Research articles

$q$ -bic forms

January 24, 2023

Mathematics

Number Theory

Algebraic Geometry

q

-bic form is a pairing

V \times V \to \mathbf{k}

that is linear in the second variable and

q

-power Frobenius linear in the first; here,

V

is a vector space over a field

\mathbf{k}

containing the finite field on

q^2

elements. ...

The Geometry of Rank Drop in a Class of Face-Splitting Matrix Products

January 24, 2023

| | |

Mathematics

Algebraic Geometry

Commutative Algebra

Combinatorics

Given

k \leq 6

points

(x_i,y_i) \in \mathbb{P}^2 \times \mathbb{P}^2

, we characterize rank deficiency of the

k \times 9

matrix

Z_k

with rows

x_i^\top \otimes y_i^\top

in terms of the geometry of the point configurations

\{x_i\}

...

Dimension drop of harmonic measure for some finite range random walks on Fuchsian Schottky groups

January 23, 2023

| | |

Mathematics

Geometric Topology

Probability

Geometric Topology

Probability

We prove that the harmonic measures of certain finite range random walks on Fuchsian Schottky groups, have dimension strictly smaller than the Hausdorff dimension of the corresponding limit set....

The Exact Solutions of Certain Linear Partial Difference Equations

January 23, 2023

| | |

Mathematics

Analysis of PDEs

Difference equations have many applications and play an important role in numerical analysis, probability, statistics, combinatorics, computer science, quantum consciousness, etc. We first prove that the partial differential equation is equ...

On K\"ahler Ricci shrinker surfaces

January 23, 2023

| | |

Mathematics

Differential Geometry

In this paper, we prove that any K\"ahler Ricci shrinker surface has bounded sectional curvature. Combining this estimate with earlier work by many authors, we provide a complete classification of all K\"ahler Ricci shrinker surfaces....

Eight-stage pseudo-symplectic Runge-Kutta methods of order (4, 8)

January 23, 2023

Mathematics

Computer Science

Numerical Analysis

Using simplifying assumptions that are related to the time reversal symmetry, a 1-dimensional family of 8-stage pseudo-symplectic Runge-Kutta methods of order (4, 8), i.e., methods of order 4 that preserve symplectic structure up to order 8...

Improved Hardness of Approximation for Geometric Bin Packing

January 23, 2023

| | |

Computer Science

Data Structures and Algorithms

Computational Complexity

Computational Geometry

Data Structures and Algorithms

Computational Complexity

Computational Geometry

The Geometric Bin Packing (GBP) problem is a generalization of Bin Packing where the input is a set of

d

-dimensional rectangles, and the goal is to pack them into unit

d

-dimensional cubes efficiently. It is NP-Hard to obtain a PTAS for ...

Hankel operators on $L^p(\mathbb{R}_+)$ and their $p$ -completely bounded multipliers

January 23, 2023

| | | | |

Mathematics

Functional Analysis

We show that for any

1<p<\infty

, the space

Hank_p(\mathbb{R}_+)\subseteq B(L^p(\mathbb{R}_+))

of all Hankel operators on

L^p(\mathbb{R}_+)

is equal to the

w^*

-closure of the linear span of the operators $\theta_u\colon L^p(\mathbb{R...

On the Convergence of the Gradient Descent Method with Stochastic Fixed-point Rounding Errors under the Polyak-Lojasiewicz Inequality

January 23, 2023

| | | | |

Statistics

Computer Science

Mathematics

Machine Learning

Numerical Analysis

Optimization and Control

Machine Learning

Numerical Analysis

Optimization and Control

When training neural networks with low-precision computation, rounding errors often cause stagnation or are detrimental to the convergence of the optimizers; in this paper we study the influence of rounding errors on the convergence of the ...

Spectrum and Fine Spectrum of Band Matrices Generated by Oscillatory Sequences

January 23, 2023

| | | | |

Mathematics

Spectral Theory

Functional Analysis

Spectral Theory

Functional Analysis

In this paper, a new class of band matrices is considered where the entries of each non-zero band form a sequence with two limit points. The compact perturbation technique is used to study the spectrum over the

\ell_{p}, (1<p<\infty)

sequ...

$q$ -bic forms

January 24, 2023

Mathematics

Number Theory

Algebraic Geometry

q

-bic form is a pairing

V \times V \to \mathbf{k}

that is linear in the second variable and

q

-power Frobenius linear in the first; here,

V

is a vector space over a field

\mathbf{k}

containing the finite field on

q^2

elements. ...

