Inconsistency of time-symmetry model

Can physical objects be in time-symmetry? Physical objects can only exist in a medium that has physical attributes, which means this medium is a type of energy. Is time energy? This article will show that time is not energy, and there is no possibility that physical objects could be in time-symmetry. Physical objects can only be in symmetry in the time-invariant space, in which they exist. In this perspective, time measured with clocks is the result of the observer’s measurement in the time-invariant space. The time-symmetry model is flawed.

Asymptotic pointwise error estimates for reconstructing shift-invariant signals with generators in a hybrid-norm space

Sampling and reconstruction of signals in a shift-invariant space are generally studied under the requirement that the generator is in a stronger Wiener amalgam space, and the error estimates are usually given in the sense of $L_{p,{1 / \omega }}$ L p , 1 / ω -norm. Since we often need to reflect the local characteristics of reconstructing error, the asymptotic pointwise error estimates for nonuniform and average sampling in a non-decaying shift-invariant space are discussed under the assumption that the generator is in a hybrid-norm space. Based on Lemma 2.1–Lemma 2.6, we first rewrite the iterative reconstruction algorithms for two kinds of average sampling functionals and prove their convergence. Then, the asymptotic pointwise error estimates are presented for two algorithms under the case that the average samples are corrupted by noise.

Frames associated with shift invariant spaces on positive half line

In this paper, we introduce the notion of Walsh shift-invariant space and present a unified approach to the study of shift-invariant systems to be frames in L2(ℝ+). We obtain a necessary condition and three sufficient conditions under which the Walsh shift-invariant systems constitute frames for L2(ℝ+). Furthermore, we discuss applications of our main results to obtain some known conclusions about the Gabor frames and wavelet frames on positive half line.

Inconsistency of Time-symmetry Model

Can physical objects be in time-symmetry? Physical objects can only exist in a medium that has physical attributes, which means this medium is a type of energy. Is time energy? This article will show that time is not energy and there is no possibility that physical objects could be in time-symmetry. Physical objects only can be in symmetry in the time-invariant space in which they exist. In this perspective time measured with clocks is the result of the observer’s measurement in the time-invariant space. The time-symmetry model is flawed.

Diffeomorphic approximation of Planar Sobolev Homeomorphisms in rearrangement invariant spaces

Let $\Omega\subseteq\mathcal{R}^2$ be a domain, let $X$ be a rearrangement invariant space and let $f\in W^{1}X(\Omega,\mathcal{R}^2)$ be a homeomorphism between $\Omega$ and $f(\Omega)$. Then there exists a sequence of diffeomorphisms $f_k$ converging to $f$ in the space $W^{1}X(\Omega,\mathcal{R}^2)$.

The primitive equations in the scaling-invariant space $$L^{\infty }(L^1)$$

Consider the primitive equations on $$\mathbb {R}^2\times (z_0,z_1)$$ R 2 × ( z 0 , z 1 ) with initial data a of the form $$a=a_1+a_2$$ a = a 1 + a 2 , where $$a_1 \in \mathrm{BUC}_\sigma (\mathbb {R}^2;L^1(z_0,z_1))$$ a 1 ∈ BUC σ ( R 2 ; L 1 ( z 0 , z 1 ) ) and $$a_2 \in L^\infty _\sigma (\mathbb {R}^2;L^1(z_0,z_1))$$ a 2 ∈ L σ ∞ ( R 2 ; L 1 ( z 0 , z 1 ) ) . These spaces are scaling-invariant and represent the anisotropic character of these equations. It is shown that for $$a_1$$ a 1 arbitrary large and $$a_2$$ a 2 sufficiently small, this set of equations admits a unique strong solution which extends to a global one and is thus strongly globally well posed for these data provided a is periodic in the horizontal variables. The approach presented depends crucially on mapping properties of the hydrostatic Stokes semigroup in the $$L^\infty (L^1)$$ L ∞ ( L 1 ) -setting. It can be seen as the counterpart of the classical iteration schemes for the Navier–Stokes equations, now for the primitive equations in the $$L^\infty (L^1)$$ L ∞ ( L 1 ) -setting.

Sign Retrieval in Shift-Invariant Spaces with Totally Positive Generator

We show that a real-valued function f in the shift-invariant space generated by a totally positive function of Gaussian type is uniquely determined, up to a sign, by its absolute values $$\{|f(\lambda )|: \lambda \in \Lambda \}$$ { | f ( λ ) | : λ ∈ Λ } on any set $$\Lambda \subseteq {\mathbb {R}}$$ Λ ⊆ R with lower Beurling density $$D^{-}(\Lambda )>2$$ D - ( Λ ) > 2 .We consider a totally positive function of Gaussian type, i.e., a function $$g \in L^2({\mathbb {R}})$$ g ∈ L 2 ( R ) whose Fourier transform factors as $$\begin{aligned} \hat{g}(\xi )= \int _{{\mathbb {R}}} g(x) e^{-2\pi i x \xi } dx = C_0 e^{- \gamma \xi ^2}\prod _{\nu =1}^m (1+2\pi i\delta _\nu \xi )^{-1}, \quad \xi \in {\mathbb {R}}, \end{aligned}$$ g ^ ( ξ ) = ∫ R g ( x ) e - 2 π i x ξ d x = C 0 e - γ ξ 2 ∏ ν = 1 m ( 1 + 2 π i δ ν ξ ) - 1 , ξ ∈ R , with $$\delta _1,\ldots ,\delta _m\in {\mathbb {R}}, C_0, \gamma >0, m \in {\mathbb {N}} \cup \{0\}$$ δ 1 , … , δ m ∈ R , C 0 , γ > 0 , m ∈ N ∪ { 0 } , and the shift-invariant space$$\begin{aligned} V^\infty (g) = \Big \{ f=\sum _{k \in {\mathbb {Z}}} c_k\, g(\cdot -k): c \in \ell ^\infty ({\mathbb {Z}}) \Big \}, \end{aligned}$$ V ∞ ( g ) = { f = ∑ k ∈ Z c k g ( · - k ) : c ∈ ℓ ∞ ( Z ) } , generated by its integer shifts within $$L^\infty ({\mathbb {R}})$$ L ∞ ( R ) . As a consequence of (1), each $$f \in V^\infty (g)$$ f ∈ V ∞ ( g ) is continuous, the defining series converges unconditionally in the weak$$^*$$ ∗ topology of $$L^\infty $$ L ∞ , and the coefficients $$c_k$$ c k are unique [6, Theorem 3.5].

Bijective model of time and J.T. Fraser’s Nowless Universe

In bijective modelling, the physical reality is represented by the set X, the model of physical reality by the set Y. Every element in the set X has exactly one correspondent element in the set Y. Set X and set X are related by the bijective function f:X→Y. Bijective modelling is confirming that time is the duration of given system entropy increasing in time-invariant space. Time-invariant space is the fundamental arena of the Nowless Universe.

From the space-time model to the time-invariant space model

Carlo Rovelli’s research on time suggests that time has no physical existence, that it is an illusion. Bijective research confirms Rovelli is right. Time is what we measure with clocks. We measure with clocks the numerical sequential order of material change, i.e. the motion running in time-invariant space. Time as the duration of change enters existence only when measured by the observer. The change runs only in time-invariant universal space. Humans are experiencing a run of changes in time-invariant space in the frame of the linear psychological time “past-present-future” that has its basis in the neurological activity of the brain. In the universe, there is neither a physical past nor physical future. There exists only what we can observe with our senses and measure with apparatuses.