scholarly journals Universality of High-Strength Tensors

2021 ◽  
Author(s):  
Arthur Bik ◽  
Alessandro Danelon ◽  
Jan Draisma ◽  
Rob H. Eggermont
Keyword(s):  
Homogeneous Polynomial ◽  
High Strength ◽  
Equivalence Classes ◽  
Dense Orbit ◽  
Linear Forms ◽  
Arbitrary Polynomial ◽  
Bounded Number ◽  
Polynomial Functors ◽  
Quasi Order

AbstractA theorem due to Kazhdan and Ziegler implies that, by substituting linear forms for its variables, a homogeneous polynomial of sufficiently high strength specialises to any given polynomial of the same degree in a bounded number of variables. Using entirely different techniques, we extend this theorem to arbitrary polynomial functors. As a corollary of our work, we show that specialisation induces a quasi-order on elements in polynomial functors, and that among the elements with a dense orbit there are unique smallest and largest equivalence classes in this quasi-order.

scholarly journals Real and Complex Rank for Real Symmetric Tensors with Low Ranks

Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Edoardo Ballico ◽  
Alessandra Bernardi
Keyword(s):  
Homogeneous Polynomial ◽  
Symmetric Tensors ◽  
Linear Forms ◽  
The Real ◽  
The Difference ◽  
Powers Of Linear Forms

We study the case of a real homogeneous polynomial whose minimal real and complex decompositions in terms of powers of linear forms are different. We prove that if the sum of the complex and the real ranks of is at most , then the difference of the two decompositions is completely determined either on a line or on a conic or two disjoint lines.

scholarly journals An Upper Bound for the Symmetric Tensor Rank of a Low Degree Polynomial in a Large Number of Variables

Geometry ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-2 ◽  
Author(s):  
E. Ballico
Keyword(s):  
Upper Bound ◽  
Homogeneous Polynomial ◽  
Symmetric Tensor ◽  
Tensor Rank ◽  
Degree Polynomial ◽  
Linear Forms ◽  
Symmetric Tensor Rank ◽  
Low Degree ◽  
Powers Of Linear Forms

Fix integers m≥5 and d≥3. Let f be a degree d homogeneous polynomial in m+1 variables. Here, we prove that f is the sum of at most d·⌈(m+dm)/(m+1)⌉d-powers of linear forms (of course, this inequality is nontrivial only if m≫d.)

scholarly journals Identifiability of homogeneous polynomials and Cremona transformations

2019 ◽  
Vol 2019 (757) ◽  
pp. 279-308 ◽  
Author(s):  
Francesco Galuppi ◽  
Massimiliano Mella
Keyword(s):  
Special Class ◽  
Homogeneous Polynomial ◽  
Projective Spaces ◽  
Homogeneous Polynomials ◽  
Additive Decomposition ◽  
Linear Forms ◽  
Cremona Transformations ◽  
General Polynomial ◽  
Powers Of Linear Forms

AbstractA homogeneous polynomial of degree d in {n+1} variables is identifiable if it admits a unique additive decomposition in powers of linear forms. Identifiability is expected to be very rare. In this paper we conclude a work started more than a century ago and we describe all values of d and n for which a general polynomial of degree d in {n+1} variables is identifiable. This is done by classifying a special class of Cremona transformations of projective spaces.

scholarly journals Orbits of homogeneous polynomials on Banach spaces

2020 ◽  
pp. 1-29
Author(s):  
RODRIGO CARDECCIA ◽  
SANTIAGO MURO
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Homogeneous Polynomial ◽  
Julia Sets ◽  
Accumulation Point ◽  
Dense Orbit ◽  
Homogeneous Polynomials ◽  
Infinite Dimensional

We study the dynamics induced by homogeneous polynomials on Banach spaces. It is known that no homogeneous polynomial defined on a Banach space can have a dense orbit. We show a simple and natural example of a homogeneous polynomial with an orbit that is at the same time $\unicode[STIX]{x1D6FF}$ -dense (the orbit meets every ball of radius $\unicode[STIX]{x1D6FF}$ ), weakly dense and such that $\unicode[STIX]{x1D6E4}\cdot \text{Orb}_{P}(x)$ is dense for every $\unicode[STIX]{x1D6E4}\subset \mathbb{C}$ that either is unbounded or has 0 as an accumulation point. Moreover, we generalize the construction to arbitrary infinite-dimensional separable Banach spaces. To prove this, we study Julia sets of homogeneous polynomials on Banach spaces.

