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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
e510fb9acbaa7a209a078bd0593729d282a29323 | subsection | 37 | 98 | Checking whether | Then,
\begin{}[label={\rm (0)}]
\item \ell _a^*=\ell _{a^*};
\item Y_j^*f =\pi (x_j^* f);
\item \ell _a Y_j = Y_j \ell _a (and hence \ell _aY_j^*=Y_j^* \ell _a);
\item
there is a unitary mapping U: \eta \otimes \eta \rightarrow \mathcal {V}_0 such that
U^*\ell _a U = a\otimes I;
\item
there exists X_j\in \operatornam... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
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0.008247588761150837,
0.0015345015563070774,
0.00838492065668106,
0.03860573098063469,
0.00... |
cf5e5459855138a15ac11251210928d781089f49 | subsection | 38 | 98 | Checking whether | Similarly, for a\in \operatorname{M}_{\eta }(,\begin{split}
\langle U^* \ell _a U (u_1\otimes v_1),(u_2\otimes v_2)\rangle _\lambda &= \operatorname{tr}\Big (((au_1)v_1^{\rm t}P^{-\frac{1}{2}}) P (P^{-\frac{1}{2}} (u_2v_2^{\rm t})^* )\Big ) \\
&= \langle au_1,u_2\rangle \, \langle v_1,v_2\rangle \\
&= \langle (a\otimes... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
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0... |
b01d5bb048838160e90bdc0e83a03bec62d8aa74 | subsection | 39 | 98 | Checking whether | For D\in \operatorname{M}_{\eta \times (\delta +\eta -\varepsilon )}(, letF_D := U (D\, (\widetilde{L}\oplus I_{\eta -\varepsilon })(X,X^*)) U^*
= \ell _{D(\widetilde{C}\oplus I)}+\sum _j \ell _{D(\widetilde{A}_j\oplus 0)} Y_j+\sum _j \ell _{D(\widetilde{B}_j\oplus 0)} Y_j^*;the second equality in (REF ) holds by Lemma... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
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0.... |
ceebfb4889386b977280664b932a302a82ddafa2 | subsection | 40 | 98 | Checking whether | If L=I+\sum _jA_jx_j+\sum _jA_j^*x_j^*, thenU((V\otimes I)L(X,X^*)(V^*\otimes I))U^*=\ell _{VV^*}+\sum _j \ell _{VA_jV^*} Y_j+\sum _j \ell _{VA_j^*V^*} Y_j^*by Lemma REF and thus\begin{split}
\langle U((V\otimes I)L(X,X^*)(V^*\otimes I))U^* v,v\rangle _\lambda &= \langle \pi (VLV^* v),v\rangle _\lambda = \lambda (v^*V ... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
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... |
915486ac943b9ee9c677f959e69f2b4645a96567 | subsection | 41 | 98 | Checking whether | Hence there is a solution with \dim (V)<\eta .We now argue by induction that,
with \delta fixed,
for each \eta \le \delta
and each \delta \times \eta affine linear pencil L^{\prime } such that
L^{\prime }(X,X^*) is full rank for every X in the interior of \mathcal {D}_L(\sigma ),
we have L^{\prime } is full rank on th... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
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0.012381992302834988,
... |
9d5e0ec76f0bded9870aa55cdd04e6a99d29195d | subsection | 42 | 98 | Checking whether | Therefore \widetilde{L} is of full rank on the interior of \mathcal {D}_L by (REF ).As a side product of Theorem REF
and the Algorithm in Subsection REF
we obtain a procedure for checking whether \mathcal {K}_f is convex.Given f\in \operatorname{M}_{{\delta }}(\mathop {<}\!x,x^*\!\mathop {>}) with f(0)=I,
we construc... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
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0.0... |
46b14fcddf6508ee8f04bb34a976a0f164a6610d | subsection | 43 | 98 | Checking whether | Hence \operatorname{Re}(D\widetilde{L})(X,X^*) is positive definite, so (D\widetilde{L})(X,X^*) is invertible. Consequently \widetilde{L}(X,X^*) has full rank.Proposition 4.3
Let \delta \ge \varepsilon . If every solution of (REF ) satisfiesP_0=0,\quad C_k=0\quad \text{for all }k,then there exists X\in \operatorname{M... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
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-0.0004426936502568424,
-0.0022027406375855207,
... |
a914d258845db97e29d9f861698a99499c5497d5 | subsection | 44 | 98 | Checking whether | Observe that\mathcal {U}\cap \mathcal {V}^{\rm h}_2 = \left\lbrace
\begin{pmatrix}\operatorname{Re}(D_1\widetilde{L}) & \widetilde{L}^*D_2^* \\ D_2\widetilde{L}& 0\end{pmatrix}
\colon D_1\in \operatorname{M}_{\varepsilon \times \delta }(,\, D_2\in \operatorname{M}_{(\eta -\varepsilon )\times \delta }( \right\rbraceand... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
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0.01950399950146675,
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0.02643265202641487,
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-0.008431893773376942,
0.005921402480453253,
... |
ae31e7efdc2a71776f02e4b36fe2c099de86c6d3 | subsection | 45 | 98 | Checking whether | Given a Hilbert space H, let \operatorname{B}(H) denote the (bounded linear) operators on H.Lemma 4.5
Suppose \lambda :\mathcal {V}_2\rightarrow is a positive linear functional in the sense that (f*f)>0 for
all fV1{0}.
Thus, the resulting scalar product f1,f2:=(f2*f1)
on V1 makes V1 a Hilbert space and V0V1
is a subsp... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.0800032913684845,
0.031137816607952118,
-0.019878773018717766,
0.019436344504356384,
0.040520355105400085,
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0.022212965413928032,
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-0.015294297598302364,
0.042168017476797104,
-0.... |
7a978f53df21c552a778a07814e0e18f0f62fee9 | subsection | 46 | 98 | Checking whether | By the definition of \langle \cdot ,\cdot \rangle _\lambda ,\begin{split}
\langle U (u_1\otimes v_1), U (u_2\otimes v_2)\rangle _\lambda & = \lambda \left((u_2v_2^{\rm t}P^{-\frac{1}{2}})^* u_1v_1^{\rm t}P^{-\frac{1}{2}} \right)\\
& = \operatorname{tr}\left((u_1v_1^{\rm t}P^{-\frac{1}{2}}) P (P^{-\frac{1}{2}} (u_2v_2^{... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.04215638339519501,
0.038218531757593155,
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0.006273089908063412,
0.02304711751639843,
... |
2dcb1746837999ec9abb7e6632e702b66f2d53a4 | subsection | 47 | 98 | Checking whether | Since \mathcal {C}+\mathcal {S} is also closed and convex
and since \mathcal {U} is a subspace, by there exists an \mathbb {R}-linear functional \lambda _0:\mathcal {V}^{\rm h}_2\rightarrow \mathbb {R} satisfying\lambda _0\left((\mathcal {C}+\mathcal {S})\setminus \lbrace 0\rbrace \right)=\mathbb {R}_{>0},\qquad \lambd... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
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-0.029055336490273476,
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0.023439599201083183,
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0.03296193853020668,
0.020463868975639343,
-0.026842614635825157,
0.01619102619588375,
0.005657702684402466,
0.0... |
50f3ddb98e28a2be1794ec7eaf4488a22a621b8f | subsection | 48 | 98 | Checking whether | ThenF_D u
& = \left(\ell _{D(\widetilde{C}\oplus I)}+\sum _j \ell _{D(\widetilde{A}_j\oplus 0)} Y_j+\sum _j \ell _{D(\widetilde{B}_j\oplus 0)} Y_j^*\right)u \\
& = \pi \left(
D(\widetilde{C}\oplus I)(I\oplus 0)+\sum _j D(\widetilde{A}_j\oplus 0)(I\oplus 0)x_j+\sum _j D(\widetilde{B}_j\oplus 0)(I\oplus 0)x_j^*
\right) \... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.04524063318967819,
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0.021246613934636116,
-0.000... |
864f918e569c618970b4199b06f136dc6b3b8df4 | subsection | 49 | 98 | Checking whether | If
\widetilde{L} is a \delta \times \varepsilon affine linear pencil
such that \widetilde{L}(X,X^*) is full rank for every X in the interior of \mathcal {D}_L(\max \lbrace d,\delta ,\varepsilon \rbrace ),
then \widetilde{L} is full rank on the interior of \mathcal {D}_L.The proof of Corollary REF given below, while not... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
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0.010712414979934692,
0.006748668849468231,
0.016... |
f3067c33ea30574861953efca99647048da12bd4 | subsection | 50 | 98 | Checking whether | Let \widetilde{L}^{\prime } denote the \delta \times \eta pencil whose coefficients are the restrictions of the
coefficients of \widetilde{L} to V. Let X satisfy L(X,X^*)\succ 0 and suppose \widetilde{L}(X,X^*)(u+u^{\prime })=0 for u\in V^\perp and u^{\prime }\in V.
