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...or, it holds that <math>\sum_{i=1}^dY_i^2=\|Y\|^2=1</math>. And due to the symmetry, all <math>\mathbf{E}[Y_i^2]</math>'s are equal. Thus, <math>\mathbf{E}[Y_i &\quad \text{(symmetry)}\\ ...

...or, it holds that <math>\sum_{i=1}^dY_i^2=\|Y\|^2=1</math>. And due to the symmetry, all <math>\mathbf{E}[Y_i^2]</math>'s are equal. Thus, <math>\mathbf{E}[Y_i &\quad \text{(symmetry)}\\ ...

...her <math>|S|\ge R(k,\ell-1)</math> or <math>|T|\ge R(k-1,\ell)</math>. By symmetry, suppose <math>|S|\ge R(k,\ell-1)</math>. By induction hypothesis, the comp By the choice of <math>n</math> and by symmetry, there exists a subset <math>S\subseteq[n-1]</math> with <math>|X|=R_t(k-1, ...

...v_3-v_2,\ldots, v_z-v_{z-1}, 1-v_z</math> are identically distributed. By symmetry, they have the same expectation, therefore As in the case of single hash function, by symmetry it holds that <math>\mathbb{E}[Y_j]=\frac{1}{z+1}</math> for every <math>j= ...

...v_3-v_2,\ldots, v_z-v_{z-1}, 1-v_z</math> are identically distributed. By symmetry, they have the same expectation, therefore As in the case of single hash function, by symmetry it holds that <math>\mathbb{E}[Y_j]=\frac{1}{z+1}</math> for every <math>j= ...

...v_3-v_2,\ldots, v_z-v_{z-1}, 1-v_z</math> are identically distributed. By symmetry, they have the same expectation, therefore As in the case of single hash function, by symmetry it holds that <math>\mathbb{E}[Y_j]=\frac{1}{z+1}</math> for every <math>j= ...

...v_3-v_2,\ldots, v_z-v_{z-1}, 1-v_z</math> are identically distributed. By symmetry, they have the same expectation, therefore As in the case of single hash function, by symmetry it holds that <math>\mathbb{E}[Y_j]=\frac{1}{z+1}</math> for every <math>j= ...

...v_3-v_2,\ldots, v_z-v_{z-1}, 1-v_z</math> are identically distributed. By symmetry, they have the same expectation, therefore As in the case of single hash function, by symmetry it holds that <math>\mathbb{E}[Y_j]=\frac{1}{z+1}</math> for every <math>j= ...

...en_and_odd_functions|even function]] so the ''y''-axis is an [[axis]] of [[symmetry]] for every constant function. ...

...her <math>|S|\ge R(k,\ell-1)</math> or <math>|T|\ge R(k-1,\ell)</math>. By symmetry, suppose <math>|S|\ge R(k,\ell-1)</math>. By induction hypothesis, the comp By the choice of <math>n</math> and by symmetry, there exists a subset <math>S\subseteq[n-1]</math> with <math>|X|=R_t(k-1, ...

...l point of view, the energy conservation law is a consequence of the shift symmetry of [[time]]; energy conservation is a result of the empirical fact that the ...

...<math>\mathcal{E}_3'</math> occurs; the second will have the same bound by symmetry. ...

Due to the symmetry. All <math>X_i</math> have the same distribution. ...

Due to the symmetry. All <math>X_i</math> have the same distribution. ...

...or, it holds that <math>\sum_{i=1}^dY_i^2=\|Y\|^2=1</math>. And due to the symmetry, all <math>\mathbf{E}[Y_i^2]</math>'s are equal. Thus, <math>\mathbf{E}[Y_i &\quad \text{(symmetry)}\\ ...

...or, it holds that <math>\sum_{i=1}^nY_i^2=\|Y\|^2=1</math>. And due to the symmetry, all <math>\mathbf{E}[Y_i^2]</math>'s are equal. Thus, <math>\mathbf{E}[Y_i &\quad \text{(symmetry)}\\ ...

which by independence between different hash functions and by symmetry is equal to: ...v_3-v_2,\ldots, v_z-v_{z-1}, 1-v_z</math> are identically distributed. By symmetry, they have the same expectation, therefore ...

# (symmetry) <math>d(x,y)=d(y,x)</math> ...

# (symmetry) <math>d(x,y)=d(y,x)</math> ...

# (symmetry) <math>d(x,y)=d(y,x)</math> ...