We are currently solving the following issue within our repository. Here is the issue text:
--- BEGIN ISSUE ---
`cupy.split` sometimes fails with a 0-sized input array
```
>>> cupy.split(cupy.ones((3, 0)), [1])
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "/home/kataoka/cupy/cupy/manipulation/split.py", line 78, in split
    return array_split(ary, indices_or_sections, axis)
  File "/home/kataoka/cupy/cupy/manipulation/split.py", line 16, in array_split
    return core.array_split(ary, indices_or_sections, axis)
  File "cupy/core/core.pyx", line 2338, in cupy.core.core.array_split
    v.data = ary.data + prev * stride
  File "cupy/cuda/memory.pyx", line 243, in cupy.cuda.memory.MemoryPointer.__add__
    assert self.ptr > 0 or offset == 0
AssertionError
```

--- END ISSUE ---

Below are some code segments, each from a relevant file. One or more of these files may contain bugs.
--- BEGIN FILES ---
Path: `cupy/math/sumprod.py`
Content:
```
1 import numpy
2 import six
3 
4 from cupy import core
5 
6 
7 def sum(a, axis=None, dtype=None, out=None, keepdims=False):
8     """Returns the sum of an array along given axes.
9 
10     Args:
11         a (cupy.ndarray): Array to take sum.
12         axis (int or sequence of ints): Axes along which the sum is taken.
13         dtype: Data type specifier.
14         out (cupy.ndarray): Output array.
15         keepdims (bool): If ``True``, the specified axes are remained as axes
16             of length one.
17 
18     Returns:
19         cupy.ndarray: The result array.
20 
21     .. seealso:: :func:`numpy.sum`
22 
23     """
24     # TODO(okuta): check type
25     return a.sum(axis, dtype, out, keepdims)
26 
27 
28 def prod(a, axis=None, dtype=None, out=None, keepdims=False):
29     """Returns the product of an array along given axes.
30 
31     Args:
32         a (cupy.ndarray): Array to take product.
33         axis (int or sequence of ints): Axes along which the product is taken.
34         dtype: Data type specifier.
35         out (cupy.ndarray): Output array.
36         keepdims (bool): If ``True``, the specified axes are remained as axes
37             of length one.
38 
39     Returns:
40         cupy.ndarray: The result array.
41 
42     .. seealso:: :func:`numpy.prod`
43 
44     """
45     # TODO(okuta): check type
46     return a.prod(axis, dtype, out, keepdims)
47 
48 
49 # TODO(okuta): Implement nansum
50 
51 
52 def _axis_to_first(x, axis):
53     if axis < 0:
54         axis = x.ndim + axis
55     trans = [axis] + [a for a in six.moves.range(x.ndim) if a != axis]
56     pre = list(six.moves.range(1, axis + 1))
57     succ = list(six.moves.range(axis + 1, x.ndim))
58     revert = pre + [0] + succ
59     return trans, revert
60 
61 
62 def _proc_as_batch(proc, x, axis):
63     trans, revert = _axis_to_first(x, axis)
64     t = x.transpose(trans)
65     s = t.shape
66     r = t.reshape(x.shape[axis], -1)
67     pos = 1
68     size = r.size
69     batch = r.shape[1]
70     while pos < size:
71         proc(pos, batch, r, size=size)
72         pos <<= 1
73     return r.reshape(s).transpose(revert)
74 
75 
76 def _cum_core(a, axis, dtype, out, kern, batch_kern):
77     if out is None:
78         if dtype is None:
79             kind = a.dtype.kind
80             if kind == 'b':
81                 dtype = numpy.dtype('l')
82             elif kind == 'i' and a.dtype.itemsize < numpy.dtype('l').itemsize:
83                 dtype = numpy.dtype('l')
84             elif kind == 'u' and a.dtype.itemsize < numpy.dtype('L').itemsize:
85                 dtype = numpy.dtype('L')
86             else:
87                 dtype = a.dtype
88 
89         out = a.astype(dtype)
90     else:
91         out[...] = a
92 
93     if axis is None:
94         out = out.ravel()
95     elif not (-a.ndim <= axis < a.ndim):
96         raise core.core._AxisError('axis(={}) out of bounds'.format(axis))
97     else:
98         return _proc_as_batch(batch_kern, out, axis=axis)
99 
100     pos = 1
101     while pos < out.size:
102         kern(pos, out, size=out.size)
103         pos <<= 1
104     return out
105 
106 
107 _cumsum_batch_kern = core.ElementwiseKernel(
108     'int64 pos, int64 batch', 'raw T x',
109     '''
110     ptrdiff_t b = i % batch;
111     ptrdiff_t j = i / batch;
112     if (j & pos) {
113       const ptrdiff_t dst_index[] = {j, b};
114       const ptrdiff_t src_index[] = {j ^ pos | (pos - 1), b};
115       x[dst_index] += x[src_index];
116     }
117     ''',
118     'cumsum_batch_kernel'
119 )
120 _cumsum_kern = core.ElementwiseKernel(
121     'int64 pos', 'raw T x',
122     '''
123     if (i & pos) {
124       x[i] += x[i ^ pos | (pos - 1)];
125     }
126     ''',
127     'cumsum_kernel'
128 )
129 
130 
131 def cumsum(a, axis=None, dtype=None, out=None):
132     """Returns the cumulative sum of an array along a given axis.
133 
134     Args:
135         a (cupy.ndarray): Input array.
136         axis (int): Axis along which the cumulative sum is taken. If it is not
137             specified, the input is flattened.
138         dtype: Data type specifier.
139         out (cupy.ndarray): Output array.
140 
141     Returns:
142         cupy.ndarray: The result array.
143 
144     .. seealso:: :func:`numpy.cumsum`
145 
146     """
147     return _cum_core(a, axis, dtype, out, _cumsum_kern, _cumsum_batch_kern)
148 
149 
150 _cumprod_batch_kern = core.ElementwiseKernel(
151     'int64 pos, int64 batch', 'raw T x',
152     '''
153     ptrdiff_t b = i % batch;
154     ptrdiff_t j = i / batch;
155     if (j & pos) {
156       const ptrdiff_t dst_index[] = {j, b};
157       const ptrdiff_t src_index[] = {j ^ pos | (pos - 1), b};
158       x[dst_index] *= x[src_index];
159     }
160     ''',
161     'cumprod_batch_kernel'
162 )
163 _cumprod_kern = core.ElementwiseKernel(
164     'int64 pos', 'raw T x',
165     '''
166     if (i & pos) {
167       x[i] *= x[i ^ pos | (pos - 1)];
168     }
169     ''',
170     'cumprod_kernel'
171 )
172 
173 
174 def cumprod(a, axis=None, dtype=None, out=None):
175     """Returns the cumulative product of an array along a given axis.
176 
177     Args:
178         a (cupy.ndarray): Input array.
179         axis (int): Axis along which the cumulative product is taken. If it is
180             not specified, the input is flattened.
181         dtype: Data type specifier.
182         out (cupy.ndarray): Output array.
183 
184     Returns:
185         cupy.ndarray: The result array.
186 
187     .. seealso:: :func:`numpy.cumprod`
188 
189     """
190     return _cum_core(a, axis, dtype, out, _cumprod_kern, _cumprod_batch_kern)
191 
192 
193 # TODO(okuta): Implement diff
194 
195 
196 # TODO(okuta): Implement ediff1d
197 
198 
199 # TODO(okuta): Implement gradient
200 
201 
202 # TODO(okuta): Implement cross
203 
204 
205 # TODO(okuta): Implement trapz
206
```
--- END FILES ---

Please first localize the bug based on the issue statement, and then generate a patch according to the `git diff` format fenced by three backticks.

