New Composite Mathematics Class 3 Pdf Free Download
Jeana Lemasters
A CompositeGate object contains a set of inner gates acting on a small set of qubits, and a mapping from this small set of qubits to the qubits of the circuit that contains the composite gate. The CompositeGate object fulfills the purpose of a subfunction in classical programming, where a set of inner gates can be packaged as a subcircuit to be used to construct an outer quantum circuit.
Name of the composite gate, specified as a string scalar. If you do not specify the name of the composite gate, the default value of this property is an empty string, "". Otherwise, the Name property value must start with a letter, followed by letters, digits, or underscores (with no white space).
When you construct a composite gate from an existing quantum circuit using the compositeGate function, the Name property of the circuit is copied to the Name property of the composite gate (unless you specify a new name when using compositeGate). This name is used in the plot of the composite gate and the function name in the generated QASM code.
Target qubits of the outer circuit containing the composite gate, returned as a numeric scalar or numeric vector of qubit indices. Each qubit of the inner gates in the Gates property is mapped to a qubit of an outer circuit containing the composite gate through the TargetQubits vector.
Create an outer circuit that contains two composite gates constructed from this inner "bell" circuit. The first composite gate acts on qubits 1 and 3 of the outer circuit containing this gate. The second composite gate acts on qubits 2 and 4 of the outer circuit containing this gate.
In a circuit diagram, each solid horizontal line represents a qubit. The top line is a qubit with index 1 and the remaining lines from top to bottom are labeled sequentially. In this example, the plotted outer circuit consists of four qubits with indices 1, 2, 3, and 4. The plot shows that qubits 1 and 3 of the outer circuit are mapped to qubits 1 and 2 of the inner circuit of the first composite gate, and qubits 2 and 4 of the outer circuit are mapped to qubits 1 and 2 of the inner circuit of the second composite gate.
A wide class of regularization problems in machine learning and statistics employ a regularization term which is obtained by composing a simple convex function omega with a linear transformation. This setting includes Group Lasso methods, the Fused Lasso and other total variation methods, multi-task learning methods and many more. In this paper, we present a general approach for computing the proximity operator of this class of regularizers, under the assumption that the proximity operator of the function \omega is known in advance. Our approach builds on a recent line of research on optimal first order optimization methods and uses fixed point iterations for numerically computing the proximity operator. It is more general than current approaches and, as we show with numerical simulations, computationally more efficient than available first order methods which do not achieve the optimal rate. In particular, our method outperforms state of the art O(1/T) methods for overlapping Group Lasso and matches optimal O(1/T2) methods for the Fused Lasso and tree structured Group Lasso.
If $G$ has no element of composite order, then there are clearly no such elements $a,b$.
If $G$ has an element $g$ of composite order, then $g^-1$ is a different element of the same composite order, so you can use $g$ and $g^-1$.
So the question is, for which values of $n$ does every nonabelian group of order $n$ have an element of composite order?
I don't know.
Certainly it's true for $n=2^m$, $m\ge3$. But I don't even know whether every nonabelian group of order $36$ has an element of composite order. (Although it shouldn't be too hard to look this up somewhere, as there are tables of small groups on the web.)
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The RePEc plagiarism page A proximal bundle method for a class of nonconvex nonsmooth composite optimization problemsLiping Pang, Xiaoliang Wang (Obfuscate( '126.com', 'xliangwang' )) and Fanyun Meng
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Liping Pang: Dalian University of Technology, Key Laboratory for Computational Mathematics and Data Intelligence of Liaoning Province
Xiaoliang Wang: Zhejiang Sci-Tech University
Fanyun Meng: Qingdao University of TechnologyJournal of Global Optimization, 2023, vol. 86, issue 3, No 3, 589-620Abstract:Abstract In this paper, a proximal bundle method is proposed for a class of nonconvex nonsmooth composite optimization problems. The composite problem considered here is the sum of two functions: one is convex and the other is nonconvex. Local convexification strategy is adopted for the nonconvex function and the corresponding convexification parameter varies along iterations. Then the sum of the convex function and the extended function is dynamically constructed to approximate the primal problem. To choose a suitable cutting plane model for the approximation function, here we consider the sum of two cutting planes, which are designed respectively for the convex function and the extended function. By choosing appropriate descent condition, our method can keep track of the relationship between primal problem and approximate models. Under mild conditions, the convergence is proved and the accumulation point of iterations is a stationary point of the primal problem. Two polynomial problems and twelve DC (difference of convex) problems are referred in numerical experiments. The preliminary numerical results show that the proposed method is effective for solving these testing problems.Keywords: Proximal bundle method; Nonconvex and nonsmooth; Redistributed strategy; DC problems; 65K05; 90C26 (search for similar items in EconPapers)
Date: 2023
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We demonstrate that an array of discrete waveguides on a slab substrate, both featuring chi2 nonlinearity, supports stable solitons composed of discrete and continuous components. Two classes of fundamental composite soliton are identified: ones consisting of a discrete fundamental-frequency (FF) component in the waveguide array, coupled to a continuous second-harmonic (SH) component in the slab waveguide, and solitons with an inverted FF/SH structure. Twisted bound states of the fundamental solitons are found, too. In contrast with the usual systems, the intersite-centered fundamental solitons and bound states with the twisted continuous components are stable over almost the entire domain of their existence.
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