diff --git "a/parse/dev/401LFvBGIb/401LFvBGIb_middle.json" "b/parse/dev/401LFvBGIb/401LFvBGIb_middle.json" new file mode 100644--- /dev/null +++ "b/parse/dev/401LFvBGIb/401LFvBGIb_middle.json" @@ -0,0 +1,28718 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 110, + 97, + 501, + 137 + ], + "lines": [ + { + "bbox": [ + 108, + 97, + 501, + 119 + ], + "spans": [ + { + "bbox": [ + 108, + 97, + 501, + 119 + ], + "score": 1.0, + "content": "Deep feedforward functionality by equilibrium-point", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 162, + 118, + 450, + 137 + ], + "spans": [ + { + "bbox": [ + 162, + 118, + 450, + 137 + ], + "score": 1.0, + "content": "control in a shallow recurrent network", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 259, + 176, + 354, + 219 + ], + "lines": [ + { + "bbox": [ + 258, + 176, + 355, + 187 + ], + "spans": [ + { + "bbox": [ + 258, + 176, + 355, + 187 + ], + "score": 1.0, + "content": "Anonymous Author(s)", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 283, + 186, + 328, + 199 + ], + "spans": [ + { + "bbox": [ + 283, + 186, + 328, + 199 + ], + "score": 1.0, + "content": "Affiliation", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 286, + 197, + 324, + 209 + ], + "spans": [ + { + "bbox": [ + 286, + 197, + 324, + 209 + ], + "score": 1.0, + "content": "Address", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 290, + 209, + 320, + 219 + ], + "spans": [ + { + "bbox": [ + 290, + 209, + 320, + 219 + ], + "score": 1.0, + "content": "email", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 283, + 248, + 328, + 261 + ], + "lines": [ + { + "bbox": [ + 281, + 247, + 331, + 263 + ], + "spans": [ + { + "bbox": [ + 281, + 247, + 331, + 263 + ], + "score": 1.0, + "content": "Abstract", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 91, + 274, + 469, + 428 + ], + "lines": [ + { + "bbox": [ + 141, + 274, + 470, + 287 + ], + "spans": [ + { + "bbox": [ + 141, + 274, + 470, + 287 + ], + "score": 1.0, + "content": "Recurrent neural network based machine learning systems are typically employed", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 285, + 470, + 298 + ], + "spans": [ + { + "bbox": [ + 141, + 285, + 470, + 298 + ], + "score": 1.0, + "content": "for their sequential functionality in handling time-varying signals, such as for", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 296, + 470, + 308 + ], + "spans": [ + { + "bbox": [ + 141, + 296, + 470, + 308 + ], + "score": 1.0, + "content": "speech processing. 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However, neurobiologists find recurrent connections in the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 142, + 307, + 470, + 319 + ], + "spans": [ + { + "bbox": [ + 142, + 307, + 470, + 319 + ], + "score": 1.0, + "content": "vision system and debate about equilibrium-point control in the motor system.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 318, + 470, + 331 + ], + "spans": [ + { + "bbox": [ + 141, + 318, + 470, + 331 + ], + "score": 1.0, + "content": "Thus, we need a deeper understanding of how recurrent dynamics can be exploited", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 140, + 328, + 470, + 342 + ], + "spans": [ + { + "bbox": [ + 140, + 328, + 470, + 342 + ], + "score": 1.0, + "content": "to attain combinational stable-input stable-output functionality. Here, we study", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 340, + 470, + 352 + ], + "spans": [ + { + "bbox": [ + 141, + 340, + 470, + 352 + ], + "score": 1.0, + "content": "how a simplified Cohen-Grossberg neural network model can realize combinational", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 142, + 351, + 470, + 363 + ], + "spans": [ + { + "bbox": [ + 142, + 351, + 470, + 363 + ], + "score": 1.0, + "content": "multi-input Boolean functionality. We place our problem within the discipline of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 142, + 362, + 471, + 375 + ], + "spans": [ + { + "bbox": [ + 142, + 362, + 471, + 375 + ], + "score": 1.0, + "content": "algebraic geometry, and solve a special case of it using piecewise-linear algebra.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 372, + 470, + 385 + ], + "spans": [ + { + "bbox": [ + 141, + 372, + 470, + 385 + ], + "score": 1.0, + "content": "We demonstrate a connectance-efficient realization of the parity function as a", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 383, + 469, + 395 + ], + "spans": [ + { + "bbox": [ + 141, + 383, + 469, + 395 + ], + "score": 1.0, + "content": "proof-of-concept. Small-scale systems of this kind can be easily built, say for", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 394, + 471, + 407 + ], + "spans": [ + { + "bbox": [ + 141, + 394, + 471, + 407 + ], + "score": 1.0, + "content": "hobby robotics, as a network of two-terminal devices of resistors and tunnel diodes.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 405, + 470, + 417 + ], + "spans": [ + { + "bbox": [ + 141, + 405, + 470, + 417 + ], + "score": 1.0, + "content": "Large-scale systems may be energy-efficiently built as an interconnected network", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 416, + 444, + 429 + ], + "spans": [ + { + "bbox": [ + 142, + 416, + 444, + 429 + ], + "score": 1.0, + "content": "of multi-electrode nanoclusters with non-monotonic transport mechanisms.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 13.5, + "bbox_fs": [ + 140, + 274, + 471, + 429 + ] + }, + { + "type": "title", + "bbox": [ + 91, + 457, + 207, + 473 + ], + "lines": [ + { + "bbox": [ + 87, + 456, + 209, + 475 + ], + "spans": [ + { + "bbox": [ + 87, + 456, + 209, + 475 + ], + "score": 1.0, + "content": "15 1 Introduction", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "index", + "bbox": [ + 90, + 487, + 505, + 608 + ], + "lines": [ + { + "bbox": [ + 90, + 487, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 90, + 489, + 100, + 499 + ], + "score": 1.0, + "content": "16", + "type": "text" + }, + { + "bbox": [ + 105, + 487, + 505, + 501 + ], + "score": 1.0, + "content": "Shallow recurrent neural networks are being investigated for more context-aware object recognition", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 90, + 500, + 99, + 510 + ], + "score": 1.0, + "content": "17", + "type": "text" + }, + { + "bbox": [ + 106, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "[25] and brain-like behaviour [23]. They can be more compact (by trading space for time) and are a", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 509, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 89, + 511, + 99, + 522 + ], + "score": 1.0, + "content": "18", + "type": "text" + }, + { + "bbox": [ + 105, + 509, + 505, + 522 + ], + "score": 1.0, + "content": "naturally robust alternative to deep neural networks (which are easily fooled by input perturbations", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 518, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 89, + 522, + 100, + 532 + ], + "score": 1.0, + "content": "19", + "type": "text" + }, + { + "bbox": [ + 104, + 518, + 505, + 535 + ], + "score": 1.0, + "content": "or transformations [18, 29, 1]) when the role of recurrent dynamics is not to produce time-varying", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 531, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 88, + 533, + 100, + 543 + ], + "score": 1.0, + "content": "20", + "type": "text" + }, + { + "bbox": [ + 105, + 531, + 505, + 544 + ], + "score": 1.0, + "content": "output but instead to produce transient (hidden) state-dynamics that facilitate deep, robust and", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 541, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 88, + 544, + 99, + 554 + ], + "score": 1.0, + "content": "21", + "type": "text" + }, + { + "bbox": [ + 105, + 541, + 506, + 555 + ], + "score": 1.0, + "content": "transformation-invariant fixed-input fixed-output functionality. To better engineer such dynamics,", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 552, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 89, + 555, + 100, + 565 + ], + "score": 1.0, + "content": "22", + "type": "text" + }, + { + "bbox": [ + 105, + 552, + 506, + 567 + ], + "score": 1.0, + "content": "we shall study equilibrium-point control, which can be defined as the process of steering to a target", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 564, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 88, + 566, + 100, + 576 + ], + "score": 1.0, + "content": "23", + "type": "text" + }, + { + "bbox": [ + 105, + 564, + 506, + 577 + ], + "score": 1.0, + "content": "in state-space by fixing the input signal, instead of driving it by a continuously varying input signal.", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 575, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 89, + 577, + 100, + 586 + ], + "score": 1.0, + "content": "24", + "type": "text" + }, + { + "bbox": [ + 105, + 575, + 505, + 587 + ], + "score": 1.0, + "content": "Historically, equilibrium-point control [14, 5] was first formulated to provide a plausible solution", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 585, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 89, + 588, + 100, + 598 + ], + "score": 1.0, + "content": "25", + "type": "text" + }, + { + "bbox": [ + 104, + 585, + 505, + 599 + ], + "score": 1.0, + "content": "to the degrees of freedom problem in motor control [3], that is, we mentally represent intermediate", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 596, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 89, + 599, + 100, + 608 + ], + "score": 1.0, + "content": "26", + "type": "text" + }, + { + "bbox": [ + 105, + 596, + 505, + 610 + ], + "score": 1.0, + "content": "destination points rather than a continuum of velocity information required to execute a movement.", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 613, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 89, + 614, + 99, + 624 + ], + "score": 1.0, + "content": "27", + "type": "text" + }, + { + "bbox": [ + 105, + 613, + 506, + 625 + ], + "score": 1.0, + "content": "Here, we shall focus on using equilibrium-point control to realize multi-input Boolean functionality,", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 624, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 89, + 626, + 100, + 636 + ], + "score": 1.0, + "content": "28", + "type": "text" + }, + { + "bbox": [ + 105, + 624, + 505, + 636 + ], + "score": 1.0, + "content": "in particular the parity function, which is a canonical proxy for nonlinear classification. Theoretical", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 635, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 89, + 637, + 99, + 646 + ], + "score": 1.0, + "content": "29", + "type": "text" + }, + { + "bbox": [ + 105, + 635, + 505, + 647 + ], + "score": 1.0, + "content": "results in circuit complexity are known already for realizing Boolean functionality out of feedforward", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 646, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 88, + 648, + 100, + 657 + ], + "score": 1.0, + "content": "30", + "type": "text" + }, + { + "bbox": [ + 105, + 646, + 505, + 658 + ], + "score": 1.0, + "content": "neural networks, with weighted-sum thresholded binary-output neurons [35]. It has been shown that", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 657, + 506, + 670 + ], + "spans": [ + { + "bbox": [ + 89, + 659, + 99, + 668 + ], + "score": 1.0, + "content": "31", + "type": "text" + }, + { + "bbox": [ + 105, + 657, + 144, + 670 + ], + "score": 1.0, + "content": "arbitrary", + "type": "text" + }, + { + "bbox": [ + 145, + 657, + 154, + 667 + ], + "score": 0.79, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 657, + 506, + 670 + ], + "score": 1.0, + "content": "-input Boolean functions can be realized in depth-3 feedforward networks with fewer", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 666, + 507, + 682 + ], + "spans": [ + { + "bbox": [ + 89, + 671, + 100, + 681 + ], + "score": 1.0, + "content": "32", + "type": "text" + }, + { + "bbox": [ + 104, + 666, + 144, + 682 + ], + "score": 1.0, + "content": "neurons", + "type": "text" + }, + { + "bbox": [ + 144, + 667, + 207, + 681 + ], + "score": 0.91, + "content": "( m = \\mathcal { O } ( 2 ^ { N / 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 666, + 268, + 682 + ], + "score": 1.0, + "content": "instead of the", + "type": "text" + }, + { + "bbox": [ + 268, + 668, + 298, + 681 + ], + "score": 0.92, + "content": "\\mathcal { O } ( 2 ^ { N } )", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 666, + 507, + 682 + ], + "score": 1.0, + "content": "in total required for depth-2). However, with the", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 679, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 88, + 682, + 100, + 691 + ], + "score": 1.0, + "content": "33", + "type": "text" + }, + { + "bbox": [ + 105, + 679, + 506, + 691 + ], + "score": 1.0, + "content": "advent of nanoelectronics, the size of an artificial neuron has been downscaled to such an extent", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 690, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 89, + 693, + 100, + 702 + ], + "score": 1.0, + "content": "34", + "type": "text" + }, + { + "bbox": [ + 105, + 690, + 506, + 703 + ], + "score": 1.0, + "content": "that it is rather the interconnect wiring that now occupies a greater area in chip design. Thus for a", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 701, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 89, + 703, + 100, + 713 + ], + "score": 1.0, + "content": "35", + "type": "text" + }, + { + "bbox": [ + 104, + 701, + 415, + 713 + ], + "score": 1.0, + "content": "fully-connected deep network, the area scales as the number of interconnects", + "type": "text" + }, + { + "bbox": [ + 415, + 701, + 471, + 713 + ], + "score": 0.94, + "content": "\\dot { m ^ { 2 } } = \\mathcal { \\bar { O } } ( 2 ^ { N } )", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 701, + 506, + 713 + ], + "score": 1.0, + "content": ". Such a", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 70, + 506, + 87 + ], + "spans": [ + { + "bbox": [ + 89, + 75, + 100, + 84 + ], + "score": 1.0, + "content": "36", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 107, + 72, + 136, + 85 + ], + "score": 0.92, + "content": "\\mathcal { O } ( 2 ^ { N } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 136, + 70, + 432, + 87 + ], + "score": 1.0, + "content": "scaling law was earlier obtained by Shannon [34] for realizing arbitrary", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 433, + 73, + 443, + 83 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 443, + 70, + 506, + 87 + ], + "score": 1.0, + "content": "-input Boolean", + "type": "text", + "cross_page": true + } + ], + "index": 0, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 89, + 86, + 99, + 95 + ], + "score": 1.0, + "content": "37", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "functions by an interconnection of input-controlled switches (or equivalently a feedforward network", + "type": "text", + "cross_page": true + } + ], + "index": 1, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 93, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 88, + 96, + 99, + 106 + ], + "score": 1.0, + "content": "38", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 93, + 506, + 108 + ], + "score": 1.0, + "content": "of 2-input Boolean gates). Thus, unless we employ higher-order neurons [16], we can say that", + "type": "text", + "cross_page": true + } + ], + "index": 2, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 105, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 88, + 108, + 99, + 118 + ], + "score": 1.0, + "content": "39", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 105, + 286, + 119 + ], + "score": 1.0, + "content": "a Shannon bottleneck limits the maximum", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 286, + 106, + 296, + 115 + ], + "score": 0.81, + "content": "N", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 297, + 105, + 506, + 119 + ], + "score": 1.0, + "content": "-input Boolean logic realizable in a given area by", + "type": "text", + "cross_page": true + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 114, + 506, + 131 + ], + "spans": [ + { + "bbox": [ + 88, + 118, + 100, + 128 + ], + "score": 1.0, + "content": "40", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 114, + 506, + 131 + ], + "score": 1.0, + "content": "(nanoscale) feedforward networks. We aim to circumvent this Shannon bottleneck by employing", + "type": "text", + "cross_page": true + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 88, + 129, + 99, + 139 + ], + "score": 1.0, + "content": "41", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "recurrent physical networks. 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An example of that kind is Harnack’s", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 384, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 89, + 385, + 100, + 394 + ], + "score": 1.0, + "content": "87", + "type": "text" + }, + { + "bbox": [ + 106, + 384, + 403, + 396 + ], + "score": 1.0, + "content": "curve theorem [19] which states that for a 2-D polynomial curve of degree", + "type": "text" + }, + { + "bbox": [ + 404, + 385, + 411, + 393 + ], + "score": 0.66, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 384, + 506, + 396 + ], + "score": 1.0, + "content": ", the maximum number", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 393, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 89, + 396, + 100, + 406 + ], + "score": 1.0, + "content": "88", + "type": "text" + }, + { + "bbox": [ + 105, + 393, + 221, + 407 + ], + "score": 1.0, + "content": "of connected components is", + "type": "text" + }, + { + "bbox": [ + 221, + 393, + 290, + 406 + ], + "score": 0.92, + "content": "( n ^ { 2 } - 3 n + 4 ) / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 393, + 506, + 407 + ], + "score": 1.0, + "content": ". Now, with the advent of computer algebra, the roots", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 89, + 407, + 100, + 416 + ], + "score": 1.0, + "content": "89", + "type": "text" + }, + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "score": 1.0, + "content": "of multivariate nonlinear equations are studied by the elimination of variables, using techniques", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 415, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 89, + 418, + 100, + 427 + ], + "score": 1.0, + "content": "90", + "type": "text" + }, + { + "bbox": [ + 104, + 415, + 506, + 429 + ], + "score": 1.0, + "content": "such as resultants [13, 38] and Groebner bases [7, 8] for polynomial systems, and as an instance", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 427, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 89, + 429, + 99, + 438 + ], + "score": 1.0, + "content": "91", + "type": "text" + }, + { + "bbox": [ + 106, + 427, + 505, + 439 + ], + "score": 1.0, + "content": "of the linear-complementarity problem [11] or equivalently as absolute-value equations [26] for", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 438, + 507, + 450 + ], + "spans": [ + { + "bbox": [ + 89, + 439, + 100, + 449 + ], + "score": 1.0, + "content": "92", + "type": "text" + }, + { + "bbox": [ + 105, + 438, + 507, + 450 + ], + "score": 1.0, + "content": "piecewise-linear systems [37]. However, computer algebra is not scalable for higher dimensions.", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 89, + 451, + 100, + 460 + ], + "score": 1.0, + "content": "93", + "type": "text" + }, + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "Thus there is a need to convey the richness in algebraic geometry using analytical expressions. While", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 459, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 89, + 461, + 100, + 471 + ], + "score": 1.0, + "content": "94", + "type": "text" + }, + { + "bbox": [ + 104, + 459, + 505, + 473 + ], + "score": 1.0, + "content": "it is unlikely that analytical expressions may be obtained for any general form of nonlinearity, we", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 470, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 89, + 473, + 100, + 482 + ], + "score": 1.0, + "content": "95", + "type": "text" + }, + { + "bbox": [ + 104, + 470, + 506, + 483 + ], + "score": 1.0, + "content": "may hope that the set of exactly solvable models can be extended well beyond linear equations, a", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 481, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 89, + 483, + 100, + 493 + ], + "score": 1.0, + "content": "96", + "type": "text" + }, + { + "bbox": [ + 105, + 481, + 506, + 494 + ], + "score": 1.0, + "content": "hope banking on our successful experience from other areas of mathematics such as integral calculus", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 493, + 325, + 505 + ], + "spans": [ + { + "bbox": [ + 89, + 494, + 100, + 504 + ], + "score": 1.0, + "content": "97", + "type": "text" + }, + { + "bbox": [ + 106, + 493, + 325, + 505 + ], + "score": 1.0, + "content": "[9, section IX] and iterated mappings [39, page 1098].", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + } + ], + "index": 12, + "bbox_fs": [ + 88, + 300, + 506, + 357 + ] + }, + { + "type": "index", + "bbox": [ + 89, + 361, + 506, + 504 + ], + "lines": [], + "index": 21, + "bbox_fs": [ + 89, + 362, + 507, + 505 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 93, + 523, + 241, + 535 + ], + "lines": [ + { + "bbox": [ + 90, + 522, + 243, + 537 + ], + "spans": [ + { + "bbox": [ + 90, + 522, + 243, + 537 + ], + "score": 1.0, + "content": "98 2.2.1 Piecewise-linear algebra", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 89, + 546, + 389, + 558 + ], + "lines": [ + { + "bbox": [ + 87, + 544, + 389, + 561 + ], + "spans": [ + { + "bbox": [ + 87, + 544, + 389, + 561 + ], + "score": 1.0, + "content": "99 In this paper, we commit to a piecewise-linear analysis by considering", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 87, + 544, + 389, + 561 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 185, + 562, + 425, + 596 + ], + "lines": [ + { + "bbox": [ + 185, + 562, + 425, + 596 + ], + "spans": [ + { + "bbox": [ + 185, + 562, + 425, + 596 + ], + "score": 0.94, + "content": "G _ { i } ( s ) = \\left\\{ \\begin{array} { l l } { g _ { i 1 } s } & { 0 \\leq s \\leq g _ { i 2 } } \\\\ { ( g _ { i 1 } + g _ { i 3 } ) g _ { i 2 } - g _ { i 3 } s } & { g _ { i 2 } \\leq s \\leq g _ { i 2 } ( 1 + \\frac { g _ { i 1 } } { g _ { i 3 } } ) } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "f81693dde742af29e7815bc0bfe1aecc8ce28ad059d980c040ebbe4f6f90bb93.jpg" + } + ] + } + ], + "index": 30.5, + "virtual_lines": [ + { + "bbox": [ + 185, + 562, + 425, + 579.0 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 185, + 579.0, + 425, + 596.0 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "index", + "bbox": [ + 87, + 600, + 504, + 635 + ], + "lines": [ + { + "bbox": [ + 86, + 599, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 86, + 603, + 100, + 612 + ], + "score": 1.0, + "content": "100", + "type": "text" + }, + { + "bbox": [ + 102, + 599, + 134, + 615 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 601, + 182, + 613 + ], + "score": 0.93, + "content": "g _ { i 1 , 2 , 3 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 599, + 214, + 615 + ], + "score": 1.0, + "content": "so that", + "type": "text" + }, + { + "bbox": [ + 215, + 601, + 227, + 612 + ], + "score": 0.9, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 599, + 438, + 615 + ], + "score": 1.0, + "content": "is a triangular peak function in a limited range of", + "type": "text" + }, + { + "bbox": [ + 438, + 603, + 444, + 611 + ], + "score": 0.76, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 599, + 506, + 615 + ], + "score": 1.0, + "content": ", thus defining", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 87, + 613, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 87, + 614, + 100, + 623 + ], + "score": 1.0, + "content": "101", + "type": "text" + }, + { + "bbox": [ + 105, + 613, + 505, + 624 + ], + "score": 1.0, + "content": "an idealized negative-differential behaviour. Shifting the state-space about its inflection points as", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 87, + 623, + 312, + 635 + ], + "spans": [ + { + "bbox": [ + 87, + 623, + 106, + 635 + ], + "score": 1.0, + "content": "102", + "type": "text" + }, + { + "bbox": [ + 106, + 624, + 155, + 635 + ], + "score": 0.91, + "content": "{ \\pmb z } \\equiv { \\pmb s } - { \\pmb g } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 623, + 312, + 635 + ], + "score": 1.0, + "content": "and then combining (5) with (4) yields", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + } + ], + "index": 33, + "bbox_fs": [ + 86, + 599, + 506, + 635 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 161, + 639, + 448, + 673 + ], + "lines": [ + { + "bbox": [ + 161, + 639, + 448, + 673 + ], + "spans": [ + { + "bbox": [ + 161, + 639, + 448, + 673 + ], + "score": 0.93, + "content": "x _ { i } = \\left\\{ \\begin{array} { l l } { - f _ { i , 1 : N } \\cdot z + g _ { i 1 } z _ { i } - f _ { i , 1 : N } \\cdot g _ { 2 } + g _ { i 1 } g _ { i 2 } } & { - g _ { i 2 } \\leq z _ { i } \\leq 0 } \\\\ { - f _ { i , 1 : N } \\cdot z - g _ { i 3 } z _ { i } - f _ { i , 1 : N } \\cdot g _ { 2 } + g _ { i 1 } g _ { i 2 } } & { 0 \\leq z _ { i } \\leq g _ { i 2 } ( \\frac { g _ { i 1 } } { g _ { i 3 } } ) } \\end{array} \\right. ,", + "type": "interline_equation", + "image_path": "668e4cbfa3a636747c9151b6e27c846a0972245bdd7b8b62f22bc4f8ef694add.