Thus,(u+u^{\prime })^* \operatorname{Re}(D\widetilde{... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
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0.03482135012745857,
-0.009857402183115482,
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0.06860508024692535,
0.00770967872813344,
0.02319388836622238,
0.0158237237483263,
0.01232938189059496,
0.0020676... |
d291d5787d9bac66ad930775b7b8e7d992dad00a | subsection | 51 | 98 | Examples | We say that a hermitian f\in \mathop {<}\!x,x^*\!\mathop {>} with f(0)=1 is a minimal degree defining polynomial for \mathcal {D}_f if \deg h\ge \deg f for every hermitian h\in \mathop {<}\!x,x^*\!\mathop {>} such that \mathcal {D}_f=\mathcal {D}_h. In this section we present examples of hermitian polynomials f such th... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.0012635330203920603,
-0.015547655522823334,
-0.012549505569040775,
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0.024595504626631737,
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-0.003055365988984704,
-0.032010775059461594,
-... |
fd08e9ca108d0fbf9d21090ebe2d58afe74439d1 | subsection | 52 | 98 | Examples | It follows that \deg h\ge 1+\deg f_1.Remark 5.2 In general, Corollary REF implies that f\in \mathop {<}\!x,x^*\!\mathop {>} with f(0)\ne 0 has convex \mathcal {K}_f if and only if it admits a complete factorization f=s_0f_1s_1\cdots f_\ell s_\ell , where \mathcal {K}_{f_k} are convex (such f_k are characterized in Sect... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
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0.02429109439253807,
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0.04275354743003845,
-0.02549649402499199,
-0.0007834145217202604,
-0.02101057581603527,
0... |
c3c3795c0da10ca62faf23c8f827a29125a6789d | subsection | 53 | 98 | Examples | Thus \mathcal {Z}_h\supseteq \mathcal {Z}_{f_1}.
Since f_1 is an atom,
h has an atomic factor of degree \deg f_1 by . Thus the degree of h exceeds two by item REF .
Hence h is not an atom by Theorem REF . It follows that \deg h\ge 1+\deg f_1.Remark 5.2 In general, Corollary REF implies that f\in \mathop {<}\!x,x^*\!\ma... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
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0.01444659661501646,
-0.022364001721143723,
-0... |
63190079697d13fb5d7c0927fdaa40399160b56d | subsection | 54 | 98 | Examples | By Lemma REFREF \mathcal {Z}_h\supseteq \mathcal {Z}_{\widetilde{L}}.
Since \mathcal {K}_{f_1} = \mathcal {D}_{\widetilde{L}}, f_1 is an atom and \widetilde{L} is minimal,
\mathcal {Z}_{f_1}=\mathcal {Z}_{\widetilde{L}}.
Thus \mathcal {Z}_h\supseteq \mathcal {Z}_{f_1}.
Since f_1 is an atom,
h has an atomic factor of de... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
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0.02661352977156639,
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-0.007321009878069162,
-0.02873467653989792,
0... |
69b337f170fa5808988cc0a783580ff6b0f51288 | subsection | 55 | 98 | Example of degree 4 | Letf_1 =1 + x + x^* - 2xx^*-(x + x^*)xx^*,
\qquad s = 1 + \frac{1}{2}(x+x^*)andL = \begin{pmatrix}1 + x + x^* & 0 & x \\ 0 & 1 & x \\ x^* & x^* & 1 \end{pmatrix}.Let us sketch how to verify the assumptions of Lemma REF . Clearly, s is an atom and
items REF and REF of Lemma REF hold. Using standard realization algorithm... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.001708736876025796,
0.018536744639277458,
-0.028377236798405647,
-0.0034022172912955284,
0.013616496697068214,
-0.051780831068754196,
0.07011923938989639,
0.029018014669418335,
0.0334729366004467,
0.010870312340557575,
-0.03325934335589409,
-0.008650480769574642,
-0.03365601226687431,
0... |
0b8e608c113b5845969720ce3f7e3882ab58ffea | subsection | 56 | 98 | Example of degree 4 | Note that\lbrace (X,X^*)\colon f(X,X^*)\succeq 0 \rbrace \ne \mathcal {D}_Lin this case.Letf_1 =1 + x + x^* - 2xx^*-(x + x^*)xx^*,
\qquad s = 1 + \frac{1}{2}(x+x^*)andL = \begin{pmatrix}1 + x + x^* & 0 & x \\ 0 & 1 & x \\ x^* & x^* & 1 \end{pmatrix}.Let us sketch how to verify the assumptions of Lemma REF . Clearly, s ... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
0.008476377464830875,
0.023117393255233765,
-0.019562045112252235,
-0.0126802334561944,
0.013092226348817348,
-0.045471835881471634,
0.055756404995918274,
0.0336308479309082,
0.0173494890332222,
0.009719986468553543,
-0.0167696475982666,
-0.019348418340086937,
-0.02053862065076828,
0.01186... |
fb286a48f947662f4c205db33e306ef67a6f7b0e | subsection | 57 | 98 | Example of degree 5 or 6 | Letf_1 &=
1 - (x + x^*) - 2 (x + x^*)^2 - 2 x^* x +
(x+x^*)^3 + 2
(x+x^*)^2x^*x,
\\
s &= 1 - (x+x^*)^2andL = \begin{pmatrix}1 - \frac{1}{2}(x+x^*) & -\sqrt{2}(x+x^*) & \frac{1}{2}(x+x^*) & x^* \\[1mm]
-\sqrt{2}(x+x^*) & 1 & 0 & 0 \\[1mm]
\frac{1}{2}(x+x^*) & 0 & 1 - \frac{1}{2}(x+x^*) & -x^* \\[1mm]
x & 0 & -x & 1
\end... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.011359705589711666,
-0.0006131876143626869,
-0.027067989110946655,
-0.02732737734913826,
0.010718862526118755,
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0.05630263686180115,
0.021803921088576317,
0.019408388063311577,
0.00433331960812211,
-0.03866419568657875,
0.001195858814753592,
-0.030150137841701508,
... |
36222f7e57b389fe04fd6f153dbba2cb56392ebd | subsection | 58 | 98 | Example of degree 5 or 6 | Note that \deg f=6, but we do not know whether f is a minimal degree defining polynomial.Of course, by taking a Schur complement of L
we obtain a quadratic 2\times 2 noncommutative polynomial q with \mathcal {D}_q=\mathcal {D}_L:q=
\begin{pmatrix}
1-\frac{x}{2}-\frac{x^*}{2}-2 x^2-2 xx^*-3 x^*x-2
(x^*)^2 &
\frac{x}{2}+... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.04735104739665985,
0.006616790313273668,
-0.003192076925188303,
-0.043140709400177,
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0.057053133845329285,
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0.018351582810282707,
0.01810750551521778,
-0.008916459046304226,
0.01845836639404297,
-0.037740495055913925,
0.... |
36fa76ac4c66e6f2dd1d3db48746d28fc94ce715 | subsection | 59 | 98 | Example of degree 5 or 6 | Note that \deg f=6, but we do not know whether f is a minimal degree defining polynomial.Of course, by taking a Schur complement of L
we obtain a quadratic 2\times 2 noncommutative polynomial q with \mathcal {D}_q=\mathcal {D}_L:q=
\begin{pmatrix}
1-\frac{x}{2}-\frac{x^*}{2}-2 x^2-2 xx^*-3 x^*x-2
(x^*)^2 &
\frac{x}{2}+... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.03944979980587959,
0.012616002932190895,
-0.011174391955137253,
-0.045429814606904984,
-0.002694439608603716,
-0.021814854815602303,
0.05543718859553337,
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0.028237270191311836,
0.017558669671416283,
-0.004954107571393251,
0.01428644172847271,
-0.04552134498953819,
... |
946c5a069a18c30ad2fb9537508cb8fc3cf26314 | subsection | 60 | 98 | High degree atoms with convex | In the previous two subsections we obtained atoms f_1 of degree 3,4
with convex \mathcal {K}_{f_1} in agreement with the degree at most four
conclusion of the main result of .