Here is an example:

```diff
diff --git a/examples/server_async.py b/examples/server_async.py
--- a/examples/server_async.py
+++ b/examples/server_async.py
@@ -313,4 +313,4 @@
 
 
 if __name__ == "__main__":
-    asyncio.run(run_async_server("."), debug=True)
+    asyncio.run(run_async_server(), debug=True)
diff --git a/examples/server_sync.py b/examples/server_sync.py
--- a/examples/server_sync.py
+++ b/examples/server_sync.py
@@ -313,5 +313,5 @@
 
 
 if __name__ == "__main__":
-    server = run_sync_server(".")
+    server = run_sync_server()
     server.shutdown()

```

diff --git a/cupy/math/sumprod.py b/cupy/math/sumprod.py
--- a/cupy/math/sumprod.py
+++ b/cupy/math/sumprod.py
@@ -1,6 +1,7 @@
 import numpy
 import six
 
+import cupy
 from cupy import core
 
 
@@ -60,6 +61,8 @@
 
 
 def _proc_as_batch(proc, x, axis):
+    if x.shape[axis] == 0:
+        return cupy.empty_like(x)
     trans, revert = _axis_to_first(x, axis)
     t = x.transpose(trans)
     s = t.shape

{"golden_diff": "diff --git a/cupy/math/sumprod.py b/cupy/math/sumprod.py\n--- a/cupy/math/sumprod.py\n+++ b/cupy/math/sumprod.py\n@@ -1,6 +1,7 @@\n import numpy\n import six\n \n+import cupy\n from cupy import core\n \n \n@@ -60,6 +61,8 @@\n \n \n def _proc_as_batch(proc, x, axis):\n+    if x.shape[axis] == 0:\n+        return cupy.empty_like(x)\n     trans, revert = _axis_to_first(x, axis)\n     t = x.transpose(trans)\n     s = t.shape\n", "issue": "`cupy.split` sometimes fails with a 0-sized input array\n```\r\n>>> cupy.split(cupy.ones((3, 0)), [1])\r\nTraceback (most recent call last):\r\n  File \"<stdin>\", line 1, in <module>\r\n  File \"/home/kataoka/cupy/cupy/manipulation/split.py\", line 78, in split\r\n    return array_split(ary, indices_or_sections, axis)\r\n  File \"/home/kataoka/cupy/cupy/manipulation/split.py\", line 16, in array_split\r\n    return core.array_split(ary, indices_or_sections, axis)\r\n  File \"cupy/core/core.pyx\", line 2338, in cupy.core.core.array_split\r\n    v.data = ary.data + prev * stride\r\n  File \"cupy/cuda/memory.pyx\", line 243, in cupy.cuda.memory.MemoryPointer.__add__\r\n    assert self.ptr > 0 or offset == 0\r\nAssertionError\r\n```\n", "before_files": [{"content": "import numpy\nimport six\n\nfrom cupy import core\n\n\ndef sum(a, axis=None, dtype=None, out=None, keepdims=False):\n    \"\"\"Returns the sum of an array along given axes.\n\n    Args:\n        a (cupy.ndarray): Array to take sum.\n        axis (int or sequence of ints): Axes along which the sum is taken.\n        dtype: Data type specifier.\n        out (cupy.ndarray): Output array.\n        keepdims (bool): If ``True``, the specified axes are remained as axes\n            of length one.\n\n    Returns:\n        cupy.ndarray: The result array.\n\n    .. seealso:: :func:`numpy.sum`\n\n    \"\"\"\n    # TODO(okuta): check type\n    return a.sum(axis, dtype, out, keepdims)\n\n\ndef prod(a, axis=None, dtype=None, out=None, keepdims=False):\n    \"\"\"Returns the product of an array along given axes.\n\n    Args:\n        a (cupy.ndarray): Array to take product.\n        axis (int or sequence of ints): Axes along which the product is taken.\n        dtype: Data type specifier.\n        out (cupy.ndarray): Output array.