Explicit optimal Hardy weight found for fractional Laplacian
The weight is null-critical and fixes the sharp constant for n^{-2σ}.
· “An optimal fractional Hardy inequality on the discrete half-line”
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Functional Analysis
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One sharp matrix bound controls Schwarz-Pick at n points
For self-maps of the disk the factor is n; for degree-d Blaschke products it drops to min(n,d).
Learning adaptive bases from data lifts spectral PDE operators
Data-dependent Parseval frames capture localized multiscale features more effectively than fixed global bases while retaining FFT speed.
· “Don't Fix the Basis -- Learn It: Spectral Representation with Adaptive Basis Learning for PDEs”
Plank theorem settles Rump's 100-euro conjecture
Deriving a weighted escape principle shows row sums of n guarantee a vector x with infinity-norm bounds and remove the prior 3+2√2 factor in
· “Plank theorems, Gaussian probabilistic estimates and Rump's 100 Euro conjecture”
Boundedness notions for functionals on n-normed spaces agree
Different definitions of bounded k-linear maps turn out to be equivalent, so all dual spaces and their norms coincide.
· “Bounded Multilinear Functionals and Multicontinuous Functions on n-Normed Spaces”
Copson inequality proves strictly improvable
For decreasing weights and several concrete weight sequences, the classical constant is not best possible.
Schäffer's matrix bound is sharp: the constant is sqrt(e)
Worst-case determinant-inverse loss grows as sqrt(e n), making Schäffer's 1970 upper bound optimal in constant.
· “Sch\"affer's matrix inequality: the exact asymptotic constant”
Verblunsky coefficients dictate bidisc core spectra
For homogeneous generators all core singular values are alpha_n; Mahler measure and a monotonicity counterexample follow.
· “Verblunsky coefficients, CMV matrices and numerical invariants of homogeneous bidisc submodules”
Vanishing disk-area integrals force f=0
Answers the area-integral problem: bounded solutions vanish for every α; smooth unbounded ones persist for α<1.
· “Variable-Radius Disk Transforms and an Area-Integral Problem of Zalcman”
Reverse H\"older equals Muckenhoupt in parabolic weights
This settles two open questions on time-lag classes and yields a John--Nirenberg inequality.
One condition decides which lattices admit smooth Gabor windows
Covolume below 1 plus an index gap turns out to be necessary and sufficient for Schwartz-class windows.
· “Tilings, packings, and the existence of Schwartz-class Gabor windows”
All invariant C*-subalgebras of PSL_n(Z) arise from normal subgroups
For n≥3, PSL_n(Z)'s reduced C*-algebra has no exotic invariant subalgebras — rigidity matches the von Neumann case.
Optimal constant found for scaled q-numerical ranges
The least C in the polynomial bound is max of 1 and 2|q|/(1+sqrt(1-|q|^2)).
Arctangent commutator is positive
A trace-class operator with trace π/2 shows a positive commutator need not fit Kato's sufficient condition.
· “A counterexample to the Kato conjecture for positive commutators”
Universal frame preservers factor through bi-Lipschitz circle maps
Maps that preserve orbit-frame generators for every normal diagonal operator split into a boundary shadow and an inner kernel.
Approximation numbers tied to Hilbert numbers: a_n ≤ e(n+1) h_n
Settles 60-year-old conjectures of Carl, Pietsch, Mityagin and Henkin up to the constant e.
· “On bounds between all s-numbers and widths of convex sets”
Frame set conjecture solved for totally positive functions
Gabor shifts form a frame exactly when lattice density is below 1, closing the irrational case.
· “Gabor Frames of Totally Positive Functions: A Complete Characterization”
Residual collapse equates ordered POVM realizations by surviving effects
Different orderings and couplings reduce to the same canonical form whose non-escape coordinates are orthogonal and sum to the identity.
Complete Crouzeix conjecture holds for every 3×3 matrix
Square-function inequalities settle the matrix-valued Crouzeix bound with the optimal constant 2 in dimension three.
· “Square Functions and the Complete Crouzeix Conjecture in Dimension Three”
Weighted sparse operators are bounded on Morrey spaces
A model theorem would let singular integrals inherit Morrey bounds from sparse sums.