Nevertheless, it is easy to construct examples of such polynomials f of high degree.For example, letf=1 + 4 (x + x^*) + 2 (x ^2 +(x^*)^2)- x x^*... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
0.015262051485478878,
0.02024262212216854,
-0.020807035267353058,
-0.03886827453970909,
0.0020002364180982113,
-0.02843424677848816,
0.0330105759203434,
0.017252754420042038,
0.0014091274933889508,
0.02465115115046501,
-0.014743401668965816,
-0.0030032149516046047,
-0.02675626054406166,
0.... |
846d060517917a4b1887acc804f95d9aee1d187c | subsection | 61 | 98 | High degree atoms with convex | Nevertheless, it is easy to construct examples of such polynomials f of high degree.For example, letf=1 + 4 (x + x^*) + 2 (x ^2 +(x^*)^2)- x x^* - 7 x x^* ( x + x^*)
- 4 x^* x ( x + x^* )\\ - x x^* ( x^2 + (x^*)^2 )
+ 2 x x^* (x x^* +
x^* x ) (x+x^*).That \mathcal {K}_f=\mathcal {D}_L, whereL=
\begin{pmatrix}
1-x-x^* &... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.013434856198728085,
0.038664311170578,
-0.026610322296619415,
-0.01377053651958704,
-0.015838025137782097,
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0.06658683717250824,
0.04022064805030823,
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-0.017592718824744225,
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-0.00868955161422491,
-0.027968304231762886,
0... |
fc7c5536211348c674d30cb6889bdb3ff4142bf3 | subsection | 62 | 98 | Counterexample to a one-term Positivstellensatz | One might hope that for polynomials whose semialgebraic sets are spectrahedra, there exists a one-term Positivstellensatz (cf. ), meaning: if \mathcal {D}_f=\mathcal {D}_L for a hermitian polynomial f with f(0)>0 and a d\times d hermitian monic pencil L, then there exists
W\in \operatorname{M}_{{d\times d}}(\mathop {<}... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.02002543769776821,
-0.003953192383050919,
-0.011401678435504436,
-0.03360976651310921,
0.018697530031204224,
-0.06581530719995499,
0.04206562787294388,
0.011485625989735126,
0.022437037900090218,
0.0045789871364831924,
-0.024741794914007187,
0.03254133462905884,
-0.0006935257697477937,
... |
bb4d72f5c96f74fb144274959d2f479f32a031df | subsection | 63 | 98 | Counterexample to a one-term Positivstellensatz | But s is a hermitian polynomial, so p divides \det W^*(\Omega ^{(n)},\Upsilon ^{(n)}) and \det W(\Omega ^{(n)},\Upsilon ^{(n)}). Therefore the left-hand side of (REF ) is divisible by p^3 but not by p^4, while the highest power of p dividing the right-hand side of (REF ) is even, a contradiction.One might hope that for... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.012276780791580677,
-0.002243240363895893,
-0.010323483496904373,
-0.0383945070207119,
0.0026895992923527956,
-0.054905977100133896,
0.058782052248716354,
0.0031779236160218716,
0.01152140460908413,
0.002433991990983486,
-0.03775358200073242,
0.04184329882264137,
-0.0052494872361421585,
... |
bbc47e7b89dc1074f465afb2558bd3f65d068151 | subsection | 64 | 98 | Counterexample to a one-term Positivstellensatz | Taking determinants of both sides of (REF ) gives\left(\det f(\Omega ^{(n)},\Upsilon ^{(n)})\right)^3
= \det W^*(\Omega ^{(n)},\Upsilon ^{(n)})\det L(\Omega ^{(n)},\Upsilon ^{(n)})\det W(\Omega ^{(n)},\Upsilon ^{(n)}).Since \det L(\Omega ^{(n)},\Upsilon ^{(n)})=\det f_1(\Omega ^{(n)},\Upsilon ^{(n)}),\left(\det f_1(\Om... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.029372233897447586,
-0.015655018389225006,
-0.012397371232509613,
-0.03710819408297539,
0.0029124286957085133,
-0.04769745469093323,
0.04510354623198509,
0.013053478673100471,
0.04873502254486084,
0.004474497400224209,
-0.0532514788210392,
0.04244859889149666,
0.0048788427375257015,
0.0... |
7c56292d367cfadb68d75138c331092a37eb4cbd | subsection | 65 | 98 | Counterexample to a one-term Positivstellensatz | However, with Example REF we shall demonstrate that (REF ) does not hold in general.Let us assume the notation of Example REF and suppose there exists W\in \mathop {<}\!x,x^*\!\mathop {>}^{3\times 3} such that\begin{pmatrix}f & 0 & 0 \\ 0 & f & 0 \\ 0 & 0 & f \end{pmatrix}= W^*LW.Let \Omega ^{(n)} and \Upsilon ^{(n)} b... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.0056224544532597065,
-0.004054727964103222,
-0.01899580843746662,
-0.04949590563774109,
-0.015089843422174454,
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0.051326826214790344,
0.018538078293204308,
0.005019776057451963,
-0.008513784036040306,
-0.029859274625778198,
0.05096064507961273,
-0.018034575507044792,
... |
9438d81276126819263ff6e2cfea1b61cc7815e3 | subsection | 66 | 98 | High degree matrix atoms defining free spectrahedra | It is fairly easy to produce examples of irreducible hermitian matrix polynomials F of arbitrary high degree such that \mathcal {D}_F is a free spectrahedron. For example, let p\in \operatorname{M}_{{\delta }}(\mathop {<}\!x,x^*\!\mathop {>})\setminus \operatorname{M}_{\delta }( be arbitrary and letF=\begin{pmatrix} I ... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
0.06707527488470078,
0.001551573514007032,
-0.025054050609469414,
-0.006999723147600889,
0.016234779730439186,
-0.09417393058538437,
0.04330291971564293,
0.05148134380578995,
0.0484602116048336,
0.026579875499010086,
-0.044462546706199646,
0.028044668957591057,
-0.0032614513766020536,
-0.0... |
7a4173fd4ee654681bb638c5da6b19005ccf60b1 | subsection | 67 | 98 | Classifying hermitian flip-poly pencils | A byproduct of investigations in earlier sections is a description of hermitian monic flip-poly pencils, which helped us construct Examples REF , REF and REF . Since it is of independent interest, we present it here in more detail.A d\times d monic pencil L=I-A\operatorname{\raisebox {1pt}{{\bigodot }}}x is called
flip... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.05310887470841408,
0.012262350879609585,
-0.020831497386097908,
-0.017199339345097542,
0.030201856046915054,
-0.04288388788700104,
0.034703902900218964,
0.03278099372982979,
0.02963719330728054,
0.011293265968561172,
-0.01970217190682888,
0.01584109477698803,
-0.013101714663207531,
0.03... |
1d3615b03d273e753099c460c0d0d6836f67f2da | subsection | 68 | 98 | Classifying hermitian flip-poly pencils | Hence by declaring N_j to be the strictly upper triangular part of u \tilde{v}_j^*-v_j u^*, we obtain matrices A_j=N_j+v_ju^* such that L=I-A\operatorname{\raisebox {1pt}{{\bigodot }}}x-A^*\operatorname{\raisebox {1pt}{{\bigodot }}}x^* is flip-poly.Thus we derived the following result.Proposition 6.3
Let L=I-A\operato... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.049499209970235825,
0.017257586121559143,
-0.030166443437337875,
0.004718751646578312,
0.026977375149726868,
-0.018096813932061195,
0.029205145314335823,
0.04696626588702202,
0.01661672070622444,
0.017135515809059143,
-0.02789289690554142,
0.033996377140283585,
-0.010627682320773602,
0.... |
957e4853e847c6091a822dcff0ce21162473fc53 | subsection | 69 | 98 | Classifying hermitian flip-poly pencils | If L=I-A\operatorname{\raisebox {1pt}{{\bigodot }}}x-A^*\operatorname{\raisebox {1pt}{{\bigodot }}}x^* is a d\times d flip-poly pencil, then by the definition above there exist jointly nilpotent matrices N_1,\dots ,N_g,\tilde{N}_1,\dots ,\tilde{N}_g and vectors u,v_1,\dots ,v_g,\tilde{v}_1,\dots ,\tilde{v}_g such thatA... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.05836578086018562,
0.03852385655045509,
-0.014354867860674858,
0.0023791228886693716,
0.03708913177251816,
-0.009386755526065826,
0.03708913177251816,
0.05024585500359535,
0.012477517127990723,
0.0022627422586083412,
-0.024344513192772865,
0.024069778621196747,
0.017018264159560204,
-0.... |
8267a47aaf851a6e0cd57a936f954440dad5a4f7 | subsection | 70 | 98 | Classifying hermitian flip-poly pencils | Then L is flip-poly if and only if there exist vectors u,v_1,\dots ,v_g such that, after a unitary change of coordinates, A_j=N_j+v_ju^*, with N_j being the strictly upper triangular part of the matrix u \tilde{v}_j^*-v_j u^*, where \tilde{v}_j is a vector satisfying\tilde{v}_j^k=
\frac{u^k \overline{v_j^k}}{\overline{... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.05912546068429947,
0.009411870501935482,
-0.02195086143910885,
0.010716108605265617,
0.021447470411658287,
