scholarly journals Sums of Linear Transformations in Higher Dimensions

2019 ◽  
Vol 70 (3) ◽  
pp. 965-984 ◽  
Author(s):  
Akshat Mudgal
Keyword(s):  
Natural Number ◽  
Linear Transformation ◽  
Finite Subset ◽  
Absolute Constant ◽  
Higher Dimensions ◽  
Dimensional Subspace ◽  
Main Term ◽  
Linear Transformations ◽  
One Dimensional

AbstractIn this paper, we prove the following two results. Let d be a natural number and q, s be co-prime integers such that 1<qs<|q|. Then there exists a constant δ>0 depending only on q, s and d such that for any finite subset A of ℝd that is not contained in a translate of a hyperplane, we have |q⋅A+s⋅A|≥(|q|+|s|+2d−2)|A|−Oq,s,d(|A|1−δ).The main term in this bound is sharp and improves upon an earlier result of Balog and Shakan. Secondly, let L∈GL2(ℝ) be a linear transformation such that L does not have any invariant one-dimensional subspace of ℝ2. Then, for all finite subsets A of ℝ2, we have |A+L(A)|≥4|A|−O(|A|1−δ), for some absolute constant δ>0. The main term in this result is sharp as well.

RIGHT SELF-INJECTIVE RINGS IN WHICH EVERY ELEMENT IS A SUM OF TWO UNITS

2007 ◽  
Vol 06 (02) ◽  
pp. 281-286 ◽  
Author(s):  
DINESH KHURANA ◽  
ASHISH K. SRIVASTAVA
Keyword(s):  
Vector Space ◽  
Linear Transformation ◽  
Classical Result ◽  
Linear Transformations ◽  
One Dimensional ◽  
Factor Ring

A classical result of Zelinsky states that every linear transformation on a vector space V, except when V is one-dimensional over ℤ2, is a sum of two invertible linear transformations. We extend this result to any right self-injective ring R by proving that every element of R is a sum of two units if no factor ring of R is isomorphic to ℤ2.

scholarly journals Difference sets in higher dimensions

2020 ◽  
pp. 1-14
Author(s):  
AKSHAT MUDGAL
Keyword(s):  
Natural Number ◽  
Absolute Constant ◽  
Difference Sets ◽  
Higher Dimensions ◽  
Image Position

Abstract Let d ≥ 3 be a natural number. We show that for all finite, non-empty sets $A \subseteq \mathbb{R}^d$ that are not contained in a translate of a hyperplane, we have $$\begin{equation*} |A-A| \geq (2d-2)|A| - O_d(|A|^{1- \delta}),\end{equation*}$$ where δ > 0 is an absolute constant only depending on d. This improves upon an earlier result of Freiman, Heppes and Uhrin, and makes progress towards a conjecture of Stanchescu.

scholarly journals DECOMPOSING ELEMENTS OF A RIGHT SELF-INJECTIVE RING

2013 ◽  
Vol 12 (06) ◽  
pp. 1350014 ◽  
Author(s):  
FEROZ SIDDIQUE ◽  
ASHISH K. SRIVASTAVA
Keyword(s):  
Vector Space ◽  
Linear Transformation ◽  
Division Ring ◽  
Linear Transformations ◽  
One Dimensional ◽  
Factor Ring ◽  
Theoretic Characterization

It was proved independently by both Wolfson [An ideal theoretic characterization of the ring of all linear transformations, Amer. J. Math.75 (1953) 358–386] and Zelinsky [Every linear transformation is sum of nonsingular ones, Proc. Amer. Math. Soc.5 (1954) 627–630] that every linear transformation of a vector space V over a division ring D is the sum of two invertible linear transformations except when V is one-dimensional over ℤ2. This was extended by Khurana and Srivastava [Right self-injective rings in which each element is sum of two units, J. Algebra Appl.6(2) (2007) 281–286] who proved that every element of a right self-injective ring R is the sum of two units if and only if R has no factor ring isomorphic to ℤ2. In this paper we prove that if R is a right self-injective ring, then for each element a ∈ R there exists a unit u ∈ R such that both a + u and a - u are units if and only if R has no factor ring isomorphic to ℤ2 or ℤ3.

scholarly journals Interconversion of Plasma Free Thyroxine Values from Assay Platforms with Different Reference Intervals Using Linear Transformation Methods

Biology ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 45
Author(s):  
Fanwen Meng ◽  
Jacqueline Jonklaas ◽  
Melvin Khee-Shing Leow
Keyword(s):  
Linear Transformation ◽  
Piecewise Linear ◽  
Equilibrium Dialysis ◽  
Accurate Method ◽  
Reference Intervals ◽  
Thyroid Stimulating Hormone ◽  
Free Thyroxine ◽  
Thyroid Function Tests ◽  
Linear Transformations ◽  
Transformation Methods

Clinicians often encounter thyroid function tests (TFT) comprising serum/plasma free thyroxine (FT4) and thyroid stimulating hormone (TSH) measured using different assay platforms during the course of follow-up evaluations which complicates reliable comparison and interpretation of TFT changes. Although interconversion between concentration units is straightforward, the validity of interconversion of FT4/TSH values from one assay platform to another with different reference intervals remains questionable. This study aims to establish an accurate and reliable methodology of interconverting FT4 by any laboratory to an equivalent FT4 value scaled to a reference range of interest via linear transformation methods. As a proof-of-concept, FT4 was simultaneously assayed by direct analog immunoassay, tandem mass spectrometry and equilibrium dialysis. Both linear and piecewise linear transformations proved relatively accurate for FT4 inter-scale conversion. Linear transformation performs better when FT4 are converted from a more accurate to a less accurate assay platform. The converse is true, whereby piecewise linear transformation is superior to linear transformation when converting values from a less accurate method to a more robust assay platform. Such transformations can potentially apply to other biochemical analytes scale conversions, including TSH. This aids interpretation of TFT trends while monitoring the treatment of patients with thyroid disorders.

scholarly journals On the eigenvalues of some class of pseudo-linear transformations

2007 ◽  
pp. 79-83
Author(s):  
Milica Andjelic
Keyword(s):  
Linear Transformation ◽  
Closed Field ◽  
Linear Transformations ◽  
Algebraically Closed Field ◽  
New Criterion

We develop a connection between the eigenvalues of a class of pseudo-linear transformation over a field K and the eigenvalues of a certain linear transformation. We give a new criterion for this class to be diagonalizable over algebraically closed field.