The Geometry of Rank Drop in a Class of Face-Splitting Matrix Products

January 24, 2023

| | |

Mathematics

Algebraic Geometry

Commutative Algebra

Combinatorics

Given

k \leq 6

points

(x_i,y_i) \in \mathbb{P}^2 \times \mathbb{P}^2

, we characterize rank deficiency of the

k \times 9

matrix

Z_k

with rows

x_i^\top \otimes y_i^\top

in terms of the geometry of the point configurations

\{x_i\}

...

Dimension drop of harmonic measure for some finite range random walks on Fuchsian Schottky groups

January 23, 2023

| | |

Mathematics

Geometric Topology

Probability

Geometric Topology

Probability

We prove that the harmonic measures of certain finite range random walks on Fuchsian Schottky groups, have dimension strictly smaller than the Hausdorff dimension of the corresponding limit set....

The Exact Solutions of Certain Linear Partial Difference Equations

January 23, 2023

| | |

Mathematics

Analysis of PDEs

On K\"ahler Ricci shrinker surfaces

January 23, 2023

| | |

Mathematics

Differential Geometry

Eight-stage pseudo-symplectic Runge-Kutta methods of order (4, 8)

January 23, 2023

Mathematics

Computer Science

Numerical Analysis

Improved Hardness of Approximation for Geometric Bin Packing

January 23, 2023

| | |

Computer Science

Data Structures and Algorithms

Computational Complexity

Computational Geometry

Data Structures and Algorithms

Computational Complexity

Computational Geometry

The Geometric Bin Packing (GBP) problem is a generalization of Bin Packing where the input is a set of

d

-dimensional rectangles, and the goal is to pack them into unit

d

-dimensional cubes efficiently. It is NP-Hard to obtain a PTAS for ...

Hankel operators on $L^p(\mathbb{R}_+)$ and their $p$ -completely bounded multipliers

January 23, 2023

| | | | |

Mathematics

Functional Analysis

We show that for any

1<p<\infty

, the space

Hank_p(\mathbb{R}_+)\subseteq B(L^p(\mathbb{R}_+))

of all Hankel operators on

L^p(\mathbb{R}_+)

is equal to the

w^*

-closure of the linear span of the operators $\theta_u\colon L^p(\mathbb{R...

On the Convergence of the Gradient Descent Method with Stochastic Fixed-point Rounding Errors under the Polyak-Lojasiewicz Inequality

January 23, 2023

| | | | |

Statistics

Computer Science

Mathematics

Machine Learning

Numerical Analysis

Optimization and Control

Machine Learning

Numerical Analysis

Optimization and Control

Spectrum and Fine Spectrum of Band Matrices Generated by Oscillatory Sequences

January 23, 2023

| | | | |

Mathematics

Spectral Theory

Functional Analysis

Spectral Theory

Functional Analysis

\ell_{p}, (1<p<\infty)

sequ...

qqq-bic forms

The Geometry of Rank Drop in a Class of Face-Splitting Matrix Products

Dimension drop of harmonic measure for some finite range random walks on Fuchsian Schottky groups

The Exact Solutions of Certain Linear Partial Difference Equations

On K\"ahler Ricci shrinker surfaces

Eight-stage pseudo-symplectic Runge-Kutta methods of order (4, 8)

Improved Hardness of Approximation for Geometric Bin Packing

Hankel operators on Lp(R+)L^p(\mathbb{R}_+)Lp(R+​) and their ppp-completely bounded multipliers

On the Convergence of the Gradient Descent Method with Stochastic Fixed-point Rounding Errors under the Polyak-Lojasiewicz Inequality

Spectrum and Fine Spectrum of Band Matrices Generated by Oscillatory Sequences

qqq-bic forms

The Geometry of Rank Drop in a Class of Face-Splitting Matrix Products

Dimension drop of harmonic measure for some finite range random walks on Fuchsian Schottky groups

The Exact Solutions of Certain Linear Partial Difference Equations

On K\"ahler Ricci shrinker surfaces

Eight-stage pseudo-symplectic Runge-Kutta methods of order (4, 8)

Improved Hardness of Approximation for Geometric Bin Packing

Hankel operators on Lp(R+)L^p(\mathbb{R}_+)Lp(R+​) and their ppp-completely bounded multipliers

On the Convergence of the Gradient Descent Method with Stochastic Fixed-point Rounding Errors under the Polyak-Lojasiewicz Inequality

Spectrum and Fine Spectrum of Band Matrices Generated by Oscillatory Sequences

$q$ -bic forms

Hankel operators on $L^p(\mathbb{R}_+)$ and their $p$ -completely bounded multipliers

$q$ -bic forms

Hankel operators on $L^p(\mathbb{R}_+)$ and their $p$ -completely bounded multipliers