-0.012256787158548832,
0.013553397729992867,
0.032491546124219894,
0.01667289063334465,
0.006238986738026142,
-0.023628827184438705,
0.012897465378046036,
0.005491528660058975,
0.0... |
c10dc942af32e311f0131b736d4562a483f8cdf0 | subsection | 71 | 98 | Classifying hermitian flip-poly pencils | If L=I-A\operatorname{\raisebox {1pt}{{\bigodot }}}x-A^*\operatorname{\raisebox {1pt}{{\bigodot }}}x^* is a d\times d flip-poly pencil, then by the definition above there exist jointly nilpotent matrices N_1,\dots ,N_g,\tilde{N}_1,\dots ,\tilde{N}_g and vectors u,v_1,\dots ,v_g,\tilde{v}_1,\dots ,\tilde{v}_g such thatA... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.05836578086018562,
0.03852385655045509,
-0.014354867860674858,
0.0023791228886693716,
0.03708913177251816,
-0.009386755526065826,
0.03708913177251816,
0.05024585500359535,
0.012477517127990723,
0.0022627422586083412,
-0.024344513192772865,
0.024069778621196747,
0.017018264159560204,
-0.... |
bb2d119afa066567dc31c2bece48513ec40dc6ed | subsection | 72 | 98 | Classifying hermitian flip-poly pencils | Then L is flip-poly if and only if there exist vectors u,v_1,\dots ,v_g such that, after a unitary change of coordinates, A_j=N_j+v_ju^*, with N_j being the strictly upper triangular part of the matrix u \tilde{v}_j^*-v_j u^*, where \tilde{v}_j is a vector satisfying\tilde{v}_j^k=
\frac{u^k \overline{v_j^k}}{\overline{... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.06727302074432373,
0.018924353644251823,
-0.004963827319443226,
0.00921799149364233,
0.009721623733639717,
-0.002449482912197709,
0.019000660628080368,
0.03955800458788872,
0.014040648937225342,
0.0033766236156225204,
-0.01976373977959156,
0.027501359581947327,
0.02045051008462906,
0.00... |
30ad524cb022f76de39645c5a8753f307aaa0f72 | subsection | 73 | 98 | Hereditary polynomials | We say that a noncommutative polynomial f is hereditary if it is a linear combination of words uv with u\in \!\mathop {<}\!x^*\!\mathop {>} and v\in \!\mathop {<}\!x\!\mathop {>}. Furthermore, f is truly hereditary if it is not analytic or anti-analytic, i.e., f\notin \mathop {<}\!x\!\mathop {>}\cup \, \mathop {<}\!x^*... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.02748602069914341,
-0.0101107656955719,
-0.01738288626074791,
0.00017300377658102661,
0.031164050102233887,
-0.06892108172178268,
0.030782511457800865,
0.035223618149757385,
0.03085881844162941,
0.005047752056270838,
-0.04123666137456894,
0.029576851055026054,
-0.02196134626865387,
0.01... |
2714f23459f65310ae65c4a4439bb8b9514902d7 | subsection | 74 | 98 | Hereditary polynomials | Since\mathcal {Z}_{L^1}\cup \cdots \cup \mathcal {Z}_{L^\ell }=\mathcal {Z}_L\subseteq \mathcal {Z}_f=\mathcal {Z}_a\cup \mathcal {Z}_h\cup \mathcal {Z}_{a^*},for each i and large enough n, the irreducible polynomial \det L^i(\Omega ^{(n)},\Upsilon ^{(n)}) divides one of the polynomials \det a^*(\Upsilon ^{(n)}), \det ... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.031078916043043137,
-0.008765260688960552,
0.006129580084234476,
-0.0371665395796299,
0.011214041151106358,
-0.027554502710700035,
0.04018746316432953,
0.025281179696321487,
0.029705766588449478,
-0.013685707934200764,
-0.027737587690353394,
0.024960778653621674,
-0.03597647696733475,
0... |
e894b95f8b4f3ca88a996a0008730396d883900c | subsection | 75 | 98 | Hereditary polynomials | For example, the composite of
an analytic polynomial (with no x^*)
with an hermitian pencil, a heavily
studied class of objects in
the geometry of free convex sets (cf. ), is hereditary.
Similarly, the hereditary functional calculus
is a powerful tool in operator theory
and complex analysis.In this section we prove th... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.04010731354355812,
0.0056047989055514336,
-0.006886767689138651,
-0.007256859913468361,
0.038184359669685364,
-0.06287752836942673,
0.04642559215426445,
-0.0054369219578802586,
0.021091442555189133,
0.0028920609038323164,
-0.05127875879406929,
0.029836300760507584,
-0.012674705125391483,
... |
50db3eb730c7f5d4e59b4c22f3d77484bd6f0b04 | subsection | 76 | 98 | Hereditary polynomials | Because the L^i are pairwise non-similar irreducible pencils, we necessarily have \ell =1, so L is irreducible. Therefore \mathcal {D}_h = \mathcal {D}_L by Proposition REFREF . Thus, h is concave of degree at most two by Theorem REF . Finally, since f is of minimal degree, a = 1 and f = h.Corollary 7.3
If q\in \matho... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.04824347048997879,
0.01623372919857502,
-0.004108018707484007,
-0.024838827550411224,
0.03268106281757355,
-0.03487810865044594,
0.05205778777599335,
0.03762441501021385,
0.03533582389354706,
0.01719493791460991,
-0.03683103621006012,
0.014570687897503376,
-0.019575070589780807,
0.02448... |
81b7172a4912140f52573c4f5dd492f5d2d7d7e3 | subsection | 77 | 98 | Hereditary polynomials | Before giving a proof of Theorem REF we record the following corollary.Corollary 7.2
Any hereditary minimal degree defining polynomial for a free spectrahedron is an atom,
and hence has degree at most 2.Let f be hereditary and minimal degree defining polynomial for \mathcal {D}_f, and let \mathcal {D}_f=\mathcal {D}_L... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.028103305026888847,
-0.004046906717121601,
0.02074945531785488,
-0.047143980860710144,
0.019803524017333984,
-0.05010383203625679,
0.03805083781480789,
-0.007166953291743994,
0.0321311429142952,
0.0002686654042918235,
-0.05349087342619896,
0.02454843558371067,
-0.021649615839123726,
0.0... |
c5229601cc4695c337830ac152193c4141900916 | subsection | 78 | 98 | Hereditary polynomials | Finally, since f is of minimal degree, a = 1 and f = h.Corollary 7.3
If q\in \mathop {<}\!x\!\mathop {>} and \mathcal {D}_{q+q^*} is a free spectrahedron, then \deg (q)\le 1.Observe that q+q^* is an atom in \mathop {<}\!x,x^*\!\mathop {>} for every non-constant q\in \mathop {<}\!x\!\mathop {>}. Therefore q+q^* is of d... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.006840359885245562,
-0.021669406443834305,
0.014672629535198212,
-0.09149982780218124,
0.021120041608810425,
-0.008187067694962025,
0.05746970698237419,
0.04651292413473129,
0.022142469882965088,
0.020097611472010612,
0.018724197521805763,
0.0014640202280133963,
-0.023531144484877586,
0... |
6ad43ebe1e3e06581c43ff9b78b243cd65dcec3f | subsection | 79 | 98 | Proof of existence of the factorization ( | Lemma 7.4
Suppose f is hereditary and f=pq. If p\notin \mathop {<}\!x^*\!\mathop {>}, then q\in \mathop {<}\!x\!\mathop {>}.
If f=a^*hb and a,b\in \mathop {<}\!x\!\mathop {>}, then h is hereditary.To prove the first statement, suppose p\notin \mathop {<}\!x^*\!\mathop {>} and q\notin \mathop {<}\!x\!\mathop {>}.
Write... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.006314022466540337,
-0.023562176153063774,
0.010544990189373493,
-0.03827327862381935,
-0.006004997994750738,
-0.029055949300527573,
0.035373784601688385,
0.05060374364256859,
0.03088720515370369,
0.013551304116845131,
-0.022936496883630753,
0.0459645576775074,
-0.06696297228336334,
-0.... |
b5c65d86bef58babdf55db6b8ff54c4efc2b69ff | subsection | 80 | 98 | Proof of existence of the factorization ( | It follows that \alpha ^{\prime } \beta ^{\prime } \gamma ^{\prime }
must appear in a^* h b (and has largest degree amongst words in a^* h b
containing an x to the left of an x^*) and thus f is not hereditary.[Proof of existence in Theorem REF ]
The hereditary polynomial p factors asf= q_0 q_1 q_2 \dots q_s q_{s+1},\qq... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
0.008822306990623474,
-0.03541133552789688,
-0.007295956369489431,
-0.054887570440769196,
-0.02523057721555233,
-0.019232019782066345,
0.03391551226377487,
0.05131591111421585,
0.024421611800789833,
-0.0032587586902081966,
-0.014874287880957127,
0.04664527624845505,
-0.0736311599612236,
0.... |
6b2444e8a2a03f3f20f79826df317fd4e49cd65d | subsection | 81 | 98 | Proof of existence of the factorization ( | Thus,\sum _{\alpha \beta =\Gamma } p_\alpha q_\beta = 0.It follows that there exists words \sigma and \tau such
that (\sigma ,\tau )\ne (\alpha ^{\prime },\beta ^{\prime }), p_\sigma \ne 0, q_\tau \ne 0 and \Gamma =\sigma \tau = \alpha ^{\prime }\beta ^{\prime }.