scholarly journals Van der Waerden/Schrijver-Valiant like Conjectures and Stable (aka Hyperbolic) Homogeneous Polynomials: One Theorem for all

10.37236/790 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Leonid Gurvits
Keyword(s):  
Control Theory ◽  
Homogeneous Polynomial ◽  
Maximum Degree ◽  
Complex Numbers ◽  
Stochastic Matrices ◽  
Homogeneous Polynomials ◽  
Linear Forms ◽  
Doubly Stochastic ◽  
Stable Polynomials ◽  
Nonnegative Coefficients

Let $p$ be a homogeneous polynomial of degree $n$ in $n$ variables, $p(z_1,...,z_n) = p(Z) , Z \in C^{n}$. We call a such polynomial $p$ H-Stable if $p(z_1,...,z_n) \neq 0$ provided the real parts $Re(z_i) > 0, 1 \leq i \leq n$. This notion from Control Theory is closely related to the notion of Hyperbolicity used intensively in the PDE theory. The main theorem in this paper states that if $p(x_1,...,x_n)$ is a homogeneous H-Stable polynomial of degree $n$ with nonnegative coefficients; $deg_{p}(i)$ is the maximum degree of the variable $x_i$, $C_i = \min(deg_{p}(i),i)$ and $Cap(p) = \inf_{x_i > 0, 1 \leq i \leq n} {p(x_1,...,x_n)\over x_1 \cdots x_n}$ then the following inequality holds $${\partial^n\over\partial x_1...\partial x_n} p(0,...,0) \geq Cap(p) \prod_{2 \leq i \leq n} \bigg({C_i-1\over C_i}\bigg)^{C_{i}-1}.$$ This inequality is a vast (and unifying) generalization of the Van der Waerden conjecture on the permanents of doubly stochastic matrices as well as the Schrijver-Valiant conjecture on the number of perfect matchings in $k$-regular bipartite graphs. These two famous results correspond to the H-Stable polynomials which are products of linear forms. Our proof is relatively simple and "noncomputational"; it uses just very basic properties of complex numbers and the AM/GM inequality.

scholarly journals Almost prime solutions to diophantine systems of high rank

2016 ◽  
Vol 13 (06) ◽  
pp. 1491-1514 ◽  
Author(s):  
Ákos Magyar ◽  
Tatchai Titichetrakun
Keyword(s):  
High Rank ◽  
Expected Number ◽  
Number Of Solutions ◽  
Linear Forms ◽  
Bounded Number ◽  
Integral Forms ◽  
Diophantine System ◽  
Linearly Independent ◽  
Integer Solutions ◽  
Almost Prime

Let [Formula: see text] be a family of [Formula: see text] integral forms of degree [Formula: see text] and [Formula: see text] be a family of pairwise linearly independent linear forms in [Formula: see text] variables [Formula: see text]. We study the number of solutions [Formula: see text] to the diophantine system [Formula: see text] under the restriction that [Formula: see text] has a bounded number of prime factors for each [Formula: see text]. We show that the system [Formula: see text] that has the expected number of such “almost prime” solutions under similar conditions was established for existence of integer solutions by Birch.

Decomposition of the metastable beta titanium alloy Ti-15-3

1983 ◽  
Vol 41 ◽  
pp. 264-265
Author(s):  
Y. L. Chen ◽  
S. Fujlshiro
Keyword(s):  
Low Cost ◽  
High Strength ◽  
Alpha Phase ◽  
Thick Sheet ◽  
Omega Phase ◽  
Beta Titanium Alloy ◽  
Beta Titanium ◽  
Beta Matrix ◽  
Metastable Beta ◽  
Very High

Metastable beta titanium alloys have been known to have numerous advantages such as cold formability, high strength, good fracture resistance, deep hardenability, and cost effectiveness. Very high strength is obtainable by precipitation of the hexagonal alpha phase in a bcc beta matrix in these alloys. Precipitation hardening in the metastable beta alloys may also result from the formation of transition phases such as omega phase. Ti-15-3 (Ti-15V- 3Cr-3Al-3Sn) has been developed recently by TIMET and USAF for low cost sheet metal applications. The purpose of the present study was to examine the aging characteristics in this alloy.The composition of the as-received material is: 14.7 V, 3.14 Cr, 3.05 Al, 2.26 Sn, and 0.145 Fe. The beta transus temperature as determined by optical metallographic method was about 770°C. Specimen coupons were prepared from a mill-annealed 1.2 mm thick sheet, and solution treated at 827°C for 2 hr in argon, then water quenched. Aging was also done in argon at temperatures ranging from 316 to 616°C for various times.