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 161, + 639, + 448, + 650.3333333333334 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 161, + 650.3333333333334, + 448, + 661.6666666666667 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 161, + 661.6666666666667, + 448, + 673.0000000000001 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 87, + 683, + 214, + 695 + ], + "lines": [ + { + "bbox": [ + 85, + 683, + 214, + 696 + ], + "spans": [ + { + "bbox": [ + 85, + 683, + 214, + 696 + ], + "score": 1.0, + "content": "103 which can be simplified to", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38, + "bbox_fs": [ + 85, + 683, + 214, + 696 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 188, + 710, + 422, + 723 + ], + "lines": [ + { + "bbox": [ + 188, + 710, + 422, + 723 + ], + "spans": [ + { + "bbox": [ + 188, + 710, + 422, + 723 + ], + "score": 0.87, + "content": "x _ { i } = - 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1 } | ) < 1 . } \\end{array}", + "type": "interline_equation", + "image_path": "1f69dfe9f62137a2d66b3beea9c0ce4b9a56a68bb4f99ecc3730d64c11d60159.jpg" + } + ] + } + ], + "index": 7.5, + "virtual_lines": [ + { + "bbox": [ + 276, + 210, + 336, + 225.5 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 276, + 225.5, + 336, + 241.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 87, + 241, + 505, + 275 + ], + "lines": [ + { + "bbox": [ + 86, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 86, + 243, + 100, + 253 + ], + "score": 1.0, + "content": "108", + "type": "text" + }, + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "However, those are not yet sufficient conditions for a unique equilibrium-point solution for (2) and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 86, + 252, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 86, + 254, + 100, + 264 + ], + "score": 1.0, + "content": "109", + "type": "text" + }, + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "score": 1.0, + "content": "(10) because the bounds in (9) were not enforced. Thus, we shall proceed to obtain a Lyapunov", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 86, + 263, + 361, + 276 + ], + "spans": [ + { + "bbox": [ + 86, + 265, + 100, + 275 + ], + "score": 1.0, + "content": "110", + "type": "text" + }, + { + "bbox": [ + 105, + 263, + 361, + 276 + ], + "score": 1.0, + "content": "function to guarantee that a stable equilibrium-point is reached.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 90, + 294, + 275, + 309 + ], + "lines": [ + { + "bbox": [ + 87, + 293, + 276, + 313 + ], + "spans": [ + { + "bbox": [ + 87, + 293, + 276, + 313 + ], + "score": 1.0, + "content": "111 2.3 Lyapunov stability analysis", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 86, + 319, + 505, + 364 + ], + "lines": [ + { + "bbox": [ + 86, + 319, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 86, + 321, + 100, + 331 + ], + "score": 1.0, + "content": "112", + "type": "text" + }, + { + "bbox": [ + 105, + 319, + 505, + 333 + ], + "score": 1.0, + "content": "Equilibrium-point stability for large complex systems is not guaranteed in general [17, 27], and the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 86, + 330, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 86, + 333, + 100, + 342 + ], + "score": 1.0, + "content": "113", + "type": "text" + }, + { + "bbox": [ + 105, + 330, + 505, + 344 + ], + "score": 1.0, + "content": "effective dimensionality of stable-input stable-output responses is richly dependent on the parameter", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 86, + 342, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 86, + 344, + 100, + 353 + ], + "score": 1.0, + "content": "114", + "type": "text" + }, + { + "bbox": [ + 106, + 342, + 505, + 354 + ], + "score": 1.0, + "content": "space [2]. However, the interaction matrix for our physical system (2) is symmetric, and hence the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 86, + 352, + 347, + 366 + ], + "spans": [ + { + "bbox": [ + 86, + 354, + 100, + 364 + ], + "score": 1.0, + "content": "115", + "type": "text" + }, + { + "bbox": [ + 105, + 352, + 347, + 366 + ], + "score": 1.0, + "content": "system is a special case of the Cohen-Grossberg model [10]", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 366, + 378, + 402 + ], + "lines": [ + { + "bbox": [ + 233, + 366, + 378, + 402 + ], + "spans": [ + { + "bbox": [ + 233, + 366, + 378, + 402 + ], + "score": 0.95, + "content": "\\dot { s _ { i } } = a _ { i } ( s _ { i } ) [ b _ { i } ( s _ { i } ) - \\sum _ { j = 1 } ^ { N } c _ { i j } d _ { j } ( s _ { j } ) ] ,", + "type": "interline_equation", + "image_path": "b775b702ac77876668eb6203cdfc27e50d96ab3a943d6abcb7a1d80873c466ec.jpg" + } + ] + } + ], + "index": 17.5, + "virtual_lines": [ + { + "bbox": [ + 233, + 366, + 378, + 384.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 233, + 384.0, + 378, + 402.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 403, + 504, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 402, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 127, + 417 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 403, + 185, + 416 + ], + "score": 0.83, + "content": "a _ { i } ( s _ { i } ) = 1 / \\tau _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 402, + 190, + 417 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 190, + 403, + 279, + 416 + ], + "score": 0.86, + "content": "b _ { i } ( s _ { i } ) = x _ { i } - G _ { i } ( s _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 402, + 283, + 417 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 283, + 404, + 330, + 416 + ], + "score": 0.89, + "content": "c _ { i j } = - f _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 402, + 350, + 417 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 350, + 403, + 401, + 416 + ], + "score": 0.92, + "content": "d _ { j } ( s _ { j } ) = s _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 402, + 506, + 417 + ], + "score": 1.0, + "content": ". Thus, it is known to be", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 414, + 311, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 311, + 427 + ], + "score": 1.0, + "content": "globally absolute stable, with a Lyapunov function", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "interline_equation", + "bbox": [ + 197, + 428, + 415, + 517 + ], + "lines": [ + { + "bbox": [ + 197, + 428, + 415, + 517 + ], + "spans": [ + { + "bbox": [ + 197, + 428, + 415, + 517 + ], + "score": 0.93, + "content": "\\begin{array} { r l r } & { } & { V = - \\displaystyle \\sum _ { i } \\int _ { 0 } ^ { s _ { i } } b _ { i } ( u ) d _ { i } ^ { \\prime } ( u ) \\mathrm { d } u + \\displaystyle \\sum _ { i , j } \\frac { c _ { i j } } { 2 } d _ { i } ( s _ { i } ) d _ { j } ( s _ { j } ) } \\\\ & { } & { \\quad = \\displaystyle \\sum _ { i } \\Big ( P _ { i } ( s _ { i } ) - x _ { i } s _ { i } - \\displaystyle \\sum _ { j > i } f _ { i j } s _ { i } s _ { j } - \\frac { f _ { i i } } { 2 } s _ { i } ^ { 2 } \\Big ) , } \\\\ & { } & { \\quad \\mathrm { w h e r e ~ o u t p u t ~ p o w e r ~ } P _ { i } ( s _ { i } ) \\equiv \\displaystyle \\int _ { 0 } ^ { s _ { i } } G _ { i } ( u ) \\mathrm { d } u . } \\end{array}", + "type": "interline_equation", + "image_path": "617c1c10f7d08e612cb314f263d3e32087eea99241d887ae6e71dbbd2746283e.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 197, + 428, + 415, + 442.8333333333333 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 197, + 442.8333333333333, + 415, + 457.66666666666663 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 197, + 457.66666666666663, + 415, + 472.49999999999994 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 197, + 472.49999999999994, + 415, + 487.33333333333326 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 197, + 487.33333333333326, + 415, + 502.1666666666666 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 197, + 502.1666666666666, + 415, + 516.9999999999999 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 86, + 522, + 506, + 614 + ], + "lines": [ + { + "bbox": [ + 86, + 523, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 86, + 525, + 100, + 534 + ], + "score": 1.0, + "content": "116", + "type": "text" + }, + { + "bbox": [ + 105, + 523, + 506, + 537 + ], + "score": 1.0, + "content": "Alternatively, since our system (5) is piecewise-linear, a piecewise-quadratic Lyapunov function may", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 86, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 86, + 536, + 100, + 546 + ], + "score": 1.0, + "content": "117", + "type": "text" + }, + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "score": 1.0, + "content": "be obtained by a piecewise-affine system [22] analysis. While this approach is more powerful and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 86, + 545, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 86, + 547, + 100, + 556 + ], + "score": 1.0, + "content": "118", + "type": "text" + }, + { + "bbox": [ + 105, + 545, + 506, + 558 + ], + "score": 1.0, + "content": "holds even for asymmetric interaction matrices, it also seems to be analytically complex. From another", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 86, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 86, + 557, + 100, + 568 + ], + "score": 1.0, + "content": "119", + "type": "text" + }, + { + "bbox": [ + 105, + 555, + 462, + 568 + ], + "score": 1.0, + "content": "angle, global asymptotic stability [21, Theorem 3] is guaranteed if the Jacobian matrix", + "type": "text" + }, + { + "bbox": [ + 462, + 556, + 469, + 566 + ], + "score": 0.69, + "content": "\\mathbf { J }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 86, + 566, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 86, + 569, + 99, + 578 + ], + "score": 1.0, + "content": "120", + "type": "text" + }, + { + "bbox": [ + 106, + 567, + 306, + 581 + ], + "score": 0.89, + "content": "J _ { i i } \\dot { + } 1 \\dot { / } 2 \\sum _ { j \\neq i } \\dot { | } J _ { i j } + J _ { j i } | < 0 \\Longleftrightarrow G _ { i } ^ { \\prime } ( s _ { i } ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 566, + 394, + 581 + ], + "score": 1.0, + "content": "because in our system", + "type": "text" + }, + { + "bbox": [ + 395, + 567, + 505, + 580 + ], + "score": 0.92, + "content": "\\begin{array} { r } { J _ { i i } = - \\sum _ { j \\neq i } f _ { i j } - G _ { i } ^ { \\prime } ( s _ { i } ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 86, + 578, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 86, + 581, + 99, + 591 + ], + "score": 1.0, + "content": "121", + "type": "text" + }, + { + "bbox": [ + 105, + 578, + 124, + 594 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 581, + 165, + 592 + ], + "score": 0.91, + "content": "J _ { i j } = f _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 578, + 412, + 594 + ], + "score": 1.0, + "content": ". Since our network employs non-monotonic functionality,", + "type": "text" + }, + { + "bbox": [ + 412, + 580, + 461, + 592 + ], + "score": 0.92, + "content": "G _ { i } ^ { \\prime } ( s _ { i } ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 578, + 506, + 594 + ], + "score": 1.0, + "content": "cannot be", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 86, + 590, + 507, + 604 + ], + "spans": [ + { + "bbox": [ + 86, + 593, + 100, + 602 + ], + "score": 1.0, + "content": "122", + "type": "text" + }, + { + "bbox": [ + 105, + 590, + 241, + 604 + ], + "score": 1.0, + "content": "guaranteed for all reachable states", + "type": "text" + }, + { + "bbox": [ + 241, + 593, + 250, + 602 + ], + "score": 0.84, + "content": "s _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 590, + 507, + 604 + ], + "score": 1.0, + "content": ", and thus the above criteria is unfortunately inapplicable. Hence,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 86, + 602, + 324, + 614 + ], + "spans": [ + { + "bbox": [ + 86, + 604, + 100, + 613 + ], + "score": 1.0, + "content": "123", + "type": "text" + }, + { + "bbox": [ + 105, + 602, + 324, + 614 + ], + "score": 1.0, + "content": "we shall proceed with the Cohen-Grossberg approach.