It follows that either \alpha ^{\prime } properly divide... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
0.016114678233861923,
-0.005756876897066832,
-0.012353060767054558,
-0.0363190658390522,
-0.006027744151651859,
-0.019334562122821808,
0.046238139271736145,
0.0614982508122921,
0.028170166537165642,
0.01957872323691845,
0.0031779182609170675,
0.03796715661883354,
-0.07440830767154694,
-0.0... |
da80f052a870fec027e9d7da4009cc9df531544d | subsection | 82 | 98 | Proof of existence of the factorization ( | If p\notin \mathop {<}\!x^*\!\mathop {>}, then q\in \mathop {<}\!x\!\mathop {>}.
If f=a^*hb and a,b\in \mathop {<}\!x\!\mathop {>}, then h is hereditary.To prove the first statement, suppose p\notin \mathop {<}\!x^*\!\mathop {>} and q\notin \mathop {<}\!x\!\mathop {>}.
Write, p=\sum p_{\alpha } \alpha and q=\sum q_\bet... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
0.012805788777768612,
-0.0448431558907032,
0.003777631325647235,
-0.036173682659864426,
-0.013385788537561893,
-0.02478736639022827,
0.03189999610185623,
0.04917789250612259,
0.02318473532795906,
0.01591947302222252,
-0.009165525436401367,
0.03831052407622337,
-0.06959999352693558,
-0.0140... |
d0a3e0e462eb2cb30b90fda7808a4d4b9676ab15 | subsection | 83 | 98 | Proof of existence of the factorization ( | It follows that \alpha ^{\prime } \beta ^{\prime } \gamma ^{\prime }
must appear in a^* h b (and has largest degree amongst words in a^* h b
containing an x to the left of an x^*) and thus f is not hereditary.[Proof of existence in Theorem REF ]
The hereditary polynomial p factors asf= q_0 q_1 q_2 \dots q_s q_{s+1},\qq... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
0.008874614723026752,
-0.02466273121535778,
0.0024437622632831335,
-0.055063821375370026,
-0.021549366414546967,
-0.030675796791911125,
0.037665605545043945,
0.04297664016485214,
0.0244185458868742,
-0.004189306870102882,
-0.02600575052201748,
0.04181675985455513,
-0.08058120310306549,
-0.... |
2fd7748b3384161c4327e2fa70e2f0048d5b7590 | subsection | 84 | 98 | Proof of uniqueness of the factorization ( | Proving uniqueness requires background from Cohn which we now introduce.Let q_1, q_2, \widehat{q}_1,\widehat{q}_2\in \mathop {<}\!x\!\mathop {>} and supposeq_1 q_2= \widehat{q}_1\widehat{q}_2.Ifq_1\mathop {<}\!x\!\mathop {>}+ \widehat{q}_1 \mathop {<}\!x\!\mathop {>}=\mathop {<}\!x\!\mathop {>}, \qquad \mathop {<}\!x\!... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
0.017215942963957787,
-0.0008589847129769623,
-0.021703077480196953,
-0.0024076374247670174,
-0.022801967337727547,
-0.01985633186995983,
0.02058892510831356,
0.021352041512727737,
-0.018818490207195282,
0.02255776897072792,
-0.019291624426841736,
-0.0072190966457128525,
-0.01630020141601562... |
d52065de74c17c5119c929d4c5427d0adb413f23 | subsection | 85 | 98 | Proof of uniqueness of the factorization ( | To say that q_1q_2q_3q_4 is a complete factorization that can be transformed to a different factorization by applying comaximal transpositions on positions (2,3), (3,4) and (1,2) (in this order) means there exists \widehat{q}_2,\widehat{q}_3,\widehat{\widehat{q}}_3,\widehat{\widehat{q}}_2 such thatq_1q_2q_3q_4=
q_1\wid... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
0.0013752724044024944,
-0.0023919439408928156,
-0.008675088174641132,
-0.012573919259011745,
-0.0019026827067136765,
-0.029023779556155205,
0.031251683831214905,
0.05032619461417198,
-0.00231564580462873,
0.025010501965880394,
-0.019440744072198868,
0.007248315028846264,
-0.00581009685993194... |
78d7cf836eebd215900103387315133962649a48 | subsection | 86 | 98 | Proof of uniqueness of the factorization ( | Letp=p_1\cdots p_k,\qquad {\widehat{p}}={\widehat{p}}_1\cdots {\widehat{p}}_{\widehat{k}},\qquad q=q_1\cdots q_\ell ,\qquad {\widehat{q}}={\widehat{q}}_1\cdots {\widehat{q}}_{\widehat{\ell }}be complete factorizations (with factors equal to 1 at the origin). Thenp_1\cdots p_khq_1\cdots q_\ell \ = \ {\widehat{p}}_1\cdot... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.024262990802526474,
-0.0026799924671649933,
-0.023469485342502594,
-0.030351629480719566,
-0.028047407045960426,
0.03906494379043579,
0.04956364631652832,
0.03714221343398094,
0.013466723263263702,
0.0023500004317611456,
-0.03961429372429848,
0.021363640204072,
-0.016465263441205025,
0.... |
9ef8d1fa0cbb92c8db0cef19ec084fe5ffe5b919 | subsection | 87 | 98 | Proof of uniqueness of the factorization ( | If, moreover, q_1, q_2, \widehat{q}_1, \widehat{q}_2 are atoms andq_1\mathop {<}\!x\!\mathop {>}\cap \; \widehat{q}_1\mathop {<}\!x\!\mathop {>}\ \mbox{is a principal right ideal in} \ \mathop {<}\!x\!\mathop {>},then (REF ) is called a comaximal transposition .Next, q_1,\widehat{q}_2 are stably associated ifI_d\otimes... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
0.025653589516878128,
-0.01724482886493206,
-0.012704707682132721,
-0.0048605999909341335,
-0.012811534106731415,
-0.03781653568148613,
0.030796516686677933,
0.0383354052901268,
0.011033638380467892,
0.025500981137156487,
-0.011369378305971622,
-0.026416635140776634,
-0.005322242621332407,
... |
ef516d975d9521c1def7744ecd66e3cfa2afea4b | subsection | 88 | 98 | Proof of uniqueness of the factorization ( | To say that q_1q_2q_3q_4 is a complete factorization that can be transformed to a different factorization by applying comaximal transpositions on positions (2,3), (3,4) and (1,2) (in this order) means there exists \widehat{q}_2,\widehat{q}_3,\widehat{\widehat{q}}_3,\widehat{\widehat{q}}_2 such thatq_1q_2q_3q_4=
q_1\wid... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
0.0013752724044024944,
-0.0023919439408928156,
-0.008675088174641132,
-0.012573919259011745,
-0.0019026827067136765,
-0.029023779556155205,
0.031251683831214905,
0.05032619461417198,
-0.00231564580462873,
0.025010501965880394,
-0.019440744072198868,
0.007248315028846264,
-0.00581009685993194... |
d9587a8fd070629290fcd7151c4a9c213777b632 | subsection | 89 | 98 | Proof of uniqueness of the factorization ( | Letp=p_1\cdots p_k,\qquad {\widehat{p}}={\widehat{p}}_1\cdots {\widehat{p}}_{\widehat{k}},\qquad q=q_1\cdots q_\ell ,\qquad {\widehat{q}}={\widehat{q}}_1\cdots {\widehat{q}}_{\widehat{\ell }}be complete factorizations (with factors equal to 1 at the origin). Thenp_1\cdots p_khq_1\cdots q_\ell \ = \ {\widehat{p}}_1\cdot... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.024262990802526474,
-0.0026799924671649933,
-0.023469485342502594,
-0.030351629480719566,
-0.028047407045960426,
0.03906494379043579,
0.04956364631652832,
0.03714221343398094,
0.013466723263263702,
0.0023500004317611456,
-0.03961429372429848,
0.021363640204072,
-0.016465263441205025,
0.... |
db24dee1d9fef7b2a19ba15ff8b6e56a11d7bd78 | subsection | 90 | 98 | Proof of uniqueness of the factorization ( | If, moreover, q_1, q_2, \widehat{q}_1, \widehat{q}_2 are atoms andq_1\mathop {<}\!x\!\mathop {>}\cap \; \widehat{q}_1\mathop {<}\!x\!\mathop {>}\ \mbox{is a principal right ideal in} \ \mathop {<}\!x\!\mathop {>},then (REF ) is called a comaximal transposition .Next, q_1,\widehat{q}_2 are stably associated ifI_d\otimes... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
0.025653589516878128,
-0.01724482886493206,
-0.012704707682132721,
-0.0048605999909341335,
-0.012811534106731415,
-0.03781653568148613,
0.030796516686677933,
0.0383354052901268,
0.011033638380467892,
0.025500981137156487,
-0.011369378305971622,
-0.026416635140776634,
-0.005322242621332407,
... |
7f0cff24d5d0c01da6eb2088cdea2dbbdf6bb597 | subsection | 91 | 98 | Proof of uniqueness of the factorization ( | To say that q_1q_2q_3q_4 is a complete factorization that can be transformed to a different factorization by applying comaximal transpositions on positions (2,3), (3,4) and (1,2) (in this order) means there exists \widehat{q}_2,\widehat{q}_3,\widehat{\widehat{q}}_3,\widehat{\widehat{q}}_2 such thatq_1q_2q_3q_4=