\n        keepdims (bool): If ``True``, the specified axes are remained as axes\n            of length one.\n\n    Returns:\n        cupy.ndarray: The result array.\n\n    .. seealso:: :func:`numpy.prod`\n\n    \"\"\"\n    # TODO(okuta): check type\n    return a.prod(axis, dtype, out, keepdims)\n\n\n# TODO(okuta): Implement nansum\n\n\ndef _axis_to_first(x, axis):\n    if axis < 0:\n        axis = x.ndim + axis\n    trans = [axis] + [a for a in six.moves.range(x.ndim) if a != axis]\n    pre = list(six.moves.range(1, axis + 1))\n    succ = list(six.moves.range(axis + 1, x.ndim))\n    revert = pre + [0] + succ\n    return trans, revert\n\n\ndef _proc_as_batch(proc, x, axis):\n    trans, revert = _axis_to_first(x, axis)\n    t = x.transpose(trans)\n    s = t.shape\n    r = t.reshape(x.shape[axis], -1)\n    pos = 1\n    size = r.size\n    batch = r.shape[1]\n    while pos < size:\n        proc(pos, batch, r, size=size)\n        pos <<= 1\n    return r.reshape(s).transpose(revert)\n\n\ndef _cum_core(a, axis, dtype, out, kern, batch_kern):\n    if out is None:\n        if dtype is None:\n            kind = a.dtype.kind\n            if kind == 'b':\n                dtype = numpy.dtype('l')\n            elif kind == 'i' and a.dtype.itemsize < numpy.dtype('l').itemsize:\n                dtype = numpy.dtype('l')\n            elif kind == 'u' and a.dtype.itemsize < numpy.dtype('L').itemsize:\n                dtype = numpy.dtype('L')\n            else:\n                dtype = a.dtype\n\n        out = a.astype(dtype)\n    else:\n        out[...] = a\n\n    if axis is None:\n        out = out.ravel()\n    elif not (-a.ndim <= axis < a.ndim):\n        raise core.core._AxisError('axis(={}) out of bounds'.format(axis))\n    else:\n        return _proc_as_batch(batch_kern, out, axis=axis)\n\n    pos = 1\n    while pos < out.size:\n        kern(pos, out, size=out.size)\n        pos <<= 1\n    return out\n\n\n_cumsum_batch_kern = core.ElementwiseKernel(\n    'int64 pos, int64 batch', 'raw T x',\n    '''\n    ptrdiff_t b = i % batch;\n    ptrdiff_t j = i / batch;\n    if (j & pos) {\n      const ptrdiff_t dst_index[] = {j, b};\n      const ptrdiff_t src_index[] = {j ^ pos | (pos - 1), b};\n      x[dst_index] += x[src_index];\n    }\n    ''',\n    'cumsum_batch_kernel'\n)\n_cumsum_kern = core.ElementwiseKernel(\n    'int64 pos', 'raw T x',\n    '''\n    if (i & pos) {\n      x[i] += x[i ^ pos | (pos - 1)];\n    }\n    ''',\n    'cumsum_kernel'\n)\n\n\ndef cumsum(a, axis=None, dtype=None, out=None):\n    \"\"\"Returns the cumulative sum of an array along a given axis.\n\n    Args:\n        a (cupy.ndarray): Input array.\n        axis (int): Axis along which the cumulative sum is taken. If it is not\n            specified, the input is flattened.\n        dtype: Data type specifier.\n        out (cupy.ndarray): Output array.\n\n    Returns:\n        cupy.ndarray: The result array.\n\n    .. seealso:: :func:`numpy.cumsum`\n\n    \"\"\"\n    return _cum_core(a, axis, dtype, out, _cumsum_kern, _cumsum_batch_kern)\n\n\n_cumprod_batch_kern = core.ElementwiseKernel(\n    'int64 pos, int64 batch', 'raw T x',\n    '''\n    ptrdiff_t b = i % batch;\n    ptrdiff_t j = i / batch;\n    if (j & pos) {\n      const ptrdiff_t dst_index[] = {j, b};\n      const ptrdiff_t src_index[] = {j ^ pos | (pos - 1), b};\n      x[dst_index] *= x[src_index];\n    }\n    ''',\n    'cumprod_batch_kernel'\n)\n_cumprod_kern = core.ElementwiseKernel(\n    'int64 pos', 'raw T x',\n    '''\n    if (i & pos) {\n      x[i] *= x[i ^ pos | (pos - 1)];\n    }\n    ''',\n    'cumprod_kernel'\n)\n\n\ndef cumprod(a, axis=None, dtype=None, out=None):\n    \"\"\"Returns the cumulative product of an array along a given axis.