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Improper clamping is one of the major causes of part deformation. Improving the fixture arrangement through force analysis of clamping points is an effective means to suppress or improve machining deformation. However, the existing research focuses on the monitoring and off-line optimization of the clamping point force, which has a certain lag on the machining deformation control, and it is difficult to predict the clamping point force due to the time-varying coupling effect of multiple factors such as process parameters, cutting force and clamping point force in the machining process. Inspired by the excellent performance of convolutional neural networks and gated recurrent networks in feature extraction and learning of temporal association laws, this paper proposes a CNN-GRU-based method for predicting the force state of clamping points under variable working conditions. Firstly, a force prediction model of clamping point during milling process with variable working conditions is established. Secondly, a convolutional neural network is designed to extract the features of dynamic coupled machining conditions. Then, a network of gated recurrent units is constructed to learn the temporal correlation law between the machining conditions and the forces on the clamping points to achieve force prediction of the clamping points during machining. Finally, it was verified by the milling process of the piston skirt. The results show that CNN-GRU can effectively predict the clamping force. In addition, CNN-GRU has higher computational efficiency and accuracy compared with CNN-LSTM, CNN-RNN and CNN-BP.

Furuta showed that if A ≥ B ≥ 0, then for each r ≥ 0 , f ( p ) = ( A r 2 B p A r 2 ) t + r p + r is decreasing for p ≥ t ≥ 0. Using this result, the following inequality ( C r 2 ( A B 2 A ) δ C r 2 ) p - 1 + r 4 δ + r ≤ C p - 1 + r is obtained for 0 p ≤ 1, r ≥ 1, ¼ ≤ δ ≤ 1 and three positive operators A, B, C satisfy ( A 1 2 B A 1 2 ) p 2 ≤ A p , ( B 1 2 A B 1 2 ) p 2 ≥ B p , ( C 1 2 A C 1 2 ) p 2 ≤ C p , ( A 1 2 C A 1 2 ) p 2 ≥ A p .

In this article, the authors study a generalized modulus of convexity, δα)(ɛ). Certain related geometrical properties of this modulus are analyzed. Their main result is that Banach space X has uniform normal structure if there exists ɛ, 0 ≤ ɛ ≤ 1, such that δ(α) (1 + ɛ) > (1-α) ɛ.

The approximate point spectrum properties of p-ω-hyponormal operators are given and proved. In fact, it is a generalization of approximate point spectrum properties of ω-hyponormal operators. The relation of spectra and numerical range of p-ω-hyponormal operators is obtained. On the other hand, for p-ω-hyponormal operators T, it is showed that if T is normal, then T is also normal.

Suppose H is a complex Hilbert space, AH (Δ) denotes the set of all analytic operator function on Δ, and the set NH (Δ)={f(z)/f(z) is an analytic operator function on the open unit disk Δ, f(z)f(w)=f(w)f(z), f*(z)f(z)=f(z)f*(z), ∀ z, w ≡ δ }. The note proves that if f(z)∈NH (Δ), (or AH (Δ)| f(z)|⩽1, ∀ z∈Δ then |f′(T)|⩽(1-|T|2)−1|I−f*(T)f(T)|1/2 |I−f(T)f*(T)|1/2 , where T∃−(H) (or T*T=TT*, respectively), |T

Suppose A, B and C are the bounded linear operators on a Hilbert space H, when A has a generalized inverse A− such that (AA−)*=AA− and B has a generalized inverse B− such that (B− B)*=B−B, the general characteristic forms for the critical points of the map Fp:X→‖AXB-C‖ p p (1

Let {X n,n≥1} be a sequence of π-mixing random variables,\(S(n) = \sum\limits_{i = 1}^n {X_i , 0 0,d=1 or −1. This note investigates that $$\sum\limits_{n = 1}^\infty {n^{q/p - 2} (\mathop {\sup (}\limits_{k \geqslant n} \left| {S(k)} \right|/(k(\varphi (k))^d )^{1/p} ) \geqslant \varepsilon )< \infty } $$ , where e is an arbitrary but fixed positive number. As application we give some results on the convergence rates for randomly indexed partial sums. Some known results are extended and generalized.