THE REGULAR PART OF A SEMIGROUP OF LINEAR TRANSFORMATIONS WITH RESTRICTED RANGE

2017 ◽  
Vol 103 (3) ◽  
pp. 402-419 ◽  
Author(s):  
WORACHEAD SOMMANEE ◽  
KRITSADA SANGKHANAN
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Dimensional Subspace ◽  
Finite Dimensional Subspace ◽  
Restricted Range ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Finite Dimensional ◽  
Idempotent Rank ◽  
Image Position

Let$V$be a vector space and let$T(V)$denote the semigroup (under composition) of all linear transformations from$V$into$V$. For a fixed subspace$W$of$V$, let$T(V,W)$be the semigroup consisting of all linear transformations from$V$into$W$. In 2008, Sullivan [‘Semigroups of linear transformations with restricted range’,Bull. Aust. Math. Soc.77(3) (2008), 441–453] proved that$$\begin{eqnarray}\displaystyle Q=\{\unicode[STIX]{x1D6FC}\in T(V,W):V\unicode[STIX]{x1D6FC}\subseteq W\unicode[STIX]{x1D6FC}\} & & \displaystyle \nonumber\end{eqnarray}$$is the largest regular subsemigroup of$T(V,W)$and characterized Green’s relations on$T(V,W)$. In this paper, we determine all the maximal regular subsemigroups of$Q$when$W$is a finite-dimensional subspace of$V$over a finite field. Moreover, we compute the rank and idempotent rank of$Q$when$W$is an$n$-dimensional subspace of an$m$-dimensional vector space$V$over a finite field$F$.

Correlation Technique for Estimating Traffic Speed from Cameras

2003 ◽  
Vol 1855 (1) ◽  
pp. 66-73 ◽  
Author(s):  
Todd N. Schoepflin ◽  
Daniel J. Dailey
Keyword(s):  
Coordinate System ◽  
Weather Conditions ◽  
Dimensional Subspace ◽  
Correlation Technique ◽  
Camera Model ◽  
One Dimensional ◽  
Congested Traffic ◽  
Adverse Weather Conditions ◽  
Adverse Weather ◽  
Favorable Weather

A new algorithm is presented for estimating speed from roadside cameras in uncongested traffic, congested traffic, favorable weather conditions, and adverse weather conditions. Individual vehicle lanes are identified and horizontal vehicle features are emphasized by using a gradient operator. The features are projected into a one-dimensional subspace and transformed into a linear coordinate system by using a simple camera model. A correlation technique is used to summarize the movement of features through a group of images and estimate mean speed for each lane of vehicles.

scholarly journals ON A GENERALIZATION OF SZEMERÉDI'S THEOREM

2006 ◽  
Vol 93 (3) ◽  
pp. 723-760 ◽  
Author(s):  
I. D. SHKREDOV
Keyword(s):  
Natural Number ◽  
Absolute Constant ◽  
Arithmetic Progressions ◽  
Two Dimensional

Let $N$ be a natural number and $A \subseteq [1, \dots, N]^2$ be a set of cardinality at least $N^2 / (\log \log N)^c$, where $c > 0$ is an absolute constant. We prove that $A$ contains a triple $\{(k, m), (k+d, m), (k, m+d) \}$, where $d > 0$. This theorem is a two-dimensional generalization of Szemerédi's theorem on arithmetic progressions.

Some Implications of Noncoaxiality in Finite Deformations

1973 ◽  
Vol 95 (2) ◽  
pp. 87-93
Author(s):  
T. C. Hsu
Keyword(s):  
Rigid Body ◽  
Linear Transformation ◽  
Metal Forming ◽  
Principal Axis ◽  
Pure Shear ◽  
Finite Deformations ◽  
Two Dimensional ◽  
Body Rotation ◽  
Linear Transformations ◽  
Rectangular Grids

Two-dimensional finite deformations are analyzed by factoring and multiplying the matrices of the linear transformations representing them. A general linear transformation consists of a pure shear, a uniform dilation, and a rigid-body rotation. Coaxiality is defined for finite deformations and its effect on the resultant distortion discussed. Tests for coaxiality are devised for use on rectangular grids which are often employed in metal forming research. Formulas are derived for the initial and final directions of the resultant major principal axis in both equal and unequal noncoaxial pure shears and, in particular, conditions are found for the constancy of distortion in the second deformation.

scholarly journals Linear Transformations Preserving the Strong $q$-log-convexity of Polynomials

10.37236/5168 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Bao-Xuan Zhu ◽  
Hua Sun
Keyword(s):  
Linear Transformation ◽  
Sufficient Condition ◽  
Linear Transformations

In this paper, we give a sufficient condition for the linear transformation preserving the strong $q$-log-convexity. As applications, we get some linear transformations (for instance, Morgan-Voyce transformation, binomial transformation, Narayana transformations of two kinds) preserving the strong $q$-log-convexity. In addition, our results not only extend some known results, but also imply the strong $q$-log-convexities of some sequences of polynomials.

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