds for every degree $n \geq 1 0^{200000}$

Teng Zhang

Sendov's conjecture, first formulated in 1958, asserts that if a complex polynomial $f$ of degree $n \geq 2$ has all its zeros in the closed unit disk ${z \in \mathbb{C} : |z| \leq…

math.OA2026

On the embedding rigidity problem for uniformly locally finite coarse spaces

Teng Zhang

In this paper, we construct countable uniformly locally finite metric spaces $X$ and $Y$ such that $C_{u}^{*} (X)$ is isomorphic to a hereditary $C^{*}$ -subalgebra of $C_{u}^{*} (Y)$ , but neit…

math.OA2026

Isomorphism rigidity of uniform Roe algebras over arbitrary uniformly locally finite coarse spaces

Teng Zhang

Let $(X, E)$ and $(Y, F)$ be uniformly locally finite coarse spaces. We prove that every $C^{*}$ -algebra isomorphism $C_{u}^{*} (X, E) ≅ C_{u}^{*} (Y, F)$ fo…

Plank theorems, Gaussian probabilistic estimates and Rump's 100 Euro conjecture

Teng Zhang

We prove Rump's 100-euro conjecture by deriving a weighted affine escape theorem from Ball's plank theorem in [Invent. Math. \textbf{104} (1991)]. More precisely, let $\mathbb{K}\i…

On a conjecture of $λ$ -Aluthge transforms and Hilbert--Schmidt self-commutators

Teng Zhang

Let $A$ be a complex square matrix, and write its polar decomposition as $A = U ∣ A ∣$ . For $0 < λ < 1$ , the $λ$ -Aluthge transform of $A$ is defined by $Δ_{λ} (A) = ∣ A ∣^{λ} U ∣ A ∣^{1 - λ} .$ In 2007…

cond-mat.soft2026

Bonded-particle model for magneto-elastic rods

Gabriel Alkuino, Joel T. Clemmer, Christian D. Santangelo +1

We develop a bonded-particle model for magneto-elastic rods that unifies large deformations, contact, and long-range magnetic interactions within a single discrete-element framewor…

math.OA2026

Metric Properties: From $S$ -Divergence to Quantum Jensen Divergence

Teng Zhang

We extend the trace-logarithmic $S$ -divergence from matrices to tracial $C^{*}$ -algebras and finite von Neumann algebras, and show that its square root defines a metric on the invert…

On interrelations among different versions of a Heron type mean and commutativity in $C^{*}$ -Algebras

Lajos Molnár, Teng Zhang

The extension of the concept of a mean of positive real numbers to noncommutative settings, e.g., for Hilbert space operators, is a widely studied question. For example, in quantum…

Angle duality and a gap principle for convex combinations of incomplete polynomials on the unit circle

Teng Zhang

In this paper, we establish an angle duality and a gap principle for convex combinations of incomplete polynomials, extending two results of Ge and Gonek in [IMRN, 2024].Our approa…

Weak majorization inequalities for the cubic and quartic coefficients of $e^{(A + B) t}$ versus $e^{A t} e^{B t}$

Teng Zhang

Let $A, B \in H_{n}$ and set $H = A + B$ . For each integer $k \geq 1$ define $Q_{k} := p = 0 \sum k (p k) A^{p} B^{k - p}, R_{k} := ℜ Q_{k} = \frac{Q _{k} + Q _{k}^{*}}{2} .$ Then $H^k=\left.\frac…

math.FA2025

Sharp perturbation bounds on the Frobenius norm of subunitary and positive polar factor

Teng Zhang, Zongben Xu

Leveraging tools from convex analysis and incorporating additional singular value information of matrices, we completely resolve the problem of establishing perturbation bounds for…

math.CO2025

Several combinatorial results generalized from one large subset of semigroups to infinitely many

Teng Zhang

In 2015, Phulara established a generalization of the famous central set theorem by an original idea. Roughly speaking, this idea extends a combinatorial result from one large subse…

From Clarkson-McCarthy inequality to Ball-Carlen-Lieb inequality

Teng Zhang

In this paper, we give two new generalizations of Clarkson-McCarthy with several operators, which depends on the unitary orbit technique developed by Bourin, Hadamard Three-lines T…

Clarkson--McCarthy type inequalities, part I: $ℓ_{p}$ -- $ℓ_{p}$ and $ℓ_{q}$ -- $ℓ_{p}$ Schatten $p$ -estimates

Teng Zhang

We characterize the matrices $U = (u_{ij})$ for which the operator square-sum identity $i = 1 \sum m j = 1 \sum n u_{ij} A_{j}^{2} = j = 1 \sum n ∣ A_{j} ∣^{2}$ holds for all Schatten-c…

Strengthening of Clarkson-McCarthy inequalities with several operators

Teng Zhang

Strengthening of two Clarkson-McCarthy inequalities with several operators is established. These not only confirm a conjecture of the author in [Israel J. Math. 2024], but also imp…