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 89, + 617, + 257, + 630 + ], + "lines": [ + { + "bbox": [ + 85, + 617, + 258, + 631 + ], + "spans": [ + { + "bbox": [ + 85, + 617, + 258, + 631 + ], + "score": 1.0, + "content": "124 The power function (21) simplifies to", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "interline_equation", + "bbox": [ + 182, + 631, + 428, + 677 + ], + "lines": [ + { + "bbox": [ + 182, + 631, + 428, + 677 + ], + "spans": [ + { + "bbox": [ + 182, + 631, + 428, + 677 + ], + "score": 0.94, + "content": "P ( s ) = \\left\\{ \\begin{array} { l l } { \\int _ { 0 } ^ { s } g _ { 1 } u \\mathrm { d } u = g _ { 1 } s ^ { 2 } / 2 } & { 0 \\le s \\le g _ { 2 } } \\\\ { g _ { 1 } g _ { 2 } ^ { 2 } / 2 + \\int _ { g _ { 2 } } ^ { s } ( g _ { 1 } + g _ { 3 } ) g _ { 2 } - g _ { 3 } u \\mathrm { d } u } & { g _ { 2 } \\le s } \\\\ { = g _ { 1 } s ^ { 2 } / 2 - g _ { \\oplus } ( s - g _ { 2 } ) ^ { 2 } , } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "800ffae7752736c21a4922a3289bf4e8558764f630301baeda6e026733c133e0.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 182, + 631, + 428, + 646.3333333333334 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 182, + 646.3333333333334, + 428, + 661.6666666666667 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 182, + 661.6666666666667, + 428, + 677.0000000000001 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 87, + 679, + 438, + 691 + ], + "lines": [ + { + "bbox": [ + 84, + 678, + 439, + 693 + ], + "spans": [ + { + "bbox": [ + 84, + 678, + 231, + 693 + ], + "score": 1.0, + "content": "125 and using the rectifier function", + "type": "text" + }, + { + "bbox": [ + 232, + 680, + 299, + 691 + ], + "score": 0.91, + "content": "[ x ) \\equiv \\operatorname* { m a x } ( x , 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 678, + 439, + 693 + ], + "score": 1.0, + "content": "may be expressed conveniently as", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "interline_equation", + "bbox": [ + 241, + 694, + 369, + 708 + ], + "lines": [ + { + "bbox": [ + 241, + 694, + 369, + 708 + ], + "spans": [ + { + "bbox": [ + 241, + 694, + 369, + 708 + ], + "score": 0.92, + "content": "P ( s ) = g _ { 1 } s ^ { 2 } / 2 - g _ { \\oplus } [ s - g _ { 2 } ) ^ { 2 } ,", + "type": "interline_equation", + "image_path": "2b9a2b8e0e8ce7e3a1917ab44ef8e3edf1e02c7ba6b658a55a75b75d1f60b808.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 241, + 694, + 369, + 708 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 87, + 711, + 304, + 723 + ], + "lines": [ + { + "bbox": [ + 85, + 710, + 304, + 724 + ], + "spans": [ + { + "bbox": [ + 85, + 710, + 304, + 724 + ], + "score": 1.0, + "content": "126 when the system is within its operational bounds.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 89, + 90, + 421, + 102 + ], + "lines": [ + { + "bbox": [ + 86, + 90, + 424, + 105 + ], + "spans": [ + { + "bbox": [ + 86, + 90, + 424, + 105 + ], + "score": 1.0, + "content": "105 The system in (7) can be expressed in the absolute-value equation normal form", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 86, + 90, + 424, + 105 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 276, + 104, + 335, + 118 + ], + "lines": [ + { + "bbox": [ + 276, + 104, + 335, + 118 + ], + "spans": [ + { + "bbox": [ + 276, + 104, + 335, + 118 + ], + "score": 0.91, + "content": "\\pmb { \\Delta z } - | \\pmb { z } | = \\pmb { b }", + "type": "interline_equation", + "image_path": "8f4de9fdca88c39df77d035b060d1144a921eda196796180f0fafca4f5e74d9b.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 276, + 104, + 335, + 118 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 262, + 135, + 346, + 149 + ], + "lines": [ + { + "bbox": [ + 262, + 135, + 346, + 149 + ], + "spans": [ + { + "bbox": [ + 262, + 135, + 346, + 149 + ], + "score": 0.72, + "content": "- g _ { 2 } \\leq z \\leq g _ { 2 } g _ { 1 } / g _ { 3 } ,", + "type": "interline_equation", + "image_path": "e19f9a3075e93e7f74935f07682d4d2b7a474b530b790b751a42958c8755cbc3.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 262, + 135, + 346, + 149 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 87, + 150, + 243, + 162 + ], + "lines": [ + { + "bbox": [ + 84, + 148, + 243, + 165 + ], + "spans": [ + { + "bbox": [ + 84, + 148, + 243, + 165 + ], + "score": 1.0, + "content": "107 Similarly, (2) can be expressed as", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3, + "bbox_fs": [ + 84, + 148, + 243, + 165 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 252, + 164, + 358, + 179 + ], + "lines": [ + { + "bbox": [ + 252, + 164, + 358, + 179 + ], + "spans": [ + { + "bbox": [ + 252, + 164, + 358, + 179 + ], + "score": 0.91, + "content": "\\tau \\dot { z } = g _ { \\oplus } ( b - { \\bf A } z + | z | ) .", + "type": "interline_equation", + "image_path": "6a9b05b23f47ef7ed37713a379d095ea37147c111bfef0e7f5f1a4e531b04813.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 252, + 164, + 358, + 179 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 186, + 504, + 209 + ], + "lines": [ + { + "bbox": [ + 106, + 185, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 185, + 505, + 199 + ], + "score": 1.0, + "content": "Two sufficient conditions are known for the absolute value equation (8) to have a unique solution", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 197, + 408, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 197, + 246, + 210 + ], + "score": 1.0, + "content": "based on the largest singular value", + "type": "text" + }, + { + "bbox": [ + 246, + 198, + 267, + 208 + ], + "score": 0.86, + "content": "\\sigma _ { \\mathrm { m i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 197, + 379, + 210 + ], + "score": 1.0, + "content": "[26] and the spectral radius", + "type": "text" + }, + { + "bbox": [ + 379, + 199, + 385, + 209 + ], + "score": 0.77, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 197, + 408, + 210 + ], + "score": 1.0, + "content": "[32]:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5, + "bbox_fs": [ + 106, + 185, + 505, + 210 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 276, + 210, + 336, + 241 + ], + "lines": [ + { + "bbox": [ + 276, + 210, + 336, + 241 + ], + "spans": [ + { + "bbox": [ + 276, + 210, + 336, + 241 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\sigma _ { \\operatorname* { m i n } } ( \\mathsf { \\pmb { A } } ) > 1 , } \\\\ { \\rho ( | \\mathsf { \\pmb { A } } ^ { - 1 } | ) < 1 . } \\end{array}", + "type": "interline_equation", + "image_path": "1f69dfe9f62137a2d66b3beea9c0ce4b9a56a68bb4f99ecc3730d64c11d60159.jpg" + } + ] + } + ], + "index": 7.5, + "virtual_lines": [ + { + "bbox": [ + 276, + 210, + 336, + 225.5 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 276, + 225.5, + 336, + 241.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "index", + "bbox": [ + 87, + 241, + 505, + 275 + ], + "lines": [ + { + "bbox": [ + 86, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 86, + 243, + 100, + 253 + ], + "score": 1.0, + "content": "108", + "type": "text" + }, + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "However, those are not yet sufficient conditions for a unique equilibrium-point solution for (2) and", + "type": "text" + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 252, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 86, + 254, + 100, + 264 + ], + "score": 1.0, + "content": "109", + "type": "text" + }, + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "score": 1.0, + "content": "(10) because the bounds in (9) were not enforced. Thus, we shall proceed to obtain a Lyapunov", + "type": "text" + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 263, + 361, + 276 + ], + "spans": [ + { + "bbox": [ + 86, + 265, + 100, + 275 + ], + "score": 1.0, + "content": "110", + "type": "text" + }, + { + "bbox": [ + 105, + 263, + 361, + 276 + ], + "score": 1.0, + "content": "function to guarantee that a stable equilibrium-point is reached.", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true + } + ], + "index": 10, + "bbox_fs": [ + 86, + 241, + 505, + 276 + ] + }, + { + "type": "title", + "bbox": [ + 90, + 294, + 275, + 309 + ], + "lines": [ + { + "bbox": [ + 87, + 293, + 276, + 313 + ], + "spans": [ + { + "bbox": [ + 87, + 293, + 276, + 313 + ], + "score": 1.0, + "content": "111 2.3 Lyapunov stability analysis", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "index", + "bbox": [ + 86, + 319, + 505, + 364 + ], + "lines": [ + { + "bbox": [ + 86, + 319, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 86, + 321, + 100, + 331 + ], + "score": 1.0, + "content": "112", + "type": "text" + }, + { + "bbox": [ + 105, + 319, + 505, + 333 + ], + "score": 1.0, + "content": "Equilibrium-point stability for large complex systems is not guaranteed in general [17, 27], and the", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 330, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 86, + 333, + 100, + 342 + ], + "score": 1.0, + "content": "113", + "type": "text" + }, + { + "bbox": [ + 105, + 330, + 505, + 344 + ], + "score": 1.0, + "content": "effective dimensionality of stable-input stable-output responses is richly dependent on the parameter", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 342, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 86, + 344, + 100, + 353 + ], + "score": 1.0, + "content": "114", + "type": "text" + }, + { + "bbox": [ + 106, + 342, + 505, + 354 + ], + "score": 1.0, + "content": "space [2]. However, the interaction matrix for our physical system (2) is symmetric, and hence the", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 352, + 347, + 366 + ], + "spans": [ + { + "bbox": [ + 86, + 354, + 100, + 364 + ], + "score": 1.0, + "content": "115", + "type": "text" + }, + { + "bbox": [ + 105, + 352, + 347, + 366 + ], + "score": 1.0, + "content": "system is a special case of the Cohen-Grossberg model [10]", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + } + ], + "index": 14.5, + "bbox_fs": [ + 86, + 319, + 505, + 366 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 366, + 378, + 402 + ], + "lines": [ + { + "bbox": [ + 233, + 366, + 378, + 402 + ], + "spans": [ + { + "bbox": [ + 233, + 366, + 378, + 402 + ], + "score": 0.95, + "content": "\\dot { s _ { i } } = a _ { i } ( s _ { i } ) [ b _ { i } ( s _ { i } ) - \\sum _ { j = 1 } ^ { N } c _ { i j } d _ { j } ( s _ { j } ) ] ,", + "type": "interline_equation", + "image_path": "b775b702ac77876668eb6203cdfc27e50d96ab3a943d6abcb7a1d80873c466ec.jpg" + } + ] + } + ], + "index": 17.5, + "virtual_lines": [ + { + "bbox": [ + 233, + 366, + 378, + 384.