· “Sparse Operators and their boundedness on Morrey-type Spaces: An Expository Note”
Subadditive weight refutes 1975 normality problem
On the algebra of all bounded operators, directed suprema survive while σ-weak limits do not.
· “A counterexample to Haagerup's problem on subadditive weights”
Ideal is analytic P iff all its circle subgroups are Polishable
The equivalence is witnessed by an explicit metric, and a dichotomy splits the subgroups into countable or Fσδ-complete.
· “Complexity and Polishability of characterized subgroups on the unit circle”
One constant sets entropy decay on every classical and quantum torus
A matrix-valued Wirtinger inequality pins the sharp rate at 1 in every dimension and deformation.
· “Sharp Complete Modified Log-Sobolev Inequalities on Classical and Quantum Tori”
Completion of a continuous inverse algebra can break the CIA property
Inversion stays continuous, but invertible elements fail to form an open set after completion.
· “The completion of a continuous inverse algebra need not be a continuous inverse algebra”
Orbit-invariant functions extend iff a block kernel is positive
A function constant on Blaschke orbits extends to H-infinity exactly when a block kernel is positive semidefinite.
· “Dynamic Nevanlinna-Pick Theory, Covariance Dilations, and Non-commutative Varieties”
One integral condition settles the dimension question
For the Möbius-invariant Laplacian on the ball, one dimension-free test controls regularity.
A mathematics paper pins down exactly when the product of two Cauchy singular integral…
For Cauchy singular integral operators on the unit circle, the semi-commutator is compact exactly when, on each support set, either the two…
· “Essentially semi-commuting singular integral operators with Cauchy kernel on L²”
P-contractions need not be enriched contractions
Two examples—one on five points, one continuous on the plane—settle the question.
· “On The Existence of \(P\)-Contractions that are not Enriched Contractions”
This paper proves a factorization theorem for normalizing unitaries in commuting squares…
A relative eigenbasis property is shown to be necessary and sufficient for regularity of the crossed-product inclusion N⋊G⊆M⋊G, while…
· “Structural and Dynamical Properties of Subfactor Commuting Squares”
Positive matrix polynomials on compact real surfaces factor as B*B
Yields bivariate Fejér-Riesz factorization and settles two open sum-of-squares questions.
· “Addendum to "Factoring non-negative operator valued trigonometric polynomials in two variables"”
Texture persistence under Gaussian blur equals Besov regularity
A scale-derivative operator turns multiscale texture responses into an equivalent norm on Sobolev and Besov spaces.
· “The pointwise multiscale texture operator: analytical foundations and functional characterization”
This paper fully describes the spectrum of a family of infinite operators built from…
For each alpha greater than 1, the Bergman-space Dirichlet series operator T_alpha has continuous spectrum [0,Lambda_alpha], no embedded…
· “Helson Forms and Integral Operators on Bergman Spaces of Dirichlet Series”
This paper proves that a Toeplitz operator on the Bergman space is invertible whenever…
For h in the disk algebra, the Bergman-space Toeplitz operator with symbol \bar z + h is invertible if |\bar z + h| is bounded below on the…
· “The Douglas question for functions of the form overline{z}+h with h in the disk algebra”
One trichotomy rules all ways of observing chaos
Absolute and distributional chaos divide generically across all observation schemes.
Singular spectra shift at almost every coupling strength
A generalized spectral-averaging identity extends the rank-one spectral-shift phenomenon to infinite rank.
· “Analysis of the Singular Spectrum for General Perturbations”
Fixed points break shadowing
On Schwartz space, the symbol's fixed points and derivative growth decide whether its operator shadows or expands.
· “Linear dynamics of composition operators on Schwartz spaces”
Global variational memory must be time-reversal complete
If a finite-memory map is a global potential gradient, every active lag forces its mirror lag to act.
No sensing design beats a rank-only stability floor
Rank-one Gaussian sensors attain it; a convex recovery rule stays stable at optimal sampling scale.
· “Optimal Condition Numbers in Low-Rank Positive Semidefinite Matrix Sensing”
Two-slope map proves type (γ,c) doesn't force type (γ)
For every c0 in (0,1), a linear φ makes the example pass at c0 and fail at another c*, so the classes differ.
· “An Affirmative Answer to Question 4 of Jachymski Concerning Mappings of Type (γ,c)”
L2 bound plus L log L force weak L1 for averaged rough integrals
Criterion: once the un-averaged operator is L2-bounded with the right constant, the averaged operator is weak type (1,1).