q_1\wid... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
0.0013752724044024944,
-0.0023919439408928156,
-0.008675088174641132,
-0.012573919259011745,
-0.0019026827067136765,
-0.029023779556155205,
0.031251683831214905,
0.05032619461417198,
-0.00231564580462873,
0.025010501965880394,
-0.019440744072198868,
0.007248315028846264,
-0.00581009685993194... |
b54bd68a4072bc98e100a1aa65182e09a44ca847 | subsection | 92 | 98 | Proof of uniqueness of the factorization ( | Letp=p_1\cdots p_k,\qquad {\widehat{p}}={\widehat{p}}_1\cdots {\widehat{p}}_{\widehat{k}},\qquad q=q_1\cdots q_\ell ,\qquad {\widehat{q}}={\widehat{q}}_1\cdots {\widehat{q}}_{\widehat{\ell }}be complete factorizations (with factors equal to 1 at the origin). Thenp_1\cdots p_khq_1\cdots q_\ell \ = \ {\widehat{p}}_1\cdot... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
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... |
20a7316bff9b9b48d15f5d4717c4146337c76dc1 | subsection | 93 | 98 | Modification of the theory: rational functions | For the reader familiar with nc rational functions as found in , , we point out that
Theorem REF extends to matrix noncommutative rational functions in a straightforward way. Assume \mathbb {r}\in x,x^* \leavevmode \vtop {
{\hfil )\cr >#\hfil \cr ^\crcr }}{\delta \times \delta } is regular
at the origin (that is, 0 is ... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
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0.0... |
c387c2f3c8a64e09ec4fefe55d5e30d927085011 | subsection | 94 | 98 | Modification of the theory: rational functions | Then \widetilde{L} is invertible on
\mathcal {D}_L if and only if
there is \varepsilon >0 such that
\widetilde{L}\widetilde{L}^*-\varepsilon is invertible on \operatorname{int}\mathcal {D}_L, and this is something that can be checked
with a sequence of SDPs (cf. Subsection REF ).We conclude with a variant of Theorem RE... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
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... |
44ddbf20fe49e267d7813cd0a6075dfb7bf855fb | subsection | 95 | 98 | Modification of the theory: rational functions | For instance, a rational function \mathbb {r}
is positive definite on the interior
of \mathcal {D}_{L} if and only if
\mathbb {r}(0)\succ 0 and
\widetilde{L} is invertible on
\operatorname{int}\mathcal {D}_{L}, where
\widetilde{L} is the minimal pencil in an FM realization of \mathbb {r}\oplus \mathbb {r}^{-1}.
The lat... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
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0.... |
131d06ee1d7c8fc38d4a5c1665b27cbb519d3b2e | subsection | 96 | 98 | Modification of the theory: rational functions | Then we define \mathcal {K}_\mathbb {r}=\bigcup _n\mathcal {K}_\mathbb {r}(n), where \mathcal {K}_\mathbb {r}(n) is the closure of the connected component of\left\lbrace (X,X^*)\in \operatorname{M}_{n}(^{2g} \colon \mathbb {r}\text{ is regular at } (X,X^*) \text{ and }\det \mathbb {r}(X,X^*)\ne 0\right\rbracecontaining... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
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... |
4fc6aace62f074aec57b5fe46072f2ab095fb235 | subsection | 97 | 98 | Modification of the theory: rational functions | Lemma REF below asserts that, given L a minimal hermitian monic pencil L,
there exists a hermitian \mathbb {s}\in x,x^* \leavevmode \vtop {
{\hfil )\cr >#\hfil \cr \crcr }} such that Ks= DL. We say that a hermitian rx,x* | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
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f58eeb3accdb6c9802ee860f0008e6ec0c25568b | abstract | 0 | 48 | Abstract | In a 2012 article in the International Journal of Non-Linear Mechanics,
Destrade et al. showed that for nonlinear elastic materials satisfying
Truesdell's so-called empirical inequalities, the deformation corresponding to
a Cauchy pure shear stress is not a simple shear. Similar results can be found
in a 2011 article o... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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9dbb397ff2b20cc529952fa1f93e11f1fd7c198b | subsection | 1 | 48 | Introduction | The term shear describes a number of closely related, but distinct concepts which play an important role in linear and nonlinear elasticity theory. While the notion of a (pure) shear stress T=se_1\otimes e_2+e_2\otimes e_1) with s\in is rather straightforward , (once the stress tensor is specified), the concept of a sh... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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fe0fa29a7c7d62675fce19f39ece30b1e4db15ea | subsection | 2 | 48 | Overview | After a short introduction and a brief discussion of the linear case in Section REF , we demonstrate in Section that non-trivial Cauchy pure shear stress tensors never correspond to simple shear deformations for arbitrary non-linear isotropic elasticity laws. This result was previously obtained by Destrade et al. in th... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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861efa7d657a45fb3a151dd13661d051dc4995fc | subsection | 3 | 48 | Different notions of shear | The classical (homogeneous) simple shear deformation with the deformation gradient tensor F=+\gamma ė_2\otimes e_1 of a unit cube with the amount of shear \gamma \in =(0,\infty ) is shown in Figure REF .
It is well known that in isotropic linear elasticity, the Cauchy stress tensor corresponding to deformations of this... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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a8ef15ef6752ff47d7f9d49fa62fd5068c287d3f | subsection | 4 | 48 | Different notions of shear | To our knowledge, the finite pure shear stretch (REF ) was first mentioned by Claude Vallée as the stretch induced by a Hencky-type logarithmic stress-strain relation under Cauchy pure shear stress.We call F\in (3) an (idealized) left finite simple shear deformation gradient if F has the form
F_\alpha = \frac{1}{\sqrt... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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6e4e83b3f775f66b9186e582877eebdf89ec9e54 | subsection | 5 | 48 | Shear in linear elasticity | In isotropic linear elasticity, the stress response is induced by the quadratic energy density function ()=\mu {}^2+\frac{\lambda }{2}()^2=\mu {}^2+\frac{\kappa }{2}()^2, where = (F-), (X)=X-\frac{1}{3}(X); here, \mu denotes the infinitesimal shear modulus, \kappa >0 is the infinitesimal bulk modulus, and \lambda is th... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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-0.0139395622536... |
e02d53fa07ff962678e3de1cea5732b0d279cd15 | subsection | 6 | 48 | Shear in linear elasticity | Every simple shear deformation (REF ) has the form (REF ) and therefore leads to an infinitesimal pure shear stress tensor \sigma _{\rm lin}:F_\gamma = {1&\gamma &0\\0&1&0\\0&0&1}=+{0&\gamma &0\\0&0&0\\0&0&0} = + \underbrace{\begin{pmatrix} 0 & \frac{\gamma }{2} & 0 \\ \frac{\gamma }{2} & 0 & 0 \\ 0 & 0 & 0\end{pmatrix... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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0.0044408... |
c4ff36815be0627ce2be3e262d576245c78a6b6e | subsection | 7 | 48 | Shear in linear elasticity | \det (+_\gamma )\ne 1.The accidental fact that the simple shear deformation (REF ) is volume preserving in the finite sense as well appears to be a major source of confusion, since it suggests that in nonlinear elasticity, a shear deformation should have the exact form F_\gamma . However, as the result by Destrade et a... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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-0.00509209511801600... |
d27152a22f116ed46faaab120f7a2137f8882026 | subsection | 8 | 48 | Shear in nonlinear elasticity | The following theorem summarizes the aforementioned result by Destrade et al. .