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 233, + 384.0, + 378, + 402.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 403, + 504, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 402, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 127, + 417 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 403, + 185, + 416 + ], + "score": 0.83, + "content": "a _ { i } ( s _ { i } ) = 1 / \\tau _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 402, + 190, + 417 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 190, + 403, + 279, + 416 + ], + "score": 0.86, + "content": "b _ { i } ( s _ { i } ) = x _ { i } - G _ { i } ( s _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 402, + 283, + 417 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 283, + 404, + 330, + 416 + ], + "score": 0.89, + "content": "c _ { i j } = - f _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 402, + 350, + 417 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 350, + 403, + 401, + 416 + ], + "score": 0.92, + "content": "d _ { j } ( s _ { j } ) = s _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 402, + 506, + 417 + ], + "score": 1.0, + "content": ". Thus, it is known to be", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 414, + 311, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 311, + 427 + ], + "score": 1.0, + "content": "globally absolute stable, with a Lyapunov function", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 402, + 506, + 427 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 197, + 428, + 415, + 517 + ], + "lines": [ + { + "bbox": [ + 197, + 428, + 415, + 517 + ], + "spans": [ + { + "bbox": [ + 197, + 428, + 415, + 517 + ], + "score": 0.93, + "content": "\\begin{array} { r l r } & { } & { V = - \\displaystyle \\sum _ { i } \\int _ { 0 } ^ { s _ { i } } b _ { i } ( u ) d _ { i } ^ { \\prime } ( u ) \\mathrm { d } u + \\displaystyle \\sum _ { i , j } \\frac { c _ { i j } } { 2 } d _ { i } ( s _ { i } ) d _ { j } ( s _ { j } ) } \\\\ & { } & { \\quad = \\displaystyle \\sum _ { i } \\Big ( P _ { i } ( s _ { i } ) - x _ { i } s _ { i } - \\displaystyle \\sum _ { j > i } f _ { i j } s _ { i } s _ { j } - \\frac { f _ { i i } } { 2 } s _ { i } ^ { 2 } \\Big ) , } \\\\ & { } & { \\quad \\mathrm { w h e r e ~ o u t p u t ~ p o w e r ~ } P _ { i } ( s _ { i } ) \\equiv \\displaystyle \\int _ { 0 } ^ { s _ { i } } G _ { i } ( u ) \\mathrm { d } u . } \\end{array}", + "type": "interline_equation", + "image_path": "617c1c10f7d08e612cb314f263d3e32087eea99241d887ae6e71dbbd2746283e.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 197, + 428, + 415, + 442.8333333333333 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 197, + 442.8333333333333, + 415, + 457.66666666666663 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 197, + 457.66666666666663, + 415, + 472.49999999999994 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 197, + 472.49999999999994, + 415, + 487.33333333333326 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 197, + 487.33333333333326, + 415, + 502.1666666666666 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 197, + 502.1666666666666, + 415, + 516.9999999999999 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "index", + "bbox": [ + 86, + 522, + 506, + 614 + ], + "lines": [ + { + "bbox": [ + 86, + 523, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 86, + 525, + 100, + 534 + ], + "score": 1.0, + "content": "116", + "type": "text" + }, + { + "bbox": [ + 105, + 523, + 506, + 537 + ], + "score": 1.0, + "content": "Alternatively, since our system (5) is piecewise-linear, a piecewise-quadratic Lyapunov function may", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 86, + 536, + 100, + 546 + ], + "score": 1.0, + "content": "117", + "type": "text" + }, + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "score": 1.0, + "content": "be obtained by a piecewise-affine system [22] analysis. While this approach is more powerful and", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 545, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 86, + 547, + 100, + 556 + ], + "score": 1.0, + "content": "118", + "type": "text" + }, + { + "bbox": [ + 105, + 545, + 506, + 558 + ], + "score": 1.0, + "content": "holds even for asymmetric interaction matrices, it also seems to be analytically complex. From another", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 86, + 557, + 100, + 568 + ], + "score": 1.0, + "content": "119", + "type": "text" + }, + { + "bbox": [ + 105, + 555, + 462, + 568 + ], + "score": 1.0, + "content": "angle, global asymptotic stability [21, Theorem 3] is guaranteed if the Jacobian matrix", + "type": "text" + }, + { + "bbox": [ + 462, + 556, + 469, + 566 + ], + "score": 0.69, + "content": "\\mathbf { J }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 566, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 86, + 569, + 99, + 578 + ], + "score": 1.0, + "content": "120", + "type": "text" + }, + { + "bbox": [ + 106, + 567, + 306, + 581 + ], + "score": 0.89, + "content": "J _ { i i } \\dot { + } 1 \\dot { / } 2 \\sum _ { j \\neq i } \\dot { | } J _ { i j } + J _ { j i } | < 0 \\Longleftrightarrow G _ { i } ^ { \\prime } ( s _ { i } ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 566, + 394, + 581 + ], + "score": 1.0, + "content": "because in our system", + "type": "text" + }, + { + "bbox": [ + 395, + 567, + 505, + 580 + ], + "score": 0.92, + "content": "\\begin{array} { r } { J _ { i i } = - \\sum _ { j \\neq i } f _ { i j } - G _ { i } ^ { \\prime } ( s _ { i } ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 578, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 86, + 581, + 99, + 591 + ], + "score": 1.0, + "content": "121", + "type": "text" + }, + { + "bbox": [ + 105, + 578, + 124, + 594 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 581, + 165, + 592 + ], + "score": 0.91, + "content": "J _ { i j } = f _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 578, + 412, + 594 + ], + "score": 1.0, + "content": ". Since our network employs non-monotonic functionality,", + "type": "text" + }, + { + "bbox": [ + 412, + 580, + 461, + 592 + ], + "score": 0.92, + "content": "G _ { i } ^ { \\prime } ( s _ { i } ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 578, + 506, + 594 + ], + "score": 1.0, + "content": "cannot be", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 590, + 507, + 604 + ], + "spans": [ + { + "bbox": [ + 86, + 593, + 100, + 602 + ], + "score": 1.0, + "content": "122", + "type": "text" + }, + { + "bbox": [ + 105, + 590, + 241, + 604 + ], + "score": 1.0, + "content": "guaranteed for all reachable states", + "type": "text" + }, + { + "bbox": [ + 241, + 593, + 250, + 602 + ], + "score": 0.84, + "content": "s _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 590, + 507, + 604 + ], + "score": 1.0, + "content": ", and thus the above criteria is unfortunately inapplicable. Hence,", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 602, + 324, + 614 + ], + "spans": [ + { + "bbox": [ + 86, + 604, + 100, + 613 + ], + "score": 1.0, + "content": "123", + "type": "text" + }, + { + "bbox": [ + 105, + 602, + 324, + 614 + ], + "score": 1.0, + "content": "we shall proceed with the Cohen-Grossberg approach.", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + } + ], + "index": 30.5, + "bbox_fs": [ + 86, + 523, + 507, + 614 + ] + }, + { + "type": "text", + "bbox": [ + 89, + 617, + 257, + 630 + ], + "lines": [ + { + "bbox": [ + 85, + 617, + 258, + 631 + ], + "spans": [ + { + "bbox": [ + 85, + 617, + 258, + 631 + ], + "score": 1.0, + "content": "124 The power function (21) simplifies to", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35, + "bbox_fs": [ + 85, + 617, + 258, + 631 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 182, + 631, + 428, + 677 + ], + "lines": [ + { + "bbox": [ + 182, + 631, + 428, + 677 + ], + "spans": [ + { + "bbox": [ + 182, + 631, + 428, + 677 + ], + "score": 0.94, + "content": "P ( s ) = \\left\\{ \\begin{array} { l l } { \\int _ { 0 } ^ { s } g _ { 1 } u \\mathrm { d } u = g _ { 1 } s ^ { 2 } / 2 } & { 0 \\le s \\le g _ { 2 } } \\\\ { g _ { 1 } g _ { 2 } ^ { 2 } / 2 + \\int _ { g _ { 2 } } ^ { s } ( g _ { 1 } + g _ { 3 } ) g _ { 2 } - g _ { 3 } u \\mathrm { d } u } & { g _ { 2 } \\le s } \\\\ { = g _ { 1 } s ^ { 2 } / 2 - g _ { \\oplus } ( s - g _ { 2 } ) ^ { 2 } , } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "800ffae7752736c21a4922a3289bf4e8558764f630301baeda6e026733c133e0.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 182, + 631, + 428, + 646.3333333333334 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 182, + 646.3333333333334, + 428, + 661.6666666666667 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 182, + 661.6666666666667, + 428, + 677.0000000000001 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 87, + 679, + 438, + 691 + ], + "lines": [ + { + "bbox": [ + 84, + 678, + 439, + 693 + ], + "spans": [ + { + "bbox": [ + 84, + 678, + 231, + 693 + ], + "score": 1.0, + "content": "125 and using the rectifier function", + "type": "text" + }, + { + "bbox": [ + 232, + 680, + 299, + 691 + ], + "score": 0.91, + "content": "[ x ) \\equiv \\operatorname* { m a x } ( x , 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 678, + 439, + 693 + ], + "score": 1.0, + "content": "may be expressed conveniently as", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39, + "bbox_fs": [ + 84, + 678, + 439, + 693 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 241, + 694, + 369, + 708 + ], + "lines": [ + { + "bbox": [ + 241, + 694, + 369, + 708 + ], + "spans": [ + { + "bbox": [ + 241, + 694, + 369, + 708 + ], + "score": 0.92, + "content": "P ( s ) = g _ { 1 } s ^ { 2 } / 2 - g _ { \\oplus } [ s - g _ { 2 } ) ^ { 2 } ,", + "type": "interline_equation", + "image_path": "2b9a2b8e0e8ce7e3a1917ab44ef8e3edf1e02c7ba6b658a55a75b75d1f60b808.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 241, + 694, + 369, + 708 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 87, + 711, + 304, + 723 + ], + "lines": [ + { + "bbox": [ + 85, + 710, + 304, + 724 + ], + "spans": [ + { + "bbox": [ + 85, + 710, + 304, + 724 + ], + "score": 1.0, + "content": "126 when the system is within its operational bounds.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41, + "bbox_fs": [ + 85, + 710, + 304, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 87, + 99, + 506, + 133 + ], + "lines": [ + { + "bbox": [ + 86, + 99, + 506, + 113 + ], + "spans": [ + { + "bbox": [ + 86, + 101, + 100, + 111 + ], + "score": 1.0, + "content": "128", + "type": "text" + }, + { + "bbox": [ + 105, + 99, + 506, + 113 + ], + "score": 1.0, + "content": "Given the Lyapunov stability result of our system, it is computationally efficient to simulate our", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 86, + 111, + 505, + 122 + ], + "spans": [ + { + "bbox": [ + 86, + 113, + 100, + 122 + ], + "score": 1.0, + "content": "129", + "type": "text" + }, + { + "bbox": [ + 105, + 111, + 505, + 122 + ], + "score": 1.0, + "content": "state-space model and probe for combinational functionality. Here, we will simulate for the simplest", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 86, + 122, + 462, + 134 + ], + "spans": [ + { + "bbox": [ + 86, + 124, + 99, + 134 + ], + "score": 1.0, + "content": "130", + "type": "text" + }, + { + "bbox": [ + 105, + 122, + 462, + 134 + ], + "score": 1.0, + "content": "proof-of-concept for deep functionality in a shallow recurrent - solving a parity problem.