\n\n    Args:\n        a (cupy.ndarray): Input array.\n        axis (int): Axis along which the cumulative product is taken. If it is\n            not specified, the input is flattened.\n        dtype: Data type specifier.\n        out (cupy.ndarray): Output array.\n\n    Returns:\n        cupy.ndarray: The result array.\n\n    .. seealso:: :func:`numpy.cumprod`\n\n    \"\"\"\n    return _cum_core(a, axis, dtype, out, _cumprod_kern, _cumprod_batch_kern)\n\n\n# TODO(okuta): Implement diff\n\n\n# TODO(okuta): Implement ediff1d\n\n\n# TODO(okuta): Implement gradient\n\n\n# TODO(okuta): Implement cross\n\n\n# TODO(okuta): Implement trapz\n", "path": "cupy/math/sumprod.py"}], "after_files": [{"content": "import numpy\nimport six\n\nimport cupy\nfrom cupy import core\n\n\ndef sum(a, axis=None, dtype=None, out=None, keepdims=False):\n    \"\"\"Returns the sum of an array along given axes.\n\n    Args:\n        a (cupy.ndarray): Array to take sum.\n        axis (int or sequence of ints): Axes along which the sum is taken.\n        dtype: Data type specifier.\n        out (cupy.ndarray): Output array.\n        keepdims (bool): If ``True``, the specified axes are remained as axes\n            of length one.\n\n    Returns:\n        cupy.ndarray: The result array.\n\n    .. seealso:: :func:`numpy.sum`\n\n    \"\"\"\n    # TODO(okuta): check type\n    return a.sum(axis, dtype, out, keepdims)\n\n\ndef prod(a, axis=None, dtype=None, out=None, keepdims=False):\n    \"\"\"Returns the product of an array along given axes.\n\n    Args:\n        a (cupy.ndarray): Array to take product.\n        axis (int or sequence of ints): Axes along which the product is taken.\n        dtype: Data type specifier.\n        out (cupy.ndarray): Output array.\n        keepdims (bool): If ``True``, the specified axes are remained as axes\n            of length one.\n\n    Returns:\n        cupy.ndarray: The result array.\n\n    .. seealso:: :func:`numpy.prod`\n\n    \"\"\"\n    # TODO(okuta): check type\n    return a.prod(axis, dtype, out, keepdims)\n\n\n# TODO(okuta): Implement nansum\n\n\ndef _axis_to_first(x, axis):\n    if axis < 0:\n        axis = x.ndim + axis\n    trans = [axis] + [a for a in six.moves.range(x.ndim) if a != axis]\n    pre = list(six.moves.range(1, axis + 1))\n    succ = list(six.moves.range(axis + 1, x.ndim))\n    revert = pre + [0] + succ\n    return trans, revert\n\n\ndef _proc_as_batch(proc, x, axis):\n    if x.shape[axis] == 0:\n        return cupy.empty_like(x)\n    trans, revert = _axis_to_first(x, axis)\n    t = x.transpose(trans)\n    s = t.shape\n    r = t.reshape(x.shape[axis], -1)\n    pos = 1\n    size = r.size\n    batch = r.shape[1]\n    while pos < size:\n        proc(pos, batch, r, size=size)\n        pos <<= 1\n    return r.reshape(s).transpose(revert)\n\n\ndef _cum_core(a, axis, dtype, out, kern, batch_kern):\n    if out is None:\n        if dtype is None:\n            