· “A criterion on weak type (1,1) bound of rough singular integrals”
Moore-Penrose inverse yields new numerical-radius bounds
A family of Cauchy-Schwarz-type estimates sharpens norm and numerical-radius inequalities for Hilbert-space operators.
· “Inner product bounds via the Moore-Penrose inverse with applications”
A third-order ODE now governs the extremal function in every dimension
Interpolation over the extremizer's zeros yields the equation and closes the even-dimension gap.
· “The H\"ormander--Bernhardsson function in higher dimensions”
Banach-space algorithm hits best proximity points strongly
No φ-p property needed: the shrinking projection scheme converges to the generalized projection of the starting point.
· “On φ-Best Proximity Points and Proximal-type Algorithms in Banach Spaces”
Orthonormal Strichartz endpoint estimate proved for d ≥ 2
Restricted-type Schatten bound settles the long-open endpoint conjecture and its transport analogue.
ℓ_p projection volumes move quadratically away from extremes
Near any extremal hyperplane, the ℓ_p-ball projection ratio deviates by at least c_p times squared distance.
· “Stability of extremal hyperplane projections of balls in ell_p^n(mathbb{R})”
q-Berezin sectorial class is strictly larger than q-sectorial
Hardy-space examples show the new class is strictly larger, which translates into sharper q-Berezin estimates.
Rational points near Brownian paths follow a Q^{5/2} law
Almost surely, the count matches the tube area times Q^3, except at one critical thickness.
A local spectral dip forces invariant subspaces
A new asymptotic quantity spots when an operator must contain a nontrivial hyperinvariant subspace.
· “Asymptotic Numerical Ranges and Invariant Subspaces of Operators”
Maximal right ideals reduce to a single row-operator test
The row-operator test is the exact obstruction behind the Dales–Żelazko conjecture for B(E).
· “Maximal right ideals of the Banach algebra of bounded operators on a Banach space”
Bourgain-Rosenthal-Schechtman spaces have the SHAI property
Every surjective homomorphism on their operator algebra is automatically injective, settling a 2022 question.
· “On the SHAI property of the Bourgain-Rosenthal-Schechtman spaces”
Quadratic-cubic unitaries generate B(H) when eigenspaces are large
Two quadratics never suffice; one quadratic and one cubic do, under a precise eigenspace-size condition.
· “Fully non-zero matrices and generators of full algebras of operators”
Shadowing conjecture resolved: false in Banach spaces
The answer to the open problem depends on whether surjectivity implies right invertibility.
· “Resolving the generalized hyperbolicity conjecture for shadowing”
AR-frames cut reconstruction into level-wise blocks
Weighted multiscale frames get optimal block preconditioners and provable stability under small perturbations.
· “Adaptive resolution frames: A multilevel framework in Hilbert spaces”
Operators that are minimum attaining on reducing subspaces are dense
Every bounded operator on a separable Hilbert space is a norm limit of such operators.
· “Minimum attaining operators on reducing subspaces: Spectral structure and density”
Convex domains pin Dirichlet-to-Neumann shifts to square-root law
Sharp bound: eigenvalue gaps never exceed sqrt(Lambda2 - Lambda1); counterexamples show geometry is essential.
Coherent-state series converge when the states are dense
Projections onto finite coherent-state spans recover any function in norm and pointwise.
Symmetric spaces on finite and infinite factors are isomorphic
Kruglov-bounded symmetric spaces over the hyperfinite II_1 and II_infty factors turn out to be linearly isomorphic.
· “Isomorphisms between symmetric spaces over infinite and finite von Neumann algebras”
Exact norm formula answers 1984 multiplication-operator question
For a factor with a trace, axb lands in any symmetric operator ideal exactly when the singular-value product does, and the norms match.
· “Norms of multiplication operators: answering Fialkow--Loebl question”
Unique solutions for accretive evolutions with moving constraints
The same viability proof also yields periodic solutions, comparison principles, and reaction-diffusion bounds.
· “Existence and qualitative theory for nonlinear accretive evolutions with Carath\'eodory forcing”
Continuous P-contractive maps on compact spaces are Picard
A short compactness argument shows every orbit converges to the unique fixed point, settling an open question.