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 87, + 137, + 505, + 204 + ], + "lines": [ + { + "bbox": [ + 86, + 137, + 506, + 149 + ], + "spans": [ + { + "bbox": [ + 86, + 139, + 100, + 149 + ], + "score": 1.0, + "content": "131", + "type": "text" + }, + { + "bbox": [ + 106, + 138, + 277, + 149 + ], + "score": 1.0, + "content": "Using a cascade of 2-input XOR gates, the", + "type": "text" + }, + { + "bbox": [ + 278, + 138, + 288, + 147 + ], + "score": 0.82, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 138, + 446, + 149 + ], + "score": 1.0, + "content": "-bit parity function can be realized with", + "type": "text" + }, + { + "bbox": [ + 446, + 137, + 506, + 149 + ], + "score": 0.91, + "content": "N / 2 + N / 4 +", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 86, + 148, + 507, + 161 + ], + "spans": [ + { + "bbox": [ + 86, + 150, + 100, + 160 + ], + "score": 1.0, + "content": "132", + "type": "text" + }, + { + "bbox": [ + 108, + 149, + 173, + 159 + ], + "score": 0.89, + "content": "\\dots + 1 = N - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 148, + 215, + 161 + ], + "score": 1.0, + "content": "gates and", + "type": "text" + }, + { + "bbox": [ + 215, + 149, + 248, + 159 + ], + "score": 0.87, + "content": "2 N - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 148, + 448, + 161 + ], + "score": 1.0, + "content": "connections. Thus its total cost in area is at least", + "type": "text" + }, + { + "bbox": [ + 448, + 150, + 481, + 159 + ], + "score": 0.86, + "content": "3 N - 2", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 148, + 507, + 161 + ], + "score": 1.0, + "content": "units.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 86, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 86, + 161, + 100, + 171 + ], + "score": 1.0, + "content": "133", + "type": "text" + }, + { + "bbox": [ + 104, + 159, + 257, + 172 + ], + "score": 1.0, + "content": "A minimally-connected network has", + "type": "text" + }, + { + "bbox": [ + 258, + 160, + 268, + 169 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 159, + 398, + 172 + ], + "score": 1.0, + "content": "input wires, 1 output wire, and", + "type": "text" + }, + { + "bbox": [ + 398, + 160, + 426, + 170 + ], + "score": 0.89, + "content": "N - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "interconnect wires", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 86, + 169, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 86, + 173, + 100, + 181 + ], + "score": 1.0, + "content": "134", + "type": "text" + }, + { + "bbox": [ + 105, + 169, + 203, + 183 + ], + "score": 1.0, + "content": "with a total area cost of", + "type": "text" + }, + { + "bbox": [ + 203, + 171, + 218, + 180 + ], + "score": 0.8, + "content": "2 N", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 169, + 506, + 183 + ], + "score": 1.0, + "content": "units, assuming that the area occupied by the remaining components is", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 86, + 181, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 86, + 183, + 100, + 193 + ], + "score": 1.0, + "content": "135", + "type": "text" + }, + { + "bbox": [ + 105, + 181, + 188, + 194 + ], + "score": 1.0, + "content": "negligible. Thus for", + "type": "text" + }, + { + "bbox": [ + 188, + 182, + 217, + 191 + ], + "score": 0.9, + "content": "N = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 181, + 505, + 194 + ], + "score": 1.0, + "content": ", while a conventional digital circuit costs 7 units, our recurrent physical", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 86, + 193, + 241, + 204 + ], + "spans": [ + { + "bbox": [ + 86, + 195, + 100, + 203 + ], + "score": 1.0, + "content": "136", + "type": "text" + }, + { + "bbox": [ + 105, + 193, + 241, + 204 + ], + "score": 1.0, + "content": "network takes just 6 wiring units.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 86, + 208, + 505, + 298 + ], + "lines": [ + { + "bbox": [ + 86, + 207, + 502, + 223 + ], + "spans": [ + { + "bbox": [ + 86, + 210, + 99, + 219 + ], + "score": 1.0, + "content": "137", + "type": "text" + }, + { + "bbox": [ + 105, + 207, + 200, + 223 + ], + "score": 1.0, + "content": "Our simple model has", + "type": "text" + }, + { + "bbox": [ + 201, + 209, + 232, + 219 + ], + "score": 0.86, + "content": "N = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 207, + 236, + 223 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 237, + 209, + 302, + 220 + ], + "score": 0.85, + "content": "f _ { 1 2 } = f _ { 1 3 } = f", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 207, + 306, + 223 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 307, + 209, + 342, + 220 + ], + "score": 0.84, + "content": "f _ { 2 3 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 207, + 346, + 223 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 346, + 210, + 385, + 220 + ], + "score": 0.84, + "content": "g _ { 1 1 } = g _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 207, + 390, + 223 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 390, + 211, + 429, + 220 + ], + "score": 0.78, + "content": "g _ { 1 3 } = g _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 207, + 434, + 223 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 434, + 211, + 502, + 220 + ], + "score": 0.85, + "content": "g _ { 2 1 } = g _ { 3 1 } = \\gamma _ { 1 }", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 85, + 219, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 85, + 221, + 100, + 232 + ], + "score": 1.0, + "content": "138", + "type": "text" + }, + { + "bbox": [ + 107, + 221, + 170, + 231 + ], + "score": 0.81, + "content": "g _ { 2 3 } = g _ { 3 3 } = \\gamma _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 219, + 174, + 233 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 174, + 222, + 211, + 231 + ], + "score": 0.85, + "content": "g _ { 1 2 } = g _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 219, + 229, + 233 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 229, + 222, + 293, + 231 + ], + "score": 0.9, + "content": "\\gamma _ { 2 2 } = \\gamma _ { 3 2 } = \\gamma _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 219, + 457, + 233 + ], + "score": 1.0, + "content": ". We find from a symbolic evaluation that", + "type": "text" + }, + { + "bbox": [ + 458, + 220, + 506, + 232 + ], + "score": 0.89, + "content": "\\sigma _ { \\mathrm { m i n } } ( \\pmb { \\mathsf { A } } ) \\neq", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 85, + 230, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 85, + 233, + 100, + 244 + ], + "score": 1.0, + "content": "139", + "type": "text" + }, + { + "bbox": [ + 106, + 231, + 153, + 244 + ], + "score": 0.91, + "content": "1 / \\rho ( | \\pmb { \\mathsf { A } } ^ { - 1 } | )", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 230, + 506, + 246 + ], + "score": 1.0, + "content": "in general, and conditions for unique stability were not obtainable (which is not surprising", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 85, + 242, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 85, + 244, + 100, + 255 + ], + "score": 1.0, + "content": "140", + "type": "text" + }, + { + "bbox": [ + 105, + 242, + 149, + 256 + ], + "score": 1.0, + "content": "due to the", + "type": "text" + }, + { + "bbox": [ + 149, + 245, + 181, + 254 + ], + "score": 0.89, + "content": "s _ { 2 } \\mathrm { ~ - ~ } s _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 242, + 506, + 256 + ], + "score": 1.0, + "content": "symmetry). Thus, parity functionality was found by trial-and-error yielding the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 85, + 253, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 85, + 255, + 100, + 266 + ], + "score": 1.0, + "content": "141", + "type": "text" + }, + { + "bbox": [ + 105, + 253, + 154, + 268 + ], + "score": 1.0, + "content": "parameters", + "type": "text" + }, + { + "bbox": [ + 154, + 254, + 486, + 266 + ], + "score": 0.78, + "content": "\\{ f = 1 . 7 5 1 , g _ { 1 } = 1 . 8 7 6 , g _ { 2 } = g _ { 3 } = 0 . 1 2 6 , \\gamma _ { 1 } = 0 . 8 7 6 , \\gamma _ { 2 } = 1 . 6 , \\gamma _ { 3 } = 0 . 7 5 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 253, + 506, + 268 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 86, + 263, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 86, + 267, + 100, + 276 + ], + "score": 1.0, + "content": "142", + "type": "text" + }, + { + "bbox": [ + 104, + 263, + 385, + 279 + ], + "score": 1.0, + "content": "simulated using Wolfram Mathematica 13 (code in Appendix). When", + "type": "text" + }, + { + "bbox": [ + 386, + 266, + 462, + 276 + ], + "score": 0.9, + "content": "x _ { 1 } = x _ { 2 } = x _ { 3 } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 263, + 506, + 279 + ], + "score": 1.0, + "content": ", the states", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 86, + 276, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 86, + 278, + 100, + 287 + ], + "score": 1.0, + "content": "143", + "type": "text" + }, + { + "bbox": [ + 105, + 276, + 505, + 288 + ], + "score": 1.0, + "content": "were forced to transition beyond the bounds in (5), so its range was extended by taking an absolute", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 86, + 286, + 264, + 300 + ], + "spans": [ + { + "bbox": [ + 86, + 289, + 100, + 298 + ], + "score": 1.0, + "content": "144", + "type": "text" + }, + { + "bbox": [ + 105, + 286, + 264, + 300 + ], + "score": 1.0, + "content": "value. The results are plotted in Fig. 2.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 87, + 320, + 194, + 336 + ], + "lines": [ + { + "bbox": [ + 82, + 318, + 197, + 338 + ], + "spans": [ + { + "bbox": [ + 82, + 318, + 197, + 338 + ], + "score": 1.0, + "content": "145 4 Discussion", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 87, + 351, + 505, + 406 + ], + "lines": [ + { + "bbox": [ + 86, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 86, + 353, + 99, + 362 + ], + "score": 1.0, + "content": "146", + "type": "text" + }, + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "Our result should be seen as a theoretical proof-of-concept and as a motivation for continued", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 86, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 86, + 363, + 100, + 374 + ], + "score": 1.0, + "content": "147", + "type": "text" + }, + { + "bbox": [ + 105, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "research in this area. Future work must extend our simulations to much higher dimensions to serve", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 86, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 86, + 375, + 99, + 384 + ], + "score": 1.0, + "content": "148", + "type": "text" + }, + { + "bbox": [ + 105, + 372, + 505, + 385 + ], + "score": 1.0, + "content": "as a practical demonstration of deep functionality by shallow recurrent networks. Moreover, the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 86, + 384, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 86, + 386, + 99, + 395 + ], + "score": 1.0, + "content": "149", + "type": "text" + }, + { + "bbox": [ + 106, + 384, + 505, + 396 + ], + "score": 1.0, + "content": "theoretical formalism introduced here is not yet fully exploited. We hope to find an analytical method", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 86, + 395, + 406, + 407 + ], + "spans": [ + { + "bbox": [ + 86, + 397, + 99, + 406 + ], + "score": 1.0, + "content": "150", + "type": "text" + }, + { + "bbox": [ + 105, + 395, + 406, + 407 + ], + "score": 1.0, + "content": "to design functionality out of piecewise-linear Cohen-Grossberg networks.