[Destrade, Murphy and Saccomandi ]
Consider an isotropic elasticity law that satisfies the empirical inequalities , , , ,\beta _0\le 0\,,\qquad \beta _1>0\,,\qquad \beta _{-1}\le 0\qquad \text{with}\qquad \sigma =\beta _0+\beta _1Ḃ+\beta _{... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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acedf5ff747ae238f0863a7ae07638125d77687f | subsection | 9 | 48 | Shear in nonlinear elasticity | Thereby, in lemPundTkommutieren, we are able to determine the general form of all B which commute
with a Cauchy pure shear stress which, in turn, allows us to compute the general form of the deformation gradient F = F_\gamma \cdot (a,b,c)\cdot Q in propFeindeutigbestimmtB. For a more detailed description regarding semi... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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5db6d88f8d3e2d9766749eb59b532af3bbb60f83 | subsection | 10 | 48 | Shear in nonlinear elasticity | Then P commutes with T if and only if P has the formP = \begin{pmatrix} p & q & 0 \\ q & p & 0 \\ 0 & 0 & r \end{pmatrix}\,.Furthermore, p=\frac{1}{2}(\mu _1+\mu _2), q=\frac{1}{2}(\mu _1-\mu _2) and r=\mu _3, where \mu _1,\mu _2,\mu _3 \in are the eigenvalues of P.
Suppose P and T commute. Then P and T are simultane... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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f921767e7ac9d0d9ec8881cfe852b9a8673a6ce8 | subsection | 11 | 48 | Shear in nonlinear elasticity | With \mu _1,\mu _2,\mu _3\in denoting the eigenvalues of P we findP &= Q\begin{pmatrix} \mu _1 & 0 & 0 \\ 0 & \mu _2 & 0 \\ 0 & 0 & \mu _3 \end{pmatrix}Q^T = \frac{1}{2}\begin{pmatrix} 1 & -1 & \;\;\,0 \\ 1 & \phantom{-}1 & \;\;\,0 \\ 0 & \phantom{-}0 & \sqrt{2} \end{pmatrix} \begin{pmatrix}\mu _1 & 0 & 0 \\ 0 & \mu _2... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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... |
4bae1ba03e9a4dbacafd5f1ec3fd9d6846f3f555 | subsection | 12 | 48 | Shear in nonlinear elasticity | Among the constitutive requirements which guarantee this bi-coaxiality of stress and strain are the (weak) empirical inequalities , , , although the (weaker) strict Baker-Ericksen inequalities \mathrm {(BE^+)} are sufficient as well .
Moreover, bi-coaxiality is equivalent to semi-invertibility (REF ). Again, recall the... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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0.00... |
f20ac4e07bfcb65d2af349c6462f682b4db59c39 | subsection | 13 | 48 | Shear in nonlinear elasticity | BecauseF F^T = \begin{pmatrix} 1 & \gamma & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 \\ \gamma & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 + \gamma ^2 & \gamma & 0 \\ \gamma & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}andF^T F = \begin{pmatrix} 1 & 0 & 0 \\ \gamma & 1 & 0 \\ 0 & 0 & 1 \end{p... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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0.0... |
51652a360fc4a6a79259ea01511edd9007d77203 | subsection | 14 | 48 | Shear in nonlinear elasticity | Let \widetilde{F}=F_\gamma (a,b,c) with a,b,c given by (REF ). Then a^2 + b^2\gamma ^2 = b^2 = p, b^2\gamma = q and c^2=r, thus\widetilde{F}\widetilde{F}^T &= \begin{pmatrix} 1 & \gamma & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix}\begin{pmatrix} a & 0 & 0... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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2663fb14732c053079ba0f266c9def4cdcd2963c | subsection | 15 | 48 | Shear in nonlinear elasticity | In order for the term \sqrt{\frac{p^2-q^2}{p}} to be well defined, the condition p>|q| must hold. This implies the upper bound |\gamma |=\frac{|q|}{p} < 1, i.e. the shear angle is always limited by 45^\circ . Note carefully that this limitation p=\frac{1}{2}\mu _1+\mu _2)>\frac{1}{2}{\mu _1-\mu _2}=q is due to the posi... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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bebb74f240340a4a2f4e8e766478b7fef44479dc | subsection | 16 | 48 | Shear in nonlinear elasticity | Note that this behaviour cannot be described by a hyperelastic material with an additive isochoric-volumetric split, i.e. an energy potential of the form W(F)=W_{\mathrm {iso}}(F/(\det F)^\frac{1}{3})+f(\det F), since in this case0=(\sigma )=f^{\prime }(\det V) \qquad \Rightarrow \qquad \det V = 1due to the usual requi... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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05325058389eb982706963a8d31c5e72454214ca | subsection | 17 | 48 | Idealized finite simple shear deformations | Of course, while any deformation gradient F\in (3) corresponding to a Cauchy pure shear stress must be of the form () regardless of the (isotropic) constitutive law of elasticity, the value of the parameters a,b,c,\gamma or, equivalently, the principal stretches \lambda _1,\lambda _2,\lambda _3, depend on the specific ... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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23e10fd7e1531bbe641af102ede6e2f8e3c74cf7 | subsection | 18 | 48 | Idealized finite simple shear deformations | Consider the elastic energy potential W and the corresponding Cauchy stress response withW(F)=\frac{1}{2}{F}^2-\log (\det F)=\frac{1}{2}\left( I_1-\log I_3\right)\,,\qquad (B)=\frac{1}{\sqrt{\det B}}\left[B-\right]\,.If \sigma =se_1\otimes e_2+e_2\otimes e_1) is a Cauchy pure shear stress, then the left Cauchy-Green de... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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9a5997d07592275102dc8c8f7b150383f32f995d | subsection | 19 | 48 | Idealized finite simple shear deformations | Then, for \lambda _1=\lambda , \lambda _2=\frac{1}{\lambda } and \lambda _3=1, eq. (REF ) yields\sqrt{P} &= \frac{1}{2}\begin{pmatrix} \lambda _1 + \lambda _2 & \lambda _1 - \lambda _2 & 0 \\ \lambda _1 - \lambda _2 & \lambda _1 + \lambda _2 & 0 \\ 0 & 0 & 2\lambda _3 \end{pmatrix}= \frac{1}{2}\begin{pmatrix} e^\alpha ... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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2f2eed0fae3ec5e7e9a93333610e58f3166f8298 | subsection | 20 | 48 | Idealized finite simple shear deformations | Then\gamma &= \frac{\lambda _1^2-\lambda _2^2}{\lambda _1^2+\lambda _2^2} = \frac{e^{2\alpha }-e^{-2\alpha }}{e^{2\alpha }+e^{-2\alpha }} = \tanh (2\alpha )\,,\qquad b = \sqrt{\frac{\lambda _1^2+\lambda _2^2}{2}} = \sqrt{\frac{e^{2\alpha } + e^{-2\alpha }}{2}} = \sqrt{\cosh (2\alpha )}\,,\\
a &= \lambda _1\lambda _2\sq... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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46fab29f923371d9df1eff1a5fda1ba17a199665 | subsection | 21 | 48 | Idealized finite simple shear deformations | F\in (3) satisfying Definition that corresponds to a (non-trivial) Cauchy pure shear stress for an isotropic law of elasticity is a left finite simple shear deformation of the formF_\alpha = \frac{1}{\sqrt{\cosh (2\alpha )}}{ 1 & \sinh (2\alpha ) & 0 \\ 0 & \cosh (2\alpha ) & 0 \\ 0 & 0 & \sqrt{\cosh (2\alpha )} }\qqua... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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3ba66645f35aa641a60c78fd1d726737de7ceb17 | subsection | 22 | 48 | Idealized finite simple shear deformations | Let \alpha \in andV_\alpha = \begin{pmatrix} \cosh (\alpha ) & \sinh (\alpha ) & 0 \\ \sinh (\alpha ) & \cosh (\alpha ) & 0 \\ 0 & 0 & 1 \end{pmatrix}
\,,\qquad R = \frac{1}{\sqrt{\cosh (2\alpha )}}\begin{pmatrix} \cosh (\alpha ) & \sinh (\alpha ) & 0 \\ -\sinh (\alpha ) & \cosh (\alpha ) & 0 \\ 0 & 0 & \sqrt{\cosh (2\... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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6fbfae3e76dd17b2a32dec1520c31056ed92b071 | subsection | 23 | 48 | Constitutive conditions for idealized shear response in Cauchy pure shear stress | As stated in Corollary , a Cauchy pure shear stress corresponds with a deformation gradient F of the general triaxial form (REF ). Theorem shows that if F_\alpha also satisfies Definition , then F must be of the form (REF ). However, it is important to note that whether or not a deformation gradient F corresponding to ... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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e479e0ba1879de526cc439d31377685265cbbcef | subsection | 24 | 48 | Pure shear stress induced by pure shear stretch | Recall from Lemma that for any left finite simple shear deformation, all eigenvectors of B=F_\alpha {F_\alpha }^T or V_\alpha are eigenvectors of a Cauchy pure shear stress (FF^T). More specifically,V_\alpha =\sqrt{F_\alpha F_\alpha ^T} = {\cosh (\alpha )&\sinh (\alpha )&0\\ \sinh (\alpha )&\cosh (\alpha )&0\\ 0&0&1} =... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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5a2608f11a70050a5cf2d38f9266c889d3dafea8 | subsection | 25 | 48 | Pure shear stress induced by pure shear stretch | In terms of this representation, equality (REF ) reads{s}{-s}{0}=\beta _0{1}{1}{1}+\beta _1{\lambda ^2}{\frac{1}{\lambda ^2}}{1}+\beta _{-1}{\frac{1}{\lambda ^2}}{\lambda ^2}{1}\,.Assume without loss of generality that \lambda \ne 1 and let s=\beta _0+\beta _1^2+\beta _{-1}{1}{\lambda ^2}. Then (REF ) is equivalent to ... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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bb2028992bbd6ba28315ec8265dcdc189b4dbd9e | subsection | 26 | 48 | Pure shear stress induced by pure shear stretch | Then the functions \beta _i can be expressed by\beta _0=\frac{2}{\sqrt{I_3}}\left(I_2{\partial W}{\partial I_2}+I_3{\partial W}{\partial I_3}\right)\,,\qquad \beta _1=\frac{2}{\sqrt{I_3}}{\partial W}{\partial I_1}\,,\qquad \beta _{-1}=-2\sqrt{I_3}{\partial W}{\partial I_2}with the matrix invariantsI_1=B=\lambda _1^2+\l... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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d2ee3d1a5a90dcb0bb313d0add86a7dd3124d2b5 | subsection | 27 | 48 | Pure shear stress induced by pure shear stretch | In order to find conditions on W which ensure that our finite pure shear stretches correspond to pure shear stresses, we use the general formula\sigma _i = \frac{\lambda _i}{\lambda _1\,\lambda _2\,\lambda _3}{W}{\lambda _i}(\lambda _1,\lambda _2,\lambda _2)\,.for the eigenvalues of (B).