Weaker contractions suffice for multivariate fractal interpolation
Matkowski and Rakotch maps still yield fractal interpolants, alpha-fractal functions, and invariant measures.
· “Generalized multivariate Fractal Interpolation Function and α-Fractal Function”
Dirichlet partial-sum norms grow at least as fast as log N
Combined with a known upper bound, this pins the operator's growth at exactly logarithmic order.
Maximal pure states force maximality in a new Choquet-type order
The amendment recovers the tight-extension rigidity theorem and matches the classical Choquet order in the commutative case.
· “Choquet-Type Relations and a State Space Level Amendment of Arveson's Hyperrigidity Conjecture”
Multi-Toeplitz spectra turn atomic on the d-letter Fock space
For T=L1+L1* the limiting eigenvalues are weighted tridiagonal atoms, matching Bernoulli random matrices.
· “Limiting eigenvalue distribution and entropy of multi-Toeplitz matrices”
Subgradient-flow errors equal their gap estimators
Initial datum fixes both optimal values, so one combined gap splits into two exact error identities.
One DLSS semigroup fits every nonnegative L² density
The square-root form of the DLSS equation has a unique maximal monotone extension, giving stable solutions in all dimensions.
Commutator estimates hold exactly for p ≤ 2
For 1≤p≤2, φ(A,B) is controlled by input commutators; for p>2 even finite-rank counterexamples block any such bound.
· “Commutator estimates for functions of noncommuting self-adjoint operators”
Two 2x2 matrices refute a Schatten norm conjecture
Explicit rank-one pair beats the conjectured constant at p=3/2; sharp bounds survive for p>=2.
· “A Counterexample to the Tang Zhang Schatten Norm Conjecture and Sharp Positive Results”
Cube-local weights prove two-weight Poincaré–Sobolev inequality
A dyadic seminorm on each cube controls the v-weighted oscillation by the w-weighted gradient in L^p, for p<q and p>q.
· “Generalization for Poincar\'e--Sobolev inequalities with local weights”
Superreflexive spans of finite-valued functions are finite-dimensional
This rigidity turns regular-language recognition into a geometric property of a canonically associated Banach space.
One orbit vector frames every exponential-solvable representation
For every ε, frame bounds (1−ε)^2 and (1+ε)^2; abelian quotients yield an orthonormal basis.
· “Near-Parseval orbit frames for irreducible unitary representations: from mixing and expansion”
λ-mean transform can expand spectral radius
On an alternating-weight shift, the transform's spectral radius exceeds T's, so the numerical-radius inequality has no spectral twin.
· “Operator Inequalities and Several Characterizations of the λ-Mean Transform”
A continuous digit map to A2-fractions is singular
The function pairs binary digits into powers of 1/2, is strictly decreasing, and has derivative zero almost everywhere.
· “A Continuous digit projector from binary representations of numbers onto A₂-representation”
Min/Max cones: Loewner convexity forces smoothness
A pure order inequality on these structured matrix cones rules out kinks and shows exactly which powers preserve convexity.
· “Entrywise Loewner Preservers on Min and Max Matrix Cones”
Averaging unitaries yields quantum channels on full state spaces
Construction works for non-normal states, extending Schrödinger-picture channels beyond finite dimensions.
· “Quantum channels on duals of von Neumann algebras in the Schr\"{o}dinger picture”
Moore-Penrose inverses sharpen commutator radius bounds
A new factor built from the operator's real and imaginary parts replaces the older ω_A factor in the 2√2 bound.
· “New perspectives on operator radius bounds in A-weighted frameworks”
Individual eventual positivity yields L^p exponential decay
Whenever the generator's spectrum lies in the left half-plane, the semigroup decays exponentially despite vector-dependent positivity times.
· “Stability of Individually Eventually Positive Semigroups on L^p-Spaces”
Unconditional bases give monomial bases for entire functions
A degree-by-degree bound n+1 makes every expansion converge uniformly on compact sets.
· “Monomial bases for holomorphic functions on Banach spaces with an unconditional basis”
Modulus isometries of vector lattices must be affine
A single modulus equality forces linearity, and order-preserving isometries preserve sup and inf.
König constant equals 1, matching Fourier bound
An explicit construction reaches the upper bound, closing a proposed route to the exact Grothendieck constant.