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 86, + 411, + 505, + 510 + ], + "lines": [ + { + "bbox": [ + 86, + 412, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 86, + 413, + 100, + 423 + ], + "score": 1.0, + "content": "151", + "type": "text" + }, + { + "bbox": [ + 106, + 412, + 505, + 424 + ], + "score": 1.0, + "content": "Our style of reasoning to circumvent the Shannon bottleneck may also be applied to other systems", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 86, + 421, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 86, + 424, + 99, + 433 + ], + "score": 1.0, + "content": "152", + "type": "text" + }, + { + "bbox": [ + 105, + 421, + 505, + 435 + ], + "score": 1.0, + "content": "such as networks of coupled oscillators [28]. Our non-modular mode of signal processing, offers", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 86, + 433, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 86, + 435, + 100, + 444 + ], + "score": 1.0, + "content": "153", + "type": "text" + }, + { + "bbox": [ + 105, + 433, + 505, + 445 + ], + "score": 1.0, + "content": "an alternative to not just circuit designers, but also to systems biologists who typically understand", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 86, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 86, + 446, + 100, + 456 + ], + "score": 1.0, + "content": "154", + "type": "text" + }, + { + "bbox": [ + 106, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "chemical reaction networks [6] as a composition of modules [20]. While, we have discussed", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 86, + 455, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 86, + 457, + 100, + 466 + ], + "score": 1.0, + "content": "155", + "type": "text" + }, + { + "bbox": [ + 105, + 455, + 506, + 467 + ], + "score": 1.0, + "content": "equilibium-point functionality in a state-space model driven by an additive input, it is also worth", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 86, + 464, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 86, + 468, + 100, + 477 + ], + "score": 1.0, + "content": "156", + "type": "text" + }, + { + "bbox": [ + 104, + 464, + 506, + 480 + ], + "score": 1.0, + "content": "investigating autonomous systems where the input is set as an initial state. An example is realizing", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 86, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 86, + 478, + 100, + 489 + ], + "score": 1.0, + "content": "157", + "type": "text" + }, + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "unboundedly-finite parity functions using just a radius-4 cellular automaton [4]. Finally, we hope", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 86, + 487, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 86, + 489, + 100, + 500 + ], + "score": 1.0, + "content": "158", + "type": "text" + }, + { + "bbox": [ + 105, + 487, + 506, + 500 + ], + "score": 1.0, + "content": "that this paper can serve as a call to action for neuromorphic engineers to look at physical reservoir", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 86, + 498, + 428, + 511 + ], + "spans": [ + { + "bbox": [ + 86, + 501, + 100, + 510 + ], + "score": 1.0, + "content": "159", + "type": "text" + }, + { + "bbox": [ + 105, + 498, + 428, + 511 + ], + "score": 1.0, + "content": "computing [36] from another angle, besides temporal input-output functionality.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27 + }, + { + "type": "title", + "bbox": [ + 91, + 533, + 174, + 547 + ], + "lines": [ + { + "bbox": [ + 89, + 531, + 177, + 550 + ], + "spans": [ + { + "bbox": [ + 89, + 538, + 100, + 547 + ], + "score": 1.0, + "content": "160", + "type": "text" + }, + { + "bbox": [ + 105, + 531, + 177, + 550 + ], + "score": 1.0, + "content": "References", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 103, + 557, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 110, + 557, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 110, + 557, + 506, + 571 + ], + "score": 1.0, + "content": "[1] M. 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Thus its total cost in area is at least", + "type": "text" + }, + { + "bbox": [ + 448, + 150, + 481, + 159 + ], + "score": 0.86, + "content": "3 N - 2", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 148, + 507, + 161 + ], + "score": 1.0, + "content": "units.", + "type": "text" + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 86, + 161, + 100, + 171 + ], + "score": 1.0, + "content": "133", + "type": "text" + }, + { + "bbox": [ + 104, + 159, + 257, + 172 + ], + "score": 1.0, + "content": "A minimally-connected network has", + "type": "text" + }, + { + "bbox": [ + 258, + 160, + 268, + 169 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 159, + 398, + 172 + ], + "score": 1.0, + "content": "input wires, 1 output wire, and", + "type": "text" + }, + { + "bbox": [ + 398, + 160, + 426, + 170 + ], + "score": 0.89, + "content": "N - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "interconnect wires", + "type": "text" + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 169, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 86, + 173, + 100, + 181 + ], + "score": 1.0, + "content": "134", + "type": "text" + }, + { + "bbox": [ + 105, + 169, + 203, + 183 + ], + "score": 1.0, + "content": "with a total area cost of", + "type": "text" + }, + { + "bbox": [ + 203, + 171, + 218, + 180 + ], + "score": 0.8, + "content": "2 N", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 169, + 506, + 183 + ], + "score": 1.0, + "content": "units, assuming that the area occupied by the remaining components is", + "type": "text" + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 181, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 86, + 183, + 100, + 193 + ], + "score": 1.0, + "content": "135", + "type": "text" + }, + { + "bbox": [ + 105, + 181, + 188, + 194 + ], + "score": 1.0, + "content": "negligible. Thus for", + "type": "text" + }, + { + "bbox": [ + 188, + 182, + 217, + 191 + ], + "score": 0.9, + "content": "N = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 181, + 505, + 194 + ], + "score": 1.0, + "content": ", while a conventional digital circuit costs 7 units, our recurrent physical", + "type": "text" + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 193, + 241, + 204 + ], + "spans": [ + { + "bbox": [ + 86, + 195, + 100, + 203 + ], + "score": 1.0, + "content": "136", + "type": "text" + }, + { + "bbox": [ + 105, + 193, + 241, + 204 + ], + "score": 1.0, + "content": "network takes just 6 wiring units.", + "type": "text" + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 207, + 502, + 223 + ], + "spans": [ + { + "bbox": [ + 86, + 210, + 99, + 219 + ], + "score": 1.0, + "content": "137", + "type": "text" + }, + { + "bbox": [ + 105, + 207, + 200, + 223 + ], + "score": 1.0, + "content": "Our simple model has", + "type": "text" + }, + { + "bbox": [ + 201, + 209, + 232, + 219 + ], + "score": 0.86, + "content": "N = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 207, + 236, + 223 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 237, + 209, + 302, + 220 + ], + "score": 0.85, + "content": "f _ { 1 2 } = f _ { 1 3 } = f", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 207, + 306, + 223 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 307, + 209, + 342, + 220 + ], + "score": 0.84, + "content": "f _ { 2 3 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 207, + 346, + 223 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 346, + 210, + 385, + 220 + ], + "score": 0.84, + "content": "g _ { 1 1 } = g _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 207, + 390, + 223 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 390, + 211, + 429, + 220 + ], + "score": 0.78, + "content": "g _ { 1 3 } = g _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 207, + 434, + 223 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 434, + 211, + 502, + 220 + ], + "score": 0.85, + "content": "g _ { 2 1 } = g _ { 3 1 } = \\gamma _ { 1 }", + "type": "inline_equation" + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 219, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 85, + 221, + 100, + 232 + ], + "score": 1.0, + "content": "138", + "type": "text" + }, + { + "bbox": [ + 107, + 221, + 170, + 231 + ], + "score": 0.81, + "content": "g _ { 2 3 } = g _ { 3 3 } = \\gamma _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 219, + 174, + 233 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 174, + 222, + 211, + 231 + ], + "score": 0.85, + "content": "g _ { 1 2 } = g _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 219, + 229, + 233 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 229, + 222, + 293, + 231 + ], + "score": 0.9, + "content": "\\gamma _ { 2 2 } = \\gamma _ { 3 2 } = \\gamma _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 219, + 457, + 233 + ], + "score": 1.0, + "content": ". We find from a symbolic evaluation that", + "type": "text" + }, + { + "bbox": [ + 458, + 220, + 506, + 232 + ], + "score": 0.89, + "content": "\\sigma _ { \\mathrm { m i n } } ( \\pmb { \\mathsf { A } } ) \\neq", + "type": "inline_equation" + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 230, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 85, + 233, + 100, + 244 + ], + "score": 1.0, + "content": "139", + "type": "text" + }, + { + "bbox": [ + 106, + 231, + 153, + 244 + ], + "score": 0.91, + "content": "1 / \\rho ( | \\pmb { \\mathsf { A } } ^ { - 1 } | )", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 230, + 506, + 246 + ], + "score": 1.0, + "content": "in general, and conditions for unique stability were not obtainable (which is not surprising", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 242, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 85, + 244, + 100, + 255 + ], + "score": 1.0, + "content": "140", + "type": "text" + }, + { + "bbox": [ + 105, + 242, + 149, + 256 + ], + "score": 1.0, + "content": "due to the", + "type": "text" + }, + { + "bbox": [ + 149, + 245, + 181, + 254 + ], + "score": 0.89, + "content": "s _ { 2 } \\mathrm { ~ - ~ } s _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 242, + 506, + 256 + ], + "score": 1.0, + "content": "symmetry). Thus, parity functionality was found by trial-and-error yielding the", + "type": "text" + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 253, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 85, + 255, + 100, + 266 + ], + "score": 1.0, + "content": "141", + "type": "text" + }, + { + "bbox": [ + 105, + 253, + 154, + 268 + ], + "score": 1.0, + "content": "parameters", + "type": "text" + }, + { + "bbox": [ + 154, + 254, + 486, + 266 + ], + "score": 0.78, + "content": "\\{ f = 1 . 7 5 1 , g _ { 1 } = 1 . 8 7 6 , g _ { 2 } = g _ { 3 } = 0 . 1 2 6 , \\gamma _ { 1 } = 0 . 8 7 6 , \\gamma _ { 2 } = 1 . 6 , \\gamma _ { 3 } = 0 . 7 5 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 253, + 506, + 268 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 263, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 86, + 267, + 100, + 276 + ], + "score": 1.0, + "content": "142", + "type": "text" + }, + { + "bbox": [ + 104, + 263, + 385, + 279 + ], + "score": 1.0, + "content": "simulated using Wolfram Mathematica 13 (code in Appendix). When", + "type": "text" + }, + { + "bbox": [ + 386, + 266, + 462, + 276 + ], + "score": 0.9, + "content": "x _ { 1 } = x _ { 2 } = x _ { 3 } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 263, + 506, + 279 + ], + "score": 1.0, + "content": ", the states", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 276, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 86, + 278, + 100, + 287 + ], + "score": 1.0, + "content": "143", + "type": "text" + }, + { + "bbox": [ + 105, + 276, + 505, + 288 + ], + "score": 1.0, + "content": "were forced to transition beyond the bounds in (5), so its range was extended by taking an absolute", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 286, + 264, + 300 + ], + "spans": [ + { + "bbox": [ + 86, + 289, + 100, + 298 + ], + "score": 1.0, + "content": "144", + "type": "text" + }, + { + "bbox": [ + 105, + 286, + 264, + 300 + ], + "score": 1.0, + "content": "value. The results are plotted in Fig. 2.", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + } + ], + "index": 1, + "bbox_fs": [ + 86, + 99, + 506, + 134 + ] + }, + { + "type": "index", + "bbox": [ + 87, + 137, + 505, + 204 + ], + "lines": [], + "index": 5.5, + "bbox_fs": [ + 86, + 137, + 507, + 204 + ], + "lines_deleted": true + }, + { + "type": "index", + "bbox": [ + 86, + 208, + 505, + 298 + ], + "lines": [], + "index": 12.5, + "bbox_fs": [ + 85, + 207, + 506, + 300 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 87, + 320, + 194, + 336 + ], + "lines": [ + { + "bbox": [ + 82, + 318, + 197, + 338 + ], + "spans": [ + { + "bbox": [ + 82, + 318, + 197, + 338 + ], + "score": 1.0, + "content": "145 4 Discussion", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "index", + "bbox": [ + 87, + 351, + 505, + 406 + ], + "lines": [ + { + "bbox": [ + 86, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 86, + 353, + 99, + 362 + ], + "score": 1.0, + "content": "146", + "type": "text" + }, + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "Our result should be seen as a theoretical proof-of-concept and as a motivation for continued", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 86, + 363, + 100, + 374 + ], + "score": 1.0, + "content": "147", + "type": "text" + }, + { + "bbox": [ + 105, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "research in this area. Future work must extend our simulations to much higher dimensions to serve", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 86, + 375, + 99, + 384 + ], + "score": 1.0, + "content": "148", + "type": "text" + }, + { + "bbox": [ + 105, + 372, + 505, + 385 + ], + "score": 1.0, + "content": "as a practical demonstration of deep functionality by shallow recurrent networks. Moreover, the", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 384, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 86, + 386, + 99, + 395 + ], + "score": 1.0, + "content": "149", + "type": "text" + }, + { + "bbox": [ + 106, + 384, + 505, + 396 + ], + "score": 1.0, + "content": "theoretical formalism introduced here is not yet fully exploited. We hope to find an analytical method", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 395, + 406, + 407 + ], + "spans": [ + { + "bbox": [ + 86, + 397, + 99, + 406 + ], + "score": 1.0, + "content": "150", + "type": "text" + }, + { + "bbox": [ + 105, + 395, + 406, + 407 + ], + "score": 1.0, + "content": "to design functionality out of piecewise-linear Cohen-Grossberg networks.", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 412, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 86, + 413, + 100, + 423 + ], + "score": 1.0, + "content": "151", + "type": "text" + }, + { + "bbox": [ + 106, + 412, + 505, + 424 + ], + "score": 1.0, + "content": "Our style of reasoning to circumvent the Shannon bottleneck may also be applied to other systems", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 421, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 86, + 424, + 99, + 433 + ], + "score": 1.0, + "content": "152", + "type": "text" + }, + { + "bbox": [ + 105, + 421, + 505, + 435 + ], + "score": 1.0, + "content": "such as networks of coupled oscillators [28]. Our non-modular mode of signal processing, offers", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 433, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 86, + 435, + 100, + 444 + ], + "score": 1.0, + "content": "153", + "type": "text" + }, + { + "bbox": [ + 105, + 433, + 505, + 445 + ], + "score": 1.0, + "content": "an alternative to not just circuit designers, but also to systems biologists who typically understand", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 86, + 446, + 100, + 456 + ], + "score": 1.0, + "content": "154", + "type": "text" + }, + { + "bbox": [ + 106, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "chemical reaction networks [6] as a composition of modules [20]. While, we have discussed", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 455, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 86, + 457, + 100, + 466 + ], + "score": 1.0, + "content": "155", + "type": "text" + }, + { + "bbox": [ + 105, + 455, + 506, + 467 + ], + "score": 1.0, + "content": "equilibium-point functionality in a state-space model driven by an additive input, it is also worth", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 464, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 86, + 468, + 100, + 477 + ], + "score": 1.0, + "content": "156", + "type": "text" + }, + { + "bbox": [ + 104, + 464, + 506, + 480 + ], + "score": 1.0, + "content": "investigating autonomous systems where the input is set as an initial state. An example is realizing", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 86, + 478, + 100, + 489 + ], + "score": 1.0, + "content": "157", + "type": "text" + }, + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "unboundedly-finite parity functions using just a radius-4 cellular automaton [4]. Finally, we hope", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 487, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 86, + 489, + 100, + 500 + ], + "score": 1.0, + "content": "158", + "type": "text" + }, + { + "bbox": [ + 105, + 487, + 506, + 500 + ], + "score": 1.0, + "content": "that this paper can serve as a call to action for neuromorphic engineers to look at physical reservoir", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 498, + 428, + 511 + ], + "spans": [ + { + "bbox": [ + 86, + 501, + 100, + 510 + ], + "score": 1.0, + "content": "159", + "type": "text" + }, + { + "bbox": [ + 105, + 498, + 428, + 511 + ], + "score": 1.0, + "content": "computing [36] from another angle, besides temporal input-output functionality.", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + } + ], + "index": 20, + "bbox_fs": [ + 86, + 351, + 505, + 407 + ] + }, + { + "type": "index", + "bbox": [ + 86, + 411, + 505, + 510 + ], + "lines": [], + "index": 27, + "bbox_fs": [ + 86, + 412, + 506, + 511 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 91, + 533, + 174, + 547 + ], + "lines": [ + { + "bbox": [ + 89, + 531, + 177, + 550 + ], + "spans": [ + { + "bbox": [ + 89, + 538, + 100, + 547 + ], + "score": 1.0, + "content": "160", + "type": "text" + }, + { + "bbox": [ + 105, + 531, + 177, + 550 + ], + "score": 1.0, + "content": "References", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "list", + "bbox": [ + 103, + 557, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 110, + 557, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 110, + 557, + 506, + 571 + ], + "score": 1.0, + "content": "[1] M. 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For all authors...", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36, + "bbox_fs": [ + 129, + 639, + 210, + 653 + ] + }, + { + "type": "text", + "bbox": [ + 147, + 656, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 146, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 146, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 162, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 162, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "contributions and scope? [Yes] The concrete result is the realization of a parity function", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 162, + 678, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 162, + 678, + 506, + 690 + ], + "score": 1.0, + "content": "by our recurrent physical network by using just 6 wiring units, while a conventional", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 162, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 162, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "digital circuit costs 7 units. That being said, the paper is written to cover a much", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 161, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 161, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "broader scope - this is a matter of taste (an earlier version of this manuscript recieved", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 161, + 711, + 438, + 723 + ], + "spans": [ + { + "bbox": [ + 161, + 711, + 438, + 723 + ], + "score": 1.0, + "content": "both positive and negative comments about the scope of this article).", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39.5, + "bbox_fs": [ + 146, + 656, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 145, + 73, + 506, + 165 + ], + "lines": [ + { + "bbox": [ + 144, + 73, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 144, + 73, + 505, + 85 + ], + "score": 1.0, + "content": "(b) Did you describe the limitations of your work? [Yes] It is mentioned that future", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 161, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 161, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "work must extend our simulations to much higher dimensions to serve as a practical", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 161, + 95, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 161, + 95, + 505, + 106 + ], + "score": 1.0, + "content": "demonstration of deep functionality by shallow recurrent networks. Also the simulation", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 161, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 161, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "parameters were found by trial and error, instead of being derived analytically from the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 162, + 117, + 454, + 128 + ], + "spans": [ + { + "bbox": [ + 162, + 117, + 454, + 128 + ], + "score": 1.0, + "content": "theoretical formalism - these limitations are mentioned in the discussion.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 146, + 130, + 470, + 142 + ], + "spans": [ + { + "bbox": [ + 146, + 130, + 470, + 142 + ], + "score": 1.0, + "content": "(c) Did you discuss any potential negative societal impacts of your work? [N/A]", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 144, + 142, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 144, + 142, + 505, + 155 + ], + "score": 1.0, + "content": "(d) Have you read the ethics review guidelines and ensured that your paper conforms to", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 161, + 153, + 214, + 166 + ], + "spans": [ + { + "bbox": [ + 161, + 153, + 214, + 166 + ], + "score": 1.0, + "content": "them? [Yes]", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 132, + 168, + 302, + 180 + ], + "lines": [ + { + "bbox": [ + 129, + 167, + 304, + 181 + ], + "spans": [ + { + "bbox": [ + 129, + 167, + 304, + 181 + ], + "score": 1.0, + "content": "2. If you are including theoretical results...", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 145, + 183, + 452, + 208 + ], + "lines": [ + { + "bbox": [ + 145, + 182, + 453, + 196 + ], + "spans": [ + { + "bbox": [ + 145, + 182, + 453, + 196 + ], + "score": 1.0, + "content": "(a) Did you state the full set of assumptions of all theoretical results? [N/A]", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 144, + 194, + 425, + 209 + ], + "spans": [ + { + "bbox": [ + 144, + 194, + 425, + 209 + ], + "score": 1.0, + "content": "(b) Did you include complete proofs of all theoretical results? [N/A]", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 131, + 211, + 241, + 222 + ], + "lines": [ + { + "bbox": [ + 128, + 209, + 242, + 225 + ], + "spans": [ + { + "bbox": [ + 128, + 209, + 242, + 225 + ], + "score": 1.0, + "content": "3. If you ran experiments...", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 146, + 226, + 505, + 352 + ], + "lines": [ + { + "bbox": [ + 145, + 225, + 506, + 238 + ], + "spans": [ + { + "bbox": [ + 145, + 225, + 506, + 238 + ], + "score": 1.0, + "content": "(a) Did you include the code, data, and instructions needed to reproduce the main ex-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 161, + 236, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 161, + 236, + 506, + 249 + ], + "score": 1.0, + "content": "perimental results (either in the supplemental material or as a URL)? [Yes] Check", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 162, + 248, + 344, + 259 + ], + "spans": [ + { + "bbox": [ + 162, + 248, + 344, + 259 + ], + "score": 1.0, + "content": "Appendix for the code to reproduce Figure 2.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 146, + 260, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 146, + 260, + 505, + 274 + ], + "score": 1.0, + "content": "(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 162, + 271, + 249, + 284 + ], + "spans": [ + { + "bbox": [ + 162, + 271, + 249, + 284 + ], + "score": 1.0, + "content": "were chosen)? [N/A]", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 146, + 284, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 146, + 284, + 506, + 297 + ], + "score": 1.0, + "content": "(c) Did you report error bars (e.g., with respect to the random seed after running experi-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 162, + 295, + 284, + 307 + ], + "spans": [ + { + "bbox": [ + 162, + 295, + 284, + 307 + ], + "score": 1.0, + "content": "ments multiple times)? 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