kind = a.dtype.kind\n            if kind == 'b':\n                dtype = numpy.dtype('l')\n            elif kind == 'i' and a.dtype.itemsize < numpy.dtype('l').itemsize:\n                dtype = numpy.dtype('l')\n            elif kind == 'u' and a.dtype.itemsize < numpy.dtype('L').itemsize:\n                dtype = numpy.dtype('L')\n            else:\n                dtype = a.dtype\n\n        out = a.astype(dtype)\n    else:\n        out[...] = a\n\n    if axis is None:\n        out = out.ravel()\n    elif not (-a.ndim <= axis < a.ndim):\n        raise core.core._AxisError('axis(={}) out of bounds'.format(axis))\n    else:\n        return _proc_as_batch(batch_kern, out, axis=axis)\n\n    pos = 1\n    while pos < out.size:\n        kern(pos, out, size=out.size)\n        pos <<= 1\n    return out\n\n\n_cumsum_batch_kern = core.ElementwiseKernel(\n    'int64 pos, int64 batch', 'raw T x',\n    '''\n    ptrdiff_t b = i % batch;\n    ptrdiff_t j = i / batch;\n    if (j & pos) {\n      const ptrdiff_t dst_index[] = {j, b};\n      const ptrdiff_t src_index[] = {j ^ pos | (pos - 1), b};\n      x[dst_index] += x[src_index];\n    }\n    ''',\n    'cumsum_batch_kernel'\n)\n_cumsum_kern = core.ElementwiseKernel(\n    'int64 pos', 'raw T x',\n    '''\n    if (i & pos) {\n      x[i] += x[i ^ pos | (pos - 1)];\n    }\n    ''',\n    'cumsum_kernel'\n)\n\n\ndef cumsum(a, axis=None, dtype=None, out=None):\n    \"\"\"Returns the cumulative sum of an array along a given axis.\n\n    Args:\n        a (cupy.ndarray): Input array.\n        axis (int): Axis along which the cumulative sum is taken. If it is not\n            specified, the input is flattened.\n        dtype: Data type specifier.\n        out (cupy.ndarray): Output array.\n\n    Returns:\n        cupy.ndarray: The result array.\n\n    .. seealso:: :func:`numpy.cumsum`\n\n    \"\"\"\n    return _cum_core(a, axis, dtype, out, _cumsum_kern, _cumsum_batch_kern)\n\n\n_cumprod_batch_kern = core.ElementwiseKernel(\n    'int64 pos, int64 batch', 'raw T x',\n    '''\n    ptrdiff_t b = i % batch;\n    ptrdiff_t j = i / batch;\n    if (j & pos) {\n      const ptrdiff_t dst_index[] = {j, b};\n      const ptrdiff_t src_index[] = {j ^ pos | (pos - 1), b};\n      x[dst_index] *= x[src_index];\n    }\n    ''',\n    'cumprod_batch_kernel'\n)\n_cumprod_kern = core.ElementwiseKernel(\n    'int64 pos', 'raw T x',\n    '''\n    if (i & pos) {\n      x[i] *= x[i ^ pos | (pos - 1)];\n    }\n    ''',\n    'cumprod_kernel'\n)\n\n\ndef cumprod(a, axis=None, dtype=None, out=None):\n    \"\"\"Returns the cumulative product of an array along a given axis.\n\n    Args:\n        a (cupy.ndarray): Input array.\n        axis (int): Axis along which the cumulative product is taken. If it is\n            not specified, the input is flattened.\n        dtype: Data type specifier.\n        out (cupy.ndarray): Output array.\n\n    Returns:\n        cupy.ndarray: The result array.\n\n    .. seealso:: :func:`numpy.cumprod`\n\n    \"\"\"\n    return _cum_core(a, axis, dtype, out, _cumprod_kern, _cumprod_batch_kern)\n\n\n# TODO(okuta): Implement diff\n\n\n# TODO(okuta): Implement ediff1d\n\n\n# TODO(okuta): Implement gradient\n\n\n# TODO(okuta): Implement cross\n\n\n# TODO(okuta): Implement trapz\n", "path": "cupy/math/sumprod.py"}]}