Again, we want to ensure that (... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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0e6f948b54e86be015ba6505ee7ef0675e14cfda | subsection | 28 | 48 | Pure shear stress induced by pure shear stretch | W\left(\lambda ,\frac{1}{\lambda +t},1\right)\right|_{t=0}\\
&=\left.\frac{\partial W}{\partial \lambda _2}\left(\lambda ,\frac{1}{\lambda +t},1\right)\frac{-1}{(\lambda +t)^2}\right|_{t=0}=-\frac{1}{\lambda ^2}\frac{\partial W}{\partial \lambda _2}\left(\lambda ,\frac{1}{\lambda },1\right)and\frac{\partial W}{\partial... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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606162a7d37847381913d7cb1d02b93adfea1113 | subsection | 29 | 48 | Pure shear stress induced by pure shear stretch | Since\frac{\partial W}{\partial \lambda _i}(\lambda _1,\lambda _2,\lambda _3)
= \frac{\partial W_{\mathrm {tc}}}{\partial \lambda _i}(\lambda _1,\lambda _2,\lambda _3) + \frac{d}{d\lambda _i}ḟ(\lambda _1\lambda _2\lambda _3)
= \frac{\partial W_{\mathrm {tc}}}{\partial \lambda _i}(\lambda _1,\lambda _2,\lambda _3) + \fr... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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e09a02abc8f16356a8656b758770f10d53b60f8d | subsection | 30 | 48 | Pure shear stress induced by pure shear stretch | Thus for every energy of the form (REF ), every finite pure shear stretch V_\alpha corresponds to a Cauchy pure shear stress (B). These energies include the classical Hencky energy(F) = \mu \,{_3 \log U}^2 + \frac{\kappa }{2}\left((\log U)\right)^2\ =\ \mu \,{\log U}^2 + \frac{\Lambda }{2}\left((\log U)\right)^2as well... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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-0.... |
993b20bd3342e51d4bf6db334fff8e285f600dc0 | subsection | 31 | 48 | Pure shear stress induced by pure shear stretch | If W_{\mathrm {iso}} is tension-compression symmetricNote that tension-compression symmetry of W(F) implies tension-compression symmetry of W_{\mathrm {iso}}(J_1,J_2) because W(F)=W(F)\iff W(\frac{I_2}{I_3},\frac{I_1}{I_3},\frac{1}{I_3})\overset{I_3=1}{\Rightarrow }W(I_1,I_2,1)=W(I_2,I_1,1)\iff W_{\mathrm {iso}}(J_1,J_... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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a761160d5959cc40706c521ea6ddf96fbd93e311 | subsection | 32 | 48 | Pure shear stress induced by pure shear stretch | It is easy to verify that a function W(3) with W(F)=\sum _{i=1}^3 w(\lambda _i) for all F with singular values \lambda _i is tension-compression symmetric if and only if w(\frac{1}{\lambda })=w(\lambda ) and that the requirement of a stress-free reference configuration implies f^{\prime }(1)=0, thus Theorem REF is dire... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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bfea06f78c866fa47de781a3defb10fea5c10f20 | subsection | 33 | 48 | Pure shear stress induced by pure shear stretch | Consider the Blatz-Ko type energyW(F)=\frac{\mu }{2}\left({F}^2+\frac{2}{\det F}-5\right)=\frac{\mu }{2}\left(I_1+\frac{2}{\sqrt{I_3}}-5\right)with the corresponding Cauchy stress response(B)=\beta _0+\beta _1Ḃ=\frac{\mu }{\sqrt{I_3}}Ḃ-\frac{\mu }{I_3}=\frac{\mu }{I_3}\left(\sqrt{I_3}Ḃ-\right)\,.For a deformation of th... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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3cf749e84c25131b55e5905318a785a759f21618 | subsection | 34 | 48 | Pure shear stretch induced by pure shear stress | We now return to the question whether (or, more precisely, under which conditions) pure shear Cauchy stress induces a pure shear stretch V_\alpha . Note again that while Theorem REF and the subsequent corollaries ensure that every pure shear stretch V_\alpha induces a Cauchy pure shear stress, additional assumptions on... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
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5b4a3568ec84e8bff3acaabebe58e1fb1fec4ff2 | subsection | 35 | 48 | Pure shear stretch induced by pure shear stress | Due to (REF ),s^\tau (\lambda ) = \lambda \cdot {W}{\lambda _1} \left(\lambda ,\frac{1}{\lambda },1\right)
\qquad \text{and}\qquad -s^\tau (\lambda ) = \frac{1}{\lambda }\cdot {W}{\lambda _2} \left(\lambda ,\frac{1}{\lambda },1\right)and thus\frac{d}{d\lambda } \, W\left(\lambda ,\frac{1}{\lambda },1\right) = {W}{\lamb... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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abcc7902fe20b61c45fa35a6c4522c19d97a83c7 | subsection | 36 | 48 | Pure shear stretch induced by pure shear stress | Then\tau (V) = Q(s^\tau ,-s^\tau ,0)Q̇^T = Q\,\tau ((\lambda ,\tfrac{1}{\lambda },1))Q̇^T = \tau (Q(\lambda ,\tfrac{1}{\lambda },1)Q̇)with Q\in (3) given by (REF ). Since Hill's strict inequality implies that the mapping \log V \mapsto \tau (V) is strictly monotone and hence injective, the mapping V \mapsto \tau (V) is... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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f152c84baf590137a7a2220d9333a679005e1739 | subsection | 37 | 48 | Conclusion | While the incompatibility between simple shear and Cauchy pure shear stress described by Destrade et al. and Moon and Truesdell as well as Mihai and Goriely is due to the difference between the principal axes of strain and stress, i.e. the eigenspaces of B=F_\gamma F_\gamma ^T and \sigma =\sigma ^s, the question whethe... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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694dca0e42de4641925418d4bd4ed74678c22ec7 | subsection | 38 | 48 | Biot pure shear stress | The following lemmas and theorems discuss the connection between Biot pure shear stress T^{\mathrm {Biot}} and right finite simple shear deformations F_\alpha analogues to Cauchy pure shear stress. For each of the results presented here, we also provide the reference to the respective counterpart for the Cauchy stress ... | {
"cite_spans": []
} | 10.1016/j.ijnonlinmec.2018.10.002 | 1806.07749 | Shear, pure and simple | [
"Christian Thiel",
"Jendrik Voss",
"Robert J. Martin",
"Patrizio Neff"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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