Spectral functions of a regular symmetric difference operator

1979 ◽  
Vol 25 (2) ◽  
pp. 130-134
Author(s):  
G. M. Il'mushkin
Keyword(s):  
Difference Operator ◽  
Symmetric Difference ◽  
Spectral Functions

scholarly journals VC Dimension and a Union Theorem for Set Systems

10.37236/8288 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Stijn Cambie ◽  
António Girão ◽  
Ross J. Kang
Keyword(s):  
Difference Operator ◽  
Negative Answer ◽  
Symmetric Difference ◽  
Vc Dimension ◽  
Compression Method ◽  
Set Systems ◽  
Theoretic Problem ◽  
Extremal Set ◽  
Positive Integers

Fix positive integers $k$ and $d$. We show that, as $n\to\infty$, any set system $\mathcal{A} \subset 2^{[n]}$ for which the VC dimension of $\{ \triangle_{i=1}^k S_i \mid S_i \in \mathcal{A}\}$ is at most $d$ has size at most $(2^{d\bmod{k}}+o(1))\binom{n}{\lfloor d/k\rfloor}$. Here $\triangle$ denotes the symmetric difference operator. This is a $k$-fold generalisation of a result of Dvir and Moran, and it settles one of their questions.  A key insight is that, by a compression method, the problem is equivalent to an extremal set theoretic problem on $k$-wise intersection or union that was originally due to Erdős and Frankl. We also give an example of a family $\mathcal{A} \subset 2^{[n]}$ such that the VC dimension of $\mathcal{A}\cap \mathcal{A}$ and of $\mathcal{A}\cup \mathcal{A}$ are both at most $d$, while $\lvert \mathcal{A} \rvert = \Omega(n^d)$. This provides a negative answer to another question of Dvir and Moran.

scholarly journals A Sauer-Shelah-Perles Lemma for Sumsets

10.37236/7945 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Zeev Dvir ◽  
Shay Moran
Keyword(s):  
Difference Operator ◽  
Symmetric Difference ◽  
Polynomial Method ◽  
Vc Dimension ◽  
Analogous Statement

We show that any family of subsets $A\subseteq 2^{[n]}$ satisfies $\lvert A\rvert \leq O\bigl(n^{\lceil{d}/{2}\rceil}\bigr)$, where $d$ is the VC dimension of $\{S\triangle T \,\vert\, S,T\in A\}$, and $\triangle$ is the symmetric difference operator. We also observe that replacing $\triangle$ by either $\cup$ or $\cap$ fails to satisfy an analogous statement. Our proof is based on the polynomial method; specifically, on an argument due to [Croot, Lev, Pach '17].

scholarly journals Low-energy behavior of spin-liquid electron spectral functions

2013 ◽  
Vol 87 (4) ◽  
Author(s):  
Evelyn Tang ◽  
Matthew P. A. Fisher ◽  
Patrick A. Lee
Keyword(s):  
Low Energy ◽  
Spin Liquid ◽  
Spectral Functions ◽  
Energy Behavior

scholarly journals Non-Debye Relaxations: Two Types of Memories and Their Stieltjes Character

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 477
Author(s):  
Katarzyna Górska ◽  
Andrzej Horzela
Keyword(s):  
Stochastic Processes ◽  
Spectral Function ◽  
Evolution Equations ◽  
Memory Function ◽  
Relaxation Phenomena ◽  
Stieltjes Function ◽  
Infinitely Divisible ◽  
Spectral Functions ◽  
Debye Relaxation ◽  
Relaxation Functions

In this paper, we show that spectral functions relevant for commonly used models of the non-Debye relaxation are related to the Stieltjes functions supported on the positive semi-axis. Using only this property, it can be shown that the response and relaxation functions are non-negative. They are connected to each other and obey the time evolution provided by integral equations involving the memory function M(t), which is the Stieltjes function as well. This fact is also due to the Stieltjes character of the spectral function. Stochastic processes-based approach to the relaxation phenomena gives the possibility to identify the memory function M(t) with the Laplace (Lévy) exponent of some infinitely divisible stochastic processes and to introduce its partner memory k(t). Both memories are related by the Sonine equation and lead to equivalent evolution equations which may be freely interchanged in dependence of our knowledge on memories governing the process.

scholarly journals Solving the 2D SUSY Gross-Neveu-Yukawa model with conformal truncation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
A. Liam Fitzpatrick ◽  
Emanuel Katz ◽  
Matthew T. Walters ◽  
Yuan Xin
Keyword(s):  
Phase Transition ◽  
Critical Point ◽  
Fine Tuning ◽  
Yukawa Model ◽  
Spectral Functions ◽  
Scalar Superfield ◽  
Local Operators ◽  
Dimensionless Coupling ◽  
Zumino Model ◽  
Yukawa Theory

Abstract We use Lightcone Conformal Truncation to analyze the RG flow of the two-dimensional supersymmetric Gross-Neveu-Yukawa theory, i.e. the theory of a real scalar superfield with a ℤ2-symmetric cubic superpotential, aka the 2d Wess-Zumino model. The theory depends on a single dimensionless coupling $$ \overline{g} $$ g ¯ , and is expected to have a critical point at a tuned value $$ {\overline{g}}_{\ast } $$ g ¯ ∗ where it flows in the IR to the Tricritical Ising Model (TIM); the theory spontaneously breaks the ℤ2 symmetry on one side of this phase transition, and breaks SUSY on the other side. We calculate the spectrum of energies as a function of $$ \overline{g} $$ g ¯ and see the gap close as the critical point is approached, and numerically read off the critical exponent ν in TIM. Beyond the critical point, the gap remains nearly zero, in agreement with the expectation of a massless Goldstino. We also study spectral functions of local operators on both sides of the phase transition and compare to analytic predictions where possible. In particular, we use the Zamolodchikov C-function to map the entire phase diagram of the theory. Crucial to this analysis is the fact that our truncation is able to preserve supersymmetry sufficiently to avoid any additional fine tuning.

scholarly journals Disentangling QCD and new physics in D → πℓ+ℓ−

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Aoife Bharucha ◽  
Diogo Boito ◽  
Cédric Méaux
Keyword(s):  
Standard Model ◽  
Branching Ratio ◽  
Branching Ratios ◽  
New Physics ◽  
Operator Product ◽  
Beyond The Standard Model ◽  
Spectral Functions ◽  
The Standard Model ◽  
Model Independent ◽  
Weak Annihilation

Abstract In this paper we consider the decay D+ → π+ℓ+ℓ−, addressing in particular the resonance contributions as well as the relatively large contributions from the weak annihilation diagrams. For the weak annihilation diagrams we include known results from QCD factorisation at low q2 and at high q2, adapting the existing calculation for B decays in the Operator Product Expansion. The hadronic resonance contributions are obtained through a dispersion relation, modelling the spectral functions as towers of Regge-like resonances in each channel, as suggested by Shifman, imposing the partonic behaviour in the deep Euclidean. The parameters of the model are extracted using e+e− → (hadrons) and τ → (hadrons) + ντ data as well as the branching ratios for the resonant decays D+ → π+R(R → ℓ+ℓ−), with R = ρ, ω, and ϕ. We perform a thorough error analysis, and present our results for the Standard Model differential branching ratio as a function of q2. Focusing then on the observables FH and AFB, we consider the sensitivity of this channel to effects of physics beyond the Standard Model, both in a model independent way and for the case of leptoquarks.

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