Effects of cooling rate on the mechanical properties of dual phase treated vanadium steels

1983 ◽  
Vol 41 ◽  
pp. 220-221
Author(s):  
L.J. Chen ◽  
H.C. Cheng ◽  
J.R. Gong ◽  
J.G. Yang
Keyword(s):  
Mechanical Properties ◽  
Cooling Rate ◽  
Automobile Industry ◽  
High Strength ◽  
Weight Percent ◽  
Low Alloy Steels ◽  
Dual Phase ◽  
Resource Requirement ◽  
Alloy Steels ◽  
Potential Weight

For fuel savings as well as energy and resource requirement, high strength low alloy steels (HSLA) are of particular interest to automobile industry because of the potential weight reduction which can be achieved by using thinner section of these steels to carry the same load and thus to improve the fuel mileage. Dual phase treatment has been utilized to obtain superior strength and ductility combinations compared to the HSLA of identical composition. Recently, cooling rate following heat treatment was found to be important to the tensile properties of the dual phase steels. In this paper, we report the results of the investigation of cooling rate on the microstructures and mechanical properties of several vanadium HSLA steels.The steels with composition (in weight percent) listed below were supplied by China Steel Corporation: 1. low V steel (0.11C, 0.65Si, 1.63Mn, 0.015P, 0.008S, 0.084Aℓ, 0.004V), 2. 0.059V steel (0.13C, 0.62S1, 1.59Mn, 0.012P, 0.008S, 0.065Aℓ, 0.059V), 3. 0.10V steel (0.11C, 0.58Si, 1.58Mn, 0.017P, 0.008S, 0.068Aℓ, 0.10V).

The Nature of Delta Phase Precipitation in Inconel 718

1982 ◽  
Vol 40 ◽  
pp. 724-725
Author(s):  
L. S. Lin ◽  
C. C. Law
Keyword(s):  
Inconel 718 ◽  
Base Alloy ◽  
High Strength ◽  
Physical Metallurgy ◽  
Nickel Base Alloy ◽  
Phase Precipitation ◽  
Weight Percentage ◽  
Delta Phase ◽  
The Subject ◽  
Nickel Base

Inconel 718, a precipitation hardenable nickel-base alloy, is a versatile high strength, weldable wrought alloy that is used in the gas turbine industry for components operated at temperatures up to about 1300°F. The nominal chemical composition is 0.6A1-0.9Ti-19.OCr-18.0Fe-3Mo-5.2(Cb + Ta)- 0.1C with the balance Ni (in weight percentage). The physical metallurgy of IN 718 has been the subject of a number of investigations and it is now established that hardening is due, primarily, to the formation of metastable, disc-shaped γ" an ordered body-centered tetragonal structure (DO2 2 type superlattice).

Physical, Chemical, and Crystallographic Characterization of Anodic Coatings on High Strength Aluminum Base Alloys

1976 ◽  
Vol 34 ◽  
pp. 636-637
Author(s):  
R. E. Herfert ◽  
N. T. McDevitt
Keyword(s):  
Sulfuric Acid ◽  
Salt Spray ◽  
High Strength ◽  
Environmental Stability ◽  
Anodic Films ◽  
Sodium Dichromate ◽  
Corrosion Resistant ◽  
Aerospace Applications ◽  
Bond Strengths ◽  
Adhesive Bonded Joints

Durability of adhesive bonded joints in moisture and salt spray environments is essential to USAF aircraft. Structural bonding technology for aerospace applications has depended for many years on the preparation of aluminum surfaces by a sulfuric acid/sodium dichromate (FPL etch) treatment. Recently, specific thin film anodizing techniques, phosphoric acid, and chromic acid anodizing have been developed which not only provide good initial bond strengths but vastly improved environmental durability. These thin anodic films are in contrast to the commonly used thick anodic films such as the sulfuric acid or "hard" sulfuric acid anodic films which are highly corrosion resistant in themselves, but which do not provide good initial bond strengths, particularly in low temperature peel.The objective of this study was to determine the characteristics of anodic films on aluminum alloys that make them corrosion resistant. The chemical composition, physical morphology and structure, and mechanical properties of the thin oxide films were to be defined and correlated with the environmental stability of these surfaces in humidity and salt spray. It is anticipated that anodic film characteristics and corrosion resistance will vary with